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.

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