For students with previous exposure to algebra but not sufficient mastery for OM013 Precalculus with Trigonometry. This course reviews and extends the topics of beginning algebra: linear equations and inequalities, absolute value, quadratic inequalities, roots and exponents, and systems of equations. Other topics include: exponential and logarithmic functions, conic sections, and arithmetic and geometric sequences.

For students who have had substantial previous exposure to algebra. The course builds on and deepens all the topics from OM011 Beginning Algebra and OM012 Honors Intermediate Algebra. Functions are studied in detail, including composition and inverses. Other topics include: the algebra of exponential and logarithmic functions, techniques of graphing and matrices, mathematical induction, sequences and series, and analytic geometry. Approximately one third of the course focuses on trigonometry and its applications.

An advanced placement course in differential and integral calculus. Topics: functions and graphs, a rigorous development limits, continuity, derivatives and differentiability, applications of the derivative, curve sketching, related rates, implicit differentiation, parametric equations, polar functions, vector functions, l’Hospital’s rule, Riemann sums, indefinite and definite integrals, techniques of integration, applications of integration, the Fundamental Theorem of Calculus, numerical approximations to definite integrals, improper integrals, differential equations, polynomial approximations, Taylor series, and convergence and divergence of infinite sequences and series. This course prepares students for the AP Calculus BC exam.

This semester-long course in theoretical mathematics develops students’ facility with abstract conceptual work and prepares students for subjects at the upper-division undergraduate level. Students are expected to have completed Honors Precalculus with Trigonometry; prior completion of AP Calculus is recommended. Students gain experience analyzing complex problem situations, formulating solutions, rigorously justifying arguments, and presenting mathematical reasoning clearly and effectively, both orally and in writing. Course topics include general guidelines for analyzing problems, proving conditional and biconditional statements, the contrapositive method, working with negations, proof by contradiction, problem-solving heuristics, understanding quantifiers, mathematical induction, the construction method, working with nested quantifiers, and special proof techniques. The course focuses on practical problem-solving and proof-construction techniques that will be invaluable in many university-level mathematics courses.

Differential calculus for functions of two or more variables. Topics: vectors and vector-valued functions in 2-space and 3-space, tangent and normal vectors, curvature, functions of two or more variables, partial derivatives and differentiability, directional derivatives and gradients, maxima and minima, optimization using Lagrange multipliers.

An introductory course in linear algebra. Topics: linear spaces, transformations, matrices, eigenvalues, eigenvectors, and linear operators.

Theory of functions of a real variable. Topics: sequences, series, limits, continuity, differentiation, integration, and basic point-set topology.

Theory of abstract algebra, with particular emphasis on applications involving symmetry. Topics: groups, rings, fields, matrix and crystallographic groups, and constructibility.

Fall only Logic provides an essential methodological framework of reasoning connecting a wide variety of disciplines in the humanities and sciences, including philosophy, mathematics, computer science, linguistics, cognitive science, and economics. This course will introduce students to logic and its applications highlighted by recent developments in these fields. We will use the open source logic course Logic in Action, which has been developed by the international team of Prof. Johan van Benthem at Amsterdam, and taught in many places, including Stanford, Amsterdam, Beijing, Seville, etc.