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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![**Problem 10:**
Determine if the function \( y(x) \) is differentiable at \( x = 0 \).
\[
y(x) =
\begin{cases}
4 - x, & x \leq 0 \\
3x, & x > 0
\end{cases}
\]
To determine if the function is differentiable at \( x = 0 \), we need to check if the function is continuous at \( x = 0 \) and whether the left-hand and right-hand derivatives at \( x = 0 \) are equal.
**Steps:**
1. **Continuity at \( x = 0 \):**
- Find \( \lim_{x \to 0^-} y(x) \).
- Find \( \lim_{x \to 0^+} y(x) \).
- Check if the above limits are equal and if they equal \( y(0) \).
2. **Differentiability:**
- Compute the derivative of \( y(x) \) for \( x \leq 0 \) and \( x > 0 \).
- Compute \( \lim_{x \to 0^-} y'(x) \) and \( \lim_{x \to 0^+} y'(x) \).
- Check if these limits are equal.
If both the function is continuous and the left-hand and right-hand derivatives are equal at \( x = 0 \), then \( y(x) \) is differentiable at \( x = 0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F883933e7-6864-477d-8bb2-712659e12b95%2Fc90dc482-cb5f-4587-82ed-dcb1e6dbe22e%2F2zvtxmi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 10:**
Determine if the function \( y(x) \) is differentiable at \( x = 0 \).
\[
y(x) =
\begin{cases}
4 - x, & x \leq 0 \\
3x, & x > 0
\end{cases}
\]
To determine if the function is differentiable at \( x = 0 \), we need to check if the function is continuous at \( x = 0 \) and whether the left-hand and right-hand derivatives at \( x = 0 \) are equal.
**Steps:**
1. **Continuity at \( x = 0 \):**
- Find \( \lim_{x \to 0^-} y(x) \).
- Find \( \lim_{x \to 0^+} y(x) \).
- Check if the above limits are equal and if they equal \( y(0) \).
2. **Differentiability:**
- Compute the derivative of \( y(x) \) for \( x \leq 0 \) and \( x > 0 \).
- Compute \( \lim_{x \to 0^-} y'(x) \) and \( \lim_{x \to 0^+} y'(x) \).
- Check if these limits are equal.
If both the function is continuous and the left-hand and right-hand derivatives are equal at \( x = 0 \), then \( y(x) \) is differentiable at \( x = 0 \).
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