Cambridge Additional Mathematics

418 Integration (Chapter 15)

What to do:

1 What is the derivative F^0 (x) of the function F(x)=5x? How does this relate to the function

f(x)?

2 Consider the simplest linear function f(x)=x.

The corresponding area function is

A(t)=

Zt

a

xdx

=shaded area in graph

=

³t+a

2

́

(t¡a)

a Write A(t) in the form F(t)¡F(a).

b What is the derivative F^0 (x)? How does it relate to the function f(x)?

3 Consider f(x)=2x+3. The corresponding area function is

A(t)=

Zt

a

(2x+3)dx

=shaded area in graph

=

³

2 t+3+2a+3

2

́

(t¡a)

a Write A(t) in the form F(t)¡F(a).

b What is the derivative F^0 (x)?

How does it relate to the function f(x)?

4 Repeat the procedure in 2 and 3 to find area functions for:

a f(x)=^12 x+3 b f(x)=5¡ 2 x

Do your results fit with your earlier observations?

5 Iff(x)=3x^2 +4x+5, predict whatF(x)would be without performing the algebraic procedure.

From theDiscoveryyou should have found that, for f(x)> 0 ,

Zt

a

f(x)dx=F(t)¡F(a) where F^0 (x)=f(x). F(x) is theantiderivativeof f(x).

The following argument shows why this is true for all functions f(x)> 0.

Consider a function y=f(x)which has antiderivativeF(x)

and an area function A(t)=

Zt

a

f(x)dx which is the area

from x=a to x=t.

A(t) is clearly an increasing function and

A(a)=0 .... (1)

a t x

y

y=2x+3

2t + 3

2a+3

t-a

O

a t x

t

y y=x

t-a

a

O

y

x

y = f(x)

at b

A(t)

O

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_15\418CamAdd_15.cdr Monday, 7 April 2014 3:58:14 PM BRIAN