MATH 407 Summer 2021 Calculus Final Exam

MATH 407 Summer 2021 Calculus Final ExamMATH 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 < inf T. (c) A sequence (sn) converges to s if and only if lim inf n→∞ sn = lim sup n→∞ . (d) The series X cos(nx)
n2
defines a continuous function on R.
(e) If f is continuous at x = a, then f is differentiable at x = a.
(f) If (sn) is a monotone and bounded sequence, then it is also Cauchy.
(g) The function f(x) = 1
x
2
is uniformly continuous on [−2, 2].
(h) The sequence of functions fn(x) = x
n
converges uniformly to the function f(x) = 0 on
[0, 1].
(i) If Z b
ag(x)
2
dx = 0, then g(x) = 0 for all x ∈ [a, b].
(j) If Z b
a
g(x) dx = 0, then g(x) = 0 for all x ∈ [a, b]
2 (20 pts.) Define a sequence (sn) by iteration:
s1 = 0 and sn+1 =

sn + 1.
(a) (5 pts.) Use induction to show that sn+1 − sn > 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] ∩ (RQ). is not integrable on R.

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