1 Drawing tangents

Look at the graph shown in Figure 1a. $A$ and $B$ are two points on the graph, and they have been joined by a straight line. The straight line segment $AB$ is known as a chord . We have lengthened the chord on both sides so that it extends beyond both $A$ and $B$ .

Figure 1a Figure 1b Figure 1c

In Figure 1b we have moved point $B$ nearer to point $A$ before drawing the extended chord. Imagine what would happen if we continue moving $B$ nearer and nearer to $A$ . You can do this for yourself by drawing additional points on the graph. Eventually, when $B$ coincides with $A$ , the extended chord is a straight line which just touches the curve at $A$ . This line is now called the tangent to the curve at $A$ , and is shown in Figure 1c.

If we know the position of two points on the line we can find the gradient of the straight line and can calculate the gradient of the tangent. We define the gradient of the curve at $A$ to be the gradient of the tangent there. If this gradient is large at a particular point, the rate at which the function is changing is large too. If the gradient is small, the rate at which the function is changing is small. This is illustrated in Figure 2. Because of this, the gradient at $A$ is also known as the instantaneous rate of change of the curve at $A$ . Recall from your knowledge of the straight line, that if the line slopes upwards as we look from left to right, the gradient of the line is positive, whereas if the line slopes downwards, the gradient is negative.

Figure 2

Task!

Draw in, by eye, tangents to the curve shown below, at points $A$ to $E$ . State whether each tangent has positive, negative or zero gradient.

Answer

In the following subsection we will see how to calculate the gradient of a curve precisely.