Introducing Functions of Several Variables into Lower Division Mathematics
Sheldon P. Gordon
The MAA’s committee on Curriculum Renewal Across the First Two Years
(CRAFTY) has been conducting a project to determine the current mathematical needs of
the partner disciplines.
The first round of this Curriculum Foundations project [1]
involved having leading educators from 17 math intensive disciplines participate in
workshops to discuss and formulate recommendations to the mathematics community on
what their students need to bring out of lower division mathematics classes. A second
round of workshops involving additional disciplines was held recently and their reports
will appear shortly [2].
Overall, there was an amazing degree of unanimity in the recommendations
across virtually all disciplines, including the need
1. to emphasis conceptual understanding over rote manipulation,
2. to emphasize realistic problem solving via mathematical modeling,
3. for the use of data, particularly statistics, which is the most important mathematical
area for virtually every discipline,
4. for the routine use of technology.
Many of the disciplines also recommended an emphasis on functions of several
variables. To a mathematician, that conjures up a request to do far more with partial
derivatives, gradients, and multiple integrals. It also likely makes us wonder what those
people really have in mind – how can we possibly include these Calculus III topics in
lower division courses? It’s just not possible or reasonable!
However, when one actually talks to people in disciplines such as physics,
economics, chemistry, and so forth, a very different picture emerges.
They intend
essentially three basic themes that are quite appropriate for students at the college algebra
level and above that have never been part of our traditional curricula. In fact, these are
the same three multiple representation themes – symbolic, numerical, and graphical – that
that have become the mainstays for looking at functions in all courses from college
algebra up through calculus. In this article, we will discuss how these themes also apply
to functions of two variables and provide a variety of illustrative examples to show
precisely the kinds of mathematics that is being requested by the partner disciplines.
Numerical Representations of Functions of Two Variables
In most fields of
endeavor, one routinely encounters information presented via tables and often those
tables are sets of values that depend on two distinct quantities – that is, are functions of
two independent variables. For instance, the wind-chill factor is an adjustment made to
temperature readings T that takes into account the effects of the wind speed W and so
provides a way of measuring how cold it feels. Thus, C is a function of both temperature
T and wind speed W. Students in courses in many different fields are expected to read
and interpret such tables and perhaps even create them. This involves a considerable
amount of mathematical understanding and reasoning.
We start with a simpler, though more artificial, example than the wind-chill
factor. Consider the following table, where the independent values are x and y. If you
carefully examine the values for the dependent variable z in the table, you will notice
that, along every row, each successive value of z is 5 more than the previous value. That
is, for each value of y, the row is a linear function of x and the slope of that line is
precisely 5. We denote this slope in the x direction by mx. In the same way, along every
column, each successive value is 30 more than the previous value. That is, for each value
of x, every column is a linear function of y and the slope of that line is precisely 30/5 = 6.
We write this slope in the y direction as my. As a result, z is a linear function of the two
variables x and y, and the resulting equation is z = 5x + 6y – 11. The constant, -11, is the
value of z that corresponds to x = y = 0 and is the vertical intercept.
x
y
0
5
10
15
20
0
-11
19
49
79
109
1
-8
22
52
82
112
2
-5
25
55
85
115
3
-2
28
58
88
118
4
1
31
61
91
121
5
4
34
64
94
124
6
7
37
67
97
127
The parallels (puns fully intended) with standard ideas about linear functions are
very clear. This is not just a matter of adding a new dimension to a college algebra or
precalculus class, but also of providing an opportunity to reinforce some of the
fundamental concepts about linearity while introducing students to notions that are
important in other subjects.
In practice, very few functions that arise via tables are linear. The following table
shows some of the values for the wind-chill factor C, with the temperature T in degrees
Fahrenheit and the wind speed W in miles per hour. (We could write C = f(T, W), but at
the level of these courses, this might not be the most effective notation for the students.)
Thus, if T = 20 and W = 25 mph, the associated wind-chill factor is C = -15. That is,
the combination of a temperature of 20F with a wind blowing at 25 mph makes it feel 15F. In comparison, the wind-chill factor associated with T = 25 and W = 20 is C = -3.
However, if you examine the rows and columns, it is clear that this function is
non-linear in both directions.
