- Just a note that there is now a link to section summaries off of the homepage:
- Your second assignment is due today.
- No "quiz opportunity" today -- but there's one tomorrow!
- Return of your homework and quiz....
- From the homework: graded 2, 8, 22, plus completeness. Please attempt every problem, even if they're tough! Perhaps you can draw a picture, or write down something you considered. A question is something....
- For the quiz: make sure that you label your graphs. If you put in a particular point, label it. Label your axes.
- Logs are, by definition, the inverses of the exponential functions:
- The laws of logarithms are reflections of the properties of
exponentials:
- The derivation of the derivative of the natural log function is
easy, given our work on inverses:
- Another way to define the log function is as the anti-derivative
of the "weird power"
:
Symmetry help us to understand the absolute value in this result: an antiderivative of the odd function 1/x is an even function ln(|x|).
- Examples:
- #7, p. 408
- 10
- 26
- 44
- 68
- We want to 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".
Let's see this process in action, on an interesting function: the example 16, p. 417.