Figure 1: z =
1
2
p
x2 + y 2
3
c=0
2
1
-3
-2
c=2
c= 3
-1
1
2
c
=
1
3
-1
-2
-3
Figure 2: x2 + y 2 = 9 − c2
1
13.1
27: f (x, y) = arccos(x + y). Domain={(x, y) ∈ R2 ; |x + y| ≤ 1}. Range=[0, π].
38: z =
cone:
1
2
p
x2 + y 2 . The graph of this function is the upper half of a elliptical
z2 =
1 2
(x + y 2 ),
4
z ≥ 0.
See figure 1.
45-48: (a)←→ (48); (b)←→ (47); (c)←→ (45); (d)←→ (46).
p
p
52: The level curves for f (x, y) = 9 − x2 − y 2 are:
9 − x2 − y 2 = c. The
level curves form a family of circles: x2 + y 2 = 9 − c2 with common center (0, 0).
Note that c needs to satisfy |c| ≤ 3. See figure 2.
1