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Calculus Practice


									                       Informal Course Descriptions

                           MATH 151—Calculus I
Slope tells something about the rate of change in a line. This is an extremely useful
concept, but has the shortcoming of only being applicable to lines. In Calculus I this
problem is overcome by the derivative, essentially a concept of slope that can be applied
to functions other than lines. Armed with the derivative, we can answer questions about
the rate of change of many functions, allowing us to find maxima or minima of functions,
study velocity and acceleration of physical bodies, chemical reactions and population
growth. We can graph complex curves and describe the relative efficiency of rival
computer algorithms. Indeed, the calculus provides a universal language to precisely
describe and compute rates of growth and corresponding changes in amount.

                          MATH 152—Calculus II
Nearly everyone knows that the area of a circle is r 2 , and so on. But few think about
where these formulas come from. In Calculus II we use the concept of the integral to
study the area under curves. This naturally generalizes to the study of volumes of solids
in space. But this same concept, combined with the derivative (from Calculus I) can be
used in many unexpected and powerful ways. Quantities as diverse as the GNP (gross
national product) and total run time of a computer program can be described as an area
under a curve on a graph. Calculus II provides tools to compute these quantities and
relate them to the functions that describe their rates of change.

It is possible for infinitely many numbers to sum to a finite value. For example, it can be
              1 1 1 1
shown that     ...  1 . The integral and derivative are used as tools to help
              2 4 8 16
us understand such infinite series. In turn, these series help us to understand several
functions better. For example, cos x and ln x can be written as infinitely long
polynomials and can be approximated reasonably well by, say, polynomials of degree
four or five.

                  Math 203 – History of Mathematics

Often, when we learn mathematics, we learn it without the story of who developed it, and
when and why. In the History of Mathematics, we look at the stories behind the

These stories take us to many places on the earth and through a long period of time. We
begin about 4000 years ago with the ancient civilizations of Egypt and Mesopotamia,
where there was already a good deal of mathematics known, particularly algebra and the
art of computation. We also explore the early mathematical discoveries of China and
India. Next we go to the amazing flowering of mathematics that occurred in ancient
Greece: geometry, astronomy, trigonometry and much more.
We see some more development of algebra with the Arabic mathematicians of Medieval
times; in fact our word, algebra, is from the Arabic. We next move to Europe to see
algebra in Italy, analytic geometry in France, logarithms in Scotland and the beginnings
of calculus almost everywhere. We follow the development of calculus and see how it
changed from around 1600 to around 1800. We then look at the surprising story of non-
Euclidean geometry in the 1800‘s. We can only survey more recent discoveries briefly
because they are more difficult and there are so many of them.
We study the biographies of a number of mathematicians, and look at the special
problems encountered by women mathematicians. Many of the students in this class
intend to become mathematics teachers so we examine the histories of specific areas of
mathematics taught in the schools, such as number systems, algebra, geometry and

                      MATH 245-Discrete Structures
The possession of logical reasoning skills is essential for anyone interested in computer
science. In this class, students enhance these skills by studying a variety of mathematical
topics related to the study of computer science, which may include propositional logic,
set theory, relations, functions, combinatorics, graph theory, and applications of these
topics. Students also learn proof-techniques such as induction (a ―domino‖ technique
that allows one to prove that a statement relating to a variable n is true for all positive
integers n) and proof by contradiction (in which one proves a desired result by showing
that if it isn‘t true, nonsensical things happen), thereby increasing their mathematical
maturity and their ability to make reasoned arguments, prerequisites for programming.
Topics vary from term to term, and may depend on student interest. Here is a sample of
things students may learn in this class:

(1) The logical difference between the statements, Not all people have red hair and All
people do not have red hair;

(2) How to show that 1+2+3+…+n= n(n+1)/2, for any positive integer n;

(3) How to show that the set of integers and the set of rational numbers have the same
―size‖, but the set of real numbers is ―bigger‖;

(4) How to compute the probability of getting a royal flush in poker.

The course is intended primarily for computer science majors and math majors.

Note: though this class has Math 152 as a prerequisite, to ensure the mathematical
preparedness of its students, its material is not directly related to that learned in the
calculus sequence.
                          MATH 253—Calculus III
Most things are related to more than just one factor. For example, your minimum
monthly credit card payment depends on the total you owe and your interest rate. The
amount you actually pay depends on the minimum payment due and the amount you have
available to pay. The growth rate of a deer population depends on the size of the
population, its age distribution, the food supply and predation. The pressure exerted by
gas in a cylinder depends on the amount of gas, its temperature and the volume of the

Other functions may only depend on one variable, but give an output that is more than
just one number. For example, a person traveling around the world has, at any given time,
a latitude and a longitude (and perhaps an altitude too if s/he is in an airplane). Thus
position can be considered a function of time but it cannot be represented by a single
value; it must be given as a doublet (or triplet) of numbers representing latitude and
longitude (and altitude). Such a doublet or triplet can be represented as a vector.

