University of Missouri Columbia Advanced Calculus Exercises

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QUIZ 6. ADVANCED CALCULUS 1, FALL 20201. (5 pts) Prove that if a function is uniformly continuous on theinterval (1,2) then there exists a finite limitlimx→1+f(x).2. (5 pts) Which two of the following statements if combined implythatf′(0) = 0? Prove your answer.(A) limx→0f(x)−2x= 0.(B)f(x)≤x3+ 2 for everyx∈[−1,1].(C)f(x)≥2−x3for everyx∈[−1,1].(D)fis continuous at zero.3. (5 pts) (i) Suppose thatfis a function on [−1,1] differentiableat 0. Prove that the sequencexn=n(f(1n)−f(0)), n∈N,has a finitelimit asn→∞.(ii) Is the reverse statement always correct? Namely, if the sequence{xn}from above has a finite limit, does it necessarily follow thatfisdifferentiable at 0?

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QUIZ 6. ADVANCED CALCULUS 1, FALL 2020
1. (5 pts) Prove that if a function is uniformly continuous on the
interval (1, 2) then there exists a finite limit
lim f (x).
x→1+
2. (5 pts) Which two of the following statements if combined imply
that f 0 (0) = 0? Prove your answer.
(A) limx→0 f (x)−2
= 0.
x
(B) f (x) ≤ x3 + 2 for every x ∈ [−1, 1].
(C) f (x) ≥ 2 − x3 for every x ∈ [−1, 1].
(D) f is continuous at zero.
3. (5 pts) (i) Suppose that f is a function on [−1, 1] differentiable
at 0. Prove that the sequence xn = n(f ( n1 ) − f (0)), n ∈ N, has a finite
limit as n → ∞.
(ii) Is the reverse statement always correct? Namely, if the sequence
{xn } from above has a finite limit, does it necessarily follow that f is
differentiable at 0?
1

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Tags:
Advanced calculus

finite limit

limit function

zero limit

function thatfis

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