** 2.(a) **
Let f(x) = x^11. Find the errors ME(n), TE(n) and SE(n)
in the Midpoint,
Trapezoid and Simpson's Rules for the interval [a,b]
(explicitly as a function of n).

** (b) ** Plot n^2 ME(n) and n^2 TE(n) (on the same graph) and
n^4 SE(n) (on a different graph) as functions of n. What can
you conclude?

** (c) ** Show that ME(n), TE(n) and SE(n) are of the
form

5 ----- (2 j) (2 j - 1) (2 j - 1) \ c[j] (b - a) (f (b) - f (a) ) ) ------------------------------------------------- / (2 j) ----- n j = 1where c[j] are constants (not involving a or b). The constants for ME, TE and SE are different.

`coeff(expand(TE(n)),n,-2*j)`

''.
** (d) ** Try it now for some other power x^m where m is an integer
>= 5). Show that ME(n), TE(n) and SE(n)
have the same form (with the same c[j]'s for
j <= 5, but
perhaps more terms).
It turns out that for any function with at least 2m+2
derivatives on [a,b], each of these errors is

/ m \ |----- (2 j) (2 j - 1) (2 j - 1) | | \ c[j] (b - a) (f (b) - f (a) )| | ) -------------------------------------------------| | / (2 j) | |----- n | \j = 1 / (- 2 m - 2) + O(n )where c[j] are constants that don't depend on n, f, a or b.

** (e) ** Compare the actual TE(n) to the sum from j=1 to 5
in the formula above, using f(x) = exp(x), a=0 and b=1.
Calculate n^12 times the difference, for n = 1 to 10.
Does this appear to approach a limit?
You should use ```Digits:=30`

'' since
the differences get very small.

** (f) ** What would the formula say about the
error in using the Trapezoid Rule to approximate

2 Pi / | 1 | ---------- dx ? | 2 + cos(x) / 0How accurate are the Midpoint, Trapezoid and Simpson's Rule approximations for n=20 in this case?

`Digits`

'' at least 20.
Do you find the result surprising?
** 3.(a) ** Find the Romberg approximations R[j,k] up to j=k=5
for

3 / | 1 | ------------ dx | 2 / x - 2 x + 5 1

1 1 / / | 11 | 41 | x dx and | x dx | | / / 0 0with n = 2, 3, 4 and 5.

** (c) ** Is R[n,n-1] - R[n,n] a good estimate of the error
in R[n,n] for
int(x^k, x=0..1) when k is large?

** (d) ** What n would be needed in order to have R[n,n] for
int(x^41, x=0 .. 1) be exactly correct?
Would it be practical to calculate
this R[n]?