Question

As a mathematics professor loads more weight in the back of his Subaru, the mileage goes down. Let $x$ be the amount of weight (in pounds) that he adds, and let $y=f(x)$ be the mileage (in mpg). a. Is $f^{\prime}(x)$ positive or negative? Explain. b. What are the units of $f^{\prime}(x) ?$

Name: Problem 62, Chapter 13: Finite Mathematics and Calculus with Applications
Uploaded: 2020-05-07T21:36:09-08:00
Duration: 1 min 52 s
Channel: Linh Vu
Description: As a mathematics professor loads more weight in the back of his Subaru, the mileage goes down. Let x be the amount of weight (in pounds) that he adds, and let y=f(x) be the mileage (in mpg). a. Is f^'(x) positive or negative? Explain. b. What are the units of f^'(x) ?

   As a mathematics professor loads more weight in the back of his Subaru, the mileage goes down. Let $x$ be the amount of weight (in pounds) that he adds, and let
$y=f(x)$ be the mileage (in mpg).
a. Is $f^{\prime}(x)$ positive or negative? Explain.
b. What are the units of $f^{\prime}(x) ?$

Finite Mathematics and Calculus with Applications

Margaret L. Lial,… 9th Edition

Chapter 13, Problem 62

Instant Answer

Solved by Expert Linh Vu

05/07/2020

Step 1

e., as $x$ increases), the mileage decreases (i.e., $y$ decreases). This means that the function $f(x)$ is decreasing as $x$ increases. Show more…

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As a mathematics professor loads more weight in the back of his Subaru, the mileage goes down. Let $x$ be the amount of weight (in pounds) that he adds, and let $y=f(x)$ be the mileage (in mpg). a. Is $f^{\prime}(x)$ positive or negative? Explain. b. What are the units of $f^{\prime}(x) ?$

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Key Concepts

Instantaneous Rate of Change

The derivative f?(x) represents how the function f(x) changes at a specific point, meaning it measures the instantaneous rate of change. In this context, it shows how the mileage (in miles per gallon) responds to a small change in weight (in pounds) added to the vehicle. Since adding weight decreases mileage, the derivative will be negative, indicating a negative slope of the relationship between weight and mileage.

Units Analysis in Derivatives

The units of the derivative f?(x) are determined by dividing the units of the function output by the units of the input. Here the output, mileage, is measured in miles per gallon (mpg), and the input, weight, is measured in pounds. Consequently, the derivative f?(x) has units of miles per gallon per pound, showing how much the mileage changes for each pound of weight added.

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