Subsections

• Purpose
• Getting Started
• Background
• Exercises

MA 1024 Lab 1: Surfaces

Purpose

The purpose of this lab is to introduce you to some of the Maple commands that can be used to plot surfaces in three dimensions.

Getting Started

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

cp ~bfarr/Surfaces_start.mws ~

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 Surfaces_start.mws, and go through it carefully. Then you can start working on the exercises.

Background

The graph of a function of a single real variable is a set of points $(x,f(x))$ in the plane. Typically, the graph of such a function is a curve. For functions of two variables, the graph is a set of points  $(x,y,f(x,y))$ in three-dimensional space. For this reason, visualizing functions of two variables is usually more difficult.

One of the most valuable services provided by computer software such as Maple is that it allows us to produce intricate graphs with a minimum of effort on our part. This becomes especially apparent when it comes to functions of two variables, because there are many more computations required to produce one graph, yet Maple performs all these computations with only a little guidance from the user.

Two common ways of representing the graph of a function of two variables are the surface plot and the contour plot. The first is simply a representation of the graph in three-dimensional space. The second draws the level curves $f(x,y)=C$ for several values of $C$ in the $x,y$ plane. We will explore how to produce these kinds of graphs in Maple, and how to use the graphs to study the functions.

Exercises

1. 1. Generate a surface plot and contour plot for the following function on the given domain.
      
      $$f(x,y) = \frac{x+y}{1+x^2+y^2}$$
      
      
      for $-3\leq x \leq 3$ and $-3 \leq y\leq 3$.

   2. What does the contour plot look like in the regions where the surface plot has a steep incline? What does it look like where the surface plot is almost flat?
   3. What can you say about the surface plot in a region where the contour plot looks like a series of nested circles?

2. A group of oceanographers is mapping the ocean floor to assist in the recovery of a sunken ship. Using sonar, they develop the model
   
   $$D = \frac{-4x}{x^2+y^2+1}\;\;\;\;\;-5 \leq x \leq 5\,,\;\;-5 \leq y \leq 5$$
   
   
   where $D$ is the depth and $x$ and $y$ are the distances in kilometer.
   1. Graph the surface corresponding to the function $D$.

   2. Plot at least $5$ contours of the function $D$ and note the values of $D$ at these contours.

   3. What is the depth of the ship if it is located at the coordinate $x=1, y = 0.5$?

   4. Use contours to approximate the slope of the ocean floor in the positive $x$ direction at the ship's location ($x=1, y = 0.5$)?