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Chapter 6 Functions of Several Variables

From a formal point of view, the first 5 chapters of this text have been concerned with functions of one variable. More realistically, we have been looking at functions of several variables all along. If we consider the formula for finding how much is in a bank account in the future, we have the formula:

\begin{equation*} \FutureAmount=\InitialDeposit*\left(1+\frac{\rate}{\ppy}\right)^{\years*\ppy}\text{,} \end{equation*}

where \(\ppy\) is the number of periods per year, indicating how often we compound the interest. From a simple point of view, the future amount is a function of 4 variables, the initial deposit, annual rate, periods per year, and number of years. To consider this as a function of a single variable, we fixed 3 of the 4 variables as constants for a particular problem. In this chapter we want to address the more realistic situation where we treat more than one quantity as a variable at a time. This approach has the added advantage that most real world functions of interest have more than one variable.

Before we look at functions of several variables, we want to create a list of tasks we have learned to accomplish with functions of one variable:

  1. Evaluate the function at a particular point with Excel.

  2. Make a table of values at a series of points with Excel.

  3. Make a reasonable graph from a table of values.

  4. Zoom in on a graph until it looks like a straight line.

  5. Find the slope of the tangent line.

  6. Give a formula for the tangent line at a point.

  7. Identify the places where the tangent line is flat.

  8. Find local extrema for the function.

  9. Find global extrema for the function.

  10. Learn applications of the derivative.

We would like to look at how to extend these tasks to functions of several variables. Most of this chapter simply notes how to modify the rules we learned for functions of a single variable to the multivariable case.