MATH 370 UCR Value Theorem and Squeeze Theorem Questionnaire

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Real Analysis，please check the file that I uploaded，My grade is 87.5% now，so I need 90%+ support tomorrow to get an A in this class，please check the file that I u’puploaded，It was our midterm，It should be very similar. Please make sure you are familiar with Real Analysis, It should be easy and finished around 1-1.5hour ，thanks.

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Math 370 Midterm Exam
March 18, 2021
There are four questions on the exam. Each question is worth ten points.
1. Do TWO of the following:
a) What does it mean for a sequence to be Cauchy?
b) State the Bolzano-Weierstrass Theorem.
c) What does it mean for a sequence {xn } to converge to a real number a?
d) State the Squeeze Theorem for sequences.
2. Are the following true or false? Justify your answers briefly.
a) If xn > 0 for all n ∈ N , and xn → x, then x > 0.
b) If {xn } is a sequence such that {x2n } converges, then {xn } converges.
c) If {xn } is Cauchy, then it is increasing.
d) For any positive number r, {rn } converges.
3. Suppose xn → a, and {yn } is a sequence with the property that |xn −yn | ≤
for all n ∈ N . Show that yn → a.
1
1
n
4. Let
f (x) =
3 sin x
.
x2 + 1
Show, using the definition, that limx→0 f (x) = 0. (HINT: x2 + 1 ≥ 1 and
| sin x| ≤ |x| for all real numbers x.)
2
Final Exam, Math 370
There are five questions on the exam. Each question is worth ten points.
1. Do TWO of the following:
a) State the Mean Value Theorem.
b) What does it mean for a function f to be integrable on [a, b]?
c) What does it mean for a function f : R → R to be uniformly continuous?
d) State the Intermediate Value Theorem.
2. Are the following true or false? Justify your answers briefly.
a) Let f be continuous on [0, 1]. Then
R1
0
f (x)dx ≥ 0.
b) Let f, g : (0, 1) → R be such that f (x) < g(x) for all x ∈ (0, 1). If limx→0+ f (x) = L, and limx→0+ g(x) = M , then L < M . c) Let f be continuous on (0, 1). If {xn } in (0, 1) is a Cauchy sequence, then {f (xn )} is a Cauchy sequence. d) If f is bounded on [0, 1], then f is integrable on [0, 1]. 3. a) Define what it means for a sequence {an } to converge to 0. b) Show carefully that if an → 0, then a2n → 0. (If you are using theorems from the book here, please state them.) 4. a) Define what it means for limx→0+ f (x) = 0. Now let 1 . f (x) = x cos x b) Show, using an epsilon-delta proof, that limx→0+ f (x) = 0. c) Conclude that f is uniformly continuous on (0, 1). 5. Let f : [0, 2] → R be a differentiable function such that f (0) = f (2) = 0, and |f 0 (x)| ≤ 1 for all x ∈ [0, 2]. a) Use the Mean Value Theorem to prove that |f (t)| ≤ t for all t ∈ [0, 1]. b) Use a similar argument to prove that |f (t)| ≤ 2 − t for all t ∈ [1, 2]. c) Conclude that Z 2 0 f (t)dt ≤ 1. Purchase answer to see full attachment Tags: value theorem Continuous function Squeeze Theorem differentiable function uniformly continuous User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.