Gradient at a Point 14

14.1 Introduction

EMBBP

In Grade 10, we investigated the idea of average gradient and saw that the gradients of most functions

varied over different intervals. In Grade 11, wediscuss the concept of average gradient further,and

introduce the idea of thegradient of a curve at apoint.

See introductory video:VMfns at http://www.everythingmaths.co.za

14.2 Average Gradient EMBBQ

We saw that the average gradient between twopoints on a curve is thegradient of the straightline

passing through the twopoints.

y

x

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A(−3;7)

C(−1;−1)

Figure 14.1: The average gradient between twopoints on a curve is thegradient of the straightline

that passes through thepoints.

What happens to the gradient if we fix the position of one point and move the second point closer to

the fixed point?

Activity: Gradient at a Single Point on a Curve

The curve shown belowis defined by y =− 2 x^2 − 5. Point B is fixed at coordinates (0;−5).

The position of point A varies. Complete the table below by calculating the y-coordinates of

point A for the given x-coordinates and then calculate the average gradient between points A

and B.