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Hughes-Hallett, Calculus: Single and Multivariable, 5/e
MATH 124/ 129/ 223 5th ed
Chapter 12. Functions of Several Variables
Reading content
12.1 Functions of Two Variables
12.2 Graphs of Functions of Two Variables
12.3 Contour Diagrams
12.4 Linear Functions
12.5 Functions of Three Variables
12.6 Limits and Continuity
12.3 Contour Diagrams
The surface which represents a function of two variables often gives a good idea of the function's general behavior
—for example, whether it is increasing or decreasing as one of the variables increases. However it is difficult to
read numerical values off a surface and it can be hard to see all of the function's behavior from a surface. Thus,
functions of two variables are often represented by contour diagrams like the weather map. Contour diagrams
have the additional advantage that they can be extended to functions of three variables.
Chapter Summary
Review Exercises and Problems for Chapter Twelve
Topographical Maps
Check Your Understanding
Projects for Chapter Twelve
Student Solutions Manual
Graphing Calculator Manual
One of the most common examples of a contour diagram is a topographical map like that shown in Figure 12.33.
It gives the elevation in the region and is a good way of getting an overall picture of the terrain: where the
mountains are, where the flat areas are. Such topographical maps are frequently colored green at the lower
elevations and brown, red, or white at the higher elevations.
Focus on Theory
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Figure 12.33 A topographical map showing the region around South Hamilton, NY
The curves on a topographical map that separate lower elevations from higher elevations are called contour lines
because they outline the contour or shape of the land.3 Because every point along the same contour has the same
elevation, contour lines are also called level curves or level sets. The more closely spaced the contours, the steeper
the terrain; the more widely spaced the contours, the flatter the terrain (provided, of course, that the elevation
between contours varies by a constant amount). Certain features have distinctive characteristics. A mountain peak
is typically surrounded by contour lines like those in Figure 12.34. A pass in a range of mountains may have
contours that look like Figure 12.35. A long valley has parallel contour lines indicating the rising elevations on
both sides of the valley (see Figure 12.36); a long ridge of mountains has the same type of contour lines, only the
elevations decrease on both sides of the ridge. Notice that the elevation numbers on the contour lines are as
important as the curves themselves. We usually draw contours for equally spaced values of z.
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Figure 12.34 Mountain peak
Figure 12.35 Pass between two mountains
Figure 12.36 Long valley
Notice that two contours corresponding to different elevations cannot cross each other as shown in Figure 12.37.
If they did, the point of intersection of the two curves would have two different elevations, which is impossible
(assuming the terrain has no overhangs).
Figure 12.37 Impossible contour lines
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Corn Production
Contour maps can display information about a function of two variables without reference to a surface. Consider
the effect of weather conditions on US corn production. Figure 12.38 gives corn production C = f (R, T) as a
function of the total rainfall, R, in inches, and average temperature, T, in degrees Fahrenheit, during the growing
season.4 At the present time, R = 15 inches and T = 76°F. Production is measured as a percentage of the present
production; thus, the contour through R = 15, T = 76, has value 100, that is, C = f (15, 76) = 100.
Figure 12.38 Corn production, C, as a function of rainfall and temperature
Example 1
Use Figure 12.38 to estimate f (18, 78) and f (12, 76) and interpret in terms of corn production.
Solution
The point with R-coordinate 18 and T-coordinate 78 is on the contour C = 100, so f (18, 78) = 100.
This means that if the annual rainfall were 18 inches and the temperature were 78°F, the country
would produce about the same amount of corn as at present, although it would be wetter and
warmer than it is now.
The point with R-coordinate 12 and T-coordinate 76 is about halfway between the C = 80 and the
C = 90 contours, so f (12, 76) ≈ 85. This means that if the rainfall fell to 12 inches and the
temperature stayed at 76°, then corn production would drop to about 85% of what it is now.
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Example 2
Use Figure 12.38 to describe in words the cross-sections with T and R constant through the point
representing present conditions. Give a common sense explanation of your answer.
