Class 12th Mathematics Part I CBSE Solution
Exercise 6.1- Find the rate of change of the area of a circle with respect to its radius r…
- The volume of a cube is increasing at the rate of 8 cm^3 /s. How fast is the…
- The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the…
- An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the…
- A stone is dropped into a quiet lake and waves move in circles at the speed of 5…
- The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate…
- The length x of a rectangle is decreasing at the rate of 5 cm/minute and the…
- A balloon, which always remains spherical on inflation, is being inflated by…
- A balloon, which always remains spherical has a variable radius. Find the rate…
- A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled…
- A particle moves along the curve 6y = x^3 +2. Find the points on the curve at…
- The radius of an air bubble is increasing at the rate of 1/2 cm/s . At what…
- A balloon, which always remains spherical, has a variable diameter 3/2 (2x+1) .…
- Sand is pouring from a pipe at the rate of 12 cm^3 /s. The falling sand forms a…
- The total cost C(x) in Rupees associated with the production of x units of an…
- R(x) = 13x^2 + 26x + 15. Find the marginal revenue when x = 7. The total…
- The rate of change of the area of a circle with respect to its radius r at r =…
- R(x) = 3x^2 + 36x + 5. The marginal revenue, when x = 15 is The total revenue…
Exercise 6.2- Show that the function given by f (x) = 3x + 17 is strictly increasing on R.…
- Show that the function given by f (x) = e2x is strictly increasing on R.…
- Show that the function given by f (x) = sin x is (a) strictly increasing in (0 ,…
- Find the intervals in which the function f given by f (x) = 2x^2 - 3x is (a)…
- Find the intervals in which the function f given by f (x) = 2x^3 - 3x^2 - 36x +…
- x^2 + 2x - 5 Find the intervals in which the following functions are strictly…
- 10 - 6x - 2x^2 Find the intervals in which the following functions are strictly…
- -2x^3 - 9x^2 - 12x + 1 Find the intervals in which the following functions are…
- 6 - 9x - x^2 Find the intervals in which the following functions are strictly…
- (x + 1)^3 (x - 3)^3 Find the intervals in which the following functions are…
- Show that y = log (1+x) - 2x/2+x , x-1 , is an increasing function of x…
- Find the values of x for which y = [x(x - 2)]^2 is an increasing function.…
- Prove that y = 4sintegrate heta /(2+costheta) - theta is an increasing function…
- Prove that the logarithmic function is strictly increasing on (0, ∞).…
- Prove that the function f given by f (x) = x^2 - x + 1 is neither strictly…
- Which of the following functions are strictly decreasing on (0 , pi /2) ? A.…
- On which of the following intervals is the function f given by f (x) = x^100 +…
- Find the least value of a that the function f given by f (x) = x^2 + ax + 1 is…
- Let I be any interval disjoint from [-1, 1]. Prove that the function f given by…
- Prove that the function f given by f (x) = log sin x is strictly increasing on…
- Prove that the function f given by f (x) = log |cos x| is strictly decreasing…
- Prove that the function given by f (x) = x^3 - 3x^2 + 3x - 100 is increasing in…
- The interval in which y = x^2 e-x is increasing isA. (- ∞, ∞) B. (- 2, 0) C.…
Exercise 6.3- Find the slope of the tangent to the curve y = 3x^4 - 4x at x = 4.…
- Find the slope of the tangent to the curve y = x-1/x-2 , x not equal 2 at x =…
- Find the slope of the tangent to curve y = x^3 - x + 1 at the point whose…
- Find the slope of the tangent to the curve y = x^3 -3x + 2 at the point whose…
- Find the slope of the normal to the curve x = acos^3 θ, y = asin^3 θ at theta =…
- Find the slope of the normal to the curve x = 1− a sinθ, y = bcos^2 θ at theta =…
- Find points at which the tangent to the curve y = x^3 - 3x^2 - 9x + 7 is…
- Find a point on the curve y = (x - 2)^2 at which the tangent is parallel to the…
- Find the point on the curve y = x^3 - 11x + 5 at which the tangent is y = x -11.…
- Find the equation of all lines having slope -1 that are tangents to the curve y…
- Find the equation of all lines having slope 2 which are tangents to the curve y…
- Find the equations of all lines having slope 0 which are tangent to the curve y…
- Find points on the curve x^2/9 + y^2/16 = 1 at which the tangents are (i)…
- Find the equations of the tangent and normal to the given curves at the…
- y = x^4 - 6x^3 + 13x^2 - 10x + 5 at (0, 5) Find the equations of the tangent…
- y = x^4 - 6x^3 + 13x^2 - 10x + 5 at (1, 3) Find the equations of the tangent…
- y = x^3 at (1, 1) Find the equations of the tangent and normal to the given…
- y = x^2 at (0, 0) Find the equations of the tangent and normal to the given…
- Find the equation of the tangent line to the curve y = x^2 - 2x +7 which is (a)…
- Show that the tangents to the curve y = 7x^3 + 11 at the points where x = 2 and…
- Find the points on the curve y = x^3 at which the slope of the tangent is equal…
- For the curve y = 4x^3 - 2x^5 , find all the points at which the tangent passes…
- Find the points on the curve x^2 + y^2 - 2x - 3 = 0 at which the tangents are…
- Find the equation of the normal at the point (am^2 , am^3) for the curve ay^2 =…
- Find the equation of the normals to the curve y = x^3 + 2x + 6 which are…
- Find the equations of the tangent and normal to the parabola y^2 = 4ax at the…
- Prove that the curves x = y^2 and xy = k cut at right angles* if 8k^2 = 1.…
- Find the equations of the tangent and normal to the hyperbola x^2/a^2 - y^2/b^2…
- Find the equation of the tangent to the curve y = root 3x-2 which is parallel…
- The slope of the normal to the curve y = 2x^2 + 3 sin x at x = 0 isA. 3 B. 1/3…
- The line y = x + 1 is a tangent to the curve y^2 = 4x at the pointA. (1, 2) B.…
Exercise 6.4- root 25.3 Using differentials, find the approximate value of each of the…
- root 49.5 Using differentials, find the approximate value of each of the…
- root 0.6 Using differentials, find the approximate value of each of the…
- (0.009)^1/3 Using differentials, find the approximate value of each of the…
- (0.999)^1/10 Using differentials, find the approximate value of each of the…
- (15)^1/4 Using differentials, find the approximate value of each of the…
- (26)^1/3 Using differentials, find the approximate value of each of the…
- (255)^1/4 Using differentials, find the approximate value of each of the…
- (82)^1/4 Using differentials, find the approximate value of each of the…
- (401)^1/2 Using differentials, find the approximate value of each of the…
- (0.0037)^1/2 Using differentials, find the approximate value of each of the…
- (26.57)^1/3 Using differentials, find the approximate value of each of the…
- (81.5)^1/4 Using differentials, find the approximate value of each of the…
- (3.968)^3/2 Using differentials, find the approximate value of each of the…
- (32.15)^1/5 Using differentials, find the approximate value of each of the…
- Find the approximate value of f (2.01), where f (x) = 4x^2 + 5x + 2.…
- Find the approximate value of f (5.001), where f (x) = x^3 - 7x2 + 15.…
- Find the approximate change in the volume V of a cube of side x metres caused by…
- Find the approximate change in the surface area of a cube of side x metres…
- If the radius of a sphere is measured as 7 m with an error of 0.02 m, then find…
- If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find…
- If f(x) = 3x^2 + 15x + 5, then the approximate value of f (3.02) isA. 47.66 B.…
- The approximate change in the volume of a cube of side x metres caused by…
Exercise 6.5- f (x) = (2x - 1)^2 + 3 Find the maximum and minimum values, if any, of the…
- f (x) = 9x^2 + 12x + 2 Find the maximum and minimum values, if any, of the…
- f(x) = - (x - 1)^2 + 10 Find the maximum and minimum values, if any, of the…
- g(x) = x^3 + 1 Find the maximum and minimum values, if any, of the following…
- f(x) = |x + 2| - 1 Find the maximum and minimum values, if any, of the…
- g(x) = -|x + 1| + 3 Find the maximum and minimum values, if any, of the…
- h(x) = sin(2x) + 5 Find the maximum and minimum values, if any, of the…
- f (x) = |sin 4x + 3| Find the maximum and minimum values, if any, of the…
- h(x) = x + 1, x ∈ (-1, 1) Find the maximum and minimum values, if any, of the…
- f (x) = x^2 Find the local maxima and local minima, if any, of the following…
- g(x) = x^3 - 3x Find the local maxima and local minima, if any, of the…
- h (x) = sinx+cosx, 0x pi /2 Find the local maxima and local minima, if any, of…
- f (x) = sin x - cos x, 0 x 2π Find the local maxima and local minima, if any,…
- f (x) = x^3 - 6x^2 + 9x + 15 Find the local maxima and local minima, if any, of…
- g (x) = x/2 + 2/x , x0 Find the local maxima and local minima, if any, of the…
- g (x) = 1/x^2 + 2 Find the local maxima and local minima, if any, of the…
- f (x) = x root 1-x , 0x1 Find the local maxima and local minima, if any, of the…
- f (x) = ex Prove that the following functions do not have maxima or minima:…
- g(x) = log x Prove that the following functions do not have maxima or minima:…
- h(x) = x^3 + x^2 + x +1 Prove that the following functions do not have maxima…
- f (x) = x^3 , x ∈ [-2, 2] Find the absolute maximum value and the absolute…
- f (x) = sin x + cos x, x ∈ [0, π] Find the absolute maximum value and the…
- f (x) = 4x - 1/2 x^2 , x in[-2 , 9/2] Find the absolute maximum value and the…
- f (x) = (x − 1)^2 + 3, x ∈ [−3, 1] Find the absolute maximum value and the…
- Find the maximum profit that a company can make, if the profit function is given…
- Find both the maximum value and the minimum value of 3x^4 - 8x^3 + 12x^2 - 48x +…
- At what points in the interval [0, 2π], does the function sin 2x attain its…
- What is the maximum value of the function sin x + cos x?
- Find the maximum value of 2x^3 - 24x + 107 in the interval [1, 3]. Find the…
- It is given that at x = 1, the function x^4 - 62x^2 + ax + 9 attains its…
- Find the maximum and minimum values of x + sin 2x on [0, 2π].
- Find two numbers whose sum is 24 and whose product is as large as possible.…
- Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.…
- Find two positive numbers x and y such that their sum is 35 and the product x^2…
- Find two positive numbers whose sum is 16 and the sum of whose cubes is…
- A square piece of tin of side 18 cm is to be made into a box without top, by…
- A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top,…
- Show that of all the rectangles inscribed in a given fixed circle, the square…
- Show that the right circular cylinder of given surface and maximum volume is…
- Of all the closed cylindrical cans (right circular), of a given volume of 100…
- A wire of length 28 m is to be cut into two pieces. One of the pieces is to be…
- Prove that the volume of the largest cone that can be inscribed in a sphere of…
- Show that the right circular cone of least curved surface and given volume has…
- Show that the semi-vertical angle of the cone of the maximum volume and of…
- Show that semi-vertical angle of right circular cone of given surface area and…
- The point on the curve x^2 = 2y which is nearest to the point (0, 5) isA.…
- For all real values of x, the minimum value of 1-x+x^2/1+x+x^2 isA. 0 B. 1 C. 3…
- The maximum value of [x (x-1) + 1]^1/3 , 0 less than equal to x less than equal…
Miscellaneous Exercise- (17/81)^1/4 Using differentials, find the approximate value of each of the…
- (33)^1/5 Using differentials, find the approximate value of each of the…
- Show that the function given by f (x) = logx/x has maximum at x = e.…
- The two equal sides of an isosceles triangle with fixed base b are decreasing at…
- Find the equation of the normal to curve x2 = 4y which passes through the point…
- Show that the normal at any point θ to the curve x = a cosθ + a θ sinθ, y = a…
- Find the intervals in which the function f given by f (x) =
- Find the intervals in which the function f given by f (x) = x^3 + 1/x^3 , x not…
- Find the maximum area of an isosceles triangle inscribed in the ellipse x^2/a^2…
- A tank with rectangular base and rectangular sides, open at the top is to be…
- The sum of the perimeter of a circle and square is k, where k is some constant.…
- A window is in the form of a rectangle surmounted by a semi - circular opening.…
- A point on the hypotenuse of a triangle is at distance a and b from the sides…
- Find the points at which the function f given by f (x) = (x - 2)^4 (x + 1)^3…
- Find the absolute maximum and minimum values of the function f given by f (x) =…
- Show that the altitude of the right circular cone of maximum volume that can be…
- Let f be a function defined on [a, b] such that f ′(x) 0, for all x ∈ (a, b).…
- Show that the height of the cylinder of maximum volume that can be inscribed in…
- Show that height of the cylinder of greatest volume which can be inscribed in a…
- A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314…
- The slope of the tangent to the curve x = t^2 + 3t - 8, y = 2t^2 - 2t - 5 at…
- The line y = mx + 1 is a tangent to the curve y^2 = 4x if the value of m isA. 1…
- The normal at the point (1,1) on the curve 2y + x^2 = 3 isA. x + y = 0 B. x - y…
- The normal to the curve x^2 = 4y passing (1,2) isA. x + y = 3 B. x - y = 3 C. x…
- The points on the curve 9y^2 = x^3 , where the normal to the curve makes equal…
- Find the rate of change of the area of a circle with respect to its radius r…
- The volume of a cube is increasing at the rate of 8 cm^3 /s. How fast is the…
- The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the…
- An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the…
- A stone is dropped into a quiet lake and waves move in circles at the speed of 5…
- The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate…
- The length x of a rectangle is decreasing at the rate of 5 cm/minute and the…
- A balloon, which always remains spherical on inflation, is being inflated by…
- A balloon, which always remains spherical has a variable radius. Find the rate…
- A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled…
- A particle moves along the curve 6y = x^3 +2. Find the points on the curve at…
- The radius of an air bubble is increasing at the rate of 1/2 cm/s . At what…
- A balloon, which always remains spherical, has a variable diameter 3/2 (2x+1) .…
- Sand is pouring from a pipe at the rate of 12 cm^3 /s. The falling sand forms a…
- The total cost C(x) in Rupees associated with the production of x units of an…
- R(x) = 13x^2 + 26x + 15. Find the marginal revenue when x = 7. The total…
- The rate of change of the area of a circle with respect to its radius r at r =…
- R(x) = 3x^2 + 36x + 5. The marginal revenue, when x = 15 is The total revenue…
- Show that the function given by f (x) = 3x + 17 is strictly increasing on R.…
- Show that the function given by f (x) = e2x is strictly increasing on R.…
- Show that the function given by f (x) = sin x is (a) strictly increasing in (0 ,…
- Find the intervals in which the function f given by f (x) = 2x^2 - 3x is (a)…
- Find the intervals in which the function f given by f (x) = 2x^3 - 3x^2 - 36x +…
- x^2 + 2x - 5 Find the intervals in which the following functions are strictly…
- 10 - 6x - 2x^2 Find the intervals in which the following functions are strictly…
- -2x^3 - 9x^2 - 12x + 1 Find the intervals in which the following functions are…
- 6 - 9x - x^2 Find the intervals in which the following functions are strictly…
- (x + 1)^3 (x - 3)^3 Find the intervals in which the following functions are…
- Show that y = log (1+x) - 2x/2+x , x-1 , is an increasing function of x…
- Find the values of x for which y = [x(x - 2)]^2 is an increasing function.…
- Prove that y = 4sintegrate heta /(2+costheta) - theta is an increasing function…
- Prove that the logarithmic function is strictly increasing on (0, ∞).…
- Prove that the function f given by f (x) = x^2 - x + 1 is neither strictly…
- Which of the following functions are strictly decreasing on (0 , pi /2) ? A.…
- On which of the following intervals is the function f given by f (x) = x^100 +…
- Find the least value of a that the function f given by f (x) = x^2 + ax + 1 is…
- Let I be any interval disjoint from [-1, 1]. Prove that the function f given by…
- Prove that the function f given by f (x) = log sin x is strictly increasing on…
- Prove that the function f given by f (x) = log |cos x| is strictly decreasing…
- Prove that the function given by f (x) = x^3 - 3x^2 + 3x - 100 is increasing in…
- The interval in which y = x^2 e-x is increasing isA. (- ∞, ∞) B. (- 2, 0) C.…
- Find the slope of the tangent to the curve y = 3x^4 - 4x at x = 4.…
- Find the slope of the tangent to the curve y = x-1/x-2 , x not equal 2 at x =…
- Find the slope of the tangent to curve y = x^3 - x + 1 at the point whose…
- Find the slope of the tangent to the curve y = x^3 -3x + 2 at the point whose…
- Find the slope of the normal to the curve x = acos^3 θ, y = asin^3 θ at theta =…
- Find the slope of the normal to the curve x = 1− a sinθ, y = bcos^2 θ at theta =…
- Find points at which the tangent to the curve y = x^3 - 3x^2 - 9x + 7 is…
- Find a point on the curve y = (x - 2)^2 at which the tangent is parallel to the…
- Find the point on the curve y = x^3 - 11x + 5 at which the tangent is y = x -11.…
- Find the equation of all lines having slope -1 that are tangents to the curve y…
- Find the equation of all lines having slope 2 which are tangents to the curve y…
- Find the equations of all lines having slope 0 which are tangent to the curve y…
- Find points on the curve x^2/9 + y^2/16 = 1 at which the tangents are (i)…
- Find the equations of the tangent and normal to the given curves at the…
- y = x^4 - 6x^3 + 13x^2 - 10x + 5 at (0, 5) Find the equations of the tangent…
- y = x^4 - 6x^3 + 13x^2 - 10x + 5 at (1, 3) Find the equations of the tangent…
- y = x^3 at (1, 1) Find the equations of the tangent and normal to the given…
- y = x^2 at (0, 0) Find the equations of the tangent and normal to the given…
- Find the equation of the tangent line to the curve y = x^2 - 2x +7 which is (a)…
- Show that the tangents to the curve y = 7x^3 + 11 at the points where x = 2 and…
- Find the points on the curve y = x^3 at which the slope of the tangent is equal…
- For the curve y = 4x^3 - 2x^5 , find all the points at which the tangent passes…
- Find the points on the curve x^2 + y^2 - 2x - 3 = 0 at which the tangents are…
- Find the equation of the normal at the point (am^2 , am^3) for the curve ay^2 =…
- Find the equation of the normals to the curve y = x^3 + 2x + 6 which are…
- Find the equations of the tangent and normal to the parabola y^2 = 4ax at the…
- Prove that the curves x = y^2 and xy = k cut at right angles* if 8k^2 = 1.…
- Find the equations of the tangent and normal to the hyperbola x^2/a^2 - y^2/b^2…
- Find the equation of the tangent to the curve y = root 3x-2 which is parallel…
- The slope of the normal to the curve y = 2x^2 + 3 sin x at x = 0 isA. 3 B. 1/3…
- The line y = x + 1 is a tangent to the curve y^2 = 4x at the pointA. (1, 2) B.…
- root 25.3 Using differentials, find the approximate value of each of the…
- root 49.5 Using differentials, find the approximate value of each of the…
- root 0.6 Using differentials, find the approximate value of each of the…
- (0.009)^1/3 Using differentials, find the approximate value of each of the…
- (0.999)^1/10 Using differentials, find the approximate value of each of the…
- (15)^1/4 Using differentials, find the approximate value of each of the…
- (26)^1/3 Using differentials, find the approximate value of each of the…
- (255)^1/4 Using differentials, find the approximate value of each of the…
- (82)^1/4 Using differentials, find the approximate value of each of the…
- (401)^1/2 Using differentials, find the approximate value of each of the…
- (0.0037)^1/2 Using differentials, find the approximate value of each of the…
- (26.57)^1/3 Using differentials, find the approximate value of each of the…
- (81.5)^1/4 Using differentials, find the approximate value of each of the…
- (3.968)^3/2 Using differentials, find the approximate value of each of the…
- (32.15)^1/5 Using differentials, find the approximate value of each of the…
- Find the approximate value of f (2.01), where f (x) = 4x^2 + 5x + 2.…
- Find the approximate value of f (5.001), where f (x) = x^3 - 7x2 + 15.…
- Find the approximate change in the volume V of a cube of side x metres caused by…
- Find the approximate change in the surface area of a cube of side x metres…
- If the radius of a sphere is measured as 7 m with an error of 0.02 m, then find…
- If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find…
- If f(x) = 3x^2 + 15x + 5, then the approximate value of f (3.02) isA. 47.66 B.…
- The approximate change in the volume of a cube of side x metres caused by…
- f (x) = (2x - 1)^2 + 3 Find the maximum and minimum values, if any, of the…
- f (x) = 9x^2 + 12x + 2 Find the maximum and minimum values, if any, of the…
- f(x) = - (x - 1)^2 + 10 Find the maximum and minimum values, if any, of the…
- g(x) = x^3 + 1 Find the maximum and minimum values, if any, of the following…
- f(x) = |x + 2| - 1 Find the maximum and minimum values, if any, of the…
- g(x) = -|x + 1| + 3 Find the maximum and minimum values, if any, of the…
- h(x) = sin(2x) + 5 Find the maximum and minimum values, if any, of the…
- f (x) = |sin 4x + 3| Find the maximum and minimum values, if any, of the…
- h(x) = x + 1, x ∈ (-1, 1) Find the maximum and minimum values, if any, of the…
- f (x) = x^2 Find the local maxima and local minima, if any, of the following…
- g(x) = x^3 - 3x Find the local maxima and local minima, if any, of the…
- h (x) = sinx+cosx, 0x pi /2 Find the local maxima and local minima, if any, of…
- f (x) = sin x - cos x, 0 x 2π Find the local maxima and local minima, if any,…
- f (x) = x^3 - 6x^2 + 9x + 15 Find the local maxima and local minima, if any, of…
- g (x) = x/2 + 2/x , x0 Find the local maxima and local minima, if any, of the…
- g (x) = 1/x^2 + 2 Find the local maxima and local minima, if any, of the…
- f (x) = x root 1-x , 0x1 Find the local maxima and local minima, if any, of the…
- f (x) = ex Prove that the following functions do not have maxima or minima:…
- g(x) = log x Prove that the following functions do not have maxima or minima:…
- h(x) = x^3 + x^2 + x +1 Prove that the following functions do not have maxima…
- f (x) = x^3 , x ∈ [-2, 2] Find the absolute maximum value and the absolute…
- f (x) = sin x + cos x, x ∈ [0, π] Find the absolute maximum value and the…
- f (x) = 4x - 1/2 x^2 , x in[-2 , 9/2] Find the absolute maximum value and the…
- f (x) = (x − 1)^2 + 3, x ∈ [−3, 1] Find the absolute maximum value and the…
- Find the maximum profit that a company can make, if the profit function is given…
- Find both the maximum value and the minimum value of 3x^4 - 8x^3 + 12x^2 - 48x +…
- At what points in the interval [0, 2π], does the function sin 2x attain its…
- What is the maximum value of the function sin x + cos x?
- Find the maximum value of 2x^3 - 24x + 107 in the interval [1, 3]. Find the…
- It is given that at x = 1, the function x^4 - 62x^2 + ax + 9 attains its…
- Find the maximum and minimum values of x + sin 2x on [0, 2π].
- Find two numbers whose sum is 24 and whose product is as large as possible.…
- Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.…
- Find two positive numbers x and y such that their sum is 35 and the product x^2…
- Find two positive numbers whose sum is 16 and the sum of whose cubes is…
- A square piece of tin of side 18 cm is to be made into a box without top, by…
- A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top,…
- Show that of all the rectangles inscribed in a given fixed circle, the square…
- Show that the right circular cylinder of given surface and maximum volume is…
- Of all the closed cylindrical cans (right circular), of a given volume of 100…
- A wire of length 28 m is to be cut into two pieces. One of the pieces is to be…
- Prove that the volume of the largest cone that can be inscribed in a sphere of…
- Show that the right circular cone of least curved surface and given volume has…
- Show that the semi-vertical angle of the cone of the maximum volume and of…
- Show that semi-vertical angle of right circular cone of given surface area and…
- The point on the curve x^2 = 2y which is nearest to the point (0, 5) isA.…
- For all real values of x, the minimum value of 1-x+x^2/1+x+x^2 isA. 0 B. 1 C. 3…
- The maximum value of [x (x-1) + 1]^1/3 , 0 less than equal to x less than equal…
- (17/81)^1/4 Using differentials, find the approximate value of each of the…
- (33)^1/5 Using differentials, find the approximate value of each of the…
- Show that the function given by f (x) = logx/x has maximum at x = e.…
- The two equal sides of an isosceles triangle with fixed base b are decreasing at…
- Find the equation of the normal to curve x2 = 4y which passes through the point…
- Show that the normal at any point θ to the curve x = a cosθ + a θ sinθ, y = a…
- Find the intervals in which the function f given by f (x) =
- Find the intervals in which the function f given by f (x) = x^3 + 1/x^3 , x not…
- Find the maximum area of an isosceles triangle inscribed in the ellipse x^2/a^2…
- A tank with rectangular base and rectangular sides, open at the top is to be…
- The sum of the perimeter of a circle and square is k, where k is some constant.…
- A window is in the form of a rectangle surmounted by a semi - circular opening.…
- A point on the hypotenuse of a triangle is at distance a and b from the sides…
- Find the points at which the function f given by f (x) = (x - 2)^4 (x + 1)^3…
- Find the absolute maximum and minimum values of the function f given by f (x) =…
- Show that the altitude of the right circular cone of maximum volume that can be…
- Let f be a function defined on [a, b] such that f ′(x) 0, for all x ∈ (a, b).…
- Show that the height of the cylinder of maximum volume that can be inscribed in…
- Show that height of the cylinder of greatest volume which can be inscribed in a…
- A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314…
- The slope of the tangent to the curve x = t^2 + 3t - 8, y = 2t^2 - 2t - 5 at…
- The line y = mx + 1 is a tangent to the curve y^2 = 4x if the value of m isA. 1…
- The normal at the point (1,1) on the curve 2y + x^2 = 3 isA. x + y = 0 B. x - y…
- The normal to the curve x^2 = 4y passing (1,2) isA. x + y = 3 B. x - y = 3 C. x…
- The points on the curve 9y^2 = x^3 , where the normal to the curve makes equal…
Exercise 6.1
Question 1.Find the rate of change of the area of a circle with respect to its radius r when
(a) r = 3 cm (b) r = 4 cm
Answer:(a) We know that area of a circle (A) is A = πr2
Then, Rate of change of the area with respect to its radius is given by,
![](data:image/png;base64,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)
When r = 3cm
Then ![](data:image/png;base64,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)
Therefore, the area of the circle is changing at the rate of 6πcm2/s when its radius is 3cm.
(b) We know that area of a circle (A) is A = πr2
Then, Rate of change of the area with respect to its radius is given by,
![](data:image/png;base64,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)
When r = 4cm
Then ![](data:image/png;base64,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)
Therefore, the area of the circle is changing at the rate of 8πcm2/s when its radius is 4cm.
Question 2.The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm?
Answer:Let a be the length of a side, V be the volume and S be the surface area of the cube.
Then, V = a3 and S = 6a2 where a is the function of time t.
Now, It is given that ![](data:image/png;base64,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)
Then, by using the chain rule, we get,
![](data:image/png;base64,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)
………………(1)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZQAAAAtCAMAAAC+qW2bAAAAAXNSR0IArs4c6QAAAJ9QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZmY6ZpC2ZpDbZrbbZrb/kDoAkGYAkGY6kLaQkLbbkNv/tmYAtmY6tmZmtpBmttv/tv//25A627Zm27aQ29uQ29u229v/2/+22////7Zm/9uQ/9u2//+2///bMc1DGQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAF00lEQVR4Xu1ba3fbNgwV02bT1i5ZvTXbOq/NlK2TvUdlW///t43gWyJFAiJ6FnvWh5ycGLgALkSQhJGmuT5XBq4M/AcM/PVWiHsmu5xYTC49b5jx99d/JDw8fffY7G5+bZr+7iM6gAIWBQpt8/kLDkL8jPPSSo7bbz8taRy+kklpDq9KkGisFBTeZUxgvGgYiwiZQSVl97UQL9/lxbXkuH1jxMan9otZenr9mc5N7kFjpaC0MtfDi2a4vPkJ494ItN8CV+Ne1n71myawlRH2N4+S7TnFM1wl2QwvTO0a2vu/ZxI7u6X0BShtFYWVgNLKXA8vGnApX+6dsK9uzs1O0n7cQsnvxLvmuLHUKnrG7ZeQnteLZcmnT4sCoW20sHq3zReXimIChZWA4qWRF80G1ZVeS+CwAy57WPf+t+Yg18w3KiklCCtpXvGmOW1Mchq1BiEdg/wx6HfY17jEi0LDmkE5ZWe2ZsXwopnQFTGKZdTTufWhCulBvuzjL/BbL+6zq8RLDrDYIAO2rsvX/dPYyb+eNkI+prBkfKJiTaCcsjOLCnxBiBfNGBlkJdLHUMxzfO/Pqio9aolpdp/EzY+ZtHhJm5TuxQe4lhidqARkVi8VawIVuAwFt3Jz4UWzOdi3QiBzAi/yneVdRXPawGZkAjs+iNvU7UNZCiR7vdjGrZAH431rtjNCUshYYVImLlcnhRfN5uQApKiNAvXsW1O+dJ2e+bSPDrgONJA0K8XsQkDYuH+Qr8bMh+WVQsZaSErSLIqGVFgMaAZWM1PepZ0Xgz6omVPPLCmNWQSJwBaJlCpQ3OM6srynkLFCKK+cNrs2KRxoxrY5Afntu+STWllwMlBlTOcG9pTT91CTlpPiJR392qh8i9XSmZcvTV7yoWJNoLxy0mwp/tnnvGiTlaL368KjC5be1NV1bIDCB3eXB0jKRqbp6A65MZaTdHeLQytVdlIXEn38Ybbj5u4pRKwplFNOmi2xMP+cF82g97Dboi6PI9wbj6rcqSjHTiZllGeu+z9byezulTwwZI5fXtLdwuGIf/soHXmSh7B/NtNNJXejJ2JNobxyyiw1Kbxo1rrqnQAzxef4Xu7G6gTbwXVCoNoAKdTsvdAoFC/0bqkXmxFYqCIBly2Q6xLryMtdYstQEQsPddmkF6Nb+A7E6VG+BClgUaBiv/cP6lo968AW41sQ4EVb68WZ6x3fCn0tm3Vg14XFi7bOh/PXOtx9NGf8oAO7OixetNVuXIBiePFSt4GqzjIv2gXQuy6EkEa47NZ1lnnRyBGZQ7M8NyObncsWGKGyYSTtBDT6lgOis5x2ei2a9RtHhb6vTJ6FwBOSxT8tcVhUzLmFdxnMexqDy1WQFFa0TFwEKsgL6OwUXFLs985VvWBeNDKZuIWGgmWEqihfvgObaGjHuIXyRUSjlS8Uq+csZN9t24Gt6yzzop0zr9Z3P6dBiKbX5xTXgXWd5WeAlg+jYkKwQjXyqYA1oRFld4QOrHgph+J8B9Z2lj8j2iSw9RPRFROCFaqJrGC/1Q5GaQjLZlmUKQo/7uiWt52uprupzojrOuQVqpGfFCyKbJkQJjQ/7hiaXEes/irXTg6VQ5hYXK+aTgrSjQqXEwEyoaVbbGa6msKrnRCUxVaYnipWvUI1ToiZ1MS4wWlXVgiCZQwzanQhaLG56epl5XlDLh5WxBhWMhWqiUViJzXLK4XTLm8UMiw97hi02Px09SKxUUMuHlZEJ6VCNbJBwaLIloNhRYvHHYPp6rwvQe8nHsEqh2EkKlQjGxQsimw5GF40ac+NO6qdejJdnfXmmhRPD3tS5BAXTAmlZkaX69dUusKnCtVLXilqCo42bjmXDoYVyzvslMsK1SgpFCyKbLl8caL5cUdaiy2S9sOKkOHfkP9SoYKtUI3IomBRZMtZYUTz44604c1IOhhWlMOS+DaHjLZCNV4qblIT/k8m7wanXd4o/LgjbXiTJl1+0a4SVwb+Bwz8C7V57WNwp87RAAAAAElFTkSuQmCC)
So, when a = 12cm, then, ![](data:image/png;base64,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)
Therefore, if the length of the edge of the cube is 12cm, then the surface area is increasing at the rate of
.
Question 3.The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.
Answer:We know that area of a circle (A) is A = πr2
Then, Rate of change of the area with respect to time is given by,
![](data:image/png;base64,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)
It is given that
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFoAAAArCAMAAADPCG9UAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmY6OmaQOma2OpDbZgAAZjoAZjo6ZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmttvbttv/tv//25A625Bm27Zm27aQ29uQ2////7Zm/9uQ/9u2//+2///bMgBuYwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABzUlEQVRYR+1Wa1PCMBDMVTAKgiCUV/FRRZqQ////vLskpE4pj5LOyIz3ocMQ2G42e5sT4narAJi2xb5oD1rJ1li3BK3GAE/ImgT/lMkqnuhKToSZk9ZK9iabh4jQWXcrkDFDP8djjEi7EeGx1rEFv01ok96jIE7ryObOk6XQL3SM0VvSrCEZfkuYKgkQ1yJR/dYqWNGknc3XI0DnVLNm5IpLK6PzTk+Q2o2auIz55CeCvbj7uJSy/33Gf6WwTKY2KmEo+MFlu0TPcNl9c+6L9GLwzokzERp3jlG5wn30liJ3OikKTpN2t2berQPNwFcQdzcCGGBMCg7LEGrYDD7ZclrhWFJ9sYfwH47uYCNREJtov/LSQZuUV9bQeztXiPC7Ahu2HtqbGm3aqT3OQ4JYosegg6k38gJ/281ShFkbHBDEmVr1UfCs9hirUhnqFo2HT+bG6/R1Vbr27N3nTM0GGrvXnzU96QVG45C9gUbuLFEcNHZ47HejZ7jAVqKKEtVs6krZa/fKQYFNfRi6UR4GKGfqMrafntbUNM3TRVSZhenpStZVLcL0FBu6NOL8Q5PwpekptiB4QfjpiYIAkyFe7acnivCkyX0cj8sfQPoBPYgpcLsuNnoAAAAASUVORK5CYII=)
Then, ![](data:image/png;base64,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)
Thus, when r = 10cm
Then ![](data:image/png;base64,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)
Therefore, the rate at which the area of the circle is increasing when the radius is 10 cm is 60π cm2/s.
Question 4.An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?
Answer:Let x be the length of a side and V be the volume of the cube, then
V = x3
…….. by chain rule
It is given that
![](data:image/png;base64,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)
Then, ![](data:image/png;base64,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)
Thus, when x = 10cm
Then ![](data:image/png;base64,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)
Therefore, the volume of the cube increasing when the edge is 10 cm long is 900 cm3/s.
Question 5.A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?
Answer:We know that area of a circle (A) is A = πr2
Then, Rate of change of the area with respect to time (t) is given by,
……………….. by chain rule
It is given that
![](data:image/png;base64,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)
Then, ![](data:image/png;base64,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)
Thus, when r = 8cm
Then ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMoAAAAhCAMAAACvOMTbAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kLbbkNv/tmYAtmY6tmZmtpA6tpBmttu2ttvbttv/tv//25A625Bm27Zm27aQ29u22////7Zm/7aQ/9uQ/9u2//+2///brRhgXAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADNElEQVRYR+1Xa3fTMAy1C+sWYGvLymBrxtYHDDbT0pDG//+fIcnPNopPH1sP4cwf0p5WVnSlq2tZiNf1j2ZAyTOMrBrSR7vXlDAU52+e2o0DolcIRX/9MRy1GsrqSkoiWDEQpjptXTo/W+o7hDCbiKLVDCvfTQzBygyqI1vAsN9AowHHHA9FXUK75IcwTD/0ItlQ/ccXYWr1aSzmHch/bVXDwXKVny5X1PIz1ma7kHT+cRlblucvVmLKv1lFFHGRyf53+SWX8FshZXctnHUQi8/Gg76XsreZc51DWWHNP0jZucZv0QuZZAQ71l0yfcggu/ZSKpA6owo670IhNxXCSobq3AAcSe9SibwEO95dCso8tEq1z/lR9h+VCZ9qWmwohG0z+zElEImyRHasuxQSFTX9vpproUwREU45yAyzLkEAqTNciCQfjnNMXJEd6462lKBVnRGwfvQrA9GiB51/orBtaNysbsGOFbXGjBgoSAiE0l3OJuX7JwvCtV8hkWC2FxNE9na8O0KS3QhUozK7mAglL8ZCgd9qGB0aVHf0oO+6TWFPXb5RDdyKoZgI6lDEAs4nt8fwDBbjztkZR5vuaJPdTQUvM2g/W/oQNHVjNcS/esKH7L4kWJqCYslnXqlsI3kodaferhEKRWiqQ1DcI/JlGTyTFz9TzcX8t04wZGmtKnFgIa91X8HOEWzdHezYAopjNQj7SeNQnyCYsH2KOatBsXceMgEobniouYvsWHfbQQm9uMh2m1MsiYg+RoxRCw2BvYIRwQ2zfF6bqkJ2rDvY4aeoRoLZQ6XswcumjW3PUg8FhCIExaBmFQgOWP9n4l+sJIwv9ohMHffBjnWH71EghvobziCh7cMxbxNJWQSlu9qhKvoeR+e3OJLgFadP2oTk0DnNKe6wwoHkZEypSJ32NOAYO9Yd/gEHCVjgyD4Kj5BkV7bVLViYeBD/DpjYgnHnYXoGa3Czw8+8/72Gso23HnEytgVgRzy63B+6jnNf8VEyY5G93Bey+5ClRvxDkT73/vimYnz7yz3oE8lae1e4EbepJGy+/yMo7nKfPgXawTpzub+GQ+g5ZKwdmF+jPFYG/gKVHW9s3TFgZQAAAABJRU5ErkJggg==)
Therefore, when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing is 80π cm2/s.
Question 6.The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?
Answer:We know that circumference of a circle (C) is C = 2πr
Then, Rate of change of circumference with respect to time (t) is given by,
……………….. by chain rule
It is given that
![](data:image/png;base64,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)
Then, ![](data:image/png;base64,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)
Therefore, the rate of increase of its circumference is 1.4π cm/s.
Question 7.The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8cm and y = 6cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.
Answer:(a) It is given that the length (x) is decreasing at the rate of 5 cm/minute and the width (y) is increasing at the rate of 4 cm/minute, then we have,
![](data:image/png;base64,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)
The perimeter (P) of a rectangle is given by:
P = 2( x + y )
![](data:image/png;base64,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)
= 2(-5 +4) = -2 cm/min
Therefore, the rate of change of the perimeter is -2cm/min.
(b) It is given that the length (x) is decreasing at the rate of 5 cm/minute and the width (y) is increasing at the rate of 4 cm/minute, then we have,
![](data:image/png;base64,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)
Now, the area (A) of the rectangle is given by
A = x × y
![](data:image/png;base64,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)
= -5y + 4x
So, when x = 8cm and y = 6cm, then,
![](data:image/png;base64,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)
Therefore, the rate of change of the area of the rectangle is 2cm2/min.
Question 8.A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.
Answer:Let V be the volume of the sphere, then
![](data:image/png;base64,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)
Therefore, Rate of change of volume (V) with respect to time (t) is given by,
…….. by chain rule
It is given that
![](data:image/png;base64,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)
Then, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Thus, when r = 15cm
Then, ![](data:image/png;base64,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)
Therefore, the rate at which the radius of the balloon increases when the radius is 15 cm is
cm/s.
Question 9.A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.
Answer:Let V be the volume of the sphere, then
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEoAAAAqCAMAAAAj+j0ZAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmaQOma2OpDbZgAAZgA6ZgBmZmZmZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY6ttu2ttv/tv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///bUAWgMAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABaklEQVRIS9WW20KDMAyGG+ahOnWChw0ZnQ7GsND3fz2TAnPimJTmxlzAVb8m+f8EhPjPkQUpU/pacqHq6IULpcKSCVU+7JlQ9VMqeFAmDgWhspW3hiW04Y+yufAUyI3iczv2a9hY+RyCV29JCGDe1mLH5GDE6Ruc9timbvA1205OskqWdFbZ3Ex8tZ9MUnBhUSWQ9erIy4CFbCDXBBxZnkk674eo6irHPUYjqi2qKU3hnI2JTapvt+1JLe+XBfY7l50ZyH1EbkN11w446RjVv1/LUJRnmn5Ag73sGNXvL1U4tr4/UCKbvT9/e8qlwF+qa3lHIo4M0roRrHn+CLS6g6kyiwo/cUmfOKZcnK4QZWJcBVoCjLTQIXOzmwNcsnydVbAWVcyyRhSJkrn087yC1CKWqJLFBwuojgAW09dZL4dCchVIG9LVQ4MNof3hHfb34+SUOKNxSlK0qMtwDd5RJThtj2wKOtfideALRtodWnsyBo0AAAAASUVORK5CYII=)
Therefore, Rate of change of volume (V) with respect to its radius (r) is given by,
…….. by chain rule
Thus, when r = 10cm
Then, ![](data:image/png;base64,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)
Therefore, the rate at which its volume is increasing with the radius when the later is 10 cm is 400π cm3/s.
Question 10.A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?
Answer:Let y be the height of the wall at which the ladder touches. Also, let the foot of the ladder be x m away from the wall.
we know that, by Pythagoras theorem,
x2 + y2 = 25
![](data:image/png;base64,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)
Then, the rate of change of height (y) with respect to time (t) is given by:
![](data:image/png;base64,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)
Now, it is given that ![](data:image/png;base64,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)
Therefore,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHEAAAAvCAMAAADjJNVlAAAAAXNSR0IArs4c6QAAAJ9QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmY6OmaQOma2OpCQOpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZjqQZmYAZrbbZrb/kDoAkDo6kDpmkGYAkGY6kLbbkNv/tmYAtmY6tmZmttv/tv//25A625Bm27Zm27aQ29uQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///bLQv/0AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACiklEQVRYR+1Yi1LDIBCEqsVnbX1rfFIVq9VQw/9/m3eQkKghuTTUGUeZ0ekoYe/27uhuGPsrK+X84odzTaMgmucjztfvSLFrESNHyc/ZYrL2SIGMhDgELNVOlwYqdsWFSTjfeNOCD2i0hBLBAkk8SoWaQ4tzZq5hm0k23hhLewIyiaxKOMUkgUpJi4NUKNyrkJkey9UHo1fj+mOyCf7D7sNf5rIbp8Bfvhw3JnE4WuyOAnFXEE0yZClmvPzCI9xSwepUEJFW2W9K5DAPODs5C02Ji8rdANlk/4Q0TCESlO0J4BWKk01CHaEGt2xx7IZI8kC1aTzrTailkWP7A1kEDjNTPhi9CAuZ0u6LEH7eRmMY7QHkyAnxpz1Hg0ZFsQuo7zoa3QC+7TbJ6PpHU4TZ5Xu9ZrFnxv+Pr4aBr3d0NBR/95cfQmfXbF3iT9Eij3DQyliNENsvOaKjKAeZtvQqGOkkyqPc+E5a4TcqYWWnES5gi0jVqnVfo0/twc5BijuBl4tyNsXaUFQAyq/S1YCMh9UurLODWzaz27wop+ZorlB1e1fTxbO4upWinMiq3oEySu9quiBanV4RrETEUt7bDm9G/GR6ZraMXRDfX/GJqW8S62pacqyYHuX6ho64uOFIZXZYqGiHpcW2aHDGpelJATDFpitFeQur81ebk5+N3NWYS3DGSd6rNfd/YXpQwDpLWYpyLcbsvmmucANTuTUpXY1NFHuifn0zPRVRDvK80emYZO3RX3He1bjaBJVng+kh3G/QnHY2kJrC1aDHaMix0fS0Q0IbzBx93tVYB76A5OufbjY97YjwJmPL8V64GobGjPO9hwAg3fQE0DPi6xpC8NQtfd9MUHEi7vsA/nZAjh/seKwAAAAASUVORK5CYII=)
And when x= 4 m, then
=![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6ADo6AGa2OgAAOgA6OjoAOmaQOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYA25Bm27Zm27aQ2////9uQ/9u2//+2///bZv7CGwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAa0lEQVQYV42OSxKAIAxDUxUV8a8IInj/Ywp14Ywrsnltms4E+MsPRK3DvXTwqkNQM3BUcZfOpzndpUMYpuhVuMRLWMF/uaJXufEvZxoqRuBed9hCs3/VTL9NCQeVTOAUM0KvY8fY2wj+++kBy4sE/+bXCNsAAAAASUVORK5CYII=)
Therefore, the height of the ladder on the wall is decreasing at the rate of
m/s.
Question 11.A particle moves along the curve 6y = x3 +2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.
Answer:It is given that equation of the curve is 6y = x3 + 2.
The rate of change of the position of the particle with respect to time (t) is given by:
![](data:image/png;base64,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)
………….. (1)
When the y- coordinate of the particle changes 8 times as fast as the x – coordinate
then, we have,
![](data:image/png;base64,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)
So, putting the value in (1), we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJsAAAArCAMAAABYdQrLAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOmY6Oma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGYAkGZmkLbbkNv/tmYAtmY6tmZmttv/tv//25A625Bm27Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///bvPw1lwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADHklEQVRYR+1Ya3fTMAy1w7bABuVZArTpIHTQZPX//3sofjtWHNnJOXyZP/R0q319LcnSlRl7Gf/JAj3nXzfaekMozajfjBvbEEqRG+qt7LYGSvz4iXgv4DY8rPHvimOe3k03Hj5wfg92azm/vXRj5MVziGRjKOJCOa27i6jVeya+j/HWVkcmGvgiPwrGEENloAyvj9PZ7e2FqQAWDdhNmrV/9SsD1E7FoOg4bWS2625ko4JkqO9VqIkm8jxhDxSKsE5fx9hsPiDrwKtyFBkOh6KS60b/hcMHvH76rJ2p/ps5cCgiCLajaEY3q3j7drzutNNl6GSOGSgaSm9c5k/vqgN7/gjcRAu26rkyGDp1aRccammV+r3Drp848erhT833Da/AblyRG2qSU/9CbnSZGocicRtzBGkiTFL+WRrX9wd2xpyxtDD63QYTZSU54JCUScGH4z+B0W90YnB+Or/hvPqShGip6Vdl64LR8j173ultrHrpqj1jZx31M6g21S3ses4SBr+9YijrQKfFmeGmYyntNSK3LotaLKAMJ7Od4ZaM9oREdIKF9UCtzxEG0yOb0HEVCTy9dL9S8tUKljHlpOW8muGP4Fpbsec4P9VwFyJFEkSW4wZm0sMscYIl/wqIxrtkTlNYbvLC6iicHgv+lhsmZb8VLAG3Oahgks/NS6Iu3sYcvJCJ0y0J8aYwlvRpe2drgQHUOTidwVI+ZU6w5DrVP5RURb2u3zqXaIuV5xBfsOSR8xsdWVCkwPBDqONvLytybyBYFDdqOQxyr4pPxS2sWTcH78hxNz7vcREIlohbWWOfOlwU+uRaHzm1qLFP3UwZCJ6koGkkNNimULSIRLWlWioBPbFL1JbYvlMoGrcZoW268dMYmkazlClGDIrGDe+eXDfucS8LNxSKxo0hPSA8NdjG3nEr6gFxKCI1P9jtEq+rdNyKemfIcvaNoKBPi98cUMCyN4eV3JDGAztsmdlWckPeuLxu3Dii9I0LgaJGm5wXv/vZblwWtUcowMVvgzFUFrf4TdV045B8ocUfy0Nef+JtH0FlUXuZbC3wD2QrUCyp5fHjAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
So, when x = +4, then, ![](data:image/png;base64,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)
And
So, when x = -4, then, y = -62/6 = -31/2
Therefore, the points required on the curve are (4,11) are (-4, -31/2).
Question 12.The radius of an air bubble is increasing at the rate of
. At what rate is the volume of the bubble increasing when the radius is 1 cm?
Answer:As, the air bubble is in the shape of a sphere.
Therefore, V be the volume of the sphere, then
![](data:image/png;base64,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)
Therefore, Rate of change of volume (V) with respect to time (t) is given by,
…….. by chain rule
It is given that
![](data:image/png;base64,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)
when, r = 1cm
Then, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALkAAAAhCAMAAACyRvpAAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjpmOjqQOmZmOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLaQkLbbkNv/tmYAtmY6tmZmtpA6tpBmttvbttv/tv//25A625Bm27Zm27aQ29u22//b2////7Zm/7aQ/9uQ/9u2//+2///bWOiDqAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADDUlEQVRYR+1Y23baMBCUSKA4DZTgNG2DKQHTpo2AFte2/v/PuivrYhsbLEWEl+jBh3Msj2ZXo9kVhLyPi2SA0SGPKJ2SNKDDizBwXjQekixEzuzqxRmk+iFfv0kOGKzCBntC4qkn4ptPD2dnnj1QihpJb1Yk/+wr5ZCJczPn0XDPv8MqPJqSxFfKUXjnZo6pLlZhg79z+O04/sDW3ZW/9cV8c0t735pIGebpzVNlaasQ8vsl2fbKgXtizud1YMUrD+/2WfQBTicY48yKbX2ySIIaPEZMP6MEnJSykwR08pPifgh3OTpYJavF1N0XxZeVTgmDYw8FomXwHWir31Wa2eJR44B/u4w0OGAOzqRKwLa71mL6CAWkY+2I6bVmnodOssjDr3Xm6eS3Kl6sO3EiMsc6i3MXKL5Jx2hr+xJPhcr4ApWg1SCZJ0A8sUpIAswrWClIqDdLKJ1tAvAp8SD5/QqaEgkMFg6csjnMs8gTScZ7wXy9Sj++GLSCeR5iKFbMY/iwjJUGKKEZII9WsCGjJRHHCgLQriiOKo8GUIEGbXKPVV6pEggGL092A/Njx+YQC2fLRBqsWBqEeJEGcLx1rhW68JA8xFdjohmqH212ACUWmTOEPcx57SsJdiQeLNkiAIUlCOmQJP3qJspv1nT0y8JgQH7F6MT8JHChWEvmysy3t7Tf2mA177A/tcRDWTtscm7MfBdY+rpkjt6kNejU0gvBiv5OY5ldKHSuHmb7pJmnY2zHW09o83bLGopk4Qj9E0WwVldzcUM5MQqPEJcBg8WgPPEfYANCkMUJrVRhaebCgyy7f9C6cBr0Mx6hV/FFANK/LvVynZhLLSIvg4X+11/ijZLOzMPkQG1KNocZEyk2uIieStPl31caOk3HsZFpD+egXX995M0NoadmWtM7bNdfzVwVgBKQvIgmdPAcnOxzLQg0764FQHVquTMv3uiLKJiEMBxfo9yu+8Ks4Ojr3OmLhdX6Fu26Fa6ZfCbmNu26I3N9EfWac/t23YG+vIh6/U/RoV13YP7+yRtn4D+SPFVlhE66/QAAAABJRU5ErkJggg==)
Therefore, the rate is the volume of the bubble increasing when the radius is 1 cm is 2πcm3/s.
Question 13.A balloon, which always remains spherical, has a variable diameter
. Find the rate of change of its volume with respect to x.
Answer:Let V be the volume of the sphere, then
![](data:image/png;base64,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)
It is given that, Diameter = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, the rate of change of its volume with respect to x is as
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the rate of change of its volume with respect to x is
cm3/s.
Question 14.Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?
Answer:Let V be the volume of the cone, then
![](data:image/png;base64,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)
It is given that, h= ![](data:image/png;base64,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)
⇒ r = 6h
Therefore, ![](data:image/png;base64,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)
Now, the rate of change of its volume with respect to time (t) is given by:
Now, the rate of change of its volume with respect to x is as
……… by chain rule
…………….(1)
Now, it is also given that
and h = 4cm
So, putting the value in equation (1), we get
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAL0AAAAuCAMAAABK++jvAAAAAXNSR0IArs4c6QAAAKJQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkGYAkGY6kLaQkLbbkNvbkNv/tmYAtmY6tmZmtpBmttv/tv//25A625Bm27Zm27aQ29uQ29v/2////7Zm/7aQ/9uQ/9u2//+2///bylOmxQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADcElEQVRoQ+1YfX/SMBBOOif1bYLgplBxUpRV0NWWfP+v5t0lbbKyJk3TCe7H/bFl3eXy5HIvycPYWU7VAznnC5bxaHWqAO24ckDP8gD0uxvaudhdc34Z6ARlq7snizgIfXnNL+5wtZTPWTmT455S2+o+Pwx9Md5kCv0I1szwIPuKttXNQgGH/V76/tua82m3WQ0thZ6+UhgGiGnLZaaI50x8objn7zYi7Rf85oppUOTA2XnMT1/eK3fJ1O3nOGNFisIQ8UC/n2Go6LjvubZeUST9Yk/v94joRYKJGyRHRJ+OIBDDxAO99JXuVoGRk1EWhcWOB3qWRbesvKkTtm/WylJVvIJfIg1E71H2xJpHk18xX8BlJ1oVMV55PEUsYRZ/8Rl7LUkAem3LE8RZ/eyB5+2Bn96JeUr+6FnUT2ULmUdJPT7m/UwWYi1u+Kp2wxS3rtcOww2LRF2im7vCv9uwPKbr963Fst3I4aQavZfXjqT8vCLn/8raxpE/XcUUP64epUOy8WawsDvsVsSmhYtIPrQ8Toq3Q9hvQ9jpXi+2bzRVJtYxPkVM0W9azYdVanT1t4mKZlwigsu2l1AwuVZI8SWTyPKfx5PfzRVyVYYNPkyr0bOrXYqY7GbRnLGt7wuhE5uWaqosBxqoKdWL3ODDDDW7a/azT4hemSCapqNUbBowaVwRkpaZRDbtZ4o6EMuq1UwlryJFvU1rNfhkp0rSKZHAFfoR04btuzDYNPedoFxS9ajTZL0qXt8p3AYJrdA/yCY6uRbJr+7l7Bx4XLaFoWnYht9g01zoseuN8VDTi69Af05waENvqMGc9nDYf1xVBPwO3sjqiV+7xYK+yaY5wm0XQ+SIhENp3MXEw9WLGFSGHJpqNvQUVOD7bAGnCEPJQmvDw6GHs50CLPIjedPie1PNhh66jZSFnCF/Pgl69I6CRR52ozfJfUvoY7yoNKfK0Al9k01rty+rBmWiZLllxsBQVpvDmmOoATA7zaNqjj5UbdgWzJpNQ7d+b2+JAhtVKU81Br5yS+GJnoWJf1ZVuaOKqRJPqzlboZySYULJblUZtidizaYxBrya7TpSImlGhYZhj7i8xRH6F/aFzV31WoMP02rM0WuJygMbdBlpGnZUkmH+bWtIrlvIMAiCrBzrjhkEWk/+F/f7gaCezXTwwF+Q5W1GMgnj+gAAAABJRU5ErkJggg==)
Therefore, the height of the sand cone increasing when the height is 4 cm is
cm/s.
Question 15.The total cost C(x) in Rupees associated with the production of x units of an item is given by
C(x) = 0.007x3 – 0.003x2 + 15x + 4000.
Find the marginal cost when 17 units are produced.
Answer:Marginal cost (MC) is the rate of change of total cost with respect to output.
Then, ![](data:image/png;base64,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)
= 0.021x2 – 0.006x +15
So, when x = 17 then,
MC = 0.021(172) – 0.006(17) +15
= 0.021(289) – 0.102 + 15
= 20.967
Therefore, the marginal cost when 17 units are produced is Rs. 20.967.
Question 16.The total revenue in Rupees received from the sale of x units of a product is given by
R(x) = 13x2 + 26x + 15.
Find the marginal revenue when x = 7.
Answer:Marginal revenue (MR) is the rate of change of total revenue with respect to the number of units sold.
Then, ![](data:image/png;base64,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)
So, when x = 7 then,
MR = 26(7) + 26 = 208
Therefore, the marginal revenue when x = 7 is Rs. 208.
Question 17.The rate of change of the area of a circle with respect to its radius r at r = 6 cm is
A. 10π
B. 12π
C. 8π
D. 11π
Answer:We know that area of a circle (A) is A = πr2
Then, Rate of change of the area with respect to its radius is given by,
![](data:image/png;base64,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)
When r = 6cm
Then ![](data:image/png;base64,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)
Therefore, the area of the circle is changing at the rate of 12πcm2/s when its radius is 6cm.
Question 18.The total revenue in Rupees received from the sale of x units of a product is given by
R(x) = 3x2 + 36x + 5. The marginal revenue, when x = 15 is
A. 116
B. 96
C. 90
D. 126
Answer:Marginal revenue (MR) is the rate of change of total revenue with respect to the number of units sold.
Then, ![](data:image/png;base64,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)
So, when x = 15 then,
MR = 6(15) + 36 = 126
Therefore, the marginal revenue when x = 15 is Rs. 126.
Find the rate of change of the area of a circle with respect to its radius r when
(a) r = 3 cm (b) r = 4 cm
Answer:
(a) We know that area of a circle (A) is A = πr2
Then, Rate of change of the area with respect to its radius is given by,
When r = 3cm
Then
Therefore, the area of the circle is changing at the rate of 6πcm2/s when its radius is 3cm.
(b) We know that area of a circle (A) is A = πr2
Then, Rate of change of the area with respect to its radius is given by,
When r = 4cm
Then
Therefore, the area of the circle is changing at the rate of 8πcm2/s when its radius is 4cm.
Question 2.
The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm?
Answer:
Let a be the length of a side, V be the volume and S be the surface area of the cube.
Then, V = a3 and S = 6a2 where a is the function of time t.
Now, It is given that
Then, by using the chain rule, we get,
………………(1)
So, when a = 12cm, then,
Therefore, if the length of the edge of the cube is 12cm, then the surface area is increasing at the rate of .
Question 3.
The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.
Answer:
We know that area of a circle (A) is A = πr2
Then, Rate of change of the area with respect to time is given by,
It is given that
Then,
Thus, when r = 10cm
Then
Therefore, the rate at which the area of the circle is increasing when the radius is 10 cm is 60π cm2/s.
Question 4.
An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?
Answer:
Let x be the length of a side and V be the volume of the cube, then
V = x3
…….. by chain rule
It is given that
Then,
Thus, when x = 10cm
Then
Therefore, the volume of the cube increasing when the edge is 10 cm long is 900 cm3/s.
Question 5.
A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?
Answer:
We know that area of a circle (A) is A = πr2
Then, Rate of change of the area with respect to time (t) is given by,
……………….. by chain rule
It is given that
Then,
Thus, when r = 8cm
Then
Therefore, when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing is 80π cm2/s.
Question 6.
The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?
Answer:
We know that circumference of a circle (C) is C = 2πr
Then, Rate of change of circumference with respect to time (t) is given by,
……………….. by chain rule
It is given that
Then,
Therefore, the rate of increase of its circumference is 1.4π cm/s.
Question 7.
The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8cm and y = 6cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.
Answer:
(a) It is given that the length (x) is decreasing at the rate of 5 cm/minute and the width (y) is increasing at the rate of 4 cm/minute, then we have,
The perimeter (P) of a rectangle is given by:
P = 2( x + y )
= 2(-5 +4) = -2 cm/min
Therefore, the rate of change of the perimeter is -2cm/min.
(b) It is given that the length (x) is decreasing at the rate of 5 cm/minute and the width (y) is increasing at the rate of 4 cm/minute, then we have,
Now, the area (A) of the rectangle is given by
A = x × y
= -5y + 4x
So, when x = 8cm and y = 6cm, then,
Therefore, the rate of change of the area of the rectangle is 2cm2/min.
Question 8.
A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.
Answer:
Let V be the volume of the sphere, then
Therefore, Rate of change of volume (V) with respect to time (t) is given by,
…….. by chain rule
It is given that
Then,
Thus, when r = 15cm
Then,
Therefore, the rate at which the radius of the balloon increases when the radius is 15 cm is cm/s.
Question 9.
A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.
Answer:
Let V be the volume of the sphere, then
Therefore, Rate of change of volume (V) with respect to its radius (r) is given by,
…….. by chain rule
Thus, when r = 10cm
Then,
Therefore, the rate at which its volume is increasing with the radius when the later is 10 cm is 400π cm3/s.
Question 10.
A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?
Answer:
Let y be the height of the wall at which the ladder touches. Also, let the foot of the ladder be x m away from the wall.
we know that, by Pythagoras theorem,
x2 + y2 = 25
Then, the rate of change of height (y) with respect to time (t) is given by:
Now, it is given that
Therefore,
And when x= 4 m, then
=
Therefore, the height of the ladder on the wall is decreasing at the rate of m/s.
Question 11.
A particle moves along the curve 6y = x3 +2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.
Answer:
It is given that equation of the curve is 6y = x3 + 2.
The rate of change of the position of the particle with respect to time (t) is given by:
………….. (1)
When the y- coordinate of the particle changes 8 times as fast as the x – coordinate
then, we have,
So, putting the value in (1), we get,
So, when x = +4, then,
And
So, when x = -4, then, y = -62/6 = -31/2
Therefore, the points required on the curve are (4,11) are (-4, -31/2).
Question 12.
The radius of an air bubble is increasing at the rate of . At what rate is the volume of the bubble increasing when the radius is 1 cm?
Answer:
As, the air bubble is in the shape of a sphere.
Therefore, V be the volume of the sphere, then
Therefore, Rate of change of volume (V) with respect to time (t) is given by,
…….. by chain rule
It is given that
when, r = 1cm
Then,
Therefore, the rate is the volume of the bubble increasing when the radius is 1 cm is 2πcm3/s.
Question 13.
A balloon, which always remains spherical, has a variable diameter . Find the rate of change of its volume with respect to x.
Answer:
Let V be the volume of the sphere, then
It is given that, Diameter =
Now, the rate of change of its volume with respect to x is as
Therefore, the rate of change of its volume with respect to x is cm3/s.
Question 14.
Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?
Answer:
Let V be the volume of the cone, then
It is given that, h=
⇒ r = 6h
Therefore,
Now, the rate of change of its volume with respect to time (t) is given by:
Now, the rate of change of its volume with respect to x is as
……… by chain rule
…………….(1)
Now, it is also given that and h = 4cm
So, putting the value in equation (1), we get
Therefore, the height of the sand cone increasing when the height is 4 cm is cm/s.
Question 15.
The total cost C(x) in Rupees associated with the production of x units of an item is given by
C(x) = 0.007x3 – 0.003x2 + 15x + 4000.
Find the marginal cost when 17 units are produced.
Answer:
Marginal cost (MC) is the rate of change of total cost with respect to output.
Then,
= 0.021x2 – 0.006x +15
So, when x = 17 then,
MC = 0.021(172) – 0.006(17) +15
= 0.021(289) – 0.102 + 15
= 20.967
Therefore, the marginal cost when 17 units are produced is Rs. 20.967.
Question 16.
The total revenue in Rupees received from the sale of x units of a product is given by
R(x) = 13x2 + 26x + 15.
Find the marginal revenue when x = 7.
Answer:
Marginal revenue (MR) is the rate of change of total revenue with respect to the number of units sold.
Then,
So, when x = 7 then,
MR = 26(7) + 26 = 208
Therefore, the marginal revenue when x = 7 is Rs. 208.
Question 17.
The rate of change of the area of a circle with respect to its radius r at r = 6 cm is
A. 10π
B. 12π
C. 8π
D. 11π
Answer:
We know that area of a circle (A) is A = πr2
Then, Rate of change of the area with respect to its radius is given by,
When r = 6cm
Then
Therefore, the area of the circle is changing at the rate of 12πcm2/s when its radius is 6cm.
Question 18.
The total revenue in Rupees received from the sale of x units of a product is given by
R(x) = 3x2 + 36x + 5. The marginal revenue, when x = 15 is
A. 116
B. 96
C. 90
D. 126
Answer:
Marginal revenue (MR) is the rate of change of total revenue with respect to the number of units sold.
Then,
So, when x = 15 then,
MR = 6(15) + 36 = 126
Therefore, the marginal revenue when x = 15 is Rs. 126.
Exercise 6.2
Question 1.Show that the function given by f (x) = 3x + 17 is strictly increasing on R.
Answer:Let x1 and x2 be any two numbers in R.
Then, we have,
x1 < x2
⇒ 3x1 < 3x2
⇒ 3x1 +17 < 3x2 +17
⇒ f(x1) < f(x2)
Therefore, f is strictly increasing on R.
Question 2.Show that the function given by f (x) = e2x is strictly increasing on R.
Answer:Let x1 and x2 be any two numbers in R.
The, we have,
x1 < x2
⇒ 2x1 < 2x2
⇒ e2x1 < e2x2
⇒ f(x1) < f(x2)
Therefore, f is strictly increasing on R.
Question 3.Show that the function given by f (x) = sin x is
(a) strictly increasing in ![](data:image/png;base64,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)
(b) strictly decreasing in ![](data:image/png;base64,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)
(c) neither increasing nor decreasing in (0, π)
Answer:(a) The function is f (x) = sin x
Then, f’(x) = cos x
Since for each x ϵ
, cos x > 0, we have f’(x) > 0
Therefore, f’ is strictly increasing in
.
(b) The function is f (x) = sin x
Then, f’(x) = cos x
Since for each
, cos x < 0, we have f’(x) < 0
Therefore, f’ is strictly decreasing in
.
(c) The function is f (x) = sin x
Then, f’(x) = cos x
Since for each x ϵ
, cos x > 0, we have f’(x) >0
Therefore, f’ is strictly increasing in
……………….(1)
Now, The function is f (x) = sin x
Then, f’(x) = cos x
Since, for each x ϵ
, cos x < 0, we have f’(x) < 0
Therefore, f’ is strictly decreasing in
…………(2)
From (1) and (2),
It is clear that f is neither increasing nor decreasing in (0, π).
