First-Year Courses in Mathematics (Division 428)
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Open courses in Mathematics
Wolverine Access Subject listing for MATH
Take me to the Fall Term '00 Time Schedule for Mathematics.
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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 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 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.
Attention Potential Elementary School Teachers: Math 385 is Offered this Spring Term
All elementary teaching certificate candidates are required to take two math courses, Math 385 and Math 489, either before or after admission to the School of Education. Math 385 is offered in the Fall Term, Math 489 in the Winter Term. Due to heavy enrollment pressure, Math 385 will be offered this Spring Term (IIIA 2000) as well. Last Fall Term a number of students were closed out of Math 385. Next Fall Term, class-size limits will be STRICTLY enforced. Anyone who can elect Math 385 in the Spring Term is urged to do so. It is the surest way to guarantee yourself a place in the course. The next Spring Term offering of Math 385 will be in 2002. For further information, contact Prof. Krause at his e-mail address, krause@math.lsa.umich.edu.
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.
Math. 110. Pre-Calculus (Self-Study).
Section 001 – Course Meets the Second Half of the Term. Students Work Independently with Guidance from Math Lab Staff. There will be no Formal Lecture.
Prerequisites & Distribution: 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).
Credits: (2).
Course Homepage: http://www.math.lsa.umich.edu/~meggin/math110.html
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 Winter and must visit the Math Lab to complete paperwork and receive course materials.
Math. 115. Calculus I.
Section – There will be Joint Evening Examinations for All sections of Math 115, 6-8 p.m., Weds, Oct 6 and Nov 10. Also a Joint Final.
Prerequisites & Distribution: 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).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~meggin/math115/
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).
Math. 115. Calculus I.
Section 100 – Students in Math 115 Section 100 Receive Individualized Self-Paced Instruction in the Mathematics Laboratory in Room B860 E H. Students Must Go to the Math Lab During the First Full Week of Classes.
Prerequisites & Distribution: 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).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~meggin/math115/
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.
Math. 116. Calculus II.
Section – There will be Joint Evening Examinations for All Sections of Math 116, 6-8 p.m., Tues., Oct. 12 and Tues., Nov 16. Also a Joint Final.
Prerequisites & Distribution: Math. 115. Credit is granted for only one course from among Math. 116, 119, 156, 176, 186, and 296. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/116/index.shtml
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.
Math. 156. Applied Honors Calculus II.
Section – There will be Joint Evening Examinations for All Sections of Math 156, Thurs, Oct 14 and Weds, Nov 17, 6:00 – 8:00 p.m. Also a Joint Final.
Prerequisites & Distribution: 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).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~krasny/math156.html
Math 156 is a 2nd semester honors calculus course
for engineering and science students.
The course is offered in the Fall semester
and
is designed for entering freshmen
who have
Advanced Placement credit for 1st semester calculus.
The course emphasizes computational skills,
conceptual understanding,
and
applications of calculus.
Math 156 provides students with
the background needed for
a variety of subsequent courses in
math, science and engineering.
It also introduces students to MAPLE,
a high-level software tool for doing math on a computer.
Prerequisite:
score of 4 or 5 on the Advanced Placement AB calculus exam
Syllabus:
- Applications of the Integral:
sigma notation,
Riemann sums,
definite integral,
fundamental theorem of calculus,
work,
improper integrals,
arclength,
surface area,
moments and center of mass,
hydrostatic force,
probability density functions
- Differential Equations:
exponential growth and decay,
Newton's law of cooling/heating,
logistic equation,
stability of equilibrium points
- Series:
sequences,
series,
geometric series,
integral and comparison tests,
alternating series,
ratio test,
power series,
radius of convergence,
Taylor series,
binomial series,
applications of Taylor approximation
Review: (as needed)
integration by parts,
trigonometric substitution,
partial fractions,
trigonometric integrals,
l'Hopital's rule
Special Topics:
asymptotic expansions,
Bessel function,
complex numbers,
Euler's formula,
Gamma function,
hyperbolic functions,
Laplace transform,
parametric curves,
polar coordinates
Exams: 2 midterm exams and a final exam
Homework:
Math 156 has weekly homework assignments.
Students may work together and discuss the homework problems
with each other,
but each student should write up and submit their own set of solutions.
After the assignment is collected,
solutions will be available in a loose-leaf book
at the Undergraduate Library Reserve Desk on the 2nd floor
of the Shapiro Library.
Sequel:
- Math 255 - Multivariable Calculus
-
Math 256 - Differential Equations and Linear Algebra
Math. 175. Combinatorics and Calculus.
Section 001 – Introduction to Cryptology and Discrete Mathematics
Prerequisites & Distribution: Permission of Honors advisor. No credit granted to those who have completed a 200-level or higher Mathematics course. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~fomin/175.html
Text (required): An Introduction to Cryptology and Discrete
Math - The Math 175 Coursepack, by C.Greene, P.Hanlon, T.Hsu, and
J.Hutchinson.
Prerequisites: Math 115 or equivalent (Single-variable
calculus) recommended.
This course gives a historical introduction to Cryptology, the
science of secrete codes. It begins with the oldest recorded codes,
taken from hieroglyphic engravings, and ends with the encryption
schemes used to maintain privacy during internet credit card transactions.
Since secret codes are based on mathematical ideas, each new kind of
encryption method leads in this course to the study of new mathematical
ideas and results.
The first part of the course deals with permutation-based codes:
substitutional ciphers, transpositional codes, Vigenere ciphers and
more complex polyalphabetic substitutions including the those created
by rotor machines such as the Enigma. The mathematical subjects treated
in this section include permutations, modular arithmetic and some
elementary statistics.
