Assigned: Thursday Aug. 29, 2013
Due: Tuesday Sept. 10, 2013
MATH 373: Homework 1
“Taylor Series, Convergence, & Round-Off Error”
Fall 2013
NOTE:
• Include a cover page.
• Always clearly label all plots (title, x-label, y-label, and legend).
• Use the subplot command from MATLAB when comparing 2 or more plots to make
comparisons easier and to save paper.
1. Compute the first 5 terms in the Taylor series (constant, linear, quadratic, cubic,
and quartic pieces) for the following functions:
(a) f (x) = 3 tan(x),
(b) f (x) = ecos(x) ,
about the point x = π/4
about the point x = 0
2. Using the results from Problem 1(b), make a single MATLAB plot which contains
all of the following:
(a) a graph of f (x) = ecos(x) versus x for x ∈ (−3, 3).
(b) a graph of p2 (x).
(c) a graph of p4 (x).
(d) a title, x-axis label, y-axis label, and a legend.
3. The quotient
g(x) =
log(1 + xex )
x
seems at first glance to be undefined at x = 0. Approximate log(1 + xex ) by a Taylor
polynomial of degree 2 about the point x = 0. Use this Taylor approximation to
determine a natural definition of g(0).
4. Consider the function
g(x) = ex .
(a) Derive the general nth order Taylor polynomial (pn (x)) as well the remainder
term (Rn (x)) for the function g(x) expanded about the point x = 0.
(b) Using the remainder term from part (b), determine the value of n needed to
guarantee that |pn (1) − g(1)| < 10−5 .
(c) Using the value of n found in part (b), compute the actual error |pn (1) − g(1)|.
5. Compute the following limits and determine the rates of convergence:
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Assigned: Thursday Aug. 29, 2013
Due: Tuesday Sept. 10, 2013
(a) limh→0
(b) limh→0
h2 −h3
1+2h3
q
1
h
+1−
q
1
h
(c) limh→0 h sin(h−1 )
(d) limh→0
3−h2
7+h+2h2
6. Determine which one of the following sequences converges to 1 faster (clearly explain
your reasoning):
sin(x2 )
x→0
x2
lim
and
(sin(x))2
.
x→0
x2
lim
7. Consider the following:
xn = 1 − cos
π n
yn = 2 sin2
π
2n
NOTE: From trigonometry we know that xn = yn since sin2 (θ) =
1
2
(1 − cos(2θ)).
(a) Subplot #1: Plot xn and yn versus n for n = 1, . . . , 1000 on a single log-log
plot using the loglog command. Use a solid line for xn and open circles for yn .
(b) Subplot #2: Plot the relative error |xn − yn |/|yn | on a log-log plot using the
loglog command.
(c) As n becomes large, which one (xn or yn ) is more accurate? Why? (HINT:
recall the 2 situations that you want to avoid when doing computer arithmetic.)
8. (from [Greebaum & Chartier, 2012]). In the seventh season of the The Simpsons,
Homer has a nightmare in which the following equation flies past him:
178212 + 184112 = 192212 .
Note that this equation, if true, would contradict Fermat’s last theorem, which
states: For n ≥ 3, there do not exist any natural numbers x, y, and z that satisfy
xn + y n = z n . Did Homer dream up a counterexample to Fermat’s last theorem?
√
(a) Compute 12 178212 + 184112 by typing the following into matlab:
format short; (1782^12 + 1841^12)^(1/12)
What result does matlab report? Now look at the answer using format long.
(b) Determine the absolute and relative errors in the approximation 178212 +
184112 ≈ 192212 . Such an example is called a Fermat near miss.
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