Question 4.Find the intervals in which the function f given by f (x) = 2x2 – 3x is
(a) strictly increasing (b) strictly decreasing
Answer:(a) It is given that function f(x) = 2x2 – 3x
⇒ f’(x) = 4x – 3
If f’(x) = 0, then we get,
![](data:image/png;base64,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)
So, the points
divides the real line into two disjoint intervals,
and ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQMAAABCCAIAAADcwCThAAAAAXNSR0IArs4c6QAABOtJREFUeF7tnU1IVFEUxzVaFAXiR9EHBaHLJENF8CNw4SS4iNroxvYj4iIDZXaCIoq51Y2bcaEgBi0SxxZC6ipaCC3VwEgKNF0YtQjs6MtpGMW57947857v/d5q3vPec8/9nfP3XOe9+8w/ODjI44BA6AlcCD0BAEDgkABKIA8ggBLIAQgcE6AmkAsQoCaQAxCgJpADEEglwOqIfIAAqyNy4NwSGB4eXl5etui+aU2Ynp4uOjrkg7gl/qWeWnQUUxDISMAoG+Ues/axvb1dWlq6vr4uH6LR6Pj4eGFh4dujQ67LRW3LdITA2QSGhoaWlpZS2xhmY54JcXElFoslLYgMpqamnFO5nuaoyUD0hUAagZNKMMzGzKuj/NOO5ubmtFIli7bd3d2mpibnekFBQcZaRgMIuCWQTMaenp6Ghgbn1E42mvyykXpUVVUlqyM5ZDkkBcEpEc51VkcqbOOJTZVmtMlYEwyzMXNNOEO1xcXFfX19kUhEZNDa2trW1lZeXl5dXV1TU9Pd3S0/dav4ELafXNgM4ayzMWXDbMznqexsREXdZuTlysJInXp7WjoE5FvK2tra+vp6W0CMaoItJ7ADAc8JUBM8DgE1weMAHA9PTfBJIHDDYwIoweMAMLxPCKAEnwQCNzwmgBI8DgDD+4SAqRL2f/1Z//rTJ5PBjXASsJKB+koQDcj90faBjyufdsIZAGbtEwJrW/tN3Stjbz6b+KOjhKQGJhe+yGeT4ekLAVsEXr/fMtGDOyWgAVthw06WCGjrwcWdNZGBrIUoAlkKIWbtErh6+eJI9H7p7SuKZl0owbEomktbkLVH7jx/fFdxPJqlEeAes3lKJD58H5leS9oRDUSqrz9tuHmj6JK6cXerI7H77NGtd6/qok/uqY9By4wEVldX5SF7edS+rKzM2QfLoUFANCD5OfbigeSnKxnIWK6V4PiHHjTidEaXiooK2YQlzwXLw+0dHR12jYfBmokGHD6uV0cnscq3ueqrsTBExdUc01ZHc3NzXV1da2v/a70ra+Fs/O3Hb5m42yKQxkqzJqRaQQa28k9kEI/HZ2dnbRkMiR3RgKEM9FdHIUGcy2nK3wktLS17e3u5HJSxkgQs1ARoWiEwPz8vO3ErKysbGxutGMSIKwIowRWu7DaWnbgDAwPyihC7b3fLrtNBsY4SfBFJeXHgxsaGuCLbc+W1URa35/pieufBCZTgiyj19vbKK0LkfsLMzMzi4qIvfAqZExa+RQ0ZMcvT5R6zZaC65qgJuuToFywCKCFY8WQ2ugRQgi45+gWLAEoIVjyZjS4BlKBLjn7BIoASghVPZqNLACXokrPUr/HhtVP/Q4Ul85hRJcD9BFVStAs2AWpCsOPL7FQJoARVUrQLNgGUEOz4MjtVAihBlRTtgk0AJQQ7vsxOlQBKUCWVg3byrhf5RjUHAzHESQIowS9ZsbOzMzg46BdvwucHSvBLzEdHRzs7O/3iTfj8QAm+iLmzcZlNmx4GAyV4CP/f0LIu6u/vl738zrmc8hq83EcFJeSeefqIExMTiUTCefpIflZSUuLs7ufIJQGeO8ol7cxjiRjk7aiZ29HCNgFqgm2ixvYoCMYIdQygBB1qWerj/HkQi8WyZB+zZxCgFpMeEDgkQE0gDyCAEsgBCBwToCaQCxCgJpADEKAmkAMQSCXwF40h8WhKHIKRAAAAAElFTkSuQmCC)
So, in interval
, f’(x) = 4x -3 >0
Therefore, the given function (f) is strictly increasing in interval
.
(b) It is given that function f(x) = 2x2 – 3x
⇒ f’(x) = 4x – 3
If f’(x) = 0, then we get,
![](data:image/png;base64,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)
So, the points
divides the real line into two disjoint intervals,
and ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADIAAAAiCAMAAAAqN3oYAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmaQOma2OpDbZgAAZgA6ZgBmZjpmZpDbZrbbZrb/kDoAkDo6kGY6kGZmkGaQkLbbkNv/tmYAtmY6tpA6trbbttu2ttv/tv//25A625Bm27Zm27aQ27a229uQ29u22////7Zm/9uQ/9u2//+2///b7UKamwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABeElEQVQ4T71U3VqDMAxt5zZxqBOdyKYbU1nnL9C+/8OZpg10QtsrzcVWPnJ6ck5CGPvHaB+rGFu9/HZTmnQXQzAmLg99kswe4OG44JP7EFLMe57yHDLVesveJiE2VdzQjc0FJfanUbb6jEpDEh3tJg9K6miaRCuBKPk0DGHC0tA/gN4J7SGj20tjhLzdsY7QA5EZGiAzK+WYREwGVwu8vkk66+LtLFFMza36OAAmABvnh6jnhF9VTG04v7Z9NxDzOxZiXrF29ZrNtl+F1SuwJC9EW6il6vLNmUoiCDfR08k7LdX4rtYuxKvFpJVTbSixxORrLZAjYIoEaUHhgY5Dc2eQsk9Tcsz0ZdhKRf4MbTTDNUzwQ+iNHUvHKvyux4IqqrteWll771dDqXaitfmLyLh1BdHmEHkRhphNpMOe6qWKQJythEe5OqgifzpZiacW9CR6KqC5AkfM2W6/LXPWGE4dLtxgYZBzesmHHoRPHHRPtC/eV3/x4gcfjiI/dVarxAAAAABJRU5ErkJggg==)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPkAAABDCAIAAADRfDTUAAAAAXNSR0IArs4c6QAABMRJREFUeF7tnU9IFFEcxzU6FAXin6I/FMR6DDK0i7rBHnYTPERdPNl9RTxkoOzZRRTzqhcv28EFMeiQuHYQcj1FB6HjrkGRFLjpwahDYL98sNq0zbw38/Y9Xr/vnHaHN7/fe5/fxzev2Zmp8fDwsAEbCDAgcIrBGDFEEPhNAK7DAy4E4DqXSmOccB0OcCEA17lUGuOE63CACwG4zqXSGCdchwNcCMB1LpXGOOE6HOBCAK5zqTTGCdfhgAMEpqeni8VixI6quZ7P51uONvpAiakHJ79G7AoOBwElAso20n2Oktvu7m4sFiuXy/QhnU7Pz883Nze/PNpoP+2UjINmIKBKYGpqamNj4+RRIWxskM9KyTKZTLU9ib64uCi+0n5PV+TDoiUIBBL42/UQNnrXMI21tr6+Ps/JhRZPe3t7yWRS7G9qalI6+6AxCMgQqMo4NjYWj8fF1/A2Bv5JVRvQWaOrq4vWMLTRooUmdTHNi/1Yw8iQzBU+yDRDGw+BmmsYVRsV1jCUXizN6S9SWE66Uz7hPcojQyA5WpRphjaBroewsZGOkTmboI0WAqknm2szPVpCsQpCV/y6u7t7e3ujjFrtmmOUTDgWBOwSwLxulD/mdaO4/0yGed0ifKQ2SgCuG8WNZBYJwHWL8JHaKAG4bhQ3klkkoOb6wfef5U/fLHYXqUEgtIGyrpPl9JvfYPbt5rsKcIOARQKlnYPk6Obci/eqfQh2vWr5s7WP9Fk1AdqDQD0IPH+9o2q8n+uwvB5FQkyNBJSM/+dvSSQ6rVgwkWssDELVj8D5s6dn0jdjV8/5pAj43ZT+bjwLo8HUtUf3rtev0/93ZPxuGr2+hTdfZvKlahyyPHXn4oP45UstZ/yDB6zXH9698uppT/r+jehdRIQqga2tLboJm27Fbm9vF08zYgtBgCwnP+ce3yI/A0Wn+MH/NqVGMD5EJXwO6ejooBuy6Q7TgYGBoaEhvcE5RFO1XDBRvveLrm76r4o4sA49Rs8aZmVlZWRkpFQ6PiOHjsznwM9ff9BgZSZyDxOpef3kMRBdl1Ukei6XW15e1hWQSRyyPITosmsYJhBNDpPW6/39/fv7+yaTMs+lPK8z56Vr+Kurq/SEbmdnZyKR0BUTcfwJwHVrhrS2tmazWXodQ/Q3Wlkbg1OJ4bqFctHL0ra3tykxPUZJr9mJ+BilhQG4mRKuW6jb+Ph4KpWi6+tLS0vr6+sWesAypfI1R5aUtA0av5tqQ6keCPO6OjMc4SYBuO5m3dBrdQJwXZ0ZjnCTAFx3s27otToBuK7ODEe4SQCuu1k39FqdAFxXZxbhiMTtCzXfcB8hJA6VJYDr67Kk0M51ApjXXa8g+i9LAK7LkkI71wnAddcriP7LEoDrsqTQznUCcN31CqL/sgTguiwp7e3ozRl0/VF7WAT8FwG4bseNSqUyOTlpJzfXrHDdTuVnZ2eHh4ft5OaaFa5bqLx4wBSP3hlGD9cNA2+g1cvExAQ9VS0S01e8+stMDeC6Gc7HWRYWFgqFgrgrhva2tbWJ56yx1ZsA7oepN2G/+KQ7/t9wYwXAvG4Mde1EmNSNFQCuG0PtTSSW6ZlMxloPmCXGOZRZwRkPF/M64+IzGzpcZ1ZwxsP9BXE2S2ZMLAWjAAAAAElFTkSuQmCC)
So, in interval
f’(x) = 4x -3 < 0
Therefore, the given function (f) is strictly decreasing in interval
.
Question 5.Find the intervals in which the function f given by f (x) = 2x3 – 3x2 – 36x + 7 is
(a) strictly increasing (b) strictly decreasing
Answer:(a) It is given that function f(x) = 2x3 – 3x2 – 36x + 7
⇒ f’(x) = 6x2 – 6x + 36
⇒ f’(x) = 6(x2 – x + 6)
⇒ f’(x) = 6(x + 2)(x – 3)
If f’(x) = 0, then we get,
⇒ x = -2, 3
So, the points x = -2 and x = 3 divides the real line into two disjoint intervals, (-∞,2), (-2,3) and (3,∞)
![](data:image/png;base64,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)
So, in interval ![](data:image/png;base64,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)
f’(x) = 6(x + 2)(x – 3) >0
Therefore, the given function (f) is strictly increasing in interval
.
(b) It is given that function f(x) = 2x3 – 3x2 – 36x + 7
⇒ f’(x) = 6x2 – 6x + 36
⇒ f’(x) = 6(x2 – x + 6)
⇒ f’(x) = 6(x + 2)(x – 3)
If f’(x) = 0, then we get,
⇒ x = -2, 3
So, the points x = -2 and x = 3 divides the real line into two disjoint intervals, (-∞,2), (-2,3) and (3,∞)
![](data:image/png;base64,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)
So, in interval![](data:image/png;base64,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)
f’(x) = 6(x + 2)(x – 3) < 0
Therefore, the given function (f) is strictly decreasing in interval
.
Question 6.Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x – 5
Answer:It is given that function f(x) = x2 + 2x – 5
f’(x) = 2x + 2
If f’(x) = 0, then we get,
⇒ x = -1
So, the point x = -1 divides the real line into two disjoint intervals, (-∞,-1) and (1,∞)
So, in interval (-∞,-1)
f’(x) = 2x + 2 < 0
Therefore, the given function (f) is strictly decreasing in interval (-∞,-1).
And in interval (1,∞)
f’(x) = 2x + 2 > 0
Therefore, the given function (f) is strictly increasing in interval (1,∞).
Thus, f is strictly increasing for x > -1.
Question 7.Find the intervals in which the following functions are strictly increasing or decreasing:
10 – 6x – 2x2
Answer:It is given that function f(x) = 10 – 6x – 2x2
f’(x) = -6 – 4x
If f’(x) = 0, then we get,
⇒ x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAgCAMAAADDlWPAAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADo6OgAAOgA6OjoAOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYA25A625Bm27Zm27aQ29u2/7Zm/9uQ/9u2//+2///bQHBKxAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAdklEQVQoU62RSw6AIAxEW8X/HwUV8f7XFDEx0IWBxFm+Tsu0AERJ5Ji0bsc5TCATToaojJBj7IxFV2jVA8yY3sTTykzhla45KI+AYOT1qOwh5iefo5Cmnz1bg1i4M3UddGNYSpJEemNMcaFgN2D3rm6/yCXf2117NQOup0T8wQAAAABJRU5ErkJggg==)
So, the point x =
divides the real line into two disjoint intervals,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALQAAAArCAMAAADmI1KWAAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmY6ZrbbZrb/kDoAkDo6kGYAkGY6kGZmkLbbkNv/tmYAtmY6tpA6tpBmttu2ttv/tv//25A625Bm27Zm27aQ29uQ29u22/+22////7Zm/9uQ/9u2//+2///bePlUKgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADXElEQVRYR+1Z23LTMBCVAm0MoTRgoJAEQmsDwRCgvuj/Pw2tLrYsKdFK9gw8RA8dz9Q6Ol6d3T1SCLmM/z8C7OdzSq8eZiM6N56XWLG4J+1uMRvrWDz2+RAfsGLJ51R0S5p1/GTPjB4PiVa+Rr7ovFY8+U5I+nQ/HopMBTFLGe3+FraI7bYps905Cg8D1jxL02WXU3r7CCvUEO/JY8BDQAkxYUZB9dDpd8wEXbZL05cDqPDCZFIDLZBrKujOE+oBL8i6uhZbnDaaTJDu8rRQO4sqvBCZ1PWkJGpe8mAUU75cIIzxAqzrxPbAoK+0O0U2FWUgN8YLkK5SM7/dZ5SulbSQ23qOywjvPGmmYxWSUQgGW4KmrSNnd/k8qyFEXVOVAELCX29ia3slOhmMGfZVZiJCZTprRdq9iq9ZzUp1XgNo0sZViHxuMt3uE7uRbin/hnRqM1I9xR8h9iWjNwfC9tpdBHfB/Hhh59dQxOn2RwZPXIbvKH2pI812uERyaKhQ+0lX1wfSvv+WX93/RuKP9Lp8ZAXIpclePAjLzZ82hH3SiTjo5HwwHBpKVl7S3RvwfY1wQ/I5OGyZCV7qD2+dorj07yBbkYeGNHc9adNxdW+hHMl4sI+adO/xxg/ye06TBp8gvUIfYEX6LCB/30NDVlZvpCXR4imslBBpdrzjzVJH2kMa2YU9NM6QJiAm/jnVhn8VLmcsTWtlcOanIx3UnEtDkj5R8njSw+1AuVrJs0lwGDsmNn/QNJCW5SJW04Q4NKSmUXUaUaIM0tBk2w+QEAJc9NwK7hvusNXj5HrKR6PaOIK02cZLXqX/5HTbgLLh5MdDXdLF+lemWIfwTv5f1WmUYepy47jNjrxPODdLCMNkSGzUEV3A0XqmMlVHRFnTkidkPwq6IW1u+aNQ7KycGHkPF3C0njG1P84iy49BWt8smURQKjMmmC4PfbXUuzyC7E5WqOz8jUax/TSiIAx+OvEgbdvnOEl7CijGjxvTUq4QbMeTeqTvaWAtlJ6QcFnjePhUf6w5xB8K0NdiwxJWa49fcyyQyNoDk6NDXSyt1j410A5g0DWgPZFGEklQG/dgU696HUAE58hbcbExrDBIT7xUdwFRpKN+vlCnhYH01J8vHEAU58tLlwh4I/AX5qJit7rIxYQAAAAASUVORK5CYII=)
So, in interval![](data:image/png;base64,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)
x < -3/2
-4x > 6
-6 - 4x > 0
f’(x) = -6 – 4x > 0
Therefore, the given function (f) is strictly increasing in interval
.
And in interval ![](data:image/png;base64,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)
f’(x) = -6 – 4x < 0
Therefore, the given function (f) is strictly decreasing in interval
.
Thus, f is strictly decreasing for x >
.
Question 8.Find the intervals in which the following functions are strictly increasing or decreasing:
–2x3 – 9x2 – 12x + 1
Answer:It is given that function f(x) = –2x3 – 9x2 – 12x + 1
⇒ f’(x) = -6x2 – 18x + 12
⇒ f’(x) = -6(x2 +3x + 6)
⇒ f’(x) = -6(x + 1)(x + 2)
If f’(x) = 0, then we get,
⇒ x = -1 and -2
So, the points x = -1 and x = -2 divides the real line into two disjoint intervals,
(-∞,-2), (-2,-1) and (-1,∞)
So, in interval (-∞,-2),(-1,∞)
f’(x) = -6(x + 1) (x +2) < 0
Therefore, the given function (f) is strictly decreasing for x < -2 and x>-1.
So, in interval (-2.-1)
f’(x) = -6(x + 1)(x+2) > 0
Therefore, the given function (f) is strictly increasing for -2 < x < -1.
Question 9.Find the intervals in which the following functions are strictly increasing or decreasing:
6 – 9x – x2
Answer:It is given that function f(x) = 6 – 9x – x2
f’(x) = -9 – 2x
If f’(x) = 0, then we get,
⇒ x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABAAAAAgCAMAAAAsVwj+AAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6AABmADo6AGa2OgAAOgA6OmZmOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkLbbkNv/tmYAtv//25A627Zm29u2/7Zm/9uQ/9u2//+2///bgcWJwgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAe0lEQVQoU6WRyQ6AIAxEp+4b4oqC+P+/6XageDAhzq2TB5lOgQAZQVHLeFtLmKxzjo5nYKqcYfIRW5/C1nSrgyIqBCMu1jajl2Lzgb1PZEDKIPSJ5RT0+Ce8CqKS19kMWCK/ClwNelKv8hb+xUmq16zPWbMbPadgxucWBwARBDTuWm1GAAAAAElFTkSuQmCC)
So, the point x =
divides the real line into two disjoint intervals,
![](data:image/png;base64,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)
So, in interval![](data:image/png;base64,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)
f’(x) = -9 – 2x > 0
Therefore, the given function (f) is strictly increasing for x <
.
And in interval![](data:image/png;base64,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)
f’(x) = -9 – 2x < 0
Therefore, the given function (f) is strictly decreasing for x>
.
Thus, f is strictly decreasing for x>
.
Question 10.Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x – 3)3
Answer:It is given that function f(x) = –(x + 1)3 (x – 3)3
⇒ f’(x) = 3(x + 1)2 (x – 3)3 +3(x + 1)3 (x – 3)2
⇒ f’(x) = 3(x + 1)2 (x – 3)2[x – 3 + x + 1]
⇒ f’(x) = 6(x + 1)2 (x – 3)2(x-1)
If f’(x) = 0, then we get,
⇒ x = -1,3 and 1
So, the points x = -1, x =1 and x = 3 divides the real line into four disjoint intervals,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQgAAAAXCAMAAADA3yE3AAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmY6ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tpA6tpBmttu2ttv/tv//25A625Bm27Zm27aQ29uQ29u22/+22////7Zm/9uQ/9u2//+2///bNJLH8wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADQElEQVRYR+1Yf3MSMRBNsK2nWEHQorZYyqHiKWo5yPf/aGZ382OTS+4O2mH8g8wwMNy+l923m01yQpzHWYGzAjkF1LfrH8erU43WEbg3XxN6vBvPgFTzd49PoamHdwH8AL4YKto17CPcVkrvTo+UeE41nxwpw2a2RGT9ir5pZPisMRmloRprchJYq1+vpbyEORrCJTzfeiGIzsOTcTrO7Yvj1sXuRlpkdcVKKsnHjLU3OajVMLQW5eBB7OYDVCLQPB1YYSvC0DF4GkCcav7yqIKoR+vKCsHdS/JxYx1NBiqMhqG1ECV4WFGiA807hDB0HJ5EEGftBOTl/bWQ12uhFlKO8v3DCcFXQ5IPIggKLwllGobW4FlJeK85VvxYCGgKPwv4pZ/eSPnWBhSkxMAx9XFwxLnFkotGdbUWu4/fp5cPf1oKxnuLotNI8uWFoHTTYBrGQuwWZm9ymuswH1UJztfFmyUVTF3cCnVvewSjc3CYphEccaYc37/H9VhAEuh3cjAhfJM4WIgkNBRiP/WVyYQz0mHAdaGDKYHLNUvnCYfbgILgaPGYki2lHYPl/gOUISmqvjghuAlK66odPaCR4guNs1CmYWNpbDArerCpuBD76UTAh9UVT4mDayGawZGAiaVBwZcXwHvCimDRN3vEVtIu74RQm1mBZwZMWEKIgMPCbWaD4LJC4DKqBsvqVmc4v6k8d49oqwiqfSYE9AjSICdEkGILNz0iDA6FSHd53YbhBLMaDvvsGlSTNA7cNTg0I4TpZaYH2h6BxqEQtE3EPSKEg4uN4JCzxzkiZ6JlNcHD/mOs2o3t0ySU7xpITdYKzlK7ObUhJxykePcJKgIjx4xXcPKaRbuGh+dcM5zdJ8skgVroFSovPmPrAyetVYrPGaNRDmoZQmt9EIWpxtSO/Tlipf/7O5V3NXQK2BgmQq3kYPy7MEpYOgfPCWE4u+8a+2l4q3LLwK4GPJBYq1a+mIp8sP/GGiYm7j5Zeu960KGx5ey8La50z2wZ5tbirNr4IqoIGmvYnLjHXcN72k0XFVmPy2qLDs/5PqIrJ31un8zVLjrSIXqL0Jr0kz18+vuIwNUeKe7zjuNk4Z8n+n8U+Aezv37zMFKbKgAAAABJRU5ErkJggg==)
So, in interval![](data:image/png;base64,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)
f’(x) = 6(x + 1)2 (x – 3)2(x-1) < 0
Therefore, the given function (f) is strictly decreasing in intervals
.
So, in interval ![](data:image/png;base64,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)
f’(x) = 6(x + 1)2 (x – 3)2(x-1) > 0
Therefore, the given function (f) is strictly increasing in intervals
.
Question 11.Show that
, is an increasing function of x throughout its domain.
Answer:Let y = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
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Now, x > -1
⇒ 1 + x >0
Also, for all x> -1, ![](data:image/png;base64,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)
for x > -1
Therefore, f is an increasing function throughout its domain.
Question 12.Find the values of x for which y = [x(x – 2)]2 is an increasing function.
Answer:It is given that y = [x(x – 2)]2, then,
![](data:image/png;base64,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)
= 4x(x - 2)(x - 1)
Now if ![](data:image/png;base64,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)
⇒ x = 0, 1,2
So, the points x = 0, x =1 and x = 2 divides the real line into four disjoint intervals,
(-∞,0), (0,1), (1, 2) and (2,∞).
So, in interval(-∞,0),(1,2)
< 0
Therefore, the given function (f) is strictly decreasing in intervals
.
So, in interval (0,1) and (2,∞)
![](data:image/png;base64,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)
Therefore, the given function (f) is strictly increasing for 0 < x < 1 and x>2
Question 13.Prove that
is an increasing function of θ in ![](data:image/png;base64,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)
Answer:We have, y = ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWQAAAAvCAMAAADjLUAJAAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmZmZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmtpBmttvbttv/tv//25A625Bm27Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///bS0RUtQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAF1ElEQVR4Xu1bDXvTOAyOd7ctB4wVVsbHrdnYwgEXtqNJm///006S7cR2/JWkyQqtn4c2bHll6ZUsf2lJcmxHBo4MJPXXi393QENx+V1K6S1Rwe5Ak/lElIytonqrs7frqBcdL9WfGVughOql6DBeYhc7SJNGDKDjHSxRYzxcxpFcZ1doWP10zdjpQ38b6+x8vVmeI7D6i/BCIj4WJ16JXayrf12QYZoiBjp3howT1URHf+urNCqSyz8oV+TsJtks+TO1OouCJyXySB/A6RlGtJCIrKd+krtYh5mGIIMuRYzi4I4oN0pER3+OwcIYliAKSHaOX4US/LEk50is6IuUlRKTZLv85Ce5g3WYGRCkiGkdHGRM7ZxHR89Wwdh/na7qjLGzdZUyt62aK1RfGySjxBPwGn5f4Ay3uYUfLNac0u2S5xxMPa3E/EpEuAl2Yh1mSkECR9MNfvxI2UJ6lVRoHaxICqKaRNeL5Sq9Seo70KTO0EXCVKsI7Xe5M12gxM1yBQzCdwavoeT6Dv4Ryep4aCSWF2v+bILdWLuVUpDEcT9W6asHGnuKCrbRG0Zp00g00TQSKCwLpK3gGSFIsqaiHskkUWYVfI8Cp7pYb5eMmkg68JokefvuQTybYDfWqmIjSOIkycT0FajSqGALpzBK2hXNL77IB69wN3j61jPFkxd4ayeNnGsNTeQZkQ6EZPrfF/bqG/WlRTKQLCSSNDC6WHXBTiyaa/bdCpJ9KiSjbEUFSbImRPTmRmGvvZOyQjIOpdInoPW9kc+0SNZ54m8+vmCnmDaUnEy6ComQM3kzSCaAC2uLpVaQxBl0KSooIaOI4r25UWNJxrjKfauMluT8XJthPSRLyp9S4ItiYAvpWmYTdcjSs2UYJIkdG85qhGsGKZfdquCaffyoQSRz1/Klwnb55oNv09zkYVrFlLRIEMlD8Y2Mcj6RIgbSMbeOtgli5cTTn7J0JKM7YDfWT3KDM0huVbCRHEYNyslg+H2yec8XvTlribPYIBngS9zcQTLM4rBe+QdmMpC2QaZpxXANzoSMCDs+jjPWyXLHZ4LdWCfJxGODo/ihD5z4FBVsq4swqh1szv5tzH2BJex/KbEcWp43Oz5KoC6SE1iSnt5jJk75ScXmFh4uMcFsrtnJW66EueODdEozpwH2YB1WckESBwt/tqIPXFiAyo0KtnVyGDVsnayqWnrWb/iebyPax7EykAdKHL6z1ZQMhZTDokE7PplXbx+86zeRfMedwjV6DziF62J7+tV8fVjIjPJwnS3uAoGMsfw7nSfHH7O2/gmcwj29Dxwl8rR5SK1/yPjPkyHfK4cNh8TkfLZWl9/tO5z5VDiInlqS68x/dHsQdExjZE+SO4cv02j1m0m1p4vmHKt9cNltefVAf0QMWaPwmJNnGDWS5II2l8vQoegxXQzxSU+Sh3RxxBzTxQwx4Ckeia4gmkzN/nsvTZVOgc9PuChfTKatQ3D9GU772J9/O34dWUE0mdb8FGEHlUniaGH77j553LfNQFxxy4Qc09l0pzIptjRJKQtqD8lGHZdNYeozkywvA8zKpFiSrQU+hfeWZwoWPTJFBREurs/WRWx95y6V1O4otMylF3REVSbJAH6cPSX7OJYVREkOSSy2qm2XHOuVeGplkq5OXGWSOIov9oljfkdO4YPXy88yxtR7Yz1xaT6PrUyi63fguIwpotxptLiEKcUtEFGvn8X/CsnKlZC55exTmcSrsfaS5FAd9lReV88H9RswNZL7VCZNpelAuWokbz98fJbbE2dlkpaTrcVFtuqi0KnMQKZGwJQKIris5vV4c7eGZLMySSM5sjKJl6jvWWsqiKgkCKt+Zm9ysutUJumri7jKJBnws1vh67AWFUQ3GRTeiCqbmRVsysL5HYDiZn1FGVWZtHcbvZnJdHY3sJbHKm9Mgc++8DGJHsNqeayqHAPZ6aEhtTx2juUfYU4SDL+40JHnydL6MX8w+oszeHDq/w9UJfcjQXrNNwAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
⟹ 8cosθ + 4 = 4 + cos2θ + 4cosθ
⟹ cos2θ - 4cosθ = 0
⟹ cosθ(cosθ-4) = 0
⟹ cosθ = 0 or cosθ = 4
Since, cosθ≠4, cosθ = 0
⟹ cosθ = 0 ⟹ θ = π/2
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
In interval,
, we have cos θ > 0. Also, 4 > cos θ
⇒ 4 – cosθ > 0
Therefore, cosθ(4 – cosθ) > 0 and also (2 + cosθ)2 > 0
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, y is strictly increasing in interval
.
Also, the given function is continuous at x = 0 and x =
.
Therefore, y is increasing in interval
.
Question 14.Prove that the logarithmic function is strictly increasing on (0, ∞).
Answer:The given function is f(x) = logx
![](data:image/png;base64,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)
It is clear that for x>0, ![](data:image/png;base64,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)
Therefore, f(x) = log x is strictly increasing in interval (0,∞).
Question 15.Prove that the function f given by f (x) = x2 – x + 1 is neither strictly increasing nor strictly decreasing on (– 1, 1).
Answer:It is given that function f(x) = x2 – x + 1
f’(x) = 2x – 1
If f’(x) = 0, then we get,
⇒ x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAWUlEQVQYV2NgwACSAiwgMSE2bjDNwCCIn5bkZxYFqWIEAg5M43CIgFQDAdHq4QpFuBkZ2RkYJLh4GYSZ+MDC4qwQWhBivTBQGsSDUGJASoyHQYITZB0Pun0AG8sCbODE2DAAAAAASUVORK5CYII=)
So, the point x =
divides the interval (-1,1) into two disjoint intervals,
![](data:image/png;base64,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)
So, in interval![](data:image/png;base64,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)
f’(x) = 2x – 1 < 0
Therefore, the given function (f) is strictly decreasing in interval ![](data:image/png;base64,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)
So, in interval![](data:image/png;base64,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)
f’(x) = 2x -1 > 0
Therefore, the given function (f) is strictly increasing in interval for
.
Therefore, f is neither strictly increasing and decreasing in interval (-1,1).
Question 16.Which of the following functions are strictly decreasing on
?
A. cos x
B. cos 2x
C. cos 3x
D. tan x
Answer:(A) Let f1(x) = cosx
![](data:image/png;base64,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)
In interval
, ![](data:image/png;base64,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)
Therefore, f1(x) = cosx is strictly decreasing in interval
.
(B) Let f2(x) = cos2x
![](data:image/png;base64,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)
Now, 0 < x < ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAAeCAMAAADJnMQBAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6AGa2OgAAOgA6OmaQOma2OpDbZgAAZgA6ZjoAZrb/kDoAkLbbkNv/tmYAtmZmttv/25A627Zm29u22////7Zm/9uQ/9vb//+2///bvb40iwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAX0lEQVQoU5WPSRKAIAwEx100igqK4v//adgOVnGhT11JppIAGexceQj7sRFU42cy9i6u+1yAFQTT9mzFhF1McfIfOPnqwZXstELXMnbvLpkaY0n7MUYlMSyGXFaEx3N8HU8DSRzETU0AAAAASUVORK5CYII=)
⇒ 0 < 2x < π
⇒ sin2x > 0
⇒ -2sin2x < 0
![](data:image/png;base64,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)
Therefore, f2(x) = cos2x is strictly decreasing in interval
.
(C) Let f3(x) = cos3x
![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
⇒ sin3x = 0
⇒ 3x = π, as xϵ![](data:image/png;base64,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)
⇒ x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAAeCAMAAADJnMQBAAAAAXNSR0IArs4c6QAAAGZQTFRFAAAAAAAAAAA6ADo6AGa2OgAAOgA6OjoAOmaQZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmZmttv/25A625Bm27Zm27aQ2////7Zm/9uQ/9u2/9vb//+2///bCtCGjgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAZklEQVQoU5WOSRaAIAxDA4qzOIDzBPe/pAhsfLIxq/+Spi0QkCqJFccwSo4ptjMB0s2TXhugco6dZYZ+y90y+t18F+aU0OqxdNtjocKnR+Lp7GpnSRJ5AlbGjaMKgcMSZua3fL+5AZnXBDVGR3G5AAAAAElFTkSuQmCC)
The point x =
divides the interval
into two distinct intervals.
i.e.
and ![](data:image/png;base64,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)
Now, in interval,
,
f3'(x) = -3sin3x < 0 as (0 < x <
=> 0 < 3x < π)
Therefore, f3 is strictly decreasing in interval
.
Now, in interval ![](data:image/png;base64,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)
f3'(x)=-3sin3x > 0 as ![](data:image/png;base64,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)
Therefore, f3 is strictly increasing in interval
.
(D) Let f4 = tanx
![](data:image/png;base64,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)
In interval
,
![](data:image/png;base64,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)
Therefore, f4 is strictly increasing in interval
.
Question 17.On which of the following intervals is the function f given by f (x) = x100 + sin x –1 strictly decreasing?
A. (0,1)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. None of these
Answer:It is given that f (x) = x100 + sin x –1
Then, f’(x) = 100x99 + cosx
In interval (0,1), cos x >0 and 100x99 > 0
⇒ f’(x)>0
Therefore, function f is strictly increasing in interval (0,1).
In interval
, cos x < 0 and 100x99 > 0.
Also, 100x99 > cos x
⇒ f’(x) > 0 in ![](data:image/png;base64,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)
Therefore, function f is strictly increasing in interval
.
In interval
, cos x < 0 and 100x99 > 0.
Also, 100x99 > cos x
⇒ f’(x) > 0 on ![](data:image/png;base64,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)
Therefore, function f is strictly increasing in interval
.
Hence, function f is strictly decreasing on none of the intervals.
Question 18.Find the least value of a that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Answer:It is given that function f(x) = x2 + ax + 1
f’(x) = 2x + a
Now, function f will be increasing in [1, 2],
if f’(x) >0 in [1, 2]
⇒ 2x +a > 0
⇒ 2x > -a
⇒ a < -2x
Therefore, we have to find the least value of a such that
⇒ a < -2x when x ϵ [1, 2]
Now, 1 ≤ x ≤ 2
⇒ -4 ≤ -2x ≤ -2
Therefore, the least value of a for f to be increasing on [1, 2] is given by
⇒ a = -4
Therefore, the least value of a is -4
Question 19.Let I be any interval disjoint from [–1, 1]. Prove that the function f given by
is strictly increasing on 1.
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHwAAAAqCAMAAABGF89mAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOgA6Ojo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZjqQZmYAZpDbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtpBmttv/tv//25A627Zm27aQ29u22////7Zm/9uQ/9u2//+2///bdwsGewAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABy0lEQVRYR+1Wa3OCMBBMbJU+bKH2ZWpFG1uD5P//v+YJCSCQiYGZlnzRgeT2bi+3CwDTGoyB7+ftYFgVoNMKXn2NBJ4t93g0cFbzBD5K4yfa/yHts7EUjm4iCOH1eyvneTI/ig10d1eqIV7uw3aKbiCMQRatJTZ6lEnIld3Kx4EWRQuAY0CkAFMU2zjZTXvD/CxLlYwXAlTlYCSAVTsaa/e1LAI5sRQJejkNldVWuq9lEX4Z1yB/FZ1WNKQQzo+YP29ohJ2cs3ZSZEyerFwtol6k7FeRAdIaGSZ8AHCKWOXq6qW66YwPtYzcncEt4qzKi1BZdP+gthXgjVfOAK8lVySr/9QiNNLOPj+K6nqD99ECzCQFnLSggWbw/OVNf3kVPb8E7W3ghc59bPNEXrQ8qciO520/33M15zRlgIQzxKavXeLK9vRhvbKHiDEvZ42TTRGcscq55LPmtyhcL8sSsX9WEOobfDZLZ23vV2/+9AkO3bYeztW6LIqXEczPtWj148p7l2kR4NDZcm88O4BhEXhobDY22iIIwyZBP4rqvGmL4HNrjfSFKW4O56VBnhkaFuEZyf04LS3C/bDnCdMiPEM5H7cswvn03zrwC+EDJjiwiqt+AAAAAElFTkSuQmCC)
Now, f’(x) =0
![](data:image/png;base64,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)
The points x =1 and x = -1 divide the real line in three disjoint intervals
(-∞,-1),(-1,1) and (1,∞)
Now in interval, (-1,1)
it is clear that -1 < x < 1
⇒ x2 < 1
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIsAAAAqCAMAAAC0h1iGAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjpmOjqQOmY6OmaQOma2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY6ttv/tv//25A627Zm27aQ29u229v/2////7Zm/9uQ/9u2//+2///bUEjOggAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB20lEQVRYR+1WaXOCMBDN2mrppUVrD6qigq0c+f9/rwsIuRATjJWZkg+MOJvdt2+PByH9uSwDu9fVZQNoe09mcLPVtr6oYTzeBF3Bgon2WOqr3fPS82KyBzrVL4OO7F26cADg9t2EyN7WCgN/o710AfC0aQZ8XHuTucU+pd5on3jNynpUe3ez4fJc2v3PykOUDWAE7A9C4keVppqdQUOnxrABmg8w2gcgxCLxA/vY8TNKUveO95HjE4+Cha6d8YnKKrB89Es9Pm38wHipzLISZVjwmbq4GA5HASNhSRYw+TGuThaMxUaaxGgFluLJnWAgope/j/xhHScsl8MvGW3sPE8kvrn3llja8YIVkQhPp1w3lDXCfjGpEVLZol9IOn9jE6vUiBx6l3UQcqjTu4jGdI4I/VqJUyLQgqxlI6Yx07Xaq7lfqJePKfUx4wi4tCNhfPMRUlpX6rkG7T22d79nAFVbFlioBwPkBRgY5EmMhPsdxuIUGQ+rciGdLklYUZm60lSeH8DQQ3xfZr3+MLxq3ZzbqNZ9n3YoCE4obbXT1+1acIITXBlKriVFaSKEEl23Y0vBKXb4dbEogmO3B4y8CYJjdNO6sSI41iNoO1QFR/uqbUNVcGxH+O/+fgEC9Sf5iLWshgAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Therefore, f’(x) = 1-
< 0 (-1,1) ~ {0}
Therefore, f is strictly decreasing on (-1,1) ~ {0}
x < -1 or 1 < x
⇒ x2 > 1
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, f’(x) = 1-
> 0 (-∞, -1) and (1,∞)
Therefore, f is strictly increasing in interval I disjoint from (-1,1)
Hence Proved.
Question 20.Prove that the function f given by f (x) = log sin x is strictly increasing on
and strictly decreasing on
.
Answer:It is given that f (x) = log sin x
![](data:image/png;base64,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)
In interval
, f’(x) = cot x >0
Therefore, f is strictly increasing in
.
In interval
, f’(x) = cot x < 0
Therefore, f is strictly decreasing in
.
Question 21.Prove that the function f given by f (x) = log |cos x| is strictly decreasing on
and strictly increasing on
.
Answer:It is given that f (x) = log |cos x|
![](data:image/png;base64,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)
In interval
, f’(x) = -tanx < 0
Therefore, f is strictly decreasing on
.
In interval
, f’(x) = -tanx > 0
Therefore, f is strictly increasing in
.
Question 22.Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Answer:We have, f (x) = x3 – 3x2 + 3x – 100
=> f’(x) = 3x2 -6x + 3
= 3(x2 -2x + 1)
= 3(x-1)2
For any x ϵ R, (x -1)2 > 0
Thus, f’(x) is always positive in R.
Therefore, the given function (f) is increasing in R.
Question 23.The interval in which y = x2 e–x is increasing is
A. (– ∞, ∞)
B. (– 2, 0)
C. (2, ∞)
D. (0, 2)
Answer:it is given that y = x2 e–x
then ![](data:image/png;base64,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)
Now if ![](data:image/png;base64,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)
⇒ x = 0 and x =2
The points x = 0 and x= 2 divide the real line into three disjoint intervals ie, (-∞,0), (0,2) and (2,∞).
In interval (-∞,0) and (2,∞),
f’(x) < 0 as e-x is always positive.
Therefore, f is decreasing on (-∞,0) and (2,∞).
In interval (0,2), f’(x)>0
Therefore, f is strictly increasing in interval (0.2).
Show that the function given by f (x) = 3x + 17 is strictly increasing on R.
Answer:
Let x1 and x2 be any two numbers in R.
Then, we have,
x1 < x2
⇒ 3x1 < 3x2
⇒ 3x1 +17 < 3x2 +17
⇒ f(x1) < f(x2)
Therefore, f is strictly increasing on R.
Question 2.
Show that the function given by f (x) = e2x is strictly increasing on R.
Answer:
Let x1 and x2 be any two numbers in R.
The, we have,
x1 < x2
⇒ 2x1 < 2x2
⇒ e2x1 < e2x2
⇒ f(x1) < f(x2)
Therefore, f is strictly increasing on R.
Question 3.
Show that the function given by f (x) = sin x is
(a) strictly increasing in
(b) strictly decreasing in
(c) neither increasing nor decreasing in (0, π)
Answer:
(a) The function is f (x) = sin x
Then, f’(x) = cos x
Since for each x ϵ , cos x > 0, we have f’(x) > 0
Therefore, f’ is strictly increasing in.
(b) The function is f (x) = sin x
Then, f’(x) = cos x
Since for each , cos x < 0, we have f’(x) < 0
Therefore, f’ is strictly decreasing in.
(c) The function is f (x) = sin x
Then, f’(x) = cos x
Since for each x ϵ , cos x > 0, we have f’(x) >0
Therefore, f’ is strictly increasing in……………….(1)
Now, The function is f (x) = sin x
Then, f’(x) = cos x
Since, for each x ϵ, cos x < 0, we have f’(x) < 0
Therefore, f’ is strictly decreasing in …………(2)
From (1) and (2),
It is clear that f is neither increasing nor decreasing in (0, π).
Question 4.
Find the intervals in which the function f given by f (x) = 2x2 – 3x is
(a) strictly increasing (b) strictly decreasing
Answer:
(a) It is given that function f(x) = 2x2 – 3x
⇒ f’(x) = 4x – 3
If f’(x) = 0, then we get,
So, the points divides the real line into two disjoint intervals,
and
So, in interval, f’(x) = 4x -3 >0
Therefore, the given function (f) is strictly increasing in interval.
(b) It is given that function f(x) = 2x2 – 3x
⇒ f’(x) = 4x – 3
If f’(x) = 0, then we get,
So, the points divides the real line into two disjoint intervals,
and
So, in interval f’(x) = 4x -3 < 0
Therefore, the given function (f) is strictly decreasing in interval.
Question 5.
Find the intervals in which the function f given by f (x) = 2x3 – 3x2 – 36x + 7 is
(a) strictly increasing (b) strictly decreasing
Answer:
(a) It is given that function f(x) = 2x3 – 3x2 – 36x + 7
⇒ f’(x) = 6x2 – 6x + 36
⇒ f’(x) = 6(x2 – x + 6)
⇒ f’(x) = 6(x + 2)(x – 3)
If f’(x) = 0, then we get,
⇒ x = -2, 3
So, the points x = -2 and x = 3 divides the real line into two disjoint intervals, (-∞,2), (-2,3) and (3,∞)
So, in interval
f’(x) = 6(x + 2)(x – 3) >0
Therefore, the given function (f) is strictly increasing in interval .
(b) It is given that function f(x) = 2x3 – 3x2 – 36x + 7
⇒ f’(x) = 6x2 – 6x + 36
⇒ f’(x) = 6(x2 – x + 6)
⇒ f’(x) = 6(x + 2)(x – 3)
If f’(x) = 0, then we get,
⇒ x = -2, 3
So, the points x = -2 and x = 3 divides the real line into two disjoint intervals, (-∞,2), (-2,3) and (3,∞)
So, in interval
f’(x) = 6(x + 2)(x – 3) < 0
Therefore, the given function (f) is strictly decreasing in interval.
Question 6.
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x – 5
Answer:
It is given that function f(x) = x2 + 2x – 5
f’(x) = 2x + 2
If f’(x) = 0, then we get,
⇒ x = -1
So, the point x = -1 divides the real line into two disjoint intervals, (-∞,-1) and (1,∞)
So, in interval (-∞,-1)
f’(x) = 2x + 2 < 0
Therefore, the given function (f) is strictly decreasing in interval (-∞,-1).
And in interval (1,∞)
f’(x) = 2x + 2 > 0
Therefore, the given function (f) is strictly increasing in interval (1,∞).
Thus, f is strictly increasing for x > -1.
Question 7.
Find the intervals in which the following functions are strictly increasing or decreasing:
10 – 6x – 2x2
Answer:
It is given that function f(x) = 10 – 6x – 2x2
f’(x) = -6 – 4x
If f’(x) = 0, then we get,
⇒ x =
So, the point x = divides the real line into two disjoint intervals,
So, in interval
-4x > 6
-6 - 4x > 0
f’(x) = -6 – 4x > 0
Therefore, the given function (f) is strictly increasing in interval.
And in interval
f’(x) = -6 – 4x < 0
Therefore, the given function (f) is strictly decreasing in interval .
Thus, f is strictly decreasing for x > .
Question 8.
Find the intervals in which the following functions are strictly increasing or decreasing:
–2x3 – 9x2 – 12x + 1
Answer:
It is given that function f(x) = –2x3 – 9x2 – 12x + 1
⇒ f’(x) = -6x2 – 18x + 12
⇒ f’(x) = -6(x2 +3x + 6)
⇒ f’(x) = -6(x + 1)(x + 2)
If f’(x) = 0, then we get,
⇒ x = -1 and -2
So, the points x = -1 and x = -2 divides the real line into two disjoint intervals,
(-∞,-2), (-2,-1) and (-1,∞)
So, in interval (-∞,-2),(-1,∞)
f’(x) = -6(x + 1) (x +2) < 0
Therefore, the given function (f) is strictly decreasing for x < -2 and x>-1.
So, in interval (-2.-1)
f’(x) = -6(x + 1)(x+2) > 0
Therefore, the given function (f) is strictly increasing for -2 < x < -1.
Question 9.
Find the intervals in which the following functions are strictly increasing or decreasing:
6 – 9x – x2
Answer:
It is given that function f(x) = 6 – 9x – x2
f’(x) = -9 – 2x
If f’(x) = 0, then we get,
⇒ x =
So, the point x = divides the real line into two disjoint intervals,
So, in interval
f’(x) = -9 – 2x > 0
Therefore, the given function (f) is strictly increasing for x < .
And in interval
f’(x) = -9 – 2x < 0
Therefore, the given function (f) is strictly decreasing for x>.
Thus, f is strictly decreasing for x>.
Question 10.
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x – 3)3
Answer:
It is given that function f(x) = –(x + 1)3 (x – 3)3
⇒ f’(x) = 3(x + 1)2 (x – 3)3 +3(x + 1)3 (x – 3)2
⇒ f’(x) = 3(x + 1)2 (x – 3)2[x – 3 + x + 1]
⇒ f’(x) = 6(x + 1)2 (x – 3)2(x-1)
If f’(x) = 0, then we get,
⇒ x = -1,3 and 1
So, the points x = -1, x =1 and x = 3 divides the real line into four disjoint intervals,
So, in interval
f’(x) = 6(x + 1)2 (x – 3)2(x-1) < 0
Therefore, the given function (f) is strictly decreasing in intervals .
So, in interval
f’(x) = 6(x + 1)2 (x – 3)2(x-1) > 0
Therefore, the given function (f) is strictly increasing in intervals.
Question 11.
Show that, is an increasing function of x throughout its domain.
Answer:
Let y =
Now, x > -1
⇒ 1 + x >0
Also, for all x> -1,
for x > -1
Therefore, f is an increasing function throughout its domain.
Question 12.
Find the values of x for which y = [x(x – 2)]2 is an increasing function.
Answer:
It is given that y = [x(x – 2)]2, then,
= 4x(x - 2)(x - 1)
Now if
⇒ x = 0, 1,2
So, the points x = 0, x =1 and x = 2 divides the real line into four disjoint intervals,
(-∞,0), (0,1), (1, 2) and (2,∞).
So, in interval(-∞,0),(1,2)
< 0
Therefore, the given function (f) is strictly decreasing in intervals .
So, in interval (0,1) and (2,∞)
Therefore, the given function (f) is strictly increasing for 0 < x < 1 and x>2
Question 13.
Prove that is an increasing function of θ in
Answer:
We have, y =
Now,
⟹ 8cosθ + 4 = 4 + cos2θ + 4cosθ
⟹ cos2θ - 4cosθ = 0
⟹ cosθ(cosθ-4) = 0
⟹ cosθ = 0 or cosθ = 4
Since, cosθ≠4, cosθ = 0
⟹ cosθ = 0 ⟹ θ = π/2
Now,
In interval,, we have cos θ > 0. Also, 4 > cos θ
⇒ 4 – cosθ > 0
Therefore, cosθ(4 – cosθ) > 0 and also (2 + cosθ)2 > 0
Therefore, y is strictly increasing in interval.
Also, the given function is continuous at x = 0 and x = .
Therefore, y is increasing in interval.
Question 14.
Prove that the logarithmic function is strictly increasing on (0, ∞).
Answer:
The given function is f(x) = logx
It is clear that for x>0,
Therefore, f(x) = log x is strictly increasing in interval (0,∞).
Question 15.
Prove that the function f given by f (x) = x2 – x + 1 is neither strictly increasing nor strictly decreasing on (– 1, 1).
Answer:
It is given that function f(x) = x2 – x + 1
f’(x) = 2x – 1
If f’(x) = 0, then we get,
⇒ x =
So, the point x = divides the interval (-1,1) into two disjoint intervals,
So, in interval
f’(x) = 2x – 1 < 0
Therefore, the given function (f) is strictly decreasing in interval
So, in interval
f’(x) = 2x -1 > 0
Therefore, the given function (f) is strictly increasing in interval for.
Therefore, f is neither strictly increasing and decreasing in interval (-1,1).
Question 16.
Which of the following functions are strictly decreasing on ?
A. cos x
B. cos 2x
C. cos 3x
D. tan x
Answer:
(A) Let f1(x) = cosx
In interval,
Therefore, f1(x) = cosx is strictly decreasing in interval.
(B) Let f2(x) = cos2x
Now, 0 < x <
⇒ 0 < 2x < π
⇒ sin2x > 0
⇒ -2sin2x < 0
Therefore, f2(x) = cos2x is strictly decreasing in interval.
(C) Let f3(x) = cos3x
Now,
⇒ sin3x = 0
⇒ 3x = π, as xϵ
⇒ x =
The point x = divides the interval
into two distinct intervals.
i.e. and
Now, in interval, ,
f3'(x) = -3sin3x < 0 as (0 < x < => 0 < 3x < π)
Therefore, f3 is strictly decreasing in interval.
Now, in interval
f3'(x)=-3sin3x > 0 as
Therefore, f3 is strictly increasing in interval.
(D) Let f4 = tanx
In interval,
Therefore, f4 is strictly increasing in interval .
Question 17.
On which of the following intervals is the function f given by f (x) = x100 + sin x –1 strictly decreasing?
A. (0,1)
B.
C.
D. None of these
Answer:
It is given that f (x) = x100 + sin x –1
Then, f’(x) = 100x99 + cosx
In interval (0,1), cos x >0 and 100x99 > 0
⇒ f’(x)>0
Therefore, function f is strictly increasing in interval (0,1).
In interval, cos x < 0 and 100x99 > 0.
Also, 100x99 > cos x
⇒ f’(x) > 0 in
Therefore, function f is strictly increasing in interval .
In interval , cos x < 0 and 100x99 > 0.
Also, 100x99 > cos x
⇒ f’(x) > 0 on
Therefore, function f is strictly increasing in interval .
Hence, function f is strictly decreasing on none of the intervals.
Question 18.
Find the least value of a that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Answer:
It is given that function f(x) = x2 + ax + 1
f’(x) = 2x + a
Now, function f will be increasing in [1, 2],
if f’(x) >0 in [1, 2]
⇒ 2x +a > 0
⇒ 2x > -a
⇒ a < -2x
Therefore, we have to find the least value of a such that
⇒ a < -2x when x ϵ [1, 2]
Now, 1 ≤ x ≤ 2
⇒ -4 ≤ -2x ≤ -2
Therefore, the least value of a for f to be increasing on [1, 2] is given by
⇒ a = -4
Therefore, the least value of a is -4
Question 19.
Let I be any interval disjoint from [–1, 1]. Prove that the function f given by is strictly increasing on 1.
Answer:
It is given that
Now, f’(x) =0
The points x =1 and x = -1 divide the real line in three disjoint intervals
(-∞,-1),(-1,1) and (1,∞)
Now in interval, (-1,1)
it is clear that -1 < x < 1
⇒ x2 < 1
Therefore, f’(x) = 1- < 0 (-1,1) ~ {0}
Therefore, f is strictly decreasing on (-1,1) ~ {0}
x < -1 or 1 < x
⇒ x2 > 1
Therefore, f’(x) = 1- > 0 (-∞, -1) and (1,∞)
Therefore, f is strictly increasing in interval I disjoint from (-1,1)
Hence Proved.
Question 20.
Prove that the function f given by f (x) = log sin x is strictly increasing on and strictly decreasing on
.
Answer:
It is given that f (x) = log sin x
In interval, f’(x) = cot x >0
Therefore, f is strictly increasing in.
In interval, f’(x) = cot x < 0
Therefore, f is strictly decreasing in.
Question 21.
Prove that the function f given by f (x) = log |cos x| is strictly decreasing onand strictly increasing on
.
Answer:
It is given that f (x) = log |cos x|
In interval, f’(x) = -tanx < 0
Therefore, f is strictly decreasing on.
In interval, f’(x) = -tanx > 0
Therefore, f is strictly increasing in.
Question 22.
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Answer:
We have, f (x) = x3 – 3x2 + 3x – 100
=> f’(x) = 3x2 -6x + 3
= 3(x2 -2x + 1)
= 3(x-1)2
For any x ϵ R, (x -1)2 > 0
Thus, f’(x) is always positive in R.
Therefore, the given function (f) is increasing in R.
Question 23.
The interval in which y = x2 e–x is increasing is
A. (– ∞, ∞)
B. (– 2, 0)
C. (2, ∞)
D. (0, 2)
Answer:
it is given that y = x2 e–x
then
Now if
⇒ x = 0 and x =2
The points x = 0 and x= 2 divide the real line into three disjoint intervals ie, (-∞,0), (0,2) and (2,∞).
In interval (-∞,0) and (2,∞),
f’(x) < 0 as e-x is always positive.
Therefore, f is decreasing on (-∞,0) and (2,∞).
In interval (0,2), f’(x)>0
Therefore, f is strictly increasing in interval (0.2).
Exercise 6.3
Question 1.Find the slope of the tangent to the curve y = 3x4 – 4x at x = 4.
Answer:The given curve y = 3x4 – 4x
Then, the slope of the tangent to the given curve at x = 4 is given by,
![](data:image/png;base64,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)
= 12(64) – 4
= 764
Therefore, the slop of the tangent is 764.
Question 2.Find the slope of the tangent to the curve
at x = 10.
Answer:The given curve is y = ![](data:image/png;base64,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)
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![](data:image/png;base64,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)
Then, the slope of the tangent
![](data:image/png;base64,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)
Therefore, the slope of the tangent is
.
Question 3.Find the slope of the tangent to curve y = x3 – x + 1 at the point whose x-coordinate is 2.
Answer:The given curve is y = x3 – x + 1
![](data:image/png;base64,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)
Then, the slope of the tangent
![](data:image/png;base64,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)
= 12 -1 = 11
Therefore, the slope of the tangent 11.
Question 4.Find the slope of the tangent to the curve y = x3 –3x + 2 at the point whose x-coordinate is 3.
Answer:The given curve is y = x3 – 3x + 2
![](data:image/png;base64,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)
Then, the slope of the tangent
![](data:image/png;base64,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)
= 27 -3 = 24
Therefore, the slope of the tangent 24.
Question 5.Find the slope of the normal to the curve x = acos3 θ, y = asin3 θ at ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADsAAAA1CAYAAAD/L5PDAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAHQSURBVGje7ZnLjcIwEIanABpAwkcKIIYWkIUoAGJx5YR8oQCTWy4pgtsWQQXpgQ7SA7tO1sEbEhOh3VU8zGGuUb55/PMwJEkC72JvA0qwBEuwBEuwBEuwBPvDtNpOIwY5ANy6jV93Wo+Dhk3T/UhM4OIHRQJ7knzNhNT7NB1pJWaMx5nrBLY8HFHWrAEHHp9dWC5Pa7SwNpJa78Yc4IoW9rBkRxc2ApYLpWfoYJtpixq2rFdghYWzaWwibRwhOdfBq3HVZ79UGKBw28tDS/oWrl6w5oP3xs2KhVSbwcA6UXwQLOMEFuVbpae9YOsUkWplJ5a2jwc/Lvq85tYIClgj521Q1gldtRAkbMzh3DZX1gIwERczqgUPe1c0D2zHgB0cbJ2q3m3itbqtyuPZplLZXwphC6wvsmGL1HvC+oB8jvi3H+1ZBk170nrg1lyV6hHtxdYzuJr19dOu/ht0n3WjYOdhJcXKRBXduGhNycWm2iyGtwj8OixWI1iCHbhVwuqfA1CAtp1qUMKWl8X6hIQcNuYsM08ifVbQoEHLIxuPs777dtDpO2fzDwOHHtakr53VUcO67zyoYd13WfSw/XZjBC/vXYZeoAiWYGnFI1hU9gnIRR/FBW7EwwAAAABJRU5ErkJggg==)
Answer:The given curve is y = acos3θ and y = asin3θ
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Then, the slope of the tangent
is given by:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARcAAAA0CAMAAACXQozKAAAAAXNSR0IArs4c6QAAAKtQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOgA6OgBmOjoAOjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmY6ZmZmZpDbZrbbZrb/kDoAkDo6kGYAkGY6kGaQkLyQkLbbkNv/tmYAtmY6tmZmttv/tv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/7aQ/9uQ/9u2/9vb//+2///b0VT9KQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAES0lEQVRoQ+1a63rTMAyNO1hgsBUWtnFpsws02xrYWEiJ3//JkOQ0tznxZfZou/hHvn7MsawT6ejIJgjGMSJgikDG2Nz0nRcxPxtxkX7nPBzjRQZMC5eEMfb64UXkyZCT+Qljh+Gcx4hGHrLJIiii3cCFX8InpnFs/JnzcBbwC+AXHiMYGcCyM7hcL/K3y8COJBJCA3k33VvCYx+Q3ZV4CQJrXIoIQ4wgxQc/h3AZcUEIKlx4vB9kRCxjvDRxwURKqF6PuADbIqEIXVdEH86AYnYJlww4045308mPYHUq9G5S1rPdiResJXl4/AdZ02zwazaZ3ocEDKK7W/GSgEc8nnwzw6Q7O8Ocer48SkFwbXz/ARVaFGn/uKAMnVKjYZf6T/v4hm/zeHpRhovnegRU/7CKyNYG4nJ32uYi6I+O1s2iE97N3/SQHbUa9HCCS68dw8Cg6asTVnKs7G0nuAjPJYM6DxEpLuKl144FLvnRLXVDPcMFLtfYxu4t+a8DIBNUSGz+M8RfQjE1dLbF/huv9NqxXNY3LiJTkEt4gr/y8N0iSKlvR1zE01m8SOz8L1zw+EoMecLU8V11pSixIFLEW724FHAghKMs4bZ2pCms2DO8U8ZLNbH+Yd8HtFyQ4YLZU1ApGoiXm1vo0ui8o29o2YGXG87prVXjIp3vgl/KisPvTkP69BQ0iIsGv6hwaW1a4P/YzkbnEeZ9fboj2JbqUREN1KMWLlp5JLMjK4XaeeQ5Xuhj1vxCuKSkX6gcynmXx5hHfzWP3ms51LFjFzDe6xFy7M13eKy+YArRUQb+GxIM6F06iRa4QKkVbUE5MJaysFKZCv967Vji0qgi3ftGJ/wC7k7m5PPviM3hooHNsRQBHiAqJx9p14RLundbwmTnSa8d8+X4JWyTvara7859oxtcNLZFR8ox5pa/ixlKW8vRyXM5Lh1dYWmq9RraJQYGulnrGscnD3noGxe1rjCESpy/CDwGuhLDVdvTi+irJS7lfSMcYsJ9I2yVZIY8rI10ha4763ip5le1VHeFoXnJsWVfWd03BtjGYIl8ZlzW/OIlj7L3D5a41PeNeBGbUvnsiRczXaH9tbEenZhfIOusX3xalLpbZ3ZzTrPfz8NDOAeocXmartDeCR1rdma7aLSxCQO4IV5Scy5vnYOsS5qIl6fqCm1gHk/k5wfmrjxeBnRZqzU32FETl+LssygLhItvXTG0yXRGPOdk2PFL474RrgPEGYDAxbOuGPI5mwr+dzLscIEurrxv5AkmI904lrj41BVDLhdnSx7PrjT7RhV4lnp3fd84i+HcjRqY8v+R+dUVg1lEpOCoMwCOsRR2PVv0qitUH9lhHqlMGf/do65Q7uU+9NUYKE0rJ/jTFUrT2zbBka7YNreV+3WpK5TGtmeCU12xPW6rdupWV6isbc/f8RzJma7YHrd1drrJukJn/77mbLKu8OXzuG6NwD86nY/zpVIEywAAAABJRU5ErkJggg==)
Then, the slope of the tangent
is given by:
![](data:image/png;base64,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)
Therefore, the slope of the tangent 1.
Question 6.Find the slope of the normal to the curve x = 1− a sinθ, y = bcos2 θ at ![](data:image/png;base64,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)
Answer:The given curve is x = 1− a sin θ and y = b cos2 θ
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Then, the slope of the tangent
is given by:
![](data:image/png;base64,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)
Then, the slope of the tangent
is given by:
![](data:image/png;base64,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)
Therefore, the slope of the tangent
.
Question 7.Find points at which the tangent to the curve y = x3 – 3x2 – 9x + 7 is parallel to the x-axis.
Answer:It is given that the curve y = x3 – 3x2 – 9x + 7
![](data:image/png;base64,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)
We know that the tangent is parallel to the x-axis if the slope of the tangent is zero.
Therefore, 3x2 -6x-9 = 0
⇒ x2 -2x-3 = 0
⇒ (x-3)(x+1) =0
⇒ x =3 and x = -1
When x = 3, y = (3)3-3(3)2-9(3) +7 = 27 – 27 -27 + 7 = -20
When x = -1, y = (-1)3-3(-1)2-9(-1) +7 = -1 – 3 + 9 + 7 = 12
Therefore, the points at which the tangent is parallel to the x- axis are (3, -20) and (-1, 12).
Question 8.Find a point on the curve y = (x – 2)2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).
Answer:We know that if a tangent is parallel to the chord joining the points (2,0) and (4,4), then
Slope of the tangent = slope of the curve………….(1)
And, the slope of the curve = ![](data:image/png;base64,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)
Now, slope of the tangent to the given curve at a point (x,y) is:
![](data:image/png;base64,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)
Now, from (1) we have,
2(x -2) = 2
⇒ x-2 = 1
⇒ x =3
So, when x = 3 then y = (3-2)2 = 1
Therefore, required points are (3,1).
Question 9.Find the point on the curve y = x3 – 11x + 5 at which the tangent is y = x –11.
Answer:It is given that equation of the curve y = x3 – 11x + 5
At which the tangent is y = x –11
⇒ Slope of the tangent = 1
Now, slope of the tangent to the given curve at a point (x,y) is:
![](data:image/png;base64,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)
⇒ 3x2 -11 = 1
⇒ 3x2 = 12
⇒ x2 = 4
⇒ x =
2
So, when x = 2 then y = (2)3 -11(2) + 5 = -9
And when x = -2 then y = (-2)3 -11(-2) + 5 = 19
Therefore, required points are (2, -9) and (-2, 19).
Question 10.Find the equation of all lines having slope –1 that are tangents to the curve ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIAAAAA1CAMAAAC+5xM/AAAAAXNSR0ICQMB9xQAAAHtQTFRFgYGBiIhmZoiIboBuf39dXX9/f25ZWW5/f25Ibm5ZSG5/SFtugFszM1uAW0hIM0ZubEYdHUZsRkYzWzMzWjMdHTNaMx5HWjMAADNaNDQdNB00SB0dHR1IMgAySBwAABxIHBwcAB0yHR0AAB0dMwAAAAAzHQAAAAAdAAAA478AaAAAACh0Uk5TAAwMGBkZJSUxMTFKTExjZHNzfX+NjZqhoaioqam7vLzFytnZ3d3s7ILB3oEAAAFiSURBVHja7VddU4MwELxgYhHBJlqlrTW0lY/9/7/QNyEQmA6NyYyTe4MZuM3e3t2GKEaMGGvi6fWqQubPv69QYSnII4AIIAKIACIA7EOmT1sAd3IQ91kYGbHPzAKA10CVMI0qcZWI1wAUr9Fl5qnfrQw81o0g4l8bh0ctUSWUN5tZAowSSBREUvWPv7G6TEyjGOTL+18WFgC83pvonFShORSjV0cxJ8Kyy9JT4lrlY01JNdsFaavkAtu4MUwA51EBTQJMAExfDhm51AClpwfd3aoBG1/3SuAoKG0bMUSULAwipt2MJaa7bf8/iarvw12xuM92bnqAazSCiEpAkQT6STScQZZ9xj+cVeBNrPlKOlsM6XZFdqARjkbwy/OakdFe7lwD0V38e5c85y/8MWD3Fz5LMPIXATRg8ReeRWjsy8nC9gHgvJTj7wGY/sI/gKm/8NyGU3/h97Y89RfOb8sxgsUPFPAwlXefecIAAAAASUVORK5CYII=)
Answer:It is given that equation of the curve y = ![](data:image/png;base64,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)
Now, slope of the tangent to the given curve at a point (x, y) is:
![](data:image/png;base64,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)
Now, if the slope of the tangent is -1, then we get,
![](data:image/png;base64,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)
⇒ (x-1)2 = 1
⇒ (x-1) =
1
⇒ x = 2, 0
So, when x = 2 then y = 1
And when x = 0 then y = 1
Therefore, required points are (0, -1) and (2, 1).
Now, the equation of the tangent (0,1) is given by:
y – (-1) = -1(x-0)
⇒ y + 1 = -x
⇒ y + x+ 1 = 0
And the equation of the tangent (2,1) is given by:
y – 1 = -1(x-2)
⇒ y - 1 = -x +2
⇒ y + x - 3 = 0
Therefore, the equations of the required lines are y + x+ 1 = 0 and y + x - 3 = 0.
Question 11.Find the equation of all lines having slope 2 which are tangents to the curve ![](data:image/png;base64,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)
Answer:It is given that equation of the curve y = ![](data:image/png;base64,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)
Now, slope of the tangent to the given curve at a point (x,y) is:
![](data:image/png;base64,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)
Now, if the slope of the tangent is 2, then we get,
![](data:image/png;base64,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)
⇒ 2(x-3)2 = -1
⇒ (x-3)2 = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABgAAAAgCAMAAAA/gEgKAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAbElEQVQ4T2NgoAKQFGDBaooQGzd2CQYGQdpLSPIzi2JzliAjEHBQwdsUGcEPcgUYMPEBDYKyKTKSnppFuBkZ2bFYKMHFyyAM9hEWIM6KQ0IQR3QIY7MCaLAgDnExoLgYD6bNEpyg8MUiQTDIAKWvApSMomtfAAAAAElFTkSuQmCC)
This is not possible since the L.H.S. is positive while the R.H.S. is negative.
Therefore, there is no tangent to the given curve having a slope 2.
Question 12.Find the equations of all lines having slope 0 which are tangent to the curve ![](data:image/png;base64,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)
Answer:It is given that equation of the curve y =
,
Now, slope of the tangent to the given curve at a point (x,y) is:
![](data:image/png;base64,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)
Now, if the slope of the tangent is 0, then we get,
![](data:image/png;base64,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)
⇒ -2(x-1)=0
⇒ x =1
So, when x = 1 then y = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEcAAAAgCAMAAAB3n5XPAAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAAAAAA6ADo6AGaQAGa2OgAAOgA6OjoAOmZmOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bLa26EgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA6ElEQVRIS+WU2RKCMAxFU1RcAVdqQQXh///RrkwtBOkTqHlgmPbm9DbpBGDyUV8XXh4RfbaOvTi4nnlxADD9j3JqOn/4NAzRM8JjOxzkqx9O/mMlFU2QEaTvZWg2PvxgxdNpI9UWv9dIhr7oWHe+3mNCNrb/bEWCBL2QkbvztYoukKtXVh8P4ntqFjpglrw9R8ulxRHJegExpXfbHKbmj/LD43ne9zVKyyWn2qkXLjJzUR5rgZJZH0fKebh+mKly4wduoXbWYcvI3flacEwh0xSnilIocY6Ru/NV3cfiQBb29N2ST+mtvwBZbQ1KSgAeoQAAAABJRU5ErkJggg==)
Now, the equation of the tangent (0,
) is given by:
y –
= 0(x-1)
⇒ y -
= 0
⇒ y = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAWUlEQVQYV2NgwACSAiwgMSE2bjDNwCCIn5bkZxYFqWIEAg5M43CIgFQDAdHq4QpFuBkZ2RkYJLh4GYSZ+MDC4qwQWhBivTBQGsSDUGJASoyHQYITZB0Pun0AG8sCbODE2DAAAAAASUVORK5CYII=)
Therefore, the equations of the required line is y =
.
Question 13.Find points on the curve
at which the tangents are
(i) parallel to x-axis (ii) parallel to y-axis.
Answer:(i) It is given that ![](data:image/png;base64,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)
Now, differentiating both sides with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAG8AAAAuCAMAAAARH7ebAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAA///b//+22///2//btv///9u229v//9uQttv/29uQkNv//7Zm27aQ27ZmkLbbZrb/kLaQZrbb25Bm25A6tpA6ZpDbOpDbOpC2tmY6OpCQtmYAkGY6kGYAOma2OmaQZmYAOmY6kDo6AGa2kDoAZjpmZjo6ZjoAOjpmADqQADpmADo6ZgAAOgAAAABmAAA6AAAAHJukvgAAAAF0Uk5TAEDm2GYAAAH8SURBVHja7ZZrV4MwDIZT6iYT5i6iOOuYG2yTSSH//9d5ymUDddCg4PGSDxw4h/RN0qR9AP622YjTngXb6I13Vv7GFjLkBE9T0vWGe4wHebjSuyT5ttAzVyMn17PlnOK4R1zLKQsQQ25KTCxdz1zPiPzsW6gVnIZeMOUc2BKnwAK1Bba+XKF32nyRWMCC+lKJVEW5pO6OD1Q9Ec/2iB6HNGTHrfUxIrfYP/Vgm5r0BBaWFSHTYwHecxjLbJ21B7p6LPDBprR1oZf6pHUCp2k7SnrKX1D6tKKXfhjbp2JGzphKqthyI3rYDqDF/g3y/NjGOjbrWadkAsNd1mICXcoI5sUzpc/hWnW4cFXo9WuwBSbelUwF7Vg/PfYoEfHlLp/gw0Q1TmIZEeoHbfv9HdlsY9UOw5frBd6yx/RU9Ve8o7Vv+r1dW113nzIn6bYz1FRUrVFQ6P+q1R35aON7O+eD2laj9zvr2XW/fPM8vJ33H8DXY3UHjXorsLqlmWjd41S9jHqyJ1kr42s9Oq7Ex4LwgkzlR77WouNKforMyFR+4msNOj55xRawWww5lcrL/NlMxyVaQjzMFPs1Uvl5PeqxYyqGJ1J5Wa+Zjj86FWlUXuJrHTquhuoDncqPfK1Fx+UTRnoc6FRe8PWcRsemfJ4Uh2GnV+k/lf9UKte3V5vXU1XvJMQBAAAAAElFTkSuQmCC)
We know that the tangent is parallel to the x –axis if the slope is 0 ie,
, which is possible if x =0
Then,
for x =0
⇒ y2 = 16
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEAAAAAXCAMAAACMPLmjAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOjqQOmaQOma2OpCQOpDbZgAAZgA6ZjpmZmZmZrb/kDoAkDo6kGYAkNv/tmYAtv//25A62////7Zm/9uQ//+2///bwcWijwAAAAF0Uk5TAEDm2GYAAACwSURBVHja7VTtCsMgDIxr7T7sNu38ivr+zzlqZ3EMwVYYDJZfKsmZu5wC/OPLEQTPduYgmwDc0Abgx3sEEIT01g2kCi0HUAxjTRC9BcC6bjIAPNpXkek0gKFZmiIpctj3U3+V6VY3cAiT3KZBECy2bfi8poAzjS0AmLrhCweVj7dA4dMHSTg/Xm56j5FW5RVhbU7ETu/xNa4Ukdbkl5SZZO0MCyHOD9r0QgU52d/+Y55HzQpo03QEvQAAAABJRU5ErkJggg==)
Therefore, the points at which the tangents are parallel to the x-axis are (0,4) and (0, -4).
(ii) It is given that ![](data:image/png;base64,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)
Now, differentiating both sides with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
We know that the tangent is parallel to the y–axis if the slope of the normal is 0 ie,
,
⇒ y = 0
Then,
for y =0
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD8AAAAXCAMAAACLfme2AAAAAXNSR0IArs4c6QAAAGZQTFRFAAAA///b//+22///tv///9u2/9uQttv/kNv//7Zm27aQ27ZmkLbbZrb/25A6OpDbtmYAkGY6Oma2OmaQZmYAOmY6kDo6AGa2kDoAZjpmOjoAADqQADpmZgAAOgAAAABmAAA6AAAAoT4peQAAAAF0Uk5TAEDm2GYAAACuSURBVHja7ZHBDoJADETfsqCLCougsorA9v9/0gMhQaKG5eCJOTVN3rTTwqb/SblsrI4PkadZzRf+ROK8Wc3XQC4ZhUijc5EsjB9M+hgKb+b9ZXxyrvaAco3O7ZutjJrmm3WjVqTSALvuWgbnBzh0fQyQLz7jLGcqFojul8Hn9/4f+F1nQd1M1NaB85WzQCoZqrDjIkG8NySu0TjxJmplocHkeZ1IqQOI71fZtF4vdvUPOZn9JM4AAAAASUVORK5CYII=)
Therefore, the points at which the tangents are parallel to the y-axis are (3,0) and (-3,0).
Question 14.Find the equations of the tangent and normal to the given curves at the indicated points:
x = cos t, y = sin t at t=π/4
Answer:It is given that equation of curve is x = cost, y = sint
![](data:image/png;base64,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)
On differentiating with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the slope of the tangent at
) is -1.
When ![](data:image/png;base64,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)
Then, the equation of the tangent is
is
y –
= -1(x –
)
⇒ x + y -
-
= 0
⇒ x + y -
= 0
Then, slope of normal at ![](data:image/png;base64,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)
=![](data:image/png;base64,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)
Now, equation of the normal at ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACYAAAAeCAMAAAB3ypxcAAAAAXNSR0IArs4c6QAAAGNQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6OjoAOmaQOma2OpDbZgAAZjoAZjpmZrbbZrb/kDoAkDo6kGaQkNv/tmYAtmY6tmZmttv/tv//25A629uQ/7Zm/9uQ/9vb//+2///b6D7hhgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAmklEQVQ4T72S2w6CMBBEF9TiFYoKSG1p//8raTEGExNnHhrntYc97bAif4y/FEuAcnh0WswOXywrFtoknZDVn7VYdRBx1R2x6dyWDNanWjYjHMhN+5R2r8pjvi/yc9r7M/YJWaRrAU6dZCCq64tSw9qyAU4xsnDdM5hp4srB2GPaTBRfj6Ftbggzy1/dIiyeM9KIPRWxtYRuRWa31Ac4J5UVZQAAAABJRU5ErkJggg==)
y –
= 1(x –
)
⇒ x = y
Question 15.Find the equations of the tangent and normal to the given curves at the indicated points:
y = x4 – 6x3 + 13x2 – 10x + 5 at (0, 5)
Answer:It is given that equation of curve is y = x4 – 6x3 + 13x2 – 10x + 5
On differentiating with respect to x, we get
= 4x3 - 18x2 +26x -10
Now, Slope of tangent will be the value of
at x = 0 i.e.
m1 = 4(0)3 - 18(0)2 + 26(0) - 10 = -10
Therefore, the slope of the tangent at (0, 5) is -10.
Then, the equation of the tangent is
y – 5 = -10(x – 0)
⇒ y – 5 = 10x
⇒ 10x +y =5
Also, slope of normal at (0,5)
![](data:image/png;base64,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)
Now, equation of the normal at (0,5)
![](data:image/png;base64,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)
⇒ 10y - 50 = x
⇒ x -10y +50 = 0
Question 16.Find the equations of the tangent and normal to the given curves at the indicated points:
y = x4 – 6x3 + 13x2– 10x + 5 at (1, 3)
Answer:It is given that equation of curve is y = y = x4 – 6x3 + 13x2– 10x + 5
On differentiating with respect to x, we get
= 4x3 - 18x2 +26x - 10
![](data:image/png;base64,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)
Therefore, the slope of the tangent at (1, 3) is 2.
Then, the equation of the tangent is
y – 3 = 2(x – 1)
⇒ y – 3 = 2x - 2
⇒ y = 2x +1
Then, slope of normal at (1,3)
=![](data:image/png;base64,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)
Now, equation of the normal at (1,3)
y – 3 =
(x – 1)
⇒ 2y -6 =- x + 1
⇒ x + 2y - 7 = 0
Question 17.Find the equations of the tangent and normal to the given curves at the indicated points:
y = x3 at (1, 1)
Answer:It is given that equation of curve is y = x3
On differentiating with respect to x, we get
= 3x2
![](data:image/png;base64,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)
Therefore, the slope of the tangent at (1, 1) is 3.
Then, the equation of the tangent is
y – 1 = 3(x – 1)
⇒ y = 3x - 2
Then, slope of normal at (1,1)
=![](data:image/png;base64,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)
Now, equation of the normal at (1,1)
y – 1 =
(x – 1)
⇒ 3y -3 =- x + 1
⇒ x + 3y - 4 = 0
Question 18.Find the equations of the tangent and normal to the given curves at the indicated points:
y = x2 at (0, 0)
Answer:It is given that equation of curve is y = x2
On differentiating with respect to x, we get
= 2x
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFgAAAAwCAMAAACi7E0xAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6OgBmOjqQOmY6OmaQOma2OpCQOpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmYAZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tpBmttv/tv//25A625Bm27aQ29uQ2////7Zm/9uQ/9u2//+2///bMY4cTgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACF0lEQVRYR9WX61LCMBCFkypVEK0iXqCiVKUqLcn7v517SUpoATtNOur+6GSG6Zft2c3mIMT/jELKeT+ZF32By7injHfAmZTydO0vTTmRchTPdYq4MpbRUqgkALiMZ0IvQGOdIq0AbhhwRjgsXn6ygscANAiRsUqugEQa40M/QcKhwTodiILEDZwxapFR24UAY5pGY+BdTkHmMGCRRy9ic8snL5MoeCCwfpXR+CsmcoF9EQrsnrACZQkNhkbjXgsOTscLk7BPV+hnKYfvrgwwKy7s5OnebjgXNqkpVXOOdQfTqDk83buDM0xWJVbTes6dwTwhD7/eEow3AgcKgMFgela/bRceJ88BNyvn08dWCk+Nm1IIUzweOSHbLceZ00e7YdlZD4q6E2rZFfs+dgN3fnWCG8l7gGub1ZzQHwcbJwT3EjihnHxnkIwrJyQyOJg6xSsqCHjrhLBNcmptC/683xnZVJiPh3Z+0XFCYIVGY3rZgHmXepTDVmQXDFZg6YCJAFdN9Mhsu9y/X31/F6ymd3ypcMYEyAfr8oy3s8vynE3C8XCcEFzSZvYTmOqoEvS3RnizrC7z4+TKCekM3i/ICznga8g2oymo7JI758ewTmiWwj2gEimvzF8Fzrg7+ODOvlIc/iQqHrQ5Fk/drOxStCveEakIoCYymoHbh7VZcrd4Rc7HBeMNamii5QE5urPPkfb6pN94+RvtnkK20Q85EQAAAABJRU5ErkJggg==)
Therefore, the slope of the tangent at (0, 0) is 0.
Then, the equation of the tangent is
y – 0 = 0(x – 0)
⇒ y = 0
Then, slope of normal at (0,0)
=![](data:image/png;base64,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)
Now, equation of the normal at (0,0)
x = 0
Question 19.Find the equation of the tangent line to the curve y = x2 – 2x +7 which is
(a) parallel to the line 2x – y + 9 = 0
(b) perpendicular to the line 5y – 15x = 13.
Answer:(a) It is given that equation of the curve is y = x2 – 2x +7
On differentiating with respect to x, we get
![](data:image/png;base64,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)
The equation of the line is 2x – y + 9 = 0
⇒ y = 2x + 9
⇒ Slope of the line = 2
Now we know that if a tangent is parallel to the line 2x – y + 9 = 0, then
Slope of the tangent = Slope of the line
⇒ 2 = 2x – 2
⇒ 2x = 4
⇒ x = 2
Now, putting x = 2, we get
y =4 -4 + 7 = 7
Then, the equation of the tangent passing through (2,7)
⇒ y – 7 = 2(x – 2)
⇒ y – 2x – 3 = 0
Therefore, the equation of the tangent line to the given curve which is parallel to line 2x – y + 9 = 0 is y – 2x – 3 = 0.
(b) It is given that equation of the curve is y = x2 – 2x +7
On differentiating with respect to x, we get
![](data:image/png;base64,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)
The equation of the line is 5y – 15x = 13
⇒ y = ![](data:image/png;base64,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)
⇒ Slope of the line = 3
Now we know that if a tangent is perpendicular to the line 5y – 15x = 13, then
![](data:image/png;base64,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)
⇒ 2x – 2=![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAgCAMAAADDlWPAAAAAAXNSR0IArs4c6QAAAGBQTFRFAAAAAAAAAAA6ADo6AGaQAGa2OgAAOgA6OjoAOmZmOpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ2////9uQ/9u2//+2///bDgGzMgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAcUlEQVQoU62Q1wqAMBRDU/ee1dbZ//9LrSK0FxQK5vGQ5A7ATWoM7YBMS0IA8UL2nF1qzoo3j9GuhmCxhgkdztwWdnPf+xlyi//ilgnzKrNJtT0mj5PyNSZk62rbMjCfEGCO9PMf7QXHahHIiEz/vukA0oUDtfY9cWEAAAAASUVORK5CYII=)
⇒ 2x = ![](data:image/png;base64,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)
⇒ x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAAA6ADqQAGa2OgAAOjo6OmaQOma2OpC2OpDbZgA6ZjoAZjo6ZrbbZrb/kDoAkDo6kNv/tmY625A627Zm29u229vb2////9uQ/9u2//+2///beWS0qAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAa0lEQVQYV41P2Q6AIAwbKoIHHoAHyv9/p0WMGo2Jfdiybt06oicsYyzRRLaNnTODl5FyPPZ8V5Lv9V57A1nz2vZJYFvAf8E1OSgx47gpEOEvO7wIlYN3XE5jl9FahR9SeJSzR01LDf+74o4NwRsEMLBHs7EAAAAASUVORK5CYII=)
Now, putting x =
, we get
y =![](data:image/png;base64,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)
Then, the equation of the tangent passing through ![](data:image/png;base64,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)
⇒ y –
=
(x –
)
⇒ ![](data:image/png;base64,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)
⇒ 36y – 217 = -2(6x -5)
⇒ 36y+12x – 227 = 0
Therefore, the equation of the tangent line to the given curve which is perpendicular to line 5y – 15x = 13 is 36y+12x – 227 = 0.
Question 20.Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and
x = – 2 are parallel.
Answer:The given curve y = 7x3 + 11
Then, the slope of the tangent to the given curve at x = 4 is given by,
![](data:image/png;base64,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)
It is cleared that the slopes of the tangents at the points where x = 2 and x = -2 are equal.
Therefore, the two tangents are parallel.
Question 21.Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
Answer:The given curve y = x3
![](data:image/png;base64,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)
Then, the slope of the tangent at the point (x, y) is given by:
![](data:image/png;base64,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)
We know that, when the slope of the tangent is equal to the y- coordinate of the point,
Then y = 3x2
Also, we have y = x3
⇒ 3x2 = x3
⇒ x2(x-3) = 0
⇒ x = 0, x = 3
When x = 0 then y = 0
and when x = 3 then y = 27
Therefore, the required points are (0, 0) and (3, 27).
Question 22.For the curve y = 4x3 – 2x5, find all the points at which the tangent passes through the origin.
Answer:The given curve y = 4x3 – 2x5
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIsAAAArCAMAAAB/24sjAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjqQOmY6OmaQOma2OpCQOpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGYAkGY6kGaQkLbbkNv/tmYAtmY6ttv/tv//25A627Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///boJMM0gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACaklEQVRYR+1Xa3eCMAxt3UP2crLNDdmzm6izSP//v1vSVmiV0g6OBz7Yc8bxTG9ym9wkhJDTcUeAUzofTHz4gLjk0XDiMhQu+SOlt9FcpJRebvKIjr56k04eJUS8gV5EerkhhPdIhTDJALWbnS3gMe4tKqSIp+Bc6gUf4rW/DJlcRDomHKPU1zHigklifdY2BkPrBWJ0PwPJ9Hey0SfZPqm+yyiKp78jvuloso4kGY6V1Pb8Qp+atATX9FreoaCLh0+ybNmcxOu1pVOo5s4FnV+16whZktpc0smbPyyrJ+lNrCAhFweOswa5aSQR75Te/Fi55BOxz4Xe+XrL9pEqRTGakG28r66lWy4lEifNNrWQxWwh0uTD59yWYn73I+cEcMEIZnsvPZmbSoWU885+Xcoonv83Wc1FckSLDI2AMUg3Byrc3Sp3t8DLFPHYRELO7RyFVaPJhaFZBveUlooYb+fjol4GihieJRI8r6PGbgI31scsVIOL6glovEmy5RUVUnGRz0Nk6bH60BShiotIVdHk0e2BTuruccClFhmWHfWrkouaY/JfYQ3O5IJ6CUdi2TbmiI11DRaz56DJYWsXYxqKdMVqF5dMvg2CRejV+pae8GqkbASypoORTi4qIbLZCzaVf2BaSafx6FRiCSnpOpCBe6N4jyBv5y9lAqewPIwgLtRHpkIS6MDY4N3I095Yn9OB7Y32gPAp8Tjfl3ujNSCO48tntdobg0eLz2Tb7839qG60tLXbBmdyCR0tbfyEYEwuXQdEiL+m3xh7Y+cB0ZUL5EXvjeGjpbNPl4Hd3pgEjpajETkZdkTgD5S3QH5u/qjoAAAAAElFTkSuQmCC)
Then, the slope of the tangent at the point (x, y) is 12x2 – 10x4
The equation of the tangent at (x,y) is given by,
Y – y = (12x2 – 10x4)(X – x) ………….(1)
When the tangent passes through the origin (0,0), then X =Y=0
Then equation (1) becomes,
-y = (12x2 – 10x4)(– x)
⇒ y = (12x3 – 10x5)
Also, we have y = 4x3 – 2x5
⇒ (12x3 – 10x5) = 4x3 – 2x5
⇒ 8x5 – 8x3 = 0
⇒ x5 – 2x3 = 0
⇒ x3(x2 – 1) = 0
⇒ x = 0 ,
1
When x = 0 then y = 0
When x = 1 then y = 2
And when x = -1 then y = -2
Therefore, the required points are (0, 0), (1,2) and (-1, -2).
Question 23.Find the points on the curve x2 + y2 – 2x – 3 = 0 at which the tangents are parallel to the x-axis.
Answer:It is given that x2 + y2 – 2x – 3 = 0
Now, differentiating both sides with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHkAAAArCAMAAABrYteHAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOmY6OmaQOma2OpCQOpDbZgAAZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6ttv/tv//25A625Bm27aQ29uQ2////7Zm/9uQ/9u2//+2///b5V/NIQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAByUlEQVRYR+1X21KDMBBN0DYtSltt02pRUZuU/P8XukEugYZwX8YZ94EZZggneztnl5B/c0VAUHqYKUJiNmTJ5vJ5HmS5oXTFDopTurhIRr0zUsol2xN1gjwrvrgQItCASZjg6QqL7j7gsUTymMTBI0AledYPdcSKtYms+JIIHQEcM3zW4Q7xuks7muYZ/H/YQaqxLPJeyXX7y2Eh1UnHMvVGPf+bJdBCV/csJtBaynQP+gmxpUrI3D81u/y1Hb/fgbfXTb183dAelfA5vFPl+j1h2I5WCOEQVXIgh1rrItt0E+UqNECVXD6HAKA4hDYOaMVy6DaqBC6kZuqmC1k7FFmpSPEsSYYqVW8H7640Gsi3V5Ns5VsPF8j9VcldYWlCHdFuo0rdo03i3ZO19IsK0zXQU5WceT6e48BCRuXxsq8qmdevpFSFUF3ConZlJimrUssdQ70wKMD755oqgqlVZ7hBaCuqhLRjWFQJadJXt6qEhlxSpXTHgFGojnc7KkTbz/Mdg+S82/bowO+KHaOedwdC2I+b83Yt706OTBw8MT646XMd746Pqv9o7BjQ51benQYY1tZsx6jl3amQsx1jz9vx7lT3+Nv//QHaWSXFyfqo+gAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
We know that the tangents are parallel to the x –axis if the slope of the tangent is 0 ie,
![](data:image/png;base64,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)
⇒ 1-x = 0
⇒ x = 1
But, x2 + y2 – 2x – 3 = 0 for x = 1
⇒ y2 = 4
⇒ y =
2
Therefore, the points at which the tangents are parallel to the x-axis are (1,2) and (1, -2).
Question 24.Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.
Answer:It is given that ay2 = x3
Now, differentiating both sides with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Then, the slope of the tangent to the given curve at (am2, am3) is
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAS4AAAAzCAMAAAAeoRWAAAAAAXNSR0IArs4c6QAAAMNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZjqQZmYAZmY6ZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kGaQkLbbkLb/kNv/tmYAtmY6tpA6tpBmtrZmttv/tv+2tv//25A625Bm27Zm27aQ29uQ29u22/+22////7Zm/9uQ/9u2//+2///bUEI90gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAGWElEQVRoQ+1bi3abRhAFtUlN47zUSnm4bVIZx3WN66bBVloLsf//VZ3HvoB9gAxKfKI9Jz7GsJeZu7Ozy95JkhzagYFvlIF/36Tp3PjeuuxBir9HlZ326P/VPSKun3/yGVW/ukhuZ5dJ+fIjPqIu+/vg7yHOjnema5OmO/ftb7vzSZH/fBeEqH68TJLqqbKPLgc1d49yle/u8mafdInb4zR9xF6LfBHxvaQHtM986W42sHnC2WMzFyG63Egac68TuZhdJNsc5hi0zXfemcjG3crUVT6mIFSXTr5sYP2As0d98knkqz+8ce1EMu9s0FWkacrGTdOKI8AtKZ5Fjr8HWqkSPYeXvnTTZYD1fXePElwMOWmZ2H5TBavPi+xU5Ni/ylIY9no5JV1kQEFhpYaJgh+YwSR6k8Fv9AOu8W+UY2jWmks/wwSs8HSPNjACxnIXInX6VdkqER9grEWOHG1wkkxO1/ac1zp6GwXZnSjw9yp7Bgth+uwiKWnYMAjYLRhw+9LDFwNrPNOjCQy9P2fhPCBNbPcriCS0qcT+JQb0xHShEy9ptku6yHeKNPkDIqmdTsnOSDPAVuRyHz+wE1MjtfrVS1xqtKniDId7YrrgDWseWxoh2Wy6FHGWK33oMsA+uhzAviFgE1tWWXRh3t3QGE5PF8QzDpOKLrF+m9Gsa1k3nC4G1ni+wYjFKYU+IvnpwrEuKFHsga4qs+jCXNM2rDMZI2uoCdIF5a7WbPaPg5c5MrHVj5dy3qbWy59OaHJMShfvTPmNkhMKsvBk5FkQbAbY4O0WXZaJbZpL3DS+5eWnoBkyOV1AzpaWYbXvwnHc/oapnliUw9rgp8fHj8CtLwEbPE2XF9g1AgapY5C4SmfzzxnxpfbYk0ZXsj2HPDXndU6+8Qr+8N8yPYVtn/3DckXu6oPhZYAVnhVcPmAnokbyG4TGy/wwLV22gfFvRn66R3DF5uq492ELwbuIqSdj0+zoiQSzpU8kxnV6dzSRzz+oxWd/0QXpK3DepbyR5127Ozd+T/hm5J32fqNrfEf2j7jP6Nq/d6O/8UDXIEoPdH0xuvCwkRsf1ozWRgMeAqSdMb9MvKt3vHHgnzyMD0SBx31D50VydzhMxkFzYEy6hoT6ICNHA7430Jh0DaLgYT58oCs4bm0V+0vQFfwYujlOZ7/DWTULIgOar0ZCAg5Ash9tqdjT0iXWINVJFVsbEfzUFmdcKxH51O4C+2okNGCYL6el0KV10jstXUW6SrbLpmwVP8jhI5zgQY4L2N+lx5mQG3DfdDnE5qj4vz1fUSSEjgnd2rOnqkIBhuLLBShVbDh5BhUb1HA4Vp02ushAmv5abFbiv0/NBuO+Z7qiMdEEVlUVHTVaA0bTVzNRaRU7QRmZpfAgXeIMMo+vEEScB27alqGibovXUtl3q9nYcc3la9FZ2wTWVRUdFVsBRuni8gTVjIqNqj9HbpCu+vXH+o1PlqGbseKDRrKktBlRs+tXlzq90gTxN5OFGVgVWTTVaBswwlczrVuyLFj0ggteFF3rd+6FG+p9/C9ZP79Lbt6HFXorRoJ0KaEWKkxwI4EtrGa3gO0aiYboawGG6WpFs00XlXFYdPlKz6p3fjLAQDyUrYAzf9OpSonXSvy/r5rdAdY27CDL8txvBrNNV33yK89Tji6fy5v3d5Uc6g4j9S93PB6hKj+opiEyjdhsVeJE1exQaUkH+L50SUANY6nYIATVSyKT6SKPt5C7F7COPb7OZhd/pbM/uWyIc5fjJu5AkfLqiX++0mZgA6havJZ09VCzsXDJ27rA96RLAZo3ahVbFFhwRkQQXbRI1stVcgUXOE1x5WwYG7qpRbiub7QXwLcZsVnm0x5qdkj8dwDrtw9SsVUvDWicUCr2KofjToychSy21OV1WJWDNNO/9tj6bgaK8+SBCYyLFptVioir2aF9lwNYO+6VxwMpFrei2KLVGZxZcENw+5RqON10BW7GaxkbdkZ39erpPuJ/iIHJ7pHDmE1KSEaKrqsf9IpXv/7bfzMJTEaXxdHdp+wU3dRPRkcUGFN9naePrtMjqKw4gu+IBf7T8f3kH//NYKp38xX57wjU6esT/40vocUNnroOCjvBjYSTr4cp/luuhKvYg9EZ3qZGA/thPuD7CIp6E/sIigIcHjgw8M0x8D/09yIHkza0bgAAAABJRU5ErkJggg==)
Then, slope of normal at (am2, am3)
=![](data:image/png;base64,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)
Therefore, equation of the normal at (am2, am3) is given by:
y - am3 =![](data:image/png;base64,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)
⇒ 3my – 3am4 = -2x + 2am2
⇒ 2x + 3my – am2(2 + 3m2) = 0
Therefore, equation of the normal at (am2, am3) is 2x + 3my – am2(2 + 3m2) = 0
Question 25.Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
Answer:It is given that the equation of the normal to the curve y = x3 + 2x + 6
Then, the slope of the tangent to the given curve at any point (x, y) is given by:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGUAAAArCAMAAABW8LbhAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOmY6OmaQOma2OpCQOpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6ttv/tv//25A627Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///boC4QuwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB/UlEQVRYR91W7VbCMAxtUagCOhWhAlYsKN3W938/k26y7qNs3Sg/yI8dziHNzedNCLk1UZSurhCTugpKzK4RS3iU+JXSKVtpTun4GDM6+gxQopgtiV5DXTQfHwlRQUCIMLax+vLuGz6TAJGQNHoGs6Yu+NEfIfJlo2g+IQoju7xYsWDKRJiOxgDyukBcTwsoTQiRoy1J3rLZFxSLFEL0Fx3Nf5iBUdhlfvIL0zb3e6K82zh92ZK9x4hBD/ds4/ihe/drPl+3hKL3j5Te10xKVzH1AdJZ1gcem7XMisBG4dX87O2yaG6NgqBLkkS+tRYYq6ysIlmqfRmlSb9bpwjwTSB9S7NgFYCowv8SijHYZ0Emm9kOBwvSZgymEQU5h4JeuQX8zaUoBNo0xcMt0Vj0WizZfjzZKn6czdyBGd9iNq3NYpNfmvdjE5WxkGwexUosGUt6ZgzVY4Yo6eK9Md8VFDHxXiRZ9KZngCXSqMnNMoo0e9gvZxoHMsHrQAt4maeunI4SiqEeo5v3dKerL9kwIGAA4XDoYL/VvazMvmmpk1af0ek2xLZW+Ksva5ww+76IJL8tbXLyT0bbi9NtWZBT25Me/xe3pZOcelitPLHvsSZyGo6AFmwUFzkNR7JRXOQ0HMW6LZ3kNBwFspTflm5yugDK/225dJPTBVBux8QfzH4rHYgn9vIAAAAASUVORK5CYII=)
Then, slope of normal to the given curve at any point (x,y)
=
Mediumy = ![](data:image/png;base64,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)
⇒ Slope of the given line = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAgCAMAAADDlWPAAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAABmADqQAGaQAGa2OgAAOjoAOmZmOma2OpDbZgAAZjoAZjpmZrbbZrb/kDoAkDo6kGaQtmYAtmY6tv//25A625Bm29uQ2////7Zm/9uQ//+2///byJLbuQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAd0lEQVQoU6WQWQ6AIAwF64Y7ruBG739NBWIVExOJ/Zy8V8oA+A2OiVuYsupBAOQLUWVgpjlWvGVu23GIF+cxqcuF38F+aXvfbfzq/9Pkd2PaEgD5xS61hNxJ3rpkzdElqhbY8t5aNMaNwiAyhPyercvvzELx6TM7Tn8FNnXPVPgAAAAASUVORK5CYII=)
We know that if the normal is parallel to the line, then we must have the slope of the normal being equal to the slope of the line.
⇒ ![](data:image/png;base64,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)
⇒ 3x2 + 2 = 14
⇒ 3x2 = 12
⇒ x2 = 4
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD8AAAAXCAMAAACLfme2AAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OmY6Oma2OpDbZgAAZjoAZjpmZmYAZrbbZrb/kDoAkDo6kNv/tmYAtv//25A627aQ2////7Zm/9uQ/9u2//+2///bboZHmAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAwElEQVQ4T+2R3RKCIBCFWUuttB8JIjH3/R/ThWESKBJvunKvGNjvnN0DY1v9LwHknTPD5xlgL1Zaz7yEG3u1u8c6AY+viFTQMQlQ9grolFEzb5u1oWQhWHSfVIr6pJkfOfk3PkIjuSLpd33eDrUdeqiPp4zZTUvgj9y5Kt/mp5LPIzcRUo2Xa/gP3+eP/WXVWxzvYmyd1NIanr8qCdcNQ0lbaAgCzMh/OFC0huVQkD9kCsS/kWdrqXQqS1tv7+kEJsPBCbFcRkK1AAAAAElFTkSuQmCC)
So, when x = 2 then y = 18
and x -2 then y = -6
Hence, there are two normals to the given curve with the slope
and passing through the points (2, 18) and (-2,-6).
Then, the equation of the normal through (2, 18) is:
y -18 = ![](data:image/png;base64,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)
⇒ 14y -252 = -x +2
⇒ x +14y-254 = 0
And, the equation of the normal through (-2, -6) is:
y –(-6) = ![](data:image/png;base64,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)
⇒ y + 6 = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEsAAAAgCAMAAABto3VBAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmZmOmaQOma2OpC2OpDbZgAAZjoAZjpmZmYAZrbbZrb/kDoAkDo6kGaQkNv/tmYAtmY6tpBmttv/tv//25A625Bm27aQ29uQ2////7Zm/9uQ/9u2//+2///bzPqS5gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABdUlEQVRIS81U61rDIAwt07l6n3azMoc61FGb938/IVwKLWB38XP50340nJ6TnKQoTj/gdXYskh93i6NhFQU/Iay2IhjPslR/xQvebjYjO8Hv3weZwC629hDog3v/FbK5Vqr84Erqoz4Bal4yOPC5IOR8rTKaS3zEQ5xFBQL1CDBSF9+VTuTTpAqgcXeEWCqHY9NyxJoSMxgh062U7sgEWKhI6I+ZmoiJ1s/k078/xGKmGCxpc4sFVPLyujDAMgLkX1XBtFO9wFLa0jfl7dx0Rwo2YUgH0hArGpaXRO0u4t3ASF6LRmC1yyffHT0sNnNk0vWyZYDVuq28qoZYaCqB5WyrpLcNeWAyQ9hZ6GtEs2NK3vjoe6BkInm5werVyzQDsTK+HzWPXte6eXR73rlFMdhrT7g9D6urruX77i+zT3kd2idhvvyxxhLz4bjtDodY7XIDtH4Zv03j/0EsXKtyx+xOxb/h9vzhGrs9/1XGV/NhVP/99g9/cR97zElPLwAAAABJRU5ErkJggg==)
⇒ 14y +84 = -x -2
⇒ x +14y+86 = 0
Therefore, the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0 are x +14y-254 = 0 and x +14y+86 = 0.
Question 26.Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).
Answer:The equation of a parabola is y2 = 4ax, then,
On differentiating it with respect to x, we get
2y![](data:image/png;base64,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)
![](data:image/png;base64,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)
Then, the slope of the tangent at (at2, 2at) is ![](data:image/png;base64,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)
Then, the equation of the tangent at (at2, 2at) is given by,
y – 2at = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAE8AAAAgCAMAAABkSNU7AAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmY6ZpDbZrb/kDoAkDo6kLbbkNv/tmYAtmY6tmZmtpA6tpBmttv/tv//25A625Bm27Zm27aQ29u229vb2/+22////7Zm/9uQ/9u2//+2///bw6d58gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABfElEQVRIS91U2VaEMAxtEcF9HBzX6oBbxQWQ/P/Hma62ZZgzIA8c+wQ9ye3NvUkImf2BIpmS49vpalI8Qvi/xvtYUbrwDPhTve3FAymjtQMI+X5lfuH5+HW42c0B4vGzF5nJKZ6lAgF2bqEHwHKZ3hzdBTnANO5OWJKVOKWWz16Y9HpvSLG1Vo1bN3jsVQdMNmJOaVyhCiF7/SqUh9LQQuiEBGr8rlVsQLBJ1XWODwPrg2NJBSKCKH5tJnBVcKCXKQAY8tuqpHzZhDti5wkR6HjEpQ1o0hOrCVavj9NkvXix6yS3dnCvQX234f0ylQVu4ufh2YD26rrXaTStIrvxs/rdr9usb+fIoC36OcVofyFHL2ozMmFrN+mSfN8IP8TXozu46LXnouo/YDRCfnYGQ0Dsu8VXJgQsaOQ3VTggw+ajO5RqPn7X1bD57eBpetiE5ozcLypd75cnbFc7x+P2n4Qz+484/LqSjLiZPV5cfY4oqzelSaPbKfFmjfUD2qwgGCRoAM4AAAAASUVORK5CYII=)
⇒ ty -2at2 = x –at2
⇒ ty = x + at2
Now, Then, slope of normal at (at2, 2at)
=![](data:image/png;base64,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)
Then, the equation of the normal at (at2, 2at) is given by:
y – 2at = -t(x-at2)
⇒ y – 2at = -tx + at3
⇒ y = -tx + 2at + at3
Therefore, the equation of the normal at (at2, 2at) is y = -tx + 2at + at3.
Question 27.Prove that the curves x = y2 and xy = k cut at right angles* if 8k2 = 1.
Answer:It is given that the curves x = y2 and xy = k
Now, putting the value of x in y = k, we get
y3 = k
![](data:image/png;base64,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)
Then, the point of intersection of the given curves is ![](data:image/png;base64,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)
On differentiating x = y2 with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Then, the slope of the tangent at xy = k at ![](data:image/png;base64,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)
is ![](data:image/png;base64,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)
As we know that two curves intersect at right angles if the tangents to the curve at the point of intersection are perpendicular to each other.
So, we should have the product of the tangent as -1.
Then, the given two curves cut at right angles if the product of the slopes of their respective tangent at
is -1.
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKgAAABwCAYAAACKJjv2AAAQzUlEQVR4Xu2defD+UxXHX5RKpTQZYgqDtEybVIzUaCJRaDCytaKFsbYRLZaRFoqyhqIQldBCVEoZpUUlLSZmIiWytEqF5lX3M/PM0/N8n/tZ7+fzPPfMfOf3x+9zP59zz30/555z7jnnLkOmLIEeS2CZHvOWWcsSIAM0g6DXEugbQB8MvA24Gvh6ryWXhrmHAocDZwM/TsNCt1/tE0AfAnwUuAH4IPBAB6J4NXBl+GYHn1vyE7G8PA74NPCewHtqvlv9fl8A+iDgY8AfgEM7AOfngbWAZwFbAhe3KuWlX16FlzWB84A9gR8k5L31T/cFoAcALwBeCfyz9VnD0wE10aU9AGhVXjYGTgFeDPy+A5kl+UQfAPo84FPAS4DfdCgFgfHTHgDUKVfl5WBA+W0P/KtD2XX2qdQAfTjwbeDkoA06m3gNULTBY1WAPjLYoR8BPtEGY6nfmRqgbwbeADwf+HvHwqgKijbYrMPLtsCHgWcDd7TBXMp3pgToo4EfAYcBZyQQQh1QNM1uHV4MPX0H+Bzw/qYZS/2+lAB9PXBI+OXfnUAQdUDRNLt1eXEXeifwTOBPTTOX8n2pAGpA/lvA9wA9+BRUFxRN8lyXFyMSvwjm0mebZCz1u1IB1AX5LvDS4CR1LQdjjysAm4VTq5uB3YEqmnwHYD3goIqTaIIX1/GL8N+j65d3EEeuONXyw1IB9O2ADpJb0p/Ls117hD+QcfplxVDNe0NEYLuKXDXFiyaTNuhTgD9W5KV3w1IA1G9+DfgL8IreSaQ8Q3UBWv6Lk0cUZoLx5Muaemnq96QA6GOAnwHG7jxzHzr1BaCaLB48fKaGudG7tUgBUON1Pwz251c7kIgAqkqTxj4WWHbkhWZfua26xY6S9uz46U7TvIx+z7X8JvBX4GVVJ9y3cSkAqlNhupj253UdCKRJUDwJ0FaNoTdOOB1rkpdJPJjlpFyfA9wbw2Tfn0kB0LcCniE/OWQv9V1G4/ypQc2+Ksj5qEF3G3vQU537Op7cOwD/lO1tHX+7lc+lAKjn7oZ3/KXrKA2d+mKDKsc9gtZ+IvDroQtW/lMA1LifW+X6c7IN9QmgRkW+ELZ47fzBUyqAPgHYCPj34CUIfQSoubWezw+eUgDUGOij5gigOn3GIN/VAzS8CPhGSP7OAK24IG7xqTWoP5BXAU8Fvg94fv23ivOpO6xJXootPmvQGqsiQDXiDYV0Ud4xzurDAihNTzNZ2loo+XhuxaPOGqKgaV4KgDqXuahVSrHFnwa4FaXy4i2UuyZUkO4DfCCUOj8DuLYO2iqMbZqX1wEfDzvD9RX46d2QFAA1bmjuYqpYnVprP+CCEHQ3uL0NsHaC2GHTvBgDPXDAMeb/+4GkAOiOwDlhm08Zq/PI1UjCkcC+iWt6muLFH5umk+/ruoSmFe2bAqAKUMckddbNa8NCbg2cHkpPWhFyxEub4MW1tIzaPIFN5yUnNAVAPSr8OfC+kNEUsX4zH/Gd5pVWKb11u7foTK9eDZSS6vDyiNAOR9PFBJa5oBQA9Rdu1s2dJfJBrf3eKZyBG+f7cgCjZSMvDCsR6+SYOa+tpua0TGIL4CvACcBeHa9qk7xo05vGqD2tfNom7ee9g7n227Y+lgKgzsXTl53DFmt62FIkmMy+Nzxl7Y3jDEILsJUBs/NNdYsFqHmo2pyCXlPDrKOTgP0b1Oix69UkL7sAxwY5/C6WgQrPuR7rhHyKNdpufFEXoG4rCsYk2TKlG57DG6ebFa9TK6jl1JLGLKUiLOS/Ckuw20grFqB664Jdnn8VkqZvBTZPEKxvkhd7NbkeWwH3VwBe7BBlrkmlvJVZq72t6gJURl1kewQZOoola7ktmjNh2bDINHILOQ74SWj05XMbhLFmjxtLLQtQ32FWv32gbCB2FfClivZr7HyXeq4JXlYKdv1bQhuhJvia9Y7jQ/OyXgPUSWhHGRwuy6hbqvXcatFp27yBbM/ui05uowA1cdg8zFGA2ttJ56sgta6act7JXcyCOeXVVcFccoBqlznhGFKrmdtZJp6o/WgTVlsI6nnG0ocANcWkLf5G4Ipg254btEkXDkMs7208Z48BnU4b/mrqTCO3f1PxPPuPJdth6nRNouQAFWzrxs4EsJWNv2S3SwXltjwro1yzwDruTSLP5bXZPKa03aCaV7t3VIMaY/WdHgbMRT5khPyNedo6SHks5RwtFzq5aA7Ekj/4o/sK0NhJjD63YcgMenxopWgIRydnGvlr1hYV0LM6YmirqR2XD2f5NluQCoC6lQtMnQR/JItA2vKXh75Mx3Q84eQa1C3B9oixZCtAY4l2C7HLxYXAWcA/ZrzABqy2/tZTn2Y/GXO7KISVTDTRZtXDN7mjAKgLZfqcvYkK7RrL+1Cfe1MIu+lN3xMxiRUBNWksuXbTynKSA1TNJwjKkGM8kSmzvRpJOAJYLRSejYdILFDTCVODGqxXaMbhDBWNe/F65dqzJwbbtgzvQ3vWJGmdR1PsjKTMIsF5U2j5M+vZ4v8Nv6065eHkADXhYJWImfiMzVMFhgHzuyLGjD/iBQpnhu3e4PUoHRXinX6j0BImPJtgLGDHw0yCU82ya9DgFdjp/RDDe+cHz91TsFiyD2tZJ8kWmZMoOUBjJq12097zRMHJVwFn8R1NilMD0PXAJQFoDf0ksiuJJ0tu+dq8xkU1FfTwC81v9zzt4nkinVFbpps3oAbtmpS7GfvuYv5QPOa8Jayd69co1Q3UCwy3dTtZ6MDUJR0gw0eGhi4JQjD7fhIZB3X7KWwkfyzapqsDt4cBhWNVl6++jNcW184XmMonBdnJb7zYsZB94+mTdQGaQkD5mwskgQzQBVrsIU41A3SIq7ZAPGeALtBiD3GqGaBDXLUF4jkDdIEWe4hTzQAd4qotEM8ZoAu02EOcagboEFdtgXjOAF2gxR7iVDNAh7hqC8RzBugCLfYQp5oBOsRVWyCeM0AXaLGHONUM0CGu2gLxnAG6QIs9xKlmgA5x1RaI5wzQBVrsIU41A3SIq7ZAPGeALtBiD3GqGaBDXLUF4rlvALURlu2rrw7NsPqwFLaXOTyUP9vsLFOHEugTQG3eYF37DaGp7AMtysEafC8u8FLb0Zbi0z5p/b116PaQurJFvvKrxyTQF4BaV+2Nb7b7O7TlGyoE52Wh9t5eUH7PVuDbzmhiu+ZIn9K5uMVtCL+GvgD0gNCtwq7HbV+PaFscLxOzWZntCncLXTG8pe2TMxZt49BN2oZntoDM1LIE+gBQLzOwlYv3JtkhuW3SxrV7iSBzey+uI9TOfHfExw8OFzBsn7BteASb8/FIaoDa3tE23ScHzdSFVDUntD29q0myl5NdmzcL7cZn8WCbSe1Qm5zZ0CxTixJIDVCvl7FPvY3HUlzdZ/MrO0HbPvs1JeSsverlX3YAvKPEuPxoSQmkBKhd2mztd1hoYV2S9dqP24jLJmV2Z/OupFmNdkc/aOjJu5q80tvLCzK1JIGUALWX6CFBC93d0vymvdZtXofIqIH3/tgI92mhR2ksK2p+++E71q7OmVqQQCqAGpD3GkP7d+rBd032uNQ5K66s2S5s82WurDE2autJgTqrv37X85ub76UCqO2rvTzBfvbFDXJdCXWPKQ5Z7E11BZ/Kzl78/utNJW0eLHQlm959JxVAvV9TB8ntscwVik0I0Ka7NrYdJxvilr0tWTNFG9QLxbq6QKsJGQzmHSkA6je9Pc7OyF4AMGRyJ7D1uDFcT6cyNSyBFABVe3lttHFEbcEh0woBoNquBw15In3lPQVAjR16TY32p5fJtk3eAlKVZo1Vfl5DaG98+/RnalgCKQC6Q0hd0/68ruH5THrdLJAtxULMWLOcnIvXMN7bwXwW6hMpAGqihufZXhVjHHLoZBzVP+dz29An0zf+UwDUc3fPvdU6067Z65ucluKnCFt5XU7j17AMSRBt8JoCoF5haLLG+nOyJRqJ8Jprt/gyV0C2sZ5z985UAPUqw40mXAg1RAEXAPX2Nc/nMzUogRQANQbqfZHzAlCvYjSvNAO0QWAWr0oBULf41BrUH4i3Mnt9t+UenqV7OW0Vyhq0itQix6QCqA6FNlvb5R2TxGCanaA0Vc48AGuh5MM75ssedfr+AqCOz7VKkcCLfSwFQE8LNxSn8uKLEg8rSPcJ5R+WgZRNFilkbC2T99mrja+PFXx+Lk4CKQBqHNQ8ylRxQzXofuHabxNEDLRvA6xdMY5pDPTAOYrrxiGno6dSAHRH4BwgddzQI1cdtSOBfWvUFwlwzRXfl6JspSOopPlMCoC6mNqAqTOAbNwgqLYGTg+lJ2VXQfldCiwLbJpzQsuKb/bzKQBqoZoVlWazm9HUBPlO80qrODlu9xbA6dWrDcuQTSBsh3NBaNlTZmx+NkICKQCqtjED6M4S+aCWZ+wEWEs02qrGshEbMEixTs7u4exczWnJxhbAV4ATgL0iZDb6iHa0qYPasBbgtU3az3sHE8liv7mnFABVqGYJ7Ry2WFPVliKdELPvjZ9aB+Q4T2wE2MqA2flmtscCVK2tzSnoNTWs6DwJ2L+CRt8FODZ82y4lbZEyWCfkMKwBbAlc3NbH+vTeugB1i3ORTNgtU7rhObwxw1mxQzWUWk4tWdQu2RWk6A7iwgl2m3rFAlRvXbDL869C0vStwOYVgvXnAcpgK+D+FhfWeWrGOEf5zACNFLZCc5FPCaGjyGFYV27RnAnLhmimkdvZcaG5gvFLaYMw1lILY6llAeo7zOq3D9RawFXAlyrYrysFW9rOJLbu6YKOB/bMAC0nam06A9Vlf9VuqZbsqkWnbfOC0rN7NZULMwpQY5gWq40C1N5ORSmxz6p1y5QSl5m5O4cFc/LYVcFcBujICmmXFVpr1sKp1cztLBNP1H7UAxZ4esGxZB8ltZZb/fgWfyNwRbBtzw2arQ3nxbp+HT1b5mheTCO3f1PxPPuPJZO4dbomUQboiFQE27qxUgVsZaNWcbt00ex5dN+M8Z4oWVO+SeS5vPbjNaH1oZpXu3dUgxpj9Z0eBrSZm2nM84yg/ZdyjpYL3VM0B2LJH9nRGaD/k0BdJ2lcjhuGzCBrz91uDeHo5EwjNYu2qICe1Z1Du1HtuHw4y785vLQAqFu5wNRh8UfSFmk/Xx6STY5p6yMZoLMB6vZke8RYsi2hsUSrNe24cSFwVkRTLvt0mrihpz7NljP+d1EIK5l/qc2qh3/tiAYVNCZs2Cep0K6xvJd5zga4hrr0pu+JGLgioCaNJZuYTSuFyVv8iBTVfIKgDDnGE5ky26ta/AhgtdDteDxcY3BeJ0wNarDeBTQmaKho3IvXK9eePXHEqSrD/6xnbdSgw2aKndGLWSQ4bwKsn48lQ16rZg06W4N6Tr1KhFR9xkauAsOA+V0RY8Yf8QKFM8N2P378eVRwhvxGobFMeDbBWMCOh5kEp1pu16DBK7AzcYghtfOD5+7JUyzZ+7Ssk2RbykmUNWis1MNzajftPU83XIgq4Cw+qUlxagC6HrgkAM+ewpNdSTxZcsvX5jUuqqmgh19ofrvnaRfXJR1AY52e1atBuybnakmJO4c/FI85bwnyUmZzS3WdJIHhtm5XDR2YuqQDZPjI0NAlYUFMy5tExkHdCgt7zR+LtunqwO1hQOFY1eFL+1fbWmDKUwpab0KBYTHfuS51rgvQFIuVv7lAEsgAXaDFHuJUM0CHuGoLxHMG6AIt9hCnmgE6xFVbIJ4zQBdosYc41f8AztNSnkefPtoAAAAASUVORK5CYII=)
Cubing both side, we get
⇒ 8k2 = 1
Therefore, the given two curves cut at right angles if 8k2 = 1.
Question 28.Find the equations of the tangent and normal to the hyperbola
at the point (x0, y0).
Answer:It is given that the equations of the tangent and normal to the hyperbola
, then,
On differentiating it with respect to x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the slope of the tangent at (x0, y0) is
![](data:image/png;base64,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)
Then, the equation of the tangent at (x0, y0) is given by:
y - y0 =![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPoAAAAXCAMAAAAV6KJTAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AAAyAABmADpmADqQAGa2OgAAOgA6OjoAOmY6OmaQOma2OpCQOpC2OpC8OpDbUTEAZgAAZgA6ZjoAZjo6ZjpmZmYAZmY6ZpDbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tpA6ttv/tv/btv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///b+9WjEQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACoklEQVRYR+2YbV/TMBDAk4qsKqhjKOqqQAUMU9du+f7fzUvSJim7u6TZG18sL+D2Y/e/50uLEKdzygCagT+fpLw8JjcIIJeZ+z3avUNCPnN/fSc21X157Aggl5n7Pdq5Q8JMZv/miNCNXwiAZip59isEU2I7RZjBVMvyoltNBMAwVRx6ke0EIZ+5OWrUIXIEwDFjx8ts8wSM+bzGqquOjRwBsMzI8ULbLAFl9nWIXW/eus3ewY9uLXQj5fm2r+Wrq0FI7D4SAHoDkxgjdfZ0K6tvwbZojUkl5doL/AQyBMq28uHoZrHVLXzcryQcSIluzreQhOreC6x5BhCYOEFVF49CgU1vWxhPdANeeIE1zhBGpvsdn0kp4y6AXWW2j1pEQnL5EYCJHhRyOKNtW4D9Ciz5Y9Jtl5MXEqHTBEpRN5OLJRoAuKHgk/4OTC/ElMMIzKWGAxIpc5Pamibzp6/fuZXjhfA3JHkcgbAeQte/P9e20aPML0Rn/IFedgLf8TSgIHThZzEMJY3BkhcIVo9ueDOqL4pmOr61ufACEwML4NsFc3x/88X1oxe4/LEEXDFkFNbZy36FRL2/ceZHgTHPAviyD7MeP0fBoLnh9wKL4Ai4YjSafb0Uu6+TUYXxk4M7g7CDN5sLovE5AKdnmkrC7fIQL1zdguEOrHtBsAyGQOjFjzQ/4Vb/u5oMu+jGHegE09O7ySKOM0oDeD0I/cOtlK+j1wZ4pqig6lJ+HIUlz+AInM90K3XjfeME+xYweVxO7C9RqnfInW/bMUr0YMjszWamzQkDJrnqB79L9YjZtGnPtR0YLvR5erq5/GFr7QXz1LHdwUWXd0r1cPo824FRogfz5jaaF6Ds5h84qft9NFuqRyR2lu2IUaqXV9/Tt/7rDPwDid5lS7cbhhkAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOMAAAAzCAMAAABSbk9RAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AAAyAABmADpmADqQAGa2OgAAOgA6OgBmOjoAOjpmOjqQOmY6OmaQOma2OpCQOpC8OpC2OpDbUTEAZgAAZgA6ZjoAZjo6ZjpmZmYAZmY6ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tpA6ttv/tv/btv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///bw4CKDAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEXklEQVRoQ+1a6WLTMAyOyxjlLlAKjGOFsYwjFGjS5P3fDNmOlcOWrNXZVmD+pW6fP0uyJF/Jstt264EkD1SLXvdfr5Xq/yaZEVi9Txr9WjpXj75349SvPmeb2Xl84B7wYhmH3yyiXp2OFKjuC2zUfVqgz3CzFvmj5/fGfyuk8+KA5VEvEg7NPj0XD8b6bUTpCF07oO+nQ7K0WY8nrZCa2AOKo5s0PVfqeFsodYoC66VmrfHVXN152QpMDSnN/3rEJZhYjjPUjYfU0KsPbNZewF92InOgbNYwMAoRI4+3oOjsvFm3Ag1vowyJ65WCRtkIWjjGITA9IzWzyW8UeC8VugYU4FoUKHw1t+YIiUnGeiWtU6Tm1fyJzRIUWCO15s1HCEIUKLhxgpyYZsz1/Ka1wqUUChyfTo9SD4oCgbahZ5qImGYUdme0rk/eWoejEA3W3AQhzJMVwq0LMSSuYC/39CvdBRmHwGqeGKwQd/VKVy4UIlFRr56dWJ84IdyhdNUFiav5h2y3ZtZ0xzgCWv32b00OPirVMkMh22lvMxmQA9o0FILDuwjriE2ddZUo2KdlHAEbzi9x02FRmsE8KvXcCUvIs+2O8xzWcraoO8VwhKUNXrZKWkYPmJ6QI0+YfQXWxICbShc5KIR8Wa/G1dCqzi7pltEDFvSiGp/GEMLaSJRryC2zcOjstQLV/CSKzGPH6AF1Lk3aQLntjto/NevFJ+NrFKix/WJoFxMyHztGD8gV1uZMqcdMrQ7rpwv3gqg5kF22HKFA2Rjwfak+tBu2UKce4xhYzcnCqv0RrNU/mGVtqpjApaNHuJnT3huMOwL6uY1os+8PjcXW74mMnLJOMDbmuhIHF9DJi7HvmOux0aau8YE9rPSb5OooaUavxEY4i7at1d/a2Nsa93TGrcPY9u63QXukAXwQl6GNRA8xtVkwqZOHyMakueI6X8k8egO6WIW6+8/GatbWnMAe4RpqzpR7E6aumnC5qbVjShuZPYA2P1hy6D1A9EWiUHj+47Gh/Rffg6bm9nKxk6CXwoIXCTyRRLCBdTnGTlJPGRPG5sid7eDUxWDDaxbLTlJPWaONjZEXiYEiHNZUPK9xPUhqfUOb3JrNQ/cyGHuRKI6+namZfRZksV3tlrJT1ERIXM5qfb3RWGdFXySKGdys2eDhsd2dlZSdok69s+qcYQ4j7IuEDWXtCjNqBDu8uJGwU9TTHZM0U+RFwtho0gyurqPYwfW2hJ2inmbH0vx8A8dX0ckZFYmmA6omZaeoJ3gLMHdlW/7iszNIbqN75hazE9QTvOmA+ubCQBj1bdJILspa/4vZCer0tzmz7MODwu5d+5QWicFCQQ3+Ilqw2h2CmD1MPcEbqzHpAlbH3ytRQhYv4LLvrmhNdtpJ2cPU6W/l0cqRAvC/ediD7bC/eYAlJu3BSXtkmmzcw7fSLulfEJlPFQ67Db4l20fVw/+WDEr2Yh/LsM/f8E1gkoG3nf9DD/wBefubWwK08nAAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Then, slope of normal at (x0, y0)
=![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPUAAAAmCAMAAAArxS3WAAAAAXNSR0IArs4c6QAAAK5QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmZmOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmaQZma2ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kLbbkLb/kNv/tmYAtmY6tpA6tpBmttv/tv//25A625Bm27Zm27aQ29uQ29vb2////7Zm/7aQ/9uQ/9u2//+2///b0tPZewAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAD2ElEQVRoQ+1ZDXuTMBBOqKVlOqdrXXUfZToHbJRWB1Ly//+Yd0cCAWwJutLZNc/DswF3l7x3xyX3lrHjOHrgMDwgbjkfLg4DizmK7CLMZhNz+YORzL70HmsRjPfsvvQy7nsFyw+zPaNOruL0um/YLNoX6vUd5xOWTTn+YT63vITzeU/494U6m96wwC6Sez0F6MEuyxr5FwY5tmfUlbmTEjWL7Fh87a2s9Yy6zODVKeca6uyTF0Go1zNu7T7NhT/qvYQS8sTyWKSfT/wR7GDCnbDVYNchjzDTd/k1baxOmcuHj1wrpakD60hPPChwuw92T0WzfRo6rLwy1IFHBTyDUr77DG8PQU8SPj+nmdIZH3o9zXmc5jk94OcHAxhQsbuMQu//+mcTRDMUXRx0lH0ZHvjrDH8Zyz+u4vV54Mdl2AJ6eWV8rt/GFK6BRbTmkX5k3TLx0lGdualGl9BhV9Q20rM22GplW5hC4Z7Hwp8z34jzSIaLFeyM1D42Ncy7ypqkvG0HhC5pd025sk1MoepAzFYMvTlOTGabGmaeKwyUUZWK7XjoUPm2rVMs1rGRKRSu9Z08CECg5eYfY5bw0QO3gGLLIPnz42vxhhiSBzgL2HE0Crh1r0nlj3O+KuH2o4P9+8qBp4NFbko9zSXRLgnLW+GSvOWljkYHNCgvcdt2DvNxZWBLMoVNk4yJOzqPA2rhjuP1FNBHA4+tLA9b8DU1o+UbPdaDUHxDLSVFKaD4qgic6dtxNr3GBl8Jyafq61DCeXAINQPaJzrXP9065ZWLbRv+IAQMC8UUNk1SIO8ggIAam0+URo4Je+8Uw0ScQOWNnuGoVUgVXzryVWgDLkJ9AtHLTcmnek1AYR01i97UmNwa5WWAGpxIS1ajYTJ3MqxHoZYrRtSFYo5avqmjLs3T4iVfpaSTU/xYlKk6ailcQb18hwmlEVw1yssgw+uoGybT9yGEa86CEcQFMtylDIcMsTy8F4/osvKNzHA7fmL4qwZcpRRm9JPiq1SsL6j0KKECNRooyS1UxNqCefXrXrjDUBFcGepXKS+DagZ86IPGjlVNUnp/hu0ashDzD1pujp8Usl1YDeDeuqEskW9Air7A1LGuUUNpKSl4LPkqSOkxXBPhoi0INplST8kAOkOSW/IW8SydsfC5/VMSXASxSnm1V3of6FGNO6iZ3FQR5P7UUjOMXtMWvNRL8ja1KN8xZCXBTTVPNZ3yMtvU/zBLG2em/WZgBG2LUDQMWYdf+IoTqUZwVSmvDifS2rJaODOVx/+KGPVFABkOZ4DOoyS4no3yOnJmnaNwAAq/AYzehXQ4Tb/hAAAAAElFTkSuQmCC)
Therefore, the equation of the normal at (x0, y0) is
y - y0 =
(x - x0)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the equation of the normal at (x0, y0) is
.
Question 29.Find the equation of the tangent to the curve
which is parallel to the line 4x − 2y + 5 = 0.
Answer:It is given that ![](data:image/png;base64,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)
Then, the equation of the tangent at any given point (x, y) is given by,
![](data:image/png;base64,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)
The equation of the given line is 4x − 2y + 5 = 0
⇒ y = 2x +![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAEhQTFRFAAAA///b//+2/9uQ29u2kNv//7Zm27ZmkLbbZrb/Zrbb25A6OpDbtmY6Oma2kDo6kDoAZjoAZgA6ZgAAOgA6OgAAAAA6AAAA2gGq0AAAAAF0Uk5TAEDm2GYAAABZSURBVHjalY45DoAwEAMnQIAlEO74/z+lWCQkKpjGR2EZ3iRJJUKyO9vTZ7f16V1YRsIcqU8jTFIZ+IwcftOsUoZq6+lK9D+HaxoB6PxmcmkztEa1S5K91y4HbAPwz6eU+AAAAABJRU5ErkJggg==)
⇒ slope of the line = 2
Now, the tangent to the given curve is parallel to the line 4x − 2y + 5 = 0
if the slope of the tangent = the slope of the line
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
When x =
,
y = ![](data:image/png;base64,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)
Then, Equation of the tangent passing through the point
is given by:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ 24y – 18 = 48x – 41
⇒ 48x -24y =23
Therefore, the equation of the required tangent is 48x -24y =23
Question 30.The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is
A. 3
B. ![](data:image/png;base64,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)
C. –3
D. ![](data:image/png;base64,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)
Answer:It I given that the slope of the normal to the curve y = 2x2 + 3 sin x,
Then, the slope of the tangent at x = 0 is
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUIAAAAuCAMAAABqMuA7AAAAAXNSR0IArs4c6QAAAKJQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmY6OmaQOma2OpCQOpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmZmZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkLb/kNv/tmYAtmY6tmZmttvbttv/tv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/7aQ/9uQ/9u2//+2///bj1JmeAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAET0lEQVRoQ+2afVubMBDACVstc2pn59ymVF1xL6hbGZDv/9V2dwkQIBSSTp/UJn/QPhSOux/3kksTBH54Am4QyBi7dkOT/dUi8wh3fXl55L1wR4YthAlj7Gizo8SDuj2/YOw0uuYxgssjFq6DcukRGvhAHl0F/AZyIY+RWwYEPUIDfnBpQuCwnKRvfsFhDue8F5owLJfncDnlQjzwFTihR2hCEGjVCHk8DzJKgt4LTRgqCDGSE5rceIQmCNH1ZC4Ech8uIR16hCYA4do0vAuKT6I7SRhGtUdoiJDfs3DxOyKGGdZkj9CQYOvyDIPaKYQpdEp703vCdEbMaJxCKOdbu/jFy93L48WNdMIdKnKKzc3I4I/HjM3GrxNirFZA+C1jJz+H9JikwHYRQaARAj3yWbW2YD2pySMtQh6rwZhgAYsnsLZGiL1qEcvEjlLMFeiJ6L6PEStsEZbLL1MQorOnU1OcjRdSo6+ugXYQTlCgJ6KHcLsQW4TJOT0ZVsuONlAIatdrW0DKJOQjuEoUYm8JnxR4xQpOLDaKBBuEJLxcVnmp64UTFBD6qSK0WUFYoRuWCLOTDSEMEjiq2HoIi9szBIarRMXyWnxi4GH48BtoMhsJFgjFmpNqhLECfREaTtIKW4S4LitGHbrlx7VYKyMSqZil0+hYUC6ZyLu0SoSf6C/IitrN/GSjSBhF2NdE2C+Olgp0RGisBV2lFTWJ5otld8JjMB8QprToc7qQ2uueHgRPEfic6M/xcfhJx3v2/gedqyWMIuw7Qdt+KwX6b0Hna2SFflgFMvwNKAamwPbkRpMLM+gl2whFtw4TnhmpVUmwRjiSC7crUAVyk061pFCINUL9u5WBHJSXn9X3o0GYR12EFdCniFZ9KwlahDBpC78Ou7msBY155gq0RQxYC7GyA8IB+DIXrtatYtaygAKephzC76qshawwDVKGhEZJStAiTOeb/N3w5JxmTIOTmkkK9ER0DG6EWHvhAEIKP55gVlR8vIMQriko16cM/rf5DkUIrqUzVJov5qoERPgIM/Y8OvoDCZwyRQlVXNgwrL9aTdr1jOO0fkQBquctET2EtRD6pbubwSoXSkHhGvqcEHyINQzbcVTcRowtqFo+RGx2hxlQnilW8OUMdG8kkBfCYnoq7qCBtV+U8YFRwDyz7rXIz9X2aFwBuKUrovskRYhk2FoOsUY4bJT1LyKQ07dV6puE0Ppp9jd2Eo57CB+OIczRsycFsj0I+zudRSjXC/9+4/FMWXmBirOtnNiDsLtT7mZQO1uXvFCkzDlP1E0q5QULr+zMfYa76t0MSl/qGsJnMPt/imx2MzSdrUdoQlj9H7nuSz1CW4R1X6pHqHZVJo947deqXlj3pXqEI13Vayc1aJ+ym6HpSwmhaVd1sAib3QxKZyu80LSrOliG1W6Gq7ovrTcKu99VOf/W3O+qXEfoeFflOj73u6o9IOhV9AReisA/CAKkaNzq1ngAAAAASUVORK5CYII=)
Therefore, the slope of the normal to the curve at x = 0 is
![](data:image/png;base64,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)
Therefore, the slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is
.
Question 31.The line y = x + 1 is a tangent to the curve y2 = 4x at the point
A. (1, 2)
B. (2, 1)
C. (1, – 2)
D. (– 1, 2)
Answer:It is given that tangent to the curve y2 = 4x
Then differentiating with respect to x, we have,
![](data:image/png;base64,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)
Then, the equation of the tangent at any given point (x,y) is given by,
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The given line is y = x + 1
⇒ Slope of the line = 1
The line y = x + 1 is a tangent to the given curve if the slope of the line is equal to the slope of the tangent.
Also, the line must intersect the curve.
Then, we have,
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⇒ y =2
Now, y = x+1
⇒ x = y -1
⇒ x = 2-1 = 1
Therefore, the line y = x+1 is a tangent to the given curve at the point (1, 2).
Find the slope of the tangent to the curve y = 3x4 – 4x at x = 4.
Answer:
The given curve y = 3x4 – 4x
Then, the slope of the tangent to the given curve at x = 4 is given by,
= 12(64) – 4
= 764
Therefore, the slop of the tangent is 764.
Question 2.
Find the slope of the tangent to the curve at x = 10.
Answer:
The given curve is y =
Then, the slope of the tangent
Therefore, the slope of the tangent is .
Question 3.
Find the slope of the tangent to curve y = x3 – x + 1 at the point whose x-coordinate is 2.
Answer:
The given curve is y = x3 – x + 1
Then, the slope of the tangent
= 12 -1 = 11
Therefore, the slope of the tangent 11.
Question 4.
Find the slope of the tangent to the curve y = x3 –3x + 2 at the point whose x-coordinate is 3.
Answer:
The given curve is y = x3 – 3x + 2
Then, the slope of the tangent
= 27 -3 = 24
Therefore, the slope of the tangent 24.
Question 5.
Find the slope of the normal to the curve x = acos3 θ, y = asin3 θ at
Answer:
The given curve is y = acos3θ and y = asin3θ
Then, the slope of the tangent is given by:
Then, the slope of the tangent is given by:
Therefore, the slope of the tangent 1.
Question 6.
Find the slope of the normal to the curve x = 1− a sinθ, y = bcos2 θ at
Answer:
The given curve is x = 1− a sin θ and y = b cos2 θ
Then, the slope of the tangent is given by:
Then, the slope of the tangent is given by:
Therefore, the slope of the tangent.
Question 7.
Find points at which the tangent to the curve y = x3 – 3x2 – 9x + 7 is parallel to the x-axis.
Answer:
It is given that the curve y = x3 – 3x2 – 9x + 7
We know that the tangent is parallel to the x-axis if the slope of the tangent is zero.
Therefore, 3x2 -6x-9 = 0
⇒ x2 -2x-3 = 0
⇒ (x-3)(x+1) =0
⇒ x =3 and x = -1
When x = 3, y = (3)3-3(3)2-9(3) +7 = 27 – 27 -27 + 7 = -20
When x = -1, y = (-1)3-3(-1)2-9(-1) +7 = -1 – 3 + 9 + 7 = 12
Therefore, the points at which the tangent is parallel to the x- axis are (3, -20) and (-1, 12).
Question 8.
Find a point on the curve y = (x – 2)2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).
Answer:
We know that if a tangent is parallel to the chord joining the points (2,0) and (4,4), then
Slope of the tangent = slope of the curve………….(1)
And, the slope of the curve =
Now, slope of the tangent to the given curve at a point (x,y) is:
Now, from (1) we have,
2(x -2) = 2
⇒ x-2 = 1
⇒ x =3
So, when x = 3 then y = (3-2)2 = 1
Therefore, required points are (3,1).
Question 9.
Find the point on the curve y = x3 – 11x + 5 at which the tangent is y = x –11.
Answer:
It is given that equation of the curve y = x3 – 11x + 5
At which the tangent is y = x –11
⇒ Slope of the tangent = 1
Now, slope of the tangent to the given curve at a point (x,y) is:
⇒ 3x2 -11 = 1
⇒ 3x2 = 12
⇒ x2 = 4
⇒ x = 2
So, when x = 2 then y = (2)3 -11(2) + 5 = -9
And when x = -2 then y = (-2)3 -11(-2) + 5 = 19
Therefore, required points are (2, -9) and (-2, 19).
Question 10.
Find the equation of all lines having slope –1 that are tangents to the curve
Answer:
It is given that equation of the curve y =
Now, slope of the tangent to the given curve at a point (x, y) is:
Now, if the slope of the tangent is -1, then we get,
⇒ (x-1)2 = 1
⇒ (x-1) = 1
⇒ x = 2, 0
So, when x = 2 then y = 1
And when x = 0 then y = 1
Therefore, required points are (0, -1) and (2, 1).
Now, the equation of the tangent (0,1) is given by:
y – (-1) = -1(x-0)
⇒ y + 1 = -x
⇒ y + x+ 1 = 0
And the equation of the tangent (2,1) is given by:
y – 1 = -1(x-2)
⇒ y - 1 = -x +2
⇒ y + x - 3 = 0
Therefore, the equations of the required lines are y + x+ 1 = 0 and y + x - 3 = 0.
Question 11.
Find the equation of all lines having slope 2 which are tangents to the curve
Answer:
It is given that equation of the curve y =
Now, slope of the tangent to the given curve at a point (x,y) is:
Now, if the slope of the tangent is 2, then we get,
⇒ 2(x-3)2 = -1
⇒ (x-3)2 =
This is not possible since the L.H.S. is positive while the R.H.S. is negative.
Therefore, there is no tangent to the given curve having a slope 2.
Question 12.
Find the equations of all lines having slope 0 which are tangent to the curve
Answer:
It is given that equation of the curve y = ,
Now, slope of the tangent to the given curve at a point (x,y) is:
Now, if the slope of the tangent is 0, then we get,
⇒ -2(x-1)=0
⇒ x =1
So, when x = 1 then y =
Now, the equation of the tangent (0,) is given by:
y – = 0(x-1)
⇒ y - = 0
⇒ y =
Therefore, the equations of the required line is y = .
Question 13.
Find points on the curve at which the tangents are
(i) parallel to x-axis (ii) parallel to y-axis.
Answer:
(i) It is given that
Now, differentiating both sides with respect to x, we get
We know that the tangent is parallel to the x –axis if the slope is 0 ie,
, which is possible if x =0
Then, for x =0
⇒ y2 = 16
⇒
Therefore, the points at which the tangents are parallel to the x-axis are (0,4) and (0, -4).
(ii) It is given that
Now, differentiating both sides with respect to x, we get
We know that the tangent is parallel to the y–axis if the slope of the normal is 0 ie,
,
⇒ y = 0
Then, for y =0
⇒
Therefore, the points at which the tangents are parallel to the y-axis are (3,0) and (-3,0).
Question 14.
Find the equations of the tangent and normal to the given curves at the indicated points:
x = cos t, y = sin t at t=π/4
Answer:
It is given that equation of curve is x = cost, y = sint
On differentiating with respect to x, we get
Therefore, the slope of the tangent at ) is -1.
When
Then, the equation of the tangent is is
y – = -1(x –
)
⇒ x + y --
= 0
⇒ x + y - = 0
Then, slope of normal at
=
Now, equation of the normal at
y – = 1(x –
)
⇒ x = y
Question 15.
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x4 – 6x3 + 13x2 – 10x + 5 at (0, 5)
Answer:
It is given that equation of curve is y = x4 – 6x3 + 13x2 – 10x + 5
On differentiating with respect to x, we get
= 4x3 - 18x2 +26x -10
Now, Slope of tangent will be the value of
m1 = 4(0)3 - 18(0)2 + 26(0) - 10 = -10
Therefore, the slope of the tangent at (0, 5) is -10.
Then, the equation of the tangent is
y – 5 = -10(x – 0)
⇒ y – 5 = 10x
⇒ 10x +y =5
Also, slope of normal at (0,5)
Now, equation of the normal at (0,5)
⇒ 10y - 50 = x
⇒ x -10y +50 = 0
Question 16.
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x4 – 6x3 + 13x2– 10x + 5 at (1, 3)
Answer:
It is given that equation of curve is y = y = x4 – 6x3 + 13x2– 10x + 5
On differentiating with respect to x, we get
= 4x3 - 18x2 +26x - 10
Therefore, the slope of the tangent at (1, 3) is 2.
Then, the equation of the tangent is
y – 3 = 2(x – 1)
⇒ y – 3 = 2x - 2
⇒ y = 2x +1
Then, slope of normal at (1,3)
=
Now, equation of the normal at (1,3)
y – 3 = (x – 1)
⇒ 2y -6 =- x + 1
⇒ x + 2y - 7 = 0
Question 17.
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x3 at (1, 1)
Answer:
It is given that equation of curve is y = x3
On differentiating with respect to x, we get
= 3x2
Therefore, the slope of the tangent at (1, 1) is 3.
Then, the equation of the tangent is
y – 1 = 3(x – 1)
⇒ y = 3x - 2
Then, slope of normal at (1,1)
=
Now, equation of the normal at (1,1)
y – 1 = (x – 1)
⇒ 3y -3 =- x + 1
⇒ x + 3y - 4 = 0
Question 18.
Find the equations of the tangent and normal to the given curves at the indicated points:
y = x2 at (0, 0)
Answer:
It is given that equation of curve is y = x2
On differentiating with respect to x, we get
= 2x
Therefore, the slope of the tangent at (0, 0) is 0.
Then, the equation of the tangent is
y – 0 = 0(x – 0)
⇒ y = 0
Then, slope of normal at (0,0)
=
Now, equation of the normal at (0,0)
x = 0
Question 19.
Find the equation of the tangent line to the curve y = x2 – 2x +7 which is
(a) parallel to the line 2x – y + 9 = 0
(b) perpendicular to the line 5y – 15x = 13.
Answer:
(a) It is given that equation of the curve is y = x2 – 2x +7
On differentiating with respect to x, we get
The equation of the line is 2x – y + 9 = 0
⇒ y = 2x + 9
⇒ Slope of the line = 2
Now we know that if a tangent is parallel to the line 2x – y + 9 = 0, then
Slope of the tangent = Slope of the line
⇒ 2 = 2x – 2
⇒ 2x = 4
⇒ x = 2
Now, putting x = 2, we get
y =4 -4 + 7 = 7
Then, the equation of the tangent passing through (2,7)
⇒ y – 7 = 2(x – 2)
⇒ y – 2x – 3 = 0
Therefore, the equation of the tangent line to the given curve which is parallel to line 2x – y + 9 = 0 is y – 2x – 3 = 0.
(b) It is given that equation of the curve is y = x2 – 2x +7
On differentiating with respect to x, we get
The equation of the line is 5y – 15x = 13
⇒ y =
⇒ Slope of the line = 3
Now we know that if a tangent is perpendicular to the line 5y – 15x = 13, then
⇒ 2x – 2=
⇒ 2x =
⇒ x =
Now, putting x = , we get
y =
Then, the equation of the tangent passing through
⇒ y – =
(x –
)
⇒
⇒ 36y – 217 = -2(6x -5)
⇒ 36y+12x – 227 = 0
Therefore, the equation of the tangent line to the given curve which is perpendicular to line 5y – 15x = 13 is 36y+12x – 227 = 0.
Question 20.
Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and
x = – 2 are parallel.
Answer:
The given curve y = 7x3 + 11
Then, the slope of the tangent to the given curve at x = 4 is given by,
It is cleared that the slopes of the tangents at the points where x = 2 and x = -2 are equal.
Therefore, the two tangents are parallel.
Question 21.
Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
Answer:
The given curve y = x3
Then, the slope of the tangent at the point (x, y) is given by:
We know that, when the slope of the tangent is equal to the y- coordinate of the point,
Then y = 3x2
Also, we have y = x3
⇒ 3x2 = x3
⇒ x2(x-3) = 0
⇒ x = 0, x = 3
When x = 0 then y = 0
and when x = 3 then y = 27
Therefore, the required points are (0, 0) and (3, 27).
Question 22.
For the curve y = 4x3 – 2x5, find all the points at which the tangent passes through the origin.
Answer:
The given curve y = 4x3 – 2x5
Then, the slope of the tangent at the point (x, y) is 12x2 – 10x4
The equation of the tangent at (x,y) is given by,
Y – y = (12x2 – 10x4)(X – x) ………….(1)
When the tangent passes through the origin (0,0), then X =Y=0
Then equation (1) becomes,
-y = (12x2 – 10x4)(– x)
⇒ y = (12x3 – 10x5)
Also, we have y = 4x3 – 2x5
⇒ (12x3 – 10x5) = 4x3 – 2x5
⇒ 8x5 – 8x3 = 0
⇒ x5 – 2x3 = 0
⇒ x3(x2 – 1) = 0
⇒ x = 0 , 1
When x = 0 then y = 0
When x = 1 then y = 2
And when x = -1 then y = -2
Therefore, the required points are (0, 0), (1,2) and (-1, -2).
Question 23.
Find the points on the curve x2 + y2 – 2x – 3 = 0 at which the tangents are parallel to the x-axis.
Answer:
It is given that x2 + y2 – 2x – 3 = 0
Now, differentiating both sides with respect to x, we get
We know that the tangents are parallel to the x –axis if the slope of the tangent is 0 ie,
⇒ 1-x = 0
⇒ x = 1
But, x2 + y2 – 2x – 3 = 0 for x = 1
⇒ y2 = 4
⇒ y = 2
Therefore, the points at which the tangents are parallel to the x-axis are (1,2) and (1, -2).
Question 24.
Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.
Answer:
It is given that ay2 = x3
Now, differentiating both sides with respect to x, we get
Then, the slope of the tangent to the given curve at (am2, am3) is
Then, slope of normal at (am2, am3)
=
Therefore, equation of the normal at (am2, am3) is given by:
y - am3 =
⇒ 3my – 3am4 = -2x + 2am2
⇒ 2x + 3my – am2(2 + 3m2) = 0
Therefore, equation of the normal at (am2, am3) is 2x + 3my – am2(2 + 3m2) = 0
Question 25.
Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
Answer:
It is given that the equation of the normal to the curve y = x3 + 2x + 6
Then, the slope of the tangent to the given curve at any point (x, y) is given by:
Then, slope of normal to the given curve at any point (x,y)
=Mediumy =
⇒ Slope of the given line =
We know that if the normal is parallel to the line, then we must have the slope of the normal being equal to the slope of the line.
⇒
⇒ 3x2 + 2 = 14
⇒ 3x2 = 12
⇒ x2 = 4
⇒
So, when x = 2 then y = 18
and x -2 then y = -6
Hence, there are two normals to the given curve with the slope and passing through the points (2, 18) and (-2,-6).
Then, the equation of the normal through (2, 18) is:
y -18 =
⇒ 14y -252 = -x +2
⇒ x +14y-254 = 0
And, the equation of the normal through (-2, -6) is:
y –(-6) =
⇒ y + 6 =
⇒ 14y +84 = -x -2
⇒ x +14y+86 = 0
Therefore, the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0 are x +14y-254 = 0 and x +14y+86 = 0.
Question 26.
Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).
Answer:
The equation of a parabola is y2 = 4ax, then,
On differentiating it with respect to x, we get
2y
Then, the slope of the tangent at (at2, 2at) is
Then, the equation of the tangent at (at2, 2at) is given by,
y – 2at =
⇒ ty -2at2 = x –at2
⇒ ty = x + at2
Now, Then, slope of normal at (at2, 2at)
=
Then, the equation of the normal at (at2, 2at) is given by:
y – 2at = -t(x-at2)
⇒ y – 2at = -tx + at3
⇒ y = -tx + 2at + at3
Therefore, the equation of the normal at (at2, 2at) is y = -tx + 2at + at3.
Question 27.
Prove that the curves x = y2 and xy = k cut at right angles* if 8k2 = 1.
Answer:
It is given that the curves x = y2 and xy = k
Now, putting the value of x in y = k, we get
y3 = k
Then, the point of intersection of the given curves is
On differentiating x = y2 with respect to x, we get
Then, the slope of the tangent at xy = k at
is
As we know that two curves intersect at right angles if the tangents to the curve at the point of intersection are perpendicular to each other.
So, we should have the product of the tangent as -1.
Then, the given two curves cut at right angles if the product of the slopes of their respective tangent at is -1.
Cubing both side, we get
⇒ 8k2 = 1
Therefore, the given two curves cut at right angles if 8k2 = 1.
Question 28.
Find the equations of the tangent and normal to the hyperbola at the point (x0, y0).
Answer:
It is given that the equations of the tangent and normal to the hyperbola , then,
On differentiating it with respect to x, we get,
Therefore, the slope of the tangent at (x0, y0) is
Then, the equation of the tangent at (x0, y0) is given by:
y - y0 =
Then, slope of normal at (x0, y0)
=
Therefore, the equation of the normal at (x0, y0) is
y - y0 = (x - x0)
Therefore, the equation of the normal at (x0, y0) is.
Question 29.
Find the equation of the tangent to the curve which is parallel to the line 4x − 2y + 5 = 0.
Answer:
It is given that
Then, the equation of the tangent at any given point (x, y) is given by,
The equation of the given line is 4x − 2y + 5 = 0
⇒ y = 2x +
⇒ slope of the line = 2
Now, the tangent to the given curve is parallel to the line 4x − 2y + 5 = 0
if the slope of the tangent = the slope of the line
When x =,
y =
Then, Equation of the tangent passing through the point is given by:
⇒ 24y – 18 = 48x – 41
⇒ 48x -24y =23
Therefore, the equation of the required tangent is 48x -24y =23
Question 30.
The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is
A. 3
B.
C. –3
D.
Answer:
It I given that the slope of the normal to the curve y = 2x2 + 3 sin x,
Then, the slope of the tangent at x = 0 is
Therefore, the slope of the normal to the curve at x = 0 is
Therefore, the slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is.
Question 31.
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
A. (1, 2)
B. (2, 1)
C. (1, – 2)
D. (– 1, 2)
Answer:
It is given that tangent to the curve y2 = 4x
Then differentiating with respect to x, we have,
Then, the equation of the tangent at any given point (x,y) is given by,
The given line is y = x + 1
⇒ Slope of the line = 1
The line y = x + 1 is a tangent to the given curve if the slope of the line is equal to the slope of the tangent.
Also, the line must intersect the curve.
Then, we have,
⇒ y =2
Now, y = x+1
⇒ x = y -1
⇒ x = 2-1 = 1
Therefore, the line y = x+1 is a tangent to the given curve at the point (1, 2).
Exercise 6.4
Question 1.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,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)
Answer:Consider y = √x.
Let x = 25 and Δx = 0.3. Then, we get
![](data:image/png;base64,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)
Δy = √25.3 - √25
Δy = √25.3 – 5
Δy = √25.3 = Δy + 5
Now, dy is approximately equal to Δy and is given by:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 0.03
Therefore, the approximate value of
is 5.03.
Question 2.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,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)
Answer:Consider y = √x.
Let x = 49 and Δx = 0.5. Then, we get
Δy = ![](data:image/png;base64,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)
= √49.5-√49
= √49.5 – 7
= √49.5 = Δy + 7
Now, dy is approximately equal to Δy and is given by:
dy = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 0.035
Therefore, the approximate value of
is 7.035.
Question 3.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,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)
Answer:Consider y =
.
Let x = 1 and Δx = -0.4. Then, we get
![](data:image/png;base64,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)
= √0.6-1
= √0.6 = Δy + 1
Now, dy is approximately equal to Δy and is given by:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMEAAAAvCAMAAABt0l5cAAAAAXNSR0IArs4c6QAAALFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZjqQZmYAZmY6ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kGZmkLbbkNv/tmYAtmY6tpBmttv/tv/btv//25A627Zm27aQ29uQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///b22S4lwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAD50lEQVRoQ+2Z+1fTMBTHm6GjPoCpU8AHVEEpIhWEdjb//x9m7s17vV2zpNTDOeQHztjam3xzX5+mWfY0Hu8O1IydPt7V48rrsRXcHF1MuyVNPqoPVods53oSBfz7lZzHU9AsEidvDq6qiRRcvoPVHzK2n5/ygrH5fZOz2QV+nTYmUlDt4vafZPxM5AEv5vciI2YX4lNyTE2joHkB2VbiuiGTcVZUVSfHwDQKSlhsu8RIgkyGP/wrqOJFahxNokC6wFHAi92sBo+M4IRJFFS4WEcBhFEpM0B+mzCmUKAWCRuvO1q7fHOsyjhmR8KYQgEUHRjV7Ee2OpI9uWR66/WvsSIqZT32/pD79C7xSzZb3OYowdagJk8II36eM8aefQlZRvw1svqvjRoiCocMrvEH/7lH4kZ1oOgAQzqI0trl2hJFHZWlVI7URKDF8+JtT341r20XDePMTpjwYnHmiCpNU0sKKF+I7TNdo7K44wjjzI5OwUUHzv7YVKwELo0UTjbPCKOyuicoWFukUdC+/zxWXbHJRRmVTuhyZt/uDZU746N6fot1qQR0rcKSrGdSGx2UUQwxgjNTFQhOVWWrTKZW02Roo8BpFGdiYRHlWQ4TEKE+aF5ei6YncU/4wHQJwuRgshgFtNFyTnNmj12twGizH/AOHUVA27Vs1U2+v+hfJWHI+QrvM6hBG/UVOJxJTxrog3YpyrTuHUP3DDlB+6DHqK/A4cykKJK0LZtDe/zJPvmkRFGPUZEHPZxJbs1Q31P7XWL84MWiY3ca+dCu+7/rWkQbRVimOTNBgWqUsHIO86qMCF24KL56wC2qH/QYxa9pziTn89o6wVIycMwSdn4Voo61S2b4O0CFy1myPGwyanuyMm05k5zLD4huTI1Bdu3HNRzZ9PztcBG4y+dMSoJP1x2WiqBrfiMOnp67h43dTQxjUww4jzNJJ3jPgR0FEUBaspNstXTPadRDtzt9yPOBTBmPM+lU1u1ZsZQPPhFPmXh4UzkPJ/zbw56d6uMIw1KZCz6xaeAmVLN3PwYP9pcMlfuWpRzwiT5tsQ9Gwh+ItMk82K+gc+LlgE/ssaPKp793MO1llwcDSvA2l6yfOkJHlKUk9tRR3rc6Z3ie+QHTYDMPbrPe7rXoBPfMzoBPpAtUCf5zh7Gka2kqD24SCZDrsJQBn9jT93JXdTCsxZU6wHR5MG3LibvhVYdhKQs+kW9AsDTUEEe82LlWSJHOgxtVw1sozVInBnwi30JJMjPcCbVU/b8lD47uqFCD6okBcVwU5t/SGdvzYOh8D3xdxV4lv8l64CUOmG89QPq/a4mcHd/HPY3HtwP/ALaejLFDzKfKAAAAAElFTkSuQmCC)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFMAAAAqCAMAAAD4QZc7AAAAAXNSR0IArs4c6QAAAHJQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6Ojo6OjqQOmaQOpC2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkGY6kNv/tmYAtpBmttv/tv//25A627aQ2////7Zm/9uQ/9u2//+2///brZw8VgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABdElEQVRIS+1W2VLDMAy0OdpAQsOd1hQ3TuL//0VsOXadVqECNMwwVI+NvDp2V1Mh/nTsHjfM/fcP8vKdF7Ortpob03V4xuSl6V/v84LZR3ZdSCmvnnkpmqLZt5LmVF1tiX3Y5q4lpna3r6RM26yO8+xayjJvSgcCuhsSDwY5J7ZZtH2TfeiKkVS9IMxkm+Vxm8YjGJkGHeqnEZPUaFcgK1K+xaFO1dQKqrhAN3XYVEzOf/eje8w4pynblKZ8IeW0GALV+TxmQHYx3G/EHhNdaKoh/YN4TfLSAS1iwrgOU8OSFIGkz2YP+zSxiT3mN2YXI0eZcKf7PCF8lHftZZRpCWYPu60RhxyWQPXpKYd1Jj2NPiIaCfORcP8AZAV6SjsNjc75yO7ci+toXJKK43SzNlLyRfR1cjPLXQIrAA0h2O7nhNUT2qB+BgnyBirLn5X4Etm0UqjQaU9ns9TS6Zk3wAyGYFx6WTCDVayY4/VjxaRP9HuZHw3BHPXSSBjGAAAAAElFTkSuQmCC)
= -0.2
Therefore, the approximate value of
is 0.8.
Question 4.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAE0AAAAtCAYAAADr/ebqAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAALHSURBVGje7ZrBTUMxDEA9AAOAhI8dgEZdoYoQA0AjbogTioQYIO2tF4bgyA5M0A04sEF3gOZ/5f+2xInzfxH040MkBHbivNiJ7Q8sFguQUTYEgkATaALtmIaz+gIB37V1FwKNMeZGXQHAJwCuBVqJp7nbszHgSqAJNIEm0ASaQBNoAu1HE1tY17kafCozvxJoUkYJtGFA8/Gtjb0cymat0ZfU5X4QaL5wVVN7NzQvmSl8jl3uvaFVLwjq1/vl8mRo0JbL+xON+BrzuPBixkYSWjXpObylTsO7+RhhVU2I41Wpy5foc2S9zd6D2lQB1xNjr1NRBOf6ratTFE9Y95twHe46vylvLNflS/Q5suGQt222ZnLt4eH04YlKXhXAR9cwjXoZqNlLarF9Yx6m+ASgPm6dO8tl2lx9riy1dv17upE4U/DS1dui2TB1ApQhQY862S76HNkANnbITWZPOEDjxXvzt53bzSB0gXs6sTDY9wpqkVJ9rmwKTGoO6qCquZVxuRD+7rJEmDXGRozIGViqz5W17maUlaP2EwE+t/Z0ex7PIwmtyyJc3VJ9ruyNs6P650gYJ8Bz/l5FXS48s4v8QWhetr2DNhCtGwUZo9Gl7qVmv3s2t/Mz7rTG0NzJxC7dDPBS/dK1dnK5jdehmj0bPXmkWju7cOJ3eOpxY0P7iw9Bdq2E5+egpUKUDS2VE+VSlS76fdaqdWkv40KrQj8JjYhxjsuyk9sC/a5rBWCpMorjjckatcSlY7mcL1liJx+Mj286r18i621v7zXac3J3Y7C56N8S6jwtv2gwPhTR4dXiQOPqc2R3XjrE94k2j9yyiHM3ssqoXL02pMEt/XrXnkMaVO158C7HsDq4B+pytCeQfwmPvnubabQWQes74dGE5iE7t4t//I2gF7QALvSWhjTsVN39yNeobXDy3bMQmgyBJtB+c3wBBFrgyryxUtYAAAAASUVORK5CYII=)
Answer:Consider y = ![](data:image/png;base64,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)
Let x = 0.008 and Δx = 0.001. Then, we get
Δy = (x+∆x)1/3 - (x)1/3
= (0.009)1/3 - (0.008)1/3 = (0.009)1/3 - 0.2
= (0.009)1/3 = Δy + 0.2
Now, dy is approximately equal to Δy and is given by:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANAAAAA3CAMAAABKOzR8AAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmY6ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kGZmkLbbkNv/tmYAtmY6tpBmttv/tv/btv//25A625Bm27Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///bf6QvjQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEZUlEQVRoQ+1a6VLbMBC2Ag1uaSFtw9EWDJSYUsxR7Fjv/2jVrk5bEpFt2UwY9IPJBHu1n/b6dpUkeV9v+ARKQs7fFrxyJECPJ6vXOagqHcNC62OyczcpIPr7lu/XAFQt4ihRHd4WEwO6+Q5gjgk5SM9pRsj8uUrJbIVfR1kTAyr20DhnCb1kMUSz+XOSlLMV+xTLAacFVH2EiM0RBiQF3B1BlrE8ZVpAOeheL9HtICnAH3oBIGkWyekmBcQNZACi2V5Sgr3imWhSQAXqbgACn8t59PBvh68pAQmdwSwihhiMr6eibmBkDV9TAoJ0BquYXSfrE84UciINI/87EFMhNhkoJuh1eXj0hswWTyki0tmtSof7HL1KCSG7v4LUGfwQrzqtVYL74eKeOO6if744iVFxKPgLxkIoy6yXLY1ZwuY5m69IQfTCkdDsmydOq8+6sAfTZsunaLa4NDDmurbGcD8bmK51tnxeUXAF02YLOeNyh8aJGeFcMI4X3/l0vDrk85IyDFBLZQ2oPvo5Qq7SQeqSz01k0+YXznVTQtUWLOdPmPJyYONFcJBusKl2JZd89EcHbY4BiFFvkRHzmERcFTq3fOCZLtqschYrCHxJ7wm2UPXpjhVfTlmZhVR5YgZriewUZwqQW34+d9Nm/x4SkFJLf8CXlMtBP1FyBlGlBwu/RIckz1cgQ5Eit/wmIIM2e/cPtVC9ZDVBFq1NL3WwkbSQR34TkEGbxRa2f2zSTVqI9xO8KtWnP3R1iuRyHvkshjy02XdomyqwBJyjs+HTjEhY/KKDTeDRf2x+wb1WZjm3fGwF3LR5ICBRswEIhb1FNHXEIR+vj66TB56WRB3yyMev3bTZt3WDbzgooKA+rO7wtfM3YxmyXhLVYfQEJWkNZwoe+ZopiF00bfZt23Qe2wFHI6cy8b80tzC4HBizSZs9iJrtg0UBR2sfHlTiD2Pb6J0N2uwzUaM7tgCNw7CZiyk8TNGAfoiHW4M2e7OC5AyCAja5WqQWvL15yfCUsaaYLeFysKMoYGJytXFCCHJKLHZrG0pkEk0BDa4Wa4zVMxP2e80aNBpcLdosuJ9qPd9qj4KhNvPWN8IouBHx5tijp7Ihr6GJzMmp4mrDDdTKyebYI0S1ns8AczcooOJqPa9T6MM+IR8EsWmN9bh/C9bRU92A1+BmS1FAzdV6XnjlMIPN0GttE1tkJkC97o/AlaSkgGeKq/W9ksSYLJCX22PKJpvpruqrvYGkVhAPc6zC88z9PplNNBqOcwLrKxzjSp7RKNUQsBeqfYiz38hSgAjgsFICMscqkntsme89pnhRKzt1Y6wiAK2vzkY+1sjisaXV1FaPLjignOxuGSDsOxQgY6yCOZCtx1F+tBLZLAZfwt5XtlfGWAUZSX20Cr9OGEPHTjIp1NQ13qOJOmSOVXguuGc/VNmetL2GK8iF/lkA6zT1WGUaptDJAl0etuj6liVrG+vrsO0uZ9712dfoh7rq+P78oBP4Dw7PkBtz9qY9AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
= 0.208
Therefore, the approximate value of
is 0.208.
Question 5.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,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)
Answer:Consider y = ![](data:image/png;base64,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)
Let x = 1 and Δx = -0.001. Then, we get
Δy = (x+∆x)1/10 - (x)1/10
Δy = (0.999)1/10 - 1
(0.999)1/10 = Δy + 1
Now, dy is approximately equal to Δy and is given by:
dy = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
= -0.0001
Therefore, the approximate value of
is 0.9999.
Question 6.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,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)
Answer:Consider y = ![](data:image/png;base64,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)
Let x = 16 and Δx = -1. Then, we get
Δy = ![](data:image/png;base64,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)
=![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒
= Δy + 2
Now, dy is approximately equal to Δy and is given by:
dy = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= -0.03125
Therefore, the approximate value of
is -0.03125.
Question 7.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,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)
Answer:Consider y = ![](data:image/png;base64,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)
Let x = 27 and Δx = -1. Then, we get
Δy = (x+∆x)1/3 - (x)1/3
![](data:image/png;base64,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)
= (26)1/3 - 3
= (26)1/3 = Δy + 3
Now, dy is approximately equal to Δy and is given by:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Therefore, the approximate value of
is 2.9629.
Question 8.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,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)
Answer:Consider y = ![](data:image/png;base64,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)
Let x = 256 and Δx = -1. Then, we get
Δy = (x+∆x)1/4 - (x)1/4
= (225)1/4 - (256)1/4
= (225)1/4 - 4
= (225)1/4 = Δy + 4
Now, dy is approximately equal to Δy and is given by:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= -0.0039
Therefore, the approximate value of (255)1/4 is 3.9961.
Question 9.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,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)
Answer:Consider y = ![](data:image/png;base64,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)
Let x = 81 and Δx = 1. Then, we get
Δy = ![](data:image/png;base64,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)
=![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒
= Δy + 3
Now, dy is approximately equal to Δy and is given by:
dy = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 0.009
Therefore, the approximate value of
is 3.009.
Question 10.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,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)
Answer:Consider y = ![](data:image/png;base64,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)
Let x = 400 and Δx = 1. Then, we get
Δy = ![](data:image/png;base64,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)
=![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒
= Δy + 20
Now, dy is approximately equal to Δy and is given by:
dy = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACYAAAAqCAMAAADoIdnnAAAAAXNSR0IArs4c6QAAAGBQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOjqQOmaQOpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrb/kDoAkGY6kNv/tmYAttv/tv//25A627aQ2////7Zm/9uQ/9u2//+2///b/BWqGwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAyklEQVQ4T+2U2RKCMAxFGzc2N5ZipUD//y9tUnAYIbajji+ah0wfDjfJTakQ34rrsfKX6g6wvnixNqlVAGZ1/tiDmWrl34IpIwDYnL17+FVAWntcBLg5Men+2eQQ7OGrRYMLfBwc7popAeKaVW8jMtEU26Yr2L+6z06EacwackZOpgQIiUJ9tlvGdNwQhjURwzyPfl+5cg5zeRamSKkrlT/F9LjyAWN7G2ccR7DqTLhJFXrBG2IBwnBIZgKSt/0RZx9gSBbteOd63ADFpQxSWJzknwAAAABJRU5ErkJggg==)
= 0.025
Therefore, the approximate value of
is 20.025.
Question 11.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,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)
Answer:Consider y = ![](data:image/png;base64,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)
Let x = 0.0036 and Δx = 0.0001. Then, we get
Δy = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHwAAAAfCAMAAAASoFl4AAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OgBmOjo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjpmZmYAZmY6ZmaQZma2ZpDbZrb/kDoAkDo6kNv/tmYAtpBmtpC2ttv/tv/btv//25A627Zm29uQ29u22////7Zm/9uQ//+2///bDD542gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB0ElEQVRYR+1V2VLDMAyMKaXcEEo5QwMFA/Xx/7+H7DhBVhTceIYpzNQvnSbaXa0kK0WxO/+0Avb96G1s6jkYTsO+fM7HiudgBvyZ0eJFkYNh5XOIcjCsuM5wnoPhxGshxMHIicvBjJT48+H2+WTsXQFP8my1ibOIvI+x1cV6Exoao48f0zBCTjG2ukxzQISe0Th9uITnMG5wBih65ASjJmzNbUV8STGlBZK9JzSHPnmEsRV/Tai4ubrdc0bxaWz8cBjyCKNn3iFcvOlaCtHZpeJq+uHrjgOTHWPII4wKfmr4xYJEHP7aylcZB9aJ5cKRY0z73lFLNFJEXMNnU/o8cWAN6dgnIfYHripHjjGynTc9Oz0P7XPLrzldmyV4VMInhwNB3MxX5rrJuodjyRGmTQ62RjRQsXNTwjCYsqnyd6BzAccsEs6HMJ24WdzgSxeLq6bdPgAFNv3Td0NLiiPHmDCQ0Lpl68zbi8VrX1flLgMKNKV7rO7X+iE0jPww5BEmXEXr6ENT++LhcrrscKB/bEqYDTSpWJ8hjzF+CdkKZsvRdCyRc1gAzZm84sANN1xEHmOSm4KvqOt1asE5U6QmFLPNrxqkt8Xv+WBZdy9+qwJfKiI5743EPTIAAAAASUVORK5CYII=)
=![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒
= Δy + 0.06
Now, dy is approximately equal to Δy and is given by:
dy = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 0.00083
Therefore, the approximate value of
is 0.06083.
Question 12.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,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)
Answer:Consider y = ![](data:image/png;base64,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)
Let x = 27 and Δx = -0.43. Then, we get
Δy = ![](data:image/png;base64,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)
=![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒
= Δy + 3
Now, dy is approximately equal to Δy and is given by:
dy = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= -0.015
Therefore, the approximate value of
is 2.984.
Question 13.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEMAAAAtCAYAAAD1NNZZAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAALlSURBVGje7ZkxbuMwEEV5gFwggOhqcwCH6yvIQpADxCbcGSmMgMAi9UJx5yYXcJcyJ9gm7Ta5gYvcIHfIZiiPJcsckpKYXXnNgoVtkeK8Gc78odlyuWRxFCNCiDAijAgDR66yIWd8k6l8eNIwHqS4Zox9MMbfTx6Gjox8dn7J+GuEEWFEGBFGhBFhRBiBBBd7L7QG+xDy4TrK8SjHI4zwMOCsZVJd9dkAJbMrW1IMAgOaG5Gq+TF4dCr4I5UUO8PQ2Zdnz7er1dkxwFitbs8yzp9NEYIVhBpWGHrhhL1QpOF38ES1VHExfTSBU2oyTFPxk3O24endfRttIBh72zfA3IbrNj3JXro4kDVZFEHB7xOVXxRRNLnA76pz9DqcbwCEBtYCxl3K7w+8KaZPNnBdjovZWOKFxSXJoWdQ1Jg2ghcrTWFo4wbjdRNPTwV76hIdzNeo0lMGGBavtIWhDYN5/Psv34pGOctXknsZeyBrPyPn4EgQHmkDoy6f9eCXr7M8P/eZR72rAEzbZ3hYvNleih6DzUHeAGPBe5hDQkUGjMVi8U1mox8lGPveqs4ySoXZbO4VGbt84Xgh1nVXJQkB43BvdALdqz61KNXfC5l7HxNqIcoDSZL8Ro/ZgISAsV9maWdRz0ghcvjsDwNDzAED8goKsl1ZBY8FzBn2Ukuf+TKCymf08dgm9qAw0LBq1ShE2DbzGwwOCaPYI51I6zD2jldtmCpfIxhUNrbpk5AwXCrTFBmtSqvPmURFWKf6t2BA4rZ1qK4i4A3Dp5qUwD7Jb4UQzJMZz6l5u7/2iCqAgKuwwOiBGK+xXGM/NJLqxivJEtHTSHS5RAn2ImMxWFfPX3XjVuFkiBATDCVHN9W5pvWtMCzl11uOu7J134dLgQbtTfo+XL1J0K61/zdeAbvWkq5bkvfytstyKdUKRohF/9kRCX3T9b/dgXaGgUCg0zsGGCoV8y+7Ha8Cif+bnOiIECIM8/gDiz1WzU+ELqgAAAAASUVORK5CYII=)
Answer:Consider y = ![](data:image/png;base64,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)
Let x = 81 and Δx = 0.5. Then, we get
Δy = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHwAAAAfCAMAAAASoFl4AAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6OgBmOjo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZgBmZjpmZmYAZmY6ZmaQZrb/kDoAkDo6kNv/tmYAtmZmtpBmtpC2ttv/tv/btv//25A625Bm27Zm29uQ29u22////7Zm/9uQ//+2///betOxAgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABzUlEQVRYR+1V2VLDMAyMoQTKGe7QcJqjsf3//4dsJ60kK6TxDFOYqV86TbS7Xl0pit35pxlwH0dvU6+eg5E03PPX1VTxHMyAPztZvChyMKJ8DlEORhQ3Gc5zMJJ4o5Q6mNhxOZiJEn8+3D0dT50V8KRPXzZxRshTjKvPl5vQ8Bgzvx+HMXKOcfXFOAdEmJLHmcMFPId2gzNAkZAzTLsv5tzVzJdWM54gnTzhd0jJCcbV8phwcXt5u+eN4hNt/HAEcoIxZXAIgzdbaqVWdrl4O/sMeceBoxUTyAmm7fw08IsFmTj8dXXIMg5sRpaLRI4x/XtPrVFLMXEDn00d7okDm1j0ZqhlJXKM0X2/mfLkrCufX37xrMqswWOrgggODOLv8048wYnkCNNfDrYGaSjq3FbQDLaKWV4HBhftBc4YaT+JHGNW7+31DR46Kt7GcocAFOjrZ++gXI/yipTI1xifxNjg7mHRO4v/yZzHqrZ+GFCgreAxjIhS8q6QyBHGi4RcOk/fFTUV74bT3w4Hdo9NuZ5QOvQCOcWEJeRq6C1bqdhSifPgLjh8xYEbbjhCTjGjm4KaQf9GF5w3xaaQY7b5VYPrbfF7PpjW3YvfysA3UtUz68WlkrwAAAAASUVORK5CYII=)
=![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒
= Δy + 3
Now, dy is approximately equal to Δy and is given by:
dy = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGkAAAAuCAMAAAAcAcfcAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6Ojo6OjpmOjqQOmaQOpC2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkDo6kGY6kNv/tmYAtpBmttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bfsm6swAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACGUlEQVRYR+1X21LCMBRMvEBVkCoqFaJWaNr8/xeanEsbsK2FZDrDSB4cH8jZ7Nk9m1SI/7u2y80o5Mtnef01BpKef+bjIFk2F6QQSS/dO5PuXY2SEWadSClvXkOaEmuv+Zh1plU+/4wFI4TJHnft1b5t0/X9KhaUyRZQyqylnDEB7TouQV59F0vkAuPXZJNdmXEU66Rhkk86KB/J1WRT2FE4AoUkAB8pFimuqRydKkVY4SNxe4+k8OvnwAWb55CoUz6SUA5eOeFgnTqce0iIB5weEnlLVlCtQtXIQ/5xNSnnEaNGMu9voszo9O1IRzfzoHukExFD/wNStO4JcgTWxsX+AJ3CF2ufO4PXLlcOUSeAW6U+/OmIPE/OdiATMFFWIp7jQfPUlZ1+blJGCPsslHNwukUql9ZRFK59GZGTaTA7zdbWAMfmaEh7cC83/xrMPko64WmEBitpHZu6CFAua1AYb393lsNk9WR5lb4gEvUFrAOCw6IB8npy8v2kFjgjrDXUr13FThqkc7/RitkOkfazi++DRsNQ71ZPG7wB6C8eq0ZtiIbOIzjJIuUrH6nxV91FzJiAVXCyrrwvDE+xpn4oEsp/0D015eNTwID5Azm1IYGhC5yt5pstVCccmT3vgZ8NZKYXllFy02rloFgdulkcEl1G4MZojytXt+/zNtbbCprYl51RKfW9gyO+gXEkh9xPAcN73lt/ACHjMJdKAsDDAAAAAElFTkSuQmCC)
= 0.0046
Therefore, the approximate value of
is 3.0046.
Question 14.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,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)
Answer:Consider y = ![](data:image/png;base64,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)
Let x = 4 and Δx = -0.032. Then, we get
Δy = ![](data:image/png;base64,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)
=![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒
= Δy + 8
Now, dy is approximately equal to Δy and is given by:
dy = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 3(-0.032)
= -0.096
Therefore, the approximate value of
is 7.904.
Question 15.Using differentials, find the approximate value of each of the following up to 3 places of decimal.
![](data:image/png;base64,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)
Answer:Consider y = ![](data:image/png;base64,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)
Let x = 32 and Δx = 0.15. Then, we get
Δy = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHwAAAAfCAMAAAASoFl4AAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OgBmOjo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZgBmZjpmZmYAZmY6Zrb/kDoAkDo6kLb/kNv/tmYAtmY6tpBmtpC2ttv/tv/btv//25A627Zm29uQ29u22////7Zm/9uQ//+2///bKIF1DgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABzUlEQVRYR+1Va1PDIBAM1hrfGusDbbRV1Ab4///Pg0A84DANM051pnzplNzu3h53UFX79U8roN9P3qamXoKhNPTqczFVvAST8acmi1dVCYaULyEqwZDissB5CYYSbxljRxM7rgQzUeLPh+uXs6mzAp7ExXobZwF5itH8arMNTRwjTx/GYRF5jNH8epwDImQdx8njJexDu8HKUCTkEaabkTXXPPIl2DwukEh24hxS8gCjOT0msbi6uT8wRvHqbfywCPIAI2vrEAZvvhGMDXZj8W7+YeuOA0dPjCAPMJ3z08IvFozE4a/mtso4sHVVk0+0fYocY/x3Qy1QS0XiEp5NYfPEga1Jx9xWmb6nyDFG+H6T9fmly9/Q9Ws4ZgEeO2aTw4FG/Hk4+ARHkiOMTw5ujaChQueqAW+q6av8HehdHGYajyLHmOG7ur3DQxeKd/1x2wAU6M5P+3NMJs37yWBcQ1b6cemdWYpQvLUF78zZokDVwLZarLPiBHmAcaOoDb071FTcDafJDgfabcUZy70NBHmIsZeQ5tBbqmF9SyXO4QLo1+wVB255wwXkIWb0pqBH2LT92AVnTKHptaMSYXb5qkF6O3zPs2Xdf/itCnwB1iU1auKXpPcAAAAASUVORK5CYII=)
=![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIEAAAAfCAMAAAD3El5RAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgBmZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkLb/kNv/tmYAtmY6tpBmtpC2ttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bID9EwQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACB0lEQVRYR+1WWVvCMBDM4kE9UTnUilIVK2iP/P9f526ybZMeaSE+qJ95CiEzO5k9QIj/9dsdkJvTN483eMIxsnz9mHko8ITrp+c+CrzhP0NB6ueBJxw9iABgvH8pesL3D/yPbHVAvlwOb+v3lYgn6yFOdtE24TK8/sTxsjkDOFrRoNnOecdxtgs6FmmAhQcj3Kfn93RAHwG6xHTT1uEyvCGSaPQospD4I7gT2fSgsCWbg96ngYqrtidKk2u5aGvwRPNH1FYxYJBqp4JN1nFDgYiP0TbnctJacBkaHR2VL09IC68WBb0muGktuOFttqxKrNKCzjQ9YI/Roqd2J9y0Fjyh1NPKpwCTwluDwFBwEehaLZJGQ62yypLSQ6syzfDyKh5tA85CqVGxsgfyAStU1SqhqRCey3okMr34+x5aE870+gEJqL6wklgqUDfSQN3QCjBu4Uk9Fz20JtwUW/BHY6vSTbZ8qgtXUaBWZWfL6qE14XxV+647QPVKot9KixVEdFJ6gJHz2bpPQRctCS/gXHOSEpyFFFu1isR4xXtjndtI3dCVkk9RTh4CdM1zB20dXuQ8W+LQvSJruapYgaRzOLwVIlvghvt18DxooSUnrZnKw6s9nV2ng2diO4ENtztvmJBeC6ifqkKqk9bh+kdsl8U/bm5IN20TvtP/A+qN7/5/sMvr/+LdLzf+QZHFHcu5AAAAAElFTkSuQmCC)
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⇒
= Δy + 2
Now, dy is approximately equal to Δy and is given by:
dy = ![](data:image/png;base64,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)
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![](data:image/png;base64,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)
= 0.00187
Therefore, the approximate value of
is 2.00187.
Question 16.Find the approximate value of f (2.01), where f (x) = 4x2 + 5x + 2.
Answer:Let x = 2 and Δx = 0.01. Then, we get,
f(2.01) = f(x + Δx) = 4(x +Δx)2 + 5 ( x + Δx) + 2
Now, Δy = f (x + Δx) – f(x)
⇒ f (x + Δx) = f(x) + Δy
≈ f(x) + f’(x).Δx (as dx = Δx)
⇒ f(2.01) ≈ (4x2 + 5x + 2) + (8x + 5) Δx
= [4(2)2 + 5(2) + 2] +[8(2) + 5] (0.01)
= (16 + 10 + 2) + (16 + 5)(0.01)
= 28 + 0.21
= 28.21
Therefore, the approximate value of f (2.01) is 28.21.
Question 17.Find the approximate value of f (5.001), where f (x) = x3 – 7x2 + 15.
Answer:Let x = 5 and Δx = 0.001. Then, we get,
f(5.001) = f(x + Δx) = (x +Δx)3 - 7 ( x + Δx)2 + 15
Now, Δy = f (x + Δx) – f(x)
⇒ f (x + Δx) = f(x) + Δy
≈ f(x) + f’(x).Δx (as dx = Δx)
⇒ f(5.001) ≈ (x3 – 7x2 + 15) + (3x2 – 14x) Δx
= [(5)2 – 7(5)2 + 15] +[3(5)2 -14(5)] (0.001)
= (125 – 175 + 15) + (75 - 70)(0.001)
= -35 + 0.005
= -34.995
Therefore, the approximate value of f (5.001) is -34.995.
Question 18.Find the approximate change in the volume V of a cube of side x metres caused by increasing the side by 1%.
Answer:It is given that the volume of a cube (V) of a side x
⇒ V = x3
![](data:image/png;base64,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)
= (3x2)Δx
= (3x2)(0.01x)
= 0.03x3
Therefore, the approximate change in the volume V of a cube of side x metres caused by increasing the side by 1% is 0.03x3.
Question 19.Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%.
Answer:It is given that the surface area of cube (S) of a side x
⇒ S = 6x2
![](data:image/png;base64,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)
= (12x)Δx
= (12x)(0.01x)
= 0.12x2
Therefore, the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1% is 0.12x2.
Question 20.If the radius of a sphere is measured as 7 m with an error of 0.02 m, then find the approximate error in calculating its volume.
Answer:Let r be the radius of the sphere and Δr be the error in measuring the radius.
Now, it is given that r = 7m and Δr = 0.02m
We know that volume of sphere (V) = ![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
= (4πr2)Δr
= 4π (72)(0.02)m3
= 3.92π m3
Therefore, the approximate error in calculating its volume is 3.92π m3.
Question 21.If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find the approximate error in calculating its surface area.
Answer:Let r be the radius of the sphere and Δr be the error in measuring the radius.
Now, it is given that r = 9m and Δr = 0.03m
We know that surface area of sphere (S) = ![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
= (8πr)Δr
= 8π (9)(0.03)m2
= 2.16π m3
Therefore, the approximate error in calculating its surface area is 2.16π m3.
Question 22.If f(x) = 3x2 + 15x + 5, then the approximate value of f (3.02) is
A. 47.66
B. 57.66
C. 67.66
D. 77.66
Answer:Let x = 3 and Δx = 0.02. Then, we get,
f(3.02) = f(x + Δx) = 3(x +Δx)2 + 15( x + Δx) + 5
Now, Δy = f (x + Δx) – f(x)
⇒ f (x + Δx) = f(x) + Δy
≈ f(x) + f’(x).Δx (as dx = Δx)
⇒ f(3.02) ≈ (3x2 + 15x + 5) + (6x + 15) Δx
= [3(3)2 + 15(3) + 5] +[6(3) + 15] (0.02)
= (27 + 45 + 5) + (18 + 15)(0.02)
= 77 + 0.66
= 77.66
Therefore, the approximate value of f (3.02) is 77.66.
Question 23.The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is
A. 0.06 x3 m3
B. 0.6 x3 m3
C. 0.09 x3 m3
D. 0.9 x3 m3
Answer:It is given that the volume of a cube (V) of a side x
⇒ V = x3
⇒ dV = ![](data:image/png;base64,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)
= (3x2)Δx
= (3x2)(0.03x)
= 0.09x3
Therefore, the approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is 0.09x3.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y = √x.
Let x = 25 and Δx = 0.3. Then, we get
Δy = √25.3 - √25
Δy = √25.3 – 5
Δy = √25.3 = Δy + 5
Now, dy is approximately equal to Δy and is given by:
= 0.03
Therefore, the approximate value of is 5.03.
Question 2.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y = √x.
Let x = 49 and Δx = 0.5. Then, we get
Δy =
= √49.5-√49
= √49.5 – 7
= √49.5 = Δy + 7
Now, dy is approximately equal to Δy and is given by:
dy =
= 0.035
Therefore, the approximate value of is 7.035.
Question 3.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y = .
Let x = 1 and Δx = -0.4. Then, we get
= √0.6-1
= √0.6 = Δy + 1
Now, dy is approximately equal to Δy and is given by:
= -0.2
Therefore, the approximate value of is 0.8.
Question 4.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y =
Let x = 0.008 and Δx = 0.001. Then, we get
Δy = (x+∆x)1/3 - (x)1/3
= (0.009)1/3 - (0.008)1/3 = (0.009)1/3 - 0.2
= (0.009)1/3 = Δy + 0.2
Now, dy is approximately equal to Δy and is given by:
= 0.208
Therefore, the approximate value of is 0.208.
Question 5.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y =
Let x = 1 and Δx = -0.001. Then, we get
Δy = (x+∆x)1/10 - (x)1/10
Δy = (0.999)1/10 - 1
(0.999)1/10 = Δy + 1
Now, dy is approximately equal to Δy and is given by:
dy =
= -0.0001
Therefore, the approximate value of is 0.9999.
Question 6.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y =
Let x = 16 and Δx = -1. Then, we get
Δy =
=
⇒ = Δy + 2
Now, dy is approximately equal to Δy and is given by:
dy =
= -0.03125
Therefore, the approximate value of is -0.03125.
Question 7.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y =
Let x = 27 and Δx = -1. Then, we get
Δy = (x+∆x)1/3 - (x)1/3
= (26)1/3 - 3
= (26)1/3 = Δy + 3
Now, dy is approximately equal to Δy and is given by:
=
Therefore, the approximate value of is 2.9629.
Question 8.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y =
Let x = 256 and Δx = -1. Then, we get
Δy = (x+∆x)1/4 - (x)1/4
= (225)1/4 - (256)1/4
= (225)1/4 - 4
= (225)1/4 = Δy + 4
Now, dy is approximately equal to Δy and is given by:
= -0.0039
Therefore, the approximate value of (255)1/4 is 3.9961.
Question 9.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y =
Let x = 81 and Δx = 1. Then, we get
Δy =
=
⇒ = Δy + 3
Now, dy is approximately equal to Δy and is given by:
dy =
= 0.009
Therefore, the approximate value of is 3.009.
Question 10.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y =
Let x = 400 and Δx = 1. Then, we get
Δy =
=
⇒ = Δy + 20
Now, dy is approximately equal to Δy and is given by:
dy =
= 0.025
Therefore, the approximate value of is 20.025.
Question 11.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y =
Let x = 0.0036 and Δx = 0.0001. Then, we get
Δy =
=
⇒ = Δy + 0.06
Now, dy is approximately equal to Δy and is given by:
dy =
= 0.00083
Therefore, the approximate value of is 0.06083.
Question 12.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y =
Let x = 27 and Δx = -0.43. Then, we get
Δy =
=
⇒ = Δy + 3
Now, dy is approximately equal to Δy and is given by:
dy =
= -0.015
Therefore, the approximate value of is 2.984.
Question 13.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y =
Let x = 81 and Δx = 0.5. Then, we get
Δy =
=
⇒ = Δy + 3
Now, dy is approximately equal to Δy and is given by:
dy =
= 0.0046
Therefore, the approximate value of is 3.0046.
Question 14.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y =
Let x = 4 and Δx = -0.032. Then, we get
Δy =
=
⇒ = Δy + 8
Now, dy is approximately equal to Δy and is given by:
dy =
= 3(-0.032)
= -0.096
Therefore, the approximate value of is 7.904.
Question 15.
Using differentials, find the approximate value of each of the following up to 3 places of decimal.
Answer:
Consider y =
Let x = 32 and Δx = 0.15. Then, we get
Δy =
=
⇒ = Δy + 2
Now, dy is approximately equal to Δy and is given by:
dy =
= 0.00187
Therefore, the approximate value of is 2.00187.
Question 16.
Find the approximate value of f (2.01), where f (x) = 4x2 + 5x + 2.
Answer:
Let x = 2 and Δx = 0.01. Then, we get,
f(2.01) = f(x + Δx) = 4(x +Δx)2 + 5 ( x + Δx) + 2
Now, Δy = f (x + Δx) – f(x)
⇒ f (x + Δx) = f(x) + Δy
≈ f(x) + f’(x).Δx (as dx = Δx)
⇒ f(2.01) ≈ (4x2 + 5x + 2) + (8x + 5) Δx
= [4(2)2 + 5(2) + 2] +[8(2) + 5] (0.01)
= (16 + 10 + 2) + (16 + 5)(0.01)
= 28 + 0.21
= 28.21
Therefore, the approximate value of f (2.01) is 28.21.
Question 17.
Find the approximate value of f (5.001), where f (x) = x3 – 7x2 + 15.
Answer:
Let x = 5 and Δx = 0.001. Then, we get,
f(5.001) = f(x + Δx) = (x +Δx)3 - 7 ( x + Δx)2 + 15
Now, Δy = f (x + Δx) – f(x)
⇒ f (x + Δx) = f(x) + Δy
≈ f(x) + f’(x).Δx (as dx = Δx)
⇒ f(5.001) ≈ (x3 – 7x2 + 15) + (3x2 – 14x) Δx
= [(5)2 – 7(5)2 + 15] +[3(5)2 -14(5)] (0.001)
= (125 – 175 + 15) + (75 - 70)(0.001)
= -35 + 0.005
= -34.995
Therefore, the approximate value of f (5.001) is -34.995.
Question 18.
Find the approximate change in the volume V of a cube of side x metres caused by increasing the side by 1%.
Answer:
It is given that the volume of a cube (V) of a side x
⇒ V = x3
= (3x2)Δx
= (3x2)(0.01x)
= 0.03x3
Therefore, the approximate change in the volume V of a cube of side x metres caused by increasing the side by 1% is 0.03x3.
Question 19.
Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%.
Answer:
It is given that the surface area of cube (S) of a side x
⇒ S = 6x2
= (12x)Δx
= (12x)(0.01x)
= 0.12x2
Therefore, the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1% is 0.12x2.
Question 20.
If the radius of a sphere is measured as 7 m with an error of 0.02 m, then find the approximate error in calculating its volume.
Answer:
Let r be the radius of the sphere and Δr be the error in measuring the radius.
Now, it is given that r = 7m and Δr = 0.02m
We know that volume of sphere (V) =
Now,
= (4πr2)Δr
= 4π (72)(0.02)m3
= 3.92π m3
Therefore, the approximate error in calculating its volume is 3.92π m3.
Question 21.
If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find the approximate error in calculating its surface area.
Answer:
Let r be the radius of the sphere and Δr be the error in measuring the radius.
Now, it is given that r = 9m and Δr = 0.03m
We know that surface area of sphere (S) =
Now,
= (8πr)Δr
= 8π (9)(0.03)m2
= 2.16π m3
Therefore, the approximate error in calculating its surface area is 2.16π m3.
Question 22.
If f(x) = 3x2 + 15x + 5, then the approximate value of f (3.02) is
A. 47.66
B. 57.66
C. 67.66
D. 77.66
Answer:
Let x = 3 and Δx = 0.02. Then, we get,
f(3.02) = f(x + Δx) = 3(x +Δx)2 + 15( x + Δx) + 5
Now, Δy = f (x + Δx) – f(x)
⇒ f (x + Δx) = f(x) + Δy
≈ f(x) + f’(x).Δx (as dx = Δx)
⇒ f(3.02) ≈ (3x2 + 15x + 5) + (6x + 15) Δx
= [3(3)2 + 15(3) + 5] +[6(3) + 15] (0.02)
= (27 + 45 + 5) + (18 + 15)(0.02)
= 77 + 0.66
= 77.66
Therefore, the approximate value of f (3.02) is 77.66.
Question 23.
The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is
A. 0.06 x3 m3
B. 0.6 x3 m3
C. 0.09 x3 m3
D. 0.9 x3 m3
Answer:
It is given that the volume of a cube (V) of a side x
⇒ V = x3
⇒ dV =
= (3x2)Δx
= (3x2)(0.03x)
= 0.09x3
Therefore, the approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is 0.09x3.
Exercise 6.5
Question 1.Find the maximum and minimum values, if any, of the following functions given by
f (x) = (2x – 1)2 + 3
Answer:It is given that f (x) = (2x – 1)2 + 3
Now, we can see that (2x – 1)2 ≥ 0 for every x ϵ R
⇒ f (x) = (2x – 1)2 + 3 ≥ 3 for every x ϵ R
The minimum value of f is attained when 2x – 1 = 0
2x -1 = 0
Then, Minimum value of ![](data:image/png;base64,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)
Therefore, function f does not have a maximum value.
Question 2.Find the maximum and minimum values, if any, of the following functions given by
f (x) = 9x2 + 12x + 2
Answer:It is given that f (x) = 9x2 + 12x + 2 = (3x + 2)2 - 2
Now, we can see that (3x + 2)2 ≥ 0 for every x ϵ R
⇒ f (x) = (3x + 2)2 - 2 ≥ -2 for every x ϵ R
The minimum value of f is attained when 3x + 2 = 0
3x +2 =0
![](data:image/png;base64,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)
Then, Minimum value of ![](data:image/png;base64,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)
Therefore, function f does not have a maximum value.
Question 3.Find the maximum and minimum values, if any, of the following functions given by
f(x) = – (x – 1)2 + 10
Answer:It is given that f (x) = –(x – 1)2 + 10
Now, we can see that (x - 1)2 ≥ 0 for every x ϵ R
⇒ f (x) = –(x – 1)2 + 10 ≤ 10 for every x ϵ R
The minimum value of f is attained when x - 1 = 0
x - 1 = 0
⇒ x = 1
Then, Maximum value of f = f(1) = -(1-1)2 + 10 = 10
Therefore, function f does not have a minimum value.
Question 4.Find the maximum and minimum values, if any, of the following functions given by
g(x) = x3 + 1
Answer:It is given that g(x) = x3 + 1
Now,
x ∈ ℝ
⇒ -∞ ≤ x ≤ ∞
⇒ -∞ ≤ x3 ≤ ∞
⇒ -∞ ≤ x3 + 1 ≤ ∞
The function g neither has a maximum value nor a minimum value.
Question 5.Find the maximum and minimum values, if any, of the following functions given by
f(x) = |x + 2| – 1
Answer:It is given that f (x) = |x + 2| – 1
Now, we can see that |x + 2| ≥ 0 for every x ϵ R
⇒ f (x) = |x + 2| – 1 ≥ -1 for every x ϵ R
The minimum value of f is attained when |x + 2| = 0
|x + 2| =0
⇒ x = -2
Then, Minimum value of f = f(-2) = |-2 + 2| - 1 = -1
Therefore, function f does not have a maximum value.
