Section 14.2 (3/23/08) Horizontal cross sections of …ashenk/Section14_2.pdfSection 14.2 (3/23/08) Horizontal cross sections of graphs and level curves Overview: In the last section

(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 = x2 + y2 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 + y2 in Figure 1 by studying its horizontalcross sections.

Solution Horizontal planes have the equations z = c with constant c. Consequently, thehorizontal cross sections of the surface z = x2 + y2 are given by the equations,

{

z = x2 + y2

z = c.

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

{

c = x2 + y2

z = c.(1)

288

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

If c is positive, then x2 + y2 = c is the circle of radius√

c in an xy-plane andthe 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 crosssection is empty (has no points in it) if c is negative. Since the radius of the circularcross section at z = c increases as c > 0 increases and the plane z = c rises, the surfacehas the bowl shape in Figure 2. �

3

6

9

x4

y

4

Level curves of f(x, y) = x2 + y2

FIGURE 2 FIGURE 3

Level curvesThe horizontal cross sections of the surface in Figure 2 are at z = 0, 1, 2, 3, . . . , 10. The lowest cross sectionis the point at the origin. The other cross sections are circles; if we drop them down to the xy-plane, weobtain the ten concentric circles in Figure 3. The point and circles are curves on which f(x, y) = x2 + y2

is constant. They are called level curves or contour curves of the function. The function has thevalue 0 at the origin and the values 1, 2, 3, . . . , 10 on the circles in Figure 3. The numbers 3, 6, and 9 onthree 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 ishorizontal in xyz-space, that the innermost circle in Figure 3 is lifted one unit to z = 1, the next circleis lifted two units to z = 2, the next circle, labeled “3” is lifted to z = 3, and so forth. This gives thehorizontal cross sections of the surface which determine its shape in Figure 3.

It is because the vertical cross sections of the surface z = x2 + y2 in Figure 2 are parabolas andits horizontal cross sections are circles, that it is called a circular paraboloid. This is a special typeof 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 thefunction 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 = y2 − x2 in Figure 5 by studying its horizontalcross sections.

FIGURE 5


Solution The horizontal cross sections of the surface z = y2 − x2 are given by the equations,

{

z = y2 − x2

z = c.

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

{

c = y2 − x2

z = c.(2)

For c = 0, the first of equations (2) reads y2 − x2 = 0 and gives y = ±x. Thelevel curve consists of the lines y = x and y = −x in Figure 6.

x

y

{

y2 − x2 = cc = 0

x

y

{

y2 − x2 = cc > 0

x

y

{

y2 − x2 = cc < 0

y = ±x y = ±√

c + x2 x = ±√

−c + y2

FIGURE 6 FIGURE 7 FIGURE 8

For c > 0, the first of equations (2) gives y2 = 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 + y2 and then x = ±√

−c + y2. The levelcurve consists of the two halves of the hyperbola in Figure 8. The level curves forc = 0,±1,±2,±4 and ±6 are shown in Figure 9.


The horizontal cross sections on the surface in Figure 10 are obtained from thelevel curves in a horizontal xy-plane in Figure 9 by raising the level curve y2 − x2 = cfor c = 0, 1, 2, 3, . . . 6 up c units and lowering the level curve for c = −1,−2,−3, . . . ,−6down |c| units. This gives the surface the shape in Figure 10. �

00

3

3

−3−3

x4

y

4

Level curves of

z = y2 − x2

FIGURE 9 FIGURE 10

The surface in Figure 10 is called a hyperbolic parabola because its vertical cross sections areparabolas and its horizontal cross sections are hyperbolas.

Example 3 Figure 11 shows the “boot-shaped” graph of the function h(x, y) = y− 112

y3− 14x2 with

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

x2 6

y

2

6

z = −1

z = 1

z = 5

z = −5

z = 1z = −1

(To be redrawn)

FIGURE 11 FIGURE 12


Solution Imagine that the surface is sitting in a container that is being filled with water. Thebottom edge of the surface is at z = −5 and has the shape of the top level curve inFigure 12. As the water rises to z = −4, z = −3, and z = −2, it reaches the next threecross sections in Figure 12, which have the shape of the next three level curves fromthe top in Figure 11.

