Let f (x) be a given function that can be evaluated at points x0 ± jh, j = 0,1,2,…, for any fixed.

Let f (x) be
a given function that can be evaluated at points x0 ± jh, j = 0,1,2,…, for
any fixed value of h, 0 < h="">

(a) Find a
second order formula (i.e., truncation error O(h2)) approximating the third
derivative f (x0). Give the formula, as well as an expression for the
truncation error, i.e., not just its order.

(b) Use your
formula to find approximations to for the function f (x) = ex employing values
h = 10_1,10_2,…,10_9, with the default MATLAB arithmetic.
Verify that for the larger values of h your formula is indeed second order accurate.
Which value of h gives the
»

Let f (x) be
a given function that can be evaluated at points x0 ± jh, j = 0,1,2,…, for
any fixed value of h, 0 < h="">

(a) Find a
second order formula (i.e., truncation error O(h2)) approximating the third
derivative f (x0). Give the formula, as well as an expression for the
truncation error, i.e., not just its order.

(b) Use your
formula to find approximations to for the function f (x) = ex employing values
h = 10_1,10_2,…,10_9, with the default MATLAB arithmetic.
Verify that for the larger values of h your formula is indeed second order accurate.
Which value of h gives the closest approximation to e0 = 1?

(c) For the
formula that you derived in (a), how does the round off error behave as a
function of h, as h _ 0?

(d) How
would you go about obtaining a fourth order formula for f (x0) in general? (You
don’t have to actually derive it: just describe in one or two sentences.) How
many points would this formula require?

»

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