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# Section 14.2 Horizontal cross sections of graphs and level by zed18012

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```									Section 14.2                                                                                       (3/23/08)

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 = x2 + 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 = x2 + 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 = x2 + y 2 are given by the equations,

z = x2 + y 2
z = c.

Setting z = c in the ﬁrst of these equation yields the equivalent equations for the cross
section,

c = x2 + y 2
(1)
z = c.

288
Section 14.2, Horizontal cross sections of graphs and level curves                                         p. 289 (3/23/08)

√
If c is positive, then x2 + 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.

y
4

9
6
3

4      x

Level curves of f (x, y) = x2 + 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) = x2 + 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 z = x2 + y 2 in Figure 2 are parabolas and
its horizontal cross sections are circles, that it is called a circular paraboloid. This is a special type
of elliptical paraboloid, whose horizontal cross sections are ellipses.
p. 290 (3/23/08)                                       Section 14.2, Horizontal cross sections of graphs and level curves

Here is a general deﬁnition of level curves.

Deﬁnition 1 The level curves (contour curves) of z = f (x, y) are the curves in the xy-plane where the
function is constant (Figure 4). They have the equations f (x, y) = c with constants c.

FIGURE 4

Example 2          Determine the shape of the surface z = y 2 − x2 in Figure 5 by studying its horizontal
cross sections.

FIGURE 5
Section 14.2, Horizontal cross sections of graphs and level curves                                 p. 291 (3/23/08)

Solution                The horizontal cross sections of the surface z = y 2 − x2 are given by the equations,

z = y 2 − x2
z = c.

Setting z = c in the ﬁrst equation gives the equivalent equations,

c = y 2 − x2
(2)
z = c.

For c = 0, the ﬁrst of equations (2) reads y 2 − x2 = 0 and gives y = ±x. The
level curve consists of the lines y = x and y = −x in Figure 6.

y 2 − x2 = c                                 y 2 − x2 = c                 y 2 − x2 = c
y                                                y                         y
c=0                                          c>0                          c<0

x                                          x                              x

√
y = ±x                                   y = ± c + x2                   x=±   −c + y 2

FIGURE 6                                     FIGURE 7                      FIGURE 8

√
For c > 0, the ﬁrst of equations (2) gives y 2 = c + x2 and then y = ± c + x2 .
The level curve consists of the two halves of the hyperbola in Figure 7.
For c < 0, we obtain x2 = −c + y 2 and then x = ± −c + y 2 . The level
curve consists of the two halves of the hyperbola in Figure 8. The level curves for
c = 0, ±1, ±2, ±4 and ±6 are shown in Figure 9.
p. 292 (3/23/08)                                             Section 14.2, Horizontal cross sections of graphs and level curves

The horizontal cross sections on the surface in Figure 10 are obtained from the
level curves in a horizontal xy-plane in Figure 9 by raising the level curve y 2 − x2 = c
for c = 0, 1, 2, 3, . . . 6 up c units and lowering the level curve for c = −1, −2, −3, . . . , −6
down |c| units. This gives the surface the shape in Figure 10.

y
4

3

0               0

−3                        −3
4   x

3

Level curves of
z = y 2 − x2

FIGURE 9                                              FIGURE 10

The surface in Figure 10 is called a hyperbolic parabola because its vertical cross sections are
parabolas and its horizontal cross sections are hyperbolas.
Example 3          Figure 11 shows the “boot-shaped” graph of the function h(x, y) = y − 12 y 3 − 4 x2 with
1      1

the vertical cross sections that we found in Section 14.1 and with its horizontal cross
sections at the integer values z = −5, −4, −3, . . . , 4, 5. The corresponding level curves
of h are in Figure 11. Explain how the level curves relate to the graph.

y
z=1
6                      z = −5

z = −1
2

2             6    x

z = −1
z=1
z=5
(To be redrawn)

FIGURE 11                                                FIGURE 12
Section 14.2, Horizontal cross sections of graphs and level curves                                   p. 293 (3/23/08)

Solution              Imagine that the surface is sitting in a container that is being ﬁlled with water. The
bottom edge of the surface is at z = −5 and has the shape of the top level curve in
Figure 12. As the water rises to z = −4, z = −3, and z = −2, it reaches the next three
cross sections in Figure 12, which have the shape of the next three level curves from
the top in Figure 11.
At z = −1 the water line is in two parts. One part goes around the “toe” of the
boot; the other is on the “instep” of the boot. The corresponding level curve is in the
two parts labeled z = −1 in Figure 12. The next two curves, at z = 0 and z = 1, are
also in two parts.
Finally, the highest cross sections, at z = 2, 3, 4, and 5, are only on the “shin”
of the boot. They correspond to the bottom four level curves in Figure 11.

