Solutions of chapter 11. Applications of Derivatives | RS Aggarwal - Mathematics

1

The side of a square is increasing at the rate of 0.2 cm/s. Find the rate of increase of the perimeter of the square.

2

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

3

The radius of a circle is increasing uniformly at the rate of 0.3 centimetre per second. At what rate is the area increasing when the radius is 10 cm?

(Take π = 3.14.)

4

The side of a square sheet of metal is increasing at 3 centimetres per minute. At what rate is the area increasing when the side is 10 cm long?

5

The radius of a circular soap bubble is increasing at the rate of 0.2 cm/s. Find the rate of increase of its surface area when the radius is 7 cm.

6

The radius of an air bubble is increasing at the rate of 0.5 centimetre per second. At what rate is the volume of the bubble increasing when the radius is 1 centimetre?

7

The volume of a spherical balloon is increasing at the rate of 25 cubic centimetres per second. Find the rate of change of its surface at the instant when its radius is 5 cm.

8

A balloon which always remains spherical is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15 cm.

9

The bottom of a rectangular swimming tank is 25 m by 40 m. Water is pumped into the tank at the rate of 500 cubic metres per minute. Find the rate at which the level of water in the tank is rising.

10

A stone is dropped into a quiet lake and waves move in circles at a speed of 3.5 cm per second. At the instant when the radius of the circular wave is 7.5 cm. how fast is the enclosed area increasing? (Take π = 22/7.)

11

A 2-m tall man walks at a uniform speed of a uniform speed of 5 km per hour away from a 6-metre-high lamp post. Find the rate at which the length of his shadow increases.

12

An inverted cone has a depth of 40 cm and a base of radius 5 cm. Water is poured into it at a rate of 1.5 cubic centimetres per minute. Find the rate at which the level of water in the cone is rising when the depth is 4 cm.

13

Sand is pouring from a pipe at the rate of 18. The falling sand forms a cone on the ground in such a way that the height of the cone is one-sixth of the radius of the base. How fast is the height of the sand cone increasing when its height is 3 cm?

14

Water is dripping through a tiny hole at the vertex in the bottom of a conical funnel at a uniform rate of 4 . When the slant height of the water is 3 cm, find the rate of decrease of the slant height of the water, given that the vertical angle of the funnel is 120°.

15

Oil is leaking at the rate of 15 mL/s from a vertically kept cylindrical drum containing oil. If the radius of the drum is 7 cm and its height is 60 cm, find the rate at which the level of the oil is changing when the oil level is 18 cm.

16

A 13-m long ladder is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 m/s. How fast is its height on the wall decreasing when the foot of the ladder is 5 m away from the wall?

17

A man is moving away from a 40-m high tower at a speed of 2 m/s. Find the rate is which the angle of elevation of the top of the tower is changing when he is at a distance of 30 metres from the foot of the tower. Assume that the eye level of the man is 1.6 m from the ground.

18

Find an angle x which increases twice as fast as its sine.

19

The radius of a balloon is increasing at the rate of 10 m/s. At what rate is the surface area of the balloon increasing when the radius is 15 cm?

20

An edge of a variable cube is increasing at the rate of 5 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

21

The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. Find the rate at which the area is increasing when the side is 10 cm.

1

Using differentials, find the approximate values of:

find the approximate values of .

2

Using differentials, find the approximate values of:

Find the approximate values of .

3

Using differentials, find the approximate values of:

Find the approximate values of

4

Using differentials, find the approximate values of:

Find the approximate values of

5

Using differentials, find the approximate values of:

Find the approximate values of

6

Using differentials, find the approximate values of:

Find the approximate values of

7

find the approximate values of

8

find the approximate values of 10.02, given that 10 = 2.3026

9

find the approximate values of log₁₀(4.04), it being given that log₁₀4 = 0.6021 and log₁₀e = 0.4343

10

find the approximate values of cos 61°, it being given that sin 60° = 0.86603 and 1° = 0.01745 radian

11

If y = sin x and x changes from to , what is the approximate change in y?

12

A circular metal plate expands under heating so that its radius increases by 2%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10 cm.

13

If the length of a simple pendulum is decreased by 2%, find the percentage decrease in its period T, where .

