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## 300-499 |
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**Elementary Mathematics Courses.
**In order to accommodate diverse backgrounds and interests, several course options are available to beginning mathematics
students. All courses require three years of high school mathematics;
four years are strongly recommended and more information is given
for some individual courses below. Students with College Board
Advanced Placement credit and anyone planning to enroll in an
upper-level class should consider one of the Honors sequences
and discuss the options with a mathematics advisor.

Students who need additional preparation for calculus are tentatively identified by a combination of the math placement test (given during orientation), college admissions test scores (SAT or ACT), and high school grade point average. Academic advisors will discuss this placement information with each student and refer students to a special mathematics advisor when necessary.

Two courses preparatory to the calculus, Math 105 and Math 110, are offered. Math 105 is a course on data analysis, functions and graphs with an emphasis on problem solving. Math 110 is a condensed half-term version of the same material offered as a self-study course through the Math Lab and directed towards students who are unable to complete a first calculus course successfully. A maximum total of 4 credits may be earned in courses numbered 110 and below. Math 103 is offered exclusively in the Summer half-term for students in the Summer Bridge Program.

Math 127 and 128 are courses containing selected topics from geometry and number theory, respectively. They are intended for students who want exposure to mathematical culture and thinking through a single course. They are neither prerequisite nor preparation for any further course. No credit will be received for the election of Math 127 or 128 if a student already has received credit for a 200- (or higher) level mathematics course.

Each of Math 115, 185, and 295 is a first course in calculus and generally credit can be received for only one course from this list. The sequence 115-116-215 is appropriate for most students who want a complete introduction to calculus. One of Math 215, 285, or 395 is prerequisite to most more advanced courses in Mathematics.

The sequences 156-255-256, 175-176-285-286, 185-186-285-286, and 295-296-395-396 are Honors sequences. All students must have the permission of an Honors advisor to enroll in any of these courses, but they need not be enrolled in the LSA Honors Program. All students with strong preparation and interest in mathematics are encouraged to consider these courses; they are both more interesting and more challenging than the standard sequences.

Math 185-285 covers much of the material of Math 115-215 with more attention to the theory in addition to applications. Most students who take Math 185 have taken a high school calculus course, but it is not required. Math 175-176 assumes a knowledge of calculus roughly equivalent to Math 115 and covers a substantial amount of so-called combinatorial mathematics (see course description) as well as calculus-related topics not usually part of the calculus sequence. Math 175 and 176 are taught by the discovery method: students are presented with a great variety of problems and encouraged to experiment in groups using computers. The sequence Math 295-396 provides a rigorous introduction to theoretical mathematics. Proofs are stressed over applications and these courses require a high level of interest and commitment. Most students electing Math 295 have completed a thorough high school calculus course. The student who completes Math 396 is prepared to explore the world of mathematics at the advanced undergraduate and graduate level.

Students with strong scores on either the AB or BC version of the College Board Advanced Placement exam may be granted credit and advanced placement in one of the sequences described above; a table explaining the possibilities is available from advisors and the Department. In addition, there are two courses expressly designed and recommended for students with one or two semesters of AP credit, Math 119 and Math 156. Both will review the basic concepts of calculus, cover integration and an introduction to differential equations, and introduce the student to the computer algebra system MAPLE. Math 119 will stress experimentation and computation, while Math 156 is an Honors course intended primarily for science and engineering concentrators and will emphasize both applications and theory. Interested students should consult a mathematics advisor for more details.

In rare circumstances and with permission of a Mathematics advisor reduced credit may be granted for Math 185 or 295 after Math 115. A list of these and other cases of reduced credit for courses with overlapping material is available from the Department. To avoid unexpected reduction in credit, students should always consult an advisor before switching from one sequence to another. In all cases a maximum total of 16 credits may be earned for calculus courses Math 115 through Math 396, and no credit can be earned for a prerequisite to a course taken after the course itself.

Students completing Math 116 who are principally interested in the application of mathematics to other fields may continue either to Math 215 (Analytic Geometry and Calculus III) or to Math 216 (Introduction to Differential Equations) – these two courses may be taken in either order. Students who have greater interest in theory or who intend to take more advanced courses in mathematics should continue with Math 215 followed by the sequence Math 217-316 (Linear Algebra-Differential Equations). Math 217 (or the Honors version, Math 513) is required for a concentration in Mathematics; it both serves as a transition to the more theoretical material of advanced courses and provides the background required for optimal treatment of differential equations in Math 316. Math 216 is not intended for mathematics concentrators.

