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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![**Problem Statement:**
3. Show that the average rate of change of the linear function \( f(x) = ax + b \) over any interval, say \([x_1, x_2]\), is the constant \( a \).
**Solution Explanation:**
To find the average rate of change of the function \( f(x) = ax + b \) over an interval \([x_1, x_2]\), we use the formula for the average rate of change:
\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]
1. Calculate \( f(x_2) \) and \( f(x_1) \):
\[
f(x_2) = a \cdot x_2 + b
\]
\[
f(x_1) = a \cdot x_1 + b
\]
2. Substitute these into the average rate of change formula:
\[
\frac{(a \cdot x_2 + b) - (a \cdot x_1 + b)}{x_2 - x_1}
\]
3. Simplify the expression:
\[
= \frac{a \cdot x_2 + b - a \cdot x_1 - b}{x_2 - x_1}
= \frac{a \cdot (x_2 - x_1)}{x_2 - x_1}
\]
4. Cancel \( (x_2 - x_1) \) from the numerator and the denominator:
\[
= a
\]
Thus, the average rate of change of the linear function \( f(x) = ax + b \) over the interval \([x_1, x_2]\) is indeed the constant \( a \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb54d9a9-2037-418c-ad8a-976bf23f5d59%2Feba90412-969f-4271-9ef7-52322fd831c7%2Fzlhdrkw.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
3. Show that the average rate of change of the linear function \( f(x) = ax + b \) over any interval, say \([x_1, x_2]\), is the constant \( a \).
**Solution Explanation:**
To find the average rate of change of the function \( f(x) = ax + b \) over an interval \([x_1, x_2]\), we use the formula for the average rate of change:
\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]
1. Calculate \( f(x_2) \) and \( f(x_1) \):
\[
f(x_2) = a \cdot x_2 + b
\]
\[
f(x_1) = a \cdot x_1 + b
\]
2. Substitute these into the average rate of change formula:
\[
\frac{(a \cdot x_2 + b) - (a \cdot x_1 + b)}{x_2 - x_1}
\]
3. Simplify the expression:
\[
= \frac{a \cdot x_2 + b - a \cdot x_1 - b}{x_2 - x_1}
= \frac{a \cdot (x_2 - x_1)}{x_2 - x_1}
\]
4. Cancel \( (x_2 - x_1) \) from the numerator and the denominator:
\[
= a
\]
Thus, the average rate of change of the linear function \( f(x) = ax + b \) over the interval \([x_1, x_2]\) is indeed the constant \( a \).
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