MATH 417: Numerical Analysis
Instructors: Prof. Wolfgang Bangerth, Prof. Guido Kanschat
Teaching Assistants: Seungil Kim, Yan Li
Homework assignment 1 – due 9/7/06 and 9/8/06
Problem 1 (Taylor series). Derive the first four terms and integral remain-
der term of the Taylor series of
a) f(x) = cos xwhen expanded around x0= 0;
b) f(x) = xcos xwhen expanded around x0=π/2;
c) f(x) = 3(x−4)2(x+ 3) when expanded around x0= 1. What happened
to the remainder term and what does this mean for the accuracy of the
Taylor expansion with only four terms?
You may use a computer algebra system like Maple to compute derivatives of
f(x), but not to generate the entire Taylor series. (4 points)
Problem 2 (Taylor series). Many important functions such as the sine can-
not be computed in a simple way, i.e. with only the four basic operations plus,
minus, multiplication and division. However, they can be approximated with
these operations.
a) Graph the first eight Taylor approximations of f(x) = cos xwhen ex-
panded around zero, i.e.
f1(x) = f(0) + f0(0)x,
f2(x) = f(0) + f0(0)x+1
2f00(0)x2,
f3(x) = f(0) + f0(0)x+1
2f00(0)x2+1
3! f000(0)x3,
etc. What do you observe? What does this mean for the approximation
of f(2π)?
b) How large is the maximal error of the approximation on the interval [0,2π].
c) Write a program to experimentally determine the number of terms you
need to approximate f(2π) = 1 to an accuracy of 10−4and 10−12.
(6 points)
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