NOTE: WL:2 for all courses.

A * maximum total of 4 credits * may be earned in Mathematics
courses numbered 110 and below. A * maximum total of 16 credits *
may be earned for calculus courses Math 112 through Math 396, and no credit can be earned for a prerequisite to a course taken
after the course itself.

**105. Data, Functions, and Graphs. *** Students with credit for Math. 103 can
elect Math. 105 for only 2 credits. No credit granted to those
who have completed any Mathematics course numbered 110 or higher.
(4). (MSA). (QR/1). *

Math 105 serves both as a preparatory class to the calculus sequences
and as a terminal course for students who need only this level
of mathematics. Students who complete 105 are fully prepared for
Math 115. This is a course on analyzing data by means of functions
and graphs. The emphasis is on mathematical modeling of real-world
applications. The functions used are linear, quadratic, polynomial, logarithmic, exponential, and trigonometric. Algebra skills are
assessed during the term by periodic testing. Math 110 is a condensed
half-term version of the same material offered as a self-study
course through the Math Lab. WL:2

Sections 170 through 175 by Permission of CSP only.

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**110. Pre-Calculus (Self-Study).
*** See Elementary Courses above. Enrollment
in Math 110 is by recommendation of Math 115 instructor and override
only. No credit granted to those who already have 4 credits for
pre-calculus mathematics courses. (2). (Excl). *

The course covers data analysis by means of functions and graphs. Math 110 serves both as a preparatory class to the calculus sequences and as a terminal course for students who need only this level of mathematics. The course is a condensed, half-term version of Math 105 (Math 105 covers the same material in a traditional classroom setting) designed for students who appear to be prepared to handle calculus but are not able to successfully complete Math 115. Students who complete 110 are fully prepared for Math 115. Students may enroll in Math 110 only on the recommendation of a mathematics instructor after the third week of classes in the Fall and must visit the Math Lab to complete paperwork and receive course materials. WL:2

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**115. Calculus I. *** Four
years of high school mathematics. See Elementary Courses
above. Credit usually is granted for only one course from among
Math. 112, 115, 185, and 295. No credit granted to those who have
completed Math. 175. (4). (MSA). (BS). (QR/1). *

The sequence Math 115-116-215 is the standard complete introduction to the concepts and methods of calculus. It is taken by the majority of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields. The emphasis is on concepts and solving problems rather than theory and proof. All sections are given a uniform midterm and final exam. The course presents the concepts of calculus from three points of view: geometric (graphs); numerical (tables); and algebraic (formulas). Students will develop their reading, writing, and questioning skills.

Topics include functions and graphs, derivatives and their applications to real-life problems in various fields, and definite integrals. Math 185 is a somewhat more theoretical course which covers some of the same material. Math 175 includes some of the material of Math 115 together with some combinatorial mathematics. A student whose preparation is insufficient for Math 115 should take Math 105 (Data, Functions, and Graphs). Math 116 is the natural sequel. A student who has done very well in this course could enter the Honors sequence at this point by taking Math 186. The cost for this course is over $100 since the student will need a text (to be used for 115 and 116) and a graphing calculator (the Texas Instruments TI-83 is recommended). WL:2

*Section 110. (RC). * This is a section of an LS&A
Math class reserved especially for Residential College students.
Since it is an LS&A Math class, it will be letter graded for
ALL students enrolled. The course will cover the same material
as other sections of Math 115 – functions, tables, and graphs;
derivatives; integrals; and applications of calculus. Students
are required to have a graphing calculator (TI-83 recommended), and will take the uniform exams along with the other sections
of LS&A Math 115. The main text is * Calculus * by Hughes-Hallet * et al. * Students whose pre-calculus skills are not strong
should consider electing Math 105.

*Sections 170 through 175 * by Permission of CSP only.

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**116. Calculus II. *** Math.
115. Credit is granted for only one course from among Math. 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1). *

See Math 115 for a general description of the sequence Math 115-116-215.

