chapter 5_extra exercise

LFSE012_Chapter 5_Extra Exercise 1

Extra Exercise – Chapter 5 – Applications of Differentiation

Extra Exercise 5.1 – Kinematics

1. A helicopter climbs vertically from the top of a 110-metre tall building, so that its height in

metres above the ground after t seconds is given by . Calculate:

a) the average velocity of the helicopter from to

b) its instantaneous velocity at time

c) its instantaneous velocity at time

d) the time at which the helicopter’s velocity is zero

e) the maximum height reached above the ground.

2. A particles moves in a straight line so that its position relative to a fixed point O in the line at

any time is given by ( ) . Find:

a) expression for the velocity and acceleration at any time .

b) the velocity and acceleration when .

3. A particle moves in a straight line so that its displacement, metres, from a point O on that

line, after seconds, is given by √ .

a) When is the displacement √ ?

b) Find expressions for the velocity and acceleration of the particle at time .

c) Find the maximum magnitude of the displacement from O.

d) When is the velocity zero?

Extra Exercise 5.2 – Curve Sketching

1. Find any turning points or stationary points of inflexion of the following functions and

determine whether they are local maximums or minimums. Sketch the curve, showing the

turning points, stationary points of inflexion, -intercept and -intercept.

a) ( )( ) b)

2. Find the asymptotes of each of the following curves and determine the asymptotic behaviour.

2 3 4a) b) c)

1 2 4 2 3

x xy y y

x x x

3. Sketch the graph of each of the following, labelling all axial intercepts, turning points and

asymptotes.

2 2

3 2a) b) c)

2 2 3

1 1d) e)

2 4 3

x x xy y y

x x x

y yx x x x


Extra Exercise 5.3 – Maxima and Minima

1. The sum of the two numbers, and is 8.

a) Write down an expression for in terms of

b) Write down an expression for , the sum of the squares of these two numbers, in terms of

c) Find the least value of the sum of their squares.

2. A piece of wire, 360 cm long, is used to make the twelve edges of a rectangular box in which

the length is twice the breadth.

a) Denoting the breadth of the box by cm, show that the volume of the box, cm3, is given

by .

b) Find the dimensions of the box which gives the greatest volume.

3. A piece of wire of length 90 cm is bent into the shape shown in the diagram below.

a) Show that the area, cm2, enclosed by the wire is given by .

b) Find the values of and for which is a maximum.

4. The diagram below illustrates a window which consists of an equilateral triangle and a

rectangle. The amount of light that comes through the window is directly proportional to the

area of the window. If the perimeter of the window must be 800 mm, find the values of and

that allow the maximum amount of light through.

5. A piece of wire 100 cm in length is to be cut into two pieces, one piece of which is to be shaped

into a circle and the other into a square. How should the wire be cut if the sum of the enclosed

areas is to be a minimum?

5𝑥 cm 5𝑥 cm

8𝑥 cm

𝑦 cm 𝑦 cm

2𝑎


6. A roll of tape 36 metres long is to be used to mark out the edges and internal lines of a

rectangular court of length metres and width metres, as shown in the diagram below.

Find the length and the width of the court for which the area is a maximum.

Extra Exercise 5.4 – Rates of Change

1. Water is being poured into a flask. The volume, , of water in the flask at time, seconds, is

given by 3

25( ) 10 ,0 20.

8 3

tV t t t

a) Find the volume of water in the flask at time:

i) ii)

b) Find the rate of flow of water into the flask.

2. According to a business magazine the expected assets, $M, of a proposed new company will be

given by 2 3200200 000 600

3M t t where stands for the number of months after the

business is set up.

a) Find the rate of growth of assets at time months.

b) Find the rate of growth of assets at time months.

c) Will the rate of growth of assets be 0 at any time?

3. As a result of survey, the marketing director of a company found that the revenue, $R, from

selling produced items at $P is given by the rule .

a) Calculate dR

dP when and

b) For what selling prices is revenue rising?

4. Oil spills out of a tanker at the rate of 550 litre per minute. The oil spreads in a circle with a

thickness of 6.8 cm. Determine the rate at which the radius of the spill is increasing when the

radius reaches (i) 30 m ii) 60 m

5. Assume that the infected area of an injury is circular. If the radius of the infected area is 3 mm

and growing at a rate of 1 mm/hr, at what rate is the infected area increasing?

