Name: \hbox to 3.0 truein{\hrulefill } \hfill Page
\folio/4
Definitions
1.
Define (ε-δ): limx → a f(x) = L.
2.
Define: limx → a− f(x) = L.
3.
Define: The function f is continuous at a.
4.
Define: The set of numbers A is bounded above.
5.
Define: The number b is the least upper bound of a
set of numbers A.
Examples
6.
Give an example of two functions f
and g such that f ° g = g °f. Be sure
to verify that the domains are the same.
7.
Give an example of a bounded function f defined for all
real numbers such that limx → 0f(x) does not
exist.
8.
Give an example of a bounded set of numbers A which has a
greatest element. Give the least upper bound of this set A.
9.
Give an example of a nonempty bounded set of numbers A
which has no greatest element. Give the least upper bound of this
set A.
10.
Give an example of a nonempty bounded set AQof
rational numbers whose least upper bound is not a rational
number .
Proofs
11.
Let f be defined on [0,1) be such that
•
f is increasing on [0,1) (If 0 ≤ x1 < x2 < 1, then f(x1) < f(x2).)
•
f is bounded above on [0,1).
Prove that
lim
x → 1−
f(x)
= L
exists.
Qualitative Properties of Functions
12.
Water drips very slowly into a circular bottle (beaker,
flask) so that the graph of the Height (in cm) as a
function of Volume (in cm3) is shown below.
Draw a side view of the bottle. Carefully explain as many features
as you can about the shape of the bottle and explain how they are
related to the Height-Volume graph.
Essay
13.
(Letter Grade: A - E) In the exam booklet, write an essay
on a topic of your choice that is very relevant to the material
considered in the course. Your essay should include at least
one substantial example and at least one substantial theorem and
its proof.
File translated from
TEX
by
TTH,
version 3.70. On 21 Nov 2005, 16:07.
