Day 2 in Mat229

• Any questions about what we went over yesterday?
• A reminder that we'll have a quiz over the Basic Classes of Functions section on Thursday, at the end of the hour. I usually allow 20 minutes for this.
• Also that on Friday we will be in MP306, for our first session with Mathematica.
• Yesterday I mentioned Gilbert Strang's "Highlights of Calculus" Lectures. He's got such a fun, quirky style! You might check them out....

• We checked out a few examples which illustrate our interest and our objectives in calculus. Calculus is about change (often represented by the symbol \(\Delta\)). These examples illustrate various quantities changing over time:
  • Duke Energy Usage
  • Your Professor's Hearing Loss
  • How Noah Lyles Won the Men's 100-Meter Gold by a Fraction
  • Keeling's Data: ppm of CO₂ in the atmosphere and Global Temperatures.
  • Eunice Foote's climate change experiments, in 1856, leading to these log models for temperature increase (from this article).

    In Eunice Foote's two page paper, she says explicitly that "An atmosphere of that gas [\(CO_2\)] would give to our earth a high temperature...."

    She also observed that "The action of the sun's rays was found to be greater in moist than in dry air." In other words, she observed (perhaps the first!) that water is also a greenhouse gas.

    Yet little is known about this phenomenal scientist! You can read more at Happy 200th birthday to Eunice Foote, hidden climate science pioneer

• Then we were working on a "one-pager review" of calculus (or so).

  We'd hit the first two concepts pretty hard, and were ready for the third and fourth:

  • \(\displaystyle{\frac{\Delta y}{\Delta x}}\)
  • \(\displaystyle{\lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}}\)
  • \(\displaystyle{\lim_{h \rightarrow 0} \frac{y(x+h)-y(x)}{h}}\)
  • \(\displaystyle{\int_{a}^{x} y'(t) dt}\)

There were several features:

1. Classes of functions (vocabulary -- know their names!):
   • Polynomials
     • e.g. linear functions

       Note that the various forms of a linear function are given, including my favorite the "point-slope" formula.

     • quadratic and cubic functions. We should recognize and know the shapes of these classes of functions, at any rate.

     • A sub-class of the polynomials is power functions (provided the powers are natural numbers).

       These featured a discussion about even and odd functions. Symmetry is an extraordinarily important idea which is frequently underplayed; I will focus on it (and will get excited whenever you point it out!:).

   • power functions with powers which are not natural numbers are also interesting, e.g. root functions like \[ x^\frac{1}{2} \equiv \sqrt{x} \] (root functions, by the way, often serve as inverses of power functions: the inverse of a power function is a power function!)
   • Rational functions (ratios of polynomials)
   • Transcendental functions (e.g. exponentials, trig functions)
   • Piecewise-defined functions

     One particularly important example is the absolute value function, \(|x|\).

2. Mathematical Models (Here's where it's so important to know the animals in the function zoo....)

3. Transformations of functions (Make sure that you're comfortable with Table 1.7.)

Let's try a few problems from the section:

• Checkpoint 1.9 -- I prefer point-slope form!
• Checkpoint 1.10 (one way to do this problem is using Wolfram Alpha!:)
• Checkpoint 1.11
• Checkpoint 1.12
• Checkpoint 1.14
• Checkpoint 1.15 (offers us the chance to use another useful piece-wise defined function, the greatest integer function, aka the floor function, \(\lfloor{x}\rfloor\)).
• Checkpoint 1.16