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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