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,

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Resuelva mediante la inversa excel x₁ + 2x2 − 3x3 = 9 2x₁ - x₂ + x3 = 0 4x1 x2 + x3 4 x1: x2: x3: -
What is linear property of the mathematical function?
1/3y+1=1/2y-3
Answer is only partly right
(1) Simplify. x-2 (SHOW YOUR WORK)
Evaluate dr. (x + 3)