Math 206A Quiz 04 page 1 Monday 10/17/2010 Name
1. Let −→
F:U⊆R2→R2be defined by −→
F((x, y)) = (px2−y, x +y+ 3) (where we’ve agreed to drop the “vector hats”
except over the name of the function −→
F).
1A. Sketch the domain of −→
F.
1B. Note that −→
Fis a vector field. Draw this much of the field, starting at the following three vectors/points: (1,−3),
(−1,0), and the origin. Put your sketch next to your answer to 1A, above.
2. In the space below, and on the same set of axes, sketch the level curves of f(x, y) = xy, for C=−2 and C= 1. Label
which curve is which.
3. Suppose the level curves for C=L−,C=L, and C=L+for some function z=f(x, y) are as in the following
figure.
3A. As (x, y) approaches (a, b), the limit of f(x, y) appears
to be L. Estimate to a tenth of a centimeter the biggest
possible δfor which 0 <||(x, y)−(a, b)|| < δ implies
|f(x)−L|< .
DRAW that collection of (x, y)’s on the graph.
Your value for δis?
3B. Explain why lim
(x,y)→(p,q)f(x, y) cannot exist.