Centered difference quotients The centered difference quotient is used to appro...

Centered difference quotients The centered difference quotient

is used to approximate f’(x) in numerical work because (1) its limit as equals 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 [-π, 2π] for h = 1, 0.5, and 0.3. 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 ƒ(x) = cos x converges to ƒ’(x) = -sin x, graph y = -sin x together with

over the interval [-π, 2π] for h = 1, 0.5 , and 0.3. Compare the results with those obtained in Exercise 64 for the same values of h.