Chapter 6. Application 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)  cm (b)  cm

Solution :  (a)  cm

The area  of a circle with radius is given by  .

Therefore, the rate of change of the area  with respect to its radius is given by

When  cm, then 

Thus, the area of the circle is changing at the rate of   .

(b)   cm

The area  of a circle with radius is given by  .

Therefore, the rate of change of the area  with respect to its radius is given by

When cm, then

Thus, the area of the circle is changing at the rate of  .

2. The volume of a cube is increasing at the rate of 8  . How fast is the surface area increasing when the length of an edge is 12 cm?

Solution : Let  be the length of a side,  be the volume and  be the surface area of the cube.

 Then,  and  , where  is a function of time  .

A/Q,  

When cm , then

Again,

 

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.

Solution : Here,   cm and  cm/s

The area  of a circle with radius  is given by  .

Therefore, the rate of change of the area  with respect to its time  is given by

 

When  cm, then

Thus, the area of the circle is changing at the rate of  .

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?

Solution :   Let  be the length of a side and  be the volume of the cube.

 Then, , where  is a function of time  .

Here,  cm and  cm/s

Now ,

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?

Solution : The area  of a circle with radius  is given by  .

Therefore, the rate of change of the area  with respect to its time  is given by

When, cm and  cm/s

Then

 

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?

Solution : The circumference  of a circle with radius  is given by  .

Therefore, the rate of change of the circumference  with respect to its time t is given by

When  then

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

Solution : Here, and

The perimeter of the rectangle  and  the area of the rectangle

(a)  We have,

 

(b) We have,

 

When  cm and  cm , then

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.

Solution : Let , the volume V of the spherical balloon with radius  is given by

Here, and

Therefore, the rate of change of volume V with respect to time  is

A/Q,  

 

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.

Solution : Let , the volume V of the spherical balloon with radius  is given by and

Therefore, the rate of change of volume V with respect to radius  is

When  , then

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 ?

Solution: Let be the height of the wall and  be the distance between the bottom of the ladder and foot of the wall .

 

Given,  m and

We have,

 

 

 

Differentiating (i) , with respect to time  , we have

 

Therefore, the height of the ladder on the wall is decreasing at the rate of  .

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

Solution : Given, the curve

 

A/Q, 

 

When x=4 , then 

When , then 

 

Therefore, the points on the curve are (4,11) and .

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?

Solution : Let  and V be the radius and the volume of the bubble respectively .

Given ,    and

We have,

Therefore, the rate of change of volume V with respect to time  is

When  , then

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

Solution:  Given, the diameter and the radius

The volume of the balloon
Therefore, the rate of change of volume V with respect to  is

 

 

14. Sand is pouring from a pipe at the rate of 12  . 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?

Solution:  Let  and  be the radius and height of the pipe respectively .

The volume of the cone

Given,

 

 

A/Q,  

When ,then

 

15. The total cost C () in Rupees associated with the production of x units of an item is given by C()  . Find the marginal cost when 17 units are produced.

Solution: The total cost

The marginal cost

When then MC

 

16. The total revenue in Rupees received from the sale of  units of a product is given by R() . Find the marginal revenue when .

Solution : The total revenue

The marginal revenue

When ,then

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  at  cm is
(A)  (B)  (C)  (D)  

Solution:  The area of the circle

The rate of change of the area of a circle with respect to its radius , then

 

When , then

 

Correct answer (B)  .

18. The total revenue in Rupees received from the sale of x units of a product is given by R() . The marginal revenue, when  is
(A) 116    (B) 96    (C) 90     (D) 126

Solution: The total revenue

When , then

The correct answer is (D) 126 .

EXERCISE 6.2

1. Show that the function given by  is strictly increasing on R.

Solution : Given, the function

Differentiating , with respect to  , we have

in every interval of R .

Therefore, the function  is increasing on R .

2. Show that the function given by   is strictly increasing on R.

Solution: Let and  be any two numbers in R .

