Math 31L.13
Calculus I with Environmental Emphasis

Fall 1997

Plan for Week 11

Our primary focus this week is on problems of optimization, that is, finding the best or worst way to do something. If "something" is quantified as the value of a function, then optimization means finding the input(s) to the function that produce the largest or smallest value. In most cases, this means finding a global maximum or minimum.

We know from last week's work that there is a strategy for reducing the problem of finding a global maximum to a finite number of steps:

The strategy for finding a global minimum is similar. Of course, if there are no critical points, only the last step matters -- and the function is either increasing on its whole domain or decreasing on its whole domain. If the domain has no end points, then only the first two steps matter.

An important aspect of optimization is modeling. The situations in which we want to find a "best" or "worst" seldom come equipped with formulas, so we need to use available information -- data, rates of change, etc. -- to develop model formulas. The end-point step in our optimization strategy makes it especially necessary to pay attention to the domain in which our function reasonable models the situation.

This week's lab builds on previous work with population dynamics to consider a stiuation in which two populations interact in ways in which the size of each affects the growth rate of the other. For example, one species might be food for the other, so the predator species reduces the numbers of the prey species, but also depends on the presence of prey for its own growth. This is another lab that starts with assumptions about growth rates (i.e., differential equations). We will see that Euler's Method works as well for two dependent variables as it does for one. There will also be connections with critical points and max/min problems.

At the end of the week we take a break to review Chapters 4 and 5 in preparation for the second take-home test, to start on November 17.

Here is the syllabus for this week:

Week 11 Date Topic Reading Activity
M 11/10 Optimization 5.5
W 11/12 More optimization 5.6
Th 11/13 Interacting species Lab: Predator-Prey Models
F 11/14 Review Chapters 4, 5

Notes

  1. You are expected to read the relevant section of the text before the class period using it.
  2. Your next homework papers will be turned in on Monday, November 17. Those papers should include solutions to the problems whose numbers are boldfaced italics in the assignment below. The assignment dates are start dates.
  3. You will want to have a copy of the Week 11 homework at hand while working on the take-home test in Week 12. Thus, you should make a copy before you turn the paper in.
  4. Remember to submit your weekly journal entry by e-mail on or before Friday, November 14.

Assignments

David A. Smith <das@math.duke.edu>

Last modified: November 1, 1997