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Schaums Outline of Theory and Problems of Differential and Integral Calculus (Third Edition) by Frank Ayres and Elliott Mendelson



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Schaums Outline of Theory and Problems of Differential and Integral Calculus (Third Edition) written by Frank Ayres Ph.D., Formerly Professor and Head Department of Mathematics, Dickinson College and Elliott Mendelson , Ph.D., Professor of Mathematics, Queens College This third edition of the well-known calculus review book by Frank Ayres, Jr., has been thoroughly revised and includes many new features. Here are some of the more significant changes: Analytic geometry, knowledge of which was presupposed in the first two editions, is now treated in detail from the beginning. Chapters 1 through 5 are completely new and introduce the reader to the basic ideas and results. Exponential and logarithmic functions are now treated in two places. They are first discussed briefly in Chapter 14, in the classical manner of earlier editions. Then, in Chapter 40, they are introduced and studied rigorously as is now customary in calculus courses. A thorough treatment of exponential growth and decay also is included in that chapter. Terminology, notation, and standards of rigor have been brought up to date. This is especially true in connection with limits, continuity, the chain rule, and the derivative tests for extreme values. Definitions of the trigonometric functions and information about the important trigonometric identities have been provided. The chapter on curve tracing has been thoroughly revised, with the emphasis shifted from singular points to examples that occur more frequently in current calculus courses. The purpose and method of the original text have nonetheless been preserved. In particular, the direct and concise exposition typical of the Schaum Outline Series has been retained. The basic aim is to offer to students a collection of carefully solved problems that are representative of those they will encounter in elementary calculus courses (generally, the first two or three semesters of a calculus sequence). Moreover, since all fundamental concepts are defined and the most important theorems are proved, this book may be used as a text for a regular calculus course, in both colleges and secondary schools. Each chapter begins with statements of definitions, principles, and theorems. These are followed by the solved problems that form the core of the book. They give step-by-step practice in applying the principles and provide derivations of some of the theorems. In choosing these problems, we have attempted to anticipate the difficulties that normally beset the beginner. Every chapter ends with a carefully selected group of supplementary problems (with answers) whose solution is essential to the effective use of this book.

Schaums Outline of Theory and Problems of Differential and Integral Calculus (Third Edition) written by Frank Ayres and Elliott Mendelson cover the following topics.


  • ABSOLUTE VALUE; LINEAR COORDINATE SYSTEMS;
    INEQUALITIES
    THE RECTANGULAR COORDINATE SYSTEM
    LINES CIRCLES
    EQUATIONS AND THEIR GRAPHS
    FUNCTIONS
    LIMITS
    CONTINUITY
    THE DERIVATIVE
    RULES FOR DIFFERENTIATING FUNCTIONS
    IMPLICIT DIFFERENTIATION
    TANGENTS AND NORMALS
    MAXIMUM AND MINIMUM VALUES
    APPLIED PROBLEMS INVOLVING MAXIMA AND MINIMA
    RECTILINEAR AND CIRCULAR MOTION
    RELATED RATES
    DIFFERENTIATION OF TRIGONOMETRIC FUNCTIONS
    DIFFERENTIATION OF INVERSE TRIGONOMETRIC FUNCTIONS
    DIFFERENTIATION OF EXPONENTIAL AND LOGARITHMIC
    FUNCTIONS
    DIFFERENTIATION OF HYPERBOLIC FUNCTIONS
    PARAMETRIC REPRESENTATION OF CURVES
    CURVATURE
    PLANE VECTORS
    CURVILINEAR MOTION
    POLAR COORDINATES
    THE LAW OFTHE MEAN
    INDETERMINATE FORMS
    DIFFERENTIALS
    CURVE TRACING
    FUNDAMENTAL INTEGRATION FORMULAS
    INTEGRATION BY PARTS
    TRIGONOMETRIC INTEGRALS
    TRIGONOMETRIC SUBSTITUTIONS
    INTEGRATION BY PARTIAL FRACTIONS
    MISCELLANEOUS SUBSTITUTIONS
    INTEGRATION OF HYPERBOLIC FUNCTIONS
    APPLICATIONS OF INDEFINITE INTEGRALS
    THE DEFINITE INTEGRAL
    PLANE AREAS BY INTEGRATION
    EXPONENTIAL AND LOGARITHMIC FUNCTIONS; EXPONENTIAL
    GROWTH AND DECAY
    VOLUMES OF SOLIDS OF REVOLUTION
    VOLUMES OF SOLIDS WITH KNOWN CROSS SECTIONS
    CENTROIDS OF PLANE AREAS AND SOLIDS OF REVOLUTION
    MOMENTS OF INERTIA OF PLANE AREAS AND SOLIDS OF EVOLUTION
    FLUID PRESSURE
    WORK
    LENGTH OF ARC
    AREAS OF A SURFACE OF REVOLUTION
    CENTROIDS AND MOMENTS OF INERTIA OF ARCS AND
    SURFACES OF REVOLUTION
    PLANE AREA AND CENTROID OF AN AREA IN POLAR
    COORDINATES
    LENGTH AND CENTROID OF AN ARC AND AREA OF A
    SURFACE OF REVOLUTION IN POLAR COORDINATES
    IMPROPER INTEGRALS
    INFINITE SEQUENCES AND SERIES
    TESTS FOR THE CONVERGENCE AND DIVERGENCE OF POSITIVE SERIES
    SERIES WITH NEGATIVE TERMS
    COMPUTATIONS WITH SERIES
    POWER SERIES
    SERIES EXPANSION O F FUNCTIONS
    MACLAURIN'S AND TAYLOR'S FORMULAS WITH REMAINDERS COMPUTATIONS USING POWER SERIES
    APPROXIMATE INTEGRATION
    PARTIAL DERIVATIVES
    TOTAL DIFFERENTIALS AND TOTAL DERIVATIVES
    IMPLICIT FUNCTIONS
    SPACE VECTORS
    SPACE CURVES AND SURFACE
    DIRECTIONAL DERIVATIVES; MAXIMUM AND MINIMUM VALUES
    VECTOR DIFFERENTIATION AND INTEGRATION
    DOUBLE AND ITERATED INTEGRALS
    CENTROIDS AND MOMENTS OF INERTIA OF PLANE AREAS
    VOLUME UNDER A SURFACE BY DOUBLE INTEGRATION
    AREA OF A CURVED SURFACE BY DOUBLE INTEGRATION
    TRIPLE INTEGRALS
    MASSES O F VARIABLE DENSITY
    DIFFERENTIAL EQUATIONS
    DIFFERENTIAL EQUATIONS OF ORDER TWO

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