Elementary Analysis - 1st Edition - ISBN: 9780080107820, 9781483137087

Elementary Analysis

1st Edition

The Commonwealth and International Library: Mathematics Division, Volume 1

Authors: K. S. Snell J. B. Morgan
Editors: W. J. Langford E. A. Maxwell
eBook ISBN: 9781483137087
Imprint: Pergamon
Published Date: 1st January 1956
Page Count: 248
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Description

Elementary Analysis, Volume 1 introduces the reader to elementary analysis in an informal manner and provides the practical experience in algebraic and analytic operations to lay a sound foundation of basic skills. The preliminary ideas are illustrated by applications to the simpler algebraic functions. Emphasis is on fundamental principles, rather than manipulative techniques. This volume is comprised of 14 chapters and begins with a discussion on number systems, covering concepts ranging from number scales to rational and real numbers, binary operations, and deductive methods. The following chapters deal with sets, vectors and congruences, and functions. Exponential and logarithmic functions, the straight line, and linear function are also considered. The remaining chapters focus on the quadratic function; the principle of mathematical induction and its applications; differentiation and the inverse process; and integration and its applications. Differential equations are presented, along with the definite integral. This book will be of particular value to teachers and students in training colleges.

Table of Contents


Preface

1. Number Systems

Number Scales

Denary Scales

Binary Scales

Rational Numbers

Geometrical Representation

Nested Intervals

Recurring Decimals

Irrational Numbers

Square Roots

Real Numbers

Geometrical Representation

Binary Operations

Axioms

Deductive Methods

Order

Inequalities

2. Sets

Definition of a Set

Equal Sets

Subsets

Empty Set

Inclusion

Operations on Sets

Algebra of Sets

Union and Intersection Tables

De Morgan's Rule

Counting Sets

Applications

3. Vectors and Congruences

Directed Lengths

Ordered Pairs

Coordinates

Vectors—Addition and Subtraction

Multiplication by a Scalar

Mid-points

The Section Formula

Homothetic Figures

Three Dimensions

Congruences

4. Functions

Relations

Graphical Representation

Mapping

Functions

Inverse Functions

Locus and Graph

Equation and Graph

5. Exponential and Logarithmic Function

Notation

The Exponential Function

Index Laws

The Logarithmic Function

An Inverse Function

The Laws of Logarithms

Logarithms to Base 10

Logarithms to Any Base

Napierian Logarithms

The Slide Rule

Graphical Solution of Equations

6. The Straight Line

Projections

The Distance Formula

Gradient

Positive and Negative Gradients

Equation of the Straight Line

Perpendicular Lines

Distance of a Point from a Line

Intersection of Two Lines

7. The Linear Function

Polynomials

The Linear Polynomial ax + b

Identity

Ordered Triples in 3-Space

Parametric Equations of a Line in 3-Space

Linear Polynomial in Two Variables

The Sign of ax + by + c

Linear Programming

8. The Quadratic Function

Graph of Quadratic Function

The Sign of(x - a)(x - b)

Graphical Solution of Inequalities

The Function y = ax2 + bx + c

The Quadratic Equation

Roots and Coefficients of a Quadratic Equation

Complex Numbers

The Algebra of Complex Numbers

9. Sequences, Series, Limits

Sequences

The Arithmetic Sequence

The Arithmetic Mean

The Geometric Sequence

The Geometric Mean

The Geometric Series

Limit of a Sequence

Limit of a Function

Formal Definition of a Limit

10. Mathematical Induction and Applications

The Principle of Mathematical Induction

Proof of the Principle

A Warning

Summation of Series

The Method of Differences

The Sigma Notation

The Limit Sum of a Series

11. Differentiation

Gradient of a Chord

Gradient of a Tangent

Gradient at a Point on a Curve

Gradient Formula

Derived Function

Differentiation of Powers of x

The Derivative of xn

The Chain Rule for Differentiation

Implicit Differentiation

12. Applications of Differentiation and the Inverse Process

Rate of Change

Stationary Values of a Function

Small Changes

Differential Equations

Velocity and Acceleration

13. Further Differentiation and the Applications

Differentiation of a Product

Differentiation of a Quotient

Implicit Differentiation

The Second Derivative

Acceleration

The Leibniz Notation

14. Integration and Applications

An Important Limit

The Definite Integral

An Example of an Integral

Area by Integration

Area Beneath a Curve

A Function Defined as a Definite Integral

The Fundamental Theorem

Square Bracket Notation

Applications of Integration

Volume of a Solid of Revolution

Mean Values

Centre of Mass

Numerical Integration

The Trapezium Rule

Simpson's Rule

Answers

Index

Details

No. of pages:
248
Language:
English
Copyright:
© Pergamon 1965
Published:
Imprint:
Pergamon
eBook ISBN:
9781483137087

About the Author

K. S. Snell

J. B. Morgan

About the Editor

W. J. Langford

E. A. Maxwell