This is the second volume of the series Calculus Illustrated, a textbook for undergraduate students.Mathematical thinking is often visual. The exposition in this book is driven by its 600 color illustrations. Another unique feature of this book is its study of incremental phenomena well in advance of their continuous counterparts. It is called “Discrete Calculus”.
Author(s): Peter Saveliev
Edition: 1
Publisher: Independently published
Year: 2020
Language: English
Pages: 544
Tags: Calculus
Preface
Limits of sequences
What is calculus about?
Infinite sequences and their long-term trends
The definition of limit
Limits under algebraic operations
Can we add infinities? Subtract? Divide? Multiply?
More properties of limits of sequences
Theorems of Analysis
Compositions
Numbers are limits
The exponential functions
The trigonometric functions
Limits and continuity
Functions
Continuity and discontinuity
Limits of functions: small scale trends
Limits and continuity under algebraic operations
The exponential and trigonometric functions
Limits and continuity under compositions
Continuity of the inverse
Comparison of limits
Global properties of continuous functions
Large-scale behavior and asymptotes
Limits and infinity: computations
Continuity and accuracy
The - definition of limit
The derivative
The Tangent Problem
The difference of a sequence and the difference of a function
The rate of change: the difference quotient
The limit of the difference quotient: the derivative
The derivative is the instantaneous rate of change
The existence of the derivative: differentiability
The derivative as a function
Basic differentiation
Basic differentiation, continued
Free fall
Differentiation
Differentiation over addition and constant multiple: linearity
Change of variables and the derivative
Differentiation over compositions: the Chain Rule
Differentiation over multiplication and division
The rate of change of the rate of change
Repeated differentiation
How to differentiate relations: implicitly
Related rates: radar gun
The derivative of the inverse function
Reversing differentiation
Shooting a cannon
The main theorems of differential calculus
Monotonicity and extreme points
Optimization of functions
What the derivative says about the difference quotient: The Mean Value Theorem
Monotonicity and the sign of the derivative
Concavity and the sign of the second derivative
Derivatives and extrema
Anti-differentiation: the derivative of what function?
Antiderivatives
What we can do with calculus
Magnitudes of functions; L'Hopital's Rule
Linear approximations
The accuracy of the best linear approximation
Solving equations numerically: bisection and Newton's method
Particle in a flow
Differential equations
Motion under forces
Optimization examples
Functions of several variables
Exercises
Exercises: Sets, logic, functions
Exercises: Background
Exercises: Sequences
Exercises: Rates of change
Exercises: Limits and continuity
Exercises: Derivatives
Exercises: Features of graphs
Exercises: Linearization
Exercises: Models
Exercises: Information from the derivatives
Exercises: Computing derivatives
Exercises: Optimization and other applications
Index