Calculus III extends the ideas of Calculus I and II by considering derivatives and
integrals of functions with more than one variable, or of vector-valued functions. Along
the way, other possible coordinate systems (such as polar coordinates) are discussed.

      MATH 317—Introduction to Proof in Mathematics
In mathematics we accept a statement as true only if we have a proof that it is true. Since
the method of proof is so basic to mathematics, anyone who seriously wants to learn
mathematics beyond a fairly elementary level must be able to understand proofs and be
reasonably proficient at constructing them. The purpose of this class is to teach you how
to understand proofs and to develop your skills at constructing proofs. Skill at proving
develops over a long period of time; this class is only a beginning. The best way to learn
to do proofs is to do them, so you will be given plenty of opportunity to practice proving

We will begin with an introduction to logic. Logic is a tool that we will use to analyze
proofs to see if they are correct and to help us to construct proofs. We will practice
writing proofs in a number of areas of mathematics: set theory, including infinite sets,
inequalities and functions. We will study the whole numbers using mathematical
induction. In addition to the usual lecture format, a good deal of class time will be spent
with students presenting their proofs to the class or constructing proofs together.

This class is good preparation for Linear Algebra (MATH 331), Abstract Algebra
(MATH 433), Analysis (MATH 455) and other advanced mathematics classes.

                           MATH 321—Geometry
The geometry most of us learned in high school is based on Euclid‘s famous 5 Postulates
and works well for describing things in or on a flat surface. However, the surface of our
world is not flat and any pilot or ship‘s navigator must understand the rules of spherical

The discovery of two-dimensional non-Euclidean geometries early in the nineteenth
century by Gauss, Bolyai and Lobachevski allowed us to ask for the first time, ―Could the
geometry of the three-dimensional universe in which we live also be non-Euclidean?‖
The work of Riemann and, later, Minkowski provide a geometric structure for Einstein‘s
theory of relativity and modern theories of cosmology where the ultimate collapse or
expansion of the universe is related to the curvature of space itself.

The discovery of two-dimensional non-Euclidean geometries also initiated a momentous
shift in our view of the entire mathematical enterprise. The question of axiomatic
foundations raised by the non-Euclidean geometries now pervades all branches of the
subject and forms the acid test of mathematical validity.

This class examines the foundations of geometry that lead to Euclid‘s geometry in the
plane and to other possible geometries, most notably spherical and hyperbolic, and
concentrates on exploring the rules of geometric logic that are universal.

                       MATH 331—Linear Algebra
Why algebra? Algebra was invented because of the limitations of our geometric intuition.
In applications ranging from business to engineering to the social sciences, it is often
useful to work with data that naturally correspond to points in the plane, or in three-
dimensional space, or even in fifty-dimensional space. Certainly we could draw pictures
or build models to avoid algebra for points in the plane or in three-dimensional space, but
what pictures or models could help us to ―see‖ in fifty dimensions? This obstacle
motivates the development of vectors and the development of algebraic rules and
techniques for manipulating them. In this course we pursue two intimately related
subjects: matrix theory and linear algebra.

Matrix theory is concerned with vectors and matrices. Vectors are the n-dimensional
generalizations of the ordered pairs representing points in the plane. We will investigate
how our geometric concepts naturally imbed in algebraic concepts. We will learn how the
geometry of lines and planes, lengths and angles is replaced by systems of equations and
operations on vectors. Further, we will see how systems of equations can be analyzed in
terms of the properties of a single algebraic object: the matrix.

Linear algebra is the study of sets of vectors and how operations on individual vectors
can be applied to entire sets. Linear algebra is the abstraction of the fundamental
properties displayed by vectors and matrices. This abstraction allows us to use the
knowledge and skills developed working with vectors and matrices to answer questions
about the behavior of wave functions in Fourier analysis or about the nature of solutions
to important families of differential equations.