14

The pressure p and the volume V of a gas are connected by the relation, , where k is a constant. Find the percentage increase in the pressure, corresponding to a diminution of 0.5% in the volume.

15

The radius of a sphere shrinks from 10 cm to 9.8 cm. Find approximately the decrease in (i) volume, and (ii) surface area.

16

If there is an error of 0.1% in the measurement of the radius of a sphere, find approximately the percentage error in the calculation of the volume of the sphere.

17

Show that the relative error in the volume of a sphere, due to an error in measuring the diameter, is three times the relative error in the diameter.

1

Verify Rolle’s theorem for each of the following functions:

2

Verify Rolle’s theorem for each of the following functions:

3

Verify Rolle’s theorem for each of the following functions:

4

Verify Rolle’s theorem for each of the following functions:

5

Verify Rolle’s theorem for each of the following functions:

6

Verify Rolle’s theorem for each of the following functions:

7

Verify Rolle’s theorem for each of the following functions:

8

Verify Rolle’s theorem for each of the following functions:

9

Verify Rolle’s theorem for each of the following functions:

10

Verify Rolle’s theorem for each of the following functions:

11

Verify Rolle’s theorem for each of the following functions:

12

Verify Rolle’s theorem for each of the following functions:

3

Verify Rolle’s theorem for each of the following functions:

14

Verify Rolle’s theorem for each of the following functions:

15

Verify Rolle’s theorem for each of the following functions:

16

Verify Rolle’s theorem for each of the following functions:

17

Verify Rolle’s theorem for each of the following functions:

18

Verify Rolle’s theorem for each of the following functions:

19

Verify Rolle’s theorem for each of the following functions:

20

Verify Rolle’s theorem for each of the following functions:

21

Verify Rolle’s theorem for each of the following functions:

Show that satisfies Rolle’s theorem on [0, 5] and that the value of c is (5/3)

22

Discuss the applicability for Rolle’s theorem, when:

, where

23

Discuss the applicability for Rolle’s theorem, when:

24

Discuss the applicability for Rolle’s theorem, when:

25

Discuss the applicability for Rolle’s theorem, when:

26

Discuss the applicability for Rolle’s theorem, when:

, where [x] denotes the greatest integer not exceeding x

27

Using Rolle’s theorem, find the point on the curve , where the tangent is parallel to the x-axis.

1

Verify Lagrange’s mean-value theorem for the following function:

2

Verify Lagrange’s mean-value theorem for the following function:

3

Verify Lagrange’s mean-value theorem for the following function:

4

Verify Lagrange’s mean-value theorem for the following function:

5

Verify Lagrange’s mean-value theorem for the following function:

6

Verify Lagrange’s mean-value theorem for the following function:

7

Verify Lagrange’s mean-value theorem for the following function:

8

Verify Lagrange’s mean-value theorem for the following function:

9

Verify Lagrange’s mean-value theorem for the following function:

10

Verify Lagrange’s mean-value theorem for the following function:

11

Verify Lagrange’s mean-value theorem for the following function:

12

Show that Lagrange’s mean-value theorem is not applicable to on .

13

Show that Lagrange’s mean-value theorem is not applicable to on

14 A

Find ‘c’ of Lagrange’s mean-value theorem for

14 B

Find ‘c’ of Lagrange’s mean-value theorem for

14 C

Find ‘c’ of Lagrange’s mean-value theorem for

15

Using Lagrange’s mean-value theorem, find a point on the curve , where the tangent is parallel to the line joining the point (1, 1) and (2, 4)

16

Find a point on the curve , where the tangent to the curve is parallel to the chord joining the points (1, 1) and (3, 27).

17

Find the points on the curve , where the tangent to the curve is parallel to the chord joining (1, −2) and (2, 2).

18

If , where c > 0, show that , where 0 < a < c < b.

1

Find the maximum or minimum values, if any, without using derivatives, of the function:

.

