Due:  Wednesday, February 12

I   (This first problem is a modification of an exercise of Professor James Herod, School of Mathematics, Georgia Tech.)

Most distance runners know that on a long, slow run there is an immediate speed up of the heart rate, but that as the run progresses, the heart rate slows again, although not to the resting level. As the run progresses, the heart rate will increase as glyogen is used up and the body turns to fat for calories. Converting fat to a usable energy source requires more oxygen than burning glyogen, resulting in an increased respiration rate and heart rate. We model this phenomenon with the function P (pulse rate) below and discover where this model predicts the maximum level for the pulse rate and the subsequent decrease, followed by a gradual increase. We also compute the maximum pulse rate predicted by the model. This type of behavior varies with individuals and depends, in part, on the level of training the individual has experienced.

Assume that the pulse rate is given by the function P=P(t) in beats-per-minute and that t is in seconds:

(A)  Find the resting pulse rate (when t = 0).

(B)  Find the limiting pulse rate (if one exists) as t increases without bound.

(C)   Graph the function P = P(t) for (i) the first minute,  (ii) the first three minutes, and (iii) the first ten minutes.

(D)   Find the maximum pulse rate and the time at which this occurs.   (Hint:  Set P'(t) = 0 to find critical points.)  Make certain that your answer conforms to what the graph suggests.

(E)    Find the minimum pulse rate that is achieved after the first 150 seconds.  When is this minimum achieved?

II  (This problem is drawn from Hughes-Hallett, Calculus, 3^rd edition, pg. 215 / problem 45.)

Consider the family of functions

(B)   On one set of axes, graph this function for several values of a.  (For example, a=1, 3, 5.)  How does varying the constant a affect the shape of the graph?  Explain your answer using the result you obtained in part (A).

(C)   Letting a = 2, find the inflection points of f.
 

III   Optimization questions have fascinated people for thousands of years.  The ancient Greeks studied a geometric problem that led to the isoperimetric theorem and inequality (in two and three dimensions) of more modern times.   Explain what this problem was and how it was solved in the nineteenth century.