009A Sample Final 2, Problem 9

A plane begins its takeoff at 2:00pm on a 2500-mile flight. After 5.5 hours, the plane arrives at its destination. Give a precise mathematical reason using the mean value theorem to explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour.

Foundations:
Mean Value Theorem
Suppose  $f(x)$  is a function that satisfies the following:
$f(x)$  is continuous on the closed interval  $[a,b].$
$f(x)$  is differentiable on the open interval  $(a,b).$
Then, there is a number  $c$  such that  $a<c<b$  and  $f'(c)={\frac {f(b)-f(a)}{b-a}}.$

Solution:

Step 1:
On average the plane flew
${\frac {2500{\text{ miles}}}{5.5{\text{ hrs}}}}\approx 454.5{\text{ miles/hr}}.$

Step 2:
In order to average this speed, the plane had to go from 0mph, up to full speed, past 454.5mph, and then it had to go back down to 0mph to land.
This means that there will be at least two times where the plane of the speed is 400mph.

Final Answer:
See above.

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