P1^

5

The gradient of a curve

123

Differentiation

Hold infinity in the palm of your hand.

William Blake

This picture illustrates one of the more frightening rides at an amusement park.

To ensure that the ride is absolutely safe, its designers need to know the gradient

of the curve at any point. What do we mean by the gradient of a curve?

The gradient of a curve

To understand what this means, think of a log on a log-flume, as in figure 5.1. If

you draw the straight line y = mx + c passing along the bottom of the log, then

this line is a tangent to the curve at the point of contact. The gradient m of the

tangent is the gradient of the curve at the point of contact.

y = mx

+ c

Figure 5.1 

5