Solution
To see what happens to corn production if the temperature stays fixed at 76°F but the rainfall
changes, look along the horizontal line T = 76. Starting from the present and moving left along the
line T = 76, the values on the contours decrease. In other words, if there is a drought, corn
production decreases. Conversely, as rainfall increases, that is, as we move from the present to the
right along the line T = 76, corn production increases, reaching a maximum of more than 110%
when R = 21, and then decreases (too much rainfall floods the fields).
If, instead, rainfall remains at the present value and temperature increases, we move up the vertical
line R = 15. Under these circumstances corn production decreases; a 2° increase causes a 10% drop
in production. This makes sense since hotter temperatures lead to greater evaporation and hence
drier conditions, even with rainfall constant at 15 inches. Similarly, a decrease in temperature leads
to a very slight increase in production, reaching a maximum of around 102% when T = 74,
followed by a decrease (the corn won't grow if it is too cold).
Contour Diagrams and Graphs
Contour diagrams and graphs are two different ways of representing a function of two variables. How do we go
from one to the other? In the case of the topographical map, the contour diagram was created by joining all the
points at the same height on the surface and dropping the curve into the xy-plane.
How do we go the other way? Suppose we wanted to plot the surface representing the corn production function
C = f (R, T) given by the contour diagram in Figure 12.38. Along each contour the function has a constant value;
if we take each contour and lift it above the plane to a height equal to this value, we get the surface in Figure
12.39.
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Figure 12.39 Getting the graph of the corn yield function from the contour diagram
Notice that the raised contours are the curves we get by slicing the surface horizontally. In general, we have the
following result:
Contour lines, or level curves, are obtained from a surface by slicing it with horizontal planes.
Finding Contours Algebraically
Algebraic equations for the contours of a function f are easy to find if we have a formula for f (x, y). Suppose the
surface has equation
A contour is obtained by slicing the surface with a horizontal plane with equation z = c. Thus, the equation for the
contour at height c is given by:
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Example 3
Find equations for the contours of f (x, y) = x2 + y2 and draw a contour diagram for f . Relate the
contour diagram to the graph of f .
Solution
The contour at height c is given by
This is a contour only for c ≥ 0, For c > 0 it is a circle of radius . For c = 0, it is a single point
(the origin). Thus, the contours at an elevation of c = 1, 2, 3, 4, … are all circles centered at the
origin of radius 1,
,
, 2, …. The contour diagram is shown in Figure 12.40. The bowl–shaped
graph of f is shown in Figure 12.41. Notice that the graph of f gets steeper as we move further away
from the origin. This is reflected in the fact that the contours become more closely packed as we
move further from the origin; for example, the contours for c = 6 and c = 8 are closer together than
the contours for c = 2 and c = 4.
Figure 12.40 Contour diagram for f (x, y) = x2 + y2 (even values of c only)
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Figure 12.41 The graph of f (x, y) = x2 + y2
Example 4
Draw a contour diagram for
and relate it to the graph of f .
Solution
The contour at level c is given by
For c > 0 this is a circle, just as in the previous example, but here the radius is c instead of . For
c = 0, it is the origin. Thus, if the level c increases by 1, the radius of the contour increases by 1.
This means the contours are equally spaced concentric circles (see Figure 12.42) which do not
become more closely packed further from the origin. Thus, the graph of f has the same constant
slope as we move away from the origin (see Figure 12.43), making it a cone rather than a bowl.
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Figure 12.42 A contour diagram for
Figure 12.43 The graph of
In both of the previous examples the level curves are concentric circles because the surfaces have circular
symmetry. Any function of two variables which depends only on the quantity (x2 + y2) has such symmetry: for
example,
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or
.
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Example 5
Draw a contour diagram for f (x, y) = 2x + 3y + 1.
Solution
The contour at level c has equation 2x + 3y + 1 = c. Rewriting this as y = -(2/3)x + (c - 1)/3, we see
that the contours are parallel lines with slope -2/3. The y-intercept for the contour at level c is
(c - 1)/3; each time c increases by 3, the y-intercept moves up by 1. The contour diagram is shown
in Figure 12.44.