Wind speed W
Temperature
T
5
35
33
30
27
25
21
20
16
15
12
10
7
5
0
0
-5
10
22
16
10
3
-3
-9
-15
-22
15
16
9
2
-5
-11
-18
-25
-31
20
12
4
-3
-10
-17
-24
-31
-39
25
8
1
-7
-15
-22
-29
-36
-44
Because this function isn’t linear, trying to predict the wind-chill factor when
values for T and W are not listed can be challenging for many students. While none of
the other disciplines mentioned interpolation, many did talk about the need for students to
know how to work with proportions and there is much discussion in mathematics
education circles about the importance of students developing the ability to do
proportional reasoning. Thus, introducing some problems of this form may be very
valuable.
In particular, suppose that one wants to estimate the wind-chill factor
corresponding to T = 18 and W = 23 mph. We use the wind-chill factor value of -17
corresponding to T = 15 and W = 20 mph. In the direction of increasing T (right to left
in the table), the slope of the function is approximately 7/5 = 1.4; that is, the wind-chill
factor increases by approximately 1.4 for every 1 increase in temperature at that point.
Therefore, corresponding to an increase of 3, the change in the wind-chill factor will be
roughly 3 × 1.4 = 4.2, assuming the wind speed is constant. Similarly, in the W
direction, the slope when T = 15 and W = 20 mph is roughly -5/5 = -1; that is, the windchill factor decreases by approximately 1 for every 1 mph increase in wind speed at that
point. Therefore, corresponding to an increase of 3 mph in the wind speed, the change in
the wind-chill factor will be roughly 3 × -1 = -3, assuming the temperature is constant.
Consequently, the total change in the wind-chill factor is approximately 4.2 - 1 = 3.2,
and so we estimate that when T = 18 and W = 23 mph, C = -17 + 3.2 = 13.8.
Most of the fundamental concepts of functions of a single variable, y = f(x),
extend in a very natural way to functions of two (or more) variables. For instance, the
formal definition of a function of two independent variables, z = f(x, y) says that for every
possible pair of values for x and y, there corresponds precisely one value of the dependent
variable z. Think about this in terms of the wind-chill factor. If two different values for
C corresponded to a value for T and a value for W, then the table would not satisfy the
definition of a function.
Furthermore, the notions of domain and range similarly extend to functions of two
(or more) variables. The domain consists of all possible pairs of values (x, y) for the
independent variables x and y, and the range consists of all the corresponding values of
the dependent variable z that arise. For instance, again think about the wind-chill factor
function. What are the possible values for the independent variables? The table shows
values of the temperature extending from 0 to 35 and wind speeds from 5 to 25 mph. If
we accept these as the only possible values for T and W, then we have the domain of this
function. The table shows that the corresponding values for the wind-chill factor C
extend from -44 to 33 and this would be the range.
Presumably, we could probably extend the values for T somewhat, say from a
minimum of -30 to a maximum of 45 (wind-chill likely wouldn’t make sense at any
warmer temperatures). Wind speeds could be extended to a minimum of 0 mph (we
certainly can’t have a negative wind-speed) and might be considered as high as 50 mph.
If that is the case, the domain would be T between -30 and 45 and W between 0 and 50
mph. The range would then consist of all the associated values for the wind-chill factor,
but we cannot specify those additional values from the table. Of course, one could ask
the students to use the web to search out the values of C for the extended range based on
T between -30 and 45 and W between 0 and 50 mph.
We can also extend some of the fundamental notions about the behavior of
functions of a single variable to talk about the behavior of a function of several variables.
With a function of one variable, it is simple to decide whether the function is increasing
or decreasing –simply look at the associated table (or graph) and decide whether the
values of the function get larger or smaller as the independent variable increases. But,
with a function of two variables, it is not that simple because there are a number of
different possibilities. A function of two variables could increase as both independent
variables increase. The function could decrease as both variables increase. However, the
function could increase as one variable increases, but decrease as the other variable
increases.
For instance, consider the table for the wind-chill factor. As the temperature rises
and the wind speed remains the same, so that we look horizontally from right to left along
each line of the table, we see that the values for C increase. Thus, C is an increasing
function of T. Alternatively, as the wind speed rises and the temperature remains the
same, we look vertically down each column and see that C gets smaller or eventually
more and more negative. Thus, C is a decreasing function of W.
