Name:
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Math 122 Lab 9: Mathematica
Introduction
In this lab, we again use
Mathematica to help visualize three-dimensional graphs. This time, the focus is
on limits and partial derivatives of functions of two variables. Part of this
work should be done by hand and part using Mathematica. You will want to refer
back to the commands covered in the previous Mathematica lab.
What to Turn In
You have two options. Work in
the space provided on this handout and copy Mathematica graphs in as best you
can, or do all of your work,
including explanations, in Mathematica. I prefer that you type it all in to
Mathematica and save it as before; if you do this, you don’t need to show every
step of your hand work (which you can do here), just the important results.
Problems
The
first main problem we explore is a complicated 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
most people, a density plot is more helpful than a 3-D graph. To do a density
plot, use the exact same format 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.
The
second part of the lab concerns partial derivatives of 
. The first set of problems will focus on the point 
. By hand, compute the values of 
, 
, 
, 
and 
.
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,-10,0). First, explain why we want to view the graph from y = -10 instead of
from y = 10. Then have Mathematica draw the graph. Match the values of and to the graphical
properties (increasing or decreasing, concave up or down) you see.
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. Match the values of and to the graphical
properties (increasing or decreasing, concave up or down) you see.
Do
a similar graphical analysis of the curve at the point (2,2).
A
reasonable (but wrong) guess as to the graphical meaning of is the concavity of
the graph in the plane y = x. To check this out graph the function using and from a viewpoint of
{50,-50,0} Describe what you see. For 
, graph the function using and 
from a viewpoint of
{50,-50,0} Describe what you see. Based on these pictures and the actual values
of and , make a conjecture as to what the mixed partial derivative
represents.