M2160-1      5th Homework Assignment      Due Mar. 24, 2006.

Do TWO out of three of the following problems.

1. For a positive integer n, let σ(n) denote the sum of its proper divisors, that is, those divisors d of n for which d<n. For example σ(12)=16 because the proper divisors of 12 are 1, 2, 3, 4, and 6. Similarly σ(52)=46 because the proper divisors of 52 are 1, 2, 4, 13, and 26. Write a program that asks the user to input n and then computes σ(n).

A positive integer is called perfect if σ(n)=n. So the sum of the proper divisors of a perfect number equals the number itself. 6 is a perfect number because σ(6)=6. Can you find any other perfect numbers?

2. Determine all real solutions (accurate up to six decimal places) of the equation

x5 + 2x4 - 3x3 - 4x2 + 4x + a = 0

where a = 0.xxx is a decimal number (xxx represent the last three digits of your student id number). Use either bisection or Newton's method. Describe in detail the procedure you used to find the answer. Be sure to explain how you know that you've found all the solutions. Enclose the programs and the final result.

3. Using the trapezoid, midpoint or Simpson's rule, find the integral of f ( x ) over the interval over for 1 ≤ x ≤ 3 to six decimal places accuracy, where

f(x)  =   ( a   +  x3)1 / 2

where a = 0.xxx is a decimal number (xxx represent the last three digits of your student id number as in problem B). Your explanation should justify the number of intervals you choose to compute your answer. You may use the error estimates for the trapezoid or Simpson's rules given in your calculus text [Varberg, Purcell & Rigdon, Calculus, 8th ed., Prentice Hall, 2000, section 11.2.]