Exercises to Hand in

1.

Examine the function f(x,y)=5+xy-x²-y² and its best linear approximation at (a,b)=(0,1). Create m-files as in the example above. Start with a 2 x 2 box around (0,1) and draw the surfaces and contour plots. Then zoom in until the tangent plane and the surface are close. Examine the contour plots after zooming as well.

• If you were to take a smooth surface, such as the graph of f, and magnify it many times near a point P such as (0,1), what would it eventually look like?
• What does the contour plot of a non-horizontal plane look like? (What about horizontal planes?) What does the contour plot of a smooth surface look like if you magnify (zoom) many times?

(Hand in your plots, answers to the questions, and any other observations you care to make.)

2.

Now consider the function g(x,y)=(x²+y²)^1/2.

• Plot the function g(x,y) on the rectangle $-1\leq x\leq1,\ -1\leq y\leq1$ (centered at (0,0)).
• Plot g(x,y) on the rectangle $-.1\leq x\leq.1,\ ,-1\leq y\leq.1$ (also centered at (0,0)). Has the graph of the function become more ``plane-like''?
• What's going on here? What is the best linear approximation? What about the tangent plane? What does the contour plot of g look like near (0,0)?