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, 128, and 147 are courses containing selected topics from geometry number theory, and financial mathematics. 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, 128, or 147 if a student has already received credit for a 200(or higher)-level mathematics course.
Each of Math 112, 115, 185, and 295 is a first course in calculus and generally credit can be received for only one course from this list. Math 112 is designed for students of business and the social sciences who require only one term of calculus. It neither presupposes nor covers any trigonometry. The sequence 115-116-215 is appropriate for most students who want a complete introduction to calculus. Math 118 is an alternative to Math 116 intended for students of the social sciences who do not intend to continue to Math 215. One of Math 215, 285, or 395 is prerequisite to most more advanced courses in Mathematics. Math 112 does not provide preparation for any subsequent course.
Students planning a career in medicine should note that some medical schools require a course in calculus. Generally either Math 112 or 115 will satisfy this requirement, although most science concentrations require at least a year of calculus. Math 112 is accepted by the School of Business Administration, but Math 115 is prerequisite to concentration in Economics and further math courses are strongly recommended.
The sequences 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 LS&A 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. 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 semester of AP credit, Math 119 and Math 186 (Fall). Both will review the basic concepts of calculus, cover integration and an introduction to differential equations, and introduce the student to the use of the computer algebra system MAPLE. Math 119 will stress experimentation and computation, while Math 186 is intended primarily for engineering and science majors, and will emphasize both applications and theory. Interested students are advised to 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 one of Math 112 or 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 a advisor before switching from one sequence to another. In all cases a maximum total of 16 credits may be earned for calculus courses Math 112 through Math 296, and no credit can be earned for a prerequisite to a course taken after the course itself.
Students completing Math 215 may continue either to Math 216 (Introduction to Differential Equations) or to the sequence Math 217-316 (Linear Algebra-Differential Equations). Math 217-316 is required for all students who intend to take more advanced courses in mathematics, particularly for those who may concentrate in mathematics. Math 217 both serves as a transition to the more theoretical material of advanced courses and provides the background required for optimal treatment of differential equations.
NOTE: WL:3 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. (4). (Excl). (QR/1).
Math 105 is a preparatory class to the calculus sequences. 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.
The course prepares students for Math 115
112. Brief Calculus. See Elementary Courses above. Credit
is granted for only one course from among Math. 112, 113, 115, 185 and 295.
(4). (N.Excl). (BS).
This is a one-term survey course that provides the basics of elementary
calculus. Emphasis is placed on intuitive understanding of concepts and
not on rigor. Topics include differentiation with application to curve sketching
and maximum-minimum problems, antiderivatives and definite integrals. Trigonometry
is not used. This course does not mesh with any of the courses in the other
calculus sequences.
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. (4). (N.Excl). (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 major 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 (Algebra and Trigonometry) or its self-paced equivalent Math 106. 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-82 is recommended).
116. Calculus II. Math. 115. Credit is granted for only one course
from among Math. 116, 119, 186, and 296. (4). (N.Excl). (BS). (QR/2).
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.
119. Calculus II Using MAPLE. Math. 115 or equivalent. Credit
is granted for only one course from among Math. 114, 116, 119, 186, and
296. (4). (Excl).
In Fall Term, 1994, new incoming students with AP credit for Math 115
were enrolled in special sections of Math 116 which met one hour per week
in a computer lab which used MAPLE, a computer software program. MAPLE is
a symbolic algebra program which aids the student in visualization, computation
and organization. Beginning Fall Term, 1995, these special sections of Calculus
II are now called Math 119. 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 4 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 and 215. In addition, students are taught to use the computer algebra system MAPLE (on the Macintosh) - 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. Students are presented with challenging unstructured problems done in groups. Learning to work well with others is an important (and satisfying) part of the course. 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 186 (Fall) is a quite similar course in the honors sequence with greater emphasis on applications to the physical sciences and engineering. Math 219 is the natural sequel. 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.
127. Geometry and the Imagination. Three years of high school
mathematics including a geometry course. No credit granted to those who
have completed a 200- (or higher) level mathematics course. (4). (NS). (BS).
(QR/1).
