MATH224: Studio 3 Assignment (Limits and Partial Derivatives)
Created By: Gus Greivel
Last Modified: 9/8/2015
For each of the following limits, (i) use Mathematica to visualize the function in 3-D
near the point (a,b) in the limit and (ii) determine the value of the limit if it exists or
prove that it does not. Please use the approaches suggested in parts (a) and (b) as
well as any other approaches you think might be useful.
1.
3x 3 y 2
5
5
( x,y)®(0,0) 2 x +y
lim
(a) Try evaluating along the lines given by y = mx
(b) Try converting the limit to polar coordinates
(c) What do you conclude about the limit?
2.
2
1-e- x -y
2
2
( x,y)®(0,0) x +y
lim
2
(a) Try evaluating along the lines given by y = mx
(b) Try converting the limit to polar coordinates
(c) What do you conclude about the limit?
3.
x 4 +x 2 y
4
2
( x,y)®(0,0) x -y
lim
(a) Try evaluating along the lines given by y = mx
(b) Try evaluating along the parabolas y = ax 2
(c) What do you conclude about the limit?
Reflection on problems 1-3:
How do the various approaches you were asked to try in these problems relate to the
definition of the limit of a function of two variables? Can you give an example of a
limit that would have different values along the paths defined by y = cx 3 through the
origin? Did the Mathematica graphs you created help inform your analyses of these
limits? How?
The graph of the function f (x, y) =10 + x 4 + y 4 - 4xy produces a simple but interesting
graph. We will study the behavior of this surface using Mathematica as well as its first
and second order partial derivatives.
4. Create a 3-D plot of the graph of this function on the domain -2 £ x £ 2, - 2 £ y £ 2 .
You may use this graph as a reference to questions below.
5. Find the first partial derivatives of f :
fx =
¶f
¶x
and
fyy =
¶2 f
¶y 2
fy =
¶f
. At what points are
¶y
fx = 0 and fy = 0 simultaneously?
6. Find the second partials of f :
fxx =
¶2 f
,
¶x 2
and
fxy =
¶2 f
. Why aren’t
¶y¶x
we asking you to find f yx ?
7. The Hessian matrix for a function f is the square matrix given by
é ¶2 f ¶2 f ù é
ù
ê ¶x 2 ¶x¶y ú ê f xx f xy ú
H =ê 2
=
¶ f
¶2 f ú ê f
f ú
êë ¶y¶x ¶y2 úû ë xy yy û
Find the determinant of H for the given function and evaluate it at each of the points
you found in question 5. What, if any, observations can you make about the points
from question 5 that result in a negative value of the determinant of H? A positive
value?
Note: Not that it is needed for this two-by-two problem, you may want to know how
to get Mathematica to do this.

Define a matrix: The following command will define a two-by-two matrix
whose first column contains “a” and “c.” These commands, of course,
generalize to higher-dimension.
A = {{a,b},{c,d}};

Take the determinant of said matrix:
Det[A]
Reflection on questions 4-7:
Based on the perspective of what you learned about derivatives and graphs in
Calculus I, please comment on any interpretations you may have developed with
respect to the information provided by the first and second partial derivatives of this
function.