Figure 12.44 A contour diagram for f (x, y) = 2x + 3y + 1
Contour Diagrams and Tables
Sometimes we can get an idea of what the contour diagram of a function looks like from its table.
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Example 6
Relate the values of f (x, y) = x2 - y2 in Table 12.4 to its contour diagram in Figure 12.45.
Table 12.4
Table of Values of f
(x, y) = x2 - y2
3
0
-5 -8 -9 -8 -5
0
2
5
0
-3 -4 -3
0
5
1
8
3
0
-1
0
3
8
y 0
9
4
1
0
1
4
9
-1
8
3
0
-1
0
3
8
-2
5
0
-3 -4 -3
0
5
-3
0
-5 -8 -9 -8 -5
0
-3 -2 -1
0
1
2
3
x
Figure 12.45 Contour map of f (x, y) = x2 - y2
Solution
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One striking feature of the values in Table 12.4 is the zeros along the diagonals. This occurs
because x2 - y2 = 0 along the lines y = x and y = -x. So the z = 0 contour consists of these two lines.
In the triangular region of the table that lies to the right of both diagonals, the entries are positive.
To the left of both diagonals, the entries are also positive. Thus, in the contour diagram, the positive
contours lie in the triangular regions to the right and left of the lines y = x and y = -x. Further, the
table shows that the numbers on the left are the same as the numbers on the right; thus, each
contour has two pieces, one on the left and one on the right. See Figure 12.45. As we move away
from the origin along the x-axis, we cross contours corresponding to successively larger values. On
the saddle-shaped graph of f (x, y) = x2 - y2 shown in Figure 12.46, this corresponds to climbing out
of the saddle along one of the ridges. Similarly, the negative contours occur in pairs in the top and
bottom triangular regions; the values get more and more negative as we go out along the y-axis.
This corresponds to descending from the saddle along the valleys that are submerged below the xyplane in Figure 12.46. Notice that we could also get the contour diagram by graphing the family of
hyperbolas x2 - y2 = 0, ±2, ±4, ….
Figure 12.46 Graph of f (x, y) = x2 - y2 showing plane z = 0
Using Contour Diagrams: The Cobb-Douglas Production
Function
Suppose you decide to expand your small printing business. How should you expand? Should you start a nightshift and hire more workers? Should you buy more expensive but faster computers which will enable the current
staff to keep up with the work? Or should you do some combination of the two?
Obviously, the way such a decision is made in practice involves many other considerations—such as whether you
could get a suitably trained night shift, or whether there are any faster computers available. Nevertheless, you
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might model the quantity, P, of work produced by your business as a function of two variables: your total
number, N, of workers, and the total value, V, of your equipment.
How would you expect such a production function to behave? In general, having more equipment and more
workers enables you to produce more. However, increasing equipment without increasing the number of workers
will increase production a bit, but not beyond a point. (If equipment is already lying idle, having more of it won't
help.) Similarly, increasing the number of workers without increasing equipment will increase production, but not
past the point where the equipment is fully utilized, as any new workers would have no equipment available to
them.
Example 7
Explain why the contour diagram in Figure 12.47 does not model the behavior expected of the
production function, whereas the contour diagram in Figure 12.48 does.
Figure 12.47 Incorrect contours for printing production
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Figure 12.48 Correct contours for printing production
Solution
Look at Figure 12.47. Fixing V and letting N increase corresponds to moving to the right on the
contour diagram. As you do so, you cross contours with larger and larger P values, meaning that
production increases indefinitely. On the other hand, in Figure 12.48, as you move in the same
direction you move nearly parallel to the contours, crossing them less and less frequently.
Therefore, production increases more and more slowly as N increases with V fixed. Similarly, if
you fix N and let V increase, the contour diagram in Figure 12.47 shows production increasing at a
steady rate, whereas Figure 12.48 shows production increasing, but at a decreasing rate. Thus,
Figure 12.48 fits the expected behavior of the production function best.
Formula for a Production Function
Production functions are often approximated by formulas of the form
where P is the quantity produced and c, α, and β are positive constants, 0 < α < 1 and 0 < β < 1.
