- Your 6.1 homework is due today.
- Your 6.2 homework is due Friday.
- We have our second exam not this Friday, but the Friday next (3/27).
- I thought of you all at least three times over spring break:
- In Wapakoneta, OH, when I realized that I had blundered on that tetrahedral volume calculation (a radical three had been bothering me);
- At my friend Michael's house, when he wondered why you can suck Ketchup back into most bottles; and
- While doing some climate research and coming across the article "Near-term acceleration in the rate of temperature change" -- RATES! -- cited by the article The oceans may be lulling us into a false sense of climate security.
- I can't linger after class: I've got to run to a talk at Miami University, entitled "Why the Struggle to Stop Climate Change Failed and What It Means for Our Future". Pretty dark stuff I study!
The error was in the calculation of the area of an equilateral triangle, with
sidelength a. We used similar triangles (with the 30/60/90 triangle), which has
side lengths 1, 2, and .
So I need to repair the area formula.
- Let's start by thinking about some real-life problems that would
require the use of the inverse of an exponential function.
- From science?
- "How long will it take my uranium-235 to degrade from one kilogram to a half a kilogram?"
- "What's the difference in energy between an earthquake of 8 and an earthquake of 9 on the Richter scale?"
- The oceans have acidified by .11 on the pH scale; by how much has the concentration of hydronium ions?
- From business? "How long will it take my money to double at 5 percent interest, compounded continuously?"
- From science?
- 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"
:
(How does symmetry help us to understand the absolute value in this result? Click on Theorem 2 to find out....)
- Examples:
- #7, p. 408
- 10
- 26
- 44
