Elementary Calculus: An Infinitesimal Approach, Third Edition by H. Jerome Keisler -- Read -- Imperial Library of Trantor

Index

Cover Page Title Page Copyright Page Dedication Preface To The Third Edition Preface To The Second Edition Preface To The First Edition Contents INTRODUCTION 1 REAL AND HYPERREAL NUMBERS

1.1 The Real Line 1.2 Functions of Real Numbers 1.3 Straight Lines 1.4 Slope and Velocity; The Hyperreal Line 1.5 Infinitesimal, Finite, and Infinite Numbers 1.6 Standard Parts Extra Problems for Chapter 1

2 DIFFERENTIATION

2.1 Derivatives 2.2 Differentials and Tangent Lines 2.3 Derivatives of Rational Functions 2.4 Inverse Functions 2.5 Transcendental Functions 2.6 Chain Rule 2.7 Higher Derivatives 2.8 Implicit Functions Extra Problems for Chapter 2

3 CONTINUOUS FUNCTIONS

3.1 How to Set Up a Problem 3.2 Related Rates 3.3 Limits 3.4 Continuity 3.5 Maxima and Minima 3.6 Maxima and Minima — Applications 3.7 Derivatives and Curve Sketching 3.8 Properties of Continuous Functions Extra Problems for Chapter 3

4 INTEGRATION

4.1 The Definite Integral 4.2 Fundamental Theorem of Calculus 4.3 Indefinite Integrals 4.4 Integration by Change of Variables 4.5 Area between Two Curves 4.6 Numerical Integration Extra Problems for Chapter 4

5 LIMITS, ANALYTIC GEOMETRY, AND APPROXIMATIONS

5.1 Infinite Limits 5.2 L’Hospital’s Rule 5.3 Limits and Curve Sketching 5.4 Parabolas 5.5 Ellipses and Hyperbolas 5.6 Second Degree Curves 5.7 Rotation of Axes 5.8 The ε, δ Condition for Limits 5.9 Newton’s Method 5.10 Derivatives and Increments Extra Problems for Chapter 5

6 APPLICATIONS OF THE INTEGRAL

6.1 Infinite Sum Theorem 6.2 Volumes of Solids of Revolution 6.3 Length of a Curve 6.4 Area of a Surface of Revolution 6.5 Averages 6.6 Some Applications to Physics 6.7 Improper Integrals Extra Problems for Chapter 6

7 TRIGONOMETRIC FUNCTIONS

7.1 Trigonometry 7.2 Derivatives of Trigonometric Functions 7.3 Inverse Trigonometric Functions 7.4 Integration by Parts 7.5 Integrals of Powers of Trigonometric Functions 7.6 Trigonometric Substitutions 7.7 Polar Coordinates 7.8 Slopes and Curve Sketching in Polar Coordinates 7.9 Area in Polar Coordinates 7.10 Length of a Curve in Polar Coordinates Extra Problems for Chapter 7

8 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

8.1 Exponential Functions 8.2 Logarithmic Functions 8.3 Derivatives of Exponential Functions and the Number e 8.4 Some Uses of Exponential Functions 8.5 Natural Logarithms 8.6 Some Differential Equations 8.7 Derivatives and Integrals Involving In x 8.8 Integration of Rational Functions 8.9 Methods of Integration Extra Problems for Chapter 8

9 INFINITE SERIES

9.1 Sequences 9.2 Series 9.3 Properties of Infinite Series 9.4 Series with Positive Terms 9.5 Alternating Series 9.6 Absolute and Conditional Convergence 9.7 Power Series 9.8 Derivatives and Integrals of Power Series 9.9 Approximations by Power Series 9.10 Taylor’s Formula 9.11 Taylor Series Extra Problems for Chapter 9

10 VECTORS

10.1 Vector Algebra 10.2 Vectors and Plane Geometry 10.3 Vectors and Lines in Space 10.4 Products of Vectors 10.5 Planes in Space 10.6 Vector Valued Functions 10.7 Vector Derivatives 10.8 Hyperreal Vectors Extra Problems for Chapter 10

11 PARTIAL DIFFERENTIATION

11.1 Surfaces 11.2 Continuous Functions of Two or More Variables 11.3 Partial Derivatives 11.4 Total Differentials and Tangent Planes 11.5 Chain Rule 11.6 Implicit Functions 11.7 Maxima and Minima 11.8 Higher Partial Derivatives Extra Problems for Chapter 11

12 MULTIPLE INTEGRALS

12.1 Double Integrals 12.2 Iterated Integrals 12.3 Infinite Sum Theorem and Volume 12.4 Applications to Physics 12.5 Double Integrals in Polar Coordinates 12.6 Triple Integrals 12.7 Cylindrical and Spherical Coordinates Extra Problems for Chapter 12

13 VECTOR CALCULUS

13.1 Directional Derivatives and Gradients 13.2 Line Integrals 13.3 Independence of Path 13.4 Green’s Theorem 13.5 Surface Area and Surface Integrals 13.6 Theorems of Stokes and Gauss Extra Problems for Chapter 13

14 DIFFERENTIAL EQUATIONS

14.1 Equations with Separable Variables 14.2 First Order Homogeneous Linear Equations 14.3 First Order Linear Equations 14.4 Existence and Approximation of Solutions 14.5 Complex Numbers 14.6 Second Order Homogeneous Linear Equations 14.7 Second Order Linear Equations Extra Problems for Chapter 14

EPILOGUE APPENDIX: TABLES

I Trigonometric Functions II Greek Alphabet III Exponential Functions IV Natural Logarithms V Powers and Roots

ANSWERS TO SELECTED PROBLEMS INDEX Back Cover