Chapter 4.2, Problem 1ES

Apply the extrapolation process described in Example 1 to determine N₃(h), an approximation to f ′(x₀), for the following functions and step sizes.

1. a. f (x) = ln x, x₀ = 1.0, h = 0.4
2. b. f (x) = x + e, x₀ = 0.0, h = 0.4
3. c. f (x) = 2^x sin x, x₀ = 1.05, h = 0.4
4. d. f (x) = x³ cos x, x₀ = 2.3, h = 0.4

Example 1 In Example 1 of Section 4.1, we used the forward-difference method with h = 0.1 and h = 0.05 to find approximations to f′(1.8) for f (x) = ln(x). Assume that this formula has truncation error O(h). Use extrapolation on these values to see if this results in a better approximation.