Problem Set 1 Calc IV (Math2443-002) Due 6/10/2002

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.