High School Component
The UMTYMP high school component is two years long. In the first year students take Algebra 1 and Algebra 2. During the second, students take Geometry in the fall semester and Math Analysis (PreCalculus) in the spring.
Number systems and properties. Polynomials and factoring. Equations and inequalities involving linear functions, polynomials, multiple variables and/or absolute values. Rational expressions and functions. Exponents and radicals. Distance, slope and equations of lines.
Functions and graph transformations. Combining functions. Polynomial division, remainder and factor theorem. Graphing rational functions. Exponential functions, inverse functions and logarithms. Linear and nonlinear systems of equations. Matrices, matrix algebra, matrix multiplication and inverse matrices. Conic sections. Sequences and Series. Probability and statistics. Permutations, combinations. Binomial Theorem. Mathematical Induction.
Proof based geometry course with emphasis on problem solving. Points, lines, and planes. Angles, triangles and congruency theorems. Perimeter and area. Similar triangles. Right triangles and Pythagorean theorem. Triangle centers, including centroid, orthocenter, incenter and circumcenter. Quadrilaters and polygonals, including classification, properties and areas. Circles and power of a point. Polyhedra and other 3D solids. Isometries. Analytic geometry. Right triangle trigonometry. Compass and straightedge constructions are used throughout.
Math Analysis (PreCalculus)
Equations and inequalities. Polynomial, rational, exponential and logarithmic functions. Unit circle trigonometry, including graphs, inverse trigonometric functions, trigonometric equations and trigonometric identities. Polar coordinates and graphs. Parametric functions and graphs. Vectors with both algebraic and geometric approaches. Conic sections. Factoring polynomials. Linear and nonlinear systems of equations. Matrices and matrix algebra. Sequences, series and mathematical induction. (Many of these topics are covered in algebra; in Math Analysis we review and extend that material.)
The Calculus component lasts for three years. Courses are named by year, not by content. This can be confusing when comparing to other calculus sequences. At most colleges and universities, for example, Calculus 1 and Calculus 2 refer to the first and second semester of single-variable calculus. In UMTYMP, single-variable calculus is covered during the fall and spring semesters of Calculus 1, sometimes refered to as UMTYMP Calculus 1A and UMTYMP Calculus 1B. In UMTYMP, Calculus 2 refers to the second year course, which actually covers linear algebra and other topics.
The UMTYMP Calculus courses cover the following material, with minor variations from year to year:
UMTYMP Calculus I
Fall (Math 1471): functions of one variable; limits; continuity; derivatives, including applications and the geometric interpretation of first and second derivatives; mean value theorem and extended mean value theorem; extreme values; linear approximations; optimization. Proofs of major results, such as the product rule, chain rule, and L’Hospital’s rule.
Spring (Math 1472): integration, including definitions, applications and techniques, with more exposure to proofs and formal reasoning. Rigorous treatment of sequences and series.
UMTYMP Calculus II
Fall (Math 1473): introduction to differential equations, including first and second order linear differential equations; systems of linear equations; logic, set theory, and methods of proof; precise definition of limits of sequences and functions; 3D coordinates; dot and cross products; equations of lines and planes in 3D; linear transformations.
Spring (Math 2471): theoretical course in linear algebra, including Euclidean space and general vector spaces, including function spaces; eigenvalues and discrete dynamical systems.
UMTYMP Calculus III
Fall (Math 2472): multivariable functions; differential geometry of curves in Euclidean space; parametric surfaces; partial and directional derivatives; total derivative matrix and linear approximations; chain rule; quadratic forms, Sylvester’s Theorem, Taylor’s Theorem, and multivariable optimization; Lagrange multipliers.
Spring (Math 2473): multiple integration; integrals on parametric curves and surfaces; classical theorems in vector analysis, stressing a conceptual and geometric approach.
In addition, we occasionally offer UMTYMP Advanced Topics (Math 4990) for student who have completed Calculus III but are still in high school. Recent Advanced Topics courses have covered Complex Analysis, Introduction to Abstract Algebra, Enumerative Combinatorics, and Discrete Geometry at a level comparable in difficulty to our department’s regular courses for math majors in those areas.