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
Find the average rate of change of g(x)=-5x-1 between the points (-4,19) and (3,-16)
![**Problem Statement:**
Find the average rate of change of the function \( g(x) = -5x - 1 \) between the points \((-4, 19)\) and \( (3, -16) \).
**Solution Explanation:**
The average rate of change of a function between two points is the change in the function's value divided by the change in the independent variable (in this case, \(x\)) over that interval. It is calculated using the formula:
\[
\text{Average Rate of Change} = \frac{g(x_2) - g(x_1)}{x_2 - x_1}
\]
Where:
- \( (x_1, g(x_1)) = (-4, 19) \)
- \( (x_2, g(x_2)) = (3, -16) \)
Plug these values into the formula:
\[
\text{Average Rate of Change} = \frac{-16 - 19}{3 - (-4)}
\]
Calculate:
1. \(-16 - 19 = -35\)
2. \(3 - (-4) = 3 + 4 = 7\)
Thus:
\[
\text{Average Rate of Change} = \frac{-35}{7} = -5
\]
The average rate of change of the function \( g(x) = -5x - 1 \) between the points \((-4, 19)\) and \( (3, -16) \) is \(-5\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F35e690b1-ffbf-4e74-a9e0-925364498c5a%2F4d045ffb-258b-496c-ad6e-033da09ee39c%2F6l889m3_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the average rate of change of the function \( g(x) = -5x - 1 \) between the points \((-4, 19)\) and \( (3, -16) \).
**Solution Explanation:**
The average rate of change of a function between two points is the change in the function's value divided by the change in the independent variable (in this case, \(x\)) over that interval. It is calculated using the formula:
\[
\text{Average Rate of Change} = \frac{g(x_2) - g(x_1)}{x_2 - x_1}
\]
Where:
- \( (x_1, g(x_1)) = (-4, 19) \)
- \( (x_2, g(x_2)) = (3, -16) \)
Plug these values into the formula:
\[
\text{Average Rate of Change} = \frac{-16 - 19}{3 - (-4)}
\]
Calculate:
1. \(-16 - 19 = -35\)
2. \(3 - (-4) = 3 + 4 = 7\)
Thus:
\[
\text{Average Rate of Change} = \frac{-35}{7} = -5
\]
The average rate of change of the function \( g(x) = -5x - 1 \) between the points \((-4, 19)\) and \( (3, -16) \) is \(-5\).
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