15
DIFFERENTIATION IN
SEVERAL VARIABLES
15.1 Functions of Two or More Variables
(ET Section 14.1)
Preliminary Questions
1.
What is the difference between a horizontal trace and a level curve? How are they related?
SOLUTION
A horizontal trace at height
c
consists of all points
(
x
,
y
,
c
)
such that
f
(
x
,
y
)
=
c
. A level curve is the
curve
f
(
x
,
y
)
=
c
in the
xy
-plane. The horizontal trace is in the
z
=
c
plane. The two curves are related in the sense that
the level curve is the projection of the horizontal trace on the
-plane. The two curves have the same shape but they are
located in parallel planes.
2.
Describe the trace of
f
(
x
,
y
)
=
x
2
−
sin
(
x
3
y
)
in the
xz
-plane.
The intersection of the graph of
f
(
x
,
y
)
=
x
2
−
sin
(
x
3
y
)
with the
-plane is obtained by setting
y
=
0in
the equation
z
=
x
2
−
sin
(
x
3
y
)
. We get the equation
z
=
x
2
−
sin 0
=
x
2
. This is the parabola
z
=
x
2
in the
-plane.
3.
Is it possible for two different level curves of a function to intersect? Explain.
Two different level curves of
f
(
x
,
y
)
are the curves in the
-plane deFned by equations
f
(
x
,
y
)
=
c
1
and
f
(
x
,
y
)
=
c
2
for
c
1
6=
c
2
. If the curves intersect at a point
(
x
0
,
y
0
)
,then
f
(
x
0
,
y
0
)
=
c
1
and
f
(
x
0
,
y
0
)
=
c
2
,wh
ich
implies that
c
1
=
c
2
. Therefore, two different level curves of a function do not intersect.
4.
Describe the contour map of
f
(
x
,
y
)
=
x
with contour interval 1.
The level curves of the function
f
(
x
,
y
)
=
x
are the vertical lines
x
=
c
. Therefore, the contour map of
f
with contour interval 1 consists of vertical lines so that every two adjacent lines are distanced one unit from another.
5.
How will the contour maps of
f
(
x
,
y
)
=
x
and
g
(
x
,
y
)
=
2
x
with contour interval 1 look different?
The level curves of
f
(
x
,
y
)
=
x
are the vertical lines
x
=
c
, and the level curves of
g
(
x
,
y
)
=
2
x
are the
vertical lines 2
x
=
c
or
x
=
c
2
. Therefore, the contour map of
f
(
x
,
y
)
=
x
with contour interval 1 consists of vertical
lines with distance one unit between adjacent lines, whereas in the contour map of
g
(
x
,
y
)
=
2
x
(with contour interval
1) the distance between two adjacent vertical lines is
1
2
.
Exercises
In Exercises 1–4, evaluate the function at the speciFed points.