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The graph (from the US Department of Energy) shows how driving speed affects gas mileage. Fuel economy $ F $ is measured in miles per gallon and speed $ v $ is measured in miles per hour.

(a) What is the meaning of the derivative $ F'(v) $?

(b) Sketch the graph of $ F'(v) $.

(c) At what speed should you drive if you want to save on gas?

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a) $\frac{\text { change in } y}{\text { change in } x}$b) answer is graphc) 55 $\mathrm{mph}$

03:40

Daniel Jaimes

04:39

David Mccaslin

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Missouri State University

Baylor University

Idaho State University

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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Alright so here we have a graph of our um for a vehicle, the MPG, how well we're able to do with respect to how fast the car is moving in MPH so let's fix that, make it green so it looks a little bit better. Alright, MPH. Okay so part A. Is we want to basically know what is the meaning of F. Prime of E. And F. Prime of V. That's basically looking at the derivative of F. With respect to V. So our units will be in notice it'll be MPG per MPH. So basically F. Prime of V. And they will try to ride it out for you is the rate of change, the instantaneous rate of change uh of our gas mileage um F with respect to to um speed um in MPG per MPH. Now the cool thing is with the units is that when you divide two fractions, you can multiply by the reciprocal, that smile. And so you can see that we're really looking at how many hours um you're able to drive per gallon. So F. Prime avi actually gives us hours per gallon in terms of when we kind of work the units out. So that's really cool. Um And so the question is, what's the best um gas mileage Or what speed is the best for the gas mileage? Well um we really want to get the best MPG so we're gonna try to find that max on the graph and it looks like to be about 55 um MPG. So it looks like um if we go well let's write it in this form, whoops okay so it looks like um We can get 55. Um Oh actually let me correct that I had their units. Right okay so we can get the max on the graph But we want the v. Of what speed and this looks like a speed of 55 MPH. So if we go a speed of 55 mph, that is our max MPG. So that's super cool. Um Next thing we want to do is to be able to draw a graph of our derivatives. So I will do that in a sec here. Alright, so we're just gonna do a rough sketch here and remember that the slope of F. With respect to V. Is um basically the slope of the derivative of F. With respect to V. Is the slope of the graph of F versus B. So notice you're like at around 20, we have a positive slope and then it kind of gets somewhat horizontal and then becomes negative. So just super rough sketch, something like positive um Going towards zero slope going towards negative, it has some shape to it but roughly we're gonna basically go from a positive slope to a negative slope. Um notice that um then our hours per gallon um is dependent on our MPH. So that's really cool. So anyway hopefully that helped to have a wonderful day.

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