This problem illustrates that the derivative of a differentiable function might not even be continuous. Let Sæ² sin(1/x), if z # 0 f(x) = 0, if r = 0. For this problem you may assume as known that sin(x) is differentiable on all of R, sin'(x) = cos(x) for all r, | sin(x)| < 1 for all a, cos(r) is continuous at all z, cos(2rn) = 1 for all n e N and cos(2rn + 7/2) = 0 for all n E N. (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 I # 0, and find a fomula for f'(x) that is valid for all æ # 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'(¤) is defined for all r. Prove f'(x) is not continuous at = 0 by showing that lim,-0 f'(x) does not exist. (Added tip: Consider lim,∞ f'(1/(2rn)) and lim, f'(1/(27n + a/2)).
This problem illustrates that the derivative of a differentiable function might not even be continuous. Let Sæ² sin(1/x), if z # 0 f(x) = 0, if r = 0. For this problem you may assume as known that sin(x) is differentiable on all of R, sin'(x) = cos(x) for all r, | sin(x)| < 1 for all a, cos(r) is continuous at all z, cos(2rn) = 1 for all n e N and cos(2rn + 7/2) = 0 for all n E N. (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 I # 0, and find a fomula for f'(x) that is valid for all æ # 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'(¤) is defined for all r. Prove f'(x) is not continuous at = 0 by showing that lim,-0 f'(x) does not exist. (Added tip: Consider lim,∞ f'(1/(2rn)) and lim, f'(1/(27n + a/2)).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Topic: Discontinuity of Derivatives**
This problem demonstrates that the derivative of a differentiable function might not be continuous.
**Function Definition:**
Let
\[
f(x) =
\begin{cases}
x^2 \sin(1/x), & \text{if } x \neq 0 \\
0, & \text{if } x = 0
\end{cases}
\]
**Background Knowledge:**
For this problem, you may assume:
- \(\sin(x)\) is differentiable on all of \(\mathbb{R}\),
- \(\sin'(x) = \cos(x)\) for all \(x\),
- \(|\sin(x)| \leq 1\) for all \(x\),
- \(\cos(x)\) is continuous for all \(x\),
- \(\cos(2 \pi n) = 1\) for all \(n \in \mathbb{N}\),
- \(\cos(2 \pi n + \pi/2) = 0\) for all \(n \in \mathbb{N}\).
**Problems to Solve:**
(a) **Differentiability for \(x \neq 0\):**
Use the Theorem about derivatives of sums, products, etc., and the Chain Rule to prove \(f\) is differentiable at all \(x \neq 0\). Find the formula for \(f'(x)\) that is valid for all \(x \neq 0\).
(b) **Differentiability at 0:**
Use the definition of the derivative to prove \(f(x)\) is differentiable at \(0\) and \(f'(0) = 0\).
(c) **Discontinuity of Derivative:**
Show \(f'(x)\) is defined for all \(x\) but is not continuous at \(x = 0\) by showing that \(\lim_{x \to 0} f'(x)\) does not exist.
**Added Tip:**
Consider \(\lim_{n \to \infty} f'(1/(2\pi n))\) and \(\lim_{n \to \infty} f'(1/(2\pi n + \pi/2))\).
This exercise underscores the intricacies of calculus and the subtle properties of differentiable functions, prompting learners to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba5a3306-2973-4825-a0c7-f3a4b662c857%2Fc2cf7a89-c0ab-4b0d-975a-e0ee0a3d11f5%2F3thztda_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Discontinuity of Derivatives**
This problem demonstrates that the derivative of a differentiable function might not be continuous.
**Function Definition:**
Let
\[
f(x) =
\begin{cases}
x^2 \sin(1/x), & \text{if } x \neq 0 \\
0, & \text{if } x = 0
\end{cases}
\]
**Background Knowledge:**
For this problem, you may assume:
- \(\sin(x)\) is differentiable on all of \(\mathbb{R}\),
- \(\sin'(x) = \cos(x)\) for all \(x\),
- \(|\sin(x)| \leq 1\) for all \(x\),
- \(\cos(x)\) is continuous for all \(x\),
- \(\cos(2 \pi n) = 1\) for all \(n \in \mathbb{N}\),
- \(\cos(2 \pi n + \pi/2) = 0\) for all \(n \in \mathbb{N}\).
**Problems to Solve:**
(a) **Differentiability for \(x \neq 0\):**
Use the Theorem about derivatives of sums, products, etc., and the Chain Rule to prove \(f\) is differentiable at all \(x \neq 0\). Find the formula for \(f'(x)\) that is valid for all \(x \neq 0\).
(b) **Differentiability at 0:**
Use the definition of the derivative to prove \(f(x)\) is differentiable at \(0\) and \(f'(0) = 0\).
(c) **Discontinuity of Derivative:**
Show \(f'(x)\) is defined for all \(x\) but is not continuous at \(x = 0\) by showing that \(\lim_{x \to 0} f'(x)\) does not exist.
**Added Tip:**
Consider \(\lim_{n \to \infty} f'(1/(2\pi n))\) and \(\lim_{n \to \infty} f'(1/(2\pi n + \pi/2))\).
This exercise underscores the intricacies of calculus and the subtle properties of differentiable functions, prompting learners to
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