aod

By Mukesh Gupta(M.Sc-Maths-IIT Delhi)

Rate of Change of Quantities

1. The radius of a circular plate is increasing at the rate of 0.2 /m s . At what rate is

the area increasing, when the radius of the plate is 25 ?cm ( )2:10 /Ans cm sπ

2. Find the rate of change of volume of sphere with respect to its radius.

( )2: 4Ans rπ

3. A cubical box which always remains cubical has a variable side 9

determine the rate of change of volume with respect to x

:15 54

4. A stone is dropped into quite lake and waves are moving in circles at the speed

of3.5 / .cm s At the instant when the radius of circular waves is 7.5cm , how fast is

the enclosed area increasing? ( )2: 52.5 /Ans cm sπ

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

of perimeter of the square ( ): 0.8 /Ans cm s .

6. The radius of a circle is increasing at the rate of 0.7 /cm s . What is the rate of

increasing of its circumference? ( ):1.4 /Ans cm sπ

7. If the volume of the air bubble is increasing at the rate of 310 /cm s , find the rate

of change of its surface area when radius is1cm ( )2: 20 /Ans cm s

8. The bottom of rectangular swimming tank is 25 40m m× . Water is pumped into

the tank at the rate of 3500 /m hr . Find the rate at which the level of water in the

tank is rising.1

Ans m hr

9. At what points of the ellipse 2 216 9 400x y+ = does the ordinate decreases at the

same rate at which the abscissa increases?16 16

: 3, & 3,3 3

Ans −

10. A particle moves along the curve 36 2y x= + .Find points on the curve at which

y co-ordinate is changing8 times as fast as the x co-

ordinate. ( )31

4,11 & 4,3

− −

11. A man, whose height is 2m , walks at a uniform speed of 6 /m minute away from a

lamp post5m high. Find the rate at which the length of his shadow increases.

( ): 4 / minAns m

For Students of S.B.D.A.V Public School Only

12. A ladder 5m in length is leaning against a wall. The bottom of the ladder is pulled

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 4m away from the

ground?0.08

Ans m s−

13. A boy is flying a kite at a height of 40 meter. A boy is carrying it horizontally at

the rate of 3 /m s . At what rate is the string being pulled out, when the length of

the string is 50m ? (Assume that the height of kite remains the same and the string

is straight)9

14. A ladder 20 ft long has one end on the ground and the other in contact with a

vertical wall. The lower end slips along the ground. Show that when the lower end

of the ladder is 16 ft away from the wall, upper end is moving4

3times as fast as

the lower end.

15. The base radius of cylindrical vessel full of oil is 3 cm. Oil is drawn at the rate

of 327000 / mincm . Find the rate at which the level of oil is falling in the

vessel30

:Ans minπ

16. Sand is pouring from a pipe at the rate of 312 /cm 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 cone increases when it is 4cm high.

17. Water is dripping out at the steady rate of 1cc/s through tiny hole at the vertex of

a conical vessel whose axis is vertical. When the slant height of the water in the

filter is 4cm, find the rate of increase of

(a) slant height of water1

: /2 3

Ans cm sπ

(b) the area of the water surface 21: /

3Ans cm s

18. An aeroplane is ascending vertically at the rate of100 /km hr . If the radius of the

earth is rkm , how fast does the area of the earth visible from the plane increases 3

minute after it started ascending?22

where r h

rAns km h

19. If 37 &y x x x= − increases at the rate of 4 units per second. How fast is the slope

of the curve changing when 2x = . ( ): 48Ans −

20. The volume of the cube is increases at the constant rate. Prove that increase in

surface area varies inversely as the length of edge of cube.

21. A ladder 13m long leans against a wall. The foot of the ladder is pulled along the

ground in away from the wall, at the rate of1.5 / secm . How fast is the

angleα between the ladder and the ground changing when the foot of the ladder is

12m away from the wall? ( ): 0.3 / secAns radian−

22. The diameter of the expanding smoke ring at time t is proportional to 3t . If

diameter is 8cm after 2 seconds, at what rate is the diameter changing? ( )2: 3Ans t

Tangents and Normals

1. Find the slope of the curve 2 4 at the point (1,2)y x=

2. Find the slope of the curve ( 2)( 3)y x x= + − at those points of the curve where it

meets x-axis. Also find the angles that the tangents to the curve at these points

make with the x-axis.

