Cambridge Additional Mathematics

Introduction to differential calculus (Chapter 13) 341

EXERCISE 13C

1 Using the graph below, find:

a f(2) b f^0 (2)

2 Using the graph below, find:

a f(0) b f^0 (0)

3 Consider the graph alongside.

Find f(2) and f^0 (2).

Discovery 3 Gradient functions

The software on the CD can be used to find the gradient of the tangent to a functionf(x)

at any point. By sliding the point along the graph we can observe the changing gradient

of the tangent. We can hence generate the gradient function f^0 (x).

What to do:

1 Consider the functions f(x)=0, f(x)=2, and f(x)=4.

a For each of these functions, what is the gradient?

b Is the gradient constant for all values ofx?

2 Consider the function f(x)=mx+c.

a State the gradient of the function. b Is the gradient constant for all values ofx?

c Use the CD software to graph the following functions and observe the gradient functionf^0 (x).

Hence verify that your answer inbis correct.

i f(x)=x¡ 1 ii f(x)=3x+2 iii f(x)=¡ 2 x+1

3aObserve the function f(x)=x^2 using the CD software. Whattypeof function is the gradient

function f^0 (x)?

b Observe the following quadratic functions using the CD software:

i f(x)=x^2 +x¡ 2 ii f(x)=2x^2 ¡ 3

iii f(x)=¡x^2 +2x¡ 1 iv f(x)=¡ 3 x^2 ¡ 3 x+6

c Whattypeof function is each of the gradient functions f^0 (x) inb?

4aObserve the function f(x)=lnx using the CD software.

b Whattypeof function is the gradient function f^0 (x)?

c What is thedomainof the gradient function f^0 (x)?

GRADIENT

FUNCTIONS

y

x

3 y = f(x)

O

y

x

4

4

O

y = f(x)

y

x

(4 5),

2

1

y = f(x)

O

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_13\341CamAdd_13.cdr Tuesday, 7 January 2014 2:35:44 PM BRIAN