Name: \hbox to 3.0 truein{\hrulefill } \hfill Page
\folio/\numpages I. Definitions
1.
(10 points) Define
(e-d): limx ® af(x) = L.
2.
(10 points) Give an example that shows that the following
definition of limx ® a f(x) = L is not correct:
For every d > 0, there is an e > 0 such
that, for all x, if 0 < |x - a| < d, then |f(x)- L | < e.
Let f(x) = 1/x, a = 1. Then limx ® 1f(x) = 1,
but the proposed definition is not satisfied for d = 1.
II. Examples
3.
(10 points) Give an example of two functions f and g
such that f / g and g / f have different nonempty
domains. Be sure to specify domain(f), domain(g),
domain(f / g), and domain(g / f).
4.
(20 points) Let
F(x)
=
Ö
x2 - 1
,
G(x)
=
Ö
x2 + 1
Describe:
·
domain(F) and domain(G).
·
domain(F + G)
·
domain(G °F)
We have that
(G °F)(x)
=
Ö
(
Ö
[`(x2 -1)])2 +1
=
Ö
(x2 -1) + 1
=
Ö
x2
= |x|.
.
The intermediate steps are defined iff x2 - 1 ³ 0, so that
domain(F°G) = {|x| ³ 1}.
·
domain(F °G)
·
domain([F/G])
·
domain([G/F])
5.
(10 points) For a > 0, find
lim
h ® 0
Ö
a + h
- Öa
h
.
6.
(10 points) Give an example of two functions f, g,
such that limx ® 0 f(x) = 0, limx ® 0 g(x) = 0,
and limx ® 0 [f(x)/g(x)] = 1. Be sure to specify
domain(f), domain(g), domain(f/g), and
domain(g/f).
III. Proofs
7.
(15 points Prove: If limx ® a f(x) = L and limx ® a g(x) = M,
then limx ® a (f + g)(x) = L + M.
8.
(15 points) Prove: If g is continuous at a, g(a) ¹ 0, then
there is a d > 0 for which (a - d,a + d) is contained
in the domain of [1/g].
9.
(15 points) Show, using only P1 - P9: For all numbers a,b,
- (a ·b)
= (-a)·b.
(§)
You may abbreviate (associative, distributive, trichotomy,
¼ ).
(a b) + (-a)b
= (a + (-a))b
(P9)
= 0 ·b
(P3)
= 0
(inclass)
The proof is finished. Optionally, add -(ab) to both sides
of the equation (a b) + (-a)b = 0.
Extra Credit: Show that for a ¹ 0,
(-a)-1 = -(a-1). Hint: Use (§).
1
= - [- (a a-1)]
= - [a (- (a-1))]
(§)
= (- a)(- (a-1))
(§)
Therefore, -(a-1) = (-a)-1.
10.
(15 points) Show by mathematical induction or otherwise:
For all natural numbers n = 1, 2, ¼,
13 + 23 + ¼+ n3 =
1
4
n2 (n + 1)2.
IV. Qualitative Properties of Functions
11.
(20 points) The graph below shows how the height of a liquid
in a Boiling Flask Z varies as water is steadily dripped into it.
Copy the graph, and on the same diagram
show the height-volume relationship for the Boiling Flask Z.
Describe the features of the graph you have drawn. Your
description should include
·
The domain of the function
[0, Volume of Flask]
·
The range of the function
[0, Height of Flask]
·
The intervals of monotonicity (Increasing,
Decreasing)
·
The intervals of constant concavity and/or
linearity
·
Other observations ¼
A person reading your description of the graph should be able to
reproduce the graph of the function (and if she's good, guess
that it came from something shaped like a Boiling Flask).
N.B. The graph does not have a "corner" at the
height at which the graph becomes linear (neck of flask).
V. Essay
12.
(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.
Good Essays!
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