3. Prove that the tangent lines to the curve 2 4y ax= at points where x a= are at

right angles to each other.

4. Find the angle between tangents to the curve 2 8 6x y= + at the points

3 50, & 4,

5. If the slope of the curve 2y ax bx= + at the point (1,2) be 3, find the values of

&a b .

6. Show that the tangents to the curve 32 3y x= − at the points where

2 & 2x x= = − are parallel.

7. Find the points on the curve 3 22 2y x x x= − + − where the gradient is zero.

8. Find the coordinates of the points on the curve 2 22 3 4 9x xy y+ + = at which slope

11− .

9. Find the coordinates of the points on the curve 4x y+ = at which tangent is

equally inclined to coordinate axes.

10. Find the points on the curve 2 2 25x y+ = at which the tangent is parallel to the

line 3 4 7 0x y− + = .

11. Find the points on the curve:

a) 22 3y x= − at which the tangent is parallel to the line 0x y+ = .

b) 2 ,y x= where the slope of the tangent is equal to the abscissa of the point.

c) 2 ,y x= where the tangent makes an angle of 45°with the x-axis.

12. Find the points on the curve:

d) 2 35 2y x x= − at which the tangent is parallel to the line 4 5y x− =

e) 2 4 32y x x= − − at which the tangent is parallel to x-axis.

f) 3 22y x x x= − − at which the tangent lines are parallel to the line 3 2y x= − .

g) 2 2 2 4 1 0x y x y+ − − + = at which the tangent is parallel to x-axis.

13. Find the point on the curve 3 11 5y x x= − + at which the tangent has the equation.

14. Prove that the curves 2 2 16 & 25x y xy− = = cut each other at right angles.

15. Find the angle of intersection of the curves 2 3&y x y x= = .

16. Do the curves 2 2 2 2 2 22 & 2x y a y x a+ = − = cut each other at right angles at any

point?

17. Show that the curves 22sin & cos 2y x y x= = intersect at 6

= . Find the angle

of intersection.

18. Prove that the curves 2and x y xy k= = cut at right angles if 28 1k = .

19. Show that the curves 2 2 2 2

2 2 2 2 2

x1and 1

a b a bλ λ λ λ+ = + =

+ + + cut each other

orthogonally.

20. Find the equation of the normal to the curve 2 23 2 1xy x y− = at the point (1,1).

21. Find the equation of the tangent and normal at 2( , 2 )at at to the curve 2 4y ax=

22. Find the equation of the tangent to the curve x y a+ = at the point 2 2

23. Find the equation of the normal to the curve 2siny x= at the point3

24. Find the equation of the tangent line to the curve 2cot 2cot 2at 4

y x x xπ

= − + =

25. Find the equation of the tangent and normal to the curve 4 3 26 13 10 5 at (1,3)y x x x x= − + − +

26. Find the equation of the tangent line to the curve

sin , 1 cos at 4

θ θ θ θ= + = + =

27. Find the equation of the tangent at 4

= to the curve sin 3 , cos 2x t y t= =

28. Prove that for all values of n the line 2x y

a b+ = touches the curve

n nx y

at point (a,b).

29. Show that the curves 2 2 24 and 6 1 0y x x y x= + − + = touch each other at the

point (1,2). Also find the equation of the common tangent and normal.

30. Prove that the curves 26 and ( 1) 2y x x y x x= + − − = + touch each other at (2,4).

Also find the equation of the common tangent.

