(a) Calculate approximations for f 0 (0.8) using the central difference formula

Question

(a) Calculate approximations for f 0 (0.8) using the central difference formula of order O(h 2 ) with h = 0.1 and h = 0.01. Carry eight or nine decimal places.
(b) Compare with the value f 0 (0.8) = cos(0.8).
(c) Compute an upper bound for the error in each of the above approximations.

Let Let E sin(r) where a is measure in radians (a) Calculate approximations for using the central difference formula of order O(h2) with h 0.1 and h 0.01. Carry eight or nine decimal places. (b) Compare with the value f (0.8) cos (0.8) (c) Compute an upper bound for the error in each of the above approximations.

Explanation / Answer

(a) f'(0.8) = [f(0.8+0.1) - f(0.8-0.1)] / 0.2

= [sin(0.9) - sin(0.7)] / 0.2

= 0.69554611

f'(0.8)

= [f(0.8+0.01) - f(0.8-0.01)] / 0.02

= [sin(0.81) - sin(0.79)] / 0.02

= 0.6966950

(b) cos 0.8 = 0.696706709

The error of approximation for h=0.1 is 0.001160599

The error of approximation for h = 0.01 is 0.000011709

So the approximation of h = 0.01 is closer

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