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## Homework Statement

Q1) Use the Taylor series of f (x), centered at x0 to show that

F1 =[ f (x + h) - f (x)]/h

F2 =[ f (x) - f (x - h) ]/h

F3 =[ f (x + h) - f (x - h) ]/2h

F4 =[ f (x - 2h) - 8 f (x - h) + 8 f (x + h) - f (x + 2h) ]/12h

are all estimates of f '(x). What is the error associated with the approximation

Fi ~ f ' (x), for i = 1; 2; 3; 4?

Example:

f (x + 3h) = f (x) + 3h f '(x) +[(9h^2)/2]*f ''(x) + ...

F5 = [f (x + 3h) - f (x)]/3h = f '(x) + (3h/2)*f ''(x) + ...

so F5 is f '(x) with an error 3h f ''=2, which is of order h1 (i.e., first-order).

## Homework Equations

Im trying to understand the example. So is x0 = x + 3h? If so, and I plug that into the taylor equation f (x + 3h) = f(x+3h) + f'(x+3h)*(3h) + f''(x+3h)*([9h^2]/2)+...

What am I doing wrong?