# Application of Derivatives

**Exercise 1**

**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

**2. **The volume of a cube is increasing at the rate of 8 cm^{3}/sec. How fast is the surface area increasing when the length of an edge is 12 cm?

**3. ** The radius of the circle is increasing uniformly at the rate of 3 cm per second. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

**4. **An edge of a variable cube is increasing at the rate of 3 cm per second. How fast is the volume of the cube increasing when the edge if 10 cm long?

**5. ** A stone is dropped into a quite lake and waves move in circles at the rate of 5 cm/sec. At the instant when radius of the circular wave is 8 cm, how fast is the enclosed area increasing?

**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?

**7. ** The length x of a rectangle is decreasing at the rate of 5 cm/minute and width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rate of change of (a) the perimeter and (b) the area of the rectangle.

**8. ** A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.

**9. ** A balloon, which always remains spherical has a variables radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.

**10. **A ladder 5 cm 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?

**11. ** A particle moves along the curve 6y = x^{3} + 2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.

**12. ** The radius of an air bubble is increasing at the rate of 1/2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

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

**14. **Sand is pouring from a pipe at the rate of 12 cm^{3}/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 4cm?

**15.** The total cost C(x) in rupees associated with the production of x units of an item given by C(x) = 0.007x^{3} – 0.003x^{2} + 15x+4000. Find the marginal cost when 17 units are produced.

**16.** The total revenue in rupees received from the sale of x units of a product is given by R(x) = 13x^{2} + 26x + 15. Find the marginal revenue when x = 7.

Choose the correct answer in Exercises 17 and 18.

**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

**18.** The total revenue in rupees received from the sale of x units of a product is given by R(x) = 3x^{2} + 36x + 15. The marginal revenue when x = 15:

(A) 116

(B) 96

(C) 90

(D) 126

**Exercise 2**

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

**20. ** Show that the function given by f(x) = e^{2x} is strictly increasing on R.

**21.** Show that the function given by f(x) = sin x is (a) strictly increasing (b) strictly decreasing in (c) neither increasing nor decreasing in (0, π).

**22.** Find the intervals in which the function f given by f(x) = 2x^{2} – 3x is (a) strictly increasing, (b) strictly decreasing.

**23.** Find the intervals in which the function f given by f(x) = 2x^{3} – 3x^{2} – 36x + 7 is (a) strictly increasing, (b) strictly decreasing.

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

(a) x^{2} + 2x – 5

(b) 10 – 6x – 2x^{2}

(c) -2x^{3} – 9x^{2} – 12x + 1

(d) 6 – 9x – x^{2}

(e) (x + 1)^{3} (x – 3)^{3}

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

**26.** Find the value of x for worth y = {x(x – 2)}^{2} is an increasing function.

**27.** Prove that is an increasing function of θ in

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

**29.** Prove that the function f given by f(x) = x – x + 1 is neither strictly increasing nor strictly decreasing on (-1, 1).

**30.** Which of the following functions are strictly decreasing on

**31.** On which of the following intervals is the function f given by f(x) = x^{100} + sin x – 1 is strictly decreasing:

**32.** Find the last value of a such that the function f given by f(x) = x + ax + 1 strictly increasing on (1, 2).

**33.** Let I be any interval disjoint from [-1, 1]. Prove that the function f given by is strictly increasing on I.

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

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

**36.** Prove that the function given by f(x) = x^{3} – 3x^{2} + 3x – 100 is increasing in R.

**37.** The interval in which y = x^{2}e^{-x} is increasing in:

(A) (-∞, ∞)

(B) (-2, 0)

(C) (2, ∞)

(d) (0, 2)

**Exercise 3**

**38.** Find the slope of tangent to the curve y = 3x – 4x at x = 4.

**39.** Find the slope of tangent to the curve

**40.** Find the slope of tangent to the curve y = x^{3} – x + 1 at the given point whose x-coordinate is 2.

**41.** Find the slope of tangent to the curve y = x^{3} – 3x + 2 at the given point whose x-coordinate is 3.

**42.** Find the slope of the normal to the curve

**43.** Find the slope of the normal to the curve

**44. ** Find the point at which the tangent to the curve y = x^{3} – 3x^{2} – 9x + 7 is parallel to the x-axis.

**45.** Find the 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).

**46.** Find the point on the curve y = x^{3} – 11x + 5 at which the tangent is y = x – 11.

**47.** Find the equation of all lines having slope -1 that are tangents to the curve

**48.** Find the equations of all lines having slope 2 which are tangents to the curve

**49.** Find the equations of all lines having slope 0 which are tangents to the curve

**50.** Find the points on the curve at which the tangents are:

(i) Parallel to x-axis

(ii) Parallel to y-axis

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

(i) y = x^{4} – 6x^{3} + 13x^{2} – 10x + 5 at (0, 5)

(ii) y = x^{4} – 6x^{3} + 13x^{2} – 10x + 5 at (1, 3)

(iii) y = x^{3} at (1, 1)

(iv) y = x^{2} at (0, 0)

**52.** Find the equation of the tangent line to curve y = x^{2} – 2x + 7 which is:

(a) parallel to the line 2x – y + 9 = 0

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

**53.** Show that the tangents to the curve y = 7x^{3} + 11 at the points where x = 2 and x = -2 are parallel.

**54.** Find the points on the curve y = x^{3 }at which the slope of the tangent is equal to the y-coordinate of the point.

