This problem illustrates that the derivative of a differentiable function might not even be continuous. Let Ja² sin(1/x), if x # 0 f(x) = if x = 0. For this problem you may assume as known that sin(x) is differentiable on all of R, sin'(x) | sin(x)| < 1 for all x, cos(x) is continuous at all x, and cos(0) = 1. (These, I believe, are the only facts concerning sin(x) you need to use, but if you think you need other facts for your solution, ask me about them.) (a) Use the Theorem about dervatives of sums, products, etc. and the Chain Rule to prove f is differentiable at all x + 0, and find a fomula for f'(x) that is valid for all x # 0. (b) Use the definition of the derivative to prove f(x) is differentiable at 0 and f'(0) = 0. (c) Parts (a) and (b) show that f'(x) is defined for all x. Prove f'(x) is not continuous at x = 0 by showing that lim,-40 f'(x) = 1. cos(x) for all x,

This problem illustrates that the derivative of a differentiable function might not even be continuous. Let Ja² sin(1/x), if x # 0 f(x) = if x = 0. For this problem you may assume as known that sin(x) is differentiable on all of R, sin'(x) | sin(x)| < 1 for all x, cos(x) is continuous at all x, and cos(0) = 1. (These, I believe, are the only facts concerning sin(x) you need to use, but if you think you need other facts for your solution, ask me about them.) (a) Use the Theorem about dervatives of sums, products, etc. and the Chain Rule to prove f is differentiable at all x + 0, and find a fomula for f'(x) that is valid for all x # 0. (b) Use the definition of the derivative to prove f(x) is differentiable at 0 and f'(0) = 0. (c) Parts (a) and (b) show that f'(x) is defined for all x. Prove f'(x) is not continuous at x = 0 by showing that lim,-40 f'(x) = 1. cos(x) for all x,

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

ChapterP: Prerequisites

SectionP.6: Analyzing Graphs Of Functions

Problem 6ECP: Find the average rates of change of f(x)=x2+2x (a) from x1=3 to x2=2 and (b) from x1=2 to x2=0.

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