31. Find the equation of the tangent to the parabola 2 8y x= which is parallel to the

line 4 3 0.x y− + =

32. Find the equation of the normal lines to the curve 2 23 8x y− = which are parallel

to the line 3 4.x y+ =

Verify Rolle’s Theorem for the following functions in the indicated intervals

1. ( )2( ) 4 3in interval [1,3]. : 2f x x x Ans c= − + =

2. 2 5( ) ( 5) in [0,5] :

3f x x x Ans c

= − =

3. 3 2 6 3( ) 6 11 6 in [1,3]. :

3f x x x x Ans c

±= − + − =

4. 9 3

( ) ( 2)( 4) in [2,4]. :3

f x x x Ans c +

= − − =

5. ( ) sin 2 in 0, :2 4

f x x Ans cπ π

6. ( )( ) cos 2 in , : 04 4

f x x Ans cπ π

= − =

7. ( ) sin cos 1 in 0, :2 4

f x x x Ans cπ π

= + − =

8. [ ]5

( ) cos sin in 0,2 : ,4 4

f x x x Ans cπ π

9. ( ) sin & 0 . :2

f x x c Ansπ

10. ( )3 2( ) 2 4 & 2 2. : 1,4f x x x x c Ans= + − − < < −

11. [ ]sin

( ) in interval 0, :4x

xf x Ans c

12. ( )5

( ) (sin cos ) in interval , :4 4

xf x e x x Ans c

π ππ

= − =

13. [ ]( ) ( 1)( 2)( 3), in interval 1,3 :na mb

f x x x x Ansm n

+ = − − −

14. ( )2 4 on [ 1,1] : 0x Ans c− − =

15. [ ]( )216 on 4, 4 : 0x Ans c− − =

16. [ ]( )2 3 4

on 1,4 : 5 65

x xAns c

− −− = −

17. [ ]( )2( 3) on 3,0 : 2

x x e Ans c−

+ − = −

Examine the applicability of Rolle’s theorm for the following functions:

1. [ ]( ) on 1,2f x x= (Ans: Not applicable).

2. [ ]( ) tan on 0,f x x π= (Ans: not applicable).

3. [ ]2

3( ) on 1,1f x x= − (Ans: not applicable).

4. [ ]2 5

( ) on 0, 42

− (Ans: not applicable).

5. [ ]3( ) on 1,1f x x= − (Ans: not applicable).

6. [ ]( ) on 2,2f x x= − (Ans: not applicable).

7. [ ]2

5( ) ( 1) on 0,3f x x= − (Ans: not applicable).

8. { [ ]22 6, 0 2( ) on 1, 4

5 4, 2 7x xf xx x

+ ≤ ≤=+ < ≤

(Ans: not applicable)

Verify LMV Theorem for the following functions in the indicated intervals:

1. 3 2 1( ) 2 3;[01] :

2f x x x x Ans c

= − − + =

2. ( )( ) ( 1)( 2)( 3);[1,4] : 3f x x x x Ans c= − − − =

3. ( )( ) log ;[1, ] Ans: 1f x x e c e= = −

4. ( )( ) ;[0,1] : log( 1)xf x e Ans c e= = −

5. ( )1 1

( ) ; , 2 : 12

f x x Ans cx

6. ( )1

( ) ;[1, 4] : 1.934 1

f x Ans cx

= =−

( ) ( 1) ;[1,2] :27

f x x Ans c

= − =

8. 1 4 3( ) tan ;[0,1] : .52f x x Ans c

π π−

− = = = =

9. 5 3

( ) sin;[ , ] :2 2 2

f x Ans cπ π π

10. ( )5

( ) cos ; , :3 3

f x x Ans cπ π

11. ( ) sin cos ; 0, :2 4

f x x x Ans cπ π

12. [ ] 1 1 33( ) sin sin 2 ; 0, : cos

8f x x x Ans cπ −

+= − =

13. [ ]( ) 2sin ; , :2

f x x x Ans cπ

= − = ±

14. { 3 52 ; 1( ) :3 ; 1 3

x xf x Ans cx x

+ ≤= = ± >

Discuss the applicability of LMV Theorem for the following functions in the

indicated intervals.

