Name:
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Math 122 Lab 8: Mathematica
Introduction
In this lab, we again use Mathematica to help visualize three-dimensional graphs.
This time, the focus is on limits of functions of two variables. You will want
to refer back to the commands covered in the previous Mathematica
lab.
What to Turn In
Do all of your work,
including explanations, in Mathematica and save it as
before in the Y: drive (use a slightly different name this time).
Problems
The
first problem is to explore a limit. In particular, 
does not exist because
(you do not need to show this) the limit along any straight line path is 0, but
the limit along 
is 1. To start
exploring this, graph 
with 
and 
. In a sentence or two, try to describe the behavior
near the origin.
You
can get a better graph using more points. Use the PlotPoints
command with 100 points and try to describe the behavior near the origin.
For
many people, a density plot is more helpful than a 3-D graph. To do a density
plot, use the exact same syntax as before except replace Plot3D with DensityPlot. Again, try to describe the behavior near the
origin. Note: white = large value, black = small value.
Do
a density plot with 100 points. Mentally sketch in the line y = x.
Starting at the right-hand corner of the graphing box, describe how the
function values increase or decrease as you move toward the origin. Explain why
the limit along this line equals 0.
Locate
the curve on the density plot,
and explain why the limit along this curve equals 1.
Locate
the curve 
on the density plot, and discuss the limit along this curve.
Mathematica
also does contour plots. The syntax is similar, but replace
DensityPlot with ContourPlot.
Try this command. Then add the option ContourShading→False. Using whichever graphic
you like better, use a contour plot to explain why the limit along y = x
equals 0 but the limit along equals 1.
Use
what you have learned to graphically determine whether each limit exists or
not.
(a) 
(b) 
The
second part of the lab gives you a preview of section 10.3 on partial
derivatives.
First,
graph 
with 
and 
. In a sentence or two, describe what this graph is doing
around (2,1).
Next,
zoom in to 
and 
. Describe what the graph is doing around (2,1).
Next,
change the domain to and 
so that the “front”
edge of the graph is the slice of the surface with y = 1. Describe this curve
as increasing or decreasing and as concave up or down at the point (2,1).
A
better picture uses and 
and a ViewPoint of {0,-50,0). First,
explain why we want to view the graph from y = -50 instead of from y = 50. Then
have Mathematica draw the graph.
Next,
change the domain to 
and (default viewpoint) so
that the “right” edge of the graph is the slice of the surface with x = 2.
Describe this curve as increasing or decreasing and as concave up or down at
the point (2,1).
A
better picture uses 
and and a ViewPoint of {50,0,0). First,
explain why we want to view the graph from x = 50 instead of from x = -50. Then
have Mathematica draw the graph.
Do
a similar graphical analysis of the curve at the point (2,2).
When
we cover section 10.3, we will learn how to compute partial derivatives to
verify these graphical properties.