# University of California Los Angeles Advanced Calculus of One Real Variable Questions

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

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QUIZ 7. ADVANCED CALCULUS 1, FALL 2020
1. (5 pts) Prove that for any x > 0, y > 0
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(x2 + y 2 ) 2 ≥ (x3 + y 3 ) 3 .
Hint: Introduce a new variable z = ( xy )2 .
2. (5 pts) Let f, g be continuous functions on [0, 1], differentiable on
(0, 1) and such that |f 0 (x)| < 2|g 0 (x)| for every x ∈ (0, 1). Which of the following are possible? (A) f (0) = −5, f (1) = 0, g(0) = −4, g(1) = 0 (B) f (0) = −5, f (1) = 0, g(0) = −4, g(1) = −1 (C) f (0) = 5, f (1) = 0, g(0) = −4, g(1) = −2 (D) f (0) = 5, f (1) = 0, g(0) = −4, g(1) = −3 3. (5 pts) Prove that the function 2 f (x) = x + x2 sin if x 6= 0, f (0) = 0 x Prove that f has positive derivative at 0. Is f increasing on any interval (−ε, ε), ε > 0?
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new variable

positive derivative

abstract metric

explore topology and convergence

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