Each problem is worth 5 points. For full credit indicate clearly how you reached your answer.
1. Find an equation of the plane tangent to f(x,y) = x2 +
y2 at the point (1,2). Have Mathematica produce a graph
of f along with the tangent plane, and make sure you get a nice view of
the point of tangency.
2. Describe (as if you were trying to convey it to someone over a telephone)
the graph of f(x,y)=ln(x2+y2). Be sure to include
an accurate description of the graph's behavior near the origin.
3. Figure out where the local minima of the function f(x,y)=x4+y4-4xy+1
occur, and produce a good graph of the surface which includes the portion
where the minima occur.
4. Consider 
.
If we approach the origin along the x- or y-axis we get a limit of 0, and
in fact approaching the origin along any other straight line gets a limit
of 0 as well. Try approaching along the curve y=x3 and see what
you get. What does this mean about the actual limit? Have Mathematica
produce a good graph of this function, and use it to explain what's going
on here.
5. Graph the function 
,
and figure out where its highest and lowest points occur. Produce a good
graph of the surface which includes all maxima and minima.