Question 6.Find the maximum and minimum values, if any, of the following functions given by
g(x) = –|x + 1| + 3
Answer:It is given that g(x) = –|x + 1| + 3
Now, we can see that –|x + 1| ≤ 0 for every x ϵ R
⇒ g(x) = –|x + 1| + 3 ≤ 3 for every x ϵ R
The maximum value of f is attained when |x + 1| = 0
|x + 1| = 0
⇒ x = -1
Then, Maximum value of g = g(-1) = -|-1 + 1| + 3 = 3
Therefore, function f does not have a minimum value.
Question 7.Find the maximum and minimum values, if any, of the following functions given by
h(x) = sin(2x) + 5
Answer:It is given that h(x) = sin(2x) + 5
Now, we can see that -1 ≤ sin2x ≤ 1
⇒ -1 + 5 ≤ sin2x + 5 ≤ 1 + 5
⇒ 4 ≤ sin2x + 5 ≤ 6
Therefore, the maximum and minimum value of function h are 6 and 4 respectively.
Question 8.Find the maximum and minimum values, if any, of the following functions given by
f (x) = |sin 4x + 3|
Answer:It is given that f(x) = |sin 4x + 3|
Now, we can see that -1 ≤ sin4x ≤ 1
⇒ 2 ≤ sin 4x + 3 ≤ 4
⇒ 2 ≤ |sin 4x + 3| ≤ 4
Therefore, the maximum and minimum value of function h are 4 and 2 respectively.
Question 9.Find the maximum and minimum values, if any, of the following functions given by
h(x) = x + 1, x ∈ (–1, 1)
Answer:It is given that h(x) = x + 1, x ∈ (–1, 1)
Now, if a point x0 is closest to -1, then,
We find
for all x0∈ (–1, 1)
Also, if a point x1 is closest to 1, then,
We find
for all x1∈ (–1, 1)
Therefore, the function h (x) has neither maximum nor minimum value in (-1, 1).
Question 10.Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
f (x) = x2
Answer:f(x) = x2
⇒ f’(x) = 2x
Now, f’(x) = 0
⇒ x = 0
⇒ x = 0 is the only critical point which could possibly be the point of local maxima or local minima of f.
⇒ f’’(0) = 2, which is positive.
Then, by second derivative test,
⇒ x = 0 is point of local maxima and local minima of f at x = 0 is f(0) = 0.
Question 11.Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
g(x) = x3 – 3x
Answer:g(x) = x3 – 3x
⇒ g’(x) = 3x2 - 3
Now, g’(x) = 0
⇒ 3x2 - 3 = 0
⇒ 3x2 = 3
⇒ x = � 1
g’’(x) = 6x
Now, g’(1) = 6>0
and g’(-1) = -6 < 0
Then, by second derivative test,
⇒ x = 1 is point of local maxima and local minima of g at x = 1 is
g(1) = 13 – 3 = 1-3 =-2
And,
x = -1 is point of local maxima and local maximum value of g at x = -1 is
g(-1) = (-1)3 – 3(-1) = -1+3 = 2
Question 12.Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
![](data:image/png;base64,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)
Answer:h(x) = sin x + cos x, ![](data:image/png;base64,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)
h’(x) = cosx - sinx
Now, h’(x) = 0
⇒ cosx - sinx = 0
⇒ cosx = sinx
⇒ tanx = 1
![](data:image/png;base64,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)
h’’(x) = -sinx – cosx = -(sinx + cosx)
![](data:image/png;base64,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)
Then, by second derivative test,
is point of local maxima and local minima of h at
is
![](data:image/png;base64,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)
Question 13.Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
f (x) = sin x – cos x, 0 < x < 2π
Answer:f (x) = sin x – cos x, 0 < x < 2π
f’(x) = cosx + sinx
Now, f’(x) = 0
⇒ cosx + sinx = 0
⇒ cosx = -sinx
⇒ tanx = -1
![](data:image/png;base64,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)
’(x) = -sinx + cosx
![](data:image/png;base64,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)
Then, by second derivative test,
is point of local maxima and the local maximum value of f at
is
![](data:image/png;base64,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)
And,
is point of local minima and the local minimum value of f at
is
![](data:image/png;base64,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)
Question 14.Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
f (x) = x3 – 6x2 + 9x + 15
Answer:f (x) = x3 – 6x2 + 9x + 15
⇒ f’(x) = 3x2 – 12x + 9
Now, f’(x) = 0
⇒ 3x2 – 12x + 9 = 0
⇒ 3(x-1)(x-3) = 0
⇒ x = 1,3
g’’(x) = 6x – 12 =6(x-2)
Now, f’(1) = 6(1-2)=-6 < 0
and f’(3) = 6(3-2) = 6 > 0
Then, by second derivative test,
⇒ x = 1 is point of local maxima and local maximum of f at x = 1 is
f(1) = 13 – 6 +9 +15 = 19
And,
x = 3 is point of local minima and local minimum value of f at x = 3 is
f(3) = 27 – 54 + 27 +15 = 15
Question 15.Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKcAAAA1CAYAAAAkuPukAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAXVSURBVHja7Zy/bttIEMb3+qSPA4+6pLeWeoOA2DhJaVwows2dz4UgLJDYgAoDod0ZuCR9uqQ6u7g2TVxe4zzBFfYT2M/gO+6SS4vUrsQ/K3FlTjGN4kgU98edb2Y/DTk5OSEYGC4G3gQMhBMDA+HEQDgxMBDOri4MIb9MB8KJ4UQcjXae94H8jKH8Twb0f/rB+BXCidEymC9iLslNDOXdVEhIaRC9QTgxWonT0/1HDMgPYOGH/dPTR+K1iAfPxGsJoPR6N4o2EM6Oajul79rQexFnWwPGf9dCu0kuCIFbxqMthLMDEUW7G5SQq0zbxQGMv8+/DjcuADGk5BvunB1MpSElXyWIm+xCpdPjkL6OYbgKePTMiXQvdk4afkXN2VGtp4qOyWT/MQM4dyWFinQPhF52addEOLUpnl57HpzR8Pi1K5p4SOGTK9eDcLYUSSpP0/tk8tiFaxr7cCB0cCeLVYQyDycA/CMLI3986ML1UJ/vdXU9EMo0ZPMbXp7xY/4k0Z9w+2J01G9TZ1IaftCleYSzQzGZ7D71wDtTBUdSgIhTGnq9O5k8XTmYAX0D/eHnYv9VHGtSyv5UHQWEs6Ro5z7dq5uCxP8dhPxtG7uCrMw3yYVI49MN+FjrHSr9uTM6er4yjRkMgvjBuJ3uveb6sP744EGAV+KQw87iApwDDT/WfaKzVk4LhUja45Rn2Krw4Aze58+2V9OI5+HgbeFM/a6N61j+A+i/okAu5XcCemmST1baHLaqSQkKHX7rqkWsCyEki3zIQr6dPJBsW0gonamlaZvjkAD7YUsDZcYHByplW2kLgZwCM+klXxeliWiX6fR9bT2gPkjXHG5ipkiPDdf+DFn2Jx3ShzYMLk0ftgTCWfNKWoDeFu+XtqWSGl2lxqH++A/dF8hoL0DU1ExhulCEs/GOdVUsrKoYXMS6eOD9JYrDOpBm/oAp/0JxoxOSzghnanTNwEj1gKoc75TxYN4HqX+va6ZY9N6mXWBe4M5px+DCebAlC5m4iBHO/Cr3NgNQs66mNc+9gc6WpXvNRLq2Aq9hplhkD5utps3R1tGfa3Ca1uQlwFnVDoA0QHu9L6TX+1e0AEWHZRGoKiPqmLn3q+bXfOEfJOk7/jJT2nLeB+nTSTUzRQLn+hhrdbu1KBZV79SlH6rZNLgcc/4kpPCxR2JIWfhuHqSN4Lzv+eWP7ZJmdB6UsnDep41qZopVwFlVGsyDq8pOXraAXKZcsW1wkZKBDd5JSUiHn3RybF62NaX8mT+SxZDH/hYXLY/LxH8qtIuqwlnVTLEIThuLWBamYiwjrS/7WmysybyHXMgDcYAg+pWmw5jcQUudgkjuAj7dSy9c3oCex76UfTNd5V/HTIGac3lhy+CioBSHMFlaX9DvNvUz1WZXzCqz2316QjNvBypTUdc1U5R5b2wl1Wv82zC4KANKriAquU7q84q7dQrtVXEz0mhOg+GgP/xcpqGa3IT6ZgrTKQLCWbKPGK+J6E3bXBMVo9FO3wP4LlpJ6vixaiRZD27Ebi0NQ6kc0GpxHdllxLyq+nQL0MRMkZy9zj5F6xbi+65SUoj1SNtEdzpJlFsT1ceuYHBRTXgbP/iLgfw14ywG3fSeM2ndZHvT7QTyy1l0EmVPd0d/lmArQkqjh/BjuPyuCezcBFoUBFs6wm05idIUc2DTSNLFMA1mWGs41RYv+1T70YYS10mbgG2bNMa8FkEVvSStd3EV2SaY6zzZLek1st/qakGn4Uy0ANuWgne6FQPe91LnrrGwbeKEb/uHXDjZzXE4uxo42Q3hdDYdJkUYTnZDOB0sIHSSoo3Jbi5NvEM4HY9VTnazYQhGODuW7lc52W0dJt4hnI6k+zYmu9kyBCOcD/WmtDzZzdWJdwinA+HCZDcXJ94hnC2HK5PdXJt4h3A6oDNdmOzm2sQ7hLNtMB2Z7ObaxDuEs22N6chkN9cm3iGcLYdLk92aGoIRTgwMhBMD4cSbgOFo/A/hn1kxg8WO5QAAAABJRU5ErkJggg==)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, g’(x) = 0
![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ x2 = 4
⇒ x = �2
Since x > 0, we take x = 2
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Then, by second derivative test,
⇒ x = 2 is point of local minima and local minimum of g at x = 2 is
![](data:image/png;base64,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)
Question 16.Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ x = 0
Now, for values close to x = 0 and to the left of 0,
g’(x) > 0.
Also, for values close to x = 0 and to the right of 0,
g’(x) < 0
Then, by first derivative test,
⇒ x = 0 is point of local maxima and local maximum of g at x = 0 is
![](data:image/png;base64,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)
Question 17.Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANcAAAAgCAYAAACB6u0UAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAUzSURBVHja7Vy9UttAEL6kTvqQyboLPT77DRhF/JVMsDVuEkLh8WgygRkXTEbQUYQHIBWkcl4gDZRpeAMyAy8Q/AwQnX6MsE6n091JxsoW2xj97N7ut/+CHB4eEiQkJPOEh4CEhOCas4Ml5Nk8E+oQwfWUwXU/z4Q6RHDNA7ju5pFQhwiupw4uNFIEF1IZ4MJzQHAhldPMQHAhuHKNBLtHivUWngWCiwsq12lvASG3YYEL43Zn0Mm81qLb1HK3VRhg97Ydd6tOAM6qt+rsrP5XRyySm3vDwII98sa+2Dk6euG59lIIMhjbrreUvG443HlpA/wE6nxj16owd3S088IGch68bzh8Wdd6q9/fbFoW/QpA/oA12K2TgQ061hoFchk4YqCXy/395v8ArDydpm7wvN4CJeQmefGBQ9cbpHGVBBdDapfCMdjuFxOMOpScEto9m3fPx6u32PmRRuOq0SBX7G91ApfXoRu+4721HXc1yEQce5U5Y9rxNuoMrIMeXcvTafomZggSBhBEN7DPVSNWVgTz37tX13pL9mznBlgcRxzZxi4h9KY3HL6ue/QS6TTldUNPFFy8l5VLxodKnYN1UQ6avF8mJw8Z9ZXieQtV5clFeVStt2YJrrLkDUGULheiUmKsK6eIrzJ1KMuDNLgeaqtgQp/YLKDX08Y+8UxTv0egu06u0bC08fHvfgoxpQzTSpHwtso8qtRbswQXV17//bryBpnGG3IR1+Y858vSfFWD3u9vLtqtxgnPgSd4vzMpUxEeCkcuGQMQHWr896B+YgImromi0nXH9d6qKEzksYouoOrwWLTeegqRqwx5JwDi6EpGjyKDbgH8Yo2RuI6TkilqhpnQoSwPxsEl45UmHUD/OaywjbuKMp6kS8mZKDV0bfgiux8narbo8KhSb8265uLJuwIwUpU3zjJ4djABl2SKH45+7FXWcQRo/ZIFhkmZVHkwCy7BofJDN71ptWAkCq1pcKXz+HJTJjGPsp9j5O0TyoCrzE9DVHVSFrjieSrruDVa9olGBFWWSZeHmYAr+awiM6wqwSXL41Qdep9l1HlbGTLgKhSZFSKgik6KZjCilHEip0W3gcAfoN1j3Y5zJNNdUZlM8DBTcAHA77j7aAJcpj27LI9xRMoCmMw+4VNoxavoRJiSKTQ0gnrJbn9ms9PAsHe8BZ3unopMpniovOZitN9fbgKsjNwD91WYF8NYZnJfVc2lwyOvtpLZJ5w1uFTlPRS24tPzrNgB56VozMD9lOx9kJLRd983+/uLRQ1cVyZdHirtFjIaDnuvW9AaxQB5aPOLB4uqXSYVUuUxK3rJfL81S3CpyJsX/eNnTEeLCHTXReaVyWbCZr/flDFwHR2a4kEaXKzbEkYO31D8yMTyV94LsoaH8TMYQOIhdKykYKMjyvWZdxBFxbKNT4fHLIDlRa3kgJ40uz+qXHJVkfehKQFjag0+iWtEuGXRIrnwrdos8dzO27gNbnUGa2XqUJcHjk6fP3K4014olV5R5zSrQ8MDQTR7eJSWpVM5/oAvZLSYx1MhHR55qWBevZV1tlVFsEfyRu/Mk5fpOGpz3+V1/VhaNZHPN0jVGdO0gYsGuCoymeQh0ulfkU6VhQ8EMbjJPvFEhhaBq6DkhkCdv99yKPXKdnh1JH2PaGCTPQrnuyYXgWcAsFr+vwzmodu2+xHBUjG4RO3YIs8IPl2BldG8ASsJsLoZRtiqtj/krf8glQSuRM69pfMlsuq9SEi1BxcSEhKCCwkJwYWEhOBCQkLi0j/JhVubdk+arAAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
f’(x) = 0
![](data:image/png;base64,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)
⇒ 2 – 3x = 0
![](data:image/png;base64,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)
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)
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8BNwcAqoroZq55FeefyqQHANCJqJBGAACkBwDQU/4PH5+9+UjWiHsAAAAASUVORK5CYII=)
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gCTBDUWf0f+quVaG2pxMsAAAAASUVORK5CYII=)
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Question 18.Prove that the following functions do not have maxima or minima:
f (x) = ex
Answer:f (x) = ex
⇒ f’(x) = ex
Now, if f’(x) = 0, then ex = 0.
But, the exponential function can never assume 0 for any value of x.
Therefore, there does not exist c ϵ R such that f’(c) = 0
Hence, function f does not have maxima or minima.
Question 19.Prove that the following functions do not have maxima or minima:
g(x) = log x
Answer:g(x) = logx
![](data:image/png;base64,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)
Since, log x is defined for a positive number x,
g’(x) > 0 for any x.
Therefore, there does not exist c ϵ R such that f’(c) = 0
Hence, function f does not have maxima or minima.
Question 20.Prove that the following functions do not have maxima or minima:
h(x) = x3 + x2 + x +1
Answer:h(x) = x3 + x2 + x +1
⇒ h’(x) = 3x2 + 2x +1
h(x) = 0
⇒ 3x2 + 2x +1 = 0
![](data:image/png;base64,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)
Therefore, there does not exist c ϵ R such that h’(c) = 0
Hence, function h does not have maxima or minima.
Question 21.Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
f (x) = x3, x ∈ [–2, 2]
Answer:It is given that f (x) = x3, x ∈ [–2, 2]
⇒ f’(x) = 3x2
Now, f’(x) = 0
⇒ x = 0
Now, we evaluate the value of f at critical point x = 0 and at end points of the interval [-2, 2].
f(0) = 0
f(-2) = (-2)3 = -8
f(2) = (2)3 = 8
Therefore, we have the absolute maximum value of f on [-2, 2] is 8 occurring at x = 2.
And, the absolute minimum value of f on [-2, 2] is -8 occurring at x =-2.
Question 22.Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
f (x) = sin x + cos x, x ∈ [0, π]
Answer:It is given that f (x) = sin x + cos x, x ∈ [0, π]
f’(x) = cosx - sinx
Now, f’(x) = 0
⇒ cosx - sinx = 0
⇒ cosx = sinx
⇒ tanx = 1
⇒ x = ![](data:image/png;base64,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)
Now, we evaluate the value of f at critical point
and at end points of the interval [0, π]
![](data:image/png;base64,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)
f(0) = sin0 +cos0 = 0+1 = 1
f(π) = sin π + cos π = 0 -1 = -1
Therefore, we have the absolute maximum value of f on [0, π] is √2 occurring at ![](data:image/png;base64,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)
And, the absolute minimum value of f on [0, π] is -1 occurring at x = π.
Question 23.Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, f’(x) = 0
⇒ x = 4
Now, we evaluate the value of f at critical point x = 0 and at end points of the interval ![](data:image/png;base64,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)
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)
Therefore, we have the absolute maximum value of f on
is 8 occurring at x = 4.
And, the absolute minimum value of f on
is -10 occurring at x = -2.
Question 24.Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
f (x) = (x − 1)2 + 3, x ∈ [−3, 1]
Answer:It is given that f (x) = (x − 1)2 + 3, x ∈ [−3,1]
⇒ f’(x) = 2(x – 1)
Now, f’(x) = 0
⇒ 2(x-1)
⇒ x = 1
Now, we evaluate the value of f at critical point x = 1 and at end points of the interval [-3, 1].
f(1) = (1 - 1)2 + 3 = 0 + 3 = 3
f(-3) = (-3 - 1)2 + 3 = 16 + 3 = 19
Therefore, we have the absolute maximum value of f on [-3, 1] is 19 occurring at x =-3.
And, the absolute minimum value of f on [-3,1] is 3 occurring at x = 1.
Question 25.Find the maximum profit that a company can make, if the profit function is given by
p(x) = 41 – 72x – 18x2
Answer:It is given that the profit function p(x) = 41 – 72x – 18x2
⇒ p’(x) = -72 – 36 x
and p’’(x) = -36
Now, g’(x) = 0
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Then, by second derivative test,
x = -2 is point of local maxima of p.
Therefore, Maximum Profit =![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 113
Therefore, the maximum profit that the company can make is 113 units.
Question 26.Find both the maximum value and the minimum value of 3x4 – 8x3 + 12x2 – 48x + 25 on the interval [0, 3].
Answer:Let f (x) = 3x4 – 8x3 + 12x2 – 48x + 25, x ∈[0,3]
⇒ f’(x) = 12x3 - 24x2 + 24x – 48
=12(x3 - 2x2 + 2x – 4)
=12[x2 (x – 2)+ 2(x – 2)]
=12(x -2)( x2+ 2)
Now, f’(x) = 0
⇒ x =2 or (x2+ 2) = 0 for which there are no real roots.
Therefore, we will only consider x = 2
Now, we evaluate the value of f at critical point x = 2 and at end points of the interval [0,3].
f(2) = 3(2)4 – 8(2)3 + 12(2)2 – 48(2) + 25
= 3(16) – 8(8) + 12(4) +25
=48 – 64 +48 – 96 + 25
= -39
f(0) = 3(0)4 – 8(0)3 + 12(0)2 – 48(0) + 25
= 0+ 0 + 0 +25
= 25
f(3) = 3(3)4 – 8(3)3 + 12(3)2 – 48(3) + 25
= 3(81) – 8(27) + 12(9) +25
=243 – 216 +108 – 144 + 25
= 16
Therefore, we have the absolute maximum value of f on [0,3] is 25 occurring at x =0.
And, the absolute minimum value of f on [0,3] is -39 occurring at x = 2.
Question 27.At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Answer:It is given that f (x) = sin2x, x ∈ [0, 2π]
f’(x) = 2cos2x
Now, f’(x) = 0
⇒ cos2x = 0
⇒ 2x = 0
⇒ x = ![](data:image/png;base64,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)
Now, we evaluate the value of f at critical point x =
and at end points of the interval [0, 2π]
f’
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEsAAAA1CAYAAAAJZhZbAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAOMSURBVGje7ZrNTttAEMc3PUPvqKxvtGc85NiekLstcIwqe5VLK+VgRZYKSD6gxnCLVOid9gK8Bu0deIEWUV6A8gqBeuLdsDFOSSC4TjKRRgobJ9g/z8d/Zs22trYYWX9GEAgWwSJYBItgkREEgjXmsOLXk7uMYMW24Vee25ydxECuexucV6NoZqJhhWFtWsyy7zGQlrKr1Hv1N5xNPKzNKixxIRu1ZnMqCsQ8B28H15vN2hRC5E59jXJWFjgJywy8fRMWyM1lgtUDlvakKKrOAGPnBKuH1R2+bsKyGT8RQTRPsHokeu1JGtaiv2EraVEaOVh40o+heSIXVhjjl9qTdBjGnrZeq1WeSoBo5KqhB3zHvKihgMIqyNhlLA9+ayDtBM/ZYUdngdzLSRyXlAguPRiWBL4dw7oYKqzEi864CFZvyQrGLhiHIzeI5h4bku9XbMeBBufsNEuyUM9n3BhmWT8ti/1CTyZY/Wq9fmEZMfskqwplJXgzzs3vF6GCPRosLN+vwfra7sU4HLu++wpARmbDG3/+jTN+qnMWfscVsIpxDm60EkjxFjg7wp6O296XWhhOjyUsD9i+PgirkQS2p9sPrFiYhJPG9qYato9Jmt1ra0HsCvnxDa4Hgq/iWl4KPOXRd1npQbA67UXsHfrH2msgG/oY3a+lpUOsvNfSYLQcyKsRVjen1Y+lq+7AsG50Db903PpSrzuA3jcILO2ZYxeGgSy/68yROD8tu3U3DWxYsAYImVynpQNVwyhw58SCtaugXTPbOzCBDRFW6z5WiJyVlgkILa5qx2kIRQ3D3HOWA84ns9Rv+Is2XvAowMo1DHWC5yC3ERi6K86XzOY2DCvPsGK2waiqiebZ7ED9g3W9lkwR4jCeFT9GRWvhNXfOO0k/XSHbDUvI90Gw+FKJypbZwCpPueh2Z9lQk4EuF78VElwc4oy9yKCyri8dstQPjuKklGDdI2f0W+InGlboV1507Upz+0R1EoUBVoiTUBLlTyq5XumqS7DSkkXtSne6iKTKXhXhGYfCwMKSXRbBh0yIGRMOylk9LOkWyLP6C0/0rJy2wEYaVpL04bgoXlVYWO1+E/gOVDeXSGfdAQqnGP8aoxAsY0SSVR0JVsqjME+ZGyTmZwTLgIFzJNxnTPeH2AYBiM9FGPEUI0e5ZTd5iib7SeWi5K//fgLGjlIPG+4TO2Or4CnBE6zJsL8cQ2UD65CGSQAAAABJRU5ErkJggg==)
f’
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGMAAAA1CAYAAABV7FV/AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAQOSURBVHja7ZuxThtBEIY3qeEBiFh30HNrl0mFjhWBEkXnFU0iubCsKwDJBYkPOpRA+pAG6NOmgRfATwASPIF5heDc7HntvfU6ssOdfXcepBFwPhszn3f+mdlZcnx8TNCyYegEhIGGMBAGWu5ghF+vdEMYM7LD+s4qo+Q2hPAsjTrtnfrhKsKYsgU+XyuX+dfayclC9Lu3EoFhj7tBsIQwpmQnJ7UFzvhnBUIHRAl5YuJoC2HMPGytOzYYpq7YDGEkLORVRi4JExemroSE2uHj3dGW79CWKQj1+o4jHV7mv2rN5mIsnC2Tm77Ax607+Jk9IIwktIOSa93B1Kl+V0COBNuiXLRAW6SesOqZDom6jX3UjBSg+KLyIdSLjgTiNg7MewAMYdVLHUZRhD6Tb0plUzYNkKuktxKCYHeJEfKIMFI2KeIWGA2X7uswHELb3A/WEEaKBk4ny/xGr0HMsGTCmLvUNul83vZ6zWZtUQoz9/di4ctj24TQJ+V8FaZAW+A5grFgrrKpKqNnukNepg3eCjgz6kXVHYACToW/QZk4HaEj/fRVy8KiOsOoTTLp8H8UqBO/mGD0NITRSQKGdGa59EOvGygt/+bC3xwCF62CB3O1SEGH7IuyW88PVrIMAuoo12UtSsm9LR2fm77PrE2m5KXSXalE7nppO8LIBJRxYZgxzYxttnhnXpv3TaJEYIB4brDSebTBE8bguveOMRGox6FZFz7+kxJ6rzQDnuNxtgdxEFJOX/BNtVEkWxpGexxhjAkDCi11E4irYORCtR4gmwEBjTKXQTYl7+n1k0ps41yJr6wTwmtZqo7HacGnvarHgtFvLXjBduwaEy2z6DJTW5vjVSqapSaez+neiM7vkJlZ21RhDHJ2+uR6jfejPh1Rm2J8GGplFcVeurrGDlPQMR0MBdD7itfwzBdMCsak/1QWQkwSq2uibAqqYq0Q6xKnepUSjOc0LdeaYaaxgymNuJPzHKZypRkuc7/o2522KY1514ypCjhsaQIQWCWhkw/0xlyzuftGNvbA8WHWpZZz1SFXamdOXYs6rGGYgza4BnieLeaXMPyH31/HiuUYDC4++v762/50n9Z8633SO/GlLFrG3rVc3kPhgPLreS/+NP/9GRUOMXRkaeWgExDGfxdZCGNGNjRFSJ02dAcQxtRByHnbjmV6sKv3zxBGytZPs3tThP3OQH+/u5jHBTKbBla4/8kKydI1RhgzslHDbQhjFuELVkYORnIKDyMSdXZb1ONl+Xmj8hANPSvy0bLcvFE58JxSWxthTGDQdrZlVwhjBjqhD0XoYQthTFEjoPcPs1dmf0oe4mf8W9Ha8pkF0fAqXjR1bj/ZWpRzfJmHEZtSsVoyU/AIAw1hIAy0iewv2V6qiw+7Cd8AAAAASUVORK5CYII=)
f’
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFUAAAA1CAYAAAAwAacAAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAASvSURBVHja7ZtNT9xGGMcfL9dAzymMT42aY/ADt6qnYIaGRMrBAtvaSytZkYWsNkvkQ5T1ckNK0nNaDmk48gkqaO5puLfKCx8gCh9h2e2M7cEvGLEsTjMOc/hLrGWzuz/P839ePAsbGxugVK8UBAVVQVVQlb5QqBbABJyUpqCOqTBsf20AHDCIw0zkI+30biioY2rNJB0Gsl8QOs9V+I+pKLSmUV/YsjxvqonhLh1UDtExYJuHOyH4p7niL7O/W6ec24Jq301lTSioTL0OvaEDfEhD/ij2UmLsW2E4XfBc3/rWILBf9Nyy8KAdRVdV+Kfyff8bl87/zAFrAAMA48AKo+kkiXmTlGh7GXhtIHw3OVc7Sl4bbxXUCm1ueleWCOxqfOUZzna8mtt4i1C7621uXokCOkvQeSrOpTPwFzHXOspTz0pcUfsqAryvCueei8uA7h95qOj2lhXUERRQcp/VqYc0iGbLUMXKTOEfKKijrlYe5mC8Lq9UVs+u56EaQF6XwSuop0G18TaQpV3uo1nX5U2aM/BSrEwB9ab/0OCvh5LUuOe+YCerFWv5Asn/22mVj7lInuRbVP5+j+zZO3lLEOFPzHsPPM/6ykaMGpn9HSRPq7xuXHF4Oi5s3fU61/nNYnCm+LF5N1g5aQdwyJLXewEuTlQE9tI6dSAS2P/RsKRii2uo1QH1MYD+oa5BR+DOr+SK/74+Zz6zg+haVUVgALwl1L9fSFys3IqvJ/iq6rpPAdT3LTRN7BICb6rKuUb32J+lA2Q3EXT9H12Hf+PWWkGtES6vl0eFmktErcQ3ip5Rlagyfzn2mvScndalhxqyJLGI+u+xv3GP8u3vEd1IgONDjQXUtwiQNyJR8WtsEzvcX9CObgcu/QEJvGJkj4ix+qsXhpOXGiofwYmTeGZ1EJ4DOi/ENImVL++SAUaW/fk5Wjpd0ufob9T9ZSnriGAoU7czzCJqBA21C0MVNd+s/eiOCO34GNrd/LCD99rlkiqd2hcAihKIvem6NJOw5Eb3R1G5yhgLalbzkcN0UFxZ4LOV+eI8UMVKv7Thz2vG4xklIcwj15xysqkL6uhhOJ6kyv5RYF8z5/RnaRgMwFjdzq/YGqH2P6Wk8FSR4fNweRYvw2py+H8WTzXR7FpeOCWOPfRvGgS0Q2z3buUrhAqo65fYU9fPTFQE3Sfpo+IJP4aF7/gAI65TQ2tabHhIqoShxpuBVfE01Lz3gB8bHk+UWMTNmC/zN6rpEo3O8fdL7LFgF0Wo1P0xCOh3adj380OK0lPPNETs7iKB3XLYnAgzsliYizZ6hVZwKNuF6uNlmKcqfQFQS7tSWk3YFiR1QjixK4UY+2m3JzVYaT8YK+dYvtQ+5pNBMsxJKg8F9ZxKyjttT+xKEc3IYrJzZSDLnqlGQeVlyzxd+6kSdsWUTEG9oJI2OXuiqqDWYQt8pUq+w7pRUJNZBP4t8yptFFQ+T+AbOfLDHQX1gjUrHy+eNY5TUM8BlG9Wq6oGFNQxgXIfzT98zLevCuoYQPnMku8dKD/q6Hh3ryPSx7KOE+X1UBudZJdf9S9RCA2k9VcpP5R4qnu66tt1qDoqlagUVKUa9R+2fExyZIXJOwAAAABJRU5ErkJggg==)
f’
= ![](data:image/png;base64,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)
f(0) = sin0, f(2π) = sin2π = 0
Therefore, we have the absolute maximum value of f on [0, 2π] is 1 occurring at
x =
and x =
.
Question 28.What is the maximum value of the function sin x + cos x?
Answer:Let f(x) = sin x + cos x,
⇒ f’(x) = cosx - sinx
Now, f’(x) = 0
⇒ cosx - sinx = 0
⇒ cosx = sinx
⇒ tanx = 1
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIsAAAA1CAYAAABx2Rj6AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAUSSURBVHja7VzNjtNWFL6GLUO3pZOTVRFdEt/kDQbHEVOkLqypY2XTqhayIosSkBejwZldpELXSCyo2IBmX6mCsuqGnweoBMwDUHiEmUl97Dg/noQ4zvV1AmfxSRMn8jnJ+eae7/wkbH9/nxEIaUAfAoHIQiCyEIgsBCILYQiDsbPsNBQiC2ECntf6RmXsMCBHfwT4oLt+hchCmEBbg05AkKMJ8OZDSkOECfiescnLVx4Ytn1+3dIOkWXah8DYGTZdUwxgnM14X6WpskeYdgD4X9qOs422ZPtBZBGlJxzjkgrs9aSeSIIftnz/wqL37nb0y2XG3g9Sz3F4L1BfG563KdMPIosQ4Wlv6KA8GwVTOYk1hcJY8LdyHD1W3ywTJMdxvrX02g0kTnRf9dDw/E3ZfhBZlkC3xa+Cbt6xe71zvqtXgDfv4fVezz6nl9jfoLU7Iu3hfevAnip4SqjNR5N+WHuy/CCyLEsci28zbv0xHiRudbeFC16/dYEz9m5WSpHlB5FlSbLE/8GDgB7mFSRXh5uMwcdpfRaZfhBZsvdCbo0HSWXwKq+mWZhqmPpq2ski0w8iS0ahq5XY8/g/OA7SlrOr4uO+4P6Ib/LvGTSeokZJ+lEfSzt5+0FkydAL2TMr18bTQnz8g3b9tm0bX5mc+1mqkIOwb3JwJnnN4nBX73Qvy/KDyCI0JbCPgeB8FwciFJbAng36Gyex4FwUSIoyv/LgB7vzHTbVgoCfx2s1y92R6QeRRWB1ojL2BnTnZrKsDhtqwF+Yrn8xk5C1ajtjTbmjclW7P+teefpBZCGsF1kwzxY9ryCsCVnwiAxy7Ntps45ZOL0E1FemXFcoOCtMluiU+NTEczraJm8C1A9wBJ9GHwTq/m007wgrwRPM0fH14DEuBf2XrBQIK0aWxIR0cQB/aTjepXmnAir8JmcPQ8KUtOe2520Mew/BKTVPxPUXInWfTqhVFLiRai//u+Xs8nlBQsI0BgO1irl3DUvKBsCTNCeKE7bK0xE4WVkQVqgaQsIEBDgu1Xf24xNjXkrCcX21Ck94q3uVAvIFkQVTEM42cCcjTfDDCeswHfU2pL7xDPos7/vn4cdKkgWJgroDU1FNt24k5x6zXg8A/6B+Aa19K00VJEqzZNFlC34eR3lhbauhGHHg01Yyu86WCtB47Hbdr1G/4EwE9Q5pls+7GorWAIG/SDvk8jxjswrVx/Hr0TYw9iFaN0zfsykaB8MF62Krrll+jK6fPrGzPLe8RjFrJp4Q89LOkChB5YPrADhJjd8cppbrGtwOey+BfklTgheNeIsNT0SutX9ZRT+aHO6F16csT2V5bvlyuVz9c5GxOfZYkmnidGopv1/1xhxWc/UwfeJS9WhSvEp+BEH/DT/LacVGlucoFwsQ9lbBeyay/KCAL1sBunqlprd//hL8oKDPLM8/PU1HrWDp+k+69WtDlk0ZfhBZFhXtYWNR3rf/irBJZBGAXWeLD9YbpQWuCJtEliWBv6PCQXkZrVDICVwRNoksAlIB9hdAt+5EfYv8A1eETSKLiH4Fjiu4dXfU5Mo/cEXYJLIISAXx+EFW4IqwSWQRULJiKog7xjICV4RNIouYkrUzPp3OO3BF2CSyiClZVVB//H181QK/JoqDznAC7rol0V91KcImkUUAZu3GDH5tqZ/Hry0VYZPIkhOKSAmUhogsRBYiC5GFMBa4aJlInm4owiaRhUBkIRBZCGuG/wE6mcEnJztx9QAAAABJRU5ErkJggg==)
Now,
If f’’(x) will be negative when (sinx + cosx) > 0, means both sinx and cosx are positive.
And, we know that sinx and cosx both are positive in the first quadrant.
Then, f’’(x) will be negative when ![](data:image/png;base64,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)
f’’(x) = -sinx – cosx = -(sinx + cosx)
Now, let us take x =![](data:image/png;base64,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)
![](data:image/png;base64,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)
Then, by second derivative test,
f will be maximum at x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAA1CAYAAABBecueAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAGgSURBVFjD7ZbNTsJAEMcHz+gdZY56p8MjENhE9MaBNr320JAegKQHogu3HsRn0JPv4Mfdj0dAXsDwCoi7pZs0atOWxURjm0yybTq/zOxMZv4wmUxgmwYFsAD+ZqB4dtIsM3Dkdo4MhBfhtEo2mtucV1KBvu/sMoQ74bSM7F2dS+tz9E6zTMCxTcfIrHMnCMrcYzUkcyq/B4FTZlW4x2ZvsHFRxha1gczrOJCscVsLqCLi3K4QwFwL2GviMA40AJ+Zx2sbAWVxWrEUFbDhjoyorUq5gGfd2ikALlREKmUR8dBxOnsWEc9U5dBZVhdgIVrjVTmFRVm30ypsHbKuMkcYRTND5vW/tBTAGyA9dj1+WEybAvgTwCxTOm1qfwYu81pRlAKY9rMY+0mKYSOgXFpJiiE3cOQ2jGgt6AN9394nhKe1FNEEynszCadSnkjloA3kXTpBsi6UFNECylTrWL+RAG2gSlUtem2gaJFBfC9rAcMWMczL+CCV8qNZhQcJ7Pj+QS5J7DHsfzdMIwW7yqVgk2wrVf57wBbCbdLd/eOJ/QGG++HAc/LNLAAAAABJRU5ErkJggg==)
And, the maximum value of f is
![](data:image/png;base64,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)
Question 29.Find the maximum value of 2x3 – 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [–3, –1].
Answer:Let f (x) = 2x3 – 24x + 107, x ∈ [1, 3]
⇒ f’(x) = 6x2 – 24
=6(x2 - 4)
Now, f’(x) = 0
⇒ 6(x2 - 4) = 0
⇒ x2 = 4
⇒ x = �2
Therefore, we will only consider the interval [1,3]
Now, we evaluate the value of f at critical point x = 2 ϵ [1,3] and at end points of the interval [1,3].
f(2) = 2(2)3 – 24(2) + 107
= 2(8) – 24(2) + 107
= 75
f(1) = 2(1)3 – 24(1)+ 107
= 2 – 24 +107
= 85
f(3) = 2(3)3 – 24(3) + 107
= 2(27) -24(3) +107
= 89
Therefore, we have the absolute maximum value of f on [1,3] is 89 occurring at x =3.
Now, we will only consider the interval [-3, -1]
Now, we evaluate the value of f at critical point x = -2 ϵ [-3, -1] and at end points of the interval [1,3].
f(-3) = 2(-3)3 – 24(-3) + 107
= 2(-27) – 24(-3) + 107
= 125
f(-1) = 2(-1)3 – 24(-1)+ 107
= -2 + 24 +107
= 129
f(-2) = 2(-2)3 – 24(-2) + 107
= 2(-8) -24(-2) +107
= 139
Therefore, we have the absolute maximum value of f on [-3, -1] is 89 occurring at x = -2.
Question 30.It is given that at x = 1, the function x4 – 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.
Answer:It is given that f(x) = x4 – 62x2 + ax + 9
Then, f’(x) = 4x3 – 124x+ a
It is given that function f attains its maximum value on the interval [0, 2] at x = 1.
⇒ f’(1) = 0
⇒ 4 – 124 + a = 0
⇒ a = 120
Therefore, the value of a is 120.
Question 31.Find the maximum and minimum values of x + sin 2x on [0, 2π].
Answer:It is given that f (x) = x + sin 2x, x ∈ [0, 2π]
f’(x) = 1+ 2cos2x
Now, f’(x) = 0
![](data:image/png;base64,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)
2x = 2π �
, n ϵ Z
⇒ x = nπ �
, n ϵ Z
![](data:image/png;base64,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)
Now, we evaluate the value of f at critical point
and at end points of the interval [0, 2π]
![](data:image/png;base64,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)
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)
f’(0) = 0 + sin 0 = 0
f’(2π) = 2π + sin 4π = 2π + 0 = 2π
Therefore, we have the absolute maximum value of f on [0, 2π] is 2π occurring at
x = 2π and absolute minimum value of f(x) in the interval [0, 2π] is 0 occuring at x = 0.
Question 32.Find two numbers whose sum is 24 and whose product is as large as possible.
Answer:Let one number be x. Then, the other number is (24 –x).
Let P(x) denote the product of the two numbers.
Then, we get,
P(x) = x(24 –x) = 24x – x2
⇒ P’(x) = 24 – 2x
⇒ P’’(x) = -2
Now, P’(x) = 0
⇒ x = 12
And
P’’(12) = -2 <0
Then, by second derivative test,
x = 12 is the point of local maxima of P.
Therefore, the product of the numbers is the maximum when the numbers are 12 and 24 – 12 = 12.
Question 33.Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
Answer:Let the two numbers are x and y such that x + y = 60
⇒ y = 60 –x
Let f(x) = xy3
⇒ f(x) = x(60-x)3
⇒ f’(x) = (60 –x)3 -3x(60 –x)2
= (60-x)2[60 – x – 3x]
= (60-x)2[60 – 4x]
And f’’(x) = -2(60 – x)(60 -4x) -4(60-x)2
= -2(60 – x)[60 -4x + 2(60-x)]
= -2(60 – x)(180 – 6x)
= -12(60 – x)(30 -x)
Now, f’(x) =0
⇒ x = 60 or x =15
When x = 60, f’’(x) = 0.
When x = 15, f’’(x) = -12(60-15)(30-15) = -12×45×15 < 0
Then, by second derivative test, x =15 is a point of local maxima of f.
Then, function xy3 is maximum when x =15 and y = 60 – 15 = 45.
Therefore, required numbers are 15 and 45.
Question 34.Find two positive numbers x and y such that their sum is 35 and the product x2 y5 is a maximum.
Answer:Let one number be x. Then, the other number is y = (35-x)
Let P(x)= x2 y5
Then, we get,
P(x) = 2x(35 –x)5 – 5x2(35-x)4
= x(35 –x)4 [2(35-x)– 5x]
= x(35 –x)4 [70-7x]
= 7x(35 –x)4(10-x)
Now, P’’(x) = 7(35 –x)4 (10-x)+7x[-(35 –x)4 – 4(35-x)3 (10-x)]
= 7(35 –x)4 (10-x) - 7x(35 –x)4 – 28x(35-x)3 (10-x)]
=7(35 –x)3 [(35-x)(10-x) - x(35 –x) – 4x(10-x)]
=7(35 –x)3 [350-45x+x2-35x+x2-40x+4x2]
=7(35 –x)3 [6x2-120x+350]
Now, P’(x) = 0
⇒ x = 0, 35, 10
When x = 0, 35 This will make the product x2y5 equal to 0.
Therefore, x= 0, 35 cannot be possible values of x.
And when x = 10
Then, we have,
P’’(x) =7(35 –10)3[6(10)2-120(10)+350]
= 7(25)3 [-250]<0
Then, by second derivative test,
x = 10 and y = 35 -10 = 25 is the point of local maxima of P.
Therefore, the required number are 10 and 25.
Question 35.Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
Answer:Let one number be x. Then, the other number is (16 – x).
Let S(x) be the sum of these number. Then,
S(x) = x3 + (16-x)3
⇒ S’(x) = 3x2 -3(16-x)2
⇒ S’’(x) = 6x + 6(16-x)
Now, S’(x) =0
⇒ 3x2 -3(16-x)2 = 0
⇒ x2 -(16-x)2 = 0
⇒ x2 – 256 - x2 + 32x = 0
⇒ x = 8
Now, S’’(8) = 6(8) + 6(16-8)
= 48 + 48 = 96 > 0
Then, by second derivative test, x = 8 is the point of local minima of S.
Therefore, the sum of the cubes of the numbers is the minimum when the numbers are 8 and 16-8 = 8.
Question 36.A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
Answer:Let the side of the square to be cut off be x, then, the length and the breadth of the box will be (18 – x) cm each and the height of the box is x cm.
Then, the volume {V(x)} of the box is given by:
V(x) = x(18-x)2
⇒ V’(x) = (18-x)2 - 2x(18-x)
= (18 - x)[18- x -2x]
= (18 - x)(18 - 3x)
Now, V’’(x) = (18 - x)(-3) + (18 - 3x)(-1)
= -3(18 - x) - (18 - 3x)
= -54 + 3x - 18 + 3x
= 6x - 72
Now, V’(x) = 0
⇒ x = 18 or 3
If x = 18 then breadth becomes 0 which is not possible
Therefore, x =3
V’’(3) = 6.3 - 72 = -ve
Then, by second derivative test, x= 3 is the point of maxima of V.
Therefore, If we remove a square of side 3cm from each corner of the square tin and make a box from the remaining sheet, then the volume of the box obtained is the largest possible.
Question 37.A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum?
Answer:Let the side of the square to be cut off be x, then, the height of the box is x and the length is 45-2x and the breadth is 24 – 2x.
Then, the volume {V(x)} of the box is given by:
V(x) = x(45-2x)(24-x)
= x(1080-90x-48x+4x2)
= 4x3 – 138x2 + 1080x
⇒ V’(x) = 12x2 – 276x + 1080
= 12(x2 - 23x + 90)
=12(x – 18)(x – 5)
Now, V’’(x) = 24x – 276 = 12(2x-23)
Now, V’(x) = 0
⇒ x = 18 or 5
It is not possible to cut off a square of side 18cm from each corner of the rectangular sheet. So, x cannot be equal to1 8.
Therefore, x =5
V’’(5) = 12(10 – 23) = -156 < 0
Then, by second derivative test, x = 5 is the point of maxima of V.
Therefore, the side of the square to be cut off to make the volume of the box maximum possible is 5cm.
Question 38.Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Answer:The figure is given below:
![](data:image/png;base64,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)
Let a rectangle of length l and breadth b be inscribed in the circle of radius a.
Then, the diagonal passes through the centre and is of length 2a cm.
Now, by Pythagoras theorem, we get,
(2a)2 = l2 + b2
⇒ b2 = 4a2 – l2
⇒ b = ![](data:image/png;base64,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)
Therefore, Area of rectangle, A = ![](data:image/png;base64,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)
.
![](data:image/png;base64,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)
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)
.
.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Gives 4a2 = 2l2
⇒ l =
a
⇒ b = ![](data:image/png;base64,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)
when l = ![](data:image/png;base64,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)
Then,
= -4 < 0
Then, by second derivative test, when l =
, then the area of the rectangle is the maximum.
Since, l = b =
,
Therefore, the rectangle is square.
Hence proved.
Question 39.Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.
Answer:Let r be the radius and h be the height of the cylinder.
Then, the surface area (S) of the cylinder is given by:
S = 2πr2 + 2πrh
⇒ h ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Let V be the volume of the cylinder. Then
V = πr2h
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPgAAAA9CAYAAACA9KQMAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAsMSURBVHja7Z3Nb9tGFsCHTpBLmmwPBYq2HhkFsqjTU8Sx0FsRFIZMK066PqgyRQjdtFjBEFQCrpLqYLh0bkYTo+f00DTdQ9a67GWB3U2cczfNfRdx3FN3gTbxf5A0Xr6R6FAUKfFbJPUOD0hgiaDmzW/mfc6Qa9euERQUlGwKDgIKCgKefCkTcowQctxBBFR2PNIhZMpZD50pHCME3LNoWu0NRoR9fRI9G5TcL1Jr8xwqOx5pFumanR4EQn6jxWYLxwgB9wW4SMR7NU17IzUDr1sVQXa03k6ZGstEB7+FgCPgEwE4wKkqUqlUki/Xt7Ze8fr9ra36K3KpdFlS1FJazF4EHAGfCMDb9fJpmdEbBaVZCQyNUqhQJt8o19unEXAUBNyl6dwfsOtwUxh23XDgntlmNe3CqHfodLgJfmzUDq3V2IUZJm+TTrJ38jgBj1KHCHiKAYdJ0Kovz+ap8MAIDuXmpJvL9dbsiihuBnn+of5sfefeplJzbdTn6vXluWKRfUkp2XMDRVOiazPF5hVCDhPrk8cFeJQ6RMBTDrjWmGeUCAeEVW8ZvjE3gwl5Qgj7KcjzNZldpHTxziifG3Zkksv9O5cj/9En56EbKMAnX6T0DpO1i5MO+HpXh0/NOlTlwkc5Qn4NqkMEPOWAVxm5bTcJNnXoBML2/T6/3a6fWqC5HS/puk2FLbkFnH++JZ3L0YWdenvr1CQDfqTDdvtN68IZRIcIeMoBh11Qmia7hNCD+cY6s/5tIce+8fv8DTl/iTDlOy/f8Qo4iMLId3l549KkAs51SMk9Jx1KubmbVh320o081nHIU5fgs5ePIeDZ3MFvAVRkuni/XK+/avZpFcY2/Dwfdm9pWthltc0LUQPOLY1pabfebiduF49xB78luNQhBD0r86xFKXmkv9sVFdwxQXfHpqX7ftKXCHjCAedmLvfVINZFD1hlVYHVPEhBiaZKeSqwB17fzQ/gvKJPoA8kVctPKuADOiw767Aqkm/1xeA5fDbHFr6RVe333Uo89igrpjwC7jBBoLSSTxKafwgRWL9pKAiueTXP/QJumOlJDLbFmSbr6fAXiw7P2unQGGc/OkLAUwi44a9VGb2uK/65PkledEFbveonDQVBH6ZsLsUFOP8eq96eZMAtOnx2pMP5QR36HWcEPCR5WbDgHPgIC3BrIYTWqp0RKfnB2Am8BrC6QR+648dk9jvxuEtApZ2k+ZBxAe5Vh2kH3NK9J6QK8C2eXiJ3+SosiPtltT0dJeAKY5rdM2An4LsABLA8gBPkvXwDntCy3bgA5zq0pMiG6TDNgAPQDYl+3rVS6JP5hsZSBXhTKnxi5I55hNsBsPCi6HTbbpC67ajkJwQ8+YB71WGaAYdYQ0FqfgL/VgF0mzhCevxsMD2J+KPdxA210EVc+d5a/621y2+JMDlY9VZc5rJp4l2Jyy3IBuA9HZb7XTonHWbFB4ffof+GtfQCzldgdjdKwCECbdQu12rtt8CvqdfLr8ps6jYhuV+9HhzhB3DSK7ZoV/If8ujuuZU/ezkNZdIBH6lD07hAfGd1nl59GYCTjqfRTNd/3++Kxcqf7LIEqfkh6415UTdHPo3YB+eFEGuKtAiBmaNmhRz7W62lnfFldXgE3EjxWE9EsVudEXBnHTZtdAh57v5xIv/sH2dxL235b3NK0M7aS0eksEOmFEn61AmUpPqdPcA7cUa0e4B35hvr4iQCPqkCHYUCle5Z51oq4G4UZ1pGD7VdLjqxgH88u3zixFlocHg9Rlfm9bMnTuzPfqwtJ2ksIAjkNZ6A4l7a7dqbcznpZqoAL5d1H6lIr3aLTMhxVS1PS5JaRsDTB3hSC3DSLOYc+LpcuBhLkG3wNA076dYGH51a0rHv5KnOkW/dnJCaVMA3/zj7Bx3wx2MA/HFUgPvtvkLAfTIyZLz7TrAV7TM84U+wRvmdPCUPu8X+TgL9uo13KkXeybMXtJMn4T74zhh88EiCbEG6rxBw74wst+qzjuPtsmOwz9cdvqKYxKHxolt5JtzrrSrP9QXnN9O52C+IYPxf3DvPyF/NnTwQpYZAgZ9IJgIeD+BBuq+GAT78soRsXZ7QHsUIsTAiBOt264vC2V8cMChO54rBqRm0qGzAhOaTm1W3jybdNNmlkvr5gNJD6ORBwOMBPIjOhgHudFmC7dxzmS5M7O49ihFLpuHlePuzfiIOqnQnQe/l71u7qsKqIkLA3QEexk6ZtS63IIGtoNaGDSO7YTMS3eSGQwZ7L2XUAY8bcPeK8S9+AffzfK+Ah7FTJhXwqHUZhbXRKy81M7IfGeBh+OCWgbhimORdAOmP1kk3BsCfRS0BAPf8/HGa6GECHsauGLUuo5A4GAnVBzcH24omk9x4eaO6yujZRRM9vT54mIBPkg/eF2wzmeRRMRKFqSSs80YJemBMMsP8gIIVKPxfEef4AfT8OCMEPFWA+9UZpsn6GeEn7RL6dICR+S4jcu9sgsQB3m3rJAdEYI8N4LoH8/ODGyBX/wJa9ngnT7HXycMr1fwfVYuAxwN4kO4rBHyAkaeEDDIiWBkxxvuD1S/8dLuF//IcNrJnNeMhPcC7pKj4g9xqnQuzkwebTQYAD73ZJGj3FQLeP1/13fqR1d3oY0RV82EwkpkBw1r0o7HAWnSUTAK+63l1G6gJhvrf8C7wQ8AtvvsYAY9a1wh4pJP6/Luvkdf+d17T3vWi8O4tlKaaYJp/WKw0LoaleAT8pXzGTm4Teukf44I7al0j4AkDnN9CKQhPTD7O0RnoYd3vhYAnA/Chuq6sfxjk5hoEPIGAGwX/Rk0wf0ardgaCGl3Fh3PNLAI+fsB7ur47VNc2xywj4CkG3HzcrFnMN4ya00xu+3fDADzobZcIuAdd924iTeJdbgh4QBPdSfixu0L/PdKue9wtu75XwE391r575BHwYLpGwDMMuHHVr/ns7C0P/bsDgKvvvX+S0J+XNO1tVxPO0m/Ne+Q99v9q2tLblJz8+T1Vex8BH2G6W3SNgCcb8P8GBRyOZqaE/atv9/bY427ZTT3n54P2yCe1JkAH/IYO+N+T8j52ukbAEwt48EkN3XQym9k2Tm91hG9E/+6gn+etbDRo7TGfuBN8+aDbOIfM6I1hukbAMwQ4BLP4BBzRJeemx90qXu/rDgo4tCAm8a7rpAAOum640DUCnhHAIUIO4NpFWu3gGdW/a7/ru6/gCgp4NU++93Mf+SQA7kXXCHgGAIfVfL2xJDKmbFgngvVgAbc97gPAwtUydGGn3t46FTXgcPh9HhYdj3eoTQLgXnSNgGcA8MNe3zoVK19bc9pQ1siYdN3wY4f2uFv6d+38cGgDdGumB+mR59+li3eT5n+PG/DDXv+1k67n5qSvkjhmCLhPwEHhqxWm8L51h7y2OTrutsfd8f1q7MIUk2+PKokM0m8Nz4YbNJMaOBoX4FzXZVZ1q2sEPAOAN5VCZfjRQP03qLjqcTfdXGm/i9M7ujnPhn0mSP8vr7emi3eSuhONC3BVLnw0UtdYyZbdIFts7wj5dN0XL9fbp8N+NlTALVD6lySnffB2UQQ804CDVBm9PlNstMJsUeymfWZalFWvJ/m3I+AIeOYBBzMcIC/IWih9yDx/LxdkgDvpQSIEHAHPPOB9kCvqSpD0DKTlVKWwkga4EXAEfGIAP5rwirRYKsmX/cAJi4RcKl2WlOZian4vAo6ABwM8v7ukqtShX1tAZccjTreUQPoPAUfAAwBO9tykvFAi36nXsn4rCQKOgoKCgKOgoEQr/wcs7tNGWTHDoQAAAABJRU5ErkJggg==)
Now, ![](data:image/png;base64,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)
If ![](data:image/png;base64,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)
So, when
then
<0
Then, by second derivative test, the volume is the maximum when ![](data:image/png;base64,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)
Now, when
. then h = ![](data:image/png;base64,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)
Therefore, the volume is the maximum when the height is the twice the radius or height is equal to diameter.
Question 40.Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
Answer:Let r be the radius and h be the height of the cylinder.
Let V be the volume of the cylinder. Then
V = πr2h = 100(given)
⇒ h = ![](data:image/png;base64,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)
hen, the surface area (S) of the cylinder is given by:
S = 2πr2 + 2πrh
![](data:image/png;base64,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)
Now,
,
<0
If ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbsAAABHCAYAAACeTunIAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAABNoSURBVHja7V3NjxvHse9ZC77EX+/0bGubQhInlm9e9hK5BUawHo3WKyd+AEORAwJxghALYkFHphIiWCy4zkmQbeRqx0D0EcB29vIu7+CVrMu7SLLvSayPHHKxtdI/EOtjM9Uzwx0Oe4YzwxlySP4M/GDtbpPsqSr2r6u6uoq98847DAAAAABmGRACAAAAALIDAAAAAJAdAAAAAOQAO4wtsJ2dBZAdAAAAMLNEV11iF7i+0WaMaSA7AAAAYOawoR/5jaaxBxbZnQbZAQAAADOLlsHfhmcHAAAAgOwAAAAAAGQHAAAAACA7AAAAAADZAQAAAEAmoKsHzRV+muvrv2Xl8mMgOwAAAGDmQFcPLI/uWwv3+Urzbb93ByEBAAAAMw8IAQAAAADZAQAAAHNOFIxpjKlrTmYFWedSkWgCsgMAAAAyIZ2WaayurlbfbJw588Q4PvPMmcYTVV1/0zBbq6pkEx8RHwpCIrLbtxi2zNhjQRWlAQAAgNlCp1F+qib4u6XqRiVNLysqWtXSz4+I6nvlTufpELL7NyWlBGAhFtnt7LCFZkU/scTZl3SHwe/eShLssemOdD13PB8CzFe4o98eyta/97Vo42ksS2UsML+66X/GHjTYU3yiqwr+nqh3X4sim6zkvm2WVonwhnl4Q+0/yqANg5/SGHtgTWrfS3Y0yXbjjaNLXLvuMmlh2fjwjUb76Mlicbve7T4Ho5mvxdS2B/Yl2YoEX/pSrzRPqBZVihasV/Q1wbVr0n64uLbS6CyPOhYI1c0Xs6yb7U79+SJjt3vPKMHvGq3u0sCcy3LOV905v9Jol9g+7MmVDxEdNwZT+JXEaMv9ll/uK83NYrjci7bch0QMN3R+6ojePD1KZDG6EZlizU923eaK4Ey7x0TtnBvL3TBLFc7YHmPiNshuvrBJ9qBpe54QwgNrk/SI7GapuvW6f/xWdel1pvE9w9w4btuOcZxrbC/uWOzIR9RNZfOnfhkGyjvO2PLg2KxhLYrtgVBWsXZuYM5lmnPhjlFprsrXVYzjBY3dmcSc84huVZzg3Pik0ek8GWV805b7t3HkTn9rVVeNKHKnMzyD84/Dvu9OQkuwRzkK2dUEu6Aite26eE1j4hbIbo5CHp3Gk8e4donr5pa78em26y8YnO3ai6plJ53O8/4dODdab3vfh8r9MFa8Xe50DycZC6Spm2afvJuWvLVA3Qwfm/kC3SkfXiq8+lG50Xk6zKt05nzL77XYESzYk20v/FO/NxxB7s+Eyd31/rjebCvkfqvc6iyGbtjaxssFfuxT63OeUkUumnI9YN9qjO+tNDaXI5OdP67qJzvJtIvsc8tVvWe5qsLPwscK4k8gu/nBtmWIJWPjl+odGbtMduL98lg78NOq8FK3ZSxxxu5RA8aIY+96xwLZ6Ybexy/vOGOzBK1XJ5fYRblG8eJndnhWfcYTZc7z7N2RV0fRuigyoDHUHdz6/yNL7rsr5fXXmWEcCvS6NYuIfKFNR+57Fgn+dthn1pbZn1Xe3aZj4/R7Is8FYZ6PXEGF4vvCPYvj4qppiobCszsnY7OL+pVyo9HH6qYQWyA7oBcB0A48/d5GadH43J/K3O3WnxMU+xe1C5HHFmsXIecMdOMLYXnlTQtJnLHjIPQCY9844dmH7pnkWqvDvZ+fpznnEdJR4dplYW6tRZHBpi33r4knenJfDJA7bayC5X7Tlnv4mWmXoob82GWVd9ezBcsxs3jqVCSyI7e0yDTLzbfDHhTyEJxd95OdY2B3nMPIe6KybiKrCRgIiSxqn9NO0Wfct1UE5lmMrkjbizEW8h6DbuwF6wotWHHGjuuZWq3W90299BYRnx2iXfqnNzzWIzTFojupOefKq6NoiSauecPaSeVOhBdP7vpQuW9vN54VGr/mjya6XmajUX56ZaXy60iFoOkFtSK7yLhx2WvA9vlIP9l5Cc+7m6JsTNzHA+TOr7lS5Mz68ni8fDdU6XpvSgJzzoLjjIW8U9PN3UB5y7CnfcYXZ+wkPJTemeTSgafmnbN/Uz7pOeeC7GKEMEPk/pmU+8sn/+JWXenJXeE1x5V7WCiTvEziIjskuh/u2XnOTNp+11BFdu5k6eKhN8NLtllAWvhcg7Kj3Hs6KhsbQmC3PGR3N8pYyDw13dydBd1Ib0JjN7yLqHsuF0h27pznlOyqgp0X5vbaKNE52/siuVOyT+ewR+57Q8guktwplGmfyan5ha4paOSs+bxEZbxT6cEpfu+/OE7hTmureNX18lQp5MB8gO7T0IE0NzZOKRchz7mc4m+90FjYuVxYGC2vBBOWGj1O3VDmWqhuFItSXyjKp5thYyf1rHam30EySuQ5z2EYk0jneIwszChyd5NRws7l4spdEic//mnQWMr6XC4YH8Qhu9PDyM4UoqvatZGXJz28KVmEgHRBiwiFQ1QZgH07OcVZW4/ArJ1b3LF5l4lzF+m+TI1udkVedXOcs0uqcyuVbqKMnSSx296nuO56DBHmbG3CqufnMe/Afn5xKQ2vNljueqDcFyLKXTVP70Zys1o6ESlBxb4jZxGVbyetIkGL1N5XfWmnbccNpOs1bDbXikKYW/5F1ls1PeiOnH2OpN0T9e3XIo696x2bV3jT/+XzTICgo+om6I5cT97m1lqSsRNZwOkMihu73gU26D5dXuY8C2Qn+UIt94H7dI7c96JmgDrz3HVDpPK9PY1bVZfZlWTnxLlverOY3N2gTEApnrzo3l+Racvy5/5kFDubk3Z1B1lewHwQHVXT4MXKH/018uRVFmG8625+ZChCY3t85eBujQyvrXDrZ3HTGzGIM3Ya4KZG50k3y8vGWbVu9vvlrYkbwbrpG3vaPzZL2Dv7nb6WMO65pOo+HVXt4Cvr6jnPa3IKeWMh4cE4cqfonl/um67cf7L+OzeB0ZW7FkPubrjVf19vaFRD9UunFuZD9/5cp1l+0VgufCjJTtMeSsa2vhimYOfdepj1eucwfXEajfIzVbFwgbHCHaO9/TJIYH6Ibr0sTJl40lcf7wD+CijyIJkumco6iuzQelXUKLPXnzQRd2xe4aZG63rl1+PMVnZ0UwvXTVOhm8Idn7y/CdbNwdhmtVSVY6vj041pkVpBvPrRG432S/Y61LCr9ZsbFeUa58xZ1mX0zrkyn15dUrIjGReW+uVuLhfOBnVJIA8sSO5RQ8epkp1NeKVfOpc0LTLTP6CalxaBfVNaNd9yd4DuxfFTpnGcElN6xaAL4v/q7e4Lk1hwp6mKub0rGq2Sd16wUSX7CGyzcZ9sR7X5ode5dsZ48Wq11f1B0GdQu4+oY3Pp0TkXn+3U6PFVfSG5DdWNIikhjrwnrZu+z6c1qKh/MGwO025PeSA7R4ZfJ5C7/ZrF4tVyc/OHceaZOtlNo2dBbYjcyutazqvis4MwHO6IzRlk5MR3jxUA8kF2xl/znolqV3nhf51bsjuovH5KVl4/FVJBPw+QVehlWAlkN29wUqM/BNkBuYo8/OLozx5//KWb5U7n2VyTcrf+3y89/viNo/Wt/4kTvZsNJQ2pil/vbKd24LwvM9dGCz1SAg/VHe1VnAfZ5S5KkHZ4WZUaDVnDnkB2ILtY6FUx950JqSrojyxou33RzaRtQOymiEfep7qjQaWuBjtKq2CfSTqL6GOUEWV/qey/YZEZUceapeMhLUcCCE2pC0qI8Pb5gpznyGMyS6uaVryRpj1lgS2b7G5MD9n9fr7IzinxcyVKBf2Bh+8ZUDysV4TJ+bEdu29WvCQY2RRR1N4Nq+vYbZZf7Ov2rYS4Xe40X6ysiDbn7AYReosa52psD4WRU9AxZS/G0HGnUX7K0cVXA7qY06LCsCelPT2TxJ4oQZDS9rO0pyQJKpNa86cyQWVnp6+EUjgU6dqjVMX3tAW5nwSUBCOLXkckPApfLvPlj4ncgsjOabR52e0mbb31w97nybCn+3PxxiuC/a8mx7D9gjj2J8qAtT0I8VVQaNRXsmoIpr+Y97h0XCuyP3t1QRlpw3QRcWc/07qatmfc9rS0SWZPxWtOmv5we9J69vRRnz0F3EcbVZYgu6xDkIYnvDMEylp+E6yKT+GuAiv8XWZ9Drk3ZV9wPfK+G2oNmhu9p9tRWj6bqL3vHe8/l3Qr20StyNEXThsmb5wr9XQs7wUN0XFcXaSqq5X+e3LTgnl4Rn9Is6AV/hbDnh6pilaHyPLbqLIcqA0Kssv5YjSc7K5kWXndPt9hDxdfrbzT6Jx5Mij00dSP9BVFjkLE0ti9NSKt8VSRXLXAqrpRAKnq+EGYjqELIA7hkT0dfvXnfxjWhFR2AB9T5/S5IrskseioyExBMSroZ/H5B2WUtIdB1Txk7TdfmSaqNKNLIqYmhy2uOniW53vOwuk+Sx7JLku7yYONyQ4OspRUsI7zSnazrpssnzFLe7JLZGkPwupw5p3sJiX3tMjuflYIDO+NeGY3yar4br3BAiv8raQfVJaJGqpxzuAeuWdwfg+PskzdsCU9S5HxLwbq/MVcYLM4I8nSbtKwsXHoOAuyS0NXUcNaY4A2hc+oZWFPtIm17an6mwj2FJnsxn1mZ/3374zkvjCTYcxRz+wIQffpVBX0/eHFUXYgv6fCupz/f5IaoMPCmJSoQp6f68m5ZOcq2O0lGHeBnbczu9R0HKHHV9pkhzO7/D3jBOwpMtnhzC7nZJdKDNytvK77quLr4VXxk2bq2Vl3lkfGi1eTngWGkR1zvAnG+D1/00mq1G4X27Z7CbrdKGbhnMhb1zRN2xiXjnFmNwfnbQmzMfvsKWJ1f5zZgewCPUTZFDPjqvjy/KZaqnJufDLKOWDoPTs36cZD1L3mh/KOnUYZWueIHNZlfU1rgZXtSoxD06q/M/LKhXy++6qeV+NEEh3Pki6AzOzp46hE4rGnR7IlzhjsCWQ3TYRn9lfQz6LzgkxFp64OI2Z3OmeNu6qzOseL+8ofRrTT4K3nowrt7fbL9uu9O8jijWktPUZdNtxwcE2wc5Ns/BtXxwe6nA1dACl7g3TdIL49fTZgTxn32UMhaAAYN9lIz7b4BcgCAMZIyqiN2e+Oy3OVnemvrAHk2pgtz1bsguwAAGQ3drKjKwLUL45qNuIAHsgSlEVbMjZ+BVmMH7PUUBgA2SUiOzsBxK7VBrIDslxsTcP4FQpZjx99DYUzPh8CQHa5JTspDKRWAxkTHZVVczNo89plfoY96oOGwiA7kB3IDmQHpA831ZrrlLbPDrVa5UXDaJUhmzEtIHZD4Wu9hsIgO5DdPJFdfxPR8mNBZOeU/JKJK2ggCoR5bkHNKKmlSX+qdeGbJJVpgPiwGwrz/obCPrKL2lA4b882rfOeMNk9l2eZO2R3M1WyazfeOGrt9q7b99bEVdMUDT/ZySaDOhqIAuEIbW4KO5nsTlnVUNhHdh27ofAXwxoK580jnNZ5T8QOWj/68ROM/8vY3PxenmXe7a59l7Pv/Ku0sfVKLBIOC2sUmXaLdnuyt1q7/oLg7Lqf7GrL/U0rZQNRWe8Sl2qBA2TR3BRIYYELaijsWYjiNBTOky6deV+atnlPzBbsKz+XRiH+WDJPTHbO1aSYrw90Q63F6SLjxmXvrtsuuDwYxky7aSUwo2ES2EmuYDcU5u8pGwp7FpLQhsKcXc7rGT5VLvHPm9Y277zHUXNyWpC0DJdS5p3Ok2qZN0eWuSzwn6CsWYA7a9dmDCK1qL8HANhJPjG0oXDArjlKQ+Hc2p41797COyXzHnsERrBzS9Wt19PYBGxbm6Q+mVtkJ6rdE6O+94bO2wuidi7u+8RalEB2AMhuNhDeULh4224oPFh8mHToayh8a1rIzvXkpmneY5eTRVAWkVxIhewGZX4zDbKrLrHz1PQ2bbI7DbIDQHazh7QaCgvGr0fpzZaD55XzPlh4+fVRwnUz+x2lFkb82KflRuepFGTeTlvmnU79+SLj15K8TyC7S6Mv1i6OQoIAADuZHgwLY8rkA0/4L6ihcN4QMm+R53lPzAY42x01lGknqrDLJHMf2Qn6OanMKXNY48Zuks4M6jekiWnsJmNL/3T7ijGnnbxMMCievOi9JzFLDUSB7AA7mV6y8zQUvjusoXCenknOuxww75+s/y6v857sprS0uiCqF5IW/PfK3N0IuWFMknm9vvZfSWTO5H3QhUQhzECyky6oXQvzIVvUr5QbjWfo7oSxXPhQkp2mPWTErmfOPCErX+hO00pZ/QIXNIFBoLnpdJMdhbc4Y3dVDYU1T0Ph3C3cwfPe7YVsEyQ7zLodrHL+sdMIW0so8z0pc8eOXI/xQObV83Hf28nC/CRpv73QP1JDTbcZamFZ/4Cao1Jli9Kq+RYRHZpWAnFCI7CTadHT4N3HXkPhlebbfb/3NhRudX+Qu2hCwLxlQ9UczzsP3p1zdvd07NduN561ZP4PkrmX0ByZf80Wi1fLzc0fxnlPKkphcP5JUq9uKNkBAAAA8wmzWDh7RG+ennT/0n15TYa3qcrPKB44lAoAAAAoPX1zuXC2RNcFJkR4kujM0kkiuqThS5AdAAAAMJzwLA+vZLZO7oyZ8Chjk47OCsvm2VGJDmQHAAAADEWrumqsrlbfHFfRdiLZqq6/aZgbx9NKHoIiAQAAgJkHhAAAAACA7AAAAABg2vEfNsZSeAK6FuQAAAAASUVORK5CYII=)
So, when
then
> 0
Then, by second derivative test, the surface area is the minimum when ![](data:image/png;base64,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)
Now, when
then h =
cm.
