Strategy for presentation:

I thought we would lead the participants through this material, so that there is a common basis for later.

Models should be intended for a purpose. So it matters what is the input and what is the output, since you typically want to transform a particular quantity into another type of quantity.

Snow Tree Cricket (from Chapter 1 of Hughes-Hallett). See Math 135 notes from 25 Jan. 2012

\( y = mx + b \), \( f(x) = mx + b \), \( \int x dx \), \( \sin(x) \), \( \sqrt{x} \), and so on.

Where do parameters get introduced in Stewart? Why would we ever write \( \sin x \)?

Students already know a lot about these: slope and intercept. Emphasize that these are parameters. The parameters specify a particular function out of the whole set of functions.

Other parameterizations (e.g. point-slope, two-points, point-angle) can be converted to slope-intercept.

This is the general purpose modeling function. All simple relationships can and are represented this way.

1. World record swim times versus year. `swim100m.csv`

2. Wage versus education. `cps.csv`

3. Natural gas usage versus temperature. `utilities.csv`

See more details on these examples

- Atmosphere pressure, water pressure
- Population of Mexico (Math 135 27 Jan 2012)
- Exponential models of data Income and housing variables

- Half-life, interest rate, exponential time constant. Which is best depends on what information you have.
- Conversion: Rule of 72.

- Half-lives of meds. How often should you take the pill to keep the drug level within 10% over the long run?
- Activity on doubling and half-lives
- Sums of exponentials. Why is cooling not a simple exponential? Fitting Stan's data

- Log axes — how to interpret. Example of log axes. Activity: Plot intervals on these scale to indicate absolute and relative precision on both log and linear axes, for instance, 10% precision on measurements of 1 and of 100 on a log axis.

- Prices versus log prices in the stock market. Which generates a better simulation of stock variation: a random walk on \( \Delta \) prices or a random walk on \( \Delta \) log prices?

- Squeezing Orange Juice … measuring the cumulative. (Make a video of the scale when squeezing orange juice.)

- Random walks. How does the typical distance scale? Explanation in terms of random vectors being perpendicular. (Later application: Explaining the power law: integration of the square distance compared to integration of the absolute distance)
- Chest circumference versus weight, tree height and thickness. (Later example in multivariate: relationships in the internal combustion engine)

Amplitude, time lag, offset.

- Day length as a function of the day of the year. (For later example, as a function of longitude.) Where does the model break down (near the arctic circle)?
- Tone and period. Model a musical scale. How does the period change from note to note? Model a chord.
- Modeling tide data. Example on Hawaiian tides
- Kepler and the planets. (See
`kepler.Rnw`

)