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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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
![**Problem Overview:**
Let \( f \) be the function defined by:
\[
f(x) =
\begin{cases}
\sqrt{x+1} & \text{for } 0 \leq x \leq 3 \\
5 - x & \text{for } 3 < x \leq 5
\end{cases}
\]
**(a)** Is \( f \) continuous at \( x = 3 \)? Explain why or why not.
**(c)** Suppose the function \( g \) is defined by:
\[
g(x) =
\begin{cases}
k\sqrt{x+1} & \text{for } 0 \leq x \leq 3 \\
mx + 2 & \text{for } 3 < x \leq 5
\end{cases}
\]
where \( k \) and \( m \) are constants. If \( g \) is differentiable at \( x = 3 \), what are the values of \( k \) and \( m \)?
---
**Instruction for Use:**
Integrate these exercises into your calculus section to enhance understanding of piecewise functions, continuity, and differentiability at given points. Encourage learners to solve the problems and explain their reasoning clearly. Provide hints, such as examining limits and using derivatives, to aid in comprehension.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc88a8d3c-414f-44f0-8ef5-044fbd549dcd%2F61c37d1c-8932-4778-89da-7841f6af45b2%2Fx2iu9s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Overview:**
Let \( f \) be the function defined by:
\[
f(x) =
\begin{cases}
\sqrt{x+1} & \text{for } 0 \leq x \leq 3 \\
5 - x & \text{for } 3 < x \leq 5
\end{cases}
\]
**(a)** Is \( f \) continuous at \( x = 3 \)? Explain why or why not.
**(c)** Suppose the function \( g \) is defined by:
\[
g(x) =
\begin{cases}
k\sqrt{x+1} & \text{for } 0 \leq x \leq 3 \\
mx + 2 & \text{for } 3 < x \leq 5
\end{cases}
\]
where \( k \) and \( m \) are constants. If \( g \) is differentiable at \( x = 3 \), what are the values of \( k \) and \( m \)?
---
**Instruction for Use:**
Integrate these exercises into your calculus section to enhance understanding of piecewise functions, continuity, and differentiability at given points. Encourage learners to solve the problems and explain their reasoning clearly. Provide hints, such as examining limits and using derivatives, to aid in comprehension.
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