Cambridge Additional Mathematics

348 Introduction to differential calculus (Chapter 13)

6 Find the gradient function of:

a f(x)=4

p

x+x b f(x)=^3

p

x c f(x)=¡

2

p

x

d f(x)=2x¡

p

x e f(x)=

4

p

x

¡ 5 f f(x)=3x^2 ¡x

p

x

g f(x)=

5

x^2

p

x

h f(x)=2x¡

3

x

p

x

7aIf y=4x¡

3

x

, find

dy

dx

and interpret its meaning.

b The position of a car moving along a straight road is given by S=2t^2 +4t metres wheretis

the time in seconds. Find

dS

dt

and interpret its meaning.

c The cost of producingxtoasters each week is given by C= 1785 + 3x+0: 002 x^2 dollars.

Find

dC

dx

and interpret its meaning.

InChapter 2we defined thecompositeof two functionsgandfas (g±f)(x) or gf(x).

We can often write complicated functions as the composite of two or more simpler functions.

For example y=(x^2 +3x)^4 could be rewritten as y=u^4 where u=x^2 +3x,oras

y=gf(x) where g(x)=x^4 and f(x)=x^2 +3x.

Example 9 Self Tutor

Find: a

b

ab

EXERCISE 13F.1

1 Find gf(x) if:

a g(x)=x^2 and f(x)=2x+7 b g(x)=2x+7and f(x)=x^2

c g(x)=

p

x and f(x)=3¡ 4 x d g(x)=3¡ 4 x and f(x)=

p

x

e g(x)=

2

x

and f(x)=x^2 +3 f g(x)=x^2 +3and f(x)=

2

x

2 Find g(x) and f(x) such that gf(x) is:

a (3x+ 10)^3 b

1

2 x+4

c

p

x^2 ¡ 3 x d

10

(3x¡x^2 )^3

F The chain rule

There are several

gf(x) if g(x)=px and f(x)=2¡ 3 x possible answers for .b

g(x) and f(x) such that gf(x)=

1

x¡x^2

.

gf(x)

=g(2¡ 3 x)

=

p

2 ¡ 3 x

gf(x)=

1

x¡x^2

=

1

f(x)

) g(x)=

1

x

and f(x)=x¡x^2

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_13\348CamAdd_13.cdr Friday, 24 January 2014 10:56:50 AM BRIAN