This course is very different from calculus. In calculus there are relatively fewer
theoretical ideas, and most of the course is devoted to applying those ideas and the
associated techniques to specific computations. In MATH 331 students learn a large
variety of new ideas and, while calculations are important, they are primarily tools for
understanding the examples that motivate the theory. Consequently much of the work in
this course is focused on explaining why certain relationships between ideas are true or
why certain sets have specified properties rather than on simply producing a slope or an
integral or a number. Calculus is a prerequisite for this course primarily because students
rarely have adequate facility with mathematical thinking, working with equations,
working extensively with symbols, thinking about exceptions or using technical
language—prior to completing the calculus sequence.

       MATH/STAT 341—Introduction to Mathematical
As the title suggests, we will apply mathematical techniques to develop some of the
fundamental ideas of statistics. So just what is statistics? Statistics is the art of extracting
patterns from data. This might consist of summarizing complicated data, whether
numerically, graphically or by constructing a simple mathematical model that connects
pieces of data to one another. Whereas mathematics uses a language of certainty,
theorems and proofs, statistics has developed precisely to deal with uncertainty,
estimates, bounds and probabilities.

In this course we will examine answers to several important questions in statistics. How
do you describe a data set so as to capture its ‗center‘ and its variation? This will lead to
topics such as the mean and the variance of a sample. What is probability and how do we
model it mathematically? This will lead to the classical distributions: binomial, Poisson,
exponential and normal. How do you decide whether your preconceptions about a large
population are in agreement with the data obtained from a sample? This will lead us to
confidence intervals and hypothesis testing.

Throughout the course we will see that statistics is much more that just the application of
mathematical techniques. We will see that, before we can apply the mathematics, we
must have good data and reasonable models. After we have done our mathematical
analysis, we must still decide whether we have enough certainty to make conclusions. In
short, we will be the lawyers, judge and jury in the court of data analysis.

We will apply the techniques of algebra and calculus to investigate probability, to
develop models and to explore their properties and understand why some estimation
techniques have better properties than others do. We will apply Minitab statistical
software to real world data sets and to simulated data sets.
Successful completion of MATH 152 is a prerequisite for this course. This course is
cross-listed under both mathematics and statistics. Students can take this course for the
mathematics major and minor, the statistics minor and the actuarial science minor.

   MATH/STAT 342—Probability and Statistical Theory
This is a continuation of MATH/STAT 341 (Introduction to Mathematical Statistics). In
this class, students will expand their basic knowledge from 341 into broader and deeper
probability and statistics theory. For instance, students will learn about conditional
distributions of multiple random variables, limiting distributions, moment generating
functions and higher moments than mean and variance. Students will learn more methods
for testing statistical hypotheses, such as the two-sample T test, the F-test and non-
parametric methods. There will also be an introduction to analysis of variance (ANOVA).

To insure that students learn more than just theoretical ideas a term project applying class
knowledge to solving real world problems is usually assigned. Minitab will be used for
the data analysis.

Students are required to complete MATH/STAT 341 prior to enrolling in this class.
MATH/STAT 342 is cross listed under mathematics and statistics. Students can take it
for the mathematics major (or minor), the statistics minor and the actuarial science minor.

    MATH/STAT 348—Applied Regression Analysis and
Regression analysis of data is a powerful statistical tool that is widely used in biology,
psychology, management, engineering, medical research, government and many other
fields. It provides a technique for building a reasonable mathematical model that relates
the mean value of a response (e.g., profit) to various independent variables or predictors
(e.g., advertising budgets, size of inventory, etc.).

Any prediction or estimation based on a random sample of data will contain a certain
unknown error. In this course, students will learn various methods to build a best
regression model for a given set of data under certain constraints so that the error is

When the relation between the dependent and independent variables is linear, we call it
linear regression. Students will also learn about nonlinear regression, where there can be
a nonlinear relationship (such as quadratic or exponential). Real world problem solving
skills are emphasized. Minitab is used extensively for the data exploration and data
analysis. A term project (with open topics) is normally assigned for students to explore
knowledge beyond the classroom.
Students are required to complete MATH/STAT 341 prior to enrolling in this class.
MATH/STAT 348 is cross-listed under mathematics and statistics. Students can take it
for the mathematics major (or minor), the statistics minor and the actuarial science minor.

                   MATH 351 Differential Equations
Differential equations are a powerful tool in constructing mathematical models for the
physical world. Their use in industry and engineering is so widespread and they perform
so well that they are among the most successful of modeling tools.

