Day 8 in MAT229

Today:

• Announcements:
  • Your assignment over 6.2 is due today.
  • Your assignment over 6.3 will be due Tuesday.
  • No class Monday, of course -- have a good Labor day!
  • At the end of class today we'll have a visit from our STEM Ambassador, Lauren Hensley. She'll be giving you some info on extra opportunities for support.

• section 6.4: Derivatives of Logarithmic Functions
  • Let's start with an ordinary function, and see how properties of logs can help us differentiate. Consider
    

  • 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 and somewhat nasty function

    

    What is its derivative? Mathematica can help...

  • Let's look at the summary section 6.4 to highlight some of the results of this section.

  • Now I'd like to go over the result following example 16 (p. 417), entitled "The Number e as a Limit". It incorporates some nice elements of limits, including that limit definition.

  This section also introduces us to a few more ways of thinking about some complicated functions and their derivatives, and we'll look at some others.

  Some example problems:

  • 13, p. 418
  • 37
  • 47

  We'll end our present study of logs by looking over this lab, to see how one might use logs to do data analysis: Logs and Exponentials as Models

• section 6.6: now we begin to study the inverse trig functions.
  • The first place to start is the geometric perspective, perhaps.

    As usual, we're focused on the undoing property: if the question for us begins as "What is sine of 30 degrees?", the undoing question is "Whose sine is ?"

    We should immediately observe that the original question has a unique solution, whereas the "undoing question" does not. Because sine is not one-to-one, we shouldn't be surprised. But this is an issue: how do we create a useful "inverse" of a non-invertible function?

  • Well, we need to worry in particular about three cases: sine, cosine, and tangent (the others -- secant, etc. -- I don't really worry about too much, but they do sometimes come into play):
    
    1. Two of these inverse functions are shown in their entirety; the other isn't. Which is which?
    2. What features of a graph would have helped us to identify these more easily?

  • The graphical definitions of these functions are easy, carried out via the mirror. But what are the properties of these inverses?

    For example, what are their derivatives? These we've carried out before, using the amazing rule for inverse functions -- do you remember it?

  • Let's look at a few problems involving these curious inverse functions:
    1. 1, p. 459
    2. 6
    3. 14
    4. 18, p. 460
    5. 22
    6. 48
    7. 49