here is my question: assume you take a car trip and you average 65 mph for the trip. Use the mean value theorem to explain what we can guarantee regarding the reading on your speedometer at least once during the trip.
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"At least once the speed will have been 65mph". As speed is a continous thing, (one can't just go from 50mph to 70mph), you would have thus passed through 65 at least once.
(it should also be noted that speed is the derivative of position, and thus subject to the MVTs rules)
In order to show work for this would this be correct:
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well, a and b are partiular times, f(b), f(a) is position, and f'(c) is your speed at a particular time c.
I'm not fully sure how one would show work for this without given specific values -- my apologies.