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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![### Finding the Average Value of a Function
#### Problem Statement:
Find the average value of the function on the given interval.
\[ f(x) = \sqrt{x + 3} \; ; \; [1, 13] \]
#### Solution:
1. **Function and Interval:**
- The function given is \( f(x) = \sqrt{x + 3} \).
- The interval over which we need to find the average value is \([1, 13]\).
2. **Formula for the Average Value of a Continuous Function:**
The average value \( \bar{f} \) of a function \( f \) over the interval \([a, b]\) is given by:
\[
\bar{f} = \frac{1}{b - a} \int_{a}^{b} f(x) \, dx
\]
3. **Calculation Steps:**
- Identify \( a \) and \( b \):
\[ a = 1, \; b = 13 \]
- Calculate \( b - a \):
\[ b - a = 13 - 1 = 12 \]
- Set up the integral for \( f(x) = \sqrt{x + 3} \):
\[
\int_{1}^{13} \sqrt{x + 3} \, dx
\]
- Use substitution to solve the integral:
Let \( u = x + 3 \):
\[
du = dx
\]
Change the limits of integration:
When \( x = 1 \), \( u = 4 \)
When \( x = 13 \), \( u = 16 \)
\[
\int_{1}^{13} \sqrt{x + 3} \, dx = \int_{4}^{16} \sqrt{u} \, du = \int_{4}^{16} u^{1/2} \, du
\]
- Integrate \( u^{1/2} \):
\[
\int u^{1/2} \, du = \frac{2}{3} u^{3/2} \Bigg|_{4}^{16}
\]
\[
\frac{2}{3} \left[ (16)^{3/2} - (4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8558f12-eb43-4f23-8df4-76c708dcf941%2F1140c2f7-6661-4063-9db6-a61acb94efe4%2F7oid83_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Average Value of a Function
#### Problem Statement:
Find the average value of the function on the given interval.
\[ f(x) = \sqrt{x + 3} \; ; \; [1, 13] \]
#### Solution:
1. **Function and Interval:**
- The function given is \( f(x) = \sqrt{x + 3} \).
- The interval over which we need to find the average value is \([1, 13]\).
2. **Formula for the Average Value of a Continuous Function:**
The average value \( \bar{f} \) of a function \( f \) over the interval \([a, b]\) is given by:
\[
\bar{f} = \frac{1}{b - a} \int_{a}^{b} f(x) \, dx
\]
3. **Calculation Steps:**
- Identify \( a \) and \( b \):
\[ a = 1, \; b = 13 \]
- Calculate \( b - a \):
\[ b - a = 13 - 1 = 12 \]
- Set up the integral for \( f(x) = \sqrt{x + 3} \):
\[
\int_{1}^{13} \sqrt{x + 3} \, dx
\]
- Use substitution to solve the integral:
Let \( u = x + 3 \):
\[
du = dx
\]
Change the limits of integration:
When \( x = 1 \), \( u = 4 \)
When \( x = 13 \), \( u = 16 \)
\[
\int_{1}^{13} \sqrt{x + 3} \, dx = \int_{4}^{16} \sqrt{u} \, du = \int_{4}^{16} u^{1/2} \, du
\]
- Integrate \( u^{1/2} \):
\[
\int u^{1/2} \, du = \frac{2}{3} u^{3/2} \Bigg|_{4}^{16}
\]
\[
\frac{2}{3} \left[ (16)^{3/2} - (4
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