The purpose of this lab is to acquaint you with techniques for finding and classifying local and global extreme values of functions of two variables.

cp /math/calclab/MA1024/least_squares.mws My_Documents

You can copy the worksheet now, but you should read through the lab
before you load it into Maple. Once you have read to the exercises,
start up Maple, load
the worksheet `least_squares.mws`, and go through it
carefully. Then you can start working on the exercises.

A crucial first step in solving such problems is being able to find and classify local extreme values of a function. What we mean by a function having a local extreme value at a point is that for values of near , for a local maximum and for a local minimum.

In single-variable calculus, we found that we could locate candidates for local extreme values by finding points where the first derivative vanishes. For functions of two dimensions, the condition is that both first order partial derivatives must vanish at a local extreme value candidate point. Such a point is called a stationary point. It is also one of the three types of points called critical points. Note carefully that the condition does not say that a point where the partial derivatives vanish must be a local extreme point. Rather, it says that stationary points are candidates for local extrema. Just as was the case for functions of a single variable, there can be stationary points that are not extrema. For example, the saddle surface has a stationary point at the origin, but it is not a local extremum.

Finding and classifying the local extreme values of a function
requires several steps. First, the partial derivatives must
be computed. Then the stationary points must be solved for, which is not
always a simple task.

Next, one must check for the presence of

If
and
then is a local minimum.

If and then is a local maximum.

If then is a saddle point.

If then no conclusion can be made.

If and then is a local maximum.

If then is a saddle point.

If then no conclusion can be made.

Notice that the function to be minimized is a function of

**Costs**- The costs that Shiela had in her ten years follows the following function:

**A)**- Enter the function.
**B)**- Plot the costs function over 120 months using the regular
`style=line`. The income function fluxuates but follows an over-all linear path. This costs function fluxuates but does not follow an over-all linear path. Discuss a possible reason for this. **C)**- To make a general comparison of normal buisness patterns at Shiela's station, what domain values would you include? Explain your reasoning. Then plot forty evenly-spaced data points in your domain.
**D)**- Enter the least-squares distance equation for the costs.
**E)**- Find a minimum point using all parts of the second-partials test to prove that the point is a minimum. Make sure to include plenty of text to keep your work clear.
**F)**- Using the minimum point that you found, write the line equation that will approximate the costs function.
**G)**- Graph data points of the original costs function along with the line equation that you just found over the full ten-year period. Make sure that your graph is clearly labeled.

**Profits**- For Shiela, to be a success, she not only wants her income to be larger than her costs but she also wants her profits to increase. Graph the difference between the income and the costs and explain whether Shiela is a success or not. Again, make sure your graph is clearly labeled.

2006-02-01