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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5.3 Q7) Hey! I need help with the following calc question, thank you!
![### Finding the Average Value of a Function
In this educational module, we will learn how to find the average value \( g_{\text{ave}} \) of a given function over a specified interval.
#### Example Problem
**Task:** Find the average value \( g_{\text{ave}} \) of the function \( g \) on the given interval.
The function provided is:
\[ g(x) = 5\sqrt[3]{x} \]
The interval given is:
\[ [1, 27] \]
#### Calculation
To find the average value of the function \( g \) over the interval \( [1, 27] \), use the following formula for the average value of a continuous function \( g \) on the interval \([a, b]\):
\[ g_{\text{ave}} = \frac{1}{b - a} \int_{a}^{b} g(x) \, dx \]
Here, \( a = 1 \) and \( b = 27 \).
The function to be integrated is:
\[ g(x) = 5\sqrt[3]{x} \]
Substitute \( g(x) \), \( a \), and \( b \) into the formula:
\[ g_{\text{ave}} = \frac{1}{27 - 1} \int_{1}^{27} 5\sqrt[3]{x} \, dx \]
Simplify the expression:
\[ g_{\text{ave}} = \frac{1}{26} \int_{1}^{27} 5\sqrt[3]{x} \, dx \]
#### Integral Calculation
To solve the integral:
\[ \int_{1}^{27} 5\sqrt[3]{x} \, dx \]
We use the antiderivative of \( 5x^{1/3} \):
\[ \int 5x^{1/3} \, dx = 5 \cdot \frac{3}{4} x^{4/3} = \frac{15}{4} x^{4/3} \]
Evaluate this antiderivative at the bounds 1 and 27:
\[ \left[ \frac{15}{4} x^{4/3} \right]_{1}^{27} \]
Calculate the definite integral:
\[ \frac{15}{4} \left( 27^{4/3} -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccd97de8-e10b-493c-b965-7d9287dd2ac4%2Fae43b927-4931-43ab-82c7-465c2e174ee8%2Fu5d6w5_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Average Value of a Function
In this educational module, we will learn how to find the average value \( g_{\text{ave}} \) of a given function over a specified interval.
#### Example Problem
**Task:** Find the average value \( g_{\text{ave}} \) of the function \( g \) on the given interval.
The function provided is:
\[ g(x) = 5\sqrt[3]{x} \]
The interval given is:
\[ [1, 27] \]
#### Calculation
To find the average value of the function \( g \) over the interval \( [1, 27] \), use the following formula for the average value of a continuous function \( g \) on the interval \([a, b]\):
\[ g_{\text{ave}} = \frac{1}{b - a} \int_{a}^{b} g(x) \, dx \]
Here, \( a = 1 \) and \( b = 27 \).
The function to be integrated is:
\[ g(x) = 5\sqrt[3]{x} \]
Substitute \( g(x) \), \( a \), and \( b \) into the formula:
\[ g_{\text{ave}} = \frac{1}{27 - 1} \int_{1}^{27} 5\sqrt[3]{x} \, dx \]
Simplify the expression:
\[ g_{\text{ave}} = \frac{1}{26} \int_{1}^{27} 5\sqrt[3]{x} \, dx \]
#### Integral Calculation
To solve the integral:
\[ \int_{1}^{27} 5\sqrt[3]{x} \, dx \]
We use the antiderivative of \( 5x^{1/3} \):
\[ \int 5x^{1/3} \, dx = 5 \cdot \frac{3}{4} x^{4/3} = \frac{15}{4} x^{4/3} \]
Evaluate this antiderivative at the bounds 1 and 27:
\[ \left[ \frac{15}{4} x^{4/3} \right]_{1}^{27} \]
Calculate the definite integral:
\[ \frac{15}{4} \left( 27^{4/3} -
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