Moreover, since C does not change in a linear manner in either direction, issues of
concavity come into play. When the wind-chill factor increases, does it increase at an
increasing or decreasing rate? When it decreases, does it decrease at an increasing or
decreasing rate?
Alternatively, one could raise the questions in a somewhat more
dynamic form as: Does the wind-chill factor increase faster at low temperatures or at high
temperatures? Does the wind-chill factor decrease faster at low wind speeds or at high
wind speeds? Questions such as these make for good classroom discussions that engage
students or for homework problems that require students to think carefully about the
mathematical issues.
Graphical Representations of Functions of
Two Variables When most mathematicians hear
the words “graph of a function of two variables”,
what invariably comes to mind is a surface plot of the function. In almost all other
disciplines, however, the standard graphical depiction is a contour diagram or contour
map. Figure 1 shows the contour diagram for the heights (in feet) of the terrain on a
hypothetical island. Each contour, or level curve, connects points that are all at the same
height. The shore around the island is at sea level, so it is represented as the closed
contour with an associated height of 0 feet. The other contours shown are all multiples of
500 feet. For instance, the 1000 contour represents all heights on the island that are 1000
feet about sea level. As such, these contour curves will be a series of closed loops.
Moreover, when you look closely at the contour map in Figure 1, you will notice
three distinct groups of closed loops, with each successive contour being smaller in size
and larger in value. Each of these nested loops represents a mountain on the island. The
peak toward the left is slightly more than 3000 feet high, the one to the right is somewhat
more than 2000 feet high, and the one in the upper center is more than 3500 feet high.
Clearly, the level curves represent a sample of heights, since it is not possible to
give the height of every point. However, one of the “rules” for creating a contour plot is
that there must be a uniform interval between contours values; this is known as the
contour interval. In Figure 1, the contour interval is 500 feet. The minimum height
displayed is 0 (sea level) and the maximum height shown is 3500 feet.
The absence of a contour can also provide useful information. In Figure 1, the
highest contour is 3500 feet. If there were any points on this island where the height
reached 4000 feet, then we would see another contour corresponding to that height.
Therefore we conclude that the maximum height must be more than 3500 and less than
4000 feet.
Figure 2 shows the contour diagram
(also known as a bathymetric map in
oceanography) for the depths of the ocean
bottom around the western end of Santa
Catalina Island off the coast of southern
California.
The depths are measured in
fathoms, which are equal to 6 feet.
Think
about the 50 fathom contour that represents
points whose depth is 50 fathoms below sea level. If you were to extend this contour as
far as possible, you should expect that it continues all the way around the island until it
comes back to complete the loop, unless it completes a full loop around some local
undersea depression or around some local bulge of the sea bottom or if there is some
sharp discontinuity under the water, like a sudden chasm or a vertical wall.
We can extract considerably more information from a contour diagram than just
the height (or depth or size of a quantity) at the points along the different contours. The
relative distances between adjacent contours also tell us a lot about the behavior of the
quantity being investigated. In particular, regions where the successive contours are far
apart indicate that the function is relatively constant there, so that any changes are slow
and therefore the slope is quite small. On the other hand, regions where successive
contours are very close together, as at the lower left of Figure 2, indicate that the function
changes very rapidly, so that the slope is very large.
Given the scale for a contour diagram, students can be asked to apply some
simple ideas on the slope of a line to estimate the slope of the bottom of the ocean or of a
terrain or the rate at which some other quantity is changing. All that is required is to
select points on two different contours, so that one knows the change in “height”,
measure the distance between the two points, and use the usual formula for the slope to
estimate its value. As such, this is another nice way to reinforce the early ideas on linear
functions.
(Incidentally, from a more sophisticated point of view, one is actually
estimating the value of the directional derivative at the initial point.)
There are other, more exploratory, activities that one can do with contour
diagrams. For instance, it is known that sediment from Catalina Island has created an
underwater trench, called the Catalina Canyon, along which silt flows to the sea floor.
Students, either individually or in small groups, can be asked to locate the head of this
“canyon” and to draw a curve tracing the path of the silt flowing down the trench. Or,
they can be asked to identify areas that might be conducive to having a beach (shallow
waters that don’t drop too suddenly) or for snorkeling (waters where there may be very
sharp drops).
