Two points determine a line. We select one point on the graph of y = f(x) by specifying a domain value a; the corresponding point on the graph has coordinates (a,f(a)). We can select another nearby point on the graph by choosing a (small) value of h. This nearby point has coordinates
Each nonzero choice of h determines a point (a+h,f(a+h)) and the secant line that joins this point to (a,f(a)). An interesting thing happens when we choose smaller and smaller values of h. We have prepared a crude animation in order to illustrate. It's so crude that you must advance the frames yourself. Just push the button and watch what happens to the secant lines!!
What's Next: We have one more thing to do before we take up the study of differentiation techniques (i.e. how to compute the derivative of a function). We investigate the relationship between differentiability and continuity.
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All rights reserved---1996
William A. Bogley