Topics include the indefinite integral, techniques of integration, introduction to differential equations, infinite series. Math
186 is a somewhat more theoretical course which covers much of the same material. Math 215 is the natural sequel. A student who
has done very well in this course could enter the Honors sequence
at this point by taking Math 285. WL:2

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**119. Calculus II Using
MAPLE. *** Math. 115 or score of 3 or higher on the AB
or BC Advanced Placement Calculus exam. Credit is granted for
only one course from among Math. 114, 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1). *

MAPLE is a symbolic algebra computer software program which aids the student in visualization, computation, and organization. Students
who complete Math 119 and continue to Calculus III should elect
Math 219 which will be a special MAPLE-oriented version of Math
215.

The sequence Math 119-219 is intended for students who have
earned a score of 3 or better on either the AB or BC version of the Advanced Placement Exam in Mathematics. No familiarity with
computers is necessary. The material covered will be approximately that of Math 116. In addition, students are taught to use the
computer algebra system MAPLE – a symbolic algebra program which
aids the student in visualization, computation, and organization
- as a tool to do routine calculations, to visualize and to explore.
MAPLE is thoroughly integrated into the course, and the use of
MAPLE is permitted (encouraged) on homework and tests. The emphasis
is on concepts and problem-solving rather than theory and proof.
Topics include applications of the definite integral, separable
differential equations, inverse functions, infinite sequences
and series, conics and parametric curves. Math 156 (Fall) is a
quite similar course in the Honors sequence with greater emphasis
on applications to the physical sciences and engineering. Math
255 is the natural sequel to Math 156. Students who complete Math
119 and continue to Calculus III should elect Math 219 which is
a special MAPLE-oriented version of Math 215. A student who has
done very well in this course could enter the Honors sequence
at this point by taking 285. WL:2

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**128. Explorations in
Number Theory. *** High school mathematics through at
least Analytic Geometry. No credit granted to those who have completed
a 200- (or higher) level mathematics course. (4). (MSA). (BS).
(QR/1). *

This course is intended for non-science concentrators and students
in the pre-concentration years with no intended concentration, who want to engage in mathematical reasoning without having to
take calculus first. Students will be introduced to elementary
ideas of number theory, an area of mathematics that deals with
properties of the integers. Students will make use of software
provided for IBM PCs to conduct numerical experiments and to make
empirical discoveries. Students will formulate precise conjectures, and in many cases prove them. Thus the students will, as a group, generate a logical development of the subject. After studying
factorizations and greatest common divisors, emphasis will shift
to the patterns that emerge when the integers are classified according
to the remainder produced upon division by some fixed number ('congruences').
Once some basic tools have been established, applications will
be made in several directions. For example, students may derive
a precise parameterization of Pythagorean triples a^{2}
+ b^{2} = c^{2}. Students who like math but don't
especially like calculus will want to enroll in this first-year
MSA seminar. Students will do hands-on experimentation with numerical
patterns and will tackle numerical riddles and brainteasers as they focus on empirical discovery and proof of theorems. Students
will write their own text in number theory and enjoy their growing
ability to think like mathematicians. WL:2

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**147. Introduction to
Interest Theory. *** Math. 112 or 115. No credit granted
to those who have completed a 200- (or higher) level mathematics
course. (3). (MSA). (BS). *

This course is designed for students who seek an introduction
to the mathematical concepts and techniques employed by financial
institutions such as banks, insurance companies, and pension funds.
Actuarial students, and other mathematics concentrators, should
elect Math 424 which covers the same topics but on a more rigorous
basis requiring considerable use of the calculus. Topics covered
include: various rates of simple and compound interest, present
and accumulated values based on these; annuity functions and their
application to amortization, sinking funds and bond values; depreciation
methods; introduction to life tables, life annuity, and life insurance
values. This course is not part of a sequence. Students should
possess financial calculators. WL:2

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**156. Applied Honors Calculus
II. *** Score of 4 or 5 on the AB or BC Advanced Placement
calculus exam. Credit is granted for only one course among Math
114, 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).
*

The sequence 156-255-256 is an Honors calculus sequence for engineering
and science concentrators who scored 4 or 5 on the AB or BC Advanced
Placement calculus exam. Topics include Riemann sums, the definite
integral, fundamental theorem of calculus, applications of integral
calculus * (e.g. * arclength, surface area, work, hydrostatic
pressure, center of mass), improper integrals, infinite sequences
and series, differential equations, complex numbers. MAPLE will
be used throughout. WL:2

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**175. Combinatorics and Calculus. *** Permission of Honors advisor. No credit
granted to those who have completed a 200-level or higher Mathematics
course. (4). (MSA). (BS). (QR/1). *