Algebraic Representations of Functions of Two Variables When faculty from other
disciplines talk about students needing to know about functions of several variables, they
often mean that students need to be able to work with certain basic formulas that arise in
their discipline that represents important relationships between three (or perhaps more)
quantities. For instance, in chemistry, one of the most important principles is the Ideal
Gas Law, PV = nRT, that relates the pressure P of a gas, its volume V, and the
temperature T. In the formula, n and R are two constants. Chemists love to assign pages
of problems in which two of the three variables are given and students have to solve for
the remaining variable. Similarly, one of the most important principles in economics is
the Cobb-Douglas model that relates production P (the monetary value of all goods
produced) to the labor input L and the capital input K via the formula P = AL K1-.
Economists then expect their students to be able to solve for one of the three variables
given values of the other two. In biology, a famous model relates the surface area S of a
human being to a person’s height H and weight W via the formula S = 15.64W 0.425H 0.725,
where W is measured in pounds and H in inches. The surface area is in square inches.
Typical questions students are expected to answer are things like: Estimate the surface
area of a person who is 70 inches tall and who weighs 180 pounds or Estimate the weight
of a person whose surface area is 3500 square inches and who is 65 inches tall. These
questions may not seem like very deep mathematics to us, but they are seemingly
challenging to students in those other courses, possibly because the variables are not x’s
and y’s.
However, students in those other fields are also expected to interpret and use
contour plots for functions such as these. While we addressed the major issues involved
in interpreting contour maps earlier, we did not indicate how one actually constructs them
based on a formula. This can actually be a lovely classroom activity at the college
algebra or precalculus level that, once more, provides the opportunity to remind students
about the properties, both behavioral and algebraic, of various important families of
functions.
Suppose that the contour diagram for the human surface area function
S= 15.64W 0.425H 0.725
is to be plotted with H horizontally and W vertically.
a. First, one must determine the shape of the contours. Typically, the level curves
associated with functions such as these might be increasing and concave up, increasing
and concave down, decreasing and concave up, or decreasing and concave down.
Students can be asked to decide which of the four possibilities is correct. As a hint to get
them started: Pick a value for S, say 3400 square inches, substitute this into the formula,
and solve algebraically for W as a function of H. Then ask them to use their knowledge of
the behavior of power functions to answer the question.
b. Second, do the successive contours get larger or smaller in a certain direction? That
is, as you move upward and toward the right, say, do the successive contours correspond
to larger or smaller values for S? A hint here is to suggest that students use the contour
corresponding to S = 3400 square inches from part (a) as a baseline and decide whether
the “next” contour in that direction is S = 3300 or S = 3500.
Alternatively, one can argue in a more dynamic, though less precise, way. What
should happen to the surface area if a person’s height and weight both increase? While
not mathematically conclusive, a good model should reflect the underlying process and
students should learn to develop their mathematical intuition to understand and predict
what will likely happen with a model.
c. Third, what is the spacing between the successive contours – do they get closer
together or further apart? Students can use the two contours from parts (a) and (b) along
with a third contour with a comparable change in S, say either S = 3200 or S =3600, to
decide whether the distances between the successive contours get smaller or larger using
a graphing calculator or a spreadsheet. A little effort to generate an appropriate window
in which all three curves can be seen clearly may be required and sometimes it can be
rather hard to decide whether the spacing is increasing or decreasing. This is the case
here, as seen in Figure 3, though the spacing appears to be increasing, albeit very slowly,
as S increases from 3200 to 3600.
Figure 3
Contour Plots of Linear Functions What does the contour plot associated with a linear
function, z = ax + by + c looks like? There are many ways to proceed. One is to use
some algebra. Let’s start with a simple application – the number of calories C in a peanut
butter and jelly sandwich. Each gram of peanut butter, P, has 6 calories and each gram of
jelly J has 2.5 calories. So, with white bread, the total number of calories is C = 150 +
6P + 2.5J. Students can then be asked to consider the contours corresponding to C = 100,
150, 200, …, say, to produce a series of normal equations for a set of lines and observe
that they are all parallel (the same slope) and appear to be equally spaced, as seen in
Figure 4.
Figure 4
Alternatively, from a graphical perspective, each of the level curves is clearly a
line. The slope in the P- (horizontal) direction is 6 and that in the J (vertical) direction is
2.5. Suppose you pick any point Q on some particular level curve. If you move one unit
to the right, the value of C increases by 6 calories and if you also move one unit vertically
upward, the value of C
increases by 2.5 calories.
