2 Calculus topics Math 222 (1) Techniques of integration • Review of 221 techniques. • Integration by parts. • Partial fractions. • Trigonometric substitutions and double angles trick. • Improper integrals (needed for the convergence tests in the section on series). • First order differential equations as an application of integration: separable and linear equations. (2) Sequences & Series • Definition and convergence of sequences. • Properties of limits and calculations. • Definition and convergence of series. • Convergence tests. (3) Taylor series • Taylor polynomials • The remainder term and Lagrange’s formula. • Convergence of Taylor series. Estimation of errors. • Taylor series and convergence for the main functions (trig, exp, log, etc). • Behavior of Taylor expansions with respect to differentiation and integration. • Calculation of Taylor expansions of different functions. • A brief overview of power series, and their radius of convergence. (4) Vector Geometry • Vectors and their algebra. • Parametric form of lines and planes. • Dot product: defining equations of lines and planes, distance to a line or plane, angles (including angles between lines and planes). • Cross product. Finding normals to planes, calculating areas, distance between lines. • An introduction to parametric curves can be included here. Comment: High school students who have passed the AP/BC calculus exam will get credit for math 222 and therefore will not take math 222 at UW Madison. The AP/BC exam does not cover vectors. Therefore instructors of any course which has math 222 as prerequisite cannot assume that their students have actually seen the topic of vectors. Since the subject is valued by client departments we include in math 222 anyway.