Harvey Mudd College
Department of Mathematics
Core Math Sample Placement Exam
Questions
The Core Mathematics program at HMC
consists of a sequence of 8
half-courses (Math 11-14 and 61-64) described below.
For an explanation of the half-course
model, see the Mathematics Core Curriculum.
A score of 5 on the AP BC-Calculus exam is sufficient to place out of
Math 11. A score of 5 on the AP Statistics exam is sufficient to place
out of Math 62. Placement out of other core math courses is determined
either by examination or by a good grade in an sufficiently rigorous
college course equivalent. Students wishing to take placement exams
should look over the following sample exams (in PDF format)
for the Core Mathematics courses.
Placement exams will have questions with a comparable
balance of topics and range of difficulty.
If you have advanced placement, but don't know what to take, you
might consult this
schedule of possible paths through the core.
Math 11: Calculus of One Real or Complex Variable
Complex numbers,
limits, epsilon-delta limit definition,
derivatives and differentiation rules;
proofs by contradiction and induction;
real and complex series, Taylor series;
integration techniques, applications of the calculus;
introduction to calculus of complex-valued functions.
Prerequisite: a year of calculus at the high school level.
2 credit hours.
Math 12: Introduction to Linear Algebra and Discrete
Dynamical Systems
Matrix
representation of systems of equations, matrix operations,
determinants, linear independence and dependence, bases, inner
products, eigenvalues and eigenvectors; examples of discrete dynamical
systems, fixed points, chaos, stability, bifurcations.
Prerequisite: Mathematics 11, or the equivalent. 2 credit hours.
Math 13: Differential Equations I
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,
applications. Prerequisite: Mathematics 11. 1.5 credit hours.
Math 14: Multivariable Calculus I
Vectors, dot and cross products; vector descriptions
of lines and planes; partial derivatives and differentiability;
gradients and directional derivatives; chain rule;
higher order
derivatives; double and triple integrals in
rectangular and other coordinate systems; line integrals; vector
fields, curl, and divergence; introduction to Green's theorem,
divergence theorem, and Stokes' theorem. Prerequisite: Mathematics
11. 1.5 credit hours.
Math 61: Multivariable Calculus II
Review of basic
multivariable calculus; optimization for multivariable functions,
Lagrange multipliers;
conservative and nonconservative vector fields; Green's theorem;
parametrized surfaces and surface integrals; divergence theorem
and applications; Stokes' theorem and applications;
unification of the major vector theorems.
Prerequisite: Mathematics 14. 1.5 credit hours.
Math 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; chi-squared goodness of fit;
simple linear regression; introduction to analysis of variance; applications
to analyzing real data sets. Prerequisite: Mathematics 11. 1.5 credit
hours.
Math 63: Linear Algebra II
Review of basic linear algebra;
diagonalization and applications;
orthogonal bases and Gram-Schmidt procedure;
linear transformations; change of basis and similarity;
orthogonal projections and least-squares approximation;
symmetric matrices; vector spaces and applications.
Prerequisite: Mathematics 12. 1.5 credit hours.
Math 64: Differential Equations II
Review of basic
ordinary differential equations; systems of linear differential
equations; nonlinear systems; phase portraits; linearization and
stability; conservative systems; Lyapunov functions; cycles and
long term behavior of solutions; series
solutions near ordinary and regular singular points; Bessel functions.
Prerequisite: Mathematics 13 and 63. 1.5 credit hours.