Therefore, the dimensions of the can which has the minimum surface area are
and h
cm.
Question 41.A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
Answer:Let a piece of length l be cut from the given wire to make a square.
Then, the other piece of wire to be made into a circle is of length (28 – l)m.
Now, side of square = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAA1CAYAAACjpdDnAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAADwSURBVFjD7daxDYJAFAbgfwAXIPHZOQA+ZkAKFxBCa0VoGOCgo2EBOzuHcAI3sGADdlDPi4GCRLy7QsMVr+RLjvvfu4eqqmCr4DC7mMj36y2vjgS6RbnwtTEhUo+BFsAdoM4Ik1XXh0W0xMUKJitmnBzmsHliw/7kpNzpN3oe+QR0qtFVUZgVbtI67KewYUCnlrsAh9nEspAKgNtUCM8I66euISbfgQ3hqhJviMVMDUWJUHuaAVYmvCOOm37p08Tk8QIKzvJjY0we7/3wGmEyBsM3Uht7xeD5n8YX5S8xFc5PQ3EcnRRa4wuYIfb/w/EBfoPHhskjsk4AAAAASUVORK5CYII=)
Let r be the radius of the circle,
Then, 2πr = 28 – l
⇒ r = ![](data:image/png;base64,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)
Therefore, the required area (a) is given by
A = (side of the square)2 + πr2
= ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALQAAAA4CAYAAABUv9KRAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAVsSURBVHja7Z0xT+MwGIY9MTGyINVsSIitNexMJULMpysR0yEmFAkxnwJbpRN/gK3b8QtuONZb+g8YGG/jP9zVSZymbRzbadzY6Tu8EqI9miaPP7/+vs8+8vT0RCCoK8JNgAA0BAFoCOoq0OPx7W7QI2+EkH+EsI/rON7Hw4C8BToaspsgivv85ytGJqQXvN2Ox7t4IJD3liOOgj4lgymiNOQ80HE0OjxnBy+U0HcRkVfeE1/vD8jgN4CGnAaag8oI+Uh9Mv2UAj2L0GwY3bh4c6Jo1B8O2XdKyTsd3j0AmC2P0PPFnxzoMAi+ueifH0N2SSh95zDzQQmgATTJF30SoO+G9IGFj5cu36AEbAANoFVAc5gFJNyeDIfRFwANeQl0+jvurYXklgRAQ95YDh8EoN2RTkEOQANob6RTkAPQANpLyQpyABpA+wm0pCAHoDcA9GKRaVn+3psmxbNe5fen/N7LCnIAekNAo7xfH/Syey8ryAFoh4HmUSgIo4tOLOjC4KIOA2VAVxXkrPscMdW6XhFUAU3Y1WSTQPPPdbXHpX5wo8+mHCwDrSrIWV6Fkk+VF3J7Fb14/XW/gynQyWfT4NXHHvGqDsskj0zpq0mkLgKsU5CDT2txRV4mUTyoimQcmgEl0/zB0sFUZk343+ORsTg4Kbt6tjFYdDoskxnPYEOHzEPLBOAcA1r1wGUzR5mtyytrs783iuJDMRjE72xAreqwFNDrWg8A7STQZ8d7ZO/vWRwfa8Eg8evidRqEsYAxB7SkHJz6/1WwxKCwta5RJQJMtt1dH+1Mdo6uJ1sHtOlIdhFoFWiy3KssKqa53RKgDaNk00DLBhqA7hjQMgD1IVqM0Lk9mUX8YjQ09bFNAy2uS+d5AWiPgS6D0si3lliVPDMwWzhyH81hpvTkl/DUbQKtkwqtDXRV6bGJ1BWArgZ6bhvMgVbtnOdZDtsZDiOgRTZEY5ZAhPYUaJOHXAZsdZov6Pd6vT8iO2Ibam2gNQbvVgBtMpu4UKHUAlpMw4ZAqwZy8npWpFnIiJR8TlOztAponc3TTgCtezNMZCNC27jOquu2BbSqPC7K9sVBnRZaUl9ta0ZzAmh4aL8idJLCY2FcByxVvrsTQMNDO+KhNXxlkqmY+eBVyEeHjAU/xIAQQUpaQWwdaHjorc9yROHpV1npe3n2nA+SGVxZrwf/nDCgsa1TX3U6LJHl2AKgdSIbh7naDpaVudPut+L7Dtj5i408tG6HZQ60zTw01D7Q61QKvbofqBRuB9C2m4ZcEXo5vAdar33U9mLNFZl02znRPlrnGNp0sXJ6L077bPOhplN/c4sm437oDv83HTobGJwCus4xtPnKPdt50ebWo/nCph2gTR+4l3bDxx0rurulRa7UhQxFAl6+takdoPNB5emeQuVgXWNPofNAu3YqUbL3Lgjjul1vTQEtoFZVAn0TP5tu3V3fzgK9TneZrevl1beqAsfK5tRSlfw7nMvR6LkcTgItrMZpEN4vJP6zRvRNW40TevKTAycDevEoV3OgcRSYzkJ8/d6hVoAuNqdwoEWE5qM4XZBt9iFzqyE+TwZ0EsGzzamJx816KfKNqzjM0Qm1AnSV3chH6obSdstTmk5PRfLdsuvrelYCQOsAXbGnzKTrbG2rUYi0pkAXj6OyuYMa8ilClwC9zt66Jn1blSdePl9tQOgUPniLga6CtnjaTxvZD9WAWrYYABpAL7y+PFW33aCjAnq5sUbMNvx7JuV7xmKcA91hoKsWd3n/b5ZrzfO8LfZxVOehV8viK+m8jjcWbR3QpsfQ8urR/P30s+0zkSuBLkTjlUVi1ouy6Rw6tKEIDUEAGoIANATN9R+Pb0QPEHp4jwAAAABJRU5ErkJggg==)
Then, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, if ![](data:image/png;base64,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)
Then, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ (π + 4)l – 112 = 0
⇒ ![](data:image/png;base64,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)
So, when ![](data:image/png;base64,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)
Then, ![](data:image/png;base64,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)
Then, by second derivative test, the area (A) is the minimum when
.
Therefore, the combined area is the minimum when the length of the wire in making the square is
cm while the length of the wire in making the circle is
28 -
=
cm.
Question 42.Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is
of the volume of the sphere.
Answer:Let r and h be the radius and height of the cone respectively inscribed in a sphere of radius R.
Let V be the volume of cone.
Then, V= ![](data:image/png;base64,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)
And height of cone h = R + ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
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)
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)
Now, if, ![](data:image/png;base64,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)
After solving this we get, ![](data:image/png;base64,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)
So, when,
, then
< 0
Then, by second derivative test, the volume of the cone is the maximum when ![](data:image/png;base64,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)
So, when,
, h = R +![](data:image/png;base64,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)
Therefore, V = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the volume of the largest cone that can be inscribed in the sphere is
the volume of the sphere.
Question 43.Show that the right circular cone of least curved surface and given volume has an altitude equal to √2 time the radius of the base.
Answer:Let r and h be the radius and height of the cone respectively.
Then, the volume (V) of the cone is given by:
V= ![](data:image/png;base64,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)
The surface area (S) of the cone
S = πrl
= ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Then, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, if ![](data:image/png;base64,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)
So, when
then
> 0
Then, by second derivative test, the surface area of the cone is the least when
.
So when
then h = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAX8AAABMCAYAAACF3bfQAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAABKUSURBVHja7V3PcxNHFu4xCacEcksluOVLspgbmpbuqS2VPDaYrA+z9ljlygK1Uy6VSrsgvDq4KDk3Z4HkmA1bFSAXFh+SSw4Bm+tCzH2rALPnDeYv4Jd3Xs+MNBrNSDPSjObXO3yVALI8r+frr1+/fv0e+fLLLwkCgUAgsgUcBAQCgUDxRyAQCASKPwKRQciEHCJbWxM4FggUfwQiI9giZGIpT36g5VoDxwOB4o9AZAS1Mr0gCOQNij8CxR+ByN4C0EDxR6D4IxAo/ggEij8CgeKPQKD4IxAo/ggEij8CgeKPQKD4IxAJA6R6rpboGi2vrmGuPwLFH4HIjtd/gRDyCqB5/xdwTBAo/ggEAoFA8UcgEIiBAkPIO36B44bijxjfBBUISW98G+L4YCO+60i49VbDgRlK8wh8Vyj+iHEIY70izc3NKWfVzc33xvm7eQE17u3Jh0b/jgNHwdjcVN9T5ubOSpX63KAFDr3R0MQ/UkE3Fn/zXeLCguKPaKryEYXRq8VKbXHcv3uzqb4/Q8k9EAaBiHtyvTk5xPMfnaHCNnwHW2md6vfZulL8I2XKVVltHukjVK7eKPJlJPEXIvz9QlWiF7X/vhQIfV5S1wv4XlD8UfjZ1LVBohkWalLxnNTYOAn/v8zIDTIp7fjZebSa8jFREJ4ScfmG559ZYaemtAUAUzjHJrzvRC3+Gw3pZFGqneMOACwCrHITvX8U/8ziQCO/5vFfo1I8UhlbdSlPibi70mp95NWbU/ITt2i5eslv2Kgm0QtT/OcOUADG4/UfxEVsNyrsNKbvovhnGi2FzVM6e3vcMX7X52mtfMQIu+tV/NerJZEK4i5j5C6EZCbySz94FXM4A5il9HZeuXzG7TNBnEUg4hHvN50FVZWPlkqLf8ZdX4LEH15cZzLqE9JKpu5/63zG/j1bW12HPrGe2INsHgVNHmvP3TFDLsM/T3CeM4i5tjU/79l7L9MGYeyeomyc5AuHQPaY0pr3EwrI0Zk7anPz/Z7FIYCziDTwJEDxF8bJJdf3Tcj/BAJ9GqprGPZJgPjDS2pW5eMiJQ8MD+K1QPO7C2pj2vTiJMq9v9fG9vKACMIbQsQndi8Stvvtz8FnqHQ3Lp6vH5tHxWUlfwbinn6ep6EuTOcpedQeY5p/VF6szgcxaSEToyJJ5/28C4jdatv3S+afK4zc9FuLR/uZG07e/6hnEWnhSQDP1xPvt3Bpt4dLB+GH4UADBCptq83m+yj+MRd/8AgLhZmvzMnXaqx8wih5qIn7s5Vm82Pryk4F8hzIRKVKyzW8wGPLwr72mctxnNB+bB7W65cmhR22snHKx/MwKgjPLZkvrwUjltsvdOJV+KvlqYZ56Ox1MdlYYaf0UI8uLNpO4BKV6hd9eYPadwiTMzv9hMDvWUQMePJACIAnAXr9XfH+vlyS1z8P2yNvNlc+LuSk7+I691H8ze235tWXi+V1+4vSJyR5wSobp61/D+lcQCSrR5i0l+/XZr/g3yOwX72KmR4iErZpubNYgsjAbkuftGxooYGtf7uAmuYl1uvypCTVZU928DMC+rCkNguqKn8gUXrbTxir/R0C/bVUXRf7/x7vZxFp4UmA4t+O9xtcuhcGlwY5GGaIaV0pzuOBb0I8f8cJqXsP+/YURTP2SybL951iufwzCpv34/XG3Wbf36PZ7yfkY02Ts4uPNmm3CaEvpHorb93WO5/B9J61LIvk++5c+tz/rAJuTNpDcDHrgN9A7j6ngbRN82eHHRcI/fQ7K/B7FhGD3QD3rKNK33UQfyFsLsHn+nHFKNT3knNFXL6B8f4Eir9JBoVN3CLMObebx2hdwhHc88ix63Hcwo9isx9o43MrKK8QvosIbM86nq2qfLzrbMARmoc34B3AHYTFEmtQSh7DTq5eKS7ysN6kdD/IXRuEfrRxveXmMVYk6VwSQgRB8ySA5/GV3+/EpabOpd1BXFpoqNOuXMHYfvLFH4itqgsFLixi+Ue3W5o89k/IvtOL5+mBrHI1KS/Gq81+QgUSpXes3tWwMM8OrEJjhojMWK4279+Ynrm+rTf/3HsY3yMG2q5AMA7nc2zmulJvfap7cOxxkIs3D5NQ6Y5d4Ic9i4iYJ7tB8CRAr99Tfj/njcEle4jIC5c+Y+QnQWhz5Z9hcSWCxTP0QnixHwhLRs8rUxCouPiNE8n1HG5I06MvStUW6/Yu6LWkhHz82OxZ6ForH4lE3A5iQvCFlLCHXV6/5kWb8Vwuqmz5WtuWSbLj+0C2wk5zAfERpgpiTEY5i4gFT06OxhM/4jRA/D3l9ztySenlEnyXGSKyZ3YZXHkLu7g0hHUsi6dXDGVzYgYEXnxN29LlCPmNk7xcXXPbynPvQFxuZ4NsJPSUv8fmkrPN4xR/8IoHlYXgk9EQbVP8/YabTPEPs52icaB7z3rQ6HgWEcBuKWyeQO0iK0/CEkFLSAdE590+4iV44xK96oVLbfF34JIp/v2SPRIo/qFXQU3cwBhpnftusWPj4Pep7v3rmRzDpAImyeZRQhx+AIdpvL/tgLIQ/AaxIdq6wJJncRT/IENhMeLJ81F44kGYXhqCc+C0AHiN9x8YBdcGcckoyWDl0l6axd8Yv5fj2MEkcoDgUo9G8D03gtf1Kn58d8Dj0zDBfaYCJs3msMUfyAii7pSxYYd1sdV3HHTXr8Ci+EfDE5+LQE/YwUu8fxQuMdKbnpsy8R9bIbxEDhAQgsBNPRch63jK+f8qSklJ0kHvsDaHKf7gpa1XT4uMVS7bJ7G9Rj4stuVJct/0zkzxNycsbPVR/OPLkyEXgHcdQhZCEFyasYR5LFxiVi6h+Acg/p36N9FnN3TyfLvrlUAaIL94NGC7aKZ9aj/6NikHvQNtHuGCyrDiDwMIJSHgwNmeZQBX9RmTrpjfSYzPWnO2za06La2uwYUshbGWF6+U30lA8R87T0ZdAPqJF+eS7M6lQkH6exeXZM6l/R4u/X71b1YuofiPKP48t7rMGnG5IdiqK5/Ci4baHwuqWgCCqKp6tMLoVcqWrwz0HM20TyH8LXC4NsuebQ5a/GGyri6yCnDCLdPAepZi3jAlhD01x7yTgaUvxF7y0M2MG/3wErJupFA6aSVV/DlPhDZPiiZPlAB4MuoC4CZenEsyW+7PperFnvnrwCUj3VPjknJT0mw3uPIWFoWwuDKmceyJ94dZCK/jKRfI94JAIH/2IA7iz6+wF3L/sJ5q53LsZ6lyYdb7xCZ3SWH5+6S8/FFt9iD+W37EHzKN+mcZdN/KNTyzx3bPE9JAoaoioeIDyMP29N66fs/guwEjiP9WvxIPceXJDMt9GwZPRvRYXVMPIRvJcljsANqVVdXmUql60bYjtHCpnh8XV6Lw+t0K4ZXk1TNBlKa2xwsbcRF/RLDY+NP0Hw4fPgFe1Ic4Hm2B+fDE4cNPp79oLeB4BOb9YwP2gMQfynXkBOG3sArhofhnRegCSPVMG9J44BuDBQCFPwDx71MI7xd9ARCfyc3msVDEv1MgCYDdb1D8UfwRiBCFvyvev24UwrMvppZCePvWcOWgQojexV9pzdcrxSXe/UYgb+B03q1SJgLFH8UfgQjO6x8EfpcDElmMm+n2Qoi8KoBAfhtU3M5R/HMF6Tvz8Mi8MOWlTV5vq8Q+SEgvzbTYhOIfrvhb68YPBu6kEzH3x/hOvYp/u4CipRAeL4ToVtyuT48E17BPl2gQ8sJLDq3RKtFTPYpBefpxQVpsQvEPV/yNuvHeeIINRRKBrl4Ag95pqXpx6Bo7Pkpg80J4AntgF/Vhitt5Fn+3uucIDPug+CMQA8XdtRKq1yqo/QrhWS+6JVb8/dayDgppt8mP+Ef1vON+p3ET/zhzPy2ciGjuv3QrhOc15GMthOf02cjFP4j4uM9SpoEh7Tb5FP9XacA4xT+I+HCEYyV44MTLtPDCq/0BvdOXet0i50J4XsQf/g3E3Sn7JzbijzF/jPljzB9j/hjz77uA2gvhvTOoHhLE+aEQnv32r3XBCUL8L9kzezDmjzF/FP/soOPtYkZSSGGgrpLX/eL9nUJ4S18bdZN6C+EZqZz9xN/tnbb/p6nKR5dE8oNeSXF1DUIYEPJoLuY/5w87Wb4vq82jSR30TnGk/hcf0moTin92xd8PT8zdf5jVVDMu/p6qoFoK4e27FsIrVxtm4bd+xe2MdP2ehcEyCboLJEEIoyfkQaW7SRSPTZ4byytLvhIEcU+uNycTL1xWm8hgm1D8syn+m7yZkdnndzBPzPCVvaAaIpwFwE38vRbCM7T7l57idpZU0Fp56q/wXfbQVCYGuyYVz5nVJ3md/0lpJ+ki6NcmFP9sin+t7MCTPrc+EWMP/wzdgH3kZ8jaoOtnGOJuUsu+DmsTin92wz7dPGG/pon7KfD+X6H4j2sC8Drh7F6qxN+DTcPU8w8Dvelz0XWNS2o9/xF5cjeO3HduWpLuCqFRV0HNnPhD2lRRqp3Pmk2tL6YXDh8+sRdlPX89qWDqa96Mg8eg6XMmr1aiqolk1PPfy0o9/7hyn/B0Rvl3Yo78u123nuZ3F9TGNJaIRvEPxuvcIhMVSTqfptAHeNJebIpa/Hl/2Ulhx5o0UFOKi7xqbKm6FsUkz5L4Gzw5F0fuw6JUKMx8Za1bzyh5IPCa9a1jaX83KP5jEP5qeaph1sWIQ5P6ICa0V5uiFv8qpKI59FPW09Doi1K1xVD8w+eJfjkoPtznrUuL5XX7omTcL9r3Uk0YkTLx740BOkGPCxolGA5B+MCpkYEsk0OrZbrG7y9o/1avy5OSVJfjapN10ro1aDBze73aFKX4G/1YnzldFGw36o7gEmFcxd8X9wc08fDLkxjtBhgVhOea+J9Coc6Y+Leq8nFt6X/kdsFBB3u20qweXyzzRgZP4BJDvVJcpAJ5zhsZGN4ENKfv13g8djZp3rGlQYOjXVDD249NUfbw7XdLnB+6TpId6/sas/jHroevV54sNNRpaxOPLo4Y6ZyOPIlxdpO58Cls4pa1Zj0iI+K/aTYsMA5/tPf/xiQv718pmH8Wn3zGyE8CaTcyuL7SaH2iX04Tn8Qpq2GgTaRjEzw3b9Bgt8ts0DCEXVGmehqe/55TZ6G2+JPekNA4Qg5xS/Vs+uAJ536nicf1riYeCcxmA9FX1YWCtvDtErH8Y5IrCqD4DysWK+yU2biYixZbvmYVCirVu24fGnUtDgir3IztTsanTUHbFaX4g42z/DYyfVFSmwWrN9dqysdECAlFcPEujuI/kCe2sgsdjiS79pb1pqrp9NCTi99oC8ARFOqMhX26yG2InzEB7lurjlonQFLqkTjYtGO3KWi7wPsWibgdlUcIogaTGjI4FtTGCYg/q6p6pCLRVlQLd9RjEgRPksZ9L4sAlDbIEfIbnExHlQmG4h8HT0hh8yaxzYPDpE8ALzalTfwBtYo0K1LywIw9a17tFUViF7W5/ZatbIz9YC/uF/7g/dt4spd28W/bBYkAcH7Bz/Xc+9AiUiz+UGbaDInoAkZ37dv0pE0ALzYFbVcs49uqfKSs7eS0Cf40CgHm/VBjXPIijdz3gwojN/lZEIp/9sQfDknLljCPOQHM6/iQ5pa0CdD0aFMYdsFkikveNNjJy9AKwmunnqTjEte4nhPxg19LmCcN3B/q/VBpG+tRZUz8+ZVv3kuAvjC9HXPrCznLqip/sCQWNsBjhDBKEiYA2HRZyZ9xtKmk26Qw1jK94KDt0mPI0R4M2uqPvy4qtcWonmU5T245hdtixJP9QTxJuvh37jR01/LhN8KpsI1dzzIo/u3ccKETEuhkjfBeB28hD5gLSZkLid6EJsbNWto2kf42wd9bBPIAJry9QcNQ4t+QTubozB21uTn2kr56Gp/8wepi+TSjwsOo7lp0POuVj/Mu4baY8GTfiSeCnfsmRxyaeCQBvJSDQJ4Smn+0oKpFPRFAPqowehXOhPCwN4viz7e55Im9Ly5kjfDCYFR8oDQaJ+1NaOKW32+3SfPeHtu9mS6b6q1PnZrrBGGXKSDjDv102ZPL/adYrvwl6q0831XR2XtxDCl440k9nyTu9+PGDMt9a7Ujl2M/S5ULsyjQGRV/REjCognIBFNuZdmjAtvhBmlUZw0IBIo/IhJPa5bS26XqOsvqGPC6MXT2Nh4kIlD8EZnz/imduZPF25P6QSL9F3r9CBR/HIRMYpnRK1PlaiMNpa29Ak5KobQxHCQiBxAo/jgImQSEf2ABKCqt+SwsACD8NaWogPBjuAeBQPHHBQAWgEp9CerBp9VOuBBVrxSXUPgRCBR/hAVQc2duTjmbRmGEBU6ZmzsrVWqYOohAoPgjEAgEij8CgUAgUPwRCAQCkXb8H7oP0pGHSxy5AAAAAElFTkSuQmCC)
Therefore, for a given volume, the right circular cone of the least curved surface has an altitude equal to √2 times the radius of the base.
Question 44.Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan-1 √2.
Answer:Let Ɵ be the semi- vertical angle of the cone.
Let r, h and l be the radius, height and the slant height of the cone respectively.
It is given that slant height is constant.
Now, r = lsinƟ and h = lcosƟ
Then, the volume of the cone (V)
V = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
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)
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.
![](data:image/png;base64,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)
Now, if ![](data:image/png;base64,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)
sin3Ɵ = 2sinƟcos2Ɵ
⇒ tan2Ɵ = 2
⇒ tanƟ = √2
⇒ ![](data:image/png;base64,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)
Now, when
, then tan2Ɵ = 2 or sin2Ɵ = 2cos2Ɵ.
Then, we get
![](data:image/png;base64,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)
= -4πl3cos3Ɵ < 0 for Ɵ ϵ ![](data:image/png;base64,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)
Then, by second derivative test, the volume (V) is the maximum when ![](data:image/png;base64,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)
Therefore, the semi-vertical angle of the cone of the maximum volume and of given slant height is
.
Hence Proved.
Question 45.Show that semi-vertical angle of right circular cone of given surface area and maximum volume is
.
Answer:We know that total surface area of the cone = S = πr(l + r) …(1)
and Volume of the cone(V) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Then by (1), we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
P = V2
Now, differentiating P with respect to r, we get,
![](data:image/png;base64,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)
Now, if,
, then
S = 4πr2
Now again differentiating with respect to r, we get ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEcAAAA4CAYAAACvxOULAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAMNSURBVGje7ZtBTuMwFIbfATgASDE7DtC6vUKwEOsR1OpqKlbIEuIAaXfdzAXYcYBZsxi2s+EGLLgBd+jkJXWmdW3HaQqNHS8soSAh+cvz/7/n/MBisYC49CtCiHAinO7BWS7vTlgCrwCwAqAf0yw7i3DWS6R0xkQ2wJ8nFJ4hYa93y+VJhKOsTLABgeFbKNXjXiGcXQ0JvJH0/tEIJ5ueDWH4p1dw7lPyWGoKrKxw8sqhqZjtapFukc9zevl0K7IL7ytnzul1HRzO2E9VbwpII/iNMKQ24TPOSCYhyefBwsHqonx+rftdIdQaCFVFdlTEDwIHNymfo+6kqfjhAqcUcPjsavVYnOf2AgW4eLMkd6ApnenglBvf1hJ1o0HBwbdPAT4I4xmW+yYom+aYVu2x6mjzaN6MogNyI4eCI/j4pqwaWJm0qnNwZKmrEFzcyg4HVkmS/JVAir9FRi+MiytvrNwEoT2c7lp2hHNIOEAnzxGOwalUBzFB6xUcXeeKVn5Jz5+qXqZBR7s5PnTVlRo3gXhPI50FB0S0XtzgmPEHFzDmwdOfCgrixq7pktcvsvs3vazegSm1M6/edX+FoPCE6I58r8BUY5HiuKXG7o4wvYJTQjAPwCq0cKpi7agmR6wMQuO0VfuitClBQBkR8lIIq2VOqwBo4JjAee84OLy63ENXd0eaJvZ/27GtOx5CGd8QAu9NL+eDhlM2peSd0Mmvfe6bTbpiO3Lbvm78jPJ9yySmnI0fJBwh5qdN4QQvyLjBfY+VrZ+RR051ukAEWQya6E6nmsCvTGK4WrmpEZR32UcbH74jiVHXBO44nrzLzqGajucRmjZ/khhHGf58SWLsL4R73CXLypFJjKDguEZRbEuXxAimctp8hbAlMXoNpy6J0Vs4LkkML+HURVGquSdv6fFZ1UMEkCp1mmRtUZTNqpDzjqkdDwqOaxSlzddQL+E0iaK0cTAv4TRJW0Q4Ec4uBJcoSu/gNImi9A7O1ixVE0VpE6b0ugm0RVHEXJzuxkzC+Z+rYN5yhBPhdGf9AwQg3Fk60apKAAAAAElFTkSuQmCC)
Therefore, P is maximum when S = 4πr2
And V is maximum when S = 4πr2
⇒ πr(l + r) = 4πr2
⇒ l = 3r
SinƟ = ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIkAAAA7CAYAAABPJqm3AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAe8SURBVHja7Z3LTxtHGMC/hYhL8+gtSspnFClVyQ17bOVSpRdnWV5JOVhmvbLSXFbIsrYiBq0qRBZuVAnqtZGqJtADhUNPvSSESy953RM1QC655PUfNAmdGXtts+zau7YBL8zhuxiv1zvfz997BpibmwMhQmqJWAQhAhIhApKjpzAACWC1o9nPWQXoYJ8lIDlkwhRraMrg4KB6XZ+fP97o58zP68fVwcHrimYM+gFOLH5IxNRTJ1WCtxNaPt2qz8xriTQS9XZKN08KSA4FID0LJGsN+XFHq6vclXT6sRJWlgz1EHUBVr3fK5TQ5rJNlU4tyAIq+Qk/79X10bgsk5uI8BLlfMGXRVFwokfOTwJsSwKSEIqlkhHEgWU/MQizChCJPI9E4AW1JNt+IWExygDiMlGtEQFJ2NyMqZ/ox8iKUpjtC3LdrEaGg0DCrykofRHsX9HN+RMCkhDJjBq9AkS7F/S6RiBhohG4F1VnrghIQmRFlG7pIcnODu0XJLPUXUndykPdNE8ISMIQixhKFCXyJGtZZ/YLEsvKniESPlEMKyogCUnA2oiraQYS2+U4A1ihkDaVDIFFos0O7zck/FqSWRSQtIGkWLHLo4DFUlIFccVp9vcDEu7mUFmpTrmFwg5AWA9mLApLXkpksUEMYmuNxCNNQ+Jyb6G0A5C8jBOSBJ8EJELqgVLwhMTF5DcIyWTQa91cnVCYhzugi3zM77xFu0BSnDWBY2Y6epVBAn1jfxSfw//8SVtBwh6IB29FZRzzai4dTGaBCwD4odHAcedzVUuqcy8h4aV1gDf0Xv9VC73PROggYd1KU0/GCUqPiw8SeUvU8UytdvU+Q3KLfqc3QXsmlUWG+05FcYll7gaAZLWZwaJGpQTJajI3HTtQSHhPQsJ3ijYxwBelkD1PADZQzk3tpYkPTUxyrXe0q+vCJg0eT+/396KB6+kLXV2bvdes0ZZAss19YMWE+jKJZvZsDGALFePGjtdZ34Ca+GTOIkch5hlP4hTK41Nu1vNQQWJxxZKNlGl9FeAXNOnm73nfAGDTWe3b28A01ekcLOaTXWBPdlWsWmXaa1vaGU8Fd5EsBa4VL8z+0Ps9hWTjACHZcIXEjoyDyniaaIj9qyndPOXHVbByMwDZctYAuC/shofQrazvpS9mo4BKPHKHKwljj7I59btYLDNrr0FBH+29TPA3BPzXBpldk5ZJgU17sb4GG0aOITzitY5Y+he3GYymFNVkCtyCmMQ9cPWKjP2KhOTxqF7orQVKCYR1gOirYcPAath0PfWlzP+2GyDHr9kfwB5BcCYGS7arY9+HQnvXbqSxNaCu8CX9tXyutnaZOPzOgGBpZSSu3LFjKUPBG+w1r4muQwdJ0w9GXU8EIi+Suhn3UlDJpWxRjj67guZQzi4zrVTMdD1xmwm1XVo0PX3VTrn5a0Sb2WXRHN+DBZociKqmG1cmwIdGilZHEpJyjEJ/cd2X03NuJrgCSQ1300R9ws8CDCA8YPeQ07kRO8Zwd4n+IWl1HHWoIWHZTk7GKZCkT24TVeXg1BuS9b2EhCtbS6Spsj9Sq/UJEF+S9LgGqZ0ZWtggCRJDHigkDBBW/6Au53liUPvR7QFLhaa1GtnNFkhkcy9jErsuI8cjv5Zg+QyxsaVqixJCSHzHjk1B0mh2Y8tPFBBE/KdelTJHgz2mGGewN51LxkoLftfTCjQZkxTT3wo8vIiH8NipfOFuWpzdUIV/5IEoSyd9tLaLc5SwUf3rrbgpfNdIKTzIAsgkebN6W2MRTulDtXtkGZALJJPOTEbEJD5dTF5NqPSBloM8ELMIDC7er6FWKJeWR6ii3vnZpdb8AkhrbP+rruun+L258slLBjizpoaR6o4CvGJARNPmVea2WOFsjIPD2u/FKilzfeVua7e8zupEAhKvtDdC/m5kOIYFkBXLFXnTyo3QtRaA7cg3DOVbVgwrF9QK1nkva4qyNuNs2LEq6S7Xh8r9VilVNPiE1IfkMPVuhAhIhIQMkt2DXKnOoKMXApJDDAmDwcylvinHWqwGhNGnrJ8mIBGQlFP5eLz/ZzvYrdSAYltZ0zwrIDnikPD6T0KedmZDdh0nyE5AAckRC1ytXJKgJL33c4yWgCRskBgXL30B+HrYss41cr0dwKqkY7FWm8MdkuFzCF+8vmhYlwQk7QxJEzv4GBzs3DTqZ55BTP6r3smKfu4tlNKG0uiG8ertHMWeGmyz8cogoPBeltgwHg5xOyckCCyltsdbDkoyN+X3Wt7IdJyLIhTSpuJ2TkjgzygofSjB+1pzw07JRGHJmQ0JhbQrJDVOQwxqkSgkm34gMc3s2SjgU+e4hlBIG8clbB632Ul87j5QWfN1Diw7ggsHHjjfKxTSzllOlgx1EHWxXv+l0rPZ2ath+4X62fyMjw3jwE+e7lh0q6kIZbS9NcHlZG665tZXy1C/5gPmGH02quvx0j6mUxrB20gyt3xmNcTr5GmhjBBYE6SxSa00lpfli4Pd5SEoNgxmbyKrG4twi4N/elVmhSJCIOwojB45V9iLM1yK88U9hVoWRyghJG6HgZKgQWwrQSnPKFNAagW2QglhA0Uzxlr179UMLTFWDxABSQglrykDrfv3anlfMYtYeCECEiECEiH7IP8DbWsRS6V4zyIAAAAASUVORK5CYII=)
Therefore, semi-vertical angle of right circular cone of given surface area and maximum volume is
.
Hence Proved.
Question 46.The point on the curve x2 = 2y which is nearest to the point (0, 5) is
A. (2√2,4)
B. (2√2,0)
C. (0, 0)
D. (2, 2)
Answer:It is given that x2 = 2y
For each value of x, the position of the point will be ![](data:image/png;base64,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)
The distance d(x) between the points
and (0,5) is given by:
![](data:image/png;base64,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)
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)
![](data:image/png;base64,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)
d’(x) = 0
⇒ x3 – 8x = 0
⇒ x(x2-8) =0
⇒ x = 0, ![](data:image/png;base64,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)
And,
.
![](data:image/png;base64,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)
So, now when x = 0, then d’’(x) =
< 0
And when, x =
, d’’(x) >0
Then, by second derivative test, d(x) is minimum at ![](data:image/png;base64,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)
So, when ![](data:image/png;base64,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)
Therefore, the point on the curve x2 = 2y which is nearest to the point (0,5) is
.
Question 47.For all real values of x, the minimum value of
is
A. 0
B. 1
C. 3
D. ![](data:image/png;base64,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)
Answer:Let ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIEAAACbCAYAAABF7fitAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAfKSURBVHja7V09btw4FOYBcoANMEw3B/DIucJECFIvMhZSbZAqUJN6IbszEPgC6dIlZ9i028wNUvgGvsOuNYI08ngkPVLvkRT5FR+Q2MZwSH187+PP+6Rubm4UkDYwCABIAIAEAEjQ4Pb204t8pX4ppf5TKrv/UFUvQYLEUG6zj3lZXdT/vsrUd7XKf326vX0BEiSKqswvtNrsU4oGePCnJKg+vNyozT8gQeKRINuWH6EJgn5Iu/VGq30j4h6hN/u8KN9yfX6R53+50gPSfYmSBE2+Vg/doPWQFdfv5n7+563+wvE5IfQlShK0yzidF1U7U+uZxLW0qwmgt5+/tLpguy3/XGpfoiXBUK4+rvH1Q7vMM8VhWfhkNtp/lu++JCsMm4fYzJ56FmdK3fcfaj3Dn/7c7SDb9gUkMAytKrv63v9ZN7t7Gz/XRfauHuBdWa2X0heQgCywnm/u9LeCa6F1+L/WP0ONAGN9AQkmw6e+G1LTxxSQ3V9e6h+uVTdnX0ACgqIfQpMCnqYFjnbPLe3OYer7mfQFJDjzcCk7e/XfrVarf00eSKh9AQlOl1hZUZFyrM5/ltflHz6WXpx9AQlOZo3Oru6eD+ZunWX5127z5VEPXOrLH63IOu7QhXNfgNoXkKCHsnj9fmirtR/uux25k/Df5fJHfeB7mUjtC0hwMmjjIuwY6vs7gO1gPhdz/lKDSV9AAgAkAEACACQAQAIAJAA8kEBiPxxAJABSJAE1kgB+gUgAQBMA0ATRw6bCGgMXGWwqrDFwEYN6cRWDFTMJiBXWGKzIIwHl7qKjL7Nb+6i2jTbvF/lbys0oaoW1k5Dk8r5cCsr+8HdZ9nUszJtUWIt36s2r7FtqRlAulH17k/rc700rrEU75bOgIgVlXxPldLbbVFgLK1O9D7n2b+nKnqt2UawTh/KvxKzgXCv7VkPMrV+UFTgDJdZlubvYbrO/tVa/kS7mKXsO30XBfKYezj3gQ4TQ+ndNABxGzVf2zcHfvDoFuVQwYcDU/g1IME/ZU8baCwkahTrOTlsS+F5xSLZvo+y7GssZ7iaCGx7jx5jSJJDyLvLd/mA7M3SByNKmdQfxSYKWkNzeRb7bBwkswjG3d5Hv9m3H2y0J2hw1wUyXmoDTu8h3+8Mks08tQZPA5N7j1MDaeBf5bh8kYFbnXN5FvtsHCWzDMaN3ke/2l0GCwIQht3eR7/YXIQxN9wlMNzlMl2jc3kW+21/EEnGKBEM+/xKmjxLeRb7bXwQJjp0P10k8JgS5bdwP9aF7CceAYA+Q2hCFE0J5BHuU7Mu3P0UEe6mkC1O4XiaKoK+XtSkBF01diMKAL5p20QApQVQPcOgu+XAV+CtnFhsFavH96s03jnTr5suiDI1/ck2UoQVFgiZ3oSCVE9SC1KBIAIQNDAIAEgAgAQASACABABIAIAEAEgAgAQASABGTILZzCu5zguhJEOOJJfeJYdQk4DLOtHlfgHx0GzauBAl64LplY/O+ABc4Z1wJEpykAYn7jFx3+EL/LlGQQOpmM/V9Ac60AcPN4sWQoFb4G632Xb2e3uyHFL9kjQP1fQGzUxDR3FMqPQVIgPMFq0OlVmPGmXNBdRWdHcWI5p4c1UbBk6Ar5c6Lqh38w7p/RK1L1T2auIqykWGCBFJ9DS4KnAu/Y24cEhXQpq6irkjAUYG8aGHYPOynkWDSC8HCUNLGVdQZCRi8CBZLgiHxR7FqcWEoCRJ4XCNT/XqkDSWdk4B5J3MhqUDfja0MKDND0lDSFQk4nMoWSYKx7WATEvQHmjOccr1oHCQYGZixzRobEkgYSiISSC4Xs6IibSxR0oGgoSRIIDQgOru6e/4gd+v+fQGqUJI2lIQwZEZZvH4/tG18OkgU40wXhpKcJBjbCEpiiVgTYFxcPQ2DFBK4MJScv/ylmXsmvU8wvnxMxzgz+W3jqTCagnFmEgdIVrMjIePMJI6SbZCScWYyl0qsw2TkxpnJXS+zSQmxG2fioik1GkScEiTfzBpXuIzUOJPTuDJqEnSDhTK0tEnQ5E4UpCZPAmAGCbguRwALJgEAEvB8GDGSAH6BSABAEwDQBABIED9s7HYwcJHBxm4HAxcxqCePGKyYSUC023HESLwIizXkE88TqHY7TtiIV+LxijrqySLVbke8UxwGkxB1A/l+xODSxG5HtFOSt2Eg6oYNLk3tdoRFCV6YLSnqzhHFxm5HrBMp3ACWjgRToo7rBrKswBm4+Ek1b0wZVFHHUYsgmM/OG0yamDemChNRx1GVJJcKJmrmKPX4qRLARNRx1CeKdIRSKWxLAt8rDsn2bUQdR6Wy4IbH+AmWNAlsjCyX0P5gOzN0gcjShmKp4iISSBhZ+m4fJLAIx9xGlr7btx1vtyQgOoq51AScRpa+2x8mmX1qCZoEJvcepwbWxsjSd/sgAbM65zKy9N0+SGAbjhmNLH23vwwSBCYMuY0sfbe/CGFouk9guslhukTjNrL03f4iloiTbyMxMG+c+xAkjCx9t78IEhw7n4bBpG8EuW3cD/UpGEz6RrAHSCkZTPpGsEfJKRlM+kawl0q6MIXrZaII+npZmxJw0dSFKJxvcCkvWpASRPUAh+6SD1eRGkx6jwKMBpduvizK0PgnF6PBpaPchYJUTnAbXGJQAZAAAAkAkACo8T9UksR1toJvRQAAAABJRU5ErkJggg==)
Then, f’(x) = 0
⇒ x2 = 1
⇒ x = �1
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
And, ![](data:image/png;base64,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)
Also, f’’(-1) = -4 < 0
Then, by second derivative test, f is minimum at x = 1 and the minimum value is given by
![](data:image/png;base64,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)
Question 48.The maximum value of
is
A. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC8AAABHCAYAAACXgB6bAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAS6SURBVGje7Zqxb9tWEIePSuAlcdotSJCjEKBFnS3ik+Atk0FTsp1CgyBThBCkQIlAEAi4ssHBSClvKuKga7u0sZfEGjp1aZy9TbK3QONMWdqm/4FbuDzaamWKlN4jGVlCaeBgG9Yzv/fe3e/duyNsbW3BtNrUgqfw79IqFTgH3W5m6uC7XcisKrCLarM1dfBNDdckCf6eSnhvAiq2UvgUPoWfEqm8p+IGqvc2wrR+oqUSAA7JUGuupenB1MEfAUgA4TkGtx8DZACOpLHB0wMtQyuVSvpds9O5GPVBnY55US+V7mqGVRq1EO7X+SATgrfNyiWd4XbBaFYTC0SjUEWmb1dM+9IQ+MMg44Y/Bs8+ZHVnaeT2uW5F0uZ+H5rC9syps6Us0x/yfFbYbcjHay54mET5P2ua5byqss8R4ddhh4pfCrNqcz1ODASvjM5WEIuPeXycVhFk+WdZhl/clT/ihacYKCI+ZrqzEnZI9fm7xAXfsc3ZRZT3tFb7psgqtA22LALvjWlpN2Vc3DPtzqzfDSk1ID+XJPxjoeEwLvj7eu42MOOR6BZGgSczGDzK6fdv+ydV0Jqf0M+Whp+F8ZwOUnfVtWvSM1ZvL40Lvu26nXRNe2ba9mzY/+VKDxxLy6HEntcd58q44B2nfoVJ+FyznJzfdUyz8r6qVj/lSswoUKO4TBz4nuv4A9eLB4DfTu6x6yPhawx2mNFeHje8N5bVdoL+1iCfR20/SPlOSZeGuOffvnHAe+6K2l4QoG3Xr+Zl7euh8OR7Cij7Ufw9Nrzv2f0av6kXVkYG7CTB919EIF/7ZuQJO2zrBOHXo2ScUVw2Njx4uT6ct6u5jwkelNVdb8sFkq4zg+9Jmj995UnqEoPfbCwoLnw3zoUjzkXFhe8uNDaVaCt/Z648M3PjwA2ay+OGdwP28o2ZmYO5O075fwv/6gzhX0WHjymVCfj8+NUmhU/hOeHDikK8xaKzhj8UsdRtUvhJhE8TszQxGyM83aQqVNr+V9cr58IKoxMFT5B2o/KRgvDjiab/JWHuRdlszU08PAV5Pr/4RS/InVb9A4bwE4Dyum7bVycWnhRCLaibfnXyJBfgT5HK28QErNNYYChJb3naQcnBW/O3LgC+WXac61Gge4Grs8wOsNq3Yj6/fB3hwpt5y7kVDT5GxYygqS/l+stLUNTvhnX6eCpmwvBRj+jjcfCD14Jx1carminVL0UmcHK6ixe8+n8JqpOLTIJ6rDLA78dlv8YG79imiutR+gKD9caQOrlIBQ0leAvAXvO6QU2B3Sh9gcHSXUB3TtRoB134Ax54qr/nAF+Idh8H4I97o/A0quuccoOQbsZAsFIrCYtPo2Szg9teZ0sZpu+Myk/+y2lO5zLU9l9EaZ+n0ErjSFqjdB8D4U8600/c3JoNXTFL/5ABHADmXpZNM09JmWma7xkMt5HVHnCqDEMsPol6hwgGc1efpGuY3HnpQV7+qv+iLcvse81YK/I8mHaIpFnkJOaC9xSA4YOs2mgl+XJPz+hli4aabfHukDA8rSxNoOAGb5ITIPCmPq8TeNwr58iDx5uAYa0m9ZqWZRRWkwAfCd+zNUMrJveaFl9MJAY/qZbCn5X9A29l8OKkqrM5AAAAAElFTkSuQmCC)
B. ![](data:image/png;base64,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)
C. 1
D. 0
Answer:Let f(x) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, if f’(x) = 0
![](data:image/png;base64,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)
Then, we evaluate the value of f at critical point
and at the end points of the interval [0, 1].
![](data:image/png;base64,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)
Therefore, we can conclude that the maximum value of f in the interval [0, 1] is 1.
Find the maximum and minimum values, if any, of the following functions given by
f (x) = (2x – 1)2 + 3
Answer:
It is given that f (x) = (2x – 1)2 + 3
Now, we can see that (2x – 1)2 ≥ 0 for every x ϵ R
⇒ f (x) = (2x – 1)2 + 3 ≥ 3 for every x ϵ R
The minimum value of f is attained when 2x – 1 = 0
2x -1 = 0
Then, Minimum value of
Therefore, function f does not have a maximum value.
Question 2.
Find the maximum and minimum values, if any, of the following functions given by
f (x) = 9x2 + 12x + 2
Answer:
It is given that f (x) = 9x2 + 12x + 2 = (3x + 2)2 - 2
Now, we can see that (3x + 2)2 ≥ 0 for every x ϵ R
⇒ f (x) = (3x + 2)2 - 2 ≥ -2 for every x ϵ R
The minimum value of f is attained when 3x + 2 = 0
3x +2 =0
Then, Minimum value of
Therefore, function f does not have a maximum value.
Question 3.
Find the maximum and minimum values, if any, of the following functions given by
f(x) = – (x – 1)2 + 10
Answer:
It is given that f (x) = –(x – 1)2 + 10
Now, we can see that (x - 1)2 ≥ 0 for every x ϵ R
⇒ f (x) = –(x – 1)2 + 10 ≤ 10 for every x ϵ R
The minimum value of f is attained when x - 1 = 0
x - 1 = 0
⇒ x = 1
Then, Maximum value of f = f(1) = -(1-1)2 + 10 = 10
Therefore, function f does not have a minimum value.
Question 4.
Find the maximum and minimum values, if any, of the following functions given by
g(x) = x3 + 1
Answer:
It is given that g(x) = x3 + 1
Now,
x ∈ ℝ
⇒ -∞ ≤ x ≤ ∞
⇒ -∞ ≤ x3 ≤ ∞
⇒ -∞ ≤ x3 + 1 ≤ ∞
The function g neither has a maximum value nor a minimum value.
Question 5.
Find the maximum and minimum values, if any, of the following functions given by
f(x) = |x + 2| – 1
Answer:
It is given that f (x) = |x + 2| – 1
Now, we can see that |x + 2| ≥ 0 for every x ϵ R
⇒ f (x) = |x + 2| – 1 ≥ -1 for every x ϵ R
The minimum value of f is attained when |x + 2| = 0
|x + 2| =0
⇒ x = -2
Then, Minimum value of f = f(-2) = |-2 + 2| - 1 = -1
Therefore, function f does not have a maximum value.
Question 6.
Find the maximum and minimum values, if any, of the following functions given by
g(x) = –|x + 1| + 3
Answer:
It is given that g(x) = –|x + 1| + 3
Now, we can see that –|x + 1| ≤ 0 for every x ϵ R
⇒ g(x) = –|x + 1| + 3 ≤ 3 for every x ϵ R
The maximum value of f is attained when |x + 1| = 0
|x + 1| = 0
⇒ x = -1
Then, Maximum value of g = g(-1) = -|-1 + 1| + 3 = 3
Therefore, function f does not have a minimum value.
Question 7.
Find the maximum and minimum values, if any, of the following functions given by
h(x) = sin(2x) + 5
Answer:
It is given that h(x) = sin(2x) + 5
Now, we can see that -1 ≤ sin2x ≤ 1
⇒ -1 + 5 ≤ sin2x + 5 ≤ 1 + 5
⇒ 4 ≤ sin2x + 5 ≤ 6
Therefore, the maximum and minimum value of function h are 6 and 4 respectively.
Question 8.
Find the maximum and minimum values, if any, of the following functions given by
f (x) = |sin 4x + 3|
Answer:
It is given that f(x) = |sin 4x + 3|
Now, we can see that -1 ≤ sin4x ≤ 1
⇒ 2 ≤ sin 4x + 3 ≤ 4
⇒ 2 ≤ |sin 4x + 3| ≤ 4
Therefore, the maximum and minimum value of function h are 4 and 2 respectively.
Question 9.
Find the maximum and minimum values, if any, of the following functions given by
h(x) = x + 1, x ∈ (–1, 1)
Answer:
It is given that h(x) = x + 1, x ∈ (–1, 1)
Now, if a point x0 is closest to -1, then,
We find for all x0∈ (–1, 1)
Also, if a point x1 is closest to 1, then,
We find for all x1∈ (–1, 1)
Therefore, the function h (x) has neither maximum nor minimum value in (-1, 1).
Question 10.
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
f (x) = x2
Answer:
f(x) = x2
⇒ f’(x) = 2x
Now, f’(x) = 0
⇒ x = 0
⇒ x = 0 is the only critical point which could possibly be the point of local maxima or local minima of f.
⇒ f’’(0) = 2, which is positive.
Then, by second derivative test,
⇒ x = 0 is point of local maxima and local minima of f at x = 0 is f(0) = 0.
Question 11.
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
g(x) = x3 – 3x
Answer:
g(x) = x3 – 3x
⇒ g’(x) = 3x2 - 3
Now, g’(x) = 0
⇒ 3x2 - 3 = 0
⇒ 3x2 = 3
⇒ x = � 1
g’’(x) = 6x
Now, g’(1) = 6>0
and g’(-1) = -6 < 0
Then, by second derivative test,
⇒ x = 1 is point of local maxima and local minima of g at x = 1 is
g(1) = 13 – 3 = 1-3 =-2
And,
x = -1 is point of local maxima and local maximum value of g at x = -1 is
g(-1) = (-1)3 – 3(-1) = -1+3 = 2
Question 12.
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
Answer:
h(x) = sin x + cos x,
h’(x) = cosx - sinx
Now, h’(x) = 0
⇒ cosx - sinx = 0
⇒ cosx = sinx
⇒ tanx = 1
h’’(x) = -sinx – cosx = -(sinx + cosx)
Then, by second derivative test,
is point of local maxima and local minima of h at
is
Question 13.
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
f (x) = sin x – cos x, 0 < x < 2π
Answer:
f (x) = sin x – cos x, 0 < x < 2π
f’(x) = cosx + sinx
Now, f’(x) = 0
⇒ cosx + sinx = 0
⇒ cosx = -sinx
⇒ tanx = -1
’(x) = -sinx + cosx
Then, by second derivative test,
is point of local maxima and the local maximum value of f at
is
And,
is point of local minima and the local minimum value of f at
is
Question 14.
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
f (x) = x3 – 6x2 + 9x + 15
Answer:
f (x) = x3 – 6x2 + 9x + 15
⇒ f’(x) = 3x2 – 12x + 9
Now, f’(x) = 0
⇒ 3x2 – 12x + 9 = 0
⇒ 3(x-1)(x-3) = 0
⇒ x = 1,3
g’’(x) = 6x – 12 =6(x-2)
Now, f’(1) = 6(1-2)=-6 < 0
and f’(3) = 6(3-2) = 6 > 0
Then, by second derivative test,
⇒ x = 1 is point of local maxima and local maximum of f at x = 1 is
f(1) = 13 – 6 +9 +15 = 19
And,
x = 3 is point of local minima and local minimum value of f at x = 3 is
f(3) = 27 – 54 + 27 +15 = 15
Question 15.
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
Answer:
Now, g’(x) = 0
=
⇒ x2 = 4
⇒ x = �2
Since x > 0, we take x = 2
Then, by second derivative test,
⇒ x = 2 is point of local minima and local minimum of g at x = 2 is
Question 16.
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
Answer:
⇒ x = 0
Now, for values close to x = 0 and to the left of 0,
g’(x) > 0.
Also, for values close to x = 0 and to the right of 0,
g’(x) < 0
Then, by first derivative test,
⇒ x = 0 is point of local maxima and local maximum of g at x = 0 is
Question 17.
Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
Answer:
f’(x) = 0
⇒ 2 – 3x = 0
Question 18.
Prove that the following functions do not have maxima or minima:
f (x) = ex
Answer:
f (x) = ex
⇒ f’(x) = ex
Now, if f’(x) = 0, then ex = 0.
But, the exponential function can never assume 0 for any value of x.
Therefore, there does not exist c ϵ R such that f’(c) = 0
Hence, function f does not have maxima or minima.
Question 19.
Prove that the following functions do not have maxima or minima:
g(x) = log x
Answer:
g(x) = logx
Since, log x is defined for a positive number x,
g’(x) > 0 for any x.
Therefore, there does not exist c ϵ R such that f’(c) = 0
Hence, function f does not have maxima or minima.
Question 20.
Prove that the following functions do not have maxima or minima:
h(x) = x3 + x2 + x +1
Answer:
h(x) = x3 + x2 + x +1
⇒ h’(x) = 3x2 + 2x +1
h(x) = 0
⇒ 3x2 + 2x +1 = 0
Therefore, there does not exist c ϵ R such that h’(c) = 0
Hence, function h does not have maxima or minima.
Question 21.
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
f (x) = x3, x ∈ [–2, 2]
Answer:
It is given that f (x) = x3, x ∈ [–2, 2]
⇒ f’(x) = 3x2
Now, f’(x) = 0
⇒ x = 0
Now, we evaluate the value of f at critical point x = 0 and at end points of the interval [-2, 2].
f(0) = 0
f(-2) = (-2)3 = -8
f(2) = (2)3 = 8
Therefore, we have the absolute maximum value of f on [-2, 2] is 8 occurring at x = 2.
And, the absolute minimum value of f on [-2, 2] is -8 occurring at x =-2.
Question 22.
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
f (x) = sin x + cos x, x ∈ [0, π]
Answer:
It is given that f (x) = sin x + cos x, x ∈ [0, π]
f’(x) = cosx - sinx
Now, f’(x) = 0
⇒ cosx - sinx = 0
⇒ cosx = sinx
⇒ tanx = 1
⇒ x =
Now, we evaluate the value of f at critical point and at end points of the interval [0, π]
f(0) = sin0 +cos0 = 0+1 = 1
f(π) = sin π + cos π = 0 -1 = -1
Therefore, we have the absolute maximum value of f on [0, π] is √2 occurring at
And, the absolute minimum value of f on [0, π] is -1 occurring at x = π.
Question 23.
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
Answer:
It is given that
Now, f’(x) = 0
⇒ x = 4
Now, we evaluate the value of f at critical point x = 0 and at end points of the interval
Therefore, we have the absolute maximum value of f on is 8 occurring at x = 4.
And, the absolute minimum value of f on is -10 occurring at x = -2.
Question 24.
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
f (x) = (x − 1)2 + 3, x ∈ [−3, 1]
Answer:
It is given that f (x) = (x − 1)2 + 3, x ∈ [−3,1]
⇒ f’(x) = 2(x – 1)
Now, f’(x) = 0
⇒ 2(x-1)
⇒ x = 1
Now, we evaluate the value of f at critical point x = 1 and at end points of the interval [-3, 1].
f(1) = (1 - 1)2 + 3 = 0 + 3 = 3
f(-3) = (-3 - 1)2 + 3 = 16 + 3 = 19
Therefore, we have the absolute maximum value of f on [-3, 1] is 19 occurring at x =-3.
And, the absolute minimum value of f on [-3,1] is 3 occurring at x = 1.
Question 25.
Find the maximum profit that a company can make, if the profit function is given by
p(x) = 41 – 72x – 18x2
Answer:
It is given that the profit function p(x) = 41 – 72x – 18x2
⇒ p’(x) = -72 – 36 x
and p’’(x) = -36
Now, g’(x) = 0
Then, by second derivative test,
x = -2 is point of local maxima of p.
Therefore, Maximum Profit =
= 113
Therefore, the maximum profit that the company can make is 113 units.
Question 26.
Find both the maximum value and the minimum value of 3x4 – 8x3 + 12x2 – 48x + 25 on the interval [0, 3].
Answer:
Let f (x) = 3x4 – 8x3 + 12x2 – 48x + 25, x ∈[0,3]
⇒ f’(x) = 12x3 - 24x2 + 24x – 48
=12(x3 - 2x2 + 2x – 4)
=12[x2 (x – 2)+ 2(x – 2)]
=12(x -2)( x2+ 2)
Now, f’(x) = 0
⇒ x =2 or (x2+ 2) = 0 for which there are no real roots.
Therefore, we will only consider x = 2
Now, we evaluate the value of f at critical point x = 2 and at end points of the interval [0,3].
f(2) = 3(2)4 – 8(2)3 + 12(2)2 – 48(2) + 25
= 3(16) – 8(8) + 12(4) +25
=48 – 64 +48 – 96 + 25
= -39
f(0) = 3(0)4 – 8(0)3 + 12(0)2 – 48(0) + 25
= 0+ 0 + 0 +25
= 25
f(3) = 3(3)4 – 8(3)3 + 12(3)2 – 48(3) + 25
= 3(81) – 8(27) + 12(9) +25
=243 – 216 +108 – 144 + 25
= 16
Therefore, we have the absolute maximum value of f on [0,3] is 25 occurring at x =0.
And, the absolute minimum value of f on [0,3] is -39 occurring at x = 2.
Question 27.
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Answer:
It is given that f (x) = sin2x, x ∈ [0, 2π]
f’(x) = 2cos2x
Now, f’(x) = 0
⇒ cos2x = 0
⇒ 2x = 0
⇒ x =
Now, we evaluate the value of f at critical point x = and at end points of the interval [0, 2π]
f’ =
f’ =
f’ =
f’ =
f(0) = sin0, f(2π) = sin2π = 0
Therefore, we have the absolute maximum value of f on [0, 2π] is 1 occurring at
x = and x =
.
Question 28.
What is the maximum value of the function sin x + cos x?
Answer:
Let f(x) = sin x + cos x,
⇒ f’(x) = cosx - sinx
Now, f’(x) = 0
⇒ cosx - sinx = 0
⇒ cosx = sinx
⇒ tanx = 1
Now,
If f’’(x) will be negative when (sinx + cosx) > 0, means both sinx and cosx are positive.
And, we know that sinx and cosx both are positive in the first quadrant.
Then, f’’(x) will be negative when
f’’(x) = -sinx – cosx = -(sinx + cosx)
Now, let us take x =
Then, by second derivative test,
f will be maximum at x =
And, the maximum value of f is
Question 29.
Find the maximum value of 2x3 – 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [–3, –1].
Answer:
Let f (x) = 2x3 – 24x + 107, x ∈ [1, 3]
⇒ f’(x) = 6x2 – 24
=6(x2 - 4)
Now, f’(x) = 0
⇒ 6(x2 - 4) = 0
⇒ x2 = 4
⇒ x = �2
Therefore, we will only consider the interval [1,3]
Now, we evaluate the value of f at critical point x = 2 ϵ [1,3] and at end points of the interval [1,3].
f(2) = 2(2)3 – 24(2) + 107
= 2(8) – 24(2) + 107
= 75
f(1) = 2(1)3 – 24(1)+ 107
= 2 – 24 +107
= 85
f(3) = 2(3)3 – 24(3) + 107
= 2(27) -24(3) +107
= 89
Therefore, we have the absolute maximum value of f on [1,3] is 89 occurring at x =3.
Now, we will only consider the interval [-3, -1]
Now, we evaluate the value of f at critical point x = -2 ϵ [-3, -1] and at end points of the interval [1,3].
f(-3) = 2(-3)3 – 24(-3) + 107
= 2(-27) – 24(-3) + 107
= 125
f(-1) = 2(-1)3 – 24(-1)+ 107
= -2 + 24 +107
= 129
f(-2) = 2(-2)3 – 24(-2) + 107
= 2(-8) -24(-2) +107
= 139
Therefore, we have the absolute maximum value of f on [-3, -1] is 89 occurring at x = -2.
Question 30.
It is given that at x = 1, the function x4 – 62x2 + ax + 9 attains its maximum value, on the interval [0, 2]. Find the value of a.
Answer:
It is given that f(x) = x4 – 62x2 + ax + 9
Then, f’(x) = 4x3 – 124x+ a
It is given that function f attains its maximum value on the interval [0, 2] at x = 1.
⇒ f’(1) = 0
⇒ 4 – 124 + a = 0
⇒ a = 120
Therefore, the value of a is 120.
Question 31.
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Answer:
It is given that f (x) = x + sin 2x, x ∈ [0, 2π]
f’(x) = 1+ 2cos2x
Now, f’(x) = 0
2x = 2π � , n ϵ Z
⇒ x = nπ � , n ϵ Z
Now, we evaluate the value of f at critical point and at end points of the interval [0, 2π]
f’(0) = 0 + sin 0 = 0
f’(2π) = 2π + sin 4π = 2π + 0 = 2π
Therefore, we have the absolute maximum value of f on [0, 2π] is 2π occurring at
x = 2π and absolute minimum value of f(x) in the interval [0, 2π] is 0 occuring at x = 0.
Question 32.
Find two numbers whose sum is 24 and whose product is as large as possible.
Answer:
Let one number be x. Then, the other number is (24 –x).
Let P(x) denote the product of the two numbers.
