- Last time we agreed to push back homework 7.2 -- I hope that you remembered that! It's due next Monday.
- Homework Section 7.1 returned: Graded #12, 44, 55, 59. I've identified a few serious problems:
- Limits of exponential functions (#12)
- The product rule!
- Finding extrema (#44)
- First derivative test
- Second derivative test
- I consulted with our chair about the possibility
of requiring attendance at the Sehnert lecture on
Monday. His comment was this: "I think the value they
would get in attending the lecture is enough to
warrant missing some or all of class."
For that reason I'm going to count attendance at that lecture like a homework. If you've got one to blow, and don't want to attend, that's up to you: on the other hand, if you attend, it will count as a perfect on a homework score. Is that okay?
By the way: your homework is still due on Monday! I'll collect it at the lecture....
- #8, p. 356
- #9-16, p. 357
- #20, p. 357
- #22, p. 357
- #31, p. 357
- #33, p. 357
- #42, p. 358
- Logs are, by definition, the inverses of the exponential functions:
- The laws of logarithms are reflections (in the mirror along
the line y=x) of the properties of exponentials:
Most importantly,
expoentials of sums are products, logs of products are sums. You should memorize all of these properties.
- 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 an anti-derivative
of the "weird power" x-1:
Symmetry help us to understand the absolute value in this result: an antiderivative of the odd function
is an even function.
- Although our laws are expressed for all bases, it's important to
realize that, for calculus, almost the only base that matters is the
base e.
- All exponential functions can be expressed using base e.
- All logarithmic functions can be expressed using ln.
- More Examples:
- 15-18, 20, p. 364
- 45-50, p. 365, and draw!
- 80, 84, 86, 90, 92, 94, p. 365
- 108, p. 366
- 109, p. 366
- Let's take a look at the inverse function lab, which concerns some of our global data, and
the "invertibility of data and the models of data".
- Some related questions:
- What are we looking for in invertible functions?
- Do we expect to see invertibility in data?
- How about models?
- What's the mathematical focus in the following graphic?
- As mentioned last time, if a variable transformed by logs appears
linear, then the original variable was probably growing
exponentially. This is the basis of lab15, concerning global temperatures.
- The lab is based on this analysis:
- Today's lab includes examples of using logs to model some interesting phenomena.
- Included is this