Calculus Illustrated. Volume 2: Differential 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