2 Finding the gradient at a specific point
In this subsection we shall consider a simple function to illustrate the calculation of a gradient. Look at the graph of the function shown in Figure 4. Notice that the gradient of the graph changes as we move from point to point. In some places the gradient is positive; at others it is negative. The gradient is greater at some points than at others. In fact the gradient changes from point to point as we move along the curve.
Figure 4
Inspect the graph carefully and make the following observations:
- is the point with coordinates .
- is the point with coordinates .
-
We can calculate the gradient of the line
from the formula
Task!
Add the point to the graph in Figure 4 where has coordinates . Draw the line and calculate its gradient.
. Would you agree that this is a better estimate of the gradient at than using ?
We now carry the last task further by introducing point at and point at as shown in Figure 5. The gradient of is found to be 3 and the gradient of is 2.5.
Figure 5
Observe that each time we carry out this procedure, and move the second point closer to , the gradient of the line drawn is getting closer and closer to the gradient of the tangent at . If we continue, the value we eventually obtain is the gradient of the tangent at whose value is 2 as we will see shortly. This procedure illustrates how we define the gradient of the curve at .