\(MATH\ 257\ Calculus\ III				Week\ of\ June\ 2, \ 2003\)
										The LAST Lab!
Lab 10 :  Max/Min on Surfaces
Name 1
Name 2
Section:

At some point during this lab, make sure you fill out a teacher evaluation form for me!  Yeah.

Partial derivatives on Mathematica

Mathematica has a couple of ways to do partial derivatives.  One way is to use the "partial" symbol (∂_□) □  on the Basic Inputs Pallette.  Example: to find  ∂ (f/∂) x (x^2) y  +  
        Sin[x*y] we can do the following:

(∂_x) ((x^2) y  +  Sin[x*y])

Note: you NEED the parentheses.  Another way to do this same thing would be to use the D[{function},x] command:

D[(x^2) y + Sin[x*y], x]

Question 1: Find the two partial derivatives (with respect to x and with respect to y) of the following functions:
(a) f(x,y)=(x/(e^y))
(b) f(x,y)=((y^3)/((x^3) + 1))

One of the big concepts in Chapter 11 in the book is trying to find maximum and minimum points on a 3D surface.  Example: Suppose we wanted to find the coordinates of the maximum point of the function f(x,y)=-(y^2) +  .2  x*y - (x^2) + x + 10.  You could try graphing it:

Plot3D[-(y^2) +  .2  x*y - (x^2) + x + 10,  {x, -5, 
        5}, {y, -5, 5}]

From the graph it's clear there's a maximum point, but it's hard to say exactly what its coordinates are.  

BUT, any maximum point should have its tangent plane be parallel to the x-y plane, right?  This means that the partial derivatives in the "x direction" and the "y direction" should be zero.  We can use this to find the max point:

Solve[
{(∂_x) (-(y^2) +  .2  x*y - 
                (x^2) + x + 10) == 
          0, 
(∂_y) (-(y^2) +  .2  x*y - 
                (x^2) + x + 10) == 0},  {x, y}]

-(y^2) +  .2  x*y - (x^2) + x + 10 /. %

So it looks like this function has a max at approximately {0.505051, 0.0505051, 10.2525}.

Be warned, though!  Just because a surface has ∂ (f/∂) x = 0 and  ∂ (f/∂) y = 0 at a point doesn't mean it has a max or a min at that point.  It could also be a saddle point.

Question 2: Find the coordinates of all local Max/Min points of the surface f(x,y)=(x^4) + (y^3) - 3 y.  Be sure to state if they are maximum or minimum points.  Double-check yourself by making a plot of the surface. (It might help to use the setting AspectRatio->1 command in your plot.)

Question 3: Do the same thing (find all local max/min points, state which is which, and plot the surface) for the function f(x,y)=(x^2) - (x^4) - (y^4).  (Again, AspectRatio->1 might help.)

Question 4: Do the same thing (find all local max/min points, state which is which, and plot the surface) for the function f(x,y)=\(\(15 \( 
                  x\^2\) \(y\^2\) e\^\(\(-x\^2\) - y\^2\)\)\/\(x\^2 + y\^2\)\). (Here you might want to play with the PlotPoints and PlotRange settings.)

Now that you're done, 
	(1) 	Clean up your work by deleting everything that's not needed.  
		There should only be the title, your names, section number, and the questions and your answers
		(with any needed explanations, graphs, etc.).
	(2) 	Save this to your disk.
	(3)	Go back to Blackboard and upload your final lab report to the Digital Drop Box, located
		in the User Tools area.  Remember that you have to first ADD your file to your digital drop box and 
		then SEND it to me!