Numerical Analysis I, Fall 2009 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 01

Assigned Sep 18, 2009. Due Sep 25, 2009. Objectives:

(1) Review Taylor polynomial and Taylor series

(2) To practice basic programming.

1. Section 1.1: Problems 3(e), 4, 10, 14.

2. Continue with Problem 14. Plot p_{3}(x), q(x) and f(x) on the same graph. Make them distinguishable by putting different colors or different plot patterns (line, dots, circles, etc). Indicate which curve corresponds to which function.

3. The program on page 8 is called eval exp tay.m on the author’s homepage. Try run it (together with the function polyeval.m), and see what it does. Then modify the code so that it displays the Taylor polynomials of cos(x), p_{0}(x), p_{2}(x), p_{4}(x), p_{6}(x), together with the errors stored in a proper file in tabular form. That is, exactly the same thing the program eval exp tay.m does to f(x) = e^{x}.

4. Derive the integral form of Taylor’s remainder term for general n, assuming the function under concern is in C^{∞}.

5. Derive Taylor’s polynomial and remainder term for f(x, y) around the point (a, b), assuming f ∈ C^{∞}(R^{2}).

Hint: define g(t) = f(a + t(x − a), b + t(y − b)). Now g(t) is function of single variable t.