Math 141 Homework #9
Due Tuesday, 10/23/07 Extra Problems
Problem #1 Consider the function
b(x) = x2+ln |x−2|
1000 .
(#1a) Without using a calculator, sketch the graph of b(x) for −10 ≤x≤10.
(#1b) Enter b(x) into your calculator and have it draw the graph. The result will probably not look like
the graph you drew in part 1 (at least if you are using a TI-83+ or something similar). Who’s right, you or
the calculator?
(#1c) Can you resolve this problem by changing the viewing window?
Problem #2 Read this article from the Lawrence Journal-World and write a sentence or two clearly ex-
plaining the mathematical basis for Sen. Haley’s proposal.
Problem #3 (Bonus problem; challenging!) The Extreme Value Theorem states that if a function f(x)
is continuous on a closed interval I, then fachieves a global maximum and a global minimum on I. In class
on Tuesday 10/16, we discussed the possibility that fhas infinitely many critical numbers in I. Let’s call
a function wild if it exhibits this behavior, and tame otherwise. As we saw in class, an example of a wild
function is
f(x) = (xsin 1
xif 0 < x ≤1
0 if x= 1
on the interval I−[0,1].
We know that if fis tame, then we can find the global minimum and maximum of fon Iby listing all
the (finitely many) critical numbers and endpoints, evaluating fat each of them, and comparing the values.
We’d like to prove that this method still works even for wild functions.
(#3a) First, explain why the range of fmust be an interval. That is, rule out the possibility that the
range is something like
[−4,0) ∪(1,3]
or
[−4,0) ∪(0,3)
or
{1,2,3,5,8,13,21}.
The next step is to figure out whether the interval is open, closed or half-and-half, and whether it is finite
or infinite.
(#3b) Second, prove the following Lemma. If {x1, x2, x3,...}is an infinite set of numbers in I, then there
is some number a∈Ithat is an “accumulation point” of the xi’s — that is, with the property that any
open1interval containing aalso contains at least one of the xi’s.
This Lemma is a key tool for the rest of the problem. If you don’t see how to prove the Lemma, that’s okay;
you can still do the rest of the problem by assuming that the Lemma is true.
1Or possibly half-open, if ais an endpoint of I, but you can ignore that case if you want to.
