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Problem 1. Finite difference approximations. Recall that forward and centered finite-difference approximations for the derivative of a function f are given by ?(1)f(x) = f(x+h)?f(x),

Problem 1. Finite difference approximations.

Recall that forward and centered finite-difference approximations for the derivative of a function f are given by ?(1)f(x) = f(x+h)?f(x), ?(0)f(x) = f(x+h)?f(x?h).

h h h 2h

(a) Other useful finite-difference approximations can be constructed in similar ways. Show that

?hf(x) = 1 ?2f(x+h)+3f(x)?6f(x?h)+f(x?2h)?, 6h

approximates f?(x) up to an error term of order O(h3).

(b) Define the finite-difference approximation errors:

e(1)(x) = f?(x) ? ?(1)f(x), hh

e(0)(x) = f?(x) ? ?(0)f(x), hh

eh(x) = f?(x) ? ?hf(x). For the function f(x) = sin(x) and x0 = 1, plot the above errors as a function of h on a log-log scale. That is,

plot log |e(1)(x0)|, log |e(0)(x0)| and log |eh(x0)| as a function of log h. hh

(c) What are the slopes in your three graphs? Are they in line with what we should expect from theoretical results?

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Problem 1. Finite difference approximations. Recall that forward and centered finite-difference approximations for the derivative of a function f are given by A ," f ( x ) = f(Ith) - f(I) 4," f ( z ) = f( x th) - f(I - h) h 2h (a) Other useful finite-difference approximations can be constructed in similar ways. Show that Anf (x) = (2f(r + h) +3f(x) -6f(x - h) + f(x - 2h)), approximates f' (x) up to an error term of order O(h3). (b) Define the finite-difference approximation errors: e(") (I) = f'(x) - Aff(x), (@ (x) = f'(1) - , f(I), en(I) = f'(x) - Anf(x). For the function f(x) = sin(x) and To = 1, plot the above errors as a function of h on a log-log scale. That is, plot log len (ro)|, log len (ro)| and log Jen (To) | as a function of log h. (c) What are the slopes in your three graphs? Are they in line with what we should expect from theoretical results

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