Find the values of a and b that make the following function continuous. f(x) { 8x if x < 1 ax² + bif 1 ≤ x ≤ 2 2 if x > 2 7x What is the value of a? (You do not need to enter the value of b.) Enter your answer as a decimal. Round to three decimal places (as needed).
Find the values of a and b that make the following function continuous. f(x) { 8x if x < 1 ax² + bif 1 ≤ x ≤ 2 2 if x > 2 7x What is the value of a? (You do not need to enter the value of b.) Enter your answer as a decimal. Round to three decimal places (as needed).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Finding the Values of \( a \) and \( b \) for Continuity**
To ensure the function \( f(x) \) is continuous, find the values of \( a \) and \( b \) that satisfy the following criteria:
\[ f(x) = \begin{cases}
8x & \text{if } x < 1 \\
ax^2 + b & \text{if } 1 \leq x \leq 2 \\
7x & \text{if } x > 2
\end{cases} \]
### Task
Determine the value of \( a \), rounding to three decimal places, and there is no need to enter the value of \( b \).
### Explanation
The piecewise function consists of three parts, each valid within different intervals:
- **\( 8x \)** for \( x < 1 \)
- **\( ax^2 + b \)** for \( 1 \leq x \leq 2 \)
- **\( 7x \)** for \( x > 2 \)
For the function to be continuous at the boundaries of these intervals (\( x = 1 \) and \( x = 2 \)), the following conditions must be met:
- The limit of \( f(x) \) as \( x \) approaches 1 from the left (\( 8x \)) must equal \( f(1) \).
- The limit of \( f(x) \) as \( x \) approaches 1 from the right (\( ax^2 + b \)) must equal \( f(1) \).
- The limit of \( f(x) \) as \( x \) approaches 2 from the left (\( ax^2 + b \)) must equal \( f(2) \).
- The limit of \( f(x) \) as \( x \) approaches 2 from the right (\( 7x \)) must equal \( f(2) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f609e33-8915-43d7-970a-e560ca4358ea%2F512a36a0-ae54-461d-97f9-34728e432603%2F8ivhbwg_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Values of \( a \) and \( b \) for Continuity**
To ensure the function \( f(x) \) is continuous, find the values of \( a \) and \( b \) that satisfy the following criteria:
\[ f(x) = \begin{cases}
8x & \text{if } x < 1 \\
ax^2 + b & \text{if } 1 \leq x \leq 2 \\
7x & \text{if } x > 2
\end{cases} \]
### Task
Determine the value of \( a \), rounding to three decimal places, and there is no need to enter the value of \( b \).
### Explanation
The piecewise function consists of three parts, each valid within different intervals:
- **\( 8x \)** for \( x < 1 \)
- **\( ax^2 + b \)** for \( 1 \leq x \leq 2 \)
- **\( 7x \)** for \( x > 2 \)
For the function to be continuous at the boundaries of these intervals (\( x = 1 \) and \( x = 2 \)), the following conditions must be met:
- The limit of \( f(x) \) as \( x \) approaches 1 from the left (\( 8x \)) must equal \( f(1) \).
- The limit of \( f(x) \) as \( x \) approaches 1 from the right (\( ax^2 + b \)) must equal \( f(1) \).
- The limit of \( f(x) \) as \( x \) approaches 2 from the left (\( ax^2 + b \)) must equal \( f(2) \).
- The limit of \( f(x) \) as \( x \) approaches 2 from the right (\( 7x \)) must equal \( f(2) \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
