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 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. 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 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 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. 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 majors 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 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 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 intending to take more advanced courses in mathematics, however, must follow the sequence 116-215-217-316. 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: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). (MSA). (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.
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110. Pre-Calculus (Self-Study). See Elementary Courses above. No credit granted to those who already have 4 credits for pre-calculus mathematics courses. (2). (Excl).
Math 110 is a preparatory course for the calculus sequence. Students who complete Math 110 are fully prepared for Math 115. The course is a condensed, half-term version of Math 105 designed for students who appear to be prepared to handle calculus but are not able to successfully complete Math 115. Students enrolling in Math 110 must visit the Math Lab to complete paperwork and receive course materials. The course covers data analysis by means of functions and graphs.
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112. Brief Calculus. See Elementary Courses above. Credit is granted for only one course from among Math. 112, 113, 115, 185 and 295. (4). (MSA). (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.
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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. (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.
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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-82 is recommended).
116. Calculus II. Math. 115. Credit is granted for only one course from among Math. 116, 119, 156, 186, and 296. (4). (MSA). (BS). (QR/1).
See Math 115 for a general description of the sequence Math 115-116-215.
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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. Credit is granted for only one course from among Math. 114, 116, 119, 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.
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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 (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 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.
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). (MSA). (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).
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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 a2 + b2 = c2. 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.
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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. The course is not part of a sequence. Students should possess financial calculators.
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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 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.
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175. Combinatorics and Calculus. Permission of Honors advisor. (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.
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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. (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.
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203. Introduction to MAPLE and MATHEMATICA. Prior or concurrent enrollment in one term of calculus. No programming experience is assumed. No credit granted to those who have completed Math. 119. (1). (Excl). (BS). Offered mandatory credit/no credit. May be repeated for a total of two credits.
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. Math 403 is a more thorough introduction to either Maple and Mathematica. This course introduces the student to a tool which can be useful in almost any course which uses mathematics.
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215. Calculus III. Math. 116 or 186. (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.
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216. Introduction to Differential Equations. Math. 215. (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.
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217. Linear Algebra. Math. 215. 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 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.
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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.
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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). (MSA). (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.
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