For example, a cup of hot coffee is initially at 170F and is left in a room with an
ambient temperature of 70F . Suppose that initially it is cooling at a rate of 20 per
                                                           0.2T  70 . This is an
minute. Then the model for the cup‘s temperature is
example of a differential equation. We are interested in predicting the temperature, T, of
the coffee at any time t. We can also ask, ―How long does it take the coffee to cool to a
temperature of, say, 110F ?"

                    MATH 356—Numerical Analysis
When one pushes the square root button on a calculator to compute the square root of 2,
one should ask, ―How does the calculator do it?‖ Numerical analysis deals with
implementing numerical methods to answer questions like this one.

While numerical methods have always been useful, since the invention of computers, the
role of numerical methods in scientific research has become essential. No modern applied
mathematician, physical scientist or engineer can be properly trained without some
understanding of numerical methods. There is more involved here than just knowing how
to use the methods. One needs to know how to analyze their accuracy and efficiency.
Numerical analysis is a broad and challenging mathematical activity, whose central
theme is the effective constructability of various kinds of approximations.

                      MATH 433—Abstract Algebra
If you can tell time, you already know some abstract algebra: you just don‘t know you
know it! Suppose you have lunch every day at 1:00pm. Then you‘ll have lunch at 1:00pm
today and at 1:00pm tomorrow. We just called both of those times ‗1:00pm‘, but they‘re
not really the same moment in time, since they‘re occurring on different days! It turns out
they both can be thought of as representatives of a coset of 24 Z in R ; this coset, in turn,
is an element of the factor group R / 24Z .

Huh, you ask? What‘s a coset? What‘s Z ? What‘s R ? What‘s a factor group?! Take this
class and find out! Abstract algebra is the study of algebraic structures such as groups,
rings and fields. (You don‘t know what these objects are yet, but if you take this class
you will!) You encounter such objects everywhere in math: the coordinate plane is an
example of a group; the set of all 2 x 2 matrices over the real numbers is an example of a
ring; the set of all real numbers is a field. By studying these structures abstractly, we can
give one proof for many results that hold for wildly different objects, instead of proving
each result for each object separately.

Abstract algebra is a beautiful and powerful area of mathematics and it is an essential part
of any mathematics curriculum. It has applications in many sciences, from physics to
chemistry, in addition to having extremely important uses in areas such as cryptography.

While the concepts in this class require minimal prerequisite knowledge of topics such as
calculus, this class is heavily proof-based and requires a large amount of mathematical
maturity. The ability to write grammatically and make logical arguments is extremely
important, while the ability to differentiate will be of little, if any, use. Conceptual
understanding, not a calculator, is at the heart of this course!

MATH/EDUC 446—Mathematics in the Secondary School
This course has been designed for prospective teachers of middle school and high school
mathematics and reflects the recommendations of the National Council of Teachers of
Mathematics (NCTM). The following excerpt is from the NCTM Principles and
Standards book:

―The Teaching Principle
Effective mathematics teaching requires understanding what students know and need to
learn and then challenging and supporting them to learn it well.

Teachers need to know and use ‗mathematics for teaching‘ that combines mathematical
knowledge and pedagogical knowledge. They must be information providers, planners,
consultants and explorers of uncharted mathematical territory. They must adjust their
practices and extend their knowledge to reflect changing curricula and technologies and
to incorporate new knowledge about how students learn mathematics. They also must be
able to describe and explain why they are aiming for particular goals.‖

The course takes the art of teaching through a series of motivational ideas suitable for
many grade levels and abilities and includes a discussion of activities, materials and
manipulatives suitable for classroom use. Problem solving and heuristics is a major
theme in the course. Other topics covered include cooperative learning, questioning
techniques, technology, lesson planning, homework options, mini-discovery lessons and
technology lessons.
               MATH 455 Mathematical Analysis
Why does calculus work? In this course we examine the foundations of calculus. What
properties of the real numbers distinguish them from the rational numbers? What role do
these differences play in the development of such fundamental concepts as limits and
convergence? What does continuity really mean, and why do we need it? Along the
way, we will study sequences, series and limits, first of numbers, and then of functions.
One consequence of our study will be a better appreciation of the central role of power
series in many of the results of calculus.

This course is strongly recommended for anyone considering a graduate degree in pure or
applied mathematics, statistics, theoretical physics or operations research. Surprisingly, a
deep understanding of the theoretical underpinnings of calculus is necessary to make
progress in such applied areas as optimization, numerical analysis, financial modeling,
probability and differential equations.

This course is almost entirely focused on formal definitions and rigorous proofs.
Students are encouraged to have as much exposure to proofs as possible prior to enrolling
in this course.


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