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These concepts can be used to set up any order of derivative and truncation errors. These are known as finite differences:

x_i + 1 − x_i forward difference x_i + 1 − x_i − 1 central difference *1cm x_i − x_i − 1 backwards difference

Figure 5 The approximation of a line using finite differences (discrete points).

Thus the first derivative is

f’(x_i) = (f(x_i + 1) − f(x_i))/(Δx) + θ(Δx)

If Δx = h and is a constant, then (Δf_i)/(h) + θ(h) where θ is order of truncation error.

Using the backwards difference, yields [(x_i − 1) = (f(x_i) − f(x_i − 1))/(h) + θ(h). With the central difference, the equations become

f(x_i + 1) = f(x_i) + f’(x_i)h f(x_i − 1) = f(x_i) − f’(x_i)h f(x_i + 1) = f(x_i − 1) + f’(x_i)h2 f’(x_i) = (f(x_i + 1 − f(x_i − 1))/(2h) + θ(h)

The second derivative of the function can be written as

f(x_i + 1) = f(x_i) + f’(x_i)h + f"(x_i)(h²)/(2) + θ(h²) f"(x_i) = (2(f(x_i + 1) − f(x_i) − f’(x_i)h))/(h²) = (2f(x_i + 1) − 2f(x_i) − (h²)/(2)⎛⎝(f(x_i + 1) − f(x_i − 1))/(2h)⎞⎠)/(h²)

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