Calculus Illustrated. Volume 1: Precalculus

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