1. ( )1

on [ 1, 2] Ans:Not applicable2 1x

2. ( )| 2 | on [-3,4] Ans:Not applicablex +

3. ( )1

3 on [ 1,1] Ans:Not applicablex −

4. ( )2 1 on [0,3] Ans: Applicablex x+ +

5. { ( )2

1 3 ; 1( ) Ans:Not applicable

2 7; 1x x

f xx x

+ ≤=

Based on finding the intervals in which function is strictly increasing or decreasing:

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

Q.No. Function Increasing Decreasing

1. 210 6 2x x− − 3,

−∞ −

− ∞

2. 3 22 12 18 15x x x− + + ( ) ( ),1 3,−∞ ∪ ∞ ( )1,3

3. 2 38 36 3 2x x x+ + − ( )2,3− ( ) ( ), 2 3,−∞ − ∪ ∞

4. 3 26 36 2x x x− − + ( ), 2 (6, )−∞ − ∞∪ ( )2,6−

5 3 22 9 12 1x x x+ + − ( ), 2 ( 1, )−∞ − − ∞∪ (-2,-1)

6 2 36 12 3 2x x x+ + − (-1,2) ( , 1) (2, )−∞ − ∞∪

7 2( 1)( 2)x x− − 4, (2, )3

−∞ ∪ ∞

8 4 22x x− ( 1,0) (1, )− ∞∪ ( , 1) (0,1)−∞ − ∪

xx −

−∞

10 2, 0

x+ ≠

( , 2) (2, )−∞ − ∞∪ ( 2,2)−

11 2 1

( 1,1)− ( , 1) (1, )−∞ − ∞∪

12 2' 1

−≠ −

( , 1) ( 1, )−∞ − − ∞∪ ∅

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

decreasing:-

1. 2( ) 2 loge

f x x x= −1 1

:decreasing in 0, & increasing in ,2 2

2. ( ) cosf x x x= + ( ): increasing Ans

3. ( )2( ) log 1f x x x= + + ( ): increasingAns

4. 5 4( ) ( 2) (2 1)f x x x= − +

1 11 1 11: increasing in - ,- , & decreasing in ,

2 8 2 18Ans

∞ ∪ ∞ −

5. ( ) 2sin (0 2 )f x x x x π= − ≤ ≤

5 5: decreasing in 0; , 2 , increasing in ,

3 3 3 3Ans

π π π ππ

6. ( ) 2sin cos 2 (0 2 )f x x x x π= + ≤ ≤

5 3 5 3: increasing in 0, , , 2 ; decreasing in , ,

6 2 6 2 6 2 6 2Ans

π π π π π π π ππ

∪ ∪ ∪

7. 2( ) , ( 0)f x x ax x a= − > 3 3

:increasing in 0, ; decreasing in ,4 4

Ans a a a

8. 4 2( ) 2 5f x x x= − −

( ) ( ) ( ) ( )( ): increasing in 1,0 1, ; decreasing in , 1 0,1Ans − ∪ ∞ −∞ − ∪

9. ( ) cosf xx

1 1 1 1: increasing in , decreasing in , ,

2n+1 2 2 2 2 1Ans n I n I

∈ ∈

10. 2 xx e

− ( ) ( ) ( )( ): decreasing in ,0 2, & increasing in 0, 2Ans −∞ ∪ ∞

11. 5 5

sin cos in[0,2 ] :increasing in 0, , 2 ; decreasing in ,4 4 4 4

x x x Ansπ π π π

Proving the functions to be strictly increasing or decreasing in given intervals

1. Prove that log x is strictly increasing whenever defined.

2. Prove that xe is strictly increasing for all real values of x .

3. Prove that tan x is strictly increasing for all points of its domain.

4. Prove that cot x is strictly decreasing for all points of its domain.

5. Prove that 1 cos x− is strictly increasing for all real values of x .

6. Prove that 3 2( ) 3 3 100f x x x x= − + − is increasing on R.

7. Prove that 1

+ is strictly increasing in any interval disjoint from ( 1,1)−

8. Prove that 1tan (sin cos )x x− + is strictly increasing function on the interval 0,4

9. Prove that 4sin

( )2 cos

xf x x

x= −

+ is strictly increasing function of x in 0,

10. Prove that 1

, 0x xx

− ≠ is strictly increasing function in its domain.

