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# Mean value theorem application

AP.CALC:
FUN‑1 (EU)
,
FUN‑1.B (LO)
,
FUN‑1.B.1 (EK)
Even if a cop never spots you while you are speeding, he can still infer when you must have been speeding... Created by Sal Khan.

## Want to join the conversation?

• So does that mean that if I were to speed, then stop my car and wait a while. (Theoretically ) would I evade the law?
• In a way, yes, the computer/camera system uses a distance and a time. By stopping your car somewhere, you are increasing the time between point A and point B. Therefore lowering your average speed, in the eyes in the computer. Although, in reality, highway patrol would probably ticket you for speeding (via radar), but that's a whole other topic.
• I still don't really see the full purpose of the mean value theorem here because if the speed limit is 55 MILES PER HOUR and the distance between the two points is 80 miles, then wouldn't it be kind of obvious that the person must have exceeded the speed limit if they went from one point to the other in just an hour?
• Exactly. That is the simplified version of what the mean value theorem is exactly telling you.

Is it obvious? Sure.
But it can be mathematically proved using the mean value theorem, and that's why it is of importance.
• I think there is a simpler way to tackle this problem.
Can not we say without bringing the notion of mean value theorem in mind that if the average speed is 80 mph, the maximum speed that the driver achieved during the journey is equal to or more than 80 mph. Therefore the maximum allowed speed limit (55 mph) must have been crossed. And this does not require the knowledge of nature of the function (continuity and differentiability).
Is there mean value theorem underlying in inferring the problem this way?
If so then we are using mean value theorem without knowing it !!
• Yes, you are right in all your affirmations. Knowing that the average speed was 80mph is enough to know that at some point the speed was equal to 80mph, and that is the essence of the mean value theorem, in a very day to day situation.
• Why doesn't Sal do a video on the proof of Mean Value Theorem?
• There are probably several reasons for this. First, Sal is great but he isn't perfect! It takes time to create videos. I'm sure there are many ideas he or others at KA would love to implement but don't have the time to implement. Also, relatedly, KA focuses on the concepts usually taught in any given class. In my own calculus class, the proof of the MVT was not taught nor was it in the book (unless it appeared significantly later). If this is common, then the majority of students would not need a video covering it. As time is limited, Sal probably considers what videos to make based on what would be the most useful to the most people.
• Did we really need the MVT for this, isn't it just common sense? There's no way you can travel 80mph at any given time under 80mph in erm... less than an hour. Ya know?
• Yes, it's just common sense.
But if you want to learn the Mean Value Theorem why not start with an easy example where you'll also be able to understand the intuition of the problem.
• why mr. khan took a gap in x-axis?
• The time axis represent a continuous increase in t values from the moment of observation which you can put at t = 0 or, as has been done in this case, start your observation at a later instant of time such as 1 PM. Since in this example we are interested in the travel time between 1 PM and 2 PM. We are not focusing on where the car was at 8 AM or 11 AM or 12 PM or even PM. The gap here shows those instants of time before the clock struck 1 PM and we started observing the car which was the position S(1).
• why did Sal make that gap/hole on the x-axis.
• Sal made the hole on the x-axis in order to show that the graph is not to scale. When looking at the distance between 1st hour and the 2nd hour on the x-axis, it is not the same length as the distance between the 0th hour and the 1st hour. Adding the gap on the x-axis shows us that part of it was omitted when drawing the graph.
• So if you were given a ticket for this, and you argued that you only went at the average speed for a short period of time, they would be right to argue that you went over the average speed?
(1 vote)
• Correct, they would be right to assume that you went either 80mph for the full hour, or more probably, went both faster and slower than 80 over the course of the trip.