2

Find the maximum or minimum values, if any, without using derivatives, of the function:

3

Find the maximum or minimum values, if any, without using derivatives, of the function:

4

Find the maximum or minimum values, if any, without using derivatives, of the function:

5

Find the maximum or minimum values, if any, without using derivatives, of the function:

6

Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions:

7

Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions:

8

Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions:

9

Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions:

10

Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions:

11

Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions:

12

Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions:

13

Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions:

14

Find the point of local maxima or local minima or local minima and the corresponding local maximum and minimum values of each of the following functions:

15

Find the maximum and minimum values of on the interval .

16

Find the maximum and minimum values of on the interval .

17

Find the maximum and minimum of

18

Show that the maximum value of is

19

Show that has a maximum and minimum, but the maximum value is less than the minimum value.

20

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

21

An enemy jet is flying along the curve . A soldier is placed at the point (3, 2). Find the nearest point between the soldier and the jet.

22

Find the maximum and minimum values of

.

1

Find two positive number whose product is 49 and the sum is minimum.

2

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

3

Divide 15 into two parts such that the square of one number multiplied with the cube of the other number is maximum.

4

Divide 8 into two positive parts such that the sum of the square of one and the cube of the other is minimum.

5

Divide a into two parts such that the product of the pth power of one part and the qth power of the second part may be maximum.

6

The rate of working of an engine is given by.

and υ is the speed of the engine. Show that R is the least when υ = 20.

7

Find the dimensions of the rectangle of area 96 whose perimeter is the least. Also, find the perimeter of the rectangle.

8

Prove that the largest rectangle with a given perimeter is a square.

9

Given the perimeter of a rectangle, show that its diagonal is minimum when it is a square.

10

Show that a rectangle of maximum perimeter which can be inscribed in a circle of radius a is a square of side .

11

The sum of the perimeters of a square and a circle is given. Show that the sum of their areas is least when the side of the square is equal to the diameter of the circle.

12

Show that the right triangle of maximum area that can be inscribed in a circle is an isosceles triangle.

13

Prove that the perimeter of a right-angled triangle of given hypotenuse is maximum when the triangle is isosceles.

14

The perimeter of a triangle is 8 cm. If one of the sides of the triangle be 3 cm, what will be the other two sides for maximum area of the triangle?

15

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

16

A square piece of tin of side 12 cm is to be made into a box without a lid by cutting a square from each corner and folding up the flaps to form the sides. What should be the side of the square to be cut off so that the volume of the box is maximum? Also, find this maximum volume.

17

An open box with a square base is to be made out of a given cardboard of area (square) units. Show that the maximum volume of the box is (cubic) units.

18

A cylindrical can is to be made to hold 1 litre of oil. Find the dimensions which will minimize the cost of the metal to make the can.

19

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

20

Find the radius of a closed right circular cylinder of volume 100 which has the minimum total surface area.

21

Show that the height of a closed cylinder of given volume and the least surface area is equal to its diameter.

22

Prove that the volume of the largest cone that can be inscribed in a sphere is of the volume of the sphere.

23

Which fraction exceeds its pth power by the greatest number possible?

24

Find the point on the curve which is nearest to the point (2, −8).

25

A right circular cylinder is inscribed in a cone. Show that the curved surface area of the cylinder is maximum when the diameter of the cylinder is equal to the radius of the base of the cone.

26

Show that the surface area of a closed cuboid with square base and given volume is minimum when it is a cube.

27

A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle so that its area is maximum. Find also this area.

28

Two sides of a triangle have lengths a and b and the angle between them is θ. What value of θ will maximize the area of the triangle?

29

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius cm is (500π).

30

A square tank of capacity 250cubic meters has to be dug out. The cost of the land is Rs. 50 per square metre. The cost of digging increases with the depth and for the whole tank, it is Rs., where h metres is the depth of the tank. What should be the dimensions of the tank so that the cost is minimum?

31

A square piece of tin of side 18 cm is to be made into a box without the top, by cutting a square piece 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? Also, find the maximum volume of the box.

32

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of the material will be least when the depth of the tank is half of its width.

33

A wire of length 36 cm is cut into two pieces. One of the pieces is turned in the form of a square and the other in the form of an equilateral triangle. Find the length of each piece so that the sum of the areas of the two be minimum.

34

Find the largest possible area of a right-angles triangle whose hypotenuse is 5 cm.

1

Show that the function is a strictly increasing function on R.

2

Show the function is a strictly decreasing function on R.

