Assignment #4
Due Friday
10/18
Exam # 1 is on Monday 10/14.
It will cover sections 1.1, 1.2, 2.1, 2.2, 2.3,
3.1, 3.2, 3.3.
The problems will be like the easier homework questions
from those sections.
Note: You may always use Matlab to do
computations onthe homework. However, when explanations, arguments,
"show that ..." are called for, you need to include your written answer
in English. Also, please try to minimize the amount of paper your solution
uses. You can copy a few numbers by hand, sometimes, to save a page-long
printout with only those few numbers. Suggestion: Print out programs
and write in the white space.
From text (Cheney&Kincaid, 4th ed.):
§2.3: Problems: # 1, 2, 5, 29
§2.3: Computer Problems: # 1
§4.1: Problems: # 1, 3, 4, 8, 16, 27 Then figure
out how to use Matlab's polyfit and polyval to do the following
problems:
(§4.1) Problem A: Plot f(x)=sin x and
the polynomial interpolant of degree 4 which has the values yj=sin
xj for x0=-1, x1=-.5, x2=0, x3=.5, x4=1. Plot
on the same graph. Make smooth graphs of both f and the
polynomial, and make the interpolating points clear. That is, match
the plot below.
(§4.1) Problem B: Plot f(x)=(1+16
x^2)^(-1) and the polynomial interpolant of degree 20 which has the
values yj=f(xj) for xj=-1+(0.1)j,
j=0,1,...,20. Make a plot like for problem A.
(§4.1) Problem C: Plot f(x)=(1+16
x^2)^(-1) and the polynomial interpolant of degree 20 which has the
values yj=f(xj) for xj=cos(j pi/20),
j=0,1,...,20. Make a plot like for problem A.
(§4.2) Problem D: Explain the difference
between problems B and C in terms of the Interpolation Errors
Theorem I on page 164.
(§1.2) Problem E: Use Taylor's Theorem to find in what
interval around c=0 the function f(x)=(1+x)^(1/2) can
be estimated to within 10^(-8) by the linear function L(x)=1
+ (1/2) x.
(§1.2) Problem F: Use Taylor's Theorem to estimate the
maximum error used in computing erf(x) by the series (2/sqrt(pi))(x-(1/3)x^3)
for x in the interval [-1,1]. Compare the
the actual error computed from Matlab. Plot both erf(x) and
the series on this interval.
Office: Chapman 301C. Office
hours online. Contact: 474-7693, ffelb@uaf.edu , www.math.uaf.edu/~bueler/