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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?

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

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So in this problem we're giving this graph of F. From the U. S. Department of Energy which is the fuel economy in MPG versus the speed or velocity in MPH. And were asked a what is the meaning of the derivative F. Prime? Uh huh. The, okay so the derivative is the rate of change. Have crime is the rate of change of F. Which is the fuel economy they call no with respect two our speed. And if we think about that, right, what are the units here? Well the units are going to be miles per gallon over MPH, which means this is hours per gallon. Okay so the next hard says to sketch the graph of F. Prime of you. Okay? So if I start here, F. Prime at you and I kind of line this up with the with the graph up here. Okay so this graph is positive, Positive, positive but it's approaching zero, isn't it? And when I get out here right about 55 MPG MPH. I hit zero. Hit the maximum, so I'm at zero there. Okay, So this is the end MPH. Okay? So at this is 50 I know that. Right about here, I'm at zero and until then I was fairly positive and coming down. What is it that? Okay? So it means that I was coming down like something like this. All right, then, what happens? Well, then I keep going negative and they keep getting more and more negative, don't I? From there on. Okay, So I need to extend this axis on down negative and can extend this going on down like that. So something like that. Okay. And then the last question says at what speed should you drive if you want to save on gas? So speed to save on gas. Well, let's think about this to save on gas means we want to get the most MPG, so we want to because we want to consume the least amount of gas. So I want to go as many miles as we can on a gallon of gas. So this is the max on this curve max of F, which happens at 55 mph. So there we go. The speed to save on gas is the maximum on this curve, Which happens there. And so that's it, like 55 miles Ferreira. There we go.

DM
Oklahoma State University

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