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 6: Determining Continuity of \( r(x) \)**
Determine which values of \( C \) make the function \( r(x) \):
- (a) left-continuous,
- (b) right-continuous,
- (c) continuous.
The piecewise function \( r(x) \) is defined as:
\[
r(x) =
\begin{cases}
x^2 + 3 & \text{for } x < -1 \\
Cx & \text{for } -1 \leq x \leq 2 \\
-x^2 + 2 & \text{for } x > 2
\end{cases}
\]
**Explanation:**
- The first piece \( x^2 + 3 \) defines \( r(x) \) for values of \( x \) less than -1.
- The second piece \( Cx \) defines \( r(x) \) for \( x \) between -1 and 2, inclusive.
- The third piece \(-x^2 + 2\) defines \( r(x) \) for values of \( x \) greater than 2.
To find the values of \( C \) that make the function \( r(x) \) left-continuous, right-continuous, or continuous, one must ensure the function values and limits match at the boundaries \( x = -1 \) and \( x = 2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae8ec10c-0c6e-4043-9bd2-93cef2c15ace%2Fcc06cf2b-34b2-413f-a907-c2c093cc22f2%2Fhn0nlq8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 6: Determining Continuity of \( r(x) \)**
Determine which values of \( C \) make the function \( r(x) \):
- (a) left-continuous,
- (b) right-continuous,
- (c) continuous.
The piecewise function \( r(x) \) is defined as:
\[
r(x) =
\begin{cases}
x^2 + 3 & \text{for } x < -1 \\
Cx & \text{for } -1 \leq x \leq 2 \\
-x^2 + 2 & \text{for } x > 2
\end{cases}
\]
**Explanation:**
- The first piece \( x^2 + 3 \) defines \( r(x) \) for values of \( x \) less than -1.
- The second piece \( Cx \) defines \( r(x) \) for \( x \) between -1 and 2, inclusive.
- The third piece \(-x^2 + 2\) defines \( r(x) \) for values of \( x \) greater than 2.
To find the values of \( C \) that make the function \( r(x) \) left-continuous, right-continuous, or continuous, one must ensure the function values and limits match at the boundaries \( x = -1 \) and \( x = 2 \).
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