Subsections

# MA 1024, Partial derivatives

## Purpose

The purpose of this lab is to acquaint you with using Maple to compute partial derivatives.

## Getting Started

To assist you, there is a worksheet associated with this lab that contains examples. You can copy that worksheet to your home directory with the following command, which must be run in a terminal window like teraterm, not in Maple.

cp /math/calclab/MA1024/Partials.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 Partials.mws, and go through it carefully. Then you can start working on the exercises.

## Background

For a function of a single real variable, the derivative gives information on whether the graph of is increasing or decreasing. Finding where the derivative is zero was important in finding extreme values. For a function of two (or more) variables, the situation is more complicated.

### Partial derivatives

A differentiable function, , of two variables has two partial derivatives: and . As you have learned in class, computing partial derivatives is very much like computing regular derivatives. The main difference is that when you are computing , you must treat the variable as if it was a constant and vice-versa when computing .

The Maple commands for computing partial derivatives are D and diff. The Getting Started worksheet has examples of how to use these commands to compute partial derivatives.

## Exercises

1. Compute the three distinct second order partial derivatives of

using the diff command and then again using the D command.

2. Given the function

do the following:
a)
Plot the function and the plane on the same graph.
b)
Find the first order partial derivative with respect to x.
c)
Find the derivative of in the plane.
d)
Find the two-dimensional equation of the intersection of the function and the plane.
e)
Graph the two-dimensional intersection of the plane.
f)
Does your two-dimensional graph look like the intersection form your tree-dimensional graph?

3. Given:

a)
Find the tangent plane at .
b)
Plot the function and the tangent plane.