This course is an alternative to Math 185 as an entry to the Honors
sequence. The sequence Math 175-176 is a two-term introduction
to Combinatorics, Dynamical Systems, and Calculus. The topics
are integrated over the two terms although the first term will
stress combinatorics and the second term will stress the development
of calculus in the context of dynamical systems. Students are
expected to have some previous experience with the basic concepts
and techniques of calculus. The course stresses discovery as a
vehicle for learning. Students will be required to experiment throughout the course on a range of problems and will participate
each term in a group project. Grades will be based on homework
and projects with a strong emphasis on homework. Personal computers
will be a valuable experimental tool in this course, and students
will be asked to learn to program in either BASIC, PASCAL, or
FORTRAN. There are two major topic areas: enumeration theory and graph theory. The section on enumeration theory will emphasize
classical methods for counting including: (1) binomial theorem
and its generalizations; (2) solving recursions; (3) generating
functions; and (4) the inclusion-exclusion principle. In the process, we will discuss infinite series. The section on graph theory will
include basic definitions and some of the more interesting and useful theorems of graph theory. The emphasis will be on topological
results and applications to computer science and will include:
(1) connectivity; (2) trees, Prufer codes, and data structures;
(3) planar graphs, Euler's formula and Kuratowski's Theorem; and (4) coloring graphs, chromatic polynomials, and orientation. This
material has many applications in the field of Computer Science.
Math 176 is the standard sequel. WL:2

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**185. Honors Analytic
Geometry and Calculus I. *** Permission of the Honors
advisor. Credit is granted for only one course from among Math.
112, 113, 115, 185, and 295. No credit granted to those who have
completed Math. 175. (4). (MSA). (BS). (QR/1). *

The sequence Math 185-186-285-286 is the Honors introduction to the calculus. It is taken by students intending to concentrate
in mathematics, science, or engineering as well as students heading
for many other fields who want a somewhat more theoretical approach.
Although much attention is paid to concepts and solving problems, the underlying theory and proofs of important results are also
included. This sequence is **not** restricted to
students enrolled in the LS&A Honors Program.

Topics covered include functions and graphs, limits, derivatives, differentiation of algebraic and trigonometric functions and applications, definite and indefinite integrals and applications. Other topics
will be included at the discretion of the instructor. Math 115
is a somewhat less theoretical course which covers much of the
same material. Math 186 is the natural sequel. WL:2

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**215. Calculus III. *** Math.
116, 119, 156, 176, 186, or 296. (4). (MSA). (BS). (QR/1). *

The sequence Math 115-116-215 is the standard complete introduction
to the concepts and methods of calculus. It is taken by the majority
of students intending to concentrate in mathematics, science, or engineering as well as students heading for many other fields.
The emphasis is on concepts and solving problems rather than theory
and proof. All sections are given a midterm and final exam. Topics
include vector algebra and vector functions; analytic geometry
of planes, surfaces, and solids; functions of several variables
and partial differentiation; line, surface, and volume integrals
and applications; vector fields and integration; Green's Theorem
and Stokes' Theorem. There is a weekly computer lab using * Maple *
software. Math 285 is a somewhat more theoretical course which
covers the same material. For students intending to concentrate
in mathematics or who have some interest in the theory of mathematics
as well as its applications, the appropriate sequel is Math 217.
Students who intend to take only one further mathematics course
and need differential equations should take Math 216. WL:2

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**216. Introduction to
Differential Equations. *** Math. 116, 119, 156, 176, 186, or 296. (4). (MSA). (BS). *

For a student who has completed the calculus sequence, there are
two sequences which deal with linear algebra and differential
equations, Math 216-417 (or 419) and Math 217-316. The sequence
Math 216-417 emphasizes problem-solving and applications and is
intended for students of engineering and the sciences. Math concentrators
and other students who have some interest in the theory of mathematics
should elect the sequence Math 217-316. After an introduction
to ordinary differential equations, the first half of the course
is devoted to topics in linear algebra, including systems of linear
algebraic equations, vector spaces, linear dependence, bases, dimension, matrix algebra, determinants, eigenvalues, and eigenvectors.
In the second half these tools are applied to the solution of
linear systems of ordinary differential equations. Topics include:
oscillating systems, the Laplace transform, initial value problems, resonance, phase portraits, and an introduction to numerical methods.
There is a weekly computer lab using * MATLAB * software.
**This course is not intended for mathematics concentrators, who should elect the sequence 217-316**. Math 286 covers
much of the same material in the Honors sequence. The sequence
Math 217-316 covers all of this material and substantially more
at greater depth and with greater emphasis on the theory. Math
404 covers further material on differential equations. Math 217
and 417 cover further material on linear algebra. Math 371 and 471 cover additional material on numerical methods. WL:2