This course introduces students to the ideas and some of the basic results
in Euclidean and non-Euclidean geometry. Beginning with geometry in ancient
Greece, the course includes the construction of new geometric objects from
old ones by projecting and by taking slices. The next topic is non-Euclidean
geometry. This section begins with the independence of Euclid's Fifth Postulate
and with the construction of spherical and hyperbolic geometries in which
the Fifth Postulate fails; how spherical and hyperbolic geometry differs
from Euclidean geometry. The last topic is geometry of higher dimensions:
coordinatization - the mathematician's tool for studying higher dimensions;
construction of higher-dimensional analogues of some familiar objects like
spheres and cubes; discussion of the proper higher-dimensional analogues
of some geometric notions (length, angle, orthogonality, etc.) This course
is intended for students who want an introduction to mathematical ideas
and culture. Emphasis on conceptual thinking - students will do hands-on
experimentation with geometric shapes, patterns and ideas. Grades based
on homework and a final project. No exams. Text: Beyond the Third Dimension
(Thomas Banchoff, 1990).
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). (NS). (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 a2
+ b2 = c2.
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). (Excl). (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.
The course is not part of a sequence. Students should possess financial
calculators.
175. Combinatorics and Calculus. Permission of Honors advisor.
(4). (N.Excl). (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.
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. (4). (N.Excl). (BS). (QR/1).
The sequence Math 185-186-285-286 is the honors introduction to the
calculus. It is taken by students intending to major 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.
186. Honors Analytic Geometry and Calculus II. Permission of the
Honors advisor. Credit is granted for only one course from among Math. 114,
116, 119, 186, and 296. (4). (N.Excl). (BS). (QR/1).
The sequence Math 185-186-285-286 is the honors introduction to the
calculus. It is taken by students intending to major 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 transcendental functions; techniques of integration; applications of calculus such as elementary differential equations, simple harmonic motion, and center of mass; conic sections; polar coordinates; infinite sequences and series including power series and Taylor series. Other topics, often an introduction to matrices and vector spaces, will be included at the discretion of the instructor. Math 116 is a somewhat less theoretical course which covers much of the same material. Math 285 is the natural sequel.
203. Introduction to MAPLE and MATHEMATICA. Prior or concurrent
enrollment in one term of calculus. No programming experience is assumed.
(1). (Excl).
This course is designed to provide the student with an introduction
to two powerful Computer Algebra Systems (MAPLE and MATHEMATICA) for doing
Algebra, Calculus and Statistical and Graphical Analysis. Recent years have
seen the development of several powerful software packages, known as Computer
Algebra Systems, for doing mathematics on the computer. These programs have
the capacity to solve problems numerically, graphically, and symbolically
in calculus, linear algebra, differential equations, statistics, and many
areas of science and engineering. This one-credit mini-course is a brief
introduction to the two most popular of these systems, Maple and
Mathematica. It will be of interest to all students whose career
interests require mathematical skills. No programming experience is assumed.
Students should have taken or be concurrently enrolled in a first course
in calculus. The elementary features of Maple and Mathematica
will be introduced and applied to various types of problems in algebra and
calculus. 403 is a more thorough introduction to either Maple or
Mathematica. This course introduces the student to a tool which can
be useful in almost any course which uses mathematics.
215. Calculus III. Math. 116 or 186. (4). (Excl). (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 major 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 major 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.
216. Introduction to Differential Equations. Math. 215. (4). (Excl).
(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.
217. Linear Algebra. Math. 215. No credit granted to those who
have completed or are enrolled in Math. 417, 419, or 513. (4). (Excl). (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 Rn; 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.
295. 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). (N.Excl). (BS). (QR/1).
The sequence Math 295-296-395-396 is a more intensive honors sequence
than 185-186-285-286. The material includes all of that of the lower sequence
and substantially more. The approach is theoretical, abstract, and rigorous.
Students are expected to learn to understand and construct proofs as well
as do calculations and solve problems. The expected background is a thorough
understanding of high school algebra and trigonometry. No previous calculus
is required, although many students in this course have had some calculus.
Students completing this sequence will be ready to take advanced undergraduate
and beginning graduate courses. This is not restricted to students
enrolled in the LS&A Honors Program. This course presents an introduction
to mathematical analysis with emphasis on proofs and theory. The precise
content may vary with the instructor, but generally will cover such topics
as Functions of one variable and their representation by graphs, set theory,
construction of the real number field, limits of sequences and functions,
continuity, elementary functions, derivatives and integrals with applications,
parametric representation, polar coordinates, applications of mathematical
induction. Additional topics may include countability, topology of the real
numbers, infinite series, and uniform continuity.