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Lesson 15: Functions of More than One Variable
Chapter 8 Material: pages 311-322 in the textbook:
Lesson 15 presents topics that extend the definition of the derivative to include functions of more than one
independent variable. There are a number of examples of functions of more than one variable throughout the
book. The Cobb-Douglas model is used throughout the book and is a function with two variables, x and y, in
the definition of the productivity function that is modeled by the Cobb-Douglas framework. In all of the prior
encounters with this mathematical model some relationship between the variables was assumed. In this lesson,
the two variables will be considered as independent inputs to the mathematical model. This is explained at the
beginning of Chapter 8.
The basic concepts that will be presented are the following:
• Definitions of functions of two and three independent variables. Functions of more than three variables
are easy to understand once the examples of functions of two and three independent variables are
understood.
• The evaluation of functions of more than one variable are not difficult. Students should be able to
evaluate functions of multiple independent variables with just a few examples.
• Another concept involves a partial evaluation which in the case of a function of two independent
variables produces a function of a single independent variable. This is how many problems related
to the Cobb-Douglas model were solved.
• Domains and ranges of functions of more than one independent variable are harder to determine.
However, the same rules apply. For example, a domain of a function cannot include points where a
division by zero would occur. A couple of examples are covered in the material in Chapter 8.
• The next concept to include would be continuity of functions of more than one independent variable.
However, continuity of functions of more than one independent variable is a rather difficult thing to
determine. So, this is skipped in this course.
• Instead of spending time on continuity, the lesson takes up the definition of partial differentiation
of functions of more than one variable. This is not very difficult if students have a firm grasp of
differentiation problems from Chapters 4 and 5.
• There is a small section on higher order derivatives. Since the second derivative test is used to test
functions for local maximum and minimum values, the higher order partial derivative concept is
important to cover. Again, I do not believe that this content is too difficult to understand.
• The last important concept is the solution of simple optimization problems in two and three dimensional problems. Students will need to be able to compute the gradient of a function of more than one
variable, set up the system of equations to solve, and then determine the location of critical points
based on these solutions. Finally, they will need to be able to apply the second derivative test to
determine if a critical point is the location of a local maximum, local minimum, a saddle point, or
none of the above.
The main idea is to extend the ideas of calculus to functions of more than one variable. Students will likely see
this in advanced economics courses in an undergraduate curriculum in business or economics. So, even though
this is a very brief introduction, it is important that students know that calculus techniques can be applied in
more general mathematical modeling.
Prerequisite Content for Lesson 15
Students must be able to do the following:
• Students must be able to use all of the differentiation rules and algebraic techniques to evaluate
functions of more than one variable and compute partial derivatives.
• Students must be able to compute higher order derivatives from the rules in Chapter 4 for elementary
functions.
• Students must be able to solve optimization problems for functions of a single independent variable
using critical points and the second derivative test.
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Goals and Objectives for Lesson 15
Students who complete this lesson should be able to:
• extend mathematical modeling techniques to functions of more than one independent variable in
some simple cases,
• evaluate functions of more than one independent variable,
• in simple cases, determine the domain and range of functions of more than one independent variable,
• compute first partial derivatives with respect to independent variables for functions of more than one
independent variable,
• compute higher order partial derivatives for functions of more than one independent variable, and
• solve simple optimization problems in two and three dimensional mathematical models.
Please keep in mind that this is a very, very brief introduction to functions of more than one variable.
The concepts are relatively simple. However, students should get some feel for how calculus techniques can be
extended to functions of more than one variable.
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1.1
Lecture Notes for Lesson 15 - Day 1
The material in this lesson is not too difficult. It is important that you present material on the basics in the first
lecture. You should be able to finish the material on the definitions of functions of more than one variable and
evaluation of functions of more than one variable at points in the domain of the functions.
Functions of Two Independent Variables:
The lesson starts with functions of two variables since the functions are much easier to illustrate and work with.
You should explain to the students that all of the ideas in this section can be applied to functions of any number
of variables.
