This preview shows pages 1–3. Sign up to view the full content.
(3/23/08)
Section 14.2
Horizontal cross sections of graphs and level curves
Overview:
In the last section we analyzed graphs of functions of two variables by studying their vertical
cross sections. Here we study horizontal cross sections of graphs and the associated level curves of
functions.
Topics:
•
Horizontal cross sections of graphs
•
Level curves
•
Estimating function values from level curves
•
Topographical maps and other contour curves
Horizontal cross sections of graphs
In Section 14.1 we determined the shape of the surface
z
=
x
2
+
y
2
in Figure 1 by studying its vertical
cross sections in planes
y
=
x
and
x
=
c
perpendicular to the
y
 and
x
axes. In the next example we look
at the cross sections of this surface in horizontal planes
z
=
c
perpendicular to the
z
axis.
FIGURE 1
Example 1
Determine the shape of the graph of
z
=
x
2
+
y
2
in Figure 1 by studying its horizontal
cross sections.
Solution
Horizontal planes have the equations
z
=
c
with constant
c
. Consequently, the
horizontal cross sections of the surface
z
=
x
2
+
y
2
are given by the equations,
b
z
=
x
2
+
y
2
z
=
c.
Setting
z
=
c
in the ±rst of these equation yields the equivalent equations for the cross
section,
b
c
=
x
2
+
y
2
z
=
c.
(1)
288
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Section 14.2, Horizontal cross sections of graphs and level curves
p. 289 (3/23/08)
If
c
is positive, then
x
2
+
y
2
=
c
is the circle of radius
√
c
in an
xy
plane and
the cross section
(1)
of the surface is the circle of radius
√
c
in the plane
z
=
c
with its
center at
c
on the
z
axis. If
c
= 0 then the curve
(1)
is the origin of
xyz
space. The cross
section is empty (has no points in it) if
c
is negative. Since the radius of the circular
cross section at
z
=
c
increases as
c >
0 increases and the plane
z
=
c
rises, the surface
has the bowl shape in Figure 2.
s
3
6
9
x
4
y
4
Level curves of
f
(
x,y
) =
x
2
+
y
2
FIGURE 2
FIGURE 3
Level curves
The horizontal cross sections of the surface in Figure 2 are at
z
= 0
,
1
,
2
,
3
,...,
10. The lowest cross section
is the point at the origin. The other cross sections are circles; if we drop them down to the
xy
plane, we
obtain the ten concentric circles in Figure 3. The point and circles are curves on which
f
(
x,y
) =
x
2
+
y
2
is constant. They are called
level curves
or
contour curves
of the function. The function has the
value 0 at the origin and the values 1
,
2
,
3
,...,
10 on the circles in Figure 3. The numbers 3, 6, and 9 on
three of the circles indicate the values of the function on those circles.
To visualize the surface in Figure 2 from the level curves, imagine that the
xy
plane in Figure 3 is
horizontal in
xyz
space, that the innermost circle in Figure 3 is lifted one unit to
z
= 1, the next circle
is lifted two units to
z
= 2, the next circle, labeled “3” is lifted to
z
= 3, and so forth. This gives the
horizontal cross sections of the surface which determine its shape in Figure 3.
It is because the vertical cross sections of the surface
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 09/13/2010 for the course MATH Math 20A taught by Professor Eggers during the Summer '08 term at UCSD.
 Summer '08
 Eggers
 Math

Click to edit the document details