3

Prove that , where a and b are constants and a > 0, is a strictly increasing function on R.

4

Prove that the function is strictly increasing on R.

5

Show that the function is

a. strictly increasing on [0, ∞[

b. strictly decreasing on [0, ∞[

c. neither strictly increasing nor strictly decreasing on R

6

Show that the function is

a. strictly increasing on ]0, ∞[

b. strictly decreasing on] − ∞, 0[

7

Prove that the function is strictly increasing on ]0, ∞[.

8

Prove that the function is strictly increasing on ]0, ∞[ when a > 1 and strictly decreasing on ]0, ∞[ when 0 < a < 1.

9

Prove that is strictly increasing on R.

10

Show that is increasing on R.

11

Show that is increasing all , where .

12

Show that is decreasing for all , where .

13

Show that is decreasing for all

14

Show that is decreasing on .

15

Show that is increasing on .

16

Prove that the function is increasing for all .

17

Let I be an interval disjoint from . Prove that the function is strictly increasing on I.

18

Show that is increasing for all , except at .

19

Find the intervals on which the function is

(a) strictly increasing

(b) strictly decreasing.

20

Find the intervals on which the function is

(a) strictly increasing (b) strictly decreasing.

21

Find the intervals on which the function is

(a) strictly increasing (b) strictly decreasing.

22

Find the intervals on which each of the following functions is (a) increasing (b) decreasing.

23

Find the intervals on which each of the following functions is (a) increasing (b) decreasing.

24

Find the intervals on which each of the following functions is (a) increasing (b) decreasing.

25

Find the intervals on which each of the following functions is (a) increasing (b) decreasing.

26

Find the intervals on which each of the following functions is (a) increasing (b) decreasing.

27

Find the intervals on which each of the following functions is (a) increasing (b) decreasing.

28

Find the intervals on which each of the following functions is (a) increasing (b) decreasing.

29

Find the intervals on which each of the following functions is (a) increasing (b) decreasing.

30

Find the intervals on which each of the following functions is (a) increasing (b) decreasing.

31

Find the intervals on which each of the following functions is (a) increasing (b) decreasing.

32

Find the intervals on which each of the following functions is (a) increasing (b) decreasing.

1

Find the slope of the tangent to the curve

i.

ii.

iii.

2

Find the equations of the tangent and the normal to the given curve at the indicated point for

3

Find the equations of the tangent and the normal to the given curve at the indicated point for

4

Find the equations of the tangent and the normal to the given curve at the indicated point for

5

Find the equations of the tangent and the normal to the given curve at the indicated point for

6

Find the equations of the tangent and the normal to the given curve at the indicated point for

7

Find the equations of the tangent and the normal to the given curve at the indicated point for

8

Find the equations of the tangent and the normal to the given curve at the indicated point for

9

Find the equations of the tangent and the normal to the given curve at the indicated point for

, where

10

Find the equations of the tangent and the normal to the given curve at the indicated point for

at the point where x = 1

11

Find the equation of the tangent to the curve

12

Show that the equation of the tangent to the hyperbola at is .

13

Find the equation of the tangent to the curve .

14

Find the equation of the normal to the curve

15

Show that the tangents to the curve at the point x = 2 and x = −2 are parallel.

16

Find the equation of the tangent to the curve , where is parallel to the line .

17

At what point on the curve , is the tangent parallel to the y-axis?

18

Find the point on the curve where the tangent is parallel to the x-axis.

19

Prove the tangent to the curve at the point (2, 0) and (3, 0) are at right angles.

20

Find the point on the curve at which the tangent passes through the origin.

21

Find the point on the curve at which the equation of tangent is .

22

Find the equation of the tangents to the curve , parallel to the line .

23

Find the equation of the tangent to the curve , which is perpendicular to the line .

24

Find the point on the curve at which the tangent is parallel to the x-axis.

25

Find the point on the parabola , where the tangent is parallel to the chord joining the point (3, 0) and (4, 1).

26

Show that the curves and cut at right angles if .

27

Show that the curves and touch each other.

28

Show that the curves and cut orthogonally.

29

Find the equation of tangent to the curve at .

30

Find the equation of the tangent at for the curve ,.