Then  

 

Therefore, the function  is strictly increasing on R .

3. Show that the function given by  is
(a) strictly increasing in  .

(b) strictly decreasing in  .
(c) neither increasing nor decreasing in

Solution: Given, the function

Differentiating the given function w.r.to x , we have

(a) Since for each , , we have  and so  is strictly increasing in  .

(b) Since for each , , we have   and so  is strictly decreasing in .

(c) When , then   in  and  in .

Therefore,  is neither increasing nor decreasing in  .
4. Find the intervals in which the function f given by  is
(a) strictly increasing (b) strictly decreasing

Solution: Given, the function

The points  divides the real line into two disjoint intervals, namely,  and  .

Interval	Sign of	Nature of function
		is strictly decreasing
		is strictly increasing

(a) The function  is strictly increasing in interval .

(b)  The function  is strictly decreasing in interval  .

5. Find the intervals in which the function  given by is
(a) strictly increasing (b) strictly decreasing

Solution:  Given, the function  

The points  and  divides the real line into three disjoint intervals, namely,  , and  .

Interval	Sign of	Nature of function
		is strictly increasing
		is strictly decreasing
		is strictly increasing

(a) The function  is strictly increasing in interval and

(b)  The function  is strictly decreasing in interval .

6. Find the intervals in which the following functions are strictly increasing or decreasing:
(a)      (b)     (c)   (d)    (e)   

Solution: (a) let  

  

Therefore, the point  divides the real line into two disjoint intervals namely, and .

Thus, the function is strictly increasing when and the functionis strictly decreasing when  .

(b)    

Let    

  

Therefore, the point  divides the real line into two disjoint intervals namely, and .

Interval	Sign of	Nature of function
		is strictly increasing
		is strictly decreasing

Thus, the function is strictly increasing when  and the functionis strictly decreasing when  .

(c)   

Let

The points  and divides the real line into three disjoint intervals, namely,  ,   and  .

  is negative for   , indicating that  is decreasing on  .

  is positive for , indicating that  is increasing on .

  is negative for  , indicating that  is decreasing on  .

Therefore, the function is strictly increasing on the interval (−2,−1) and strictly decreasing on the intervals   and   .

(d)   
Let

Therefore, the point  divides the real line into two disjoint intervals namely, and .

Interval	Sign of	Nature of function
		is strictly increasing
		is strictly decreasing

Thus, the function is strictly increasing when  and the functionis strictly decreasing when  .

(e) 

Let

or  

or

The points and divides the real line into four disjoint intervals, namely,  , and  .

  is negative for , indicating that  is decreasing on  .

  is negative for , indicating that  is decreasing on .

  is positive for  , indicating that  is increasing on (1,3) .

  is positive for  , indicating that  is increasing on .

Therefore, the function is strictly increasing on the interval (1,3) and , and strictly decreasing on the intervals and .

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

Solution: We have,

Now, we want to show that   for  .

 Since  , the numerator  is positive.

The denominator  is also positive forsince each term in the factorization is positive.

Therefore,   for  .

 This implies that is an increasing function for  .

8. Find the values of  for which  is an increasing function.

Solution: We have,

 

 

The points and   divides the real line into four disjoint intervals, namely,  , and  .

 is negative for  , so  is decreasing on  .

 ' is positive for  , so  is increasing on  .

'  is negative for  , so  is decreasing on  .

is positive for  , so  is increasing on  .

Therefore,is increasing on the intervals  and   .

9. Prove that  is an increasing function of  in

Solution: We have,

Since, , , so  .

Therefore, for   

Since in the given interval , then  is an increasing function of  in  .

10. Prove that the logarithmic function is strictly increasing on

Solution: let

When 

Therefore, for all .

So, the given function is strictly increasing on

11. Prove that the function  given by   is neither strictly increasing nor strictly decreasing on (– 1, 1).

Solution: Given, the function

The points   divides the real line into two disjoint intervals, namely,  and  .

