University of Missouri Columbia Advanced Calculus Exercises

Question Description

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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