Advanced Placement for Core Mathematics Classes

The Core Mathematics program at HMC consists of a sequence of eight half-courses (Math 11–14 and 61–64) described below. For an explanation of the half-course model, see the Mathematics Core Curriculum.

A year of calculus at the high school level is an entrance requirement for HMC, so our courses are taught with that in mind. In the 2009–2010 academic year, a new Math 11M course is being offered to interested students who qualify, which may be taken as an alternative to Math 11. Naturally, we wish to place you in the math course that will provide you with the appropriate challenge and ensure you have mastered the foundational mathematics required of scientists, engineers, and mathematicians.

What is the difference between Math 11 and Math 11M?

Math 11 Math 11M
Every HMC student is entitled to take Math 11. Students must place into Math 11M.
A score of 5 on the AP Calculus BC Exam (or a score of 6 on the International Baccalaureate Mathematics Exam) will allow you to place out of Math 11 and into Math 11M. A score of 5 on the AP Calculus BC Exam (or a score of 6 on the International Baccalaureate Mathematics Exam) will allow you to place into Math 11M but not out of it.
Assumes some foundation in single-variable calculus. Assumes an already strong foundation in single-variable calculus.
Focuses on strengthening foundations in single-variable calculus while also introducing multivariable calculus topics. Focuses on multivariable topics and exploring more theory behind selected single-variable topics (not all single-variable topics will be covered).
Meets 4 days/week. Meets 3 days/week.
Meets in the second half semester (Math 12 is taken in the first half of the Fall semester). Meets in the first half semester (Math 12 is taken in the second half of the Fall semester).

Taking Math 11 over Math 11M or vice versa will not affect your eligibility for course placement beyond the first semester. Both courses will prepare you equally well for the rest of the math core. Based on our past experience, we expect about 70% of students to choose Math 11, about 25% to choose Math 11M, and about 5% to place out of both.

Placing Into Math 11M

Students who earn a score of 5 on the AP Calculus BC Exam, a score of 6 on the International Baccalaureate Mathematics Exam, or a grade of B or higher in a year-long college-level calculus course are entitled to place into Math 11M, but must request to do so. Students may also place into Math 11M by taking a placement exam during Orientation Week. Note that students who are eligible to place out of Math 11 may still wish to take Math 11 (and many do).

Placing Out of Both Math 11 and Math 11M

Students who have earned a grade of B or higher in a Multivariable Calculus class are entitled to place out of Math 11 and 11M, but must request to do so (see next page of the survey). Students may also place out of Math 11 and 11M by taking a placement exam during Orientation Week.

Placing Out of Other Math Courses

Students who have earned a B or higher in a college level course in either Linear Algebra or Differential Equations are entitled to place out of Math 12 or 13, respectively. It is possible, for example, to place out of Math 13 without placing out of Math 12. Students who receive a grade of 5 on the AP Statistics Exam are entitled to credit in Math 62. Placing out of any other math course requires a meeting with the chair of the math department or taking a suitable placement examination.

Course Descriptions

Mathematics 11: Calculus

Limits, derivatives and differentiation rules; partial derivatives; gradients and directional derivatives; introduction to calculus of complex-valued functions; infinite series, Taylor series, convergence tests; fundamental theorem of calculus; techniques of integration; double and triple integrals.

Prerequisites: One year of calculus at the high school level.

Mathematics 11M: Calculus

An alternative to Mathematics 11 for advanced students. With an emphasis on multivariable calculus topics: partial derivatives, gradients and directional derivatives, double and triple integrals. Also, various topics from single variable calculus studied from a more theoretical perspective.

Prerequisites: Strong background in calculus at the high school level (at least one year).

Mathematics 12: Introduction to Linear Algebra

Complex numbers; proofs by contradiction and induction; matrix representation of systems of equations, matrices, operations, determinants, row and column spaces; vectors, dot and cross products; vector descriptions of lines and planes; linear independence and dependence, bases; eigenvalues and eigenvectors; examples of discrete dynamical systems.

Prerequisites: One year of calculus at the high school level.

Mathematics 13: Differential Equations

Modeling physical systems, first-order ordinary differential equations, existence, uniqueness, and long-term behavior of solutions; bifurcations; approximate solutions; second-order ordinary differential equations and their properties, applications; first-order systems of ordinary differential equations.

Prerequisites: Math 11 or Math 11M, or the equivalent.

Mathematics 14: Multivariable Calculus I

Review of basic multivariable calculus; optimization and the second derivative test; higher order derivatives and Taylor approximations; line integrals; vector fields, curl, and divergence; Green's theorem, divergence theorem and Stokes' theorem, outline of proof and applications.

Prerequisites: Math 11 or Math 11m, or the equivalent.

Mathematics 61: Multivariable Calculus II

Constrained optimization using Lagrange multipliers; conservative and nonconservative vector fields; Green's theorem; parameterized surfaces and surface integrals; divergence theorem, outline of proof and applications; Stokes' theorem, outline of proof and applications, unification of major vector theorems.

Prerequisites: Math 14.

Mathematics 62: Introduction to Probability and Statistics

Sample spaces, events, axioms for probabilities; conditional probabilities and Bayes' theorem; random variables and their distributions, discrete and continuous; expected values, means and variances; covariance and correlation; law of large numbers and central limit theorem; point and interval estimation; hypothesis testing; simple linear regression; applications to analyzing real data sets.

Prerequisites: Math 12.

Mathematics 63: Linear Algebra II

General vector spaces and linear transformations; rank-nullity theorem; orthogonal expansion and Fourier coefficients; change of basis and similarity; generalized eigenvalues and eigenvectors; diagonalization of symmetric matrices; applications of eigenvalues to systems of ordinary differential equations; LU, QR, and singular value decomposition theorems; Jordan canonical forms.

Prerequisites: Math 12.

Mathematics 64: Differential Equations II

Linear systems of homogeneous ordinary differential equations, matrix exponential and non-homogeneous linear systems; non-linear systems; equilibrium points and their stability; phase portraits.

Prerequisites: Math 12 and Math 13.