Math 210 Sec. 201 Assignment 3 - due March 1, 1996

1. Find the length of the polar curve r = cos^{2}(θ), −π/2 ≤ θ ≤ π/2. Maple needs help to do the integral: make a substitution.

2.(a) Let f(x) = x^{11}. Find the errors ME(n), T E(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}T E(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), T E(n) and SE(n) are of the form

∑_{j=1}^{5} c_{j} \frac{(b − a)^{2j} (f^{(2j−1)}(b) − f^{(2j−1)}(a))}{n^{2j}}

where c_{j} are constants (not involving a or b). The constants for ME, T E and SE are different. Hint: to find the coefficient of n^{−2j}, you can use “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), T E(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

∑_{j=1}^{m} c_{j} \frac{(b − a)^{2j} (f^{(2j−1)}(b) − f^{(2j−1)}(a))}{n^{2j}} + O (n^{−2m−2})

where c_{j} are constants that don’t depend on n, f, a or b.

(e) Compare the actual T E(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

∫_{0}^{2π} \frac{dx}{2 + cos x}? How accurate are the Midpoint, Trapezoid and Simpson’s Rule approximations

for n = 20 in this case? Note: make “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 ∫_{1}^{3} \frac{dx}{x^{2} − 2x + 5}.

(b) Calculate the error in R_{n,n} and compare to R_{n,n−1} − R_{n,n}, for the integrals ∫_{0}^{1} x^{11} dx and

∫_{0}^{1} x^{41} dx with 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 ∫_{0}^{1} x^{k} dx when k is large?

(d) What n would be needed in order to have R_{n,n} for ∫_{0}^{1} x^{41} dx be exactly correct? Would it be

practical to calculate this R_{n}?