Graphics Reference
In-Depth Information
Section 4.4
Consider the curve
C
defined by the equation x + y
2
4.4.1.
+ cos xy = 0.
(a)
Can
C
be parameterized by a function of the form y = f(x) in a neighborhood of
(0,0)?
(b)
Can
C
be parameterized by a function of the form x = g(y) in a neighborhood of
(0,0)?
4.4.2.
The point
p
= (1,2,1) lies on the set
X
defined by
xy
-+=
++-=
4
xz
yz
0
40.
xyz
x
z
Determine which of the variables can be solved for in terms of the other two at
p
.
Section 4.5
Discuss the nature of the critical points of the function f(x,y) = 2x
4
+ y
4
- x
2
- 2y
2
.
4.5.1.
Consider the function f(x,y) = x
2
+ 2xy - 4x + 8y. Find the maxima, minima, and saddle
points of f in the rectangle bounded by the lines x =-5, x = 1, y = 0, and y = 7.
4.5.2.
4.5.3.
Find the extreme value of the function f(x,y,z) = xyz subject to the constraints
111
xyz
c
++=>
0
and x, y, z > 0.
4.5.4.
(a)
Prove Case 1 for Theorem 4.5.13.
(b)
Fill in the details left out of the proof of Case 2 for Theorem 4.5.13.
Section 4.8
4.8.1.
Let
X
be the region of the plane defined by
{
}
X
=
(
)
2
2
xy
,
4
£+£
x
y
16
,
3
y x
-≥
0
,
3
x y
-≥
0
.
If f(x,y) = x, compute the integral Ú
X
f by a change of variables to polar coordinates.
Section 4.9
4.9.1.
If f(x,y) = xy + sin xy,
v
= (-2,1), and
p
= (2,3), find df(
p
)(
v
p
).
Define f :
R
2
Æ
R
by f(x,y) = xy + 3y. Find f
*
(
v
p
), where f
*
:T
p
(
R
2
) Æ T
f(
p
)
(
R
),
v
= (-2,1),
and
p
= (3,5).
4.9.2.
If w=xy dx + y
2
dy, compute dw.
4.9.3.
(a)
(b)
If w=xy dxdy, compute dw.
