Math 31L.13
Calculus I with Environmental Emphasis

Fall 1997

Plan for Week 6

We have seen that we can calculate "instantaneous" rates of change -- to a high degree of accuracy -- by computing difference quotients with very small step sizes. Our computers and calculators can automate this process and do it so fast that they can draw pictures of derivative functions. However, for many functions defined by formulas, it is more efficient to find an exact formula for the derivative. This week we begin the process of finding formulas -- the "short-cuts" in the title of Chapter 4. In particular, we will find formulas for differentiating the simplest and most commonly encountered functions, exponential and polynomial functions.

We interrupt this development in mid-week to focus on the raison d'etre for differential calculus, the concept of differential equation. We have already met two important differential equations,

The first of these represents "natural" population growth or radioactive decay, depending on whether k is positive or negative. The solutions in either case are exponential functions. The second represents limited population growth. We saw in the Week 4 lab that we could generate graphical solutions of this equation, even though we have no idea of their symbolic form yet. This week we will learn how to draw a "picture" of a differential equation -- a picture that contains a lot of information about possible solutions. This picture is called a slope field (in Maple, a direction field).

The idea of slope field is quite simple -- and it's essentially the same idea that we used in the Limited Population Growth lab. At each point (t,P) in the plane, the right-hand side of the differential equation gives us a formula for slope of the (unknown) solution curve. If we draw little line segments with the given slopes at a grid of points on the plane, any solution has to pass through this "field of slopes" in the directions they indicate. As we will see, this makes it easy to do freehand sketches of solutions -- or to test proposed formulas for solutions.

Here is the syllabus for this week:

Week 6 Date Topic Reading Activity
M 10/6 Power Rule 4.1, 4.2 Start Take-Home Test 1
W 10/8 Slope fields 9.1, 9.2 Take-Home Test 1 due
Th 10/9 Lab: Slope Fields
F 10/10 Exponential Rules 4.3

Notes

  1. Your first take-home test will be handed out at the end of class on Monday. It will be due at the start of class on Wednesday.
  2. There will be no instructor's office hours this week. Help Room hours continue as usual.
  3. Friday's class will be led by Professor Stephanie Fitchett.
  4. If you use an HP-48 or TI-86 calculator, you have a handy tool (in addition to Maple) that will draw slope fields. If you use a TI-82, 83, or 85, your instructor can provide a program to draw slope fields.
  5. You are expected to read the relevant section of the text before the class period using it.
  6. Your next homework papers will be turned in on Wednesday, October 15, right after Fall Break. Those papers should include solutions to the problems whose numbers are boldfaced italics in the assignment below. The assignment dates are start dates.
  7. Remember to submit your weekly journal entry by e-mail on or before Friday, October 10.

Assignments

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

Last modified: October 4, 1997