Cambridge Additional Mathematics

DEMO

As B approaches A, the gradient of

AB approaches orconvergesto. 2

A() 11 ,¡

y

x

f(x) = x¡¡ 2

O

B,(x x )¡ 2

A() 11 ,¡

y

x

f(x) = x¡¡ 2

O

A tangent at A

B 1

B 2

B 3

B 4

Introduction to differential calculus (Chapter 13) 339

Thegradient of the tangentto y=f(x) at x=a is theinstantaneous rate of changein f(x) with

respect toxat that point.

Discovery 2 The gradient of a tangent#endboxedheading

Given a curve f(x), we wish to find the gradient of the

tangent at the point (a,f(a)).

In this Discovery we find the gradient of the tangent to

f(x)=x^2 at the point A(1,1).

What to do:

1 Suppose B lies on f(x)=x^2 , and B has coordinates (x,x^2 ).

x Point B gradient of AB

5 (5,25) 6

3

2

1 : 5

1 : 1

1 : 01

1 : 001

a Show that the chord AB has gradient

x^2 ¡ 1

x¡ 1

.

b Copy and complete the table shown.

c Comment on the gradient of AB asxgets closer to 1.

2 Repeat the process lettingxget closer to 1 , but from the left of A. Use the points where

x=0, 0 : 8 , 0 : 9 , 0 : 99 , and 0 : 999.

3 Click on the icon to view a demonstration of the process.

4 What do you suspect is the gradient of the tangent at A?

Fortunately we do not have to use a graph and table of values each time we wish to find the gradient of a

tangent. Instead we can use an algebraic and geometric approach which involveslimits.

FromDiscovery 2, the gradient of AB=

x^2 ¡ 1

x¡ 1

.

As B approaches A, x! 1 and

the gradient of AB!the gradient of the tangent at A.

So, the gradient of the tangent at the point A is

mT= lim

x! 1

x^2 ¡ 1

x¡ 1

= lim

x! 1

(x+ 1)(x¡1)

x¡ 1

= lim

x! 1

(x+1) since x 6 =1

=2

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_13\339CamAdd_13.cdr Tuesday, 7 January 2014 2:35:13 PM BRIAN