Estimating function values from level curves
Level curves of a function, as in Figure 13, show where the function has each of the z-values for the given
curves, and we can estimate the function’s values at other points from values on nearby level curves,

y

3

80

−6   −3            3   60   6   x
Level curves of P
FIGURE 13                                                                       40
−3

Example 4             The level curves in Figure 13 are of the function z = P (x, y). What are the approximate
values of (a) P (0, −1) and (b) P (−2, 1)?
Solution              (a) The level curves in Figure 14 are where P has the values 40, 50, 60, 70, and 80.
The point (0, −1) on the negative y-axis is on the the level curve where P = 50, so
P (0, −1) = 50.
(b) The point (−2, 1) on the upper left in Figure 15 is about halfway between the
level curves where P = 70 and P = 80. We conclude that P (−2, 1) ≈ 75. (Diﬀerent

y

3

80

−6   −3            3   60   6   x

FIGURE 14                                                                       40
−3
p. 294 (3/23/08)                                                 Section 14.2, Horizontal cross sections of graphs and level curves

Topographical maps and other contour curves
The next example involves a topographical map showing level curves of the elevation on the mountain.
Example 5            Figure 15 shows a topographical map of Mt. Shasta in northern California and
Figure 16 is a photograph of the mountain. The numbers on the contour curves are the
elevation above sea level. What is the approximate elevation of the highest point on
the mountain?
Solution             The contour curves drawn with heavy lines in Figure 14 are 2,500 feet apart, and the
other curves represent elevations 500 feet apart. The summit, therefore, is approxi-
mately 14,000 feet above sea level.

FIGURE 15                                                 FIGURE 16

Weather maps show isotherms, which are level curves of temperature, and isobars, which are
level curves of barometric pressure.

Interactive Examples 14.2
Interactive solutions are on the web page http//www.math.ucsd.edu/˜ashenk/.†
1.        Label the level curves of P (x, y) = x2 + 4y 2 in Figure 17.

y
4

5         x

FIGURE 17

† In
the published text the interactive solutions of these examples will be on an accompanying CD disk which can be run by
any computer browser without using an internet connection.
Section 14.2, Horizontal cross sections of graphs and level curves                                                           p. 295 (3/23/08)

2.         Draw and label several level curves of the function B(x, y) = x2 y.
3.         Figure 18 shows level curves of z = R(x, y). What are the approximate values of (a) R(15, 0)
and (b) R(0, 20)?

y

20                       1
2

3
4

−10                                20               x
FIGURE 18

Exercises 14.2
A                          O                                    C
Answer provided.           Outline of solution provided.        Graphing calculator or computer required.
CONCEPTS:

1.         Draw and label the level curves (a) where f (x, y) = x2 + y 2 has the values 1 and 4, (b) where
2
+y 2
g(x, y) =  x2 + y 2 has the values 1 and 2, and (c) where h(x, y) = ex                                       has the values e
4
and e . (d) How are the three sets of curves related?

2.         Figure 19 shows level curves of a function z = M (x, y). How can you convert it into a sketch of
level curves of z = 10M (x, y)?

y

2

6
x
4

FIGURE 19

BASICS:

3.O                                                            1
Draw the level curves y − 1 x = c of N (x, y) = y − 2 x for c = 0, ±1, and ±2.
2
4.O        Draw and label the level curves of S(x, y) = y − sin x where it has the values 0, ±2, ±4.
A                                                      √
5.         Draw and label the level curves of T (x, y) = 3 x − y where it has the values 0, ±1, and ±2.
Draw and label level curves of the functions in Exercises 6 through 8.
6.A         W (x, y) = y − 1 x2 .
2
7.         V (x, y) = 4x + y 2
8.         A(x, y) = x + 3y.
p. 296 (3/23/08)                                         Section 14.2, Horizontal cross sections of graphs and level curves

9.O       What are the values of L(x, y) = |x| + |y| on its three level curves in Figure 20?

z                                                        y

2
2

2      y                                                2            x

FIGURE 20                                             FIGURE 21
1     1
10.       What are the values of K(x, y) =      + 2 on its eight level curves in Figure 21?
x2   y
11.       Figure 22 shows level curves of the function F (x, y) = Ax + By + C. What are the values of the
constants A, B, and C?
y

