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While we agree with Edwin E. Moise's sentiment that it is "unrealistic to judge a curriculum by its general outline, or to judge a course by its syllabus" (Moise, E.E. Activity and motivation in mathematics, Amer. Math. Monthly 72, 1965, 407-412.), the Transfer Evaluation Committee for the Department of Mathematics evaluates courses only on content, not on how effective the course was at having the students learn that content. We record here the minimum content requirements for standard introductory courses to transfer to Michigan as something other than departmental credit. These minimums reflect the core material that these courses cover in order to satisfy the requirements of courses and academic programs at the University of Michigan for which they are prerequisites. Because courses at other colleges and universities are structured around a different set of requirements, many will be very good college- or university-level courses that do not cover the material indicated here. Such courses will be given departmental credit.
Data, Functions, and Graphs (Math 105)
In order for a course to transfer to the University of Michigan as Math 105, it must thoroughly cover the following topics:
- Mathematical modeling throughout the course.
- The following types of functions: linear, exponential, logarithmic, trigonometric, polynomial, and rational.
- Inverse trigonometric functions and trigonometry in radians.
- Transformations, compositions, and inverses of functions.
Calculus I (Math 115)
In order for a course to transfer to the University of Michigan as Math 115, it must thoroughly cover the following topics:
- Limits and their computation.
- Derivative as a limit.
- Derivatives of transcendental (i.e., exponential and logarithmic) and trigonometric functions.
- Differentiation rules, implicit differentiation.
- Curve sketching with derivatives.
- Optimization with modeling.
- Linear approximation
- Related rates or strong mathematical modeling content.
- The Mean Value Theorem.
- Approximating definite integrals using sums of areas.
- Fundamental Theorem of Calculus or Net Change Theorem in an applied setting; specifically, interpretation in a real-world context of the definite integral of a rate of change (other than velocity or acceleration).
Calculus II (Math 116)
In order for a course to transfer to the University of Michigan as Math 116, it must thoroughly cover the following topics:
- Volumes of revolution by slicing.
- Integration by parts and substitution.
- Improper integrals.
- Sequences and series including p-series, ratio test, alt series test, integral test, comparison test, limit comparison test.
- Taylor series and power series plus radius of convergence and applications.
- Polar and parametric coordinates, and polar and parametric integrals including arc length and polar areas.
Multivariable Calculus (Math 215)
In order for a course to transfer to the University of Michigan as Math 215, it must thoroughly cover the following topics:
- Partial differentiation.
- Optimization, including one constraint Lagrange multipliers.
- Multiple integration, including cylindrical and spherical coordinates.
- Line integrals and surface integrals.
- The theorems of Green, Stokes, and Gauss (Divergence).
Differential Equations (Math 216)
In order for a course to transfer to the University of Michigan as Math 216, it must thoroughly cover the following topics:
- First-order ODEs: linear and separable.
- Homogeneous and non-homogeneous higher-order linear equations, method of undetermined coefficients and variation of parameters.
- Linear systems of differential equations; real and complex eigenvalues.
- Analysis of autonomous nonlinear systems by linearization, phase plane analysis.
- Laplace transforms in the solution of linear differential equations, including partial fractions, translation of transforms, step functions and impulses.
Linear Algebra (Math 214/417)
In order for a course to transfer to the University of Michigan as Math 214/417, it must thoroughly cover the following topics:
- Gauss-Jordan elimination.
- Linear independence, span, basis.
- The dictionary between linear transformations and matrices.
- Image, kernel, rank.
- Eigenvalues, eigenvectors, diagonalization.
- Orthogonality, Gram-Schmidt, QR decomposition.
- Spectral theorem for self-adjoint operators