Example 8
Show that the contours of the function P = cNαVβ have approximately the shape of the contours in
Figure 12.48.
Solution
The contours are the curves where P is equal to a constant value, say P0, that is, where
Solving for V we get
Thus, V is a power function of N with a negative exponent, so its graph has the shape shown in
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Figure 12.48.
The Cobb-Douglas Production Model
In 1928, Cobb and Douglas used a similar function to model the production of the entire US economy in the first
quarter of this century. Using government estimates of P, the total yearly production between 1899 and 1922, of
K, the total capital investment over the same period, and of L, the total labor force, they found that P was well
approximated by the Cobb-Douglas production function
This function turned out to model the US economy surprisingly well, both for the period on which it was based,
and for some time afterward.
Exercises and Problems for Section 12.3
Exercises
For each of the surfaces in Exercises 1, 2, 3 and 4, sketch a possible contour diagram, marked with reasonable zvalues. (Note: There are many possible answers.)
1.
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2.
3.
4.
In Exercises 5, 6, 7, 8, 9, 10, 11, 12 and 13, sketch a contour diagram for the function with at least four labeled
contours. Describe in words the contours and how they are spaced.
5. f (x, y) = x + y
6. f (x, y) = 3x + 3y
7. f (x, y) = x2 + y2
8. f (x, y) = -x2 - y2 + 1
9. f (x, y) = xy
10. f (x, y) = y - x2
11. f (x, y) = x2 + 2y2
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12.
13.
14. Find an equation for the contour of f (x, y) = 3x2y + 7x + 20 that goes through the point (5, 10).
15.
(a) For z = f (x, y) = xy, sketch and label the level curves z = ±1, z = ±2.
(b) Sketch and label cross-sections of f with x = ±1, x = ±2.
(c) The surface z = xy is cut by a vertical plane containing the line y = x. Sketch the cross-section.
16. Match the surfaces (a)–(e) in Figure 12.49 with the contour diagrams (I)–(V) in Figure 12.50.
Figure 12.49
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Figure 12.50
17. Match Tables 12.5, 12.6, 12.7 and 12.8 with the contour diagrams (I)–(IV) in Figure 12.51.
Table 12.5
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y
\x
-1 0
1
1
2 1
2
0
1 0
1
1
2 1
2
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Table 12.6
y
\x
-1 0
1
1
0 1
0
0
1 2
1
1
0 1
0
Table 12.7
y
\x
-1 0
1
1
2 0
2
0
2 0
2
1
2 0
2
Table 12.8
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y
\x
-1 0
1
1
2 2
2
0
0 0
0
1
2 2
2
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Figure 12.51
18. Total sales, Q, of a product is a function of its price and the amount spent on advertising. Figure 12.52 shows
a contour diagram for total sales. Which axis corresponds to the price of the product and which to the amount
spent on advertising? Explain.
Figure 12.52
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Problems
19. Figure 12.53 shows contours of f (x, y) = 100ex - 50y2. Find the values of f on the contours. They are equally
spaced multiples of 10.
Figure 12.53
20. Figure 12.54 shows the level curves of the temperature H in a room near a recently opened window. Label
the three level curves with reasonable values of H if the house is in the following locations.
(a) Minnesota in winter (where winters are harsh).
(b) San Francisco in winter (where winters are mild).
(c) Houston in summer (where summers are hot).
(d) Oregon in summer (where summers are mild).
Figure 12.54
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21. Figure 12.55 shows a contour map of a hill with two paths, A and B.
(a) On which path, A or B, will you have to climb more steeply?
(b) On which path, A or B, will you probably have a better view of the surrounding countryside?
(Assuming trees do not block your view.)
(c) Alongside which path is there more likely to be a stream?
Figure 12.55
22. Figure 12.56 is a contour diagram of the monthly payment on a 5-year car loan as a function of the interest
rate and the amount you borrow. The interest rate is 13% and you borrow $6000.
(a) What is your monthly payment?
(b) If interest rates drop to 11%, how much more can you borrow without increasing your monthly
payment?
(c) Make a table of how much you can borrow, without increasing your monthly payment, as a function
of the interest rate.
