Calculus Illustrated. Volume 2: Differential Calculus

Description

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”. 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