2nd Edition - January 1, 1981

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  • Author: Stanley I. Grossman
  • eBook ISBN: 9781483262437

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Calculus, Second Edition discusses the techniques and theorems of calculus. This edition introduces the sine and cosine functions, distributes ?-? material over several chapters, and includes a detailed account of analytic geometry and vector analysis. This book also discusses the equation of a straight line, trigonometric limit, derivative of a power function, mean value theorem, and fundamental theorems of calculus. The exponential and logarithmic functions, inverse trigonometric functions, linear and quadratic denominators, and centroid of a plane region are likewise elaborated. Other topics include the sequences of real numbers, dot product, arc length as a parameter, quadric surfaces, higher-order partial derivatives, and Green's theorem in the plane. This publication is a good source for students learning calculus.

Table of Contents

  • Preface

    To the Instructor

    One Preliminaries

    1.1 Sets of Real Numbers

    1.2 Absolute Value and Inequalities

    1.3 The Cartesian Plane

    1.4 Lines

    1.5 The Equation of a Straight Line

    1.6 Circles

    1.7 Functions

    1.8 Operations with Functions

    1.9 Shifting the Graphs of Functions

    1.10 Second-Degree Equations

    Review Exercises for Chapter One

    Two Limits And Derivatives

    2.1 Introduction to Limits

    2.2 The Calculation of Limits

    2.3 The Limit Theorems

    2.4 Infinite Limits and Limits at Infinity

    2.5 One-Sided Limits

    2.6 A Trigonometric Limit

    2.7 Tangent Lines and Derivatives

    2.8 Tangent Lines and Derivatives (Continued)

    2.9 The Derivative as a Rate of Change

    2.10 Continuity

    2.11 The Theory of Limits (Optional)

    Review Exercises for Chapter Two

    Three More About Derivatives

    3.1 Some Differentiation Formulas

    3.2 The Product and Quotient Rules

    3.3 The Derivative of Composite Functions—The Chain Rule

    3.4 The Derivative of a Power Function

    3.5 The Derivatives of Sines and Cosines

    3.6 Implicit Differentiation

    3.7 Higher-Order Derivatives

    3.8 Approximation and Differentials

    Review Exercises for Chapter Three

    Four Applications Of The Derivative

    4.1 Related Rates of Change

    4.2 The Mean Value Theorem

    4.3 Elementary Curve Sketching I: Increasing and Decreasing Functions and the First Derivative Test

    4.4 Elementary Curve Sketching II: Concavity and the Second Derivative Test

    4.5 The Theory of Maxima and Minima

    4.6 Maxima and Minima: Applications

    4.7 Some Applications in Economics

    4.8 Newton's Method for Solving Equations

    Review Exercises for Chapter Four

    Five The Integral

    5.1 The Area Problem

    5.2 The Σ Notation

    5.3 Approximations to Area

    5.4 The Definite Integral

    5.5 The Antiderivative

    5.6 The Fundamental Theorems of Calculus

    5.7 Integration by Substitution

    5.8 The Area Between Two Curves

    5.9 Work, Power, and Energy (Optional)

    5.10 Additional Integration Theory (Optional)

    Review Exercises for Chapter Five

    Six Exponentials And Logarithms

    6.1 The Exponential and Logarithmic Functions

    6.2 The Derivatives and Integrals of logax and ax

    6.3 The Exponential and Logarithmic Functions II

    6.4 Differentiation and Integration of More General Exponential and Logarithmic Functions

    6.5 Differential Equations of Exponential Growth and Decay

    6.6 Applications in Economics (Optional)

    6.7 Inverse Functions

    Review Exercises for Chapter Six

    Seven The Trigonometric And Hyperbolic Functions

    7.1 Differentiation of Trigonometric Functions

    7.2 Integration of Trigonometric Functions

    7.3 The Inverse Trigonometric Functions

    7.4 Periodic Motion (Optional)

    7.5 The Hyperbolic Functions

    7.6 The Inverse Hyperbolic Functions (Optional)

    Review Exercises for Chapter Seven

    Eight Techniques Of Integration

    8.1 Review of the Basic Formulas of Integration

    8.2 Integration by Parts

    8.3 Integrals of Certain Trigonometric Functions

    8.4 The Idea Behind Integration by Substitution

    8.5 Integrals Involving √a2 — x2, √a2 + x2, and √x2 — a2; Trigonometric Substitutions

    8.6 The Integration of Rational Functions I: Linear and Quadratic Denominators

    8.7 The Integration of Rational Functions II: The Method of Partial Fractions

    8.8 Other Substitutions

    8.9 Using the Integral Tables

    8.10 Numerical Integration

    Review Exercises for Chapter Eight

    Nine Further Applications Of The Definite Integral

    9.1 Volumes

    9.2 Arc Length

    9.3 Surface Area

    9.4 Center of Mass and the First Moment

    9.5 The Centroid of a Plane Region

    9.6 Moments of Inertia and Kinetic Energy (Optional)

    9.7 Fluid Pressure (Optional)

    Review Exercises for Chapter Nine

    Ten Topics In Analytic Geometry

    10.1 The Ellipse and Translation of Axes

    10.2 The Parabola

    10.3 The Hyperbola

    10.4 Second-Degree Equations and Rotation of Axes

    Review Exercises for Chapter Ten

    Eleven Polar Coordinates

    11.1 The Polar Coordinate System

    11.2 Graphing in Polar Coordinates

