Engineering Mathematics 4 Episode 11 pps

Some applications of differentiation 46.1 Rates of change If a quantity y depends on and varies with a quantity x then the rate of change of y with respect to x A rate of change with res

ln 2x[3.82]

This is known as the ‘function of a function’ rule

(or sometimes the chain rule).

For example, if y D 3x  19 then, by making the

substitution u D 3x  1, y D u9, which is of the

Since y is a function of u, and u is a function of x,

then y is a function of a function of x

dt D12t

23 and dy

dt D6u

5Using the function of a function rule,

Problem 22 Find the differential coefficient

Exercise 156 Further problems on the

function of a function

In Problems 1 to 8, find the differential

coefficients with respect to the variable

7 6 tan3y C 1 [18 sec23y C 1]

8 2etan  [2 sec2etan ]

9 Differentiate:  sin



 3

When a function y D fx is differentiated with

respect to x the differential coefficient is written as

dy

dx or f

0x If the expression is differentiated again,

the second differential coefficient is obtained and is

dy

dx D12x

3,d

Since y D cos x  sin x, dy

dx D sin x  cos x and

d2y

dx2 D cos x C sin xWhen d

Since y D 4 sec 2, then

dy

d D42 sec 2 tan 2 (from Problem 15)

D8 sec 2 tan 2 (i.e a product)

d2y

d2 D8 sec 22 sec22

Ctan 2[82 sec 2 tan 2]

D16 sec32 C 16 sec 2 tan22

When  D 0,

d2y

d2 D16 sec30 C 16 sec 0 tan20

D161 C 1610 D 16

Now try the following exercise

Exercise 157 Further problems on

succes-sive differentiation

1 If y D 3x4C2x33x C 2 find(a) d

2y

dx2 (b) d

3y

dx3[(a) 36x2C12x (b) 72x C 12]

(a) 4sin2x cos2x (b) 482x  32

5 Evaluate f00when  D 0 given

Some applications of differentiation

46.1 Rates of change

If a quantity y depends on and varies with a quantity

x then the rate of change of y with respect to x

A rate of change with respect to time is usually

just called ‘the rate of change’, the ‘with respect to

time’ being assumed Thus, for example, a rate of

change of current, i, is d i

d t and a rate of change of

temperature, , is d 

d t, and so on.

Problem 1 The length l metres of a certain

metal rod at temperature °C is given by:

l D1 C 0.00005 C 0.00000042 Determine

the rate of change of length, in mm/°C, when

the temperature is (a) 100°C and (b) 400°C

The rate of change of length means d l

Problem 2 The luminous intensity I

candelas of a lamp at varying voltage V is

given by: I D 4 ð 104 V2 Determine the

voltage at which the light is increasing at a

rate of 0.6 candelas per volt

The rate of change of light with respect to voltage

V D 0.6

8 ð 104 D0.075 ð 10C4 D750 volts

Problem 3 Newtons law of cooling isgiven by:  D 0ekt, where the excess oftemperature at zero time is 0°C and at time

tseconds is °C Determine the rate of

change of temperature after 40 s, given that

Using the product rule,

d s

d t Dae

kt 2f cos 2ft

Csin 2ft akekt

(Note that cos 10 means ‘the cosine of 10

radi-ans’, not degrees, and cos 10  cos 2 D 1).

Now try the following exercise

Exercise 158 Further problems on rates

of change

1 An alternating current, i amperes, is given

by i D 10 sin 2ft, where f is the

fre-quency in hertz and t the time in seconds

Determine the rate of change of current

when t D 20 ms, given that f D 150 Hz

[3000 A/s]

2 The luminous intensity, I candelas, of

a lamp is given by I D 6 ð 104 V2,

where V is the voltage Find (a) the rate of

change of luminous intensity with voltage

when V D 200 volts, and (b) the voltage

at which the light is increasing at a rate

of 0.3 candelas per volt

[(a) 0.24 cd/V (b) 250 V]

3 The voltage across the plates of a

capac-itor at any time t seconds is given by

vDV et/CR, where V, C and R are

con-stants Given V D 300 volts,

C D0.12 ð 106 farads and

R D4 ð 106 ohms find (a) the initial rate

of change of voltage, and (b) the rate of

change of voltage after 0.5 s

[(a) 625 V/s (b) 220.5 V/s]

4 The pressure p of the atmosphere at

height h above ground level is given by

p D p0 eh/c, where p0 is the pressure at

ground level and c is a constant

Deter-mine the rate of change of pressure with

height when p0 D 1.013 ð 105 Pascals

and c D 6.05 ð 104 at 1450 metres

[1.635 Pa/m]

46.2 Velocity and acceleration

When a car moves a distance x metres in a time t

seconds along a straight road, if the velocity v is

υt !0, the chord AB becomes a tangent, such that

at point A, the velocity is given by:vD d x

d

Figure 46.3

The acceleration a of the car is defined as the

rate of change of velocity A velocity/time graph is

shown in Fig 46.3 If υv is the change inv and υt

the corresponding change in time, then a D υv

υt As

υt !0, the chord CD becomes a tangent, such that

at point C, the acceleration is given by: a D d v

d t

Hence the acceleration of the car at any instant is

given by the gradient of the velocity/time graph If

an expression for velocity is known in terms of time

tthen the acceleration is obtained by differentiating

coefficient of distance x with respect to time t

Summarising, if a body moves a distance x

metres in a time t seconds then:

(i) distance x =f t /

(ii) velocityvDf
t /or dx

dt, which is the

gradi-ent of the distance/time graph

(iii) acceleration a = d v

dt = f

or d2x

d t2, which

is the gradient of the velocity/time graph

Problem 5 The distance x metres moved

by a car in a time t seconds is given by:

2x

d x2 D18t  4 m/s2(a) When time t D 0,

velocity vD90 240 C 4 D 4 m = s

and acceleration a D 180  4 D −4 m = s2

(i.e a deceleration)(b) When time t D 1.5 s,velocityvD91.5 241.5 C 4 D 18.25 m = s

and acceleration a D 181.5 2 D6.339 cm

Hence for the least surface area, a cylinder of

volume 200 cm 3 has a radius of 3.169 cm and

height of 6.339 cm.

Problem 18 Determine the area of the

largest piece of rectangular ground that can

be enclosed by 100 m of fencing, if part of

an existing straight wall is used as one side

Let the dimensions of the rectangle be x and y as

shown in Fig 46.9, where PQ represents the straight

Since the maximum area is required, a formula for

area A is needed in terms of one variable only

When y D 25 m, x D 50 m from equation (1)

Hence the maximum possible area

Dxy D 50 25 D 1250 m2

Problem 19 An open rectangular box with

square ends is fitted with an overlapping lid

which covers the top and the front face

Determine the maximum volume of the box

if 6 m2 of metal are used in its construction

A rectangular box having square ends of side x and

length y is shown in Fig 46.10

2x5



D 6x

5 

2x35

V D x2y D 1 2

45



D 4

5 m 3

Problem 20 Find the diameter and height

of a cylinder of maximum volume which can

be cut from a sphere of radius 12 cm

A cylinder of radius r and height h is shownenclosed in a sphere of radius R D 12 cm inFig 46.11

Volume of cylinder, V D r2h 1