Interval	Sign of	Nature of function
-		is strictly decreasing
		is strictly increasing

Since the function  changes its behavior at (from decreasing to increasing), it is neither strictly increasing nor strictly decreasing on (−1,1) .

Thus, the function  is neither strictly increasing nor strictly decreasing on (– 1, 1).

12. Which of the following functions are strictly decreasing on  ?
(A)    (B)   (C)   (D)
13. On which of the following intervals is the function  given by  strictly decreasing ?
(A) (0,1)    (B)  (C)    (D) None of these
14. Find the least value of a such that the function  given by  is strictly increasing on (1, 2).
15. Let be any interval disjoint from (–1, 1). Prove that the function  given by  is strictly increasing on .
16. Prove that the function  given by  is strictly increasing on  and strictly decreasing on  .
17. Prove that the function  given by  is strictly decreasing on  and strictly increasing on   .
18. Prove that the function given by  is increasing in R.
19. The interval in which   is increasing is
(A)   (B) (– 2, 0)   (C)   (D) (0, 2)

EXERCISE 6.3

1. Find the maximum and minimum values, if any, of the following functions given by
(i)      (ii)       (iii)       (iv)
2. Find the maximum and minimum values, if any, of the following functions given by
(i)                      

(ii)
(iii)                     

(iv)
(v) 
3. 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)     (ii)
(iii)
(iv) 
(v)          

(vi) 
(vii) 
(viii)
4. Prove that the following functions do not have maxima or minima:
(i)   (ii) 
(iii) 
5. Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
(i)           (ii)
(iii)      (iv)
6. Find the maximum profit that a company can make, if the profit function is given by .

7. Find both the maximum value and the minimum value of  on the interval [0, 3].
8. At what points in the interval  does the function  attain its maximum value?
9. What is the maximum value of the function  ?
10. Find the maximum value of  in the interval [1, 3]. Find the maximum value of the same function in [–3, –1].

11. It is given that at  , the function  attains its maximum value, on the interval [0, 2]. Find the value of .
12. Find the maximum and minimum values of  on [0, 2π].
13. Find two numbers whose sum is 24 and whose product is as large as possible.
14. Find two positive numbers  and  such that  and  is maximum.
15. Find two positive numbers  and  such that their sum is 35 and the product  is a maximum.
16. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
17. 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.
18. 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 ?
19. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
20. 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.
21. 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?
22. 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?
23. Prove that the volume of the largest cone that can be inscribed in a sphere of radius  is of the volume of the sphere.
24. Show that the right circular cone of least curved surface and given volume has an altitude equal to  time the radius of the base.
25. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is  .

26. Show that semi-vertical angle of right circular cone of given surface area and maximum volume is  .

Choose the correct answer in the Exercises 27 and 29.
27. The point on the curve  which is nearest to the point (0, 5) is
(A) ( ,4)   (B) ( ,0)    (C) (0, 0)     (D) (2, 2)
28. For all real values of , the minimum value of  is
(A) 0     (B) 1     (C) 3     (D) 1/3
29. The maximum value of  is
(A)    (B)     (C) 1    (D) 0

Miscellaneous Exercise on Chapter 6

1. Show that the function given by has maximum at   .
2. The two equal sides of an isosceles triangle with fixed base  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  given by  is (i) increasing (ii) decreasing.
4. Find the intervals in which the function  given by  is: (i) increasing (ii) decreasing.
5. Find the maximum area of an isosceles triangle inscribed in the ellipse   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  . 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 , where  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 .
10. Find the points at which the function  given by  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
12. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius  is   .
13. Let  be a function defined on [a, b] such that , for all  Then prove that  is an increasing function on
14. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius  is  . 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  is one-third that of the cone and the greatest volume of cylinder is   .
Choose the correct answer in the Exercises from 19 to 24.
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    (B) 0.1       (C) 1.1     (D) 0.5