Then, we get,
P(x) = x(24 –x) = 24x – x2
⇒ P’(x) = 24 – 2x
⇒ P’’(x) = -2
Now, P’(x) = 0
⇒ x = 12
And
P’’(12) = -2 <0
Then, by second derivative test,
x = 12 is the point of local maxima of P.
Therefore, the product of the numbers is the maximum when the numbers are 12 and 24 – 12 = 12.
Question 33.
Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.
Answer:
Let the two numbers are x and y such that x + y = 60
⇒ y = 60 –x
Let f(x) = xy3
⇒ f(x) = x(60-x)3
⇒ f’(x) = (60 –x)3 -3x(60 –x)2
= (60-x)2[60 – x – 3x]
= (60-x)2[60 – 4x]
And f’’(x) = -2(60 – x)(60 -4x) -4(60-x)2
= -2(60 – x)[60 -4x + 2(60-x)]
= -2(60 – x)(180 – 6x)
= -12(60 – x)(30 -x)
Now, f’(x) =0
⇒ x = 60 or x =15
When x = 60, f’’(x) = 0.
When x = 15, f’’(x) = -12(60-15)(30-15) = -12×45×15 < 0
Then, by second derivative test, x =15 is a point of local maxima of f.
Then, function xy3 is maximum when x =15 and y = 60 – 15 = 45.
Therefore, required numbers are 15 and 45.
Question 34.
Find two positive numbers x and y such that their sum is 35 and the product x2 y5 is a maximum.
Answer:
Let one number be x. Then, the other number is y = (35-x)
Let P(x)= x2 y5
Then, we get,
P(x) = 2x(35 –x)5 – 5x2(35-x)4
= x(35 –x)4 [2(35-x)– 5x]
= x(35 –x)4 [70-7x]
= 7x(35 –x)4(10-x)
Now, P’’(x) = 7(35 –x)4 (10-x)+7x[-(35 –x)4 – 4(35-x)3 (10-x)]
= 7(35 –x)4 (10-x) - 7x(35 –x)4 – 28x(35-x)3 (10-x)]
=7(35 –x)3 [(35-x)(10-x) - x(35 –x) – 4x(10-x)]
=7(35 –x)3 [350-45x+x2-35x+x2-40x+4x2]
=7(35 –x)3 [6x2-120x+350]
Now, P’(x) = 0
⇒ x = 0, 35, 10
When x = 0, 35 This will make the product x2y5 equal to 0.
Therefore, x= 0, 35 cannot be possible values of x.
And when x = 10
Then, we have,
P’’(x) =7(35 –10)3[6(10)2-120(10)+350]
= 7(25)3 [-250]<0
Then, by second derivative test,
x = 10 and y = 35 -10 = 25 is the point of local maxima of P.
Therefore, the required number are 10 and 25.
Question 35.
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
Answer:
Let one number be x. Then, the other number is (16 – x).
Let S(x) be the sum of these number. Then,
S(x) = x3 + (16-x)3
⇒ S’(x) = 3x2 -3(16-x)2
⇒ S’’(x) = 6x + 6(16-x)
Now, S’(x) =0
⇒ 3x2 -3(16-x)2 = 0
⇒ x2 -(16-x)2 = 0
⇒ x2 – 256 - x2 + 32x = 0
⇒ x = 8
Now, S’’(8) = 6(8) + 6(16-8)
= 48 + 48 = 96 > 0
Then, by second derivative test, x = 8 is the point of local minima of S.
Therefore, the sum of the cubes of the numbers is the minimum when the numbers are 8 and 16-8 = 8.
Question 36.
A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
Answer:
Let the side of the square to be cut off be x, then, the length and the breadth of the box will be (18 – x) cm each and the height of the box is x cm.
Then, the volume {V(x)} of the box is given by:
V(x) = x(18-x)2
⇒ V’(x) = (18-x)2 - 2x(18-x)
= (18 - x)[18- x -2x]
= (18 - x)(18 - 3x)
Now, V’’(x) = (18 - x)(-3) + (18 - 3x)(-1)
= -3(18 - x) - (18 - 3x)
= -54 + 3x - 18 + 3x
= 6x - 72
Now, V’(x) = 0
⇒ x = 18 or 3
If x = 18 then breadth becomes 0 which is not possible
Therefore, x =3
V’’(3) = 6.3 - 72 = -ve
Then, by second derivative test, x= 3 is the point of maxima of V.
Therefore, If we remove a square of side 3cm from each corner of the square tin and make a box from the remaining sheet, then the volume of the box obtained is the largest possible.
Question 37.
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum?
Answer:
Let the side of the square to be cut off be x, then, the height of the box is x and the length is 45-2x and the breadth is 24 – 2x.
Then, the volume {V(x)} of the box is given by:
V(x) = x(45-2x)(24-x)
= x(1080-90x-48x+4x2)
= 4x3 – 138x2 + 1080x
⇒ V’(x) = 12x2 – 276x + 1080
= 12(x2 - 23x + 90)
=12(x – 18)(x – 5)
Now, V’’(x) = 24x – 276 = 12(2x-23)
Now, V’(x) = 0
⇒ x = 18 or 5
It is not possible to cut off a square of side 18cm from each corner of the rectangular sheet. So, x cannot be equal to1 8.
Therefore, x =5
V’’(5) = 12(10 – 23) = -156 < 0
Then, by second derivative test, x = 5 is the point of maxima of V.
Therefore, the side of the square to be cut off to make the volume of the box maximum possible is 5cm.
Question 38.
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Answer:
The figure is given below:
Let a rectangle of length l and breadth b be inscribed in the circle of radius a.
Then, the diagonal passes through the centre and is of length 2a cm.
Now, by Pythagoras theorem, we get,
(2a)2 = l2 + b2
⇒ b2 = 4a2 – l2
⇒ b =
Therefore, Area of rectangle, A =
.
.
.
Gives 4a2 = 2l2
⇒ l = a
⇒ b =
when l =
Then, = -4 < 0
Then, by second derivative test, when l =, then the area of the rectangle is the maximum.
Since, l = b =,
Therefore, the rectangle is square.
Hence proved.
Question 39.
Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.
Answer:
Let r be the radius and h be the height of the cylinder.
Then, the surface area (S) of the cylinder is given by:
S = 2πr2 + 2πrh
⇒ h
Let V be the volume of the cylinder. Then
V = πr2h
Now,
If
So, when then
<0
Then, by second derivative test, the volume is the maximum when
Now, when . then h =
Therefore, the volume is the maximum when the height is the twice the radius or height is equal to diameter.
Question 40.
Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
Answer:
Let r be the radius and h be the height of the cylinder.
Let V be the volume of the cylinder. Then
V = πr2h = 100(given)
⇒ h =
hen, the surface area (S) of the cylinder is given by:
S = 2πr2 + 2πrh
Now, ,
<0
If
So, when then
> 0
Then, by second derivative test, the surface area is the minimum when
Now, when then h =
cm.
Therefore, the dimensions of the can which has the minimum surface area are and h
cm.
Question 41.
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
Answer:
Let a piece of length l be cut from the given wire to make a square.
Then, the other piece of wire to be made into a circle is of length (28 – l)m.
Now, side of square =
Let r be the radius of the circle,
Then, 2πr = 28 – l
⇒ r =
Therefore, the required area (a) is given by
A = (side of the square)2 + πr2
=
Then,
Now, if
Then,
⇒ (π + 4)l – 112 = 0
⇒
So, when
Then,
Then, by second derivative test, the area (A) is the minimum when .
Therefore, the combined area is the minimum when the length of the wire in making the square is cm while the length of the wire in making the circle is
28 - =
cm.
Question 42.
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is of the volume of the sphere.
Answer:
Let r and h be the radius and height of the cone respectively inscribed in a sphere of radius R.
Let V be the volume of cone.
Then, V=
And height of cone h = R +
Now,
Now, if,
After solving this we get,
So, when, , then
< 0
Then, by second derivative test, the volume of the cone is the maximum when
So, when, , h = R +
Therefore, V =
Therefore, the volume of the largest cone that can be inscribed in the sphere is the volume of the sphere.
Question 43.
Show that the right circular cone of least curved surface and given volume has an altitude equal to √2 time the radius of the base.
Answer:
Let r and h be the radius and height of the cone respectively.
Then, the volume (V) of the cone is given by:
V=
The surface area (S) of the cone
S = πrl
=
Then,
Now, if
So, when then
> 0
Then, by second derivative test, the surface area of the cone is the least when .
So when then h =
Therefore, for a given volume, the right circular cone of the least curved surface has an altitude equal to √2 times the radius of the base.
Question 44.
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is tan-1 √2.
Answer:
Let Ɵ be the semi- vertical angle of the cone.
Let r, h and l be the radius, height and the slant height of the cone respectively.
It is given that slant height is constant.
Now, r = lsinƟ and h = lcosƟ
Then, the volume of the cone (V)
V =
.
Now, if
sin3Ɵ = 2sinƟcos2Ɵ
⇒ tan2Ɵ = 2
⇒ tanƟ = √2
⇒
Now, when, then tan2Ɵ = 2 or sin2Ɵ = 2cos2Ɵ.
Then, we get
= -4πl3cos3Ɵ < 0 for Ɵ ϵ
Then, by second derivative test, the volume (V) is the maximum when
Therefore, the semi-vertical angle of the cone of the maximum volume and of given slant height is .
Hence Proved.
Question 45.
Show that semi-vertical angle of right circular cone of given surface area and maximum volume is .
Answer:
We know that total surface area of the cone = S = πr(l + r) …(1)
and Volume of the cone(V) =
Then by (1), we get,
P = V2
Now, differentiating P with respect to r, we get,
Now, if, , then
S = 4πr2
Now again differentiating with respect to r, we get
Therefore, P is maximum when S = 4πr2
And V is maximum when S = 4πr2
⇒ πr(l + r) = 4πr2
⇒ l = 3r
SinƟ =
Therefore, semi-vertical angle of right circular cone of given surface area and maximum volume is .
Hence Proved.
Question 46.
The point on the curve x2 = 2y which is nearest to the point (0, 5) is
A. (2√2,4)
B. (2√2,0)
C. (0, 0)
D. (2, 2)
Answer:
It is given that x2 = 2y
For each value of x, the position of the point will be
The distance d(x) between the points and (0,5) is given by:
d’(x) = 0
⇒ x3 – 8x = 0
⇒ x(x2-8) =0
⇒ x = 0,
And, .
So, now when x = 0, then d’’(x) = < 0
And when, x = , d’’(x) >0
Then, by second derivative test, d(x) is minimum at
So, when
Therefore, the point on the curve x2 = 2y which is nearest to the point (0,5) is .
Question 47.
For all real values of x, the minimum value of is
A. 0
B. 1
C. 3
D.
Answer:
Let
Then, f’(x) = 0
⇒ x2 = 1
⇒ x = �1
Now,
And,
Also, f’’(-1) = -4 < 0
Then, by second derivative test, f is minimum at x = 1 and the minimum value is given by
Question 48.
The maximum value of is
A.
B.
C. 1
D. 0
Answer:
Let f(x) =
Now, if f’(x) = 0
Then, we evaluate the value of f at critical point and at the end points of the interval [0, 1].
Therefore, we can conclude that the maximum value of f in the interval [0, 1] is 1.
Miscellaneous Exercise
Question 1.Using differentials, find the approximate value of each of the following:
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Answer:Let us consider y = x1/4 and
and ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADoAAAAgCAMAAAB0KJvnAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6OgBmOmY6OmZmOmaQOma2OpDbZgAAZgA6ZjoAZjpmZmYAZmY6ZrbbZrb/kDoAkDo6kNv/tmYAtmY6tv/btv//25A625Bm27Zm29uQ2////7Zm/9uQ/9u2//+2///bnMc5+AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABCElEQVRIS+VU2w6CMAzdvOENES9Mp06Fwf7/D207EDUx6Yg+GPsCS3p6Ts+6CvEb4Q6jjkLPs7QrVAjzP1Cnh3lrsY0WbL+NhGjTjRw8FGJXgcRquentQwBtbjG4kmItgR0EZewyTmVOkWIN5HBiI4WdnIQhxVjBNBaAhjred4ODVXjTbDSN33Pea/kfMCkBhVXiB9Ozc6PwbfZP6PVqTV8MhmBNUgv01e32DT2H2I5JIkIcVqm75kBpsDD6RyV7wPo4ZpwCX88pUynnaGrwfnJqIcoE/AjfTzQBmgYgdD85FeclwsOhAnuN6WWHslbpFl5IJ8E2aqDP+4lzs5fIX87LfuJAP5ZzA7DwENr/TrcKAAAAAElFTkSuQmCC)
Then, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, ![](data:image/png;base64,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)
Now, dy is approximately equal to Δy and is equal to
dy = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the approximate value of
is ![](data:image/png;base64,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)
Question 2.Using differentials, find the approximate value of each of the following:
![](data:image/png;base64,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)
Answer:Let us consider y =
and x=32 and ![](data:image/png;base64,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)
Then, ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, ![](data:image/png;base64,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)
Now, dy is approximately equal to Δy and is equal to
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the approximate value of
is ![](data:image/png;base64,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)
Question 3.Show that the function given by
has maximum at x = e.
Answer:It is given that f(x) = ![](data:image/png;base64,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)
Then, f’(x) = ![](data:image/png;base64,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)
Now, f’(x) = 0
⇒ 1 - logx =0
⇒ log x =1
⇒ log x = log e
⇒ x = e
Now, f’’(x) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, f’’(e)=![](data:image/png;base64,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)
Therefore, by second derivative test, f is the maximum at x = e.
Question 4.The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?
Answer:Let ΔABC be isosceles where BC is the base of fixed length b.
Let the length of the two equal sides of ΔABC be a.
Now, draw a perpendicular AD to BC. Then, we have
In ΔADC, by using Pythagoras theorem,
AD = ![](data:image/png;base64,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)
Then, Area of triangle (A) = ![](data:image/png;base64,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)
The rate of change of the area with respect to time (t) is given by:
![](data:image/png;base64,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)
It is given that the two equal sides of the triangle are decreasing at the rate of 3cm per second.
![](data:image/png;base64,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)
Then ![](data:image/png;base64,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)
And when a = b we have,
![](data:image/png;base64,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)
Therefore, if the two sides are equal to the base, then the area of the triangle is decreasing at the rate of
cm2/s.
Question 5.Find the equation of the normal to curve x2 = 4y which passes through the point (1, 2).
Answer:It is given that curve is y2 = 4x.
Differentiating with respect to x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, the slope of the normal at point (1,2) is ![](data:image/png;base64,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)
Therefore, Equation of the normal at (1,2) is y - 2 = - 1(x – 1)
⇒ y - 2 = - x + 1
⇒ x + y - 3 = 0
Question 6.Show that the normal at any point θ to the curve
x = a cosθ + a θ sinθ, y = a sinθ – aθ cosθ is at a constant distance from the origin.
Answer:We have x = a cosθ + a θ sinθ,
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUUAAAAhCAMAAAB5uEmXAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZjoAZjo6ZmY6ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkLb/kNv/tmYAtmY6tmZmtpA6ttvbttv/tv//25A625Bm27Zm27aQ2/+22////7Zm/7aQ/9uQ/9u2//+2///b424ymwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADPUlEQVRoQ+1ZW3ubMAy10yUL6yVrsrWhuyTd1pJ2ZBD+/4+bJIMxtgPG8NAN80DygX10dJBkgRkLR1DgDSmQ8OUbYvPPUtkHFUd4dklQcaCK+YZzyOiXaLbLokU6EG2i04t4mRZfIBaPizRZBRH9wiD7sGOMMjp5d+cHEWaxWsXDZchn34A4rVdpHr9P2Z/vRTx/8oXR5xVfOR9eH/qiuIx3GUPeuAyUY44Rv/nJ7w7Rsthz72g88q0qJFbbfN175R+I4mLVZYwQ0cEFlzF94lLz/ziDaksn5SjihtIWeBcUcO8cjNWqZsZlDE1xGegypo+K2tg9BnUWNd3tVtEFpUVFq1UnTIuro4LZpSxeLqHuwb38gfPZKj1yyGg8HSK8DJEO907r2/ZY9EIxVaxgVKsZdMEzeIj4e43VviRqZ6YmTOnZKGDtcUhN5h4StogX0G1C4FHcZdHVjiWgp6AgzvWhx6IfiqGiAiOtZtE9y9dbYAS/8cWzJGpnpnKUng0H28P7jDi02qaIgsJRuGXXUkWS8hYui8ntKhJWbxR7RgsYaZXSEQ76QKCa0JjZ/ewPJuUq/5wLQtMehd8jv/pViSECEoU90fKsxuLZx9ILhbXC1FarWiJ+6VwSNZlZHBYqVi4MBGvJ6uL1U4SlkDEoI/PnciWRKrrWRR8US10sYWqrTcfFdUG0uy6OCNaR0ViK5BL8GgHJsi6Wscgoo05QmLrqYn8Ua10UMNKqJXyACBFtMjP8rD3rD9a+lph3qU9C4lgSiZimYkL9IlT1NhX9UAwVJQyTVqtagoufRtTKrGY5KliHrLiC5J9RRVwDN/hFCNObTngPqwq8uzQbHcN9P5TzMIrVhN+z4scOGAFRlFIStTKr3ZWcxgDrCs5H6BZ/r/k2f4DyeIMkoUrSCRdBUA++Ws4+aihG1+2FYtbFCka1Cn3r/BtWQ+CE7/MVUTszhag/mNh3wfnh+2JX/LTcx7YquXgycnAA5ASnwlfaIoY8TLy/6UxQtIbL5b4LtSX6qjp1bZz9r/ZdxKuR1ps4o0x9YLVjEGJxSCRUKoa6OERFue+Ca/RG65OHAE9rbrnvIjZrpuV68DYo8N8p8Bd1JJfYocwRkgAAAABJRU5ErkJggg==)
And y = a sinθ – aθ cosθ
⇒ ![](data:image/png;base64,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)
So, ![](data:image/png;base64,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)
Then, Slope of the normal at any point θ is
.
The equation of the normal at a given point (x,y) is:
y - a sinθ + aθ cosθ =
(x - a cosθ - a θ sinθ)
⇒ ysinθ – asin2θ + aθ sinθ cosθ = - x cosθ + acos2θ + aθ sinθ cosθ
⇒ xcosθ + ysinθ – a(sin2θ + cos2θ ) = 0
⇒ xcosθ + ysinθ –a = 0
Now, the perpendicular distance of the normal from the origin is
, which is independent of θ .
Therefore, the perpendicular distance of the normal from the origin is constant.
Question 7.Find the intervals in which the function f given by
![](data:image/png;base64,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)
is (i) strictly increasing (ii) strictly decreasing.
Answer:(i) It is given that f(x) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGMAAAAhCAMAAAD6deXAAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOmaQOma2OpDbZgAAZgA6ZjoAZjpmZpC2ZpDbZrbbZrb/kDoAkDo6kGY6kGaQkLbbkLb/kNv/tmYAtmaQtpBmttv/tv/btv//25A625Bm27Zm29uQ29u22/+22////7Zm/7aQ/9uQ/9u2//+2///bcRjLOAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABlklEQVRIS+1WDU/CMBC9DtmKCIjCpuLwYwVxo6z//9d5va5AZ7bMGmIwXNKStrf3eu+AN4D/GSLMT1KY5PEet41jM2Vs5HcD9TA8cLRAlLcLWAepF4mYJ5pjxRmbFSzMcbzyIIY1D1LJXe3kIP22b/PUI2NhtsNib1BvgjNHAMVIaQ45iEGMQWslggUs8bMIczFy24MZ+ERt36xVEuXy+h3n3SSycHQE5V2mkvkTpgT3iEcc1QBxNQMoJ4xC17o27ajv01qXWM2ih5gEZ1IFAWhFNlMUyOFYDV2l8EoU9X1aH3Ps4fappBVWpZJKq6qO7bNK+m+HHhdIUWBqfd+sy0mUq5dPnHdJZOHoSCN88F4GW+xRP5M8WEjOIhzjFY/Ukio0YUSLsZvuvl3LKeungDPDJho4c+T1Tbw8dN4KmB/YSeO8Bbrc/gwV+IV9d63WtW/6/z9JGN/RceBotG3Ptyayb5ej2bb9Xswq+3YsvcW2faS09u1o1W7bP6Wx9u1q1WzbR87flcrad73njbbt2fOu9/n7vC8TGTgdJz0V9AAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
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![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQMAAAAxCAMAAADpyjzrAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjpmZmYAZmZmZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmtpBmttvbttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///b33PGTgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAD8klEQVRoQ+1Z61qcMBANVqWtW62rbV2sLr1SLbe8/8t1ZnIBQm5WdpVvyQ/YD7LnzJyZDJBhbBmLAnNR4O91klxMZuy0aJOZ5Qdqr+7Zw9F2IrZp0SYyKgqmfjuVBkgXjVY8W/oIBP7wLkmOw/4Vl1FSyUlD4irZGH+ORavT52pgQeCZYU5+dM+aLMj04CwHI0QM89D0kQZutKFW7fpL0LLAMrYgjDU4BZBiFCgzbu6KaNEgZHoRW1/zy0poUENRPoLo4Xn1Cy40t3DhosyT5KQsErf5GqHnkC1qjOVvfnuIGKvA6GrDrIwWREUsDa3QRjz8SenpItE0nDOS1aoUGtTpDWvWG3HOwFienZT860nJcrhv94lQNUKfwza/uTtHad1E7TqBAWGwMY4RFbE2tE4xhOnZlhJOo4UcaK+2TGiQg7c4ckxaRGvXUJ7qVUliuItLh+DXAE06Rwo3kQawMY400MTaUKkBCdEvrX4HeAZzQYNiQ0AwxJmO35Kzn3StTj+olQVpKocsIh2C9mA0R915TCG9fEQaosdIcTFZ4VqPWBnaaaBIJKABN1wUsHrEMDTgGWYDPM6OcQEzz7OvQxgg29dOlVwaGhhEGmPMaCL2iaWhTg18DghKWguW8MCtxxS1aD99pmLmHLKq9u7bNcAU9RFpAAujFVETk6FODYIOCA1EQPCMdQHRsBTQ4uW323Yt7jpGWAPKXCjaGx+RRrcxujXQhro0CDsgE6VIbhj/DhUS8rVBIah+X58ynmPNgKvuEc5cjq9HBAs120HUSWBjtGpAxMpQklgcBjUxwgFYWFTf4Kl6fI9VIIWPN7C1uYUf5yXP4DYWdbcICsG3Fpo7Cesk6iSwMto0EMTSUHAc6hodBuZqB3yZvNxbFFgUWBRYFFgUOEwFnF8yhylH57X+HjuAH4cea+n/shaWRFgU2IsC/MfKu83yNCMK2n2FMaeeIc8+wuc2f4R9+YhmUliQ+j31aObUMxTbRLD3ifvz5r6bZ1/eLUbXKYzuGYaV3emMSvhN+/KjZtJ/acAKudfPYnuGO3UwDK42I2nmqOs31iCmpaXCH9szDFu52xm0mamGaKj1xkiDqJaWXF7RPcPdehhG7+8rD/Sgv477r1EtLVpYqmcYtuGlZ/Q0kOHTSWHpEMW2tECprmf40i4G+Qud/oPKIP9n5sFQA2dLS/Ulg+yvY0KXB/mp6Nz66kFkS2uuGtADrTI6EWYeRLa0qB7MZ6g6SM8zaut4nwtRLS2jdfzqxVCRld/nIQ2iWlpzeT3UwZHviVMGS78nTgm6SyzjiTgB1ezSAN+D8LtxuiG/G6cD3AfSjvYP9mH6wnFYCvwDHVaJ+arVTC8AAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, if f’(x) =0
⇒ cos x = 0 or cosx = 4
But, cosx = 4 is not possible
Therefore, cosx =0
⇒ x = ![](data:image/png;base64,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)
Now, x =
divides (0,2π) into three disjoints intervals
![](data:image/png;base64,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)
In the intervals
and
, f’(x)>0
Therefore, f(x) is increasing for 0< x <
and
< x < 2π.
In interval
, f’(x)<0
Therefore, f(x) is decreasing for
< x <
.
Question 8.Find the intervals in which the function f given by
is
(i) Increasing (ii) decreasing.
Answer:It is given that f(x) =![](data:image/png;base64,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)
![](data:image/png;base64,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)
Then, f’(x) =0
⇒ 3x6 - 3 = 0
⇒ x6 = 1
⇒ x =
1
Now, the points x =1 and x = - 1 divide the real line into three disjoint intervals
( - ∞, - 1), ( - 1,1) and (1,∞).
In interval ( - ∞, - 1) and (1,∞) when x < - 1 and x > 1 then f’(x) >0
Therefore, when x < - 1 and x > 1, f is increasing.
And, in interval ( - 1,1) when - 1< x < 1 then f’(x) < 0.
Therefore, when - 1 < x < 1, f is decreasing.
Question 9.Find the maximum area of an isosceles triangle inscribed in the ellipse
with its vertex at one end of the major axis.
Answer:It is given that ellipse ![](data:image/png;base64,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)
Let the major axis be along the x – axis.1).
Let ABC be the triangle inscribed in the ellipse where vertex C is at (a,0).
Since, the ellipse is symmetrical w.r.t. x - axis and y - axis, we can assume the coordinates of A to be ( - x1,y1) and the coordinates of B to be ( - x1, - y1).
Now, we have y1 = ± ![](data:image/png;base64,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)
Therefore, Coordinates of A
and the coordinates of B![](data:image/png;base64,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)
As the point(x1,y1) lies on the ellipse, the area of triangle ABC (A) is given by:
A = ![](data:image/png;base64,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)
……..(1)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOQAAAA0CAMAAACtt2SQAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmZmOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmYAZmY6ZmZmZrbbZrb/kDoAkDo6kDpmkLbbkNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///bFi1CRgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADsUlEQVRoQ+1ZbX+TMBCHaV111dUyO5+KOqpmnRZKvv93M7mEEOAIseRof1pe7MU4/nf/3F3uoVF0eS4ncDmB//YEWHyaZ8oD518eplR3Gl3FYn8axVNqza8tbb/Wcbz00s6/L372CLLbH+ZNU0zD2wJeyvyFeLrBhDPrv+Xdt2h35Qpfpt/y9G07APiTOKHn8uPipsJsihn4WsDffD9JnGT5ruWQ4qWDZDFXJHm66ijN4s/RIXkm4SoMREy9cirxo4NL4SQ7Kcm65hu8MvmkSObApflkMu5ZDE5kM/AzIqbhtcAYPnDYu1fN/MJJglJLdrd0KM5WOZDkqZ3I9ge5Iqk8VYkh8GFcKRTseWbnF05yK6yxZJmLY77YK5LFHE1v8SZTLlZxqsUweCzej3SrMSYz9b51rdQpCbK54Jj3ESjvHiJFUv1FnlqhdLUt1oaH4A7yNE4c9WRdQKRsmcjD0CTbBwOHL+xmG2M9KgKWZzIp2yRt+GwWgiF/up8bg1UIIS4SFstXbVnMgLyKhk3EkHunmapAshLD4IOQlInQzB2MpOrpurK9p+wM10yo1I/tSRQ+CEmwxhGuh3vpQ1VAurJHkYSykKsKBCmnwxWFD5KTxXwVHT7at2DDk+x2DSZB+ndle0mqjge9XaEq8AxIlol1u2LwSmD0sxVV8ndS3SJdOKhouqcbkjVfi7yULNE6qW8hML5ZJxH4MHWyyQopITyd7fnXviZ74Iyxjsf+pL/j0VKBOp7BUBCRd/SYNVDLHb2rMovCkSjhMrl29anuQ0KmkPqDvinESNBNIR2r2dXN8UuBo+bJKlatgbO26rGv0xqMSZdAqeaic3msK1tcIbO92D2FoM2C9Y9BDqoayAWYnCx6hvwgqqYDUa2s/Zi+X1793Qujd8yYzubxmnhq8qeYv14OA55ml/o3WrscLJKRFbvDZM9YwhGuUfn+gx6/t/XF8S+Eq+U8MR2JMi4c9PhmfV6348igsUoItPl5DG3wmZWAkSTrZoCnYgyQwSxZTkDSs7EZyc/1OQ3JehMuZyqvRTkhRyJPWptwz0U5JUeevaD4QcrahHsuyglJwo+nQZYIiJGwN/BclBNypIWGTbjnopzWEjp0Rc9zUU5nBimyvnAqkgOLclJTyMCrXPRclJPZQQpcbcJ7wrW9KCe1hQrcbMJxkp1FOZUdlLj1JtxzUU5pDBV2vQn3XJRTGTIRrueifCJriNR4LsqJtE8F67kon8ocIj2nnyeJiF1gw53AHxNzbR48Abt+AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGkAAAAnCAMAAAA7DpYUAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOmY6OmZmOma2OpDbZgAAZjoAZjo6ZjpmZmYAZmY6ZrbbZrb/kDoAkDo6kNv/tmYAtmY6tv//25A625Bm27Zm27aQ2/+22////7Zm/9uQ/9u2//+2///bjl3XhgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABbElEQVRYR+2W7XbCIAyGic7ZfVnn5ph0qGtrc/93OEo/6NFhE3rkl/lbwsMbkpcKcQ92BfDwDPDGTuMnoFzmqGY7fmZQRpl8BeXxk+KQ8PieAETQVN+TiKKpqJshCqlMVuL0GaUjMjNNv2mMi+J36j2DUIF9hBlrjjFofQWwyDV7vsnmrZ3v1haMkinSb95VCmfRo1AaTavzkhupbfjfApopoJz/tLuXyQvlDbtkO5JTcdkkA5IYlJLQTc0Sr3lfqZ6oNh+NPsyWPWmkenTzHsjA712V1oj969qRRsSRzXtwl6hMNxRgW0KTSb15G3FXD+UmFyXMjCawKDpJdOY9RvIcg0HqdqhSO4l4XAM80P+NAkjZ1jIVbMUp7QdnrIVRPeZjazzfVX1bmvoSGvNrris0CiopFOBmkFy9iSiaDU6E2PaTUyrPOEDgYDEI3VJl/mKjhF4YUBGhfuWT8QdrobeO9pGJQLq1kn/2/wM/8xcexisBsgAAAABJRU5ErkJggg==)
But, x1 cannot be equal to a.
⇒ x1 = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAeCAMAAADXVfSyAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADpmADqQAGa2OgA6Ojo6Oma2OpDbZgAAZgA6ZjoAZrbbZrb/kDoAkGY6kLbbkNv/tmY6tpBmtv//25A627Zm29u22////7Zm/9uQ/9u2//+2YkLolwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAY0lEQVQYV42OSQ6AMAwDY8q+lL2FQP//TVIqcShCwpdR4kQ2UaxzACoiV3c0p/vtsqctACEnExm1kuuhFmSv968Fgn7fP4dbC5TSpxnJSrbXkQca6SmyYvspgAWspb+P03HeBalyA2GH2v98AAAAAElFTkSuQmCC)
y1 = ![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZQAAABHCAMAAAAXxkGnAAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmY6ZmZmZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY6tpA6tpBmttv/tv/btv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///b3wvDQAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAHrUlEQVR4Xu1d63rbNgy1sqWb03WXxs2unZtGTbcpyRY54vs/2kSKpGgSEA8l2lFc+UebzwEg4IAkQJBQVqvlsyCwIJANAXF/cZNN2CKIRKA+/5iEjNie/5XEsBCPQOBhfZnAJbavHhPIF9KRCOxSvFKdLWvXSJzT2Goc6GaTMq3S1FioXQTE9lsUkAT/oSIXOhoBfE0ql4hyrEEER5VmA8+pY+l+ss+BsYa9d7JQHc8wsf3qH+hpdfEnRLcQZUCgBPMvPvj8e1UUP8U1Acnigo5PcVDVKeFopGed17z7uLqPe5YiE5/fYNM05ofqx0OWGkALY0rSvyeFV8yyVBX765rrlADMHVYTk2QOfmL7NleJYPfdyMUVHRegheMco4S7A4tzyqpinRKCWWH7SkVm8RNbkguduvv2jwSNGBfioV2Oz/3iBWjhOKd0wp2Ble6UEMx7IKS0T9VkBr+aTDF26/hSSJle+dsosY1PHmpclMX71dPG0w20kPXJoDJGeD+wYKfYZS4As8J8Ysk6/OhiQrP5fZxTgqmCOIUaF6Xcj3mogBby82RImV64HVgDTvn7ujj7wzyoNE4xYIr7113WVbf/1MyoLIvi1WNVSN6erMNvt6Z4yktVznH4hlYEq4P0sbcY+jgQIvkik9wA9AyghUZTRyv7VTBtKfjswLJg+8ZXZ21K07usKrTRGszWoEcho3+zKdoPt1RICgWPQ9bhR9bS6jeP3feWb9AnRgdJpEa48wkGZyiSHhdKmFy+DANoYe8AjcyQMiR8dmCxqa4KuP2G3zrFBZO3qtewnSl+QFf4UU5p3t3o7+XhDRZcjQ5+bS5wSiiSrbF2MjEdaCoPGXr5CuDTA0tsuSW8y76sqXWhR+KQU9oZrz9W6m79fRBylFCd3bksaqS08qsWEoKPkO4sg65TOEpPFWOKT25GLKl7YCJF5S7OtDJy1IZO6bKVxs8z7JzznLJb6/TGpMri4ec1v2z1U5dIcRV+xDCtjcFynCKpsatDdKaEIr2s31l/9I+IDqFYAplwppDwaRuaDVeR95xiCTWYclFkgvXe0t788muXXopPdtHnnKIYtXyKz48vezpEY8oqEMksX2VrmfogOoRUFDLEWkrBp52yW3MVeR1TTDywU0pbov4DYsqHmy4u3f1w1TtF/sTxdvIFxec7xdUhOBgNcQhE0k5RiWnd2g3pEFJRyATK0PDpgWVDhW9wm3i1ScStjQ22nKzBlKX8p9/ItNbNOkoZJbrErTJO6fDj8lE1GATFF6jo6hDdpxAiyXGhBEliTIeQikImcAoJnxlYGrDA3hbDt9d75QaTphkwP7W7lP82wwV9sS3O2plSKK9Yp2j86B19G1fObmi+UElHh9iOnhJJjgsdly8xHSgqApkwplDwmYHF7h0DBGzuTINJuNX7yjpF48fUvsIpixx5ptW+jCqwKVb3QTMxKl6EGVgjnIKC6T/cqGzxA6vEiKmJVWIjEjYF0cFZC+IjlKKwwCQ4xdboQDC954ryG5XUOPhBdXPDN2Ro4nlKLxI0BdFBBqLOwpGfHhj8lNfZB0Bg+ouX3IO0oSURvzbf6PgyflyRkCmYDhgVa4cDTDyp1VLgmZ4RvS9VFIx1WiT9UuHMZHeQSjJy/Q1zpscvYigEmo2quvNVM8Ukbs/zXHFYnAAhUF+8p+hsMbRof+uULfvvl58OgIBxRV28inpP3IJ39qKSFgIEgfpC9nNFlq/wYA8RvdCMREDHlCg3nDtHJc2M4O61cxNhJrqh2VdCJ8tMLMPUEB+gC56YsExUeH/WCfenzG0Phq9K6oLHSX6ersn88/lsTah9jbsk93ymoU8ui69n5pSEKvGpOmW1eogdoKL+zUTHO8W/54c2smRS7Fhi5FUzfA0/jlbscbC9wqf1gFu+jqN3vqfcrWeXEvNn9NJsZwjxF8Ty4bNI6hDgrxgFTjmRlu2D9srlGVb24mMgzrvCN+y9PMqkSoGODT2hpp0t9QQ0Vbcp9OwNSf+eH39BbMrjJ/F6B+z4FFCbxcS7FpM0TWRmQ4V/hW84+CQ+NQu5d2ia0C6q29lUm4wuv2fRKJcQ9ta9f4XPtkLkevJkOcSlLaxeYtrZ0LLfZE2TBbDbD+8KH77LTFZhHANVIYU6Wmw7G+bCcdpN42I7uTyxKN00bRK4TbrubHL5dlGyza9b//BQlKDcRFJ0BsxuQ++0ZKg2v/ZK2UBDLNXmpzryEkLRRKQT2F+6U5SpatYMtYvSrXLmNGJ269iLdYrbhiWdstcuGp5us21+corlvYGZMCUY0hfrFNPxM7XNz750YTqW2SSgTkHpsikWEzS2zc+R210wnPzmgpim6b/Hei1lO1b8rRrpT5/AobOv1DY/p/eyOwofDEUT9JvCip7yzu7NeHqfktrm5/Reqvg+/OaCKciO54XfAA0TjtclkVPv6Me3+c12R49PAHSdS4R2PDncMOA9IugoG6/CgTjxtw3jl5EOpGogFmzD8vm0U+ZbJU5ZlJj3MxzLBcRzxpynmM7EGZ+n4BNFNb5Pael7RuftPXpiZ+LhzXhKen99c3UCf6pjYmfi4X3ysMbeaWc0EZ+xF3geXvPTfULqH7U5XSQWyxYEFgReLAL/A6OL7/Tj9UJ1AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
Also, when x1 =
, then,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
< 0
Then, the area is the maximum when x1 =
.
Therefore, Maximum area of the triangle is given by:
A = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 10.A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?
Answer:Let l, b, and h be the length, breadth and height of the tank respectively.
then, we have h = 2m
Volume of the tank = 8 m3
Volume of the tank = l × b × h
⇒ 8 = l × b × 2
⇒ lb = 4
⇒ b = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAABmADqQAGaQAGa2OgAAOjoAOma2OpDbZjpmZrbbZrb/kDoAkDo6kGaQtmYAtmY6tv//25A629uQ2////7Zm/9uQ//+2///bXMqRXwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAVUlEQVQYV2NgwATibDwgQUk+djAtzM0LosU4JEG0BJeQJC83P4MwIwgwgxWC5RkYRNiYhLAYh10IrJ2RkWj1QAtZIYoFoLQwkbQgiyhImxgjIyembQC0QwLShQK/0gAAAABJRU5ErkJggg==)
Now, area of the base = lb = 4
Area of the 4 walls (A) = 2h(l + b)
![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
⇒
= 0
⇒ l2 = 4
⇒ l = ±2
Since, length cannot be negative therefore l =2.
⇒ b = 2
Now, ![](data:image/png;base64,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)
When l =2, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIEAAAAjCAMAAACEqpmHAAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjqQOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjpmZmZmZmaQZma2ZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kLbbkNv/tmYAtmY6tpA6tpBmttu2ttv/tv//25A625Bm27Zm27aQ29u229v/2////7Zm/9uQ/9u2//+2///bleqgRgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACQUlEQVRYR+1Wa1PCMBBM8EVVKFV8WxRoVayoTZv8/5/mXVJS2pJIoMgXbwYKw5Hb7O3dLCH/oRkQI0qP3vdJCL+c8eFgXwgS2sXS/EpxwAP59U8jwpLZTaqKsvMDczs+zmjnHpK+hpT67YFMAAG7TTM8moint+DRdLYYTci8MyX8Qj3biBwuA13gATykDphPJCfGyE5V5cVzSxQi7KbiebliPCXM0gaSjx9UzUQC3jrkRbALi8g8IIMa20AieqgQzFuSQQMB3gyIsVzt00N8SUsAoP9+mocnxRgQkksVxiaR8YspyRABqIUwM1MuvWEe7b9QOQUQIqRQnFF6rDFVD/vw5DRK4Vp6hSTZR0WMKe3NXJC65mZeiSC+a9xHhMfAvU3xrgXr+Ty4LhGI+NT/rmYw/JXZWdwOQzSQNRYhXr1+hfIIr7/L/c96aQUBKGzuHU1KRNAERGCSm06MUG8ybLJqZuHI1BDgBjkrVgkqHmvLd12i9sHQg/WyRAhLBRAk9XHV87GEYLtuG/4N06yiiqDBwRo62LQLiMyqA1IoEbjaXdhnIUF+djqNlZ24Yh/gGCgxVKNFvwpa0POzYicS9CX9EgA6RXzt0a+iK1LOaOFXLQKpwW9HSuhRpE/RftV8Lk57HrSp4sIpSgTar1ouxtFC2K2kGy3aKQKC0q9azhAhmBqznXarjtnapy27ResxSJpvcDDu9TdAwIcPvzhJRxjaKcalW7QekXktIyCFU4QVsqa+52Dol5aJ443/0xsM/ABdfzlP1okPrwAAAABJRU5ErkJggg==)
Then, by second derivative test, the area is the minimum when l =2.
We have, l =b=h=2
Therefore, Cost of building the base = Rs 70 × (lb) = Rs 70 (4) = Rs 280.
Cost of building the walls = Rs 2h (l + b) × 45 = Rs 90(2)(2 + 2)
= Rs 8(90) = Rs 720.
Required total cost = Rs(280 + 720) = Rs 1000.
Therefore, the total cost of the tank will be Rs 1000.
Question 11.The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.
Answer:Let r be the radius of the circle and a be the side of the square.
Then, 2πr + 4a = k (where k is constant)
![](data:image/png;base64,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)
The sum of the areas of the circle and square (A) is
= πr2 + a2 = πr2 + ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZkAAAArCAMAAACdDgU2AAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjo6OjpmOjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZmZmZpCQZpDbZra2ZrbbZrb/kDoAkGYAkGY6kLaQkLbbkNv/tmYAtmY6tpBmttv/tv/btv//25A625Bm27Zm27aQ29uQ29v/2////7Zm/7aQ/9uQ/9u2//+2///bQ3pNAAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAE70lEQVR4Xu1b61rUMBBtF5F6QXBXQRFWhCJSWaWW7fs/mpNr0zRN02Sytp+bP1w2czlzkkk6nU2S/dhHYB+B4AiUaXoVrGSEgvr78UORLm5HiChTi5N7+RfRNKxFlRiePXoGHpyu6ZIzU69fPo12zEmg3pyl6SHjol6/ByullRl1vm6geiPWEdPUGkbBRsLJ2XGT8OAY7FYZw1p6L+QhMHl6mTyvDsgCr9dLasvKjDK/q7p6JSimmlrDLCgkhtz0+BwPjsm4YCb/nHWxenjbFcmP4H8F3Zol5WeImWa+wX7BtjbX1GbGLMglULC0leDB0Z2rIM28Y3tm+/FHtHTGtiSYqdckdoyZAjJcc2h0wkbm19cpH0tyHP7MyFZjW0Bo6oZbE4QJqJtG8QkRToeY7DKpv7JzpjhKCrag44ycKJeZE0IMsaWLv2eQ+Xe31esHLlRlby83hBSWQYSmrrQuKFMoDq62T/wgCIaj+5aTxMBuAPX6yoI2HBSLpDhfyM/y2EYMj7zKjMi1NDn2nlQdQZhMJdBG4xMeHM257apZfRXEqT9DBKPSjkqA9CiIyUXGUm8gfH6iMiMuZWw98TuELt0VJMyg3jqNzEg4iQnPIBwLM5D1YcRKZ4JzkS/h1Eht0ZJrpJeZnsxrEIzIDB6cfma2H8ixKh5tgrdIJ23yI0XZ/lVmSTG5OIIG94xmySAYkRk8OBoMtsAoHSVdwdsVakKW5ui1tYTU2UrM/bUHOZ/ejdnJ0Rz5lnPGJIh+zjQ+4cHRl3KxuEmez8n1lN1mt+xxEHvQW2udAzPNUy0EO+973Gzm0+tilS3/wIVBEKmejpqnRkECC/VBrfEJD44e8vouXZz+ytIvK5r2t/AjBjX8TITw8BUAx8zitl6nqTlizfyEXIHr9eIC6BGzLc8zJkGy3XjZAGnFSZ8Q4SC5FqDG9OQ+Up2lBtCjKXoNYCSC1vR4zo3zSlwfx0mps211M7NW5C2jGsGD4x8QyERKxdhfj6FCPEqZtdZs1GSvNQeiwoMzKgrtydYKsDKVVBMsw+2tSp8C9Pczrqj6HMKD40+NUjG2Khlgxt+BKJKuqKIYx1TaWwFujMyLGeq3AyrMIJp1wRXU80Uv1ddbAZ41Mw6opsOMsc4onhRNFWDp+pT3jD+q+MyoFmSRV/3Fenqzd8XGCjDbUqa6sdHMrv9pu5NYUemQojgeyru1AjyLPWOKgBuq0NgNyRekHvK8cnhdYVr91grwLJjxRzUU2dDP3ZkxWLJXgGfBjD+q0MjHlLdXgOd6N3NFFTOyvboduzTtFeC5MuOK6p8wE+9VZjiczbl4+qrvMofjMdwinoYi6MmRXYJ5lyaeU0ians/k+6IyO/2NpHVHairaHRc2pspMdXIveyCgM25eY7v6FMaM6NJseiAnFQDOTKwGhYhY86W9eXvIdCW7NGUP5JDITj/nzERr6okGBvocw5hpujShDyKam/6KOTP5wTfowD598le0Y0nSJRbEjNKHMs3DhjEDzRzwXZnNJNeOkXL6FhqYKby/NDYbZuiFGbcJNuYuIj2odPwnzET9rgI+UUHZTOnSnHI2oy/s5rRnKM1BzCSyS3OihQD+HA1N0E/Jo39uwN8QDhrDagCiS/Oq6YF0sLmjKfU1NGamLy7AHHnsOrzZkV0cM7QRFUfVXss+AvsIaBH4CwvSxJKyiLEcAAAAAElFTkSuQmCC)
Now, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADIAAAAhCAMAAACsowi2AAAAAXNSR0IArs4c6QAAAHJQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6Ojo6OjqQOmaQOma2OpDbZgAAZjoAZpC2ZpDbZrb/kDoAkDpmkGYAkGY6kNv/tmYAtmY6tpA6tpBmttu2ttv/tv//25A62////7Zm/9uQ/9u2//+2///bQp0tKgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABC0lEQVQ4T71U0VKDMBDMIYKlSKuxQm1tCDT//4veJQRxHIaFh+4DMMMut7ewKPUwGMpk1r3yJwyN59rd0w3jM8uIxH1cq3dI0h+IvDFbqjBtCU5nrTsJ9VwrCznrXupgrMt5GiHORol55XU04uxelW2vn9ver35OeOYibE77LzpqYrYlSts5hfskKr4XnzchOJ2yHSifqLLi2UL5REkjA1Z9U+JLJPO7NvKSPIYMg8Qfx1vxYiaSiQQNLRqbf9f/jKlhff5GYBjJd13IElZwh0OKtR8VQ/lxOTOxOv55pC8/jqH8ltJLjm03lt8kdVdAgfw2GRvB9jdIYvmVgaeoUP43/imtiw0PeAvzB+TYEp5D7v3BAAAAAElFTkSuQmCC)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKMAAAArCAMAAAAjUcgHAAAAAXNSR0IArs4c6QAAAH5QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OgBmOjo6OjpmOjqQOmaQOpC2OpDbZgAAZgA6ZjoAZmZmZpDbZrbbZrb/kDoAkGY6kNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///bGJIRwAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACM0lEQVRYR+1YaVfDIBCEesSjmnq0MVFLrRHD//+DshwNacOakETpe+VDj1c2O8wOyxRCTiN+BsTbfM3oLA9Dyu7ewwL7RIns/pOQEsUoto+Unrevgt+s+qQLmSuyFMJwjAVdku/F2bo1Ab8KLEFnuKXO/AvGSzmFUQ9h7EIWYuQhXqgZKREZpNcYmawnIq5SYnRCS0pXmwQkMgWRrzm/XhOeAC36VWOUeDFCCsl4M/R2uYU6G7mMzGSN0dYY3ss5BtGsxl2eUrIcha7FuKMV48cOYmG14HQkS9ZhCSTGCQRJ6kTMbFapLoplsrp1Qq1MJsfo1JonSMkKK9U/4xEajqahoUdfc5F9BzguQX91qMPjFHqECvMk/crrfQ34Ct9ho7qLKABjHQq9SI1qYTfPmLsG2ojIZs/QNxQHUo6zXGSUtmczmwh+3IXyxM6eoj82VmvOmSEMTHHONPAMb8C/04hZkk7kaN8TPjr4HtSSdMoM/rHTxNZJPv+4cQyIOoe8liQ899DIujOZJ3ktydBMA+LZXiPzWpIBOfqGVovdKW8+NEAilkQ2tBaD0Dd/0HyROcYdsyRtT99f7yjfDxO5GFFLEsRAUBBaa9SS/F+t3T2DW5IgSsYIcnsPbknGyBb2jEYP14r3WJKw5x9V1H4fjhA8V3+fox7V4il6jEWK37BEQLC8EogdY/Ugby7irrUyARIjm/x2MVwycHGhRsQY1eoir/XRYDyCc0bdXoRL+hQZwMAPczY1DSJ2gk8AAAAASUVORK5CYII=)
8r = k - 2πr
⇒ (8 + 2π)r = k
⇒ r = ![](data:image/png;base64,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)
Now,
> 0
Therefore, When r =
,
> 0
⇒ The sum of the area is least when r = ![](data:image/png;base64,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)
So, when r = ![](data:image/png;base64,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)
Then a = ![](data:image/png;base64,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)
Therefore, it is proved that the sum of their areas is least when the side of the square is double the radius of the circle.
Question 12.A window is in the form of a rectangle surmounted by a semi - circular opening.
The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
Answer:Let x and y be the length and breadth of the rectangular window.
Radius of the semi - circular opening = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAeCAMAAADXVfSyAAAAAXNSR0IArs4c6QAAAGNQTFRFAAAAAAAAAAA6ADo6ADpmADqQAGa2OgAAOgA6OjoAOma2OpDbZgAAZgA6ZjoAZpDbZrb/kDoAkLbbkLb/kNv/tmYAttv/25A625Bm27Zm29u22////7Zm/7aQ/9uQ//+2///bvcEmbgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAYklEQVQYV42NSQ6AIAADC4oo7isguPz/lbJcDCd6mTSdpEASzel2cwZYZqQwbpXFGBzVMF+v9Z3LA4pX706clxkSk2n/tHMgRABPv0DTLQx3HSnbAO1m3yKsg53wdP5uSv8+BBwD99uHPb8AAAAASUVORK5CYII=)
It is given that the perimeter of the window is 10m.
⇒ x + 2y + ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, Area of the window (A) is given by
= ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now,
, then
![](data:image/png;base64,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)
=0
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Then, when x =
then
< 0.
Therefore, by second derivative test, the area is maximum when length
x =
m.
Now, y = ![](data:image/png;base64,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)
Therefore, the required dimensions of the window to admit maximum light is given by length =
m and breadth =
m.
Question 13.A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.
Show that the maximum length of the hypotenuse is ![](data:image/png;base64,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)
Answer:Let ΔABC be right - angled at B. Let AB = x and BC = y.
Let P be a point on the hypotenuse of the triangle such that P is at a distance of a and b from the sides AB and Bc respectively.
Let <C = θ .
Now, we have,
Ac = ![](data:image/png;base64,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)
Now, PC = b cosecθ
And AP = a secθ
⇒ AC = AP + PC
⇒ AC = a secθ + b cosecθ
![](data:image/png;base64,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)
Now, if ![](data:image/png;base64,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)
⇒ asecθ tanθ = bcosecθ cotθ
⇒ ![](data:image/png;base64,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)
⇒ asin3θ =bcos3θ
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
………..(1)
So, it is clear that
< 0 when ![](data:image/png;base64,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)
Therefore, by second derivative test, the length of the hypotenuse is the maximum when ![](data:image/png;base64,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)
Now, when
, we get,
Ac = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the maximum length of the hypotenuses is
.
Question 14.Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has
(i) local maxima (ii) local minima
(iii) point of inflexion
Answer:It is given that function is f (x) = (x – 2)4 (x + 1)3
⇒ f’(x) = 4(x - 2)3 (x + 1)3 + 3(x + 1)2(x - 2)4
=(x - 2)3(x + 1)2[4(x + 1) + 3 (x - 2)]
=(x - 2)3(x + 1)2(7x - 2)
Now, f’(x) =0
⇒ x = - 1 and x =
or x = 2
Now, for values of x close to
and to the left of ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAsAAAAgCAMAAADzGXLhAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGa2OgA6Oma2OpDbZgAAZgA6ZjoAZrbbZrb/kDoAkDo6kDqQkLbbkNv/tmY625A627Zm29u22////7Zm/9uQ//+2///bWFK0RwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAaUlEQVQoU6WPSw6AIAxEi6io4A8U6f0vagkEMMGFcVYv0+mkBXjTMTLWhaEbVjDNlpJXm1n3yTYxTobOaAmtDD2CkQL/kW8hfa5AxdNOyU5IQOUrOewTAC4n4Jx/9C9EPeyirUzX7fq1N3EqA37DwQi6AAAAAElFTkSuQmCC)
f’(x) > 0.
Also, for values of x close to
and to the right of
, f’(x) < 0.
Then, x =
is the point of local maxima.
Now, for values of x close to 2 and to the left of 2, f’(x) < 0.
Also, for values of x close to 2 and to the right of 2. f’(x) > 0.
Then, x = 2 is the point of local minima.
Now, as the value of x varies through - 1, f’(x) does not changes its sign.
Then, x = - 1 is the point of inflexion.
Question 15.Find the absolute maximum and minimum values of the function f given by
f (x) = cos2 x + sin x, x ∈ [0, π]
Answer:It is given that f (x) = cos2 x + sin x, x ∈ [0, π]
f’(x) = 2cosx( - sinx) + cosx
= - 2sinxcosx + cosx
Now, if f’(x) = 0
⇒ 2sinxcosx = cosx
⇒ cosx(2sinx - 1)=0
⇒ sin x =
or cosx = 0
⇒ x = ![](data:image/png;base64,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)
Now, evaluating the value of f at critical points x =
and x =
and at the end points of the interval [0,π], (ie, at x = 0 and x =π), we get,
f![](data:image/png;base64,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)
f(0)=![](data:image/png;base64,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)
f(π)=cos2π + sinπ = ( - 1)2 + 0 =1
f![](data:image/png;base64,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)
Therefore, the absolute maximum value of f is
occurring at x =
and the absolute minimum value of f is 1 occuring at x =1,
and π.
Question 16.Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is
.
Answer:Let R and h be the radius and the height of the cone respectively.
![](data:image/png;base64,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)
The volume (V) of the cone is given by;
V = ![](data:image/png;base64,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)
Now, from the right triangle BCD, we get,
BC = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEUAAAAaCAMAAADWi2E5AAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOma2OpDbZgAAZgA6ZgBmZjoAZjqQZrbbZrb/kDoAkDo6kDpmkGY6kLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///bHf8xdAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABIklEQVRIS+2TXXeCMAyGG52izuE26b5QK+habf///1uTRrB4YBz10lzRc968SZ4EIR7xDwEFV0ft7D5+7gDavv/ewUWPa5PdG8C8h+elLs+qNLv4FsWgZUAnCd9sLcSlzn1uouJm0obJpr7rMuEqsc48x1jUS9tETuLsCkLvsQ5fGiDbhiJFO5bgooHKVLrjHp8rzDXJbFniKKqDLfdCxU66wxegtX1FLCYJc2hvomva8WzkUpKy0u32+dAbhD2bhFJtiltgl7y6aOZJS6Lvcx01oCiHXbovBXs5pE/xSoWTww2ff28XD/fsSAMFyHjP/V2cbF6lk6OCqZ5gdM1EV+eHHzX+OwVTxgLhCroC4aJNHgjXYVNc082hmqhudnwYIIE/308U1JkbtcsAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
V = ![](data:image/png;base64,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)
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Now, if
, then,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMAAAAAxCAMAAAC7zFaJAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjpmOjqQOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjqQZpDbZrbbZrb/kDoAkDo6kDpmkGY6kLyQkLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29u229v/2////7Zm/7aQ/9uQ/9u2/9vb//+2///b1lp/cQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADcklEQVRoQ+1ZYVfbIBSFOm1cnbbOrcZlNtq0UbNk4f//ucEDAySEUEoaz5l88OT0PC7vPi7keYPQ5/h/K/Byh/GNN/3dAs9+es8OMbG+fUT72cYTijwcM9tz0e608tKXAMPis0lOd/L8GJx2XiZEEmM2rp714GzZWwuyX3Ty0lGqZM1mp3iNqtXZNlhVzYj1ao5QHumS2VuOQDp7RFXckpiKkuIvnAAFRhm+D0jAhEji7q+Z7Qgb89JR8qjJughJAErBEEkCuqFjifjSBX0qML7fsa0oaP6FvXApVYYZpb7doFISYIFhB0N82pRft2IZUTsmiTK6WueXm3rFqNkIVMk1PTM9KLtIuUYVKoFoCMQWgTxih7aEv4ODEbx+g9tGLwNHUQeJnRAHl5QB74jq0rTc/Egay5W+q00EAVYegTKMKEo2fHdDjgZRr121Ood8pHSHFmVnpr0DAkWZms5howKOBrG1+QVmpTqAAFebEUWmm13Q/IuQIpKIBZUAzxc2hcC97kSAi5DfjkaUhgC8j0kakICCmAGB5Z8NglcQfabVcrqzgWsVs+IiI0pDQJyegAQURLjHY9o1siaAMUjx7HfE3guDo0po3A2I24QSsvcZzOUzwKcCPe2jD9REc4zt40S5eC1rah+9gE45aaB9PGUqfmvZ20c/zBPPGmwfjU3XiZPUlmuaQAw/u7ePEqXFSSKe4qldu2Pbxyn3orMD9NXfbR8/nIRaJfNvHyevPU/Av338IASOaB8PYZD5n+/OMiYH7JBcfGLJQ8Bu2OiA+WR1wJz6R8B/YsM7c8NMCsVFcDXZ7XHhjS3JwqDRVNpZVpNd6emtcdwBG2t0NUp+6T5gv8mu9/Q9cdIBG4dCV6PlN/0I9Jvsek/fHyccsHEIAKqmUZZIY/EiZDHZpSVMMWxmPHfARhuNRv++sjWe2CUqLF767rSY7IolbI1z9Vv9GDYarRKwVurv3GDkNbOa7ECAm7l9caoD5pefyyyu0ZdXUBK/RIU/p5nsnU4RriGwhHvNeNUBc0nFM0ZoFMqewSXqaDDO6fcvsIR7h+KAeSbnMk0IhsRnW9FHOBMQPb3LKqPE6Bqlvqi4RN0JcEt4sqFrlMQXe3F4XT7LKZbwZASQrtEML8QRcLB4FUt4yj3QalcH/SQ9xbZkob9nTUEi4Jr/AOsvZ7Yx7MxJAAAAAElFTkSuQmCC)
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![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
Now, when
, it can be shown that
< 0.
Therefore, the volume is the maximum when
.
When
,
Height of the cone = r +
.
Therefore, it can be seen that the altitude of the circular cone of maximum volume that can be inscribed in a sphere of radius r is
.
Question 17.Let f be a function defined on [a, b] such that f ′(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
Answer:Since, f’(x) > 0 on (a,b)
Then, f is a differentiating function (a,b)
Also, every differentiable function is continuous,
Therefore, f is continuous on [a,b]
Let x1, x2 ϵ (a,b) and x2 > x1 then by LMV theorem, there exists c ϵ (a,b) s.t.
f’(c) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAH0AAAAXCAMAAAARMbArAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjpmZmYAZpC2ZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm29u229v/2////7Zm/9uQ//+2///b1rPNRgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABwklEQVRIS+1V21LCMBBNULF4Q/BKFQoaq6TN//+e2+wmTdK0Dcw4ozPmhWnI2XP2ztj/+cURULur9448cfP2s5LVK+cLxlR+u48QVZerIXoCs2Olq/yciQWQg4LYqWabfnoCHykd7FaZdk6edMOOrGJqYrJ9CqND4MOkfzjRlLz5aJzoOa3zajubf3mvEJwkvcWRZA3MOAcTdFNwPt2L5qI9rmNql7llSGCSHgNH8yYmNpkoX9JFAb8q9wutcMOiyuxsbaUh2DgTAWvp9RI89I6l99lVDr4H9VfYxCNreXH6TPwJ4FhGVW6qDA0I81ll1/OgAEJ2eG20++wQhASwrjKf3UTesWw1jPpupbeyPHBy5Fn98Kh1fd5zTkEYzbuVHgMHadM9HFadKRz1sqmXkKr6bs1KfFQv2zoIap4Fee+CmScdI+J0HBmgplEFMMlm8DavcMqN97uZORGwK91kw5k2Urc7DQyV8wn4ruc+BAhFDMw6AqP0KHhwTrfV3R2WJaY9Bd8d0wR2pAeNFHyGO04Q+fCOQyMd6QROkk4mvP0ugVyuWOJ+D6QTmI2s595wYIsOrnYP6+13A06UPpyTv/3vNy5ANQPWlj1HAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
Therefore, f is an increasing function.
Question 18.Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is
. Also find the maximum volume.
Answer:Let r and h be the radius and the height of the cylinder respectively.
Now, h = ![](data:image/png;base64,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)
The volume (V) of the cylinder is given by:
V = πr2h =2 πr2![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, if ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, we can see that at
, ,
< 0.
Therefore, the volume is the maximum when
.
When
, the height of the cylinder is
.
Therefore, the volume of the cylinder is the maximum when the height of the cylinder is
.
Hence Proved.
Question 19.Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle α is one - third that of the cone and the greatest volume of cylinder is ![](data:image/png;base64,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)
Answer:The figure is given below:
![](data:image/png;base64,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)
Let VAB be a given cone of height h, semi-vertical angle α and let x b ethe radius of the base of the cylinder A'B'DC which is inscribed in the cone VAB.