Problems on Local Maxima and Minima

Find all the points of local maxima and minima and the corresponding local maximum

or minimum values of the following functions, if any

1. 3 2( ) 6 9 15f x x x x= − + + ( : max at 1, value 19,min at 3, value=15)Ans x x= = =

2. 3 2( ) 2 21 36 20f x x x x= − + −

( : max at 1, value 3,min at 6, value=-128)Ans x x= = − =

3. 4 3 23 45( ) 8 105

4 2f x x x x= − − − +

231 295: max at 0, value 105,min at 3, value= ,max at 5, value

4 4Ans x x x

= = = − = − =

4. 4 2( ) 62 120 9f x x x x= − + +

( : max at 1, value 68, min at 5, 6, values=-1647,-316)Ans x x= = = −

5. 2( ) ( 1)( 2)f x x x= − +

( : max at 2, value 0, min at 0, value=-4)Ans x x= − = =

6. 3 2( ) 12 5f x x x= − + −

( : max at 8, value 251,min at 0, value=-5)Ans x x= = =

2 2( ) , 0f x x

x x= − >

1( : max at 2, value )

2Ans x = =

8. 3 2( ) ( 1) ( 1)f x x x= − +

1 -3456

( : max at 1, value 0, min at , value= )5 3125

Ans x x= − = = −

Find the points of local maxima and minima for the following functions:

1. ( ) sin cos ,02

f x x x xπ

= + < < Local Max at 4

2. 4 4( ) sin cos ,02

f x x x xπ

= + < < : local min at 4

Ans xπ

3. ( ) sin cos ,0 2f x x x x π= − < <3

,point of local max.4

( ) sin 2 ,0 , point of locla max, , point of local minima4 4

f x x x x xπ π

= < < = =

5. ( ) cos ,0f x x x π< < ( ): None in the interval(0, )Ans π

6. ( ) sin 2 ,2 2

f x x x xπ π

= − − ≤ ≤

is point of local max & is point of local min6 6

x xπ π

= = −

7. ( ) 2sin ,2 2

f x x x xπ π

= − − ≤ ≤

-: , pointof local max.,x= , pointof local min.

3 3Ans x

π π =

( ) sin cos 2 ,02 2

f x x x xπ

= + ≤ ≤

: Local max at &local min at 6 2

Ans x xπ π

9. ( )( ) : Local min. at 1xf x xe Ans x= = −

10. ( )2( ) 32 , 5 5 :local max. at 4; local min. at 4f x x x x Ans x x= − − ≤ ≤ = = −

11. ( )2

( ) , 0, 0 :local max. at , local min. at a

f x x a x Ans x a x ax

= + > ≠ = − =

12. ( )2( ) 2 2 2 : local max. at 1, local min. at 1f x x x x Ans x x= − − ≤ ≤ = = −

( ) 1 , 1 : local max. at 4

f x x x x Ans x

= + − ≤ =

Absolute Maximum & Minimum

Find the absolute maximum and minimum values of the following functions in the

indicated intervals. Also find the points of absolute max and minima.

1. ( )2( ) ( 1) 3 in [ 3,1] Ans:Max 19 at 3 & min 3 at 1f x x x x= − + − = − =

2. 2 1 6 1( ) 1 , 1, Ans:Max at , Min is 0 at 1,0

2 8 2f x x x x x

= + − = = −

3. ( )2

1( ) [0,2] Ans: Max is 2 at 1, Min is 1 at 0

xf x x x

+= = =

4. ( )( ) sin [0,2 ] Ans:Max 2 at 2 , Min is 0 at 0f x x x x xπ π π= + = =

5. ( ) sin cos [0, ] Ans:Max value is 2 at , min is -1 at 4

f x x x x xπ

= + = =

6. [ ]( )( ) 3 1 2,3 Ans:Max is 7 at 3,Min is 3 at 1f x x in x x= + + − = = −

Word Problem

1. Amongst all pairs of positive numbers with product 64, find those whose sum is

the least. (Ans: 8,8)

2. Amongst all pairs of positive numbers with sum 24, find those whose product is

maximum. (Ans: 12,12)

3. Determine two positive numbers whose sum is 15 and the sum of those whose

squares is minimum. (Ans: 15 15

4. Divide 15 into two parts such that the square of one multiplied with the cube of

the other is minimum.(Ans: 6,9)

5. Find two positive numbers &x y such that 360 &x y xy+ = is maximum.(Ans:

15,45)

6. Prove that if product of 2 numbers is constant, their sum will be minimum when

the 2 numbers are equal.

