Application of Derivatives NCERT Solutions Math Class 12 CBSE

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

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?

NCERT Question Solutions

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.

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?

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?

Question 6

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 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

Question 7

A particle moves along the curve. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.

Question 8

The radius of an air bubble is increasing at the rate of cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

Question 9

A balloon, which always remains spherical, has a variable diameter Find the rate of change of its volume with respect to x.

Question 10

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?

Question 11

The total cost C (x) in Rupees associated with the production of x units of an item is given by

Find the marginal cost when 17 units are produced.

Question 12

The total revenue in Rupees received from the sale of x units of a product is given by

Find the marginal revenue when x = 7.

Question 13

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

Question 14

The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

Question 15

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 = 8 cm and y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

Question 16

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.

Question 17

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.

Question 18

The total revenue in Rupees received from the sale of x units of a product is given by

. The marginal revenue, when is

(A) 116 (B) 96 (C) 90 (D) 126

Question 19

A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic mere 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

Question 20

Show that the function given by f(x) = 3x + 17 is strictly increasing on R.

Question 21

Show that the function given by f(x) = e2x is strictly increasing on R.

Question 22

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)

Question 23

Find the intervals in which the function f given by f(x) = 2x2 − 3x is

(a) strictly increasing (b) strictly decreasing

Question 24

Find the intervals in which the function f given by f(x) = 2x3 − 3x2 − 36x + 7 is

(a) strictly increasing (b) strictly decreasing

Question 25

Find the intervals in which the following functions are strictly increasing or decreasing:

(a) x2 + 2x − 5 (b) 10 − 6x − 2x2

(c) −2x3 − 9x2 − 12x + 1 (d) 6 − 9xx2

(e) (x + 1)3 (x − 3)3

Question 26

Show that, is an increasing function of x throughout its domain.

Question 27

Find the values of x for whichis an increasing function.

Question 28

Prove that is an increasing function of θ in.

Question 29

Prove that the logarithmic function is strictly increasing on (0, ∞).

Question 30

Prove that the function f given by f(x) = x2x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).

Question 31

Which of the following functions are strictly decreasing on?

(A) cos x (B) cos 2x (C) cos 3x (D) tan x

Question 32

On which of the following intervals is the function f given by strictly decreasing?

(A) (B)

(C) (D) None of these

Question 33

Find the least value of a such that the function f given is strictly increasing on (1, 2).

Question 34

Let I be any interval disjoint from (−1, 1). Prove that the function f given by

is strictly increasing on I.

Question 35

Prove that the function f given by f(x) = log sin x is strictly increasing on and strictly decreasing on

Question 36

Prove that the function f given by f(x) = log cos x is strictly decreasing on and strictly increasing on

Question 37

Prove that the function given by is increasing in R.

Question 38

The interval in which is increasing is

(A) (B) (−2, 0) (C) (D) (0, 2)

Question 39

Find the intervals in which the function f given by

is (i) increasing (ii) decreasing

Question 40

Find the intervals in which the function f given byis

(i) increasing (ii) decreasing

Question 41

Find the slope of the tangent to the curve y = 3x4 − 4x at x = 4.

Question 42

Find the slope of the tangent to the curve, x ≠ 2 at x = 10.

Question 43

Find the slope of the tangent to curve y = x3x + 1 at the point whose x-coordinate is 2.

Question 44

Find the slope of the tangent to the curve y = x3 − 3x + 2 at the point whose x-coordinate is 3.

Question 45

Find the slope of the normal to the curve x = acos3θ, y = asin3θ at.

Question 46

Find the slope of the normal to the curve x = 1 − a sin θ, y = b cos2θ at .

Question 47

Find points at which the tangent to the curve y = x3 − 3x2 − 9x + 7 is parallel to the x-axis.

Question 48

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).

Question 49

Find the equation of all lines having slope −1 that are tangents to the curve .

Question 50

Find the equation of all lines having slope 2 which are tangents to the curve.

Question 51

Find the equations of all lines having slope 0 which are tangent to the curve .

Question 52

Find points on the curve at which the tangents are

(i) parallel to x-axis (ii) parallel to y-axis

Question 53

Find the equations of the tangent and normal to the given curves at the indicated points:

(i) y = x4 − 6x3 + 13x2 − 10x + 5 at (0, 5)

(ii) y = x4 − 6x3 + 13x2 − 10x + 5 at (1, 3)

(iii) y = x3 at (1, 1)

(iv) y = x2 at (0, 0)

(v) x = cos t, y = sin t at

Question 54

Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is

(a) parallel to the line 2xy + 9 = 0

(b) perpendicular to the line 5y − 15x = 13.

Question 55

Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and x = −2 are parallel.

Question 56

Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.

Question 57

For the curve y = 4x3 − 2x5, find all the points at which the tangents passes through the origin.

Question 58

Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis.

Question 59

Find the equation of the normal at the point (am2, am3) for the curve ay2 = x3.

Question 60

Find the point on the curve y = x3 − 11x + 5 at which the tangent is y = x − 11.

Question 61

Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.

