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.
Author(s): Peter Saveliev
Edition: 1
Publisher: Independently published
Year: 2019
Language: English
Pages: 514
Tags: Calculus
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
