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Neil Antrassian

on 4 November 2015

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What is the Mean Value Theorem?
The Mean Value Theorem states that if y= f(x) is continuous on [a, b] and differentiable on (a, b), then there is a "c" (at least one point) in (a, b) where f'(c)= (f(b)- f(a)) / (b-a). Or, there is at least one point where the slope of the secant line of the function is the same as the slope of a tangent line of the same function. Or, that at one point on the function, the instantaneous velocity (tangent line, also the velocity at a specific point on the interval) is equal to the average velocity (secant line, the total distance divided by the total time).
A Real Life Application of The Mean Value Theorem
I used The Mean Value Theorem to test the accuracy of my speedometer. If I (my mom) set the cruise control of our car to 70 mph, and I timed how long it took us to travel one mile (mile marker to mile marker), then this information could be used to test the accuracy of our speedometer. I took the average of two one mile trials and used this number for my calculations.
Trial 1: 53.76 seconds
Trial 2: 52.05 seconds
53.76 sec + 52.05 sec = 105.81 sec

105.81 sec / 2 = 52.91 seconds
What Does This Time Mean?
Using the time that it took for me to travel one mile I can calculate my average velocity. We already know that:
avg velocity = total distance / total time

and so my average velocity can be calculated by doing

avg velocity = (1 mile / 52.91 sec) x (3600 sec / 1 hour) = 68.04 mph

My original value was given in miles per second and so I had to multiply it by 3600 sec / 1 hour to get it in miles per hour.
There are chances of error in this experiment. Sources of these errors could be human error in the timing, mile markers that arent placed at exactly one mile, or the needle of the speedometer not being set at exactly 70 mph. My percent error can be calculated by doing:
An Example of The Mean Value Theorem
A trucker travels 163 miles on a toll road with a speed limit of 70 miles per hour. The trucker completes the 163 mile journey in 2 hours. At the end of the toll road the trucker is issued a speeding ticket. Why?

Well, since the truck's position is continuous on the closed interval, differentiable on the open interval, there were no discontinuities on the position graph (the truck passed through every point on the way from toll booth to toll booth, there were no wormholes or teleporters to create a hole. And a cusp in a position graph would not be fun if you were traveling at 70 miles per hour, I would not want to be in that car) the Mean Value Theorem applies. The Mean Value Theorem states that at one point the average velocity of the trucker must be equal to the instantaneous velocity of the trucker. And since
avg velocity= total distance (displacement) / total time

163 miles / 2 hours = 81.5 mph
Then at least once in time while on the toll road, the trucker was going 81.5 miles per hour, well over the speed limit.
How Does The Mean Value Theorem Apply Here?
Since my position function over time in the car was continuous on the closed interval and differentiable on the open interval, a sudden change, cusp, or hole in my interval would be very troublesome (again, there were no discontinuities on the position graph, the car went through every point on the mile and I did not use any wormholes or teleporters in this experiment) then the Mean Value Theorem applies to this situation. And since the Mean Value Theorem states that there is at least one point along the interval where my instantaneous velocity is equal to my average velocity. This means that if my average velocity on the interval was 68. 04 miles per hour, then there is at least one point along the interval where my instantaneous velocity was 68.04 miles per hour.
In conclusion, The Mean Vale Theorem has told me that at least once along my interval I was traveling at 68.04 mph. But, if my cruise control was set at 70, and my velocity never changed, then it can be concluded that there is at least one point where my speedometer was wrong- it told me I was traveling 70 miles per hour when really, I was only traveling 68.04 miles per hour.
= l 70 mph - 68.04 mph / 70 l x 100 =
5.4 % Error
A presentation by Carl Antrassian
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