MATH 407 Summer 2021 Calculus Final Exam

Posted: May 7th, 2022

MATH 407 Summer 2021 Calculus Final Exam
MATH 407 Summer 2021 Final Exam – June 24, 2021
Due 11:45 am. Late submissions will NOT be accepted!
1 (20 pts.) Answer TRUE or FALSE to each statement. No explanation is required. Each part
is worth 2 pts.
(a) The sequence sn = (−1)n
1
1
n
has a convergent subsequence.
(b) If S and T are nonempty bounded subsets of R such that S ⊂ T, then inf S 0 for all n ∈ N.
(b) (5 pts.) Use induction to show that sn 6 3 for all n ∈ N.
(c) (5 pts.) Using parts (a) and (b) explain why (sn) converges.
(d) (5 pts.) Find the limit of (sn). (Hint: If sn → s then also sn 1 → s.)
3 (10 pts.) This problem consists of five parts. Make sure that you justify your answer!
(a) (2 pts.) Give an example of a divergent series Pan such that the series Pa
2
n
converges.
(b) (2 pts.) Give an example of a convergent series Pan such that the series Pa
2
n diverges.
(c) (2 pts.) Give examples of a convergent series Pan and a divergent series P
P
bn such that
anbn converges.
(d) (2 pts.) Give examples of a convergent series Pan and a divergent series P
P
bn such that
anbn diverges.
(e) (2 pts.) Give examples of a convergent series Pan and a convergent series P
P
bn such that
anbn diverges
4 (15 pts.) Consider the sequence of functions
fn(x) = nx
1 n2x
2
defined on the whole real line R.
(a) (5 pts.) Find the function f such that fn → f pointwise as n → ∞.
(b) (10 pts.) Does fn converge to f from part (a) uniformly? Prove your answer.
5 (15 pts.) Consider the function f on R given by the formula
f(x) = (
x
2
if x > 1,
x if x < 1.
(a) (5 pts.) Prove that the function f is continuous at x = 1.
(b) (10 pts.) Prove that the function f is not differentiable at x = 1.
Pick any TWO of the following five problems:
6 (10 pts.) Consider the function
f(x) = x
x 2
.
(a) (6 pts.) Prove that the function f is uniformly continuous on the interval (2, 4).
(b) (4 pts.) Is the function f uniformly continuous on the interval (−4, 4). Justify your
answer.
7 (10 pts.) This problem consists of two parts.
(a) (2 pts.) Write down the definition of limn→∞
sn = s.
(b) (8 pts.) Using the definition in part (a) show that limn→∞
2n 5
5n − 9
=
3
4
.
8 (10 pts.) Show that the function f(x) = 1
x
is integrable on [1, 5].
9 (10 pts.) Suppose f is a differentiable function on R such that
|f
0
(x)| 6 28 ∀ x ∈ R.
Prove that f is uniformly continuous on R. (Hint: Use the Mean Value Theorem.)
10 (10 pts.) Show that the function
f(x) = (
x if x ∈ [0, 1] ∩ Q,
0 if x ∈ [0, 1] ∩ (R\Q).
is not integrable on R.