1. The function. (a) (10 %) Given x > 0, let U(x) be the average value of e-t on [0,x]. Find all critical numbers of U. 3 2 (b) (10 %) Let V(x) be an anti-derivative of the function + Find all inflection points of V. √1 - x2 1+x2 (c) Suppose that w(x) = x³ if x is rational and w(x) = 0 if x is irrational. i. (10%) Prove or disprove that w(x) is continuous at 0. ii. (10%) Prove or disprove that w(x) is differentiable at 0. (d) (10 %) Let p(x) = 7√√x² + cos. Find all slant asymptotes of p(x). 2x (e) (10%) Let q(x) = 0 if x = 0 and q(x) = x ln |x| if x = 0. Prove or disprove that q(x) has neither a local maximum value nor a local minimum value.

1. The function. (a) (10 %) Given x > 0, let U(x) be the average value of e-t on [0,x]. Find all critical numbers of U. 3 2 (b) (10 %) Let V(x) be an anti-derivative of the function + Find all inflection points of V. √1 - x2 1+x2 (c) Suppose that w(x) = x³ if x is rational and w(x) = 0 if x is irrational. i. (10%) Prove or disprove that w(x) is continuous at 0. ii. (10%) Prove or disprove that w(x) is differentiable at 0. (d) (10 %) Let p(x) = 7√√x² + cos. Find all slant asymptotes of p(x). 2x (e) (10%) Let q(x) = 0 if x = 0 and q(x) = x ln |x| if x = 0. Prove or disprove that q(x) has neither a local maximum value nor a local minimum value.

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter2: Functions

Section2.4: Average Rate Of Change Of A Function

Problem 4.2E: bThe average rate of change of the linear function f(x)=3x+5 between any two points is ________.

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