- Today we're catching up from last time.
- Your assignment over 6.3 won't be due until Wednesday.
- I'm going to consolidate two sections from chapter 9 into tomorrow. Take a look at sections 9.3 and 9.4, reading more for a big picture understanding than for detail. Then we'll look at some of the modelling that we do with logs in class.
- You have a quiz opportunity over sections 9.3 and 9.4 tomorrow. You might be asked about an example treated in those sections.
- We should spend a minute recognizing that we can turn any
exponential or logarithm of base b into another with base
e:
- We also want to realize that the natural log, ln(x), as the inverse of an increasing function, e^x, is also increasing; but, using the mirror, the increase in the exponential function, being extremely rapid, is reflected as an extremely slow growth in the natural logarithm.
- This section features the logarithm taking care of some tricky
cases that we couldn't handle previously, through a process called
"logarithmic differentiation" (which I outline in the highlights for this section).
- We begin with the derivative of the composition of a log with another function:
From there, we realize that
So if the derivative of the log of the function is simple to calculate, then we're in business....
- Let's see this process in action, on an interesting function: the
example 16, p. 417.
- This section also introduces us to a few more ways of thinking about
some complicated functions and their derivatives.
- 13 (in two ways: directly, and using properties of logarithms)
- #47, 51, 53, p. 419
- 75, 78
- 37
- We'll end by looking over this lab: Logs and Exponentials as Models