Question

Help! please! (:

true or false. If Fancy Mary drive 70 miles in 1 hour, then at some point during that hour she was traveling 70 miles per hour.

Answer 1

Are you familiar with the Mean Value Theorem of Calculus? If so, that'd give you some valuable insight into the solution of the problem you've posted.

Answer 2

How do i incorporate the MVT into my answer?

Answer 3

@mathmale

Answer 4

First, are you familiar with the Mean Value Theorem? If so, think about how you would apply this Theorem to the problem at hand.

Answer 5

Yes I'm familiar with the MVT, but I don't understand how you can apply it to the problem,

Answer 6

@mathmale

Answer 7

share with me what the MVT says, because this info is crucial to solving the problem that you've posted.

Answer 8

MVT: Suppose y=f(x) is continious on a closed interval [a,b] and differentiable on the interval's interior (a,b). Then there is at least on point c in (a,b) which
\[\frac{ f(b)-f(a) }{ b-a } = f'(c)\]

Answer 9

@mathmale

Answer 10

that's better. If Mary started her drive 0 miles from home and ended it 70 miles from home, how far did she drive? Once you have that , divide that result by 1 hour. What do you get? Hint: This is the slope of a SECANT line.

Answer 11

she drove 70 miles from home. and you get 70/60? haha I'm really confused. @mathmale

Answer 12

sorry you're confused. But I asked you to divide 70 miles by 1 hour.

Answer 13

31.2928 m / s ? @mathmale I don't get how this proves the question true or false...

Answer 14

70 miles
------- = 70 mph, right? That's all I asked you for.
1 hour

Is 70 mph equivalent to 70 miles per hour?

Answer 15

ohhh. yes, they're the same... so how does the MVT apply to it? @mathmale

Answer 16

\[\frac{ f(b)-f(a) }{ b-a } = f'(c)\] says that if certain conditions are met (what conditions?), the average rate of change of a function (which in this case is distance traveled) is equal to the derivative (the instantaneous rate of change) at some time c which is in between times a and b.

You've found that the average rate of change here is 70mph; you are asked whether the car is ever traveling at 70 miles per hour, right? So, what's your conclusion?

Answer 17

True, because 70 mph equivalent to 70 miles per hour, and the mean vaule thm. proves it? f'(c) would be the the distance traveled over time? @mathmale

Answer 18

sorry, but that is NOT what I said.

Answer 19

You said previously that you understood the MVT.

What did I say that f '(c) represents?

Answer 20

f'(c) is the rate of change? @mathmale

Answer 21

As before, f '(c) represents the INSTANTANEOUS rate of change of distance with respect to time, and is equivalent to "at some point in time she was traveling at 70 miles per hour." do you know what INSTANTANEOUS means? This word is an essential feature in learning calculus.

Answer 22

A description which may horrify the purists, but I think gets across the idea:

If she averages 70 miles per hour, one of two things must happen:

1) she drives exactly 70 miles per hour for the entire trip (not really practical, but never mind that)
2) she spends some of the time driving less than 70 miles per hour, and some of the time driving more than 70 miles per hour, such that after 1 hour has elapsed, exactly 70 miles have been driven

If 1), then we're done — she spent the whole trip driving 70 mph, so clearly she did drive 70 mph at some point!

If 2), then in order for her speed to go from a number less than 70 mph to a number greater than 70 mph (which it must have done, or the average could not be 70 mph), there must be an instant at which the speed is exactly 70 mph, because the instantaneous speed is a continuous function.

It's analogous to having a continuous function which is positive at x = a, and negative at x = b — you know that somewhere between a and b the function must equal 0 for that to be true.