# 3.7Assignment 3 - due March 1, 1996

1. Find the length of the polar curve r = cos^2(theta), -Pi/2 <= theta <= Pi/2. Maple needs help to do the integral: make a substitution.

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 = 1

```
where c[j] are constants (not involving a or b). The constants for ME, TE 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), 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)
/
0
```
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

```                         3
/
|        1
|  ------------ dx
|   2
/   x  - 2 x + 5
1
```
(b) Calculate the error in R[n,n] and compare to R[n,n-1] - R[n,n], for the integrals
```                      1              1
/              /
|   11         |   41
|  x   dx and  |  x   dx
|              |
/              /
0              0
```
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 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]?