ST 552

Calculus on a Computer

 

Assignment 8 – More Max-Min problems

 

1.                  Suppose the second derivative of a function is such that every time x increases by one, the slope increases by one.  Start at x=0 with a slope of -2 and approximate (by hand) the function with pieces of straight line.  What deos your sketch look like?

 

2.                  Must a function with positive second derivative go up?

 

3.                  Consider the function, on the interval [-2, 2].  Find the critical point.  What does the second derivative test say?  Plot the function and explain.

 

4.                  A pyramid consists of 4 isosceles triangles around a square base.  If this is to be cut and folded out of a single square piece of paper, Maximize the volume.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.                  What is the longest pipe that can be carried horizontally around a corner from a five foot wide corridor into an eight foot wide one?

 

6.                  We are told that if 60 grapevines are planted per acre, each vine will produce 120 lbs. of grapes per year, but that each additional vine per acre reduces the yield by 3 lbs.  How many vines should be planted to maximize the yield?

 

7.                  A 12 foot high movie screen starts 5 feet above the floor.  Sitting down, your eyes are approximately 4 feet from the floor.  How far back should you sit to maximize the viewing angle?

 

8.                  On a straight road, the speed of traffic is a function of density.  If is the density of traffic (in cars per mile), suppose speed is given by.  What density maximizes the flow of cars?