    11.3 Points of Intersection of Graphs of Polar Equations

    11.4 Derivatives and Tangent Lines

    11.5 Areas in Polar Coordinates

    Review Exercises for Chapter Eleven

    Twelve Indeterminate Forms And Improper Integrals

    12.1 The Indeterminate Form 0/0 and L'Hôpital's Rule

    12.2 Proof of L'Hôpital's Rule (Optional)

    12.3 Other Indeterminate Forms

    12.4 Improper Integrals

    Review Exercises for Chapter Twelve

    Thirteen Taylor Polynomials And Approximation

    13.1 Taylor's Theorem and Taylor Polynomials

    13.2 A Proof of Taylor's Theorem and Estimates on the Remainder Term (Optional)

    13.3 Approximation Using Taylor Polynomials

    Review Exercises for Chapter Thirteen

    Fourteen Sequences And Series

    14.1 Sequences of Real Numbers

    14.2 Bounded and Monotonic Sequences

    14.3 Geometric Series

    14.4 Infinite Series

    14.5 Series with Nonnegative Terms I: Two Comparison Tests and the Integral Test

    14.6 Series with Nonnegative Terms II: The Ratio and Root Tests

    14.7 Absolute and Conditional Convergence: Alternating Series

    14.8 Power Series

    14.9 Differentiation and Integration of Power Series

    14.10 Taylor and Maclaurin Series

    Review Exercises for Chapter Fourteen

    Fifteen Vectors In The Plane

    15.1 Vectors and Vector Operations

    15.2 The Dot Product

    15.3 Some Applications of Vectors (Optional)

    Review Exercises for Chapter Fifteen

    Sixteen Vector Functions, Vector Differentiation, And Parametric Equations

    16.1 Vector Functions and Parametric Equations

    16.2 The Equation of the Tangent Line to a Parametric Curve

    16.3 The Differentiation and Integration of a Vector Function

    16.4 Some Differentiation Formulas

    16.5 Arc Length Revisited

    16.6 Arc Length as a Parameter

    16.7 Velocity, Acceleration, Force, and Momentum (Optional)

    16.8 Curvature and the Acceleration Vector (Optional)

    Review Exercises for Chapter Sixteen

    Seventeen Vectors In Space

    17.1 The Rectangular Coordinate System in Space

    17.2 Vectors in R3

    17.3 Lines in R3

    17.4 The Cross Product of Two Vectors

    17.5 Planes

    17.6 Quadric Surfaces

    17.7 Vector Functions and Parametric Equations in R3

    17.8 Cylindrical and Spherical Coordinates

    Review Exercises for Chapter Seventeen

    Eighteen Differentiation Of Functions Of Two And Three Variables

    18.1 Functions of Two and Three Variables

    18.2 Limits and Continuity

    18.3 Partial Derivatives

    18.4 Higher-Order Partial Derivatives

    18.5 Differentiability and the Gradient

    18.6 The Chain Rule

    18.7 Tangent Planes, Normal Lines, and Gradients

    18.8 Directional Derivatives and the Gradient

    18.9 Conservative Vector Fields and the Gradient (Optional)

    18.10 The Total Differential and Approximation

    18.11 Exact Differentials or How to Obtain a Function from Its Gradient

    18.12 Maxima and Minima for a Function of Two Variables

    18.13 Constrained Maxima and Minima—Lagrange Multipliers

    Review Exercises for Chapter Eighteen

    Nineteen Multiple Integrations

    19.1 Volume Under a Surface and the Double Integral

    19.2 The Calculation of Double Integrals

    19.3 Density, Mass, and Center of Mass (Optional)

    19.4 Double Integrals in Polar Coordinates

    19.5 Surface Area

    19.6 The Triple Integral

    19.7 The Triple Integral in Cylindrical and Spherical Coordinates

    Review Exercises for Chapter Nineteen

    Twenty Introduction To Vector Analysis

    20.1 Vector Fields

    20.2 Work, Line Integrals in the Plane, and Independence of Path

    20.3 Green's Theorem in the Plane

    20.4 Line Integrals in Space

    20.5 Surface Integrals

    20.6 Divergence and Curl of a Vector Field in R3

    20.7 Stokes' Theorem

    20.8 The Divergence Theorem

    20.9 Changing Variables in Multiple Integrals and the Jacobian

    Review Exercises for Chapter Twenty

    Twenty-One Ordinary Differential Equations

    21.1 Introduction

    21.2 First-Order Equations—Separation of Variables

    21.3 First-Order Linear Equations

    21.4 Second-Order Linear, Homogeneous Equations with Constant Coefficients

    21.5 Second-Order Nonhomogeneous Equations with Constant Coefficients: The Method of Undetermined Coefficients

    21.6 Vibratory Motion (Optional)

    Review Exercises for Chapter Twenty-One

    Appendix 1 Review Of Trigonometry

    1.1 Angles and Radian Measure

    1.2 The Trigonometric Functions and Basic Identities

    1.3 Other Trigonometric Functions

    1.4 Triangles

    Appendix 2 Mathematical Induction

    Appendix 3 Determinants

    Appendix 4 The Binomial Theorem

    Appendix 5 The Proofs Of Some Theorems On Limits, Continuity, And Differentiation

    Appendix 6 Complex Numbers

    Tables A.1 Exponential Functions

    A.2 Natural Logarithms

    A.3 Common Logarithms

    A.4 Trigonometric Functions

    A.5 Hyperbolic Functions

    A.6 Integrals

    Answers to Odd-Numbered Problems and Review Exercises


Product details

  • No. of pages: 1174
  • Language: English
  • Copyright: © Academic Press 1981
  • Published: January 1, 1981
  • Imprint: Academic Press
  • eBook ISBN: 9781483262437

About the Author

Stanley I. Grossman

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