In the second part of the course, the subject moves to bit stream
encryption methods. These include block cipher schemes such as the
Data Encryption Standard (DES). The mathematical concepts introduced
here are recurrence relations and some more advanced statistical results.
Public key encryption is the subject of the final part of the course.
We learn the mathematical underpinnings of Diffie-Hellman key exchange,
RSA and Knapsack codes. A substantial number of results from elementary
number theory are needed and proved in this section of the course.
There is considerable development of problem-solving skills in Math 175.
So, students taking the course should have significant mathematical
experience and sophistication. We recommend that students who sign up
for this course have credit for Math 115 or have taken a course at least
to the level of Math 115.
There are no quizzes and no exams in the course. The course is based
in large part on a series of homework sets in which students are asked to
solve problems according to Steps 1-6 above. There will also be weekly
computer labs and a final project which is an elaborate cryptanalysis
problem which serves as a capstone experience the course.
Math. 185. Honors Calculus I.
Prerequisites & Distribution: 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).
Credits: (4).
Course Homepage: No Homepage Submitted.
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.
Math. 214. Linear Algebra and Differential Equations.
Prerequisites & Distribution: Math 115 and 116. Credit can be earned for only one of Math. 214, 217, 417, or 419. Two credits granted to those who have completed or are enrolled in Math. 216. No credit granted to those who have completed or are enrolled in Math 513. (4). (MSA). (BS).
Credits: (4).
Course Homepage: No Homepage Submitted.
This course is intended for second-year students who might otherwise take Math 216 (Introduction to Differential Equations) but who have a greater need or desire to study Linear Algebra. This may include some Engineering students, particularly from Industrial and Operations engineering (IOE), as well as students of Economics and other quantitative social sciences. Students intending to concentrate in Mathematics must continue to elect Math 217.
While Math 216 includes 3-4 weeks of Linear Algebra as a tool in the study of Differential Equations, Math 214 will include roughly 3 weeks of Differential Equations as an application of Linear Algebra. The textbook is Linear Algebra and its Applications by David Lay.
The following is a tentative outline of the course:
- Systems of linear equations, matrices, row operations, reduced row echelon form, free variables, basic variables, basic solution, parametric description of the solution space. Rank of a matrix.
- Vectors, vector equations, vector algebra, linear combinations of vectors, the linear span of vectors.
- The matrix equation Ax = b. Algebraic rules for multiplication of matrices and vectors.
- Homogeneous systems, principle of superposition.
- Linear independence.
- Applications, Linear models
- Matrix algebra, dot product, matrix multiplication.
- Inverse of a matrix.
- Invertible matrix theorem.
- Partitioned matrices.
- 2-dimensional discrete dynamical systems.
- Markov process, steady state.
- Transition matrix, eigenvector, steady state lines (affine hulls).
- Geometry of two and three dimensions: affine hulls, linear hulls, convex hulls, half planes, distance from point to a plane, optimization.
- Introduction to linear programming.
- The geometry of transition matrices in 2 dimensions (rotations, shears, ellipses, eigenvectors).
- Transition matrices for 3-D (rotations, orthogonal matrices, symmetric matrices)
- Determinants.
- 2- and 3-dimensional determinant as area and volume.
- Eigenvectors and Eigenvalues
- Eigenvectors
- Complex numbers including Euler's formula.
- Complex eigenvalues and their geometric meaning
- Review of ordinary differential equations.
- Systems of ordinary differential equations in 2 dimensions.
Regular problem sets and exams.
Math. 215. Calculus III.
Section – There will be Joint Evening Examinations for All Sections of Math 215, 6-8 p.m., Thurs Oct 14, And Weds Nov 18. Also a Joint Final.
Prerequisites & Distribution: Math. 116, 119, 156, 176, 186, or 296. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS). (QR/1).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/215/
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.
Math. 216. Introduction to Differential Equations.
Section – There will be Joint Evening Examinations for All Sections of Math 216, 6-8 p.m., Mon, Oct 11 And Nov 15. Also a Joint Final.
Prerequisites & Distribution: Math. 116, 119, 156, 176, 186, or 296. Credit can be earned for only one of Math. 216, 256, 286, or 316. No credit granted to those who have completed or are enrolled in Math 214. (4). (MSA). (BS).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/courses/216/
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.
Math. 285. Honors Calculus III.
Section 002.
Prerequisites & Distribution: Math. 176 or 186, or permission of the Honors advisor. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~meggin/math285/index.html
Required Text: Calculus by Stewart, fourth edition, Brooks/Cole, 1999.
Background and Goals: The sequence Math 185-186-285-286 is the honors introduction to 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.
Content: 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's Theorem. Additional topics may be added at the discretion of the instructor.
Alternatives: Math 215 (Calculus III) is a somewhat less theoretical course which covers much of the same material.
Subsequent Courses: Math 216 (Introduction To Differential Equations), Math 286 (Honors Differential Equations) or Math 217 (Linear
Algebra).
Math. 285. Honors Calculus III.
Section 003.
Prerequisites & Distribution: Math. 176 or 186, or permission of the Honors advisor. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS).
Credits: (4).
Course Homepage: http://www.math.lsa.umich.edu/~varolin/285.html
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.
Math. 285. Honors Calculus III.
Section 004.
Instructor(s): Robert L Griess Jr (rlg@umich.edu)
Prerequisites & Distribution: Math. 176 or 186, or permission of the Honors advisor. Credit can be earned for only one of Math. 215, 255, or 285. (4). (MSA). (BS).
Credits: (4).
Course Homepage: No Homepage Submitted.
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.
Math. 295. Honors Mathematics I.
Prerequisites & Distribution: 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).
Credits: (4).
Course Homepage: No Homepage Submitted.
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.
This page was created at 8:01 AM on Wed, Oct 4, 2000.
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