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 y ( x ) = x 2 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

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Inspect the graph carefully and make the following observations:

  1. A is the point with coordinates ( 1 , 1 ) .
  2. B is the point with coordinates ( 4 , 16 ) .
  3. We can calculate the gradient of the line A B from the formula
     gradient  =  difference between  y  coordinates  difference between  x  coordinates
    Therefore the gradient of chord A B is equal to 16 1 4 1 = 15 3 = 5 . The gradient of A B is not the same as the gradient of the graph at A but we can regard it as an approximation, or estimate of the gradient at A . Is it an over-estimate or under-estimate ?
Task!

Add the point C to the graph in Figure 4 where C has coordinates ( 3 , 9 ) . Draw the line A C and calculate its gradient.

9 1 3 1 = 4 . Would you agree that this is a better estimate of the gradient at A than using A B ?

We now carry the last task further by introducing point D at ( 2 , 4 ) and point E at ( 1.5 , 2.25 ) as shown in Figure 5. The gradient of A D is found to be 3 and the gradient of A E is 2.5.

Figure 5

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Observe that each time we carry out this procedure, and move the second point closer to A , the gradient of the line drawn is getting closer and closer to the gradient of the tangent at A . If we continue, the value we eventually obtain is the gradient of the tangent at A whose value is 2 as we will see shortly. This procedure illustrates how we define the gradient of the curve at A .