Theory and ExamplesA caution about centered difference
Chapter 3, Problem 66E(choose chapter or problem)
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.
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