**55.** For the curve y = 4x^{3} – 2x^{5}, find all points at which the tangent passes through the origin.

**56.** Find the points on the curve x^{2} + y^{2} – 2x – 3 = 0 at which the tangents are parallel to x-axis.

**57.** Find the equation of the normal at the point (am^{2}, am^{3}) for the curve ay^{2} = x^{3}.

**58.** Find the equations of the normal to the curve y = x^{3} + 2x + 6 which are parallel to the line x + 14y + 4 = 0.

**59.** Find the equation of the tangent and normal to the parabola y^{2} = 4ax at the point (at^{2}, 2at).

**60.** Prove that the curves x = y^{2} and xy = k cut at right angles if 8k^{2} = 1.

**61.** Find the equations of the tangent and normal to the hyperbola at the point (x_{0}, y_{0}).

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

Choose the correct answer in Exercises 63 and 64.

**63.** The slope of the normal to the curve y = 2x^{2} + 3 sin x at x = 0 is:

**64. ** The line y = x + 1 is a tangent to the curve y^{2} = 4x at the point:

(A) (1, 2)

(B) (2, 1)

(C) (1, -2)

(D) (-1, 2)

**Exercise 4**

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

**66.** Find the approximate value of f(2.01) where f(x) = 4x^{2} + 5x + 2.

**67.** Find the approximate value of f(5.001) where f(x) = x^{3} – 7x^{2} + 15.

**68.** Find the approximate change in the volume of a cube of side x meters caused by increasing the side by 1%.

**69.** Find the approximate change in the surface area of a cube of side x meters caused by decreasing the side by 1%.

**70.** 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.

**71.** 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.

**72.** If f(x) = 3x^{2} + 15x + 5, then the approximate value of f(3.02) is:

(A) 47.66

(B) 57.66

(C) 67.66

(D) 77.66

**73.** The approximate change in the volume of a cube of side x meters caused by increasing the side by 3% is:

(A) 0.06 x^{3} m^{3}

(B) 0.6 x^{3} m^{3}

(C) 0.09 x^{3} m^{3}

(D) 0.9 x^{3} m^{3}

**Exercise 5**

**74.** Using differentials, find the approximate value of each of the following:

**75.** Show that the function given by has maximum value at x = e.

**76.** 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?

**77.** Find the equation of the normal to the curve y^{2} = 4x at the point (1, 2).

**78.** Show that the normal at any point θ to the curve x = a cos θ + a θ sin θ, y = a sin θ a θ cos θ is at a constant distance from the origin.

**79.** Find the intervals in which the function f given by is (i) increasing (ii) decreasing.

**80.** Find the intervals in which the function f given by is (i) increasing (ii) decreasing.

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

**82.** 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 m^{3}. If building of tank costs 70 per sq. meter for the base and 45 per square meter for sides. What is the cost of least expensive tank?

**83.** 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.

**84.** A window is in the form of a 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.

**85. **A point on the hypotenuse of a triangle is at distances and from the sides of the triangle. Show that the maximum length of the hypotenuse is

**86. ** 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.

**87.** Find the absolute maximum and minimum values of the function f given by f(x) = cos^{2} x + sin x, x ∈ [0, π].

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

**89.** 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).

**90. ** 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.

**91.** Show that the height of the cylinder of greatest volume which can be inscribed in a right circular cone of height and having semi-vertical angle is one-third that of the cone and the greatest volume of the cylinder is

Choose the correct answer in the Exercises 92 to 97:

**92. **A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic meter per hour. Then the depth of 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

**93.** The slope of the tangent to the curve x = t^{2} + 3t – 8, y = 2t^{2} – 2t – 5 at the point (2, -1) is:

**94.** The line y = mx + 1 is a tangent to the curve y^{2} = 4x if the value of m is:

(A) 1

(B) 2

(C) 3

(D) 1/2

**95. ** The normal at the point (1, 1) on the curve 2y + x^{2} = 3 is:

(A) x + y = 0

(B) x – y = 0

(C) x + y + 1 = 0

(D) x – y = 1

**96.** The normal to the curve x^{2} = 4y passing through (1, 2) is:

(A) x + y = 3

(B) x – y = 3

(C) x + y = 1

(D) x – y = 1

**97.** The points on the curve 9y^{2} = x^{3}, where the normal to the curve make equal intercepts with axes are: