The forward-difference approximation to is

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Scientific programming for physicists

Computational Physics, Ch Letting the computer do the algebra

Problem

The forward-difference approximation to

f'(x)

D_h f(x) = \dfrac{f(x+h)-f(x)}{h}

. Take

f(x) = e^{-x}

. (a) Expand

f(x+h)

as a Taylor series in

h

about

x

, keeping every term up to and including the one in

h^2

, and use it to show that

D_hf(x) - f'(x) = \tfrac{1}{2}hf''(x) + \mathcal{O}(h^2).

(b) Given

f''(x) = e^{-x}

, evaluate the coefficient of

h

in the error at

x=1

. (c) A ratio table built by halving

h

for

f(x)=e^{-x}

x=1

gives a measured coefficient (the number multiplying

h

) of

0.184

. Compare it with the symbolic value.