This week we return to the central task of Chapter 4: developing formulas ("shortcuts") for calculating derivatives. In fact, we take up the most important -- and least obvious -- rules for generating formulas, the Product Rule and the Chain Rule. We will see that, in combination with the rules we know already, these two rules enable us to generate a derivative formula for almost any combination of functions, if we know formulas for differentiating the individual parts of the combination.
At the end of Week 7, we learned how to solve any differential equation
of the form
dy/dt = -k(y - b)
-- that is, we learned how to find a formula for y(t), which revealed that y must decay exponentially to a stable equilibrium at y = b. Now suppose we have data that appear to follow such a decay pattern -- how can we tell if that's really true? If b = 0, we know from the Radioactive Decay lab that we can plot the data on a semilog graph and look for a straight line. What if b is not 0 and is not readily accessible by observation? An example might be an aggressive pollution control program that is driving down concentrations of volatile organic compounds -- but the natural "background" level of these compounds is unknown. A moment's reflection shows that a semilog plot isn't going to solve this problem. This is the problem we address in lab this week.
At the end of the week, we go back to Chapter 1 to review some properties of trigonometric functions, in preparation for finding their rates of change in Week 9. These functions are the primary building blocks for modeling cyclic (or periodic) phenomena, such as solar energy input to a standing body of water.
Here is the syllabus for this week:
| Week 8 | Date | Topic | Reading | Activity |
| M | 10/20 | Product Rule | 4.4 | |
| W | 10/22 | Chain Rule | 4.5 | |
| Th | 10/23 | Newton's Law of Cooling | 9.5 | Lab: Fitting Exponential Decay |
| F | 10/24 | Trigonometry review | 1.10 | |
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Last modified: October 13, 1997