7. Divide 50 into two parts such that their product is maximum.(Ans:25,25)

8. If 4,x y+ = find the maximum value of 2 3x y .

2 2 42

9. Ifθ ψ α+ = (constant), show that sin .sinθ ψ has maximum value when θ ψ= .

10. Show that among all rectangles with given area square has least perimeter.

11. Prove that the height and diameter of the base of a right circular cylinder of

maximum volume are equal when the total surface area is given.

12. Prove that the area of the right angled triangle drawn on a given side as

hypotenuse is maximum when the triangle is isosceles.

13. The sum of the perimeter of a square and a circle is constant. If the sum of their

areas is minimum, find the ratio of the side of the square and the radius of the

circle.(Ans: 2:1)

14. Show that a right circular conical tent of given volume will require the least

amount of canvas if its height is 2 times the radius of its base.

15. A square piece of tin of side 24cm is to be made into a box without top by cutting

a square from each corner and folding up the flaps to form a box. What should be

the side of the square to be cut off so that the volume of the box is minimum?

Also find this maximum volume.(Ans:4cm,1024cm 3 )

16. Show that the surface area of a closed cuboid with square base and given volume

is minimum, when it is a cube.

17. An open box with a square base is to be made out of a given quantity of card

board of area 2c square units. How that the maximum volume of the box

ccube units.

18. Show that rectangle of maximum perimeter which can be inscribed in circle of

radius a is a square of side 2a .

19. Prove that the area of a right-angled triangle of a given hypotenuse is maximum

when the triangle is isosceles.

20. A rectangle is inscribed in a semi-circle of radius r with one of it’s so that its area

is maximum. Find also the area. 2: , 2 ,area2

rAns r r

21. Find the volume of the largest cylinder that can be inscribed in a sphere of radius

r cm.34

22. Show that the height of the cylinder of maximum volume that can be inscribed in

a sphere of radius a is2

23. Show that a cylinder of a given volume which is open at the top, has minimum

volume, is equal to the diameter of its base.

24. Show that the height of the closed cylinder of given surface and maximum

volume, is equal to the diameter of its base.

25. Show that the volume of the largest cone that can be inscribed in a sphere of

radius R is8

27of the volume of the sphere.

26. Show that the volume of the greatest cylinder which can be inscribed in a cone of

height h and semi-vertical angleα is 3 24tan .

27rh α

27. A figure consists of a semi-circle with a rectangle on its diameter. Given the

perimeter k of the figure, Find its dimensions in order that the area may be

maximum.2

Ans:radius ,length ,breadth4 4 4

π π π

28. A beam of length l is supported at one end. If W is the uniform load per unit

length, the bending moment M at distance x from the end is given

by 21 1

2 2M lx Wx= − . Find the point on the beam at which the bending moment

has the maximum value.1

(at a distance of units from the supporting end).

29. A manufacturer can sell x items at a price of Rs(250-x)each. The cost of

producing x items is Rs 2(2 50 12)x x− + . Determine the number of items to be

sold so that he can maximum profit.(Ans:50)

30. The cost of fuel for running a bus is proportional to the square of the speed

generated in /km hr . It costs Rs48/hr when the bus is moving with a speed of

20km/hr. What is the most economical speed if the fixed charges are Rs108 for

one hour over and above the running charges? ( : 30 / )Ans km hr

31. A wire of length of 25m is to be cut into two pieces. One of the wires 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?100 25

: ,4 4

Ans m mπ

32. Find the point on the curve 2 4y x= which is nearest to the point (2,-8).

( : (4, 4))Ans −

33. Find the point on the curve 2 4y x= which is nearest to the point (2,1)?(Ans:(1,2))

34. A jet of an enemy is flying along the curve 2 2y x= + . A soldier is placed at the

point (3,2). What is the nearest distance between the soldier and the

jet?(Ans: 5 units)

35. Show that of all the rectangles that can be inscribed in a given circle, the square

has the maximum area.