Evaluation of Functions of Two Variables:
In think that this is an easy concept and the examples are clear. Students will have a few problems involving
evaluation of functions of more than one variable on the homework for the lesson. As an instructor, I would only
spend a few minutes on this concept.
Partial Evaluation of Functions of More Than One Variable:
The idea of this section is to relate a lot of the work done in the first seven chapters of the book to functions of
more than one variable. Although there are many examples that could be used, one of the mathematical modeling
threads that runs through the entire book is the use of Cobb-Douglas functions of productivity. These functions
have two (or more) input variables. Prior to this chapter the problems involving Cobb-Douglas functions required
assumptions about the input variables. In all cases up to this point, the variables were made to be dependent
on each other. The result was always a function of one variable or an implicit expression relating the variables
in the model. This is essentially the same as a function of one variable.
Indicate that the idea of fixing all but one variable is essential to the definition of partial derivatives. This is
why it is important to introduce the concept of a partial derivative in the first lesson.
Determination of the Domain and Range of Functions of More Than One Variable:
Determination of the domain and range can be very difficult. There are a couple of examples in the textbook
that show some ideas on how to proceed. In general, this can be difficult.
A Word or Two on Continuity of Functions of More than One Variable:
In this section, continuity of functions of more than one variable is discussed in very brief terms. You should
indicate to the students that the definition of continuity of functions of more than one variable is difficult at best
to use. In this course the topic is basically skipped. This type of description should be enough in the last week
of class.
Partial Derivative
I would expect that you will be able to introduce the idea of computing a partial derivative on the first class in
this lesson. You should present the definition of partial derivatives and work some simple examples. Make sure
that the students understand the idea of fixing all of the variables but one. Then the definition is about the
same as the definition of a derivative for functions of one independent variable.
The section above on the partial evaluation of functions of more than one variable was included in this lesson
since the idea should help students understand fixing all of the variables except one. I think if you can work one
or two of the examples at the beginning of section 8.3 by the end of the first lecture, that should be enough for
this lesson.
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1.2
Lecture Notes for Lesson 15 - Day 2
Start this lecture with some examples of partial differentiation for functions of two variables. This should help
with the continuity of the lesson.
Partial Derivatives
Examples 244 and 245 in the textbook can be used to show how to compute partial derivatives. This should not
be too difficult.
Higher Order Partial Derivatives
The main concepts in this section should involve the idea of computing derivatives of derivatives. This is the
same idea as presented for higher order derivatives of functions of a single variable. Again, this is a relatively
simple extension and students should be able to compute partial derivatives quickly.
Make sure that you show students that the number of higher order partial derivatives grows based on the
number of independent variables. For example, there are two first partial derivatives for a function of two
independent variables and then four second partial derivatives, followed by eight third partial derivatives and so
on.
Functions of Three Independent Variables
The extensions here are not difficulty and there are a number of examples that can be presented. Present the
definition of the partial derivatives in this case and compare the definition to the definition of partial derivatives
of a function of two variables. From these two definitions, the general definition of partial differentiation is not
too difficult to understand.
Note that the number of higher order partial derivatives for functions of three variables can be compared to
the number of higher order partial derivatives in the case of a function of two independent variables. In the case
of three variables there are three first partial derivatives, followed by nine second partial derivatives, followed by
27 third partial derivatives.
A Review of Optimization for a Function of One Variable
I think it is a
problem from
the use of the
one variable.
through 214.
good idea to review optimization problems for
Chapter 5 involving maximizing profit. There
second derivative test since this is the primary
There are a couple of examples at the end of
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a function of a single variable. You can present a
are lots of examples. Make sure that you present
tool used in this lesson for functions of more than
Chapter 5 in Section 1. These are on pages 212
1.3
Lecture Notes for Lesson 15 - Day 3
This lecture will cover finding solutions of optimization problems for functions of two independent variables
Finding Critical Points for Functions of More than One Variable
Present Defintiion 90 and work through the examples that follow the definition. This should give enough
information on how to compute critical points for functions of more than one variable.