2

2            x

4          0           −4
FIGURE 22

EXPLORATION:
x + 2y
12.A      Describe the level curve N (x, y) = 1 of N (x, y) =          .
3x + y
13.A      Level curves of G(x, y) = Ay 3 − cos(Bx) are in Figure 23. What are the constants A and B?

y

0                          2
1

1                x
−1
−2

FIGURE 23

14.       Match the graphs of (a) z = − 9 x3 sin y in Figure 24 (b) z = sin y in Figure 25,
1

(c) z = − sin x sin y in Figure 26, (d) z = sin y − 1 x3 in Figure 27, (e) z = 3e−x/5 sin y in
9
Figure 28, and (f ) z = 1 x2 + sin2 y in Figure 29 with the level curves in Figures 30 through 35.
2
Section 14.2, Horizontal cross sections of graphs and level curves               p. 297 (3/23/08)

FIGURE 24                                   FIGURE 25

FIGURE 26                                   FIGURE 27

FIGURE 28                                   FIGURE 29
p. 298 (3/23/08)               Section 14.2, Horizontal cross sections of graphs and level curves

FIGURE 30                    FIGURE 31

FIGURE 32                    FIGURE 33

FIGURE 34                    FIGURE 35
Section 14.2, Horizontal cross sections of graphs and level curves                                            p. 299 (3/23/08)

1
15.      (a) Explain why the horizontal cross sections of z = ln(x2 +y 2 ) and of z =           are circles.
x2 + y 2
(b) Match the surfaces in Figures 36 and 37 to their equations in part (a). Explain your choices.

FIGURE 36                                               FIGURE 37
16.      Large earthquakes with epicenters near coastlines in an ocean generate tsunami waves which can
travel up to 600 miles per hour in very deep water and can be up to 90 feet high near shores.
Figure 38 shows level curves of T = T (P ), where for points P in the Paciﬁc Ocean and on its
shores, T (P ) is the time it took the tsunami wave created by an earthquake in 1964 to reach P .
The dot in the drawing is the epicenter of the earthquake.(1) Approximately how much longer did
it take the wave to reach New Zealand than to reach Japan?

FIGURE 38

(1)
Adapted from A. Strahler and A. Strahler, Environmental Geoscience, Santa Babara: Hamilton Publishing Co., 1973.
p. 300 (3/23/08)                                        Section 14.2, Horizontal cross sections of graphs and level curves

C
17.O      Generate the level curves where p(x, y) = xy+ 4 x3 has the values −4, 0, and 4 by solving for y and
1

generating the graphs of the resulting functions of x. Use the window −5 ≤ x ≤ 5, −7 ≤ y ≤ 7.
Copy the curves on your paper and label them with the corresponding values of p.
C
18.A      Generate the level curves where q(x, y) = ey − cos(πx) has the values 0, 1, 2, 3, 4 by solving for
y and generating the graphs of the resulting functions of x. Use the window −3 ≤ x ≤ 3, −2 ≤
y ≤ 2. Copy the curves on your paper and label them with the corresponding values of q.
C
19.       Generate the level curves where r(x, y) = ey − ex has the values 0, ±0.5, ±1, ±1.5 by solving for
y and generating the graphs of the resulting functions of x. Use a window with equal scales on
the axes. Copy the curves on your paper and label those where r = 0, ±1.
C
20        Find the values of a such that the curve with parametric equations x = a cos3 t, y = a sin3 t are
the level curves where r(x, y) = x2/3 + y 2/3 has the values 1, 2, and 3. Then generate the three
curves in a window with equal scales on the axes and copy and label them.
w
21.       A person’s body-mass index is the number I(w, h) =        , where w is his or her weight, measured
h2
in kilograms, and h is his or her height, measured in meters. (a) What is your body-mass index?
(A kilogram is 2.2 pounds and a meter is 39.37 inches.) (b) A study of middle-aged men found
that those with a body-mass index of over 29 had twice the risk of death than those whose
body-mass index was less than 19. Suppose a man is 1.5 meters tall and has a body-mass index
of 29. How much weight would he have to lose to reduce his body-mass index to 19?
ln y
22.       (a) What is the domain of z =         ? (b) Draw some of its level curves.
x
23.A      (a) What is the domain of z = ln(xy)? (b) How are its level curves related to those of the
function xy?
−y
24.       Figure 39 shows the graph of the function f (x, y) =             and Figure 40 shows its level
2
x + y2 + 1
curves. Show that one of the level curves is a line and the others are circles. (For c = 0, set
c = 2/k and complete the square.)

FIGURE 39                                           FIGURE 40

(End of Section 14.2)

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