Sample Midterm M350 Fall 1999

Instructions: You are allowed a calculator and a single sided sheet of notes on 8.5 by 11 paper. All books should be closed. You have 50 minutes. All questions carry equal weight. Read the entire exam before beginning. Problem 1. Find the Talyor polynomial of degree two for the function

$$f(x) = exp(x^2) \sin x$$

about x=0. Problem 2. Propose two fixed point iteration schemes to solve

x⁴-2x²-3=0

Prove that at least one of these methods converges on the interval [1,2] with p₀=1. Which of your methods is superior and why? Problem 3. Write down Newton's method for computing a zero of a function. Show graphically how a single iteration proceeds. Problem 4. Find the interpolating polynomial P₁(x) of degree one through the points (1,2), (2, 3). Modify this polynomial to be a second degree polynomial P₂(x) which interpolates the point (3,2) in addition to the above points. Specifically, identify N(x) where

P₂(x) = P₁(x) + N(x)

Problem 5. Compare the bounds on the error for P₁(1.5) and P₂(1.5) from the previous problem.

Problem 6. Evaluate the polynomial

6x⁴ + 3x³+2x²+7x+2=0

using 3 digit rounding arithmetic in the most efficient manner. What is the minimum number of add/multiplies required? Compare your result with direct evaluation of the polynomial and compute the absolute and relative errors.