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7.1 Introduction: the Obvious Approximation: f '(x) ~ (f(x+d) - f(x)) / d

Suppose we have a given function, f , and we seek its derivative at argument x 0 .

One way to estimate it is to evaluate f at two points, x 1 and x 2 , and examine the slope of the line from ( x 1 , f ( x 1 ) ) to ( x 2 , f ( x 2 ) ) . But what should we use for x 1 and x 2 and what will we learn about f ' ( x 0 ) ?

The choice that first occurs to people is to set x 1 = x 0 , and x 2 = x 0 + d for some very small d . So one can compute

f ( x 0 + d ) − f ( x 0 ) d

This is not a horrible thing to do, but it is not very good, as we shall see.

What's wrong with it?

Well, if d is too big, the linear approximation won't be accurate and if d is too small, the inaccuracy of your calculation tools may screw up your answer. And the transition from being too big to too small may be difficult to find.

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