36. If the sum of the lengths of the hypotenuse and side of a right triangle is given,

show that the area of the triangle is maximum when the angle between them is 3

37. Show that the triangle of maximum area that can be inscribed in a given circle is

an equilateral triangle.

38. Show that the height of a cylinder, which is open at the top, having a given

surface area and greatest volume, is equal to the radius of the base.

39. Show that the semi-vertical angle of a cone of maximum volume and given slant

height is 1tan 2− .

40. Show that the semi-vertical angle of a cone of given surface area and maximum

volume is 1 1sin

41. Prove that the radius of the right circular cylinder of greatest curved surface area

which can be inscribed in a given cone is half of that of the cone.

42. The combined resistance R of two resistors 1 2&R R 1 2( , 0)R R > is given

R R R= + . If 1 2R R C+ = (a constant), show that the maximum resistance R

is obtained by choosing 1 2R R= .

43. 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.

44. A window is in the form of a rectangle surmounted by a semi-circular opening.

The total perimeter of the window is 10m. Find the dimensions of the rectangular

window to admit the maximum light through the whole

opening.20 10

lenght ,4 4π π

Approximation

1. If 3 4 &y x x x= − changes from 2 to1.99, find the approximate change in the

value of .y ( Ans: -0.08)

2. If the radius of a circle increases from 5cm to5,1cm, find the increase in

area.(Ans: 2cmπ= ).

3. Find the value of 10log (10.1) , given that 10log 0.4343.e = (Ans: =1.004343)

4. Given sin 30 0.5 & cos30 0.866,° = ° = find an approximate value of

sin31° .(Take 3.1416)π = (Ans: 0.5151).

5. In the following problems, use differentials to find the approximate values:

(i) 401 (ii) 25.2 (iii)

4(255) (iv) 0.0037 (v) Cube root of .009 (vi) Fourth

root of 15 (vii) Fourth root of 80. (Ans: 20.025, 5.02, 3.9961, .0608, 0.208,

1.968,2.9907)

6. If sin &y x x= changes from22

to ,2 14

πwhat is the approximate change in

y ?(Ans: No change)

7. A circular metal plate expands under heating so that its radius increases by 2%.

Find the approximate increase in the area of the pate if the radius of the plate

before heating is 10cm. 2( : 4 )Ans cmπ

8. If the error in measuring the radius of the circle is 0.01%, find the corresponding

error in computing the area of the circle.(Ans:0.02%)

9. If the side of the cube is increased by 0.1%, find the corresponding increase in the

volume of the cube. (Ans: 0.3%)

10. In a ABC∆ , the side c and the angle C are constants. Show that

0cos cos

A B+ = .

11. The time T of a complete oscillation of a simple pendulum of length l is given by

the equation 2l

π= where g is a constant. What is the percentage error in T

when l is increased by 1%? (Ans:0.5%)

12. Using differentials find the approximate value of tan 46∂ , if it si given that

1 0.01745° = radians.

13. If a triangle ABC, inscribed in a fixed circle, be slightly varied in such a way as to

have its vertices always on the circle, then show that 0cos cos cos

da db dc

A B C+ + =

14. Find the percentage error in calculating the surface area of a cubical box if an

error of 1% is made in measuring the lengths of the edges of the cube. (Ans:2%)

15. If there is an error of 0.1% in the measurement of the radius of the sphere, find

approximately the percentage error in the calculation of the volume. (Ans:0.3%)

16. The pressure & volume p v of a gas are connected by the relation 1

4 constantpv =

Find the percentage error in p corresponding to a decrease of ½% in v .(Ans:0.7%)

17. The height of a cone increases by k%, its semi-vertical angle remaining the same.

What is the approximate % increase in (i) total surface area and (ii) the volume,

assuming that k is constant. (Ans:2k%,3k%)

18. Show that relative error in computing the volume of a sphere, due to an error in

measuring the radius, is approximately equal to three times the relative error in

the radius.

19. By how much (approximately) does the side of a square change if its area

increases from 2 29 to 9.1cm cm ?(0.016m)

20. Without use of tables and any standard value of logarithm, find the value of

log 7e

(Ans:1.95 approximately)

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