2005 Handbooks

Complex numbers: arithmetic, geometry, polar coordinates, modulus, argument, de Moivre, extracting n-th roots, roots of unity. Congruences and arithmetic modulo n. Vectors: arithmetic, scalar and vector products, geometric interpretations on the plane and 3-space, area and volume. Equations of lines and planes in 3-space. Matrices and linear equations, addition and multiplication, elementary row operations. Square matrices and their determinants, inverse and transpose, relationship to solvability of linear equations.

Functions, domain, codomain. Limits and continuity, intermediate-value theorem. Differentiation, geometric interpretation, derivatives of some simple functions, sum rule, product rule, quotient rule, chain rule, local maxima and minima, curve sketching, l'Hopital's rule. Implicit differentiation and related problems. Rolle's theorem and mean-value theorem. One-to-one functions, onto functions, inverse functions. Exponential and logarithmic functions. Techniques of integration, substitution, parts, partial fractions. The Riemann integral, fundamental theorem of calculus. Application to evaluation of area and volume.