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Figure 12.56
23. Describe in words the level surfaces of the function g(x, y, z) = cos(x + y + z).
24. Match the functions (a)–(d) with the shapes of their level curves (I)–(IV). Sketch each contour diagram.
(a) f (x, y) = x2
(b) f (x, y) = x2 + 2y2
(c) f (x, y) = y - x2
(d) f (x, y) = x2 - y2
I.
Lines
II. Parabolas
III. Hyperbolas
IV. Ellipses
25. Figure 12.57 shows the density of the fox population P (in foxes per square kilometer) for southern England.
Draw two different cross-sections along a north-south line and two different cross-sections along an eastwest line of the population density P.
Figure 12.57
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26. A manufacturer sells two goods, one at a price of $3000 a unit and the other at a price of $12,000 a unit. A
quantity q1 of the first good and q2 of the second good are sold at a total cost of $4000 to the manufacturer.
(a) Express the manufacturer's profit, π, as a function of q1 and q2.
(b) Sketch curves of constant profit in the q1q2-plane for π = 10,000, π = 20,000, and π = 30,000 and the
break-even curve π = 0.
27. Match each Cobb-Douglas production function (a)–(c) with a graph in Figure 12.58 and a statement (D)–(G).
(a) F (L, K) = L 0.25K 0.25
(b) F (L, K) = L 0.5K 0.5
(c) F (L, K) = L 0.75K 0.75
(D) Tripling each input triples output.
(E) Quadrupling each input doubles output.
(G) Doubling each input almost triples output.
Figure 12.58
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28. A general Cobb-Douglas production function has the form
What happens to production if labor and capital are both scaled up? For example, does production double if
both labor and capital are doubled? Economists talk about
• increasing returns to scale if doubling L and K more than doubles P,
• constant returns to scale if doubling L and K exactly doubles P,
• decreasing returns to scale if doubling L and K less than doubles P.
What conditions on α and β lead to increasing, constant, or decreasing returns to scale?
29. Figure 12.59 is the contour diagram of f (x, y). Sketch the contour diagram of each of the following functions.
(a) 3f (x, y)
(b) f (x, y) - 10
(c) f (x - 2, y - 2)
(d) f (-x, y)
Figure 12.59
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30. Figure 12.60 shows part of the contour diagram of f (x, y). Complete the diagram for x < 0 if
(a) f (-x, y) = f (x, y)
(b) f (-x, y) = -f (x, y)
Figure 12.60
31. Values of
are in Table 12.9.
(a) Find a pattern in the table. Make a conjecture and use it to complete Table 12.9 without computation.
Check by using the formula for f .
(b) Using the formula, check that the pattern holds for all x ≥ 1 and y ≥ 1.
Table 12.9
y
x
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1
2
3
4
5
6
1
1
3
6
10
15
21
2
2
5
9
14
20
3
4
8
13
19
4
7
12
18
5
11
17
6
16
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32. The temperature T (in °C) at any point in the region -10 ≤ x ≤ 10, -10 ≤ y ≤ 10 is given by the function
(a) Sketch isothermal curves (curves of constant temperature) for T = 100°C, T = 75°C, T = 50°C, T = 25°
C, and T = 0°C.
(b) A heat-seeking bug is put down at a point on the xy-plane. In which direction should it move to
increase its temperature fastest? How is that direction related to the level curve through that point?
33. Use the factored form of f (x, y) = x2 - y2 = (x - y)(x + y) to sketch the contour f (x, y) = 0 and to find the
regions in the xy-plane where f (x, y) > 0 and the regions where f (x, y) < 0. Explain how this sketch shows
that the graph of f (x, y) is saddle-shaped at the origin.
34. Use Problem 33 to find a formula for a “monkey-saddle' surface z = g(x, y) which has three regions with g(x,
y) > 0 and three with g(x, y) < 0.
35. Use the contour diagram for f (x, t) = cos t sin x in Figure 12.61 to describe in words the cross-sections of f
with t fixed and the cross-sections of f with x fixed. Explain what you see in terms of the behavior of the
string.
Figure 12.61
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