In triangle VO'A',
![](data:image/png;base64,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)
VO' = x cotα
OO' = VO - VO' = h - x cotα
Let V be the volume of the cylinder. Then,
V = π(O'B')2 (OO')
V = πx2(h - x cotα)
Differentiating with respect to x, we get,
![](data:image/png;base64,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)
Now, putting dV/dx = 0, for maxima or minima, we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWIAAACpCAYAAAARBDe5AAAgAElEQVR4Xu2dBbQuPXWGH4pbcXd3d5eiRYq7F4q7FPefYsXd3R2KFHd3d3d3h66nTWiYfz6dOZ+c781ad917z5nJzLxJ3uy8e2fnIKQEgSAQBILAWhE4yFqfnocHgSAQBIIAIeJ0gkkInA24MPBz4HXA1wNVEAgCe4NAiHhvcN3mWg8O3Ax4E/AZ4JTAM4E3APcC/rLNH5d3DwKbiECIeBNbZb3vdAPgPcDnmtc4HfA+4FbAU9f7enl6ENh/CISI91+bDvmigwLPBZ4N/FdTkVayRPxb4ALAn4c8JPcGgSDw9wiEiNMjWgQOBXwEODygFfyz5pdvAY4HnL4QcpALAkFgJAS2hYhPApwMOG75bpfHfx0Jg1Tz9wicHzgc8NrmxxLzR4HvABeMTpwuEwTGRaBLxC5N1QhvDJwB+DHwVuCRZWk67tPnr+3qwBmBfytL5mvNf+vGXvkPwBWAIwJHAw4NHB94AfD6DXvrS5TIiasBL9ywd8vrBIGtR6AlYonhAOC7xUtu2JKW6GNLGNNty7/X9dHHLg6kWwDPWtdLjPRccTcyQR32a02dhosZnXB/4D4jPWtoNYcBlCWULMR+lVETVy6Ow28P/YjcHwQ2GYGWiC8LHBJ4ceeFtdh01JwAOC/w4TV9kFbZq4t22Xr01/Q6gx57qrLU93uuCfyh1Cb+HwJOVFYAXxr0lOE32z+cnI8O3BT44/AqF6rBcLmnAd9c6K5cHAS2DIGWiF8EXBK4PvCSznfcrVhpDwdut6ZvfBggGZ8V+M2a3mGsx54TeC/wjTKx/KJUfLDyc7/R6IR3jPXAJeuRfJVL7gH8ack6lr3NFZqW+LVDxMtCmPu2BYGWiF8FXAZ4InCTzgdcHnhZ+XPFNXycBPVu4CuAevG2F0nG79DSa8lWnVgJwOiE03Zki1V/86WKg1T/QHWM6jC1DVYRvmZftM+dOES86qbP81aNQEvEWj7KEzqLftR5EUnjeUWbvW7zO6UKveg6+aaVVwAfby44BOBAP2nxzn+6/O4cpT5J1z+VACQmd3ndusS5XrQs3b8AOIH8btXA7dHzdEh+EHhOcZp2I0NWhbftcBrg6Z3oFJ2kxhm372UfOnNpNycYZSw3hPSRtdeepVj7Qvj2InW19R0VuCpwX+DIgKsw/RUWLeR3TsD+KID9wj5lhMcbG8lnj5or1QaBcRCYN3ztScCNinShR98B92Dg9nO8hrriv5ZNAl5+DOB6hWx+CDwOeA3we+AHwGeB5wMvB55R6neCeCWg88bBqZPLUCqjO/yZpL7tZOymCZ2QEspVOjG8q8RbfVpdVpJsCdJViW0pQdai38AVlH3Cv9W6jbYxuuI/ge8310qQ6s1O9La3fe/ShXSVvqoeLqlrDUvY/j2LiA21u2cJuXtgsZ6dsHR4ajRUZ6j6u31POSglCGwUAvMQ8XGKhWHugesUrfCGRTtUtzXo3wHjklXL1phTifbRPV+qJXxL4FGN4+cagPXokVebNkTNQX3vJnLg8UW71iL2d7W4bP0yoAX3gRGQPXWxxHVQLlvcfabGqwU/q7iSUAbSGeY9OkLv1+MUWxXeDwDeDJxvwov7rhV/SU1ZRevdd67lPGWV4grKtracsFiz+h8k+LYYKaLlrSbeRpA48WsAuFKb5KyznV5a2l4ybyM61LWNidbvob6tLm9/V5tPCQIbhcAsIvb3Wqx6+S9XSFeL6WIdQrxLiTXWieaOLPVNrdpuuXiRPdrIizsBdwQkQS1kZQgHjwP5V4CW4vsBQ5jUqlunkQTsUngsIl5n4xhLrPWnxS+BOcFYVo33vBioHTsxmxSotXy1Rl3NOJEa+mgfsi0lY3/XlSy0tJUyXAm1stcsInYSMxmR/e3cwK87L36uMlEo92ggiOlTsitw3ubNdatEYBYRO9BcZmq11u2uDgAtj7ps1SrRGpFMLQ6mr07w+DtoXYK2ZGoI12GBi0yIUdXq/WTRTLubCbSiDXFyoujq2qvEcaxniYOTjphKWlqIq8Z7nm9RDlCHdeL0PafFFh+rkKzx0Wq/fcV2dZK2HY1jt8wiYvuLOrAE+5ieSk8OfL5M3oYE6svQek8JAhuHwDQivhBghIQW67RwMa1U5Yi6ycKlq1LCx+b42iMUJ55LUJfFfcXBq+Wj86haiV7nuzu4JXU14v2y5fkhwB1KljMliW7Za7znaLb/XbWotbrrUmlhWjl7mVzUnfUV9BW3rLujs13ZzCJiZSw1ZnNf6LTtlvqO+hHMoeHzXWGlBIGNQ2ASEesocYC5VG6tV5fJWru1eL8andKE1odFK0ULeR7NVj3xXcWqMkqir6gf6gBykLYbCrSUjcTQKpaot6m4MhBfLbQaQ1zfvxKQy2nbQSfmKvGeB8djAl8smraW5rSi800pymiLSVvTbWP9Cn6v4XuWPiLW+laKcpVgnTUKo89RW4nYnBk6lrva9DzfmWuCwEoQ6CNivdtaw1op7ZLTGFcHh462Wk5RSNDB6GCoGxLuXqzVWR8hgbu01Nr9aXOxDrsnl51+ErohSzfvVObGkjsXjVCN0ugJT5IYYvWswlmnw1L5Qe1SL78YtKUSkHKMDry6685r9hrveVcV6vZOoEooEm1Xn/Vd1bpt05owyIn6nyd0CMnSbztTMzH1EbFYPaKsfiRsrXI3+fSVSsSutDQMUoLAxiLQJWLzOWh1uM25OygdJFqmNaTMj5KUHWh6wy2ViHXWdKUGn6WEYFSBnnmvfVshez3m9XlaujpftBYdnJ8qJGssci11g4dWo0ta/69l7CDd9CKJGSsslnXCad/ZsCuJw6gTE7G3Za/xXgQ7Jz53Y6rV2p5t0WL1GwxX0zmn/8CJUxlBXbktOik/UdruQc0v3PptH3AVVqMpbOMnlFWaKzH7no7jvuKKw/fqw3iR78y1QWDPEWiJ+EjFivWssknFzq0uaJG0lQbU4NwBZbE+g+6Ne/Xa1kMukWvdKkH4OyMjlBv8d002rhPoNsBDi4UtwUs+WqptCJNWpWSmRWl0hlqh1pEDehuKGvBPim7Zvq8JdsRIS9Nwq281v1wF3otgZxsbGWE0jVapcd216OC1PYyGsDj56FBzsnUFVCUmf+6EY7+RZFvpyRhlHYJaxpKuOq/hc3Wy1RL3567e2rA3/Q7q6EpoTgRu+FGa0OHsiqldeS3yvbk2COwZAi0Rq+HZWScVtWI90VUjNqJCq9cldhuxYNYwrRkHSqt/SjJaPD5Tp5vWrHqyFouxokZluIOqhq35Hncty1UHdjfsyU0exi8br+sfHXfbUiQgnaCSmadheDCn5Cuehv45AXUT/qwC72Xw0yK2fYx2sP1sc1cxlYRrnba77aXzTilDUjZKxAnVyJk+WUTJyyx1JnmSQO0bbaicmrK6s+lalcacDOwnbv7x/4a2OVmoK9tH2lNHlvnW3BME9gSBWeFr0x6qVepA6loYk36+Jx+w5ZW6kcPVgBKM5ORqw1VGHykF7y1v7Lx+EJiEwBAiDqpBIAgEgSAwAgIh4hFATBVBIAgEgSEIhIiHoJd7g0AQCAIjIBAiHgHEVBEEgkAQGIJAiHgIerk3CASBIDACAiHiEUBMFUEgCASBIQiEiIegl3uDQBAIAiMgECIeAcRUEQSCQBAYgkCIeAh6uTcIBIEgMAICIeIRQEwVQSAIBIEhCISIh6CXe4NAEAgCIyAQIh4BxFQRBIJAEBiCQIh4CHq5NwgEgSAwAgIh4hFATBVBIAgEgSEIhIiHoJd7g0AQCAIjIBAiHgHEVBEEgkAQGIJAiHgIerk3CASBIDACAiHiEUBMFUEgCASBIQiEiIegl3uDQBAIAiMgECIeDuJJgJMBxy1VPXXCmXPDn5QagkAQ2JcIhIj7m/WgwA2AGwNnKKcEe7DnI8uJwO1dVy8nWXsatacEe6pwShAIAkFgbgRCxAeGyiPuDyjHwz+zHBGv1ftY4MLAbcu/2zuPXY58vwXwrLnRz4VBIAgEASBEfOBucFngkMCLO786YrGGTwCcF/hw8/tLAK8GTlcIOZ0rCASBIDA3AiHiA0P1IuCSwPWBl3R+fTfg/sDDgds1v3sYIBmfFfjN3OjnwiAQBIJALOLePvAq4DLAE4GbdK64PPCy8ueK5XcHA94NfAVQL04JAkEgCCyEQCziA8N1fEB54gXAjzq/lmifV3Tg65bfHQ/4DHBr4LnARYvz7guApP67hVokFweBILBzCISIF2vyJwE3KtLF68utkvYrgSsDRy6RE98pERf+7FIh48VAztVBYNcQCBHP3+LHAT4KvAm4DvCncuvji56sRaycUcuJgS8D5wA+MP9jcmUQCAK7hkCIeL4WF6fHAacCLgf8rNx2cOD9wLcB9eNKzv5aAn5fiHg+gHNVENhlBELE87W+FvDVgGs0JOydWr2fLJs/XtipypjiexXy7mrN8z01VwWBILATCISIZzfzhQAjJO7UE5p2VcBNH6cpMkStTVzfUCxkNeK/zn5MrggCQWBXEQgRT2/5s5TddMYNt7LDiYCvAk8vW6CVIf7Y0Yc/DmgVS9QpQSAIBIGJCISIJ3eOkwJawybx+Utz2aFL5IQRFDrh3gncvFONmz3uXHbafb9EVLwO+NWUvmiExXuK3pwuGwSCwA4hECLub2xzR7iN2W3OXVnhTMUKfi/wqUKyr2iqqRs8Pl20Y/+vZfyIGf1KPflpwDd3qP/lU4NAEMjOut4+cKSi755tSg8x+c8JgUcBp+6Q5yGADwIPBJ4PXBr4BvCJKfWZaOgtwLVDxBmXQWD3EIhFfOA2d3ec0RGTilrxyct2Zq1joyn+3LnYTR5uk3bHnX903E0rXuvWaaMwYhHv3jjMF+84ApOIWKtOb7866WsBl9kWnVIXLLkVzK+QaIBhHeiogJEX9y278nQK/rxUqYWs/twWN5VcHDgW8DHgzT279g5TtlibqN76nwP8EjhP+fOTkinuezNe/Shlu7Z9wI0sbwT+MOxzc3cQCAJ9CPQR8TGA65UB/MOykeE1wO+BHwCfLUvulwPPCKyDEDhzsZyNztAqnkTEtpO5LSRUJ0bb4nzAfwB3Bd7WvMXRi/PwYsA5gVOUyA+v+XzRviVn82boHOyWwwH3BPxbeUULXb3crHO+w9fKDaYKta8ou6QEgSAwAIEuEWsJ37JonzUcy2W6aR51OJkW0pMo3Mp7b+A+A56ttqpVbZ7fZctvS+pJl//bXMxfYRSGCYf6pAl39H2oTHxthMYVgCcUDLqEeJFixd6hRH7U3YDi5EkiFlc9bbEtXlqiQUz52UaL3AM4f8mzoTwjydt/dFqmBIEgMACBLhG77HUXWJv03I0MdyxOKS1ks42Zr9csZNPCsQa81kpvNVphXaVOZLOI2GT0yhRawzdtJCGJ++vANUt7tN8hab69TKyP6XygIXnq2238s8dDGfNscvtzA7/u3HMu4B1F9lCqcsJ+CuBkmBIEgsAABLpEfErgS53NC548cVhAC6u1kAY8dqNu3QYibgFTEtBCdkVxWuAuZZXy5A6qlYjV9CXktkjE/v70DZFWC1qC7RK39+qgVNowp4bWufUqcaQEgSAwEIFZURNHANwh5rL5AQOfldsnIzDLIvZOrd+7l78lXTVfdVw1W+WiSUR8gWLJziJiZSdD7SRncyl3iysh5Q8PVD1UiXneDyui9MsgsHYEZhGxnvZ3FWeNem7K3iDQR8Q6yMzeph6rXPDfJSG9zrkaLlfJcSgRa2UrR9kfdBz2JbOvz1IeeXCPlb03yKTWILADCMwiYpe9LlVNavPTBo868IeEr8VZ9/+A9hGx2LsbzwgJ5SFlI3XdXzTt0CXiM5QVjJdUaWIei1gL9yPF4vXsvb5Sn+XKSEdeShAIAiMh0BKx/9aLrvPF+FS35rr8VRd2MFfSddOBzpzogyM1QnG2iafJhGp4mFEqRkSY81h56LtFl20nv7piqROj91R9dxEi9kuMljg8YNhbX3E3of2iz/oeD4nUFAR2EIGWiLW2TGKjBOGgMzJCr7r/lohdDqtJ3gZ4aI7/GbW3GBXhpgktYwlRC9WDS2t+CjX6K5X2+GJ5sm1xA+BaJWLCa29bYpG9xDZzIu2ziM0ap/TROuuMafbZJjqqk4H16CfQQWe2uQPKOXxKE4Y1msioXSmNCkoqCwK7gkBLxO7IelDRCT3ixxAld1NpAekoMg7VXV/7JWxt2TY2L4Txu8bcHg0wG5v4eNhoPcdumbqNQrgZ8LlCbuJs5jaLGq7xw4aQ2S7KFWrHErQTqDvztKhNUm/bWY+TqDKDMcPmvnCziNKFFq8bM3x3oyf8XU3VqT4ssf+4TLSevecE7Jl86sZq1caPq10bu13jkZf53twTBIJAQWCWRhyg/h4B8ZLkJKDWanTVYD4Jd58N2eQSvINAENhBBELEizW68btKCDrP3ERRcy9osRpbq8Z7xhKLvVjNuToIBIGdRSBEvFjTu63XLb3G07pMrxEMOjb9+VknaLKLPSVXB4EgsFMIhIgXa271YZPlmA/C7b61qBMb/mWIl7vdWtlisSfk6iAQBHYOgRDxOE2uHKHTS4eZkQxD4qvHeaPUEgSCwNYgECIe3lTG+T4LMH/vVUp0yfBaU0MQCAI7g0CIeLmmNlOZcb7m/lUXdnvw/TonOS9Xc+4KAkFg5xAIEQ9vcuNxjdE9ctkObhxvShAIAkFgbgRCxHNDNfVC04S+v2zycMdanHXj4JpagsBOIBAiHq+ZHwLU0zBuOF61qSkIBIH9jkCIeLEWNgOau+iMjmizoFlLzaDmFmS3CrsNOSUIBIEgMBOBEPFMiP52gef5KT8YquahmqapbEslYvM96MDLicfzY5srg8BOIxAinr/5DVMzVtjEOX2pIM0zYZ7eRwO3mr/aXBkEgsCuIxAiXqwHqAH/pBwT1N5p5jpTiJqRzW3Q31qs2lwdBILALiMQIl6s9bWKPdXarc7PLicoS76eWuHW5usn4c9igObqIBAE/u+MspTFEXAjhwnXT1GOnX9rOUUjW5sXxzJ3BIGdRyBEvPNdIAAEgSCwbgRCxOtugTw/CASBnUcgRLzzXSAABIEgsG4EQsTrboE8PwgEgZ1HIES8810gAASBILBuBELE626BPD8IBIGdRyBEvPNdIAAEgSCwbgRCxOtugTw/CASBnUcgRLzzXSAABIEgsG4EQsTrboE8PwgEgZ1HIES8810gAASBILBuBELE626BPD8IBIGdRyBEvPNdIAAEgSCwbgRCxOtugTw/CASBnUcgRLzzXSAABIEgsG4EQsTrboE8PwgEgZ1HIES8810gAASBILBuBELE626BPD8IBIGdRyBEvPNdIAAEgSCwbgRCxKtrgbMBFwZ+DryuHDy6uqfnSUEgCGwsAiHivW8aT36+GfAm4DPAKYFnAm8A7gX8Ze9fIU8IAkFgkxEIEe9969wAeA/wueZRpwPeB9wKeOrev0KeEASCwCYjECLe29Y5KPBc4NnAfzWP0kqWiH8LXAD4896+RmoPAkFgkxEIEe9t6xwK+AhweEAr+GfN494CHA84fSHkvX2T1B4EgsDGIhAi3vumOT9wOOC1zaMk5o8C3wEuGJ147xshTwgCm4xAiHg9rXOJEjlxNeCF63mFPDUIBIFNQSBEvPqWOAygLKFkcYtYw6tvgDwxCGwaAiHi8VrkEMBZgBMAxwFeCnytU714HwAcHbgp8Mfye+WLC/W8yseAV/b8/DzARTo/fxHw2fE+JzUFgSCwKgRCxOMhfWzgRsD5gPMWJ9wXOtVLvscH7gH8qfmdpCoZXxU4OeB9LwbeDryx5xW9/ibAFYHPAy8HngJ8ebzPSU1BIAisCoEQ8fhIPwz4J8CddH9oqr8UcDLgkcBfy8/9/1ea8LW7AfcvksVjZ7zaqcq11wZ+M/5npMYgEARWhUCIeFykjRvWitVK/dem6nMApwGe3pCwv75WiTOuxHxN4DmAZH77Ka9mu2lVPwn43rifkNqCQBBYNQIh4nERV55wB50SRY2GOBHwtELQlXB96sGKRnzf5hWUNN4JvBq47JRXU08+apEvxv2C1BYEgsDKEQgRjwu58oNOMzdvKDm4g+7NRTfue5I67xObX5wU+GLZdScp9+24Mwb57oAyRqszj/slqS0IBIGVIRAiXh5qsTtr2ZBhtIIbNh5eSFcpokZELPKEIwKfLAQsmf+y5+bbAa8pDr1F6s61QSAIbCgCIeLlGua4hXS1dpUdDFe7KHBD4MMlNG2Zmg2B+xBwkhI98e1OJW6HNrriMctUnnuCQBDYTARCxIu3iyQsAT+4kznNXXLPBy4PvGLxav92x+uBiwNnBz7Y1CNJ3we4X6IkBqCbW4PABiIQIl6sUdR8XwYcqWzAaOUHnWsS8akHJn03V/F1gH8BXtW83nVLNIZZ21KCQBDYRwiEiBdrzJoj4sYldKy92/jgcwHnHuhEu20JX/PvR5QHmKXN0LYHLva6uToIBIFtQGAaEZsTwZ1gZg6rxdy6X+r5ML39EkVbfge4KeFX2wDEnO/4EuDSPbvmtJTfX0LPbj1nXZMuu3KJvHhCwf8fihxhbPGPB9ad2ycjoC7vBhulJ4sJ+9tww2AXBPYMgWlELAHfvOTSvV5xSNk5dUh1ix5+NdIrlY4sYZkn4fGAhLwfSs0tbEiZ0RK/bz7qxCXaQQyMAR5S1IYldaMwDIf7Z8BnK4nsanGjjCeduBI5Q5mQ3lp2KY4l1VwdOCPwbyWJv5tt9nMxJ4qYGplzaOAX5Sivvi31+xmHjfi2eaSJI5QOLyG4CeG0JY9u3wdIxFqHaqX7rdhZP1E6q/ptWxzE7nLzPDojHRzEYrDMyRsOEFcdTmSXBO5QYoaXqWs/tIErAhMlfbec9efhq1qvrrY8jFUJZ9Z28HlxqBtyzIr3rHlv2sLrzFWi49ft8ca7W5zgDL90R+hXt/Cb5nllV5seW9aNRprn3j29Zh4iNmfCCUt87C0B41htsL7iGWxu0f3Jnr71eiqv25dNrKPjrBYnHp1qOvDMiuZkpeX2qCVf8x+LdW3biKXhcX1y0JLVb91tOkEP2bOL0JhrrWEnLje/GDY4tOgDcEXjCq89Y3BovZt0v99mZI5G03ubF/vvEoLp0V3v2KQXHvFdPKzX8fTNEescpap5iFjrwGWyUsUHykYCNyx43lpbtFzuDPzHPtbWjGbwRGZjeU3oY0jZ9QHxcLLSQpMUxNWtyssUidwBovzROuyWqWs/3ONORVcG4qzk1ZaaJEnDQANhaFGHl4zFfj8mUtJoeF0Zn4ZItieImwvlnGUlsMxmpKHY7/X98pN5wF0FbCURmw3MZYxLY4+Ad1njwHBWbYu5D9RI9/NmAxvTPBI6J39YzqAzXaVOHSMaXPJ8v6SkbDv5op3Mg0a1jN0ksl809kUxqNe72rhM2QrulvC2GLOtdu4fU4IOKU6A7y5LdaWm/VhqLhNXtvt5nPa1nX3IfqI/Z+uI2LwGtyleez+uhm85OC7XsXxNRKOVaGeeVY5S6jIhjidV6CDozsIuPSX2H5UNEm1ehUqITgw1+brEpQYkMep42Oaixiwun9nmjxjp3c3frDzxgtIX2molzOcVK66ViyQczwJUTppW3Hjz8XKBIYLibdSL0UFOgjrvzA1tf98PE6IWv6ssc2a/a4H2WQZPq3fFqMNZw8VV9afLM11B2j5yhX/a6BRXk2cuv3ecKz+p6/b5SLzWwxiUUyxmPlSiauvTQDTPt8m1jlxkVf0MFi3kSStXOco+4Lt7vqQc1aa1XQC+2ZfOkib8SGcQyc2iw0qPvj/zd6Z7rMUOrIPjp1Mea0jcvYvMoQXpzGSD3BMwMuPr5d5jFPCeXCYBZRDTPtaiziVROTG4y82i/mPdSiljOW9mI5gr1omADlJXKHWF5sB1x+O0FKL1fZ34dUw9u/xAsvc0FCdzB6yrEg93Ve/3ZxLKNpOx2Bhpoqzm4QOG6jn5+K0aMZJSN2JiCJ6OYce0fg5Xj48rOVKMNvpBOU1Gh7aHGjyjtIF6v0mwXG37t8SnE1GD7D/LarO2nwSpE9cJ2twrcpmhpZKuklX1q0jqWsPylX8rY00jYiVY+ci/K0c5EakMONlXw0+/hd/4jTEGwCwiNo5Yq7N6Vn1m3XBgh//38hLz6MNauB4fpM4sUO3SXceWjeBS05+r9xlHq05nDl+tIpOt1+LgeABgcnQb1aLH29nNBvTdUoYh4A5BrRXbbdniBKreuheWvfk9tFTeVHYiumIytNK+ouX3szLw7LtaYq7uJIZHT/gYQy3VoTUo2ox4Gh06aLXi7LvbWiQOrUW1YCcg/RD1aK3DAjrr1OBbR/yyeGoJK384rutK9xqlXTSUfI5hguKs8aT0KanpJJS43cZfiw5wVySufKzToj/GicP20gpui34aOUMruT2qzAnbidv+MUmamMZRGoJOYk769jX1dPmzdXgu3TdmEbEAORO0soGE96my/HfpZoc/WrEanPX6iktEt+5qybrz7NediwTJXWSnKM9y5vLoHyMRzEbmIHH2q8VZ1KVD9yh6ZRTBH5LroT5DC3vbi+23H4v91r7mRKxEZh9U5rpYh0TvUkIvndDte4Ze9oVW1g05avwaA60MJgG7PN52Iq7hl1qSOtUf1OkYWv2uaCUbc5wMwVNHoJJiG8lyJ+COJQWAFrLWuKQmwbrpy52pOsMNAdXPUovWqONdwnala9t7j2TclypWrV8pw0mmlatmEfEsjnLXrBOFnOfE7qQgR3WDFpYab9OIuKsPtw9wEGgt1xMmnIV8oUmzgwNEy3qSk0D9xhnH7GK1EbR0XWa4iaSVQVwyaAlpXduh2qJmqNUyxtltIeKlutRKbnLAulzVypKELQ4kV1NVH9S6ceXl4Lc4KI2P7QvNqhty3OBQE/rXD9GCsy9I+pLLkCL5+B6ztOtJz1DPXtbIqBuS/Iq7BSoAAAmISURBVA7HqzJFW5yojJNXJtDSHIKnZKo00E5ohgVqeevs7zqy65iWoCXXaY7uYxWSlU/kh75iGzoZ+K3Gn1tmEbHvpTQziaOUc5RinajNkKgRqPU+SplGxGorBs5Xfbh9oIQp4dlwAqc3W4u3ai/dl5u0Nbhep5CvA8GPrcD5bup0dqC28dSM3OwguXc1LXcCuvzYj+E3ozT4PqhEp7ARElpY00LMHDAaE3VjhstgV1b2nW5xQNt/Xba3k7h90AEvoagRD93yvE4iVj5UunNF6vh1pdkW3029U9+Ly+7uGFoEzy6+bgpzEnFsKil2S322k4OTxLRSd54aD9weR9beo/HmpNquYmYR8SyOqu+oLCon+fzR0jdMI2LJVd2o1Yfrx9qoNZRN4ASnu9Sp11Ztyv9r2bZbg/2ZEoMaoqTexjYqd/hzhXMHUC0C4bOUMdoljEsvNcD22n3AO/mEBgH7j/1NHbO1tlxGt7vB7NeumJQmqkPZSVsLuU/nVVN0gu8m9NdSlkC0iiXqbS86tpx0phGxy+7uOF0Uzy5O6rxGaWi09UVVHbOcTKOUoaU5rWggep2RLZO2oduecoHf4cRi6SNi30fZSevfOmsURp9TthKxRqM+qK42PahvTCNil2POXpOsyxrDqRyhiT5JH65LImdboxy6xWRBLofaCAivUZNR62nB9OcOCC3nrj6ktqX2PMYOq0Gg7pObN81Zp7apNay10y5dnYAdZO1ORidp+4mD2kFVN8l4xJQGRFvsn5Kz1qIrqrboNFb+ctnupK+O6oaI0SyhFfcVJQctuT69WzlBXdUoJMO2Wut/ETz7PskJ0SW/K442qkqHnZFRto9ErZwk0XZ9SNZpdIf3uspRmnSCNe1CX5EsfeczNaGsfUTse+mb8lsl7Ekc5TMqEcuJTuijlklE7McKXN8yor6Auo6kJyl2k5h3X1LrxDqVE9oi8M6Q6l4Okrbx9Xq+rTgNWu+nIUbW1S5htNB11DkYtZQU8rWAnN3VjUcR1EdFPpUtgoAOYifeunmmvdfBpjVbQ6D8nf3APiLxWCoR6/Tp9mkHrM5nSbbVX+sGD/uQy1z/r2VcU5Mu8v6bcm11tPsdOrzaUg2fSo7t7xbBU05RxnHMSeri5jh28nRM1zHuakOZpOqsNeugMmQNSa3voMUq+emwN55Y3d9JUsteXbktrqRdXdtO7Sq9npDu6qnyiTgYnSVnTOKoWrd843v14TO4fScRsZ5oRfFZy3zjNSVsB8IkfdiXdJbzQ7VoKgjqRlo3zmxa392DMJUsJHrDiSTfWo9A2KCSfz3TTevcQzcdUBYHjnVb+nYBDgZuDRWY88POIM5aZTXmeg2vstJHGjmjFev3Tyqt80myUU5QwqoZ6+znhjs5YXttuzlAspZoXAG0YU2GYBk9YCypkRbGqGoxOci3udy15EQxbrrF4SHl5xJhq70viqcToysMDSyxdvxpgftvx63P1IjTcHpoE5tt2xgZIfe4OjaGuxYds2LvCtlilIs7AyV1+aeu2v254Ym2tyTbruYNj9WS1jKWi1wJKb/WibWPo3yWPCW/KH05ERhKpzSho9hxOG3fxNz9pCVitVwjISTVq5RUg3bAGvvbV6lk7ctJfLOK8aTOSubUNVBba1gLpT0OqFuHg0Oyd4DYOSQfrSK95koWRlZ8rwCsJ7MWnYwOXpc4Buzb4Nta7FzmtzBeVs3cJaTLbr/PCWzIVuptwEQt0E4/qTiBuyqrGrF9Q6vXMKM2ysFQPvufA67deSkxSR4O9u7uLcnKUEpx909X1tgG/LrvaH8yH4ySjX9LVo5fCVhppvW7eO+ieLppS0tUbtHx6YrC1akGlDG8RrloTNSwte77+R62hU57r7OtNLAqCdfrrd+20SBznEvKRmXIJ0Zo9DlWlaocSyZ0kkB9h/Z75RR1ZzlKfJwM7BMagv5ficrJQl3Z/mAwwShlVhzxKA9ZYyVKFA6mZTOhrfHV//ZoB4mdsM0GZoewM5jtrlr+m/Cum/AOWrIOyK6lMunnm/DO63gHd9YpFWoZKh2okfaRV/BcQevsdyJ2OemAnCf/xQrgXvgRamNahFr17eyrVSMRq8PV5d7CleeGIBAENgOB/UzEkpiyiUvSbXXW1YgTHU9awXXzgr1HzVNPrg6Lbf2+zRgFeYsgsGYE9jMRK8qrKRsLverick69SQeBORHU2dvIj0Xex7A8nRuG5NRSQ3jUsLrbvBepO9cGgSCwAQjsVyKWuPTStsfRrxJuPc1OBO4WNOxKq9V0imOVmo5Up0Z3S+5Yz0g9QSAIrAiB/UrEK4Jv5mPMAmbWOEOvxsplqldaWULniiE6+z1qYibIuSAIbDsCIeK9a8F6xp1x0pP2xC/6dNtL3fvoJdQwOTUWRTDXB4ENRCBEvHeNUk8EVqIYSz4wzttYTDPVdTfA7N2XpOYgEAT2FIEQ8d7B6zZPD7402qEvcdKiT7Y+Yz/N21rjPf2/dfcdI7No/bk+CASBNSEQIh4HeHF056ARDCZOMcLBDGE667oZvXziomd5WYcJU8wq1Qbd13zQQ9MzjoNCagkCQWApBELES8H2dzcdt5CuCUHMbGW4mtmrPGbGXBnKCW1Z9Cwvk5RYr2n3WsI1mYoasUn1U4JAENhiBELEwxpPEpaATQLSbjU2rMw8HSYLaTN6LXqWlzkTrF/Luq+YtKQ9X23Y1+TuIBAE1oJAiHh52N1mbHYvs4OZVa6NYDC/hURs0qI2S9oyZ3kt/4a5MwgEga1AIES8fDPVTRWmW/QImLboUDO/q/lW2+iGRc/yWv7tcmcQCAJbg0CIePmmmnTGVT0R2BMfzKU8rcw6y2v5t8udQSAIbA0CIeLlmqom4zFszGiJ9hy+eiKwOrF5UaeVWWd5Lfd2uSsIBIGtQiBEvFxzeU6aJzWYHPpfOlV4NJNShTLEt0uiafXivljfWWd5JSxtufbJXUFgqxAIES/XXHX7sicQXLepQlnCREM68LR2DTFTQzYx/bJneS33hrkrCASBrUEgRLx8U3mEjMeumKbShD6Gpnn+mZsvPBnE7G9mXhNj9eJlz/Ja/g1zZxAIAluBQIh4+WbysEPzSHjMuyfJmrS9njLsgZPKEp6H9ZSSIW3oWV7Lv2nuDAJBYKMRCBFvdPPk5YJAENgFBELEu9DK+cYgEAQ2GoEQ8UY3T14uCASBXUAgRLwLrZxvDAJBYKMRCBFvdPPk5YJAENgFBELEu9DK+cYgEAQ2GoH/AUorTyJEqQTrAAAAAElFTkSuQmCC)
Putting the value of x, we get,
![](data:image/png;base64,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)
Therefore, there is maxima at x = 2h/3 tanα
Hence, putting the value of x, in formula of volume, we get,
![](data:image/png;base64,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)
Hence, Proved.
Question 20.A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of
A. 1 m/h
B. 0.1 m/h
C. 1.1 m/h
D. 0.5 m/h
Answer:Let V be the volume of the cylinder
V = πr2h
= π(10)2h
⇒ V =100πh
Differentiating w.r.t. t we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGwAAAArCAMAAACq5Z0rAAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOgA6OgBmOjo6OjpmOjqQOmY6OmaQOma2OpDbZgAAZgA6ZgBmZjoAZjo6ZpDbZrbbZrb/kDoAkGYAkGY6kLbbkNvbkNv/tmYAtmY6tmZmttu2ttv/tv//25A627aQ29uQ2////7Zm/7aQ/9uQ/9u2//+2///b7wYlXQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACEUlEQVRYR91Xb1+CMBDeMKW/JqUlWWIpJU32/T9edzdgQ5dIHL3wXsCcP55ne7jdPQhx3pFJOe9jh4SbymBZA8/kXMcSZ/E22HAxA64Q2R6ZCmEyoUkdD7dcXIJwvWS0DJFHjIoekKmplHe4gjwaAVnGJWKFmwVvKyknKJcKn4V+wU0ZAROa7R4WN5M3a23eUYIMRsEUJvKHeur8mdXimizBSx7hTkhbuE5ExpQeDi4lCDG4ZKgjl4qNZCIdfDwynTEvmY4pBU0FUeE1/uIIB9fKCFnxKnYzQwblg+2QWVybIEKvZHD/FRoWSiGeqHChQAZLFfZUfnkW++8onzNz6PVCytt1fcS8mN206D94JHcxllE7YuZS43Vq6jSls+1WxdFhpoNzT2QJ3qhNVCMU1gQWQJYwZKZB5NFwa0erpbraFFW2PVVSrtQ1ES4ZEhkyc20kqwDdwZGFnUrmw22937qMo0pQrKmNOzvCdkTGMi0gFYoEwaToQuZdR5GNadV97Yjsi2nDTIHWAaKeiIX9w4VAt/9uMhaneWK9gOItL56ADWqJHFOLsCNUVMcB/l2GH7enGlC25bqupLW6bFKg9cvw4nqNcmvowwd+9cTgWzm/KkTpiV1cx7vufW9025kX1/GurGQ+3H2H12079mkv7vmS7XtXLhn9uNa74mfMO9u59uI6nhjcMV/R7guXS3pGnB+zCEhNhoK9EgAAAABJRU5ErkJggg==)
The tank is being filled with wheat at the rate of 314 cubic meter per hour.
![](data:image/png;base64,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)
Then, we have,
314 = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the depth of wheat is increasing at the rate of 1 m/h.
Question 21.The slope of the tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2,– 1) is
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:It is given that curve x = t2 + 3t – 8, y = 2t2 – 2t – 5
Then, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANwAAAAhCAMAAACFiDV0AAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjpmOjqQOmY6OmaQOma2OpDbZgAAZgA6ZjoAZjo6ZjpmZmY6ZmZmZpDbZrbbZrb/kDoAkDpmkGYAkGY6kLbbkLb/kNv/tmYAtmY6tmZmtpA6ttv/tv//25A625Bm27Zm27aQ29uQ29vb2/+22////7Zm/7aQ/9uQ/9u2//+2///bFRV44gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADBElEQVRYR+2YDXPTMAyG7cK6mgFtAwO2hm3EZZCuQEoS//+fhiQ7X7XXOBBfxt18111vteXnlWT5g7Hn9h97IOWL6emDQcgnII6FgkifgrgQEMV7ziEtH8QsycU8myY/DYTks+TA+WYsCBUvMnUDkTvMs3Q1kbYKoojWjG3hM1LLXyWMUUakL69GsjnYTA2RzjP1+fvg8Y8NaMTtLqZKSlZDlO+SdLzAsTJaZUV8nrHfX1R89m00p/Ubai2uGoLJ8w/jBQ7WmuDLe361Ewsl+aOxU3uoO2eQwV5NPVx49D40lcNAMJaLvsClsz6KYawgSPJrVkQvwKuUQp2m4uPyJmd3rIj7KHJhl8WyL3A5VHWY3aZokBpWr0iAOCo56OqDxewQV/c+Zd4Wt036SmUZfaL5bYqWOK/Zj8kwj7awK3KMYNNscfSbrHpRkq4ACHawncBv4HnI8rdW5KT+8USTa5LloDga1Mp5v/gRrk/kwFxxuzTViTYxiUi5eJ3o8Ofimqmb4bv14U2m5z8Vua5rLWkSw0OtnYM6jbzElRHny6xll8aaP1AyJFauPufaFLBTmPn7xGnWWob+ciKEKqZC1jXrdgP224t28jbiSjiF4MdopQk9IYgA5k83LQongWH1S0joBdlFfb0iRx25qepqfynowEgKXeI8IWDR6tYW5xpbsTrturwhFzrNvMVVWxauuSojB4lzZ4XPmqtYPZ3G4NQHwnRaHI2xqqVOimpJ0YBuWmrP9q05N5qHuJrVUxxtmkoCNAbka0efQxzoKWJz3MEBxUfMSFJDAU1xl78cXi2BVp9QbIpGSMPqKc6kCEZky2fdk4W9zxW3sMjq+xNsSqtfEd/kuPKwjq6ZAiOrH+Iv1MG6I3UWRSOkxYr/DPZ+4ek66hYMAk9XIa75Q8TRES8EBNoM9UDjLTAEhHm/uIf9Y7LbKgsEUT+iTBm5UBD1FX9KcaEgQtn1XmrYMRRE/X4BJ/Wfg4hG7BwMonq/gAv8ZG971UsOnDwmhBgxWs+m/skDfwDJCFZ6l5SroQAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
The given points is (2, - 1)
At x = 2, we get
t2 + 3t – 8 = 2
⇒ t2 + 3t – 10 = 0
⇒ (t – 2)(t + 5) = 0
⇒ t = 2 and - 5
At y = - 1, we get
2t2 - 2t – 5 = - 1
⇒ 2t2 - 2t – 4 = 0
⇒ 2(t2 - t – 2) = 0
⇒ (t – 2)(t + 1) = 0
⇒ t = 2 and - 1
Therefore, the common value of t is 2.
Hence, the slope of the tangent to the given curve at point (2, - 1) is
![](data:image/png;base64,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)
Question 22.The line y = mx + 1 is a tangent to the curve y2 = 4x if the value of m is
A. 1
B. 2
C. 3
D. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABMAAAA1CAYAAACjpdDnAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAFQSURBVFjD7dcxasMwFADQ3z0XKPR7a/f4kxsUIWgyBiqLbiWDMV4yhqB4M4TcoTlHewKfIB16guQMSS27LW0aOVbtDiEy/MWG54/1/5cMSZJAWwEO+x8svy4aY2E49BmjKSK8IovGf8ZmkvrgeSvPg1We2a4R9gN1mMPOEtMNrgQNNAYULI81vPGBinkXAdY5sP0eVRm6SeuwOpguStuowra24VbzLLGqmrLCJuHwxkfIiimrA/2MiejOGpuEt7kDm70CLVASalAbS9NRh1/BC3KpRmnaKUe4uOYIzyVIbw9KXdbC9OwnFj+aXgKAGx6rbuPVDAiWVpmZ4iszkk+N66zc+vzMlJUVFhAuSM76jTsgYjhu7axxaHWtsaJMSKq6PxlmKD+wIAWL/f7UbUbE558FfRSLRE98tNPuUJi+368bsezdV28iuG6lA0530r4DYd6fxbCpPSgAAAAASUVORK5CYII=)
Answer:It is given that the equation of the tangent to the given curve is
y = mx + 1
Now, substituting the value of y in y2 = 4x, we get
⇒ (mx + 1)2 = 4x
⇒ m2x2 + 1 + 2mx - 4x =0
⇒ m2x2 + x(2m - 4) + 1 = 0………………..(1)
Since, a tangent touches the curve at one point, the root of equation (1) must be equal.
Thus, we get
Discriminant = 0
(2m - 4)2 – 4(m2)(1) = 0
⇒ 4m2 + 16 - 16m - 4m2 =0
⇒ 16 – 16m = 0
⇒ m =1
Therefore, the required value of m is 1.
Question 23.The normal at the point (1,1) on the curve 2y + x2 = 3 is
A. x + y = 0
B. x – y = 0
C. x + y + 1 = 0
D. x – y = 1
Answer:It is given that the equation of curve is 2y + x2 = 3
Differentiating w.r.t. x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The slope of the normal to the given curve at point (1,1) is
![](data:image/png;base64,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)
Then, the equation of the normal to the curve at (1,1) is
⇒ y – 1 =1(x - 1)
⇒ y - 1 = x – 1
⇒ x – y = 0
Question 24.The normal to the curve x2 = 4y passing (1,2) is
A. x + y = 3
B. x – y = 3
C. x + y = 1
D. x – y = 1
Answer:It is given that the equation of curve is x2 = 4y
Differentiating w.r.t. x, we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEoAAAArCAMAAADopu68AAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjqQOmY6OmaQOma2OpCQOpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmZmZrbbZrb/kDoAkDo6kGYAkLbbkNv/tmYAtmY6tv//25A627aQ29uQ2////7Zm/9uQ/9u2//+2///bCsvNvQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABmklEQVRIS+1V7VLDIBCEaIvWmKjVNNaiaKDw/k/oHfmiZgSK/HJ6M81kprll79i9I+TfhaR0l6somQ9KsWysMkGpR0o3bGdaSledYrRI7ppiDTF76JVpVx0hsjgkQ3ELgG0XVx/wWCcj6bqCXNsrfJjXdFIOlGnXRCLHxHCgsEL+B1EglaFXRNf3W2hXcojijRyferVzio1LDvNOi/KLWSyJd5gnZLoSXAIgg0kJ5hMccB0rC/FT1aYt9yMpThtyrCOrVWwJRe9GTXHEFHGDR9cvQa+hmThaXHgHI6/CtuVYIIcTTevRrbztglD9CMNxISaxAckhxpr0wyE4TEzbAyi2KX9Xmv0KWAk8djpnfsHM3k8Yi6t2gWG/9OFpAV8Pt6m3z7MolgUirr9Xws5DIA+q1XXAA14odQNKN7yyPxixfpN7WzAUUsHqKIAV9WJBvxyNXrZz5DgctnOMdQOI03YOWzfEbd7Op9YN5S3/d/eg37pBbBfKb92zoE6sG8xcfOBs5xjreg+YtnOUdb1Q43ZuYqx7ftXZMr4Bsl0jZiuVAmYAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
The slope of the normal to the given curve at point (h,k) is
![](data:image/png;base64,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)
Then, the equation of the normal to the curve at (h,k) is
⇒ y – k =![](data:image/png;base64,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)
Now, it is given that the normal passes through the point (1,2)
Thus, we get,
⇒ 2 – k =![](data:image/png;base64,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)
⇒ k =
………………(1)
Since (h,k) lies on the curve x2 = 4y, we have h2 = 4k
⇒ k = ![](data:image/png;base64,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)
Now putting the value of of k in (1), we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ h3 = 8
⇒ h = 2
Therefore, the equation of the normal is given as:
⇒ y – 1 = ![](data:image/png;base64,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)
⇒ y - 1 = - (x - 2)
⇒ x + y = 3
Question 25.The points on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with the axes are
A. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEQAAAA7CAYAAADCZyymAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAATBSURBVGje7Zs7TxtBEMcXChpIWkTCIlFAoOTWiJIGWcuroLDAXllIRHFhWSfxkK4gykGHIqCngzSBJiUN0KWJv0JMRxX8FQK53Xv4OO/d7RJb7BojjQS2Ze5+nvnPY8dgf38fdK1hXQhdIC0A4vz0UNPyBiWvXegNTYIXMCYbpYODAZ1gHByUBgjGG5iYC6JQEp+0rNIbguDhNDFXdQ4Dk0yvQkQOS5b15tlAKIwCgkeouLeoaCj0hi3t9XtFtAhR4SjNU2LDhMKA2NxSEcZuOTeOIPjlXOdfZtCo5sq746meguEWzFa2k6BwH7TzaBnC+XMVNWO3PGdkMvirf222mR9z4aDbom0PpWnKPITnKG8vCwOhoYIhvMCmPaWiSGKEP0c/KNvEUxCAOiJ7S2nvwV4L8UWcnnC9AyByqpNoUq9xgNyL6h1B4DTOS5o/gWFwLUJaJXEtIHAm8yHuEbQEhvE1TxKa3QkY1bRYVKXgKpdzyHGNKkD4h0hKDe7TLg4ZAFZ5sqBluDBPhuDKyzIPjj1Co3AsAyUubJ78QV1Pp3ChYFjR5egHg5Kt7EiFDSqcxQJxqauZXYQyB4OSnnqbsk1ERyJxZVzpoB88Y8IqAyTmfjsGSCULdwDEV6LFZDqQGBdStYdpKiadckGm1YiTCK2AsDIdgJrbu5QRBeN35LSblc9UCgHxBje90jeRGTkJmjrHIMxcYrI5/7zU3UYgspMpFvMSItieWiYJyPrESl/fZM25wMFnito2ba0leo/6SwJxRHVwsq+vNrFur7woEMsqvvNmGQ+vHkhj8ES+0KygNpD/1BARIGzw5GQDv6tWW0MkgPgZImxUJGkvEX3cF1oaKhmY+U4BdBwQOpsMp75Q1/kQfZwObfxQ8f95xwGRDRn2XKiKfNVAWIo1CsfhMCqVcm8dIDcUSM6y3osWaJxwjDVlgcSEVzDYcX9Hv0U8JeZ9uKZ0yHAvptNDhh0CCXabSUBCGaxHGyC8tJtgPaJAGo/DOspWPumedrnGOytpDImfagcd2jSGx6jWznBqe3PXaiMI2e0E0vZeppVzEuqt09j8qHVz1yo3dhdc0gc+HB3r6UgPETF/dwUC8CeYnKHCkeTpXWcA8QfKdMKeN+2xfW/WysSYntcKQukYIOzEzUnL0ewgsxLRUUDclA/rc+VdI3KDQwiA29YBMWdm+wG8W7LtUZWBBMeWRuGbHx4sM9GDeqmDqqVRCPrvZkx7lg9Eo5M7enLPmkN2PmN9cFfAMpe+pgh6SPLJnW6H3fRwyuuWpVchgrFE0mG3Tz5pIU0VoyHi3hD86aXeB1kotDvn7cI0KzhnZ0I1GGx+621JBimXhtAwvhGFErcL0yxYCRt6SogqFU8nTMI3wypcT1dEZjLCK1V+F6py2Lh7IM11SDA6EPDwpIzEL3wUDhtv9PBIP7Rw/xJUsAI7ckmrY9wGi277RgsfZULGLcBqjpfcZ/OVRX9oTTC0RWYorhjHb2nzy2O6KK+wljAhjaxFjGTwSVod4m9pJy34Jub5tEV5nczLTttpizXJswkHSjRWdYVhkuk19p2ZlNI+fWDjfoFoTeevmInCSAXi2ybB83p/xUx85Up7bWi5R3UhdIEk2j9SEaTEIVxX9gAAAABJRU5ErkJggg==)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEQAAAA7CAYAAADCZyymAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAATBSURBVGje7Zs7TxtBEMcXChpIWkTCIlFAoOTWiJIGWcuroLDAXllIRHFhWSfxkK4gykGHIqCngzSBJiUN0KWJv0JMRxX8FQK53Xv4OO/d7RJb7BojjQS2Ze5+nvnPY8dgf38fdK1hXQhdIC0A4vz0UNPyBiWvXegNTYIXMCYbpYODAZ1gHByUBgjGG5iYC6JQEp+0rNIbguDhNDFXdQ4Dk0yvQkQOS5b15tlAKIwCgkeouLeoaCj0hi3t9XtFtAhR4SjNU2LDhMKA2NxSEcZuOTeOIPjlXOdfZtCo5sq746meguEWzFa2k6BwH7TzaBnC+XMVNWO3PGdkMvirf222mR9z4aDbom0PpWnKPITnKG8vCwOhoYIhvMCmPaWiSGKEP0c/KNvEUxCAOiJ7S2nvwV4L8UWcnnC9AyByqpNoUq9xgNyL6h1B4DTOS5o/gWFwLUJaJXEtIHAm8yHuEbQEhvE1TxKa3QkY1bRYVKXgKpdzyHGNKkD4h0hKDe7TLg4ZAFZ5sqBluDBPhuDKyzIPjj1Co3AsAyUubJ78QV1Pp3ChYFjR5egHg5Kt7EiFDSqcxQJxqauZXYQyB4OSnnqbsk1ERyJxZVzpoB88Y8IqAyTmfjsGSCULdwDEV6LFZDqQGBdStYdpKiadckGm1YiTCK2AsDIdgJrbu5QRBeN35LSblc9UCgHxBje90jeRGTkJmjrHIMxcYrI5/7zU3UYgspMpFvMSItieWiYJyPrESl/fZM25wMFnito2ba0leo/6SwJxRHVwsq+vNrFur7woEMsqvvNmGQ+vHkhj8ES+0KygNpD/1BARIGzw5GQDv6tWW0MkgPgZImxUJGkvEX3cF1oaKhmY+U4BdBwQOpsMp75Q1/kQfZwObfxQ8f95xwGRDRn2XKiKfNVAWIo1CsfhMCqVcm8dIDcUSM6y3osWaJxwjDVlgcSEVzDYcX9Hv0U8JeZ9uKZ0yHAvptNDhh0CCXabSUBCGaxHGyC8tJtgPaJAGo/DOspWPumedrnGOytpDImfagcd2jSGx6jWznBqe3PXaiMI2e0E0vZeppVzEuqt09j8qHVz1yo3dhdc0gc+HB3r6UgPETF/dwUC8CeYnKHCkeTpXWcA8QfKdMKeN+2xfW/WysSYntcKQukYIOzEzUnL0ewgsxLRUUDclA/rc+VdI3KDQwiA29YBMWdm+wG8W7LtUZWBBMeWRuGbHx4sM9GDeqmDqqVRCPrvZkx7lg9Eo5M7enLPmkN2PmN9cFfAMpe+pgh6SPLJnW6H3fRwyuuWpVchgrFE0mG3Tz5pIU0VoyHi3hD86aXeB1kotDvn7cI0KzhnZ0I1GGx+621JBimXhtAwvhGFErcL0yxYCRt6SogqFU8nTMI3wypcT1dEZjLCK1V+F6py2Lh7IM11SDA6EPDwpIzEL3wUDhtv9PBIP7Rw/xJUsAI7ckmrY9wGi277RgsfZULGLcBqjpfcZ/OVRX9oTTC0RWYorhjHb2nzy2O6KK+wljAhjaxFjGTwSVod4m9pJy34Jub5tEV5nczLTttpizXJswkHSjRWdYVhkuk19p2ZlNI+fWDjfoFoTeevmInCSAXi2ybB83p/xUx85Up7bWi5R3UhdIEk2j9SEaTEIVxX9gAAAABJRU5ErkJggg==)
Answer:It is given that the equation of the curve is 9y2 = x3
Differentiating w.r.t. x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The slope of the normal to the given curve at point (x1,y1) is
![](data:image/png;base64,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)
Then, the equation of the normal to the curve at (x1,y1)is
⇒ y – y1 =![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Now, it is given that the normal makes equal intercepts with the axes.
Thus, we get,
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADMAAAAhCAMAAABDYWOIAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6ADo6ADpmADqQAGaQAGa2OgAAOgBmOjoAOjo6OmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZpDbZrb/kDoAkDpmkLb/kNv/tmYAtmY6tpC2ttv/tv//25A625Bm27Zm29u229vb2////7Zm/7aQ/9uQ/9u2//+2///bkEQ+gQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABB0lEQVQ4T9VUiw6CMAxcUXQ+8P1AN9SBokj///scAj4GFEmMiUtIyHrXa5vuGPvRCbglIm6HpJwES5wB1jnobIfKoSnsOhkz5ukvP6q1qu1J2SFuDk+Y308qw6D3cmcmiadCJTIZ6rJFt73H3XFGcJjsLHU4Q/m8ixK0UkxyIp52846iOXEiU+BEhI4n8qG9oSQAdKvmJ8FJQySqdvomIMmWHkuUkR/h+09F+o9AjUu7d0rV1jzjPzCCxbDm2ZldoDdqyGBMVS5alr1oGugOFwO6toJpRNw5HV1ayzQN/dh1fbRdmaaBrhMirVM0jescgLS4D02jfKto0yjnUKZRsbvftYO6B3IDuLMVxCz/1UAAAAAASUVORK5CYII=)
⇒
…………………….(1)
Since, the point (x1,y1)lies on the curve
…………(2)
From (1) and (2), we get
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEwAAAAXCAMAAACWAFktAAAAAXNSR0IArs4c6QAAAGNQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOjqQOmY6OmZmOma2OpDbZgAAZgA6ZjoAZjpmZmYAZmZmZrb/kDoAkDo6kNv/tmYAtmY6tv//25A625Bm2////7Zm/9uQ//+2///bXoKRCAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA00lEQVQ4T+2T2RKCMAxFG1xwAVGpVMvS/P9XGvYOZmh1eNExr82c3HuTCvGvn0hAB9liPqqQg+XpJwNMfOVgVTjQFMC60AAeeBWVrE3LvKIGlB6scle0MBPDpIYZKElZNLVNgrvqG80pE7wy0rJ6dIAq3B/dEaKkgQTTjAcLJrwWXvZK0zmbwpwvrUq8bweFrzbrJ+cC8JaZuKbkh2SE8b5ZmHUaqOosoNmAdsHYQMajRQkBKYOG5oJRbv7/yQVz79zqWBKGalO8NXymmT5oG93X1hP+JwurwWDXrgAAAABJRU5ErkJggg==)
Now putting the value of x1 in (2), we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the required points are ![](data:image/png;base64,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)
Using differentials, find the approximate value of each of the following:
Answer:
Let us consider y = x1/4 and and
Then,
Therefore,
Now, dy is approximately equal to Δy and is equal to
dy =
Therefore, the approximate value of is
Question 2.
Using differentials, find the approximate value of each of the following:
Answer:
Let us consider y = and x=32 and
Then,
=
Therefore,
Now, dy is approximately equal to Δy and is equal to
Therefore, the approximate value of is
Question 3.
Show that the function given by has maximum at x = e.
Answer:
It is given that f(x) =
Then, f’(x) =
Now, f’(x) = 0
⇒ 1 - logx =0
⇒ log x =1
⇒ log x = log e
⇒ x = e
Now, f’’(x) =
Now, f’’(e)=
Therefore, by second derivative test, f is the maximum at x = e.
Question 4.
The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?
Answer:
Let ΔABC be isosceles where BC is the base of fixed length b.
Let the length of the two equal sides of ΔABC be a.
Now, draw a perpendicular AD to BC. Then, we have
In ΔADC, by using Pythagoras theorem,
AD =
Then, Area of triangle (A) =
The rate of change of the area with respect to time (t) is given by:
It is given that the two equal sides of the triangle are decreasing at the rate of 3cm per second.
Then
And when a = b we have,
Therefore, if the two sides are equal to the base, then the area of the triangle is decreasing at the rate of cm2/s.
Question 5.
Find the equation of the normal to curve x2 = 4y which passes through the point (1, 2).
Answer:
It is given that curve is y2 = 4x.
Differentiating with respect to x, we get,
Now, the slope of the normal at point (1,2) is
Therefore, Equation of the normal at (1,2) is y - 2 = - 1(x – 1)
⇒ y - 2 = - x + 1
⇒ x + y - 3 = 0
Question 6.
Show that the normal at any point θ to the curve
x = a cosθ + a θ sinθ, y = a sinθ – aθ cosθ is at a constant distance from the origin.
Answer:
We have x = a cosθ + a θ sinθ,
⇒
And y = a sinθ – aθ cosθ
⇒
So,
Then, Slope of the normal at any point θ is .
The equation of the normal at a given point (x,y) is:
y - a sinθ + aθ cosθ = (x - a cosθ - a θ sinθ)
⇒ ysinθ – asin2θ + aθ sinθ cosθ = - x cosθ + acos2θ + aθ sinθ cosθ
⇒ xcosθ + ysinθ – a(sin2θ + cos2θ ) = 0
⇒ xcosθ + ysinθ –a = 0
Now, the perpendicular distance of the normal from the origin is
, which is independent of θ .
Therefore, the perpendicular distance of the normal from the origin is constant.
Question 7.
Find the intervals in which the function f given by
is (i) strictly increasing (ii) strictly decreasing.
Answer:
(i) It is given that f(x) =
Now, if f’(x) =0
⇒ cos x = 0 or cosx = 4
But, cosx = 4 is not possible
Therefore, cosx =0
⇒ x =
Now, x = divides (0,2π) into three disjoints intervals
In the intervals and
, f’(x)>0
Therefore, f(x) is increasing for 0< x < and
< x < 2π.
In interval, f’(x)<0
Therefore, f(x) is decreasing for < x <
.
Question 8.
Find the intervals in which the function f given by is
(i) Increasing (ii) decreasing.
Answer:
It is given that f(x) =
Then, f’(x) =0
⇒ 3x6 - 3 = 0
⇒ x6 = 1
⇒ x = 1
Now, the points x =1 and x = - 1 divide the real line into three disjoint intervals
( - ∞, - 1), ( - 1,1) and (1,∞).
In interval ( - ∞, - 1) and (1,∞) when x < - 1 and x > 1 then f’(x) >0
Therefore, when x < - 1 and x > 1, f is increasing.
And, in interval ( - 1,1) when - 1< x < 1 then f’(x) < 0.
Therefore, when - 1 < x < 1, f is decreasing.
Question 9.
Find the maximum area of an isosceles triangle inscribed in the ellipse with its vertex at one end of the major axis.
Answer:
It is given that ellipse
Let the major axis be along the x – axis.1).
Let ABC be the triangle inscribed in the ellipse where vertex C is at (a,0).
Since, the ellipse is symmetrical w.r.t. x - axis and y - axis, we can assume the coordinates of A to be ( - x1,y1) and the coordinates of B to be ( - x1, - y1).
Now, we have y1 = ±
Therefore, Coordinates of A and the coordinates of B
As the point(x1,y1) lies on the ellipse, the area of triangle ABC (A) is given by:
A =
……..(1)
Now,
But, x1 cannot be equal to a.
⇒ x1 =
y1 =
Now,
Also, when x1 = , then,
< 0
Then, the area is the maximum when x1 = .
Therefore, Maximum area of the triangle is given by:
A =
Question 10.
A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?
Answer:
Let l, b, and h be the length, breadth and height of the tank respectively.
then, we have h = 2m
Volume of the tank = 8 m3
Volume of the tank = l × b × h
⇒ 8 = l × b × 2
⇒ lb = 4
⇒ b =
Now, area of the base = lb = 4
Area of the 4 walls (A) = 2h(l + b)
Now,
⇒ = 0
⇒ l2 = 4
⇒ l = ±2
Since, length cannot be negative therefore l =2.
⇒ b = 2
Now,
When l =2,
Then, by second derivative test, the area is the minimum when l =2.
We have, l =b=h=2
Therefore, Cost of building the base = Rs 70 × (lb) = Rs 70 (4) = Rs 280.
Cost of building the walls = Rs 2h (l + b) × 45 = Rs 90(2)(2 + 2)
= Rs 8(90) = Rs 720.
Required total cost = Rs(280 + 720) = Rs 1000.
Therefore, the total cost of the tank will be Rs 1000.
Question 11.
The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.
Answer:
Let r be the radius of the circle and a be the side of the square.
Then, 2πr + 4a = k (where k is constant)
The sum of the areas of the circle and square (A) is
= πr2 + a2 = πr2 +
Now,
8r = k - 2πr
⇒ (8 + 2π)r = k
⇒ r =
Now, > 0
Therefore, When r = ,
> 0
⇒ The sum of the area is least when r =
So, when r =
Then a =
Therefore, it is proved that the sum of their areas is least when the side of the square is double the radius of the circle.
Question 12.
A window is in the form of a rectangle surmounted by a semi - circular opening.
The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
Answer:
Let x and y be the length and breadth of the rectangular window.
Radius of the semi - circular opening =
It is given that the perimeter of the window is 10m.
⇒ x + 2y +
Therefore, Area of the window (A) is given by
=
Now, , then
=0
Then, when x = then
< 0.
Therefore, by second derivative test, the area is maximum when length
x = m.
Now, y =
Therefore, the required dimensions of the window to admit maximum light is given by length = m and breadth =
m.
Question 13.
A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.
Show that the maximum length of the hypotenuse is
Answer:
Let ΔABC be right - angled at B. Let AB = x and BC = y.
Let P be a point on the hypotenuse of the triangle such that P is at a distance of a and b from the sides AB and Bc respectively.
Let <C = θ .
Now, we have,
Ac =
Now, PC = b cosecθ
And AP = a secθ
⇒ AC = AP + PC
⇒ AC = a secθ + b cosecθ
Now, if
⇒ asecθ tanθ = bcosecθ cotθ
⇒
⇒ asin3θ =bcos3θ
⇒
⇒
………..(1)
So, it is clear that < 0 when
Therefore, by second derivative test, the length of the hypotenuse is the maximum when
Now, when , we get,
Ac =
Therefore, the maximum length of the hypotenuses is .
Question 14.
Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has
(i) local maxima (ii) local minima
(iii) point of inflexion
Answer:
It is given that function is f (x) = (x – 2)4 (x + 1)3
⇒ f’(x) = 4(x - 2)3 (x + 1)3 + 3(x + 1)2(x - 2)4
=(x - 2)3(x + 1)2[4(x + 1) + 3 (x - 2)]
=(x - 2)3(x + 1)2(7x - 2)
Now, f’(x) =0
⇒ x = - 1 and x = or x = 2
Now, for values of x close to and to the left of
f’(x) > 0.
Also, for values of x close to and to the right of
, f’(x) < 0.
Then, x = is the point of local maxima.
Now, for values of x close to 2 and to the left of 2, f’(x) < 0.
Also, for values of x close to 2 and to the right of 2. f’(x) > 0.
Then, x = 2 is the point of local minima.
Now, as the value of x varies through - 1, f’(x) does not changes its sign.
Then, x = - 1 is the point of inflexion.
Question 15.
Find the absolute maximum and minimum values of the function f given by
f (x) = cos2 x + sin x, x ∈ [0, π]
Answer:
It is given that f (x) = cos2 x + sin x, x ∈ [0, π]
f’(x) = 2cosx( - sinx) + cosx
= - 2sinxcosx + cosx
Now, if f’(x) = 0
⇒ 2sinxcosx = cosx
⇒ cosx(2sinx - 1)=0
⇒ sin x = or cosx = 0
⇒ x =
Now, evaluating the value of f at critical points x = and x =
and at the end points of the interval [0,π], (ie, at x = 0 and x =π), we get,
f
f(0)=
f(π)=cos2π + sinπ = ( - 1)2 + 0 =1
f
Therefore, the absolute maximum value of f is occurring at x =
and the absolute minimum value of f is 1 occuring at x =1,
and π.
Question 16.
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is .
Answer:
Let R and h be the radius and the height of the cone respectively.
The volume (V) of the cone is given by;
V =
Now, from the right triangle BCD, we get,
BC =
V =
Now, if , then,
Now,
Now, when , it can be shown that
< 0.
Therefore, the volume is the maximum when .
When,
Height of the cone = r + .
Therefore, it can be seen that the altitude of the circular cone of maximum volume that can be inscribed in a sphere of radius r is.
Question 17.
Let f be a function defined on [a, b] such that f ′(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
Answer:
Since, f’(x) > 0 on (a,b)
Then, f is a differentiating function (a,b)
Also, every differentiable function is continuous,
Therefore, f is continuous on [a,b]
Let x1, x2 ϵ (a,b) and x2 > x1 then by LMV theorem, there exists c ϵ (a,b) s.t.
f’(c) =
Therefore, f is an increasing function.
Question 18.
Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is . Also find the maximum volume.
Answer:
Let r and h be the radius and the height of the cylinder respectively.
Now, h =
The volume (V) of the cylinder is given by:
V = πr2h =2 πr2
Now, if
Now,
Now, we can see that at, ,
< 0.
Therefore, the volume is the maximum when.
When , the height of the cylinder is
.
Therefore, the volume of the cylinder is the maximum when the height of the cylinder is .
Hence Proved.
Question 19.
Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle α is one - third that of the cone and the greatest volume of cylinder is
Answer:
The figure is given below:
Let VAB be a given cone of height h, semi-vertical angle α and let x b ethe radius of the base of the cylinder A'B'DC which is inscribed in the cone VAB.
In triangle VO'A',
VO' = x cotα
OO' = VO - VO' = h - x cotα
Let V be the volume of the cylinder. Then,
V = π(O'B')2 (OO')
V = πx2(h - x cotα)
Differentiating with respect to x, we get,
Now, putting dV/dx = 0, for maxima or minima, we get,
Putting the value of x, we get,
Therefore, there is maxima at x = 2h/3 tanα
Hence, putting the value of x, in formula of volume, we get,
Hence, Proved.
Question 20.
A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of
A. 1 m/h
B. 0.1 m/h
C. 1.1 m/h
D. 0.5 m/h
Answer:
Let V be the volume of the cylinder
V = πr2h
= π(10)2h
⇒ V =100πh
Differentiating w.r.t. t we get,
The tank is being filled with wheat at the rate of 314 cubic meter per hour.
Then, we have,
314 =
Therefore, the depth of wheat is increasing at the rate of 1 m/h.
Question 21.
The slope of the tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2,– 1) is
A.
B.
C.
D.
Answer:
It is given that curve x = t2 + 3t – 8, y = 2t2 – 2t – 5
Then,
The given points is (2, - 1)
At x = 2, we get
t2 + 3t – 8 = 2
⇒ t2 + 3t – 10 = 0
⇒ (t – 2)(t + 5) = 0
⇒ t = 2 and - 5
At y = - 1, we get
2t2 - 2t – 5 = - 1
⇒ 2t2 - 2t – 4 = 0
⇒ 2(t2 - t – 2) = 0
⇒ (t – 2)(t + 1) = 0
⇒ t = 2 and - 1
Therefore, the common value of t is 2.
Hence, the slope of the tangent to the given curve at point (2, - 1) is
Question 22.
The line y = mx + 1 is a tangent to the curve y2 = 4x if the value of m is
A. 1
B. 2
C. 3
D.
Answer:
It is given that the equation of the tangent to the given curve is
y = mx + 1
Now, substituting the value of y in y2 = 4x, we get
⇒ (mx + 1)2 = 4x
⇒ m2x2 + 1 + 2mx - 4x =0
⇒ m2x2 + x(2m - 4) + 1 = 0………………..(1)
Since, a tangent touches the curve at one point, the root of equation (1) must be equal.
Thus, we get
Discriminant = 0
(2m - 4)2 – 4(m2)(1) = 0
⇒ 4m2 + 16 - 16m - 4m2 =0
⇒ 16 – 16m = 0
⇒ m =1
Therefore, the required value of m is 1.
Question 23.
The normal at the point (1,1) on the curve 2y + x2 = 3 is
A. x + y = 0
B. x – y = 0
C. x + y + 1 = 0
D. x – y = 1
Answer:
It is given that the equation of curve is 2y + x2 = 3
Differentiating w.r.t. x, we get,
The slope of the normal to the given curve at point (1,1) is
Then, the equation of the normal to the curve at (1,1) is
⇒ y – 1 =1(x - 1)
⇒ y - 1 = x – 1
⇒ x – y = 0
Question 24.
The normal to the curve x2 = 4y passing (1,2) is
A. x + y = 3
B. x – y = 3
C. x + y = 1
D. x – y = 1
Answer:
It is given that the equation of curve is x2 = 4y
Differentiating w.r.t. x, we get,
The slope of the normal to the given curve at point (h,k) is
Then, the equation of the normal to the curve at (h,k) is
⇒ y – k =
Now, it is given that the normal passes through the point (1,2)
Thus, we get,
⇒ 2 – k =
⇒ k = ………………(1)
Since (h,k) lies on the curve x2 = 4y, we have h2 = 4k
⇒ k =
Now putting the value of of k in (1), we get
⇒ h3 = 8
⇒ h = 2
Therefore, the equation of the normal is given as:
⇒ y – 1 =
⇒ y - 1 = - (x - 2)
⇒ x + y = 3
Question 25.
The points on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with the axes are
A.
B.
C.
D.
Answer:
It is given that the equation of the curve is 9y2 = x3
Differentiating w.r.t. x, we get,
The slope of the normal to the given curve at point (x1,y1) is
Then, the equation of the normal to the curve at (x1,y1)is
⇒ y – y1 =
⇒
⇒
⇒
⇒
Now, it is given that the normal makes equal intercepts with the axes.
Thus, we get,
⇒
⇒
⇒ …………………….(1)
Since, the point (x1,y1)lies on the curve …………(2)
From (1) and (2), we get
Now putting the value of x1 in (2), we get
Therefore, the required points are