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Elementary Calculus: An Infinitesimal Approach (2nd Edition) by H. Jerome Keisler



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Elementary Calculus: An Infinitesimal Approach (2nd Edition) by H. Jerome Keisler , University of Wisconsin. This book is concerned with the infinitesimal approach originally set forth by Newton and Leibnitz. The author has moved the theoretical material from Chapter One to an Appendix in this edition. A new chapter on differential equations has been added and the transcendental functions have been fully integrated into the first section. This book should be of interest to first and second year undergraduate mathematics students.

Elementary Calculus: An Infinitesimal Approach (2nd Edition) by H. Jerome Keisler cover the following topics.


INTRODUCTION
  • 1. REAL AND HVPERREAL 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 I

  • 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 e, 8 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 547
    9.11 Taylor Series 554
    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 I0

  • 11. PARTIAL DIFFERENTIATION
    II.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 II

  • 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


  • APPENDIX: Tables
    I Trigonometric Functions
    II Greek Alphabet
    III Exponential Functions
    IV Natural Logarithms
    V Powers And Roots
    ANSWERS TO SELECTED PROBLEMS
    Index

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