Cambridge International AS and A Level Mathematics Pure Mathematics 1

Differentiation

124

P1^

5

One method of finding the gradient of a curve is shown for point A in figure 5.2.

ACTIVITY 5.1  Find the gradient at the points B, C and D using the method shown in figure 5.2.

(Use a piece of tracing paper to avoid drawing directly on the book!) Repeat the

process for each point, using different triangles, and see whether you get the

same answers.

You probably found that your answers were slightly different each time, because

they depended on the accuracy of your drawing and measuring. Clearly you need

a more accurate method of finding the gradient at a point. As you will see in this

chapter, a method is available which can be used on many types of curve, and

which does not involve any drawing at all.

Finding the gradient of a curve

Figure 5.3 shows the part of the graph y = x^2 which lies between x = −1 and x = 3.

What is the value of the gradient at the point P(3, 9)?

C

D

B

A

1.5

5.5

A

Figure 5.2 

(^) Gradient = –––––y step
x step
5.5 = –––
1.5
= 3.7
y
3
6
–1 1
gradient 3
gradient 5
O 2 3
9
x
gradient 4
y = x^2
P
(1, 1)
(2, 4)
(3, 9)
Figure 5.3 
The line OP is called
a chord. It joins two
points on the curve,
in this case (0, 0)
and (3, 9).