Calculus Illustrated. Volume 1: Precalculus

Description

Mathematics is a science; it studies the inner workings of the Universe.This is the first volume of the series Calculus Illustrated, a textbook for undergraduate students. Only the topics necessary for calculus are included. The book's website hosts the drafts of volumes 2-5. Preface Calculus of sequences What is calculus about? The real number line Sequences Repeated addition and repeated multiplication How to find nth-term formulas for sequences The algebra of exponents The Binomial Formula The sequence of differences: velocity The sequence of the sums: displacement Sums of differences and differences of sums: motion The algebra of sums and differences Sets and functions Sets and relations Functions Sequences are numerical functions How numerical functions emerge: optimization Set building The xy-plane: where graphs live... Linear relations Relations vs. functions A function as a black box Give the function a domain... The graph of a function Linear functions Algebra creates functions The image: the range of values of a function Compositions of functions Operations on sets Piecewise-defined functions Numerical functions are transformations of the line Functions with regularities: one-to-one and onto Compositions of functions The inverse of a function Units conversions and changes of variables Transforming the axes transforms the plane Changing a variable transforms the graph of a function The graph of a quadratic polynomial is a parabola The main classes of functions The simplest functions Monotonicity and the extreme values of functions Functions with symmetries Quadratic polynomials The polynomial functions The rational functions The root functions The exponential functions The logarithmic functions The trigonometric functions Algebra and geometry The arithmetic operations on functions The algebra of compositions Solving equations The algebra of logarithms The Cartesian system for the Euclidean plane The Euclidean plane: distances Trigonometry and the wave function The Euclidean plane: angles From geometry to calculus Solving inequalities Exercises Exercises: Algebra Exercises: Sequences Exercises: Sets and logic Exercises: Coordinate system Exercises: Linear algebra Exercises: Polynomials Exercises: Relations and functions Exercises: Graphs Exercises: Compositions Exercises: Transformations Exercises: Basic models Index