Introducing Functions of Multiple Variables

To judge from the conventional curriculum, one needs to master functions of one variable before tackling functions of two variables.

The June 29 Derecho

The composite parameter, used to predict derechos:

A graph showing the probability that a meso-scale disturbance will maintain itself:

Once a mesoscale convective system (MCS) develops, a common parameter to analyze is the MCS maintenance parameter. This parameter uses many variables to analyze whether an MCS which has already formed will maintain itself. In the image capture below, the probability was over 90% and indeed this derecho maintained itself all the way to the Atlantic Ocean!

How often do derechos occur?

Here's a relationship shown between 3 input variables (latitude, longitude, time) and an output variable (cloudiness).

How to introduce functions of multiple variables.

Strategy:

Activity: Describe your Landscape

In this activity, each student generates his or her own landscape in two dimensions. Then they describe it using geographical terms.

The landscape is generated with the rfun() operator, which generates a “random” function based on the numerical seed provided. It's easy to have students use their birthday as the seed, e.g., April 10, 1994 is 19940410.

f = rfun(~x + y, seed = 19940410)
plotFun(f(x, y) ~ x + y, x.lim = c(-5, 5), y.lim = c(-5, 5))

plot of chunk April10

After describing the landscape, add a path and ask for a description of the slope walking along the path.

Additional Path-Related Problems

Here are some nice problems from Hughes-Hallett et al., Applied Calculus 4/e (p.360):

In answering Problem 2, students need to think about how the demand for orange juice depends on the price of orange juice and on the price of apple juice. They generally have an easier time thinking about this verbally without reference to the graph.

Some settings

==Two Variables==
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