The situation

In an oversimplified Mom and Pop store, only two items are being sold: x=#radios sold each day, y=#of tapedecks.

We also pretend that the simple function  calculates the weekly profit in $  as a function of these sales-numbers.

(P(x,y) plays the role of "Elevation" and the location (x,y) of height P(x,y) has an interpretation in terms of sales.

Let's take a look at the surface and its contour map:

Let's look at a few of these profit values:

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P(10,7);

weekly profit goes up $275 if we sell 8 tapedecks daily instead of 7.

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P(10,9);

weekly profit goes up $325 if we sell 9 tapedecks daily instead of 8.

The impact of an extra tape-deck on the weekly profit is not the same! the profit rises more significantly in the second case! This means that the surface rises by more in the second case and must therefore be steeper!

To "examine steepness" is, of course, to "examine slopes"!

Estimating slopes in x- and y-direction using contour maps

Let's look at a section of the graph of the surface near the point x=10, y=7:

Next, we draw the paths in x- and y-direction through the point (10,7) onto the surface:

By looking at the contour map, we can tell in which direction the profit rises more quickly:

Using cross-sectional models  we can calculate these slopes exactly.

Visualizing and calculating Slopes using cross-sections:

We "slice" the surface with a vertical plane (as shown), mark the path in y-direction that goes through the point (10,7) on that plane, and then remove the surface...

Similarly, we "slice" the surface with a vertical plane in y-direction, mark the path in y-direction that goes through the point (10,7) on that plane, and then remove the surface...

Looking at the green and the brown plane we see that the heights of the surface along a path in x-direction or y-direction are given as a functions of only one variable (x or y, resp.), and therefore calculating the slope of a surface in x- or y-direction reduces to a MATH150 problem:

To calculate the slopes at the indicated point, we simply need a formula for the cross-sectional models! (Sec. 9.1)

To get the path in the green plane:

Setting y=7 in P(x,y) gives us the cross-sectional model in x-direction! (x is then the only input variable left!) and finding the slope of that curve at the red dot (i.e. at x=10) is now a MATH150 problem!!

To get the path in the brown plane:

So these slopes are easily calculated! (done in class)

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