Question 62

Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).

Question 63

Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1. [Hint: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]

Question 64

Find the equations of the tangent and normal to the hyperbola at the point.

Question 65

Find the equation of the tangent to the curve which is parallel to the line 4x − 2y + 5 = 0.

Question 66

The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is

(A) 3 (B) (C) −3 (D)

Question 67

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)

Question 68

Find the equation of the normal to curve y2 = 4x at the point (1, 2).

Question 69

Show that the normal at any point θ to the curve

is at a constant distance from the origin.

Question 70

The slope of the tangent to the curveat the point (2, −1) is

(A) (B) (C) (D)

Question 71

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)

Question 72

The normal at the point (1, 1) on the curve 2y + x2 = 3 is

(A) x + y = 0 (B) xy = 0

(C) x + y + 1 = 0 (D) x y = 1

Question 73

The normal to the curve x2 = 4y passing (1, 2) is

(A) x + y = 3 (B) xy = 3

(C) x + y = 1 (D) xy = 1

Question 74

The points on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with the axes are

(A) (B)

(C) (D)

Question 75

1. Using differentials, find the approximate value of each of the following up to 3 places of decimal

(i) (ii) (iii)

(iv) (v) (vi)

(vii) (viii) (ix)

(x) (xi) (xii)

(xiii) (xiv) (xv)

Question 76

Find the approximate value of f (2.01), where f (x) = 4x2 + 5x + 2

Question 77

Find the approximate value of f (5.001), where f (x) = x3 − 7x2 + 15.

Question 78

Find the approximate change in the volume V of a cube of side x metres caused by increasing side by 1%.

Question 79

Find the approximate change in the surface area of a cube of side x metres caused by decreasing the side by 1%

Question 80

If the radius of a sphere is measured as 7 m with an error of 0.02m, then find the approximate error in calculating its volume.

Question 81

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 in surface area.

Question 82

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

Question 83

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

Question 84

Using differentials, find the approximate value of each of the following.

(a) (b)

Question 85

Find the maximum and minimum values, if any, of the following functions given by

(i) f(x) = |x + 2| − 1 (ii) g(x) = − |x + 1| + 3

(iii) h(x) = sin(2x) + 5 (iv) f(x) = |sin 4x + 3|

(v) h(x) = x + 4, x (−1, 1)

Question 86

Find the maximum and minimum values, if any, of the following functions given by

(i) f(x) = (2x − 1)2 + 3        (ii) f(x) = 9x2 + 12x + 2

(iii) f(x) = −(x − 1)2 + 10    (iv) g(x) = x3 + 1

Question 87

Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].

Question 88

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:

(i). f(x) = x2 (ii). g(x) = x3 − 3x

(iii). h(x) = sinx + cos, 0 < (iv). f(x) = sinx − cos x, 0 < x < 2π

(v). f(x) = x3 − 6x2 + 9x + 15

(vi).

(vii).

(viii).

Question 89

Prove that the following functions do not have maxima or minima:

(i) f(x) = ex (ii) g(x) = logx

(iii) h(x) = x3 + x2 + x + 1

Question 90

Find the maximum profit that a company can make, if the profit function is given by

p(x) = 41 − 24x − 18x2

Question 91

Find two numbers whose sum is 24 and whose product is as large as possible.

Question 92

Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.

Question 93

Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum

Question 94

Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

Question 95

A square piece of tin of side 18 cm is to 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?

Question 96

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 the maximum possible?

Question 97

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Question 98

Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.

Question 99

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?

Question 100

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?

Question 101

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.

Question 102

Show that the right circular cone of least curved surface and given volume has an altitude equal to time the radius of the base.

Question 103

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is.

Question 104

The point on the curve x2 = 2y which is nearest to the point (0, 5) is

(A) (B)

(C) (0, 0) (D) (2, 2)

Question 105

For all real values of x, the minimum value of is

(A) 0 (B) 1

(C) 3 (D)

Question 106

The maximum value of is

(A) (B)

(C) 1 (D) 0

Question 107

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.

Question 108

Show that the function given byhas maximum at x = e.

Question 109

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?

Question 110

Find the maximum area of an isosceles triangle inscribed in the ellipse with its vertex at one end of the major axis.

Question 111

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 meters for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?

Question 112

A window is in the form of rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

Question 113

A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle.

Show that the minimum length of the hypotenuse is

Question 114

Find the points at which the function f given byhas

(i) local maxima (ii) local minima

(ii) point of inflexion

Question 115

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is.

Question 116

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.

Question 117

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 istan2α.

Question 118

Find the maximum value of 2x3 − 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [−3, −1].

Question 119

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:

(i) (ii)

(iii)

(iv)

Question 120

Find both the maximum value and the minimum value of

3x4 − 8x3 + 12x2 − 48x + 25 on the interval [0, 3]

Question 121

What is the maximum value of the function sin x + cos x?

Question 122

At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?

Question 123

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.

Question 124

Find the maximum and minimum values of x + sin 2x on [0, 2π].

Question 125

Find the absolute maximum and minimum values of the function f given by

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