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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Question
Plz explain the answer in decimal form.
![**Problem Statement:**
Find the average value of the function \( f(x) = e^{0.2x} \) on the interval \( 0 \leq x \leq 3 \). Express your answer in exact form.
**Explanation:**
This problem involves calculating the average value of an exponential function over a specified interval. The function given is \( f(x) = e^{0.2x} \), and we are interested in finding its average value between \( x = 0 \) and \( x = 3 \). The notation \( 0 \leq x \leq 3 \) defines the closed interval on the number line.
To find the average value of a continuous function \( f(x) \) over the interval \([a, b]\), we use the formula:
\[
\text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx
\]
In this case, \( a = 0 \) and \( b = 3 \).
**Steps for Solution:**
1. Integrate \( f(x) = e^{0.2x} \) with respect to \( x \) from 0 to 3.
2. Divide the result by the length of the interval \((b-a)\), which is \(3 - 0 = 3\).
3. Simplify the expression to find the average value in its exact form.
This problem does not contain graphs or diagrams, so no visual explanation is necessary.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08bb2a94-da17-43f2-90ab-4ad444df37b3%2F585dea92-511f-424c-a7c4-134736507be6%2F0elgo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the average value of the function \( f(x) = e^{0.2x} \) on the interval \( 0 \leq x \leq 3 \). Express your answer in exact form.
**Explanation:**
This problem involves calculating the average value of an exponential function over a specified interval. The function given is \( f(x) = e^{0.2x} \), and we are interested in finding its average value between \( x = 0 \) and \( x = 3 \). The notation \( 0 \leq x \leq 3 \) defines the closed interval on the number line.
To find the average value of a continuous function \( f(x) \) over the interval \([a, b]\), we use the formula:
\[
\text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx
\]
In this case, \( a = 0 \) and \( b = 3 \).
**Steps for Solution:**
1. Integrate \( f(x) = e^{0.2x} \) with respect to \( x \) from 0 to 3.
2. Divide the result by the length of the interval \((b-a)\), which is \(3 - 0 = 3\).
3. Simplify the expression to find the average value in its exact form.
This problem does not contain graphs or diagrams, so no visual explanation is necessary.
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