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DVANCED CALCULUS 1,

Basic topology of the real line, numerical sequences and series, continuity, differentiability, Riemann integration, uniform convergence, power series.

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EXAM 1. ADVANCED CALCULUS I, FALL 2020

Name:

In Questions 1-5 prove your answer. In this test you can use

without proofs theorems that were proved in the lectures or in the

book, just give a reference.

1.(10 pts) Let S be a subset of R, and u ∈ R. Two of the following

statements, if combined, imply that u = inf(S). Which two statements?

A: u ≤ s for every s ∈ S.

B: There exists > 0 and s ∈ S so that s < u + .
C: For every > 0, the number −u − is not an upper bound of the

set −S = {x ∈ R : −x ∈ S}.

D: There exists > 0 so that u + is not a lower bound of S.

1

2

2. (10 pts) Let {xn } and {yn } be two sequences,

lim xn = L1 6= 0,

lim yn = L2 6= 0,

and let > 0. Suppose that |xn | < M for every n. Select one of the
following sets of constants 1 and 2 for which the inequalities
|xn − L1 | < 1
imply |xn yn − L1 L2 | < .
A: 1 =
B: 1 =
C: 1 =
D: 1 =
, 2 = 2|L 1 | .
2M
, 2 = 2M
.
2|L1 |
, 2 = 2M .
2|L2 |
, 2 = 2|L 2 | .
2M
and
|yn − L2 | < 2
3
3. (10 pts) Which of the following statements imply that a sequence
{xn } does not have a limit:
A: for every > 0 there exist n, m ∈ N such that |xn − xm | > .

B: there exist a natural number K and > 0 such that for every

n > K we have |xn − xK | > .

C: {xn } is an increasing sequence and |xn | > 1000 for every n > 1000.

D: there exists > 0 such that for every natural number K there

exist m, n > K with |xm − xn | > .

4

4.(10 pts) Which two of the following statements combined imply

that limx→3 f (x) = 2?

A: limn→∞ f (3 + n1 ) = limn→∞ f (3 − n1 ) = 2.

B: limx→3+ f (x) = 2

C: limx→3+ f (x) and limx→3− f (x) exist

D: For every sequence xn → 3 with xn ≥ 3 for every n, one has

lim f (xn ) = 2.

n→∞

5

5. (10 pts) Which two of the following statements combined imply

that the equation f (x) = 1 has a solution in the interval [1, 3]?

A: 0 < f (1) and f (3) < 2.
B: f is continuous on [1, 3].
C: f (1) < 0 and f (3) > 2.

D: f is monotone on [1, 3].

6

6. (10 pts) Suppose that lim |xn | = 3, but {xn } does not have a

subsequence with limit 3. Prove that lim xn = −3.

7

7. (10 pts). Suppose {xn } and {yn } are bounded sequences and for

every n ∈ N

xn + yn+1 ≤ xn+1 + yn ,

and

xn + yn ≥ xn+1 + yn+1 .

Prove that both sequences {xn } and {yn } converge.

8

8. (10 pts) Prove using the definition that the following limit is equal

to −∞ :

x+1

= −∞.

x→2− x2 − 4

lim

9

9. (10 pts) Prove that if a function f is non-negative and continuous

on the interval [1, ∞), and limx→∞ f (x) = 0, then there exists xM ∈

[1, ∞) such that f (xM ) ≥ f (x) for every x ∈ [1, ∞).

10

10. (10 pts) Prove that the function f (x) = x1/3 is uniformly continuous on [1, ∞).

Hint: Use x − y = (x1/3 − y 1/3 )(x2/3 + x1/3 y 1/3 + y 2/3 ).

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Tags:

Continuity

Advanced calculus

numerical sequences

differentiability

Riemann integration

uniform convergence

power series

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