𝑥 𝑥 𝑥

𝑦


6. Suppose that the average yearly cost per item for producing items of a business product is

100( ) 10C x

x . If the current production is and production is increasing at a rate of 2

items per year, find the rate of change of the average cost.

7. The volume of a spherical balloon increases at a rate of 50 cm3 per second. Find the rate of

increase in the total surface area of the balloon when the radius of the balloon is 10 cm.

Extra Exercise 5.5 – Small Increments

1. a) Write down an expression relating the area, of a square to its length, .

b) Determine the rate of change of area with respect to side length.

c) Use dA

A LdL

to estimate the increase in the area of the square if increases from 3 to

3.01 cm.

2. The time taken (the period, seconds) for one oscillation of a pendulum of length, metres, is 1

22T L .

a) Use the approximate method to find the increase in the period as increases from 1.0 m to

1.01 m.

b) My pendulum clock, whose pendulum is 1.0 m long, runs fast. By counting the number of

oscillations per minute, I calculate that I need to increase the period by 0.015 seconds. By how

much should the pendulum’s length be changed?

3. A balloon has been inflated so that its radius is 200 mm. if an additional puff of air adds 0.1

litres to the volume of the balloon, how much will the radius increase?

4. The surface area of a sphere is . If the radius of the sphere is increased from 10 cm to

10.1 cm, what is the approximate increase in surface area?

5. The height of a cylinder is 10 cm and its radius is 4 cm. Find the approximate increase in

volume when the radius increases to 4.02 cm.

6. An error of 3 % is made in measuring the radius of the sphere. Find the percentage error in

volume.

7. Given that y x , use differentiation to obtain an approximate value of 25.1 .

8. Given that 3y x , use differentiation to obtain an approximate value of 3 8.01 .

9. One side of a rectangle is three times the other. If the perimeter increases by 2%, what is the

percentage increase in area?


Answers:

Extra Exercise 5.1

1. a) 44 m/s b) c) 44 m/s d) 5 s e) 247.5 m

2. a)

2

4 4,

1 1v a

t t

b) m/s , m/s2

3. a) 2 s b)

32

2 2

9,

9 9

tv a

t t

c) 3 m d)

Extra Exercise 5.2

1. a) local minimum: 2 22

1 , 143 27

; local maximum: ( )

b) local minimum: ( ) ; stationary point of inflexion: ( )

2.

Vertical Asymptote Asymptotic behaviour Horizontal Asymptote Asymptotic behaviour

a)

b)

c)

3,

2

3,

2

x y

x y

𝑥

𝑦 𝑥(𝑥 )(𝑥 )

( )

6. 𝑦

𝑥

( )

0

𝑦 𝑥 𝑥 b)

a) 𝑦


Extra Exercise 5.3

1. a) b) ( ) c) 32

2. b) 20 cm 40 cm 30 cm 3. b) max when

4. 5. circle : 44 cm ; square: 56 cm

6. width 4.5 m ; length 7.2 m

𝑥

𝑦 𝑥

𝑥

𝑥

𝑦

𝑦 8. 𝑦

𝑥

𝑥

( )

𝑦

b) 𝑦 𝑥

𝑥

𝑦

𝑥

𝑥

𝑦

c) 𝑦

𝑥

𝑥

𝑦

𝑥

𝑥

𝑥

d)

𝑦

𝑥 𝑥

𝑦

𝑥

𝑥

𝑥

e)

𝑦

𝑥 𝑥

3 a)


Extra Exercise 5.4

1. a) (i) 0 (ii) 1

8333

b) 2520

8t t

2. a) b) 1800 c)

3. a) b) 10 , 10 c)

4. i) 4.291 cm/min ii) 2.145 cm/min

5. 6 mm/hr

6. 2

7. 10 cm2/s

Extra Exercise 5.5

1. a) b) 2dA

LdL

c) 0.06 cm2

2. a) 0.01 s b) 0.015 m

3. 1

16 or 0.02 cm

4. 8 cm2 5. 1.6 cm3

6. 9 % 7. 5.01

8. 2.00083 9. 4%

chapter 5_extra exercise

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