Day 6 in MAT229

Today:

• Announcements:
  • Just a note that there is now a link to section summaries off of the homepage.
  • Your first assignment is due today.

• Today we'll look at some examples from section 6.3, and then move on into section 6.4.

• section 6.3: Logarithmic Functions
  • 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:

    

    We can demonstrate this easily given our theorem about the derivatives of inverses.

  • Another way to define the log function is as the anti-derivative of the "weird power" :

    

    Symmetry help us to understand the absolute value in this result: an antiderivative of the odd function 1/x is an even function ln(|x|).

    This is a very general and beautiful result, which you should think about carefully.

    For those who don't remember:

    • An even function satisfies
    • An odd function satisfies
    We can demonstrate this result using the limit definition of the derivative. Let's show that the derivative of an even function is an odd function.

  • Examples:
    • #7, p. 408 (started this one last time)
    • 10
    • 26
    • 44
    • 67
    • 73

  • 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, ln(x), 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.