Define: The number b is the least upper bound of a
set of numbers A.
6.
Define: The domain of a function f.
7.
(See chap5cproj) Is the following "definition" correct?
Explain your answer!
Definition HH.
lim
x → a
f(x) = L
means: For any ϵ > 0, there is a δ > 0 such that, for all x, |f(x) − L| < ϵ and 0 < |x −a| < δ.
Examples
8.
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.
9.
Give an example of a bounded function f defined for all
real numbers such that limx → 0f(x) does not
exist.
10.
Give an example of a bounded set of numbers A which has a
greatest element. Give the least upper bound of this set A.
11.
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.
12.
Give an example of a nonempty bounded set AQof
rational numbers whose least upper bound is not a rational
number .
13.
Give an example of a function f with domain (0,1)
such that
•
f is continuous on (0,1) and
•
f is not bounded above on (0,1).
14.
Give an example of a function f with domain
[0,1] such that
•
f is continuous on [0,1] except at x = [1/2] and
•
f is not bounded above on [0,1].
15.
Give an example of a function f with domain
[0,1] such that
•
f is continuous on [0,1] except at x = [1/2] and
•
f is bounded above on [0,1], and
•
f does not assume a maximum value on
[0,1].
16.
Find the decimal and binary expansions of x = [1/5].
17.
Express as a rational number x = [p/q], p, q
natural numbers.
•
x = 0.bin―01 (Binary or Base 2)
•
x = 0.―01 (Decimal or Base 10)
Proofs
18.
Using (P1 - P9), show that
− (a ·b) = (−a) ·b.
19.
Using (P1 - P12), show that
a ≤ b
if and only if:
Forevery ϵ > 0, a < b + ϵ.
20.
Prove by mathematical induction (PMI) or otherwise:
13 + …+ n3 = (1 + …+ n)2.
21.
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.
Hint: State precisely the version of (P13) that
you use.
22.
Prove (ϵ− δ):
Theorem. If
lim
x → a
f(x)
= L and
lim
x → a
g(x)
= M,
then
lim
x → a
(f(x) + g(x))
= L + M.
23.
Let A and B be nonempty bounded sets. Define
A+B
= { x | x = a + b, a ∈ A, b ∈ B}
Show that
inf
(A + B)
=
inf
A +
inf
B.
Qualitative Properties of Functions
24.
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.
Construct your own (not too complicated)
Height-Volume Graph!
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
25.
(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.
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