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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Question
![### Continuity of Piecewise Functions
**Problem Statement:**
Find the values of \( K \) that make each function continuous over the given interval.
**Function \( f(x) \):**
\[
f(x) =
\begin{cases}
x + 3 & \text{if} \, 0 \leq x \leq 4 \\
K/x & \text{if} \, 4 < x \leq 8
\end{cases}
\]
**Solution:**
To ensure the continuity of \( f(x) \) at \( x = 4 \), the left-hand limit (from \( 0 \leq x \leq 4 \)) and the right-hand limit (from \( 4 < x \leq 8 \)) must be equal at \( x = 4 \).
Evaluating the left-hand limit at \( x = 4 \):
\[
\lim_{x \to 4^-} f(x) = 4 + 3 = 7
\]
Evaluating the right-hand limit at \( x = 4 \):
\[
\lim_{x \to 4^+} f(x) = \frac{K}{4}
\]
Set the left-hand limit equal to the right-hand limit to find \( K \):
\[
7 = \frac{K}{4}
\]
Solving for \( K \):
\[
K = 7 \times 4 = 28
\]
**Conclusion:**
The value of \( K \) that makes the function continuous is \( K = 28 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2133091-a753-411c-beef-444bf7f4574e%2Fb213f24e-7bed-40f8-a108-ace7738c048d%2Fbpca6cd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Continuity of Piecewise Functions
**Problem Statement:**
Find the values of \( K \) that make each function continuous over the given interval.
**Function \( f(x) \):**
\[
f(x) =
\begin{cases}
x + 3 & \text{if} \, 0 \leq x \leq 4 \\
K/x & \text{if} \, 4 < x \leq 8
\end{cases}
\]
**Solution:**
To ensure the continuity of \( f(x) \) at \( x = 4 \), the left-hand limit (from \( 0 \leq x \leq 4 \)) and the right-hand limit (from \( 4 < x \leq 8 \)) must be equal at \( x = 4 \).
Evaluating the left-hand limit at \( x = 4 \):
\[
\lim_{x \to 4^-} f(x) = 4 + 3 = 7
\]
Evaluating the right-hand limit at \( x = 4 \):
\[
\lim_{x \to 4^+} f(x) = \frac{K}{4}
\]
Set the left-hand limit equal to the right-hand limit to find \( K \):
\[
7 = \frac{K}{4}
\]
Solving for \( K \):
\[
K = 7 \times 4 = 28
\]
**Conclusion:**
The value of \( K \) that makes the function continuous is \( K = 28 \).
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