Chapter 2 | Limits
125
Compare the graphs of these three functions with the graph of k ( x ) = x 2 ( Figure2.3 ). The graph of k ( x ) = x 2 starts from the left by decreasing rapidly, then begins to decrease more slowly and level off, and then finally begins to increase—slowly at first, followed by an increasing rate of increase as it moves toward the right. Unlike a linear function, no single number represents the rate of change for this function. We quite naturally ask: How do we measure the rate of change of a nonlinear function?
Figure 2.3 The function k ( x ) = x 2 does not have a constant rate of change.
We can approximate the rate of change of a function f ( x ) at apoint ⎛ ⎝ x , f ( x ) ⎞ ⎠ on the graph of f ( x ), drawing a line through the two points, and calculating the slope of the resulting line. Such a line is called a secant line. Figure 2.4 shows a secant line to a function f ( x ) at a point ⎛ ⎝ a , f ( a ) ⎞ ⎠ . ⎝ a , f ( a ) ⎞ ⎠ on its graph by taking another point ⎛
Figure 2.4 The slope of a secant line through a point ⎛ ⎝ a , f ( a ) ⎞ ⎠ estimates the rate of change of the function at the point ⎛ ⎝ a , f ( a ) ⎞ ⎠ .
We formally define a secant line as follows:
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