See Figure 5.
So, for
every possible increase of
1 gram of peanut butter
(move one unit over) and
1 gram of jelly (move one
unit up), the effective
change is to move the
point and the vertical and
horizontal lines in the
figure as a group along the lower line, and the function increases by a total of 8.5
calories. The result is a new line that is parallel to the original line, but with the constant
term 8.5 calories larger. That is, the contours consist of a collection of parallel and
uniformly spaced lines.
A third approach might be to work with a considerably simper equation, say z =
2x + 3y + 1. Students can be asked to create a table such as the first one in this article,
(although a slightly larger table might actually be more effective), plot all the points on a
sheet of graph paper along with the associated values of the function z, and then connect
the points corresponding to the same values of z. (Hopefully, if the table is extended far
enough, most of the values for z will occur at least three times to give a much more
dramatic impact.) The results will again be a series of parallel, equally spaced lines.
Multivariate Linear Regression There is one other aspect of functions of several
variables that came up in many of the Curriculum Foundations reports, particularly those
from the social sciences. Many of them mentioned the importance of multivariate linear
regression. Just as linear regression involves finding the line that comes closest to all the
points in a set of (x, y) data in the least squares sense, one can also find the plane that
comes closest to all the points in a set of (x, y, z) data in the least squares sense; if there
are more than two independent variables, then it is the hyperplane that comes closest to
all the data points. While not a routine built into graphing calculators, this can be done
quite easily using Excel.
To perform multivariate regression, Excel requires that the dependent variable be
denoted by Y and its values listed in a single column or a single row. The various
independent variables are denoted by X1, X2, … and their values are also listed in separate
columns or rows. That is fine is the data values are given in such a format. However,
suppose we want to find the linear function that is the best fit to our data on wind-chill
factors. The above table has 40 entries in a 5 × 8 array, but the format is inappropriate
for performing the regression computations. Instead, it is necessary to re-write the values
in a 40 × 3 array, if we choose to work with columns, in which all the wind-chill factor
values are the first column (Y), all the temperature readings are in the second column (X1)
and all the wind speed values are in the third column (X2).
When multivariate linear regression is performed, the equation of the resulting
plane is Y=1.315X1 -1.55X2 - 4.817, which is equivalent to WC=1.315T -1.55W - 4.817.
Notice that the slope of this plane in the increasing T direction is 1.315, so that the windchill factor, on averages, increases by about 1.3 for each 1 increase in temperature.
Similarly, the slope of the plane in the W direction is -1.55, so that the wind-chill factor,
on average, decreases by 1.55 for each additional 1 mph that the wind is blowing.
Another, more mathematical, application of these ideas that can be introduced
into courses at the college algebra or precalculus level has to do with the way that
polynomial regression may be performed on a calculator or in Excel. Suppose we have a
set of (x, y) data and wish to fit a cubic polynomial, say, to the values. That is, we wish
to find the function c0 + c1 x + c2 x2 + c3 x3 that is the best fit to the data. Instead of
thinking of this as a polynomial, the usual computational approach is to think of it as a
linear function of the three variables x, x2, and x3 and to perform multivariate linear
regression on the associated values. That is, there is one column for the given y-values,
another for the given x-values, a third for the squares of each of the x-values, and a fourth
for the cubes of the x-values.
The resulting linear function is the desired cubic
polynomial.
Conclusions The overwhelming majority of the students who take our introductory
courses do so to fulfill requirements from the partner disciplines. As such, it is in all of
our interests to offer courses that meet the needs of both the students and our colleagues.
This is particularly helpful if we can introduce such topics in ways that allow us to revisit
fundamental mathematical topics to reinforce them in our students’ minds in the process
of extending them in new directions.
References
1. Ganter, Susan and William Barker, Eds., The Curriculum Foundations Project:
Voices of the Partner Disciplines, MAA Reports, Mathematical Association of America,
2004.
2.
Ganter, Susan L. and W.E. Haver (Eds.), Partner Discipline Recommendations for
Introductory College Mathematics and the Implications for College Algebra, MAA Reports,
Mathematical Association of America, 2011 (to appear).