EXAM
Exam 3
Final Exam
Math 4350-201, Summer II, 2013
August 8, 2013
• Write all of your answers on separate sheets of paper.
You can keep the exam questions when you leave.
You may leave when finished.
• You must show enough work to justify your answers.
Unless otherwise instructed,
give exact answers, not
√
approximations (e.g., 2, not 1.414).
• This exam has 4 problems. There are 380 points
total.
Good luck!
130 pts.
Problem 1. State the following. All sequence are assumed to be in R.
(1.) The Well-Ordering Property of N.
(2.) The definitions of “denumerable set” and “countable set.”
(3.) Let A ⊆ R be nonempty. State the definitions of “a lower bound for A”
and inf(A).
(4.) The Archimedean Property.
(5.) The Characterization Theorem for intervals.
(6.) The Nested Intervals Property.
∞
(7.) The definition of the limit of a sequence {xn }n=1 .
(8.) The Bolzano-Weierstrass Theorem.
(9.) The definition of a Cauchy Sequence.
(10.) The Cauchy Convergence Criterion.
(11.) Assume that f : R ⊇ A → R. State the definition of “c is a cluster point of
A.” If c is a cluster point of A, give the definition of
lim f (x) = L,
x→c
L ∈ R.
(12.) The Maximum-Minimum Theorem.
(13.) The Interior Extremum Theorem.
(14.) Rolle’s Theorem.
(15.) The Mean Value Theorem.
(16.) Darboux’s Theorem.
150 pts.
Problem 2. In each part, decide if the given statement is True or False. Assume that f : R ⊇ A → R and that c is a cluster point of A. All
sequences refered to are sequences in R.
(1.) If there is a sujection N → S, then S is countable.
(2.) If S ⊆ N, S is countable.
(3.) If S ⊆ R is a nonempty set which is bounded below then there is a sequence
in S that converges to inf(S).
(4.) Let S ⊆ R be a nonempty set. If b is an upper bound for S and b ∈ S, then
b = sup(S).
1
(5.) Every sequence has a bounded subsequence.
(6.) Every sequence has a convergent subsequence.
(7.) Every sequence has a monotone increasing subsequence.
(8.) If f (xn ) → L for every sequence {xn } ⊆ A such that xn 6= c for all n, then
lim f (x) = L.
x→c
(9.) If f is a differentible function on [a, b], and f acheives its maximum at
c ∈ [a, b], then f 0 (c) = 0.
(10.) A differentible function is continuous.
(11.) If f is a differentible function on an interval I, the derivative f 0 can have
a jump discountinuity.
(12.) If f is a differentible function on an open interval I, c ∈ I, and f 0 (c) > 0,
then f is increasing on an interval around c.
(13.) If f is an injective differentiable function on an interval I, and J = f (I) is
the image interval, then the inverse function f −1 is differentible at every
point of J.
(14.) If f and g are uniformly continous functions on R, the product f g is uniformly continuous.
(15.) If f is a differentible function on R and f 0 is bounded, then f is uniformly
continuous.
50 pts.
50 pts.
Problem 3. Let f : (0, ∞) → R be the function defined by f (x) = 1/x.
Give a careful ε-δ proof that f is continuous at 1. Given ε > 0, find a
specification for δ.
Problem 4. Let f : R → R be the function defined by
(
x2 , x ≥ 0
f (x) =
x3 x < 0.
Use the definition of derivative to show that f is differentible at 0. (It is not
necessary to give an ε-δ proof.)
Is f 0 differentible at 0?
2