Extensions of the Second Derivative Test for Functions of More than One Variable
The definition needed for the second derivative test for functions of two variables is Definition 91. Work the
example after the definition to explain how to use the test.
This should be the end of the course. There is no need to continue on into Definitions 92 and 93 or continue
to optimization in three dimensions.
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1.4
Homework
Identifying the Number of Independent Variables in an Expression
Problem 1: Determine if the following relationships define a function of one independent variable, two independent variables, or three independent variables. Explain your answer in one sentence.
a.
x2 + 3 · x · y − y 2 = 7
Show your work below:
b.
f = x2 + 3 · x · y − y 2 − 7
Show your work below:
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c.
w + x2 + 3 · x · y − y 2 = 7
Show your work below:
d.
z = w + x2 + 3 · x · y − y 2 − 7
Show your work below:
e.
f (x, y) = cosh(2 · x · y) − sinh2 (2 · x · y)
Show your work below:
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Evaluation of Functions of Two and Three Independent Variables
Problem 2: Determine the values of the functions at the given input values. Use
f (x, y) = 4 · x2 + 2 · x · y + y 2 − 400
and
g(x, y, z) = ln(1 + x2 + y 2 + z 2 )
a.
f (1, 2)
Show your work below:
b.
f (−1, −1)
Show your work below:
c.
f (r, 3)
Show your work below:
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d.
f (−2, t)
Show your work below:
e.
f (s, z)
Show your work below:
f.
g(0, 0, 0)
Show your work below:
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g.
g(1, −1, 1)
Show your work below:
h.
g(r, t, 1)
Show your work below:
h.
f (1, 1) + 2 · g(−1, −1, 0)
Show your work below:
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Domain and Range of Functions of Two Variables
Problem 3: Determine the domain and range of each of the following functions of two variables.
a.
f (x, y) = x · y · e−x
2
Show your work below:
b.
g(x, y) =
x2 − y 2
x−y
Show your work below:
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Partial Derivatives for Functions of Two Variables
Problem 4: Compute the first partial derivatives of each of the following functions.
a.
f (x, y) = cosh(x · y)
Show your work below:
b.
g(x, y) = ln(1 + x + 3 · y)
Show your work below:
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Partial Derivatives for Functions of Three Variables
Problem 5: Compute the first partial derivative of the following functions.
a.
h(x, y, z) = x · y + y · z + z · x
Show your work below:
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b.
w=
x2
x·y·z
+ y2 + z2
Show your work below:
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Higher Order Derivatives in Two Variables
Problem 6: Compute the second partial derivatives of the functions from Problem 4. You may use the first
derivatives you have computed in the solution of Problem 4.
a.
f (x, y) = cosh(x · y)
Show your work below:
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b.
g(x, y) = ln(1 + x + 3 · y)
Show your work below:
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Higher Order Derivatives for Functions of Three Variables
Problem 7: Compute the second partial derivatives of the following functions. You may use the first partial
derivatives computed in Problem 5.
a.
h(x, y, z) = x · y + y · z + z · x
Show your work below:
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b.
w=
x2
x·y·z
+ y2 + z2
Show your work below:
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Critical Points for Functions of Two Independent Variables
Problem 8: Compute the critical points for the following function of two independent variables.
P (x, y) = 4 · x2 − 5 · x · y + 2 · y 2 − 3 · x + 4 · y + 1000
Show your work below:
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Critical Points for Functions of Three Independent Variables
Problem 9: Compute the average value of the function on the given interval. Hint: You may use an antiderivative from a previous problem.
P (x, y, z) = 4 · x2 − 8 · x · y + 2 · y 2 + 8 · y · z − 6 · z 2
Show your work below:
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Optimization for a Function of Two Independent Variables
Problem 10: For the function following function of two variables compute the location of all critical points in
the domain of the function and determine whether these critical points are the location of a local maximum,
local minimum, or saddle point using the second derivative test.
P (x, y) = x3 − y 3 − 3 · x + 9 · y + 42
Make sure you account for all combinations of the points you determine.
Show your work below:
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