At z = −1 the water line is in two parts. One part goes around the “toe” of theboot; the other is on the “instep” of the boot. The corresponding level curve is in thetwo parts labeled z = −1 in Figure 12. The next two curves, at z = 0 and z = 1, arealso 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 curvesLevel curves of a function, as in Figure 13, show where the function has each of the z-values for the givencurves, and we can estimate the function’s values at other points from values on nearby level curves,

40

60

80

x−3 3

y

3

−3

−6 6Level curves of P

FIGURE 13

Example 4 The level curves in Figure 13 are of the function z = P (x, y). What are the approximatevalues 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, soP (0,−1) = 50.(b) The point (−2, 1) on the upper left in Figure 15 is about halfway between thelevel curves where P = 70 and P = 80. We conclude that P (−2, 1) ≈ 75. (Differentestimates give different answers.) �

40

60

80

x−3 3

y

3

−3

−6 6

FIGURE 14


Topographical maps and other contour curvesThe 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 andFigure 16 is a photograph of the mountain. The numbers on the contour curves are theelevation above sea level. What is the approximate elevation of the highest point onthe mountain?

Solution The contour curves drawn with heavy lines in Figure 14 are 2,500 feet apart, and theother 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 arelevel 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 + 4y2 in Figure 17.

x5

y

4

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.


2. Draw and label several level curves of the function B(x, y) = x2y.

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)?

4

3

2

1

x20−10

y

20

FIGURE 18

Exercises 14.2AAnswer provided. OOutline of solution provided. CGraphing calculator or computer required.

CONCEPTS:

1. Draw and label the level curves (a) where f(x, y) = x2 + y2 has the values 1 and 4, (b) where

g(x, y) =√

x2 + y2 has the values 1 and 2, and (c) where h(x, y) = ex2+y2

has the values e

and e4. (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 oflevel curves of z = 10M(x, y)?

2

4 6x

y

FIGURE 19

BASICS:

3.O Draw the level curves y − 12x = c of N(x, y) = y − 1

2x for c = 0,±1, and ±2.

4.O Draw and label the level curves of S(x, y) = y − sin x where it has the values 0,±2,±4.

5.A 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 − 12x2.

7. V (x, y) = 4x + y2

8. A(x, y) = x + 3y.


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

y2

z

2

x2

y

2

FIGURE 20 FIGURE 21

10. What are the values of K(x, y) =1

x2+

1

y2on its eight level curves in Figure 21?

11. Figure 22 shows level curves of the function F (x, y) = Ax + By + C. What are the values of theconstants A, B, and C?

0 −44

x2

y

2

FIGURE 22

EXPLORATION:

12.A Describe the level curve N(x, y) = 1 of N(x, y) =x + 2y

3x + y.

13.A Level curves of G(x, y) = Ay3 − cos(Bx) are in Figure 23. What are the constants A and B?

−2

−1

0

1

2

x1

y

FIGURE 23

14. Match the graphs of (a) z = − 19x3 sin y in Figure 24 (b) z = sin y in Figure 25,

(c) z = − sin x sin y in Figure 26, (d) z = sin y − 19x3 in Figure 27, (e) z = 3e−x/5 sin y in

Figure 28, and (f) z = 12x2 +sin2 y in Figure 29 with the level curves in Figures 30 through 35.


FIGURE 24 FIGURE 25

FIGURE 26 FIGURE 27

FIGURE 28 FIGURE 29


FIGURE 30 FIGURE 31

FIGURE 32 FIGURE 33

FIGURE 34 FIGURE 35


15. (a) Explain why the horizontal cross sections of z = ln(x2+y2) and of z =1

√

x2 + y2are circles.

(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 cantravel 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 itsshores, 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 didit 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.


C 17.O Generate the level curves where p(x, y) = xy+ 14x3 has the values −4, 0, and 4 by solving for y and

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 fory 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 fory and generating the graphs of the resulting functions of x. Use a window with equal scales onthe 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 + y2/3 has the values 1, 2, and 3. Then generate the threecurves in a window with equal scales on the axes and copy and label them.

21. A person’s body-mass index is the number I(w,h) =w

h2, where w is his or her weight, measured

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 foundthat those with a body-mass index of over 29 had twice the risk of death than those whosebody-mass index was less than 19. Suppose a man is 1.5 meters tall and has a body-mass indexof 29. How much weight would he have to lose to reduce his body-mass index to 19?

22. (a) What is the domain of z =ln y

x? (b) Draw some of its level curves.

23.A (a) What is the domain of z = ln(xy)? (b) How are its level curves related to those of thefunction xy?

24. Figure 39 shows the graph of the function f(x, y) =−y

x2 + y2 + 1and Figure 40 shows its level

curves. Show that one of the level curves is a line and the others are circles. (For c 6= 0, setc = 2/k and complete the square.)

FIGURE 39 FIGURE 40

(End of Section 14.2)

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Section 14.2 (3/23/08) Horizontal cross sections of …ashenk/Section14_2.pdfSection 14.2 (3/23/08) Horizontal cross sections of graphs and level curves Overview: In the last section