chillimath.com

chillimath

Menu

Ncert Solutions for Class 12

Applications of Derivatives

Applications of Derivatives

Miscellaneous Exercise

1. Show that the function given by \(f\left(x\right)=\frac{log{x}}{x}\) has maximum at x=e.

2. 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 ?

3. Find the intervals in which the function f given by \(f\left(x\right)=\frac{4sin{x}-2x-xcos{x}}{2+cos{x}}\) is
(i) increasing
(ii) decreasing.

4. Find the intervals in which the function f given by \(f\left(x\right)=x^3+\frac{1}{x^3},x\neq0\) is
(i) increasing
(ii) decreasing.

5. Find the maximum area of an isosceles triangle inscribed in the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) with its vertex at one end of the major axis.

6. 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?

7. 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.

8. 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.

9. 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 \((a^{2/3}+b^{2/3}\ )^{3/2}\).

10. 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

11. Find the absolute maximum and minimum values of the function f given by \(f(x)=\cos^2{x}+\sin{x},\ x\in[0,\pi]\).

12. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is \(\frac{4r}{3}\).

13. Let f be a function defined on [a,b] such that f’(x)>0, for all \(x\in(a,b)\). Then prove that f is an increasing function on (a,b).

14. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is \(\frac{2R}{\sqrt3}\). Also find the maximum volume.

15. 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 \(\frac{4}{27}\pi\ h^3{tan}^2{\alpha}\).

16. 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) \(1  \ m^3/h\)
(B) \(0.1 \  m^3/h\)
(C) \(1.1   \ m^3/h\)
(D) \(0.5  \ m^3/h\)

Applications of Derivatives

Exercise 6.1

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 \(8 cm^3/s\). How fast is the surface area increasing when the length of an edge is 12 cm?

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.

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?

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?

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 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.

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.

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.

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 ?

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 \(\frac{3}{2}(2x+1)\). 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 4 cm?

15. The total cost C(x) in Rupees associated with the production of x units of an item is 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 the 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 \(R(x)=3x^2+36x+5\). The marginal revenue, when x=15 is:
(A) 116  (B) 96  (C) 90  (D) 126