1. The speed limit along a stretch of road is 60 mph. A traffic camera located a

Question

1. The speed limit along a stretch of road is 60 mph. A traffic camera located at mile marker 355 records your speed at 55 mph at noon.Another traffic camera located at mile market 377 records your speed at 55 mph at 12:20 pm. A police officer pulls you over and issues you a speeding ticket. you know the officer only has the evidence recorded at the traffic cameras, so you exclaim, "Officer, I was only going 55 miles per hour the whole way!" What theorem does the officer cite that justifies your new speeding ticket? Explain you answer using the theorem, and the appropriate difference quotients and derivatives.

2. If a box with a square base and an open top is to have volume 32000 cubic meters, find the dimensions of the box that minimize the material needed to manufacture it. Height=??? Length=Width=???

3. Find all vertical and horizontal asymptotes of graph of f(x)=3x/sqrt(4x^2 -1)

Vertical Asymptotes=???

Horizontal Asymptotes=???

Explanation / Answer

Average speed was (377-355)/1/3= 66mph. So according to the mean value theorem since you averaged 66mph, you must have been going 66mph at least once during the trip (over the speed limit)

Minimize x^2 +4xy subject to x^2y=32,000

y=32,000/x^2

So mimimize x^2 +4x*32,000/x^2

Take derivative to get 2x -128,000/x^2

Se equal to zero and solve for x to get x= 40

Sub in x^2y=32,000 to get y=20

So lengrh and width =40 height =20.

3x/(4x^2-1)^1/2

Vertical asymptotes are where the denominator=0 so x= +1/2. -1/2

Horizontal are limits as x approaches pos or neg infinity so 3x/2x= 3/2

y=3/2 is horizontal asymptote.

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