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...
Related questions
Question
![**Continuity of Piecewise Functions**
In calculus, it's essential to understand how to make piecewise functions continuous at the junction points.
Consider the following piecewise function \( f \):
\[ f(x) = \begin{cases}
x^2 - c & \text{for } x < 5 \\
3x + 5c & \text{for } x \geq 5
\end{cases} \]
We are tasked with finding the value of \( c \) that ensures \( f \) is continuous across its entire domain.
---
To check for continuity at \( x = 5 \):
1. Ensure the left-hand limit is equal to the right-hand limit at \( x = 5 \).
2. Ensure the function value at \( x = 5 \) matches these limits.
\[ \lim_{{x \to 5^-}} f(x) = \lim_{{x \to 5^+}} f(x) = f(5) \]
Let's compute these limits and function values:
### Left-hand limit as \( x \) approaches 5:
\[ \lim_{{x \to 5^-}} f(x) = \lim_{{x \to 5^-}} (x^2 - c) = 5^2 - c = 25 - c \]
### Right-hand limit as \( x \) approaches 5:
\[ \lim_{{x \to 5^+}} f(x) = \lim_{{x \to 5^+}} (3x + 5c) = 3(5) + 5c = 15 + 5c \]
### Ensure the function is continuous:
Set the left-hand limit equal to the right-hand limit:
\[ 25 - c = 15 + 5c \]
### Solve for \( c \):
\[ 25 - 15 = 5c + c \]
\[ 10 = 6c \]
\[ c = \frac{10}{6} = \frac{5}{3} \]
Therefore, the value of \( c \) that makes the function continuous is \( \frac{5}{3} \).
---
This approach illustrates how you can analytically find the necessary conditions to maintain the continuity of piecewise functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6bcaed0f-2f01-4310-ad37-5698e9f2c4bd%2F3dd39dcf-887a-4a90-bc08-25a40eebdcc7%2Fhv45xkg.jpeg&w=3840&q=75)
Transcribed Image Text:**Continuity of Piecewise Functions**
In calculus, it's essential to understand how to make piecewise functions continuous at the junction points.
Consider the following piecewise function \( f \):
\[ f(x) = \begin{cases}
x^2 - c & \text{for } x < 5 \\
3x + 5c & \text{for } x \geq 5
\end{cases} \]
We are tasked with finding the value of \( c \) that ensures \( f \) is continuous across its entire domain.
---
To check for continuity at \( x = 5 \):
1. Ensure the left-hand limit is equal to the right-hand limit at \( x = 5 \).
2. Ensure the function value at \( x = 5 \) matches these limits.
\[ \lim_{{x \to 5^-}} f(x) = \lim_{{x \to 5^+}} f(x) = f(5) \]
Let's compute these limits and function values:
### Left-hand limit as \( x \) approaches 5:
\[ \lim_{{x \to 5^-}} f(x) = \lim_{{x \to 5^-}} (x^2 - c) = 5^2 - c = 25 - c \]
### Right-hand limit as \( x \) approaches 5:
\[ \lim_{{x \to 5^+}} f(x) = \lim_{{x \to 5^+}} (3x + 5c) = 3(5) + 5c = 15 + 5c \]
### Ensure the function is continuous:
Set the left-hand limit equal to the right-hand limit:
\[ 25 - c = 15 + 5c \]
### Solve for \( c \):
\[ 25 - 15 = 5c + c \]
\[ 10 = 6c \]
\[ c = \frac{10}{6} = \frac{5}{3} \]
Therefore, the value of \( c \) that makes the function continuous is \( \frac{5}{3} \).
---
This approach illustrates how you can analytically find the necessary conditions to maintain the continuity of piecewise functions.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning