Theory and ExamplesA caution about centered difference

Chapter 3, Problem 66E

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Problem 66E

Theory and Examples

A caution about centered difference quotients (Continuation of Exercise 65. ) The quotient  may have a limit as when ƒ has no derivative at x. As a case in point, take  and calculate  As you will see, the limit exists even though has no derivative at x = 0.Moral: Before using a centered difference quotient, be sure the derivative exists.

Centered difference quotients The centered difference quotient is used to approximate f '(x) in numerical work because (1) its limit as h → 0equals f '(x) when f '(x) exists, and (2) it usually gives a better approximation of f '(x) for a given value of h than the difference quotient

 See the accompanying figure.

a. To see how rapidly the centered difference quotient for f(x) = sin x converges to f '(x) = cos x, graph y = cos x together with  over the interval  Compare the results with those obtained in Exercise 63 for the same values of h.

b. To see how rapidly the centered difference quotient for f(x) = cos x converges to f '(x) = –sin x, graph y = –sin x together withover the interval . Compare the results with those obtained in Exercise 64 for the same values of h.