Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Determine f(0), where
![The image shows a piecewise function defined as follows:
\[
f(x) =
\begin{cases}
x \sin \left( \frac{1}{x} \right) & \text{if } x \neq 0 \\
0 & \text{if } x = 0
\end{cases}
\]
Explanation:
- The function \( f(x) \) has two different expressions depending on the value of \( x \).
- For all values of \( x \) except zero (\( x \neq 0 \)), the function is given by \( x \sin \left( \frac{1}{x} \right) \). This indicates a dependence on both \( x \) and the sine of the reciprocal of \( x \).
- For \( x = 0 \), the function is simply 0.
This piecewise definition allows the function to be continuous and differentiable around specific points, a common approach in calculus for handling points where typical definitions might not hold.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12714285-7303-4fe0-8e46-6968fffe450d%2Fa35a3e3a-d151-4573-9094-873ab4e2ee5f%2F5hky3j.jpeg&w=3840&q=75)
Transcribed Image Text:The image shows a piecewise function defined as follows:
\[
f(x) =
\begin{cases}
x \sin \left( \frac{1}{x} \right) & \text{if } x \neq 0 \\
0 & \text{if } x = 0
\end{cases}
\]
Explanation:
- The function \( f(x) \) has two different expressions depending on the value of \( x \).
- For all values of \( x \) except zero (\( x \neq 0 \)), the function is given by \( x \sin \left( \frac{1}{x} \right) \). This indicates a dependence on both \( x \) and the sine of the reciprocal of \( x \).
- For \( x = 0 \), the function is simply 0.
This piecewise definition allows the function to be continuous and differentiable around specific points, a common approach in calculus for handling points where typical definitions might not hold.
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