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Chapter 3 | Derivatives
Figure 3.3 We can calculate the slope of a secant line in either of two ways.
In Figure 3.4 (a) we see that, as the values of x approach a , the slopes of the secant lines provide better estimates of the rate of change of the function at a . Furthermore, the secant lines themselves approach the tangent line to the function at a , which represents the limit of the secant lines. Similarly, Figure 3.4 (b) shows that as the values of h get closer to 0, the secant lines also approach the tangent line. The slope of the tangent line at a is the rate of change of the function at a , as shown in Figure 3.4 (c).
Figure 3.4 The secant lines approach the tangent line (shown in green) as the second point approaches the first.
You can use this site (http://www.openstax.org/l/20_diffmicros) to explore graphs to see if they have a tangent line at a point.
In Figure 3.5 we show the graph of f ( x ) = x and its tangent line at (1, 1) in a series of tighter intervals about x =1. As the intervals become narrower, the graph of the function and its tangent line appear to coincide, making the values on the tangent line a good approximation to the values of the function for choices of x close to 1. In fact, the graph of f ( x ) itself appears to be locally linear in the immediate vicinity of x =1.
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