Day 7 in MAT229

Today:

• Announcements:
  • Your first assignment is returned. Randomly graded
    • 8
    • 22
    • 32
    and checked for completeness.

    Your score is out of 8 -- 2 points each. Some comments:

    • 8: "yes" and "no" are not sufficient (and I got both!). How much should each of those be worth?
    • 22:
      1. Could someone please explain the difference between "mass" and "rest mass"?
      2. Algebra, algebra, algebra!
      3. Don't forget the (and then explain how you choose the "proper" one!)
      4. Don't forget to answer "the rest of the question" -- what does this mean? That's really the important part! We want to learn, reflect, think about things....
      5. "Units" can tell you when you don't have the right answer. Use them!

    • 32:
      1. draw both graphs, and , so that you can admire the beautiful symmetry.
      2. Label your graphs!
      3. Make sure that any points that appear on give rise to points that appear on .

    • By the way, we did 48 in class, but most of you couldn't do it (I just glanced at them). Are you taking notes? I actually assume that you'll remember things from lectures, and be able to refer to them!

• Section 6.3: Logarithmic Functions
  • Two more examples:
    • 44, p. 409
    • 67

  • 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 remember that the natural log,, as the inverse of an increasing function, , 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.

• section 6.4: Derivatives of Logarithmic Functions
  • 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.

  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 by looking over this lab, to see how one might use logs to do data analysis: Logs and Exponentials as Models