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**217. Linear Algebra.
*** Math. 215, 255, or 285. No credit granted to those
who have completed or are enrolled in Math. 417, 419, or 513.
(4). (MSA). (BS). (QR/1). *

For a student who has completed the calculus sequence, there are
two sequences which deal with linear algebra and differential
equations, Math 216-417 (or 419) and Math 217-316. The sequence
Math 216-417 emphasizes problem-solving and applications and is
intended for students of Engineering and the sciences. Math concentrators
and other students who have some interest in the theory of mathematics
should elect the sequence Math 217-316. These courses are explicitly
designed to introduce the student to both the concepts and applications
of their subjects and to the methods by which the results are
proved. Therefore the student entering Math 217 should come with
a sincere interest in learning about proofs. The topics covered
include: systems of linear equations; matrix algebra; vectors, vector spaces, and subspaces; geometry of R^{n}; linear
dependence, bases, and dimension; linear transformations; eigenvalues
and eigenvectors; diagonalization; inner products. Throughout there will be emphasis on the concepts, logic, and methods of theoretical mathematics. Math 417 and 419 cover similar material
with more emphasis on computation and applications and less emphasis
on proofs. Math 513 covers more in a much more sophisticated way.
The intended course to follow Math 217 is 316. Math 217 is also
prerequisite for Math 412 and all more advanced courses in mathematics.
WL:2

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**256. Applied Honors Calculus
IV. *** Math. 255. (4). (MSA). (BS). *

Linear algebra, matrices, systems of differential equations, initial
and boundary value problems, qualitative theory of dynamical systems, nonlinear equations, numerical methods, MAPLE. WL:2

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**285. Honors Analytic
Geometry and Calculus III. *** Math. 176 or 186, or permission
of the Honors advisor. (4). (MSA). (BS). *

See Math. 185 for a general description of the sequence Math 185-186-285-286.

Topics include vector algebra and vector functions; analytic
geometry of planes, surfaces, and solids; functions of several
variables and partial differentiation, maximum-minimum problems;
line, surface, and volume integrals and applications; vector fields
and integration; curl, divergence, and gradient; Green's Theorem
and Stokes' Theorem. Additional topics may be added at the discretion
of the instructor. Math 215 is a less theoretical course which
covers the same material. WL:2

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**289. Problem Seminar.
*** (1). (Excl). (BS). May be repeated for credit with
permission. *

One of the best ways to develop mathematical abilities is by solving
problems using a variety of methods. Familiarity with numerous
methods is a great asset to the developing student of mathematics.
Methods learned in attacking a specific problem frequently find
application in many other areas of mathematics. In many instances
an interest in and appreciation of mathematics is better developed
by solving problems than by hearing formal lectures on specific
topics. The student has an opportunity to participate more actively
in his/her education and development. This course is intended
for superior students who have exhibited both ability and interest
in doing mathematics, but it is not restricted to Honors students.
This course is excellent preparation for the Putnam exam. Students
and one or more faculty and graduate student assistants will meet
in small groups to explore problems in many different areas of
mathematics. Problems will be selected according to the interests
and background of the students. WL:2

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**295(195). Honors Mathematics
I. *** Prior knowledge of first year calculus and permission
of the Honors advisor. Credit is granted for only one course from
among Math. 112, 113, 115, 185, and 295. (4). (MSA). (BS). (QR/1).
*

Math 295-296-395-396 is the main Honors calculus sequence. It
is aimed at talented students who intend to major in mathematics, science, or engineering. The emphasis is on concepts, problem
solving, as well as the underlying theory and proofs of important
results. Students interested in taking advanced mathematical courses
later should consider seriously starting with this sequence. The
expected background is high school trigonometry and algebra (previous
calculus not required). This sequence is not restricted to students
enrolled in the LS&A Honors program. Real functions, limits, continuous functions, limits of sequences, complex numbers, derivatives, indefinite integrals and applications, some linear algebra. Math
175 and Math 185 are less intensive Honors courses. Math 296 is the intended sequel. WL:2

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