Theory and ExamplesCentered difference quotients The

Chapter 3, Problem 65E

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Problem 65E

Theory and Examples

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.