Math 31L.13
Calculus I with Environmental Emphasis

Fall 1997

Plan for Week 10

We have now finished Chapter 4 on shortcuts for finding derivatives. At the end of this week we take a break from new material to check on proficiency in the basic skills of calculating derivative functions. Instead of having a lab this week, we will have a practice session on those skills and on checking skills -- so you can know you have the right answer without having to consult an external "authority." In brief, the rules for the Gateway test are: You must get seven out of eight right to pass, and you must take the test as many times as necessary to pass. (See General Information for more details.)

We noted that the most important "rule" in Chapter 4 is the Chain Rule. We start this week by seeing how the Chain Rule enables us to solve simple differential equations, such as the ones we have encountered in labs. We know how to represent differential equations graphically via slope fields -- and thereby generate geometric solutions. We know how to use the same idea numerically -- in the form of Euler's method -- and thereby generate numerical solutions. Now we will see how (in some very special cases) to generate symbolic or algebraic solutions. The technique for this is called separation of variables, and the Chain Rule is what makes it work.

At midweek we start on Chapter 5, the theme of which is putting derivatives to work to solve interesting problems. Actually, we have been doing that most of the semester, so it should come as no surprise that a good deal of this will be organizing, summarizing, and practicing things you have already observed. Wednesday's lesson will summarize observations about the relationships between derivatives (first and second) and shapes of curves.

About half of Chapter 5 is relegated to Supplementary Reading, which you will be prepared to do at any time after this week. Our major emphasis in the chapter -- and the primary theme next week (Week 11) -- will be on problems of optimization.

Here is the syllabus for this week:

Week 10 Date Topic Reading Activity
M 11/3 Separation of variables 9.4
W 11/5 Using derivatives 5.1, 5.2
Th 11/6 Review of differentiation Practice for Gateway Test
F 11/7 Gateway Test
Suppl.:
5.3, 5.4,
5.7

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 10. Those papers should include solutions to the problems whose numbers are boldfaced italics in the assignment below. The assignment dates are start dates.
  3. In Section 9.4 you will encounter two things we have not taken up yet (because they're in Chapter 6). The first is the word "integrate" -- which you should take to mean find an antiderivative. The second is an elongated S-shaped symbol -- which means exactly the same thing. This section is about finding antiderivatives. The connections with "integration" and "S" will be made when we get to Chapter 6.
  4. Your instructor will be out of town on Friday, November 7. Angelika Langen will hand out the gateway test, and you may return the completed test to her office, 214 Physics.
  5. Remember to submit your weekly journal entry by e-mail on or before Friday, November 7.

Assignments

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

Last modified: November 1, 1997