Real Analysis Exam 2 Review Problems and Solutions

Main Real Analysis topics: 1) limit of a function, 2) continuity, 3) Intermediate Value Theorem, 4) Extreme Value Theorem, 5) uniform continuity, 6) differentiability, 7) Mean Value Theorem, 8) basics of Riemann integrability. https://amzn.to/3GgFjcc ("Real Analysis", by Russell Gordon)

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(0:00) Introduction
(0:21) Limit of a function (epsilon delta definition)
(4:52) Continuity at a point (epsilon delta definition)
(7:17) Riemann integrable definition
(13:12) Intermediate Value Theorem
(15:33) Extreme Value Theorem
(17:52) Uniform continuity on an interval
(20:46) Uniform Continuity Theorem
(22:11) Mean Value Theorem
(25:46) Definition of the derivative calculation (f(x)=x^3 has f'(x)=3x^2)
(30:11) Chain Rule calculation
(32:10) Set of discontinuities of a monotone function
(33:41) Monotonicity and derivatives
(35:14) Riemann integrability and boundedness
(37:33) Riemann integrability, continuity, and monotonicity
(38:25) Intermediate value property of derivatives (even when they are not continuous)
(41:37) Global extreme values calculation (find critical points and compare function values including at the endpoints of the closed and bounded interval [a,b])
(47:38) epsilon/delta proof of limit of a quadratic function
(56:33) Prove part of the Extreme Value Theorem (a continuous function on a compact set attains its global minimum value). The Bolzano-Weierstrass Theorem is needed for the proof.
(1:04:27) Prove (1+x)^(1/5) is less than 1+x/5 when x is positive (Mean Value Theorem required)
(1:08:56) Prove f is uniformly continuous on R when its derivative is bounded on R
(1:12:59) Prove a constant function is Riemann integrable (definition of Riemann integrability required)

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