- Your 6.1 homework isn't graded yet -- sorry.
- Your 6.2 homework is due Friday.
- We have our second exam not this Friday, but the Friday next (3/27).
- 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
- 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.
- Let's look at the summary section
6.4 for some of the results of this section.
Now I'd like to go over the result following example 16, entitled "The Number e as a Limit". It incorporates some nice elements of limits, including that limit definition.
- Examples:
- #11, p. 418
- #44
- #78