Snell / Morgan / Langford Elementary Analysis

The Commonwealth and International Library: Mathematics Division, Volume 1
1. Auflage 2014
ISBN: 978-1-4831-3708-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 248 Seiten, Web PDF

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.

Autoren/Hrsg.

Snell, K. S.

Morgan, J. B.

Langford, W. J.

Maxwell, E. A.

Inhaltsverzeichnis

1;Front Cover;1
2;Elementary Analysis;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;8
6;CHAPTER 1. NUMBER SYSTEMS;10
6.1;Number scales;10
6.2;Rational numbers;12
6.3;Geometrical representation of rational numbers;13
6.4;Nested intervals;14
6.5;Irrational numbers;15
6.6;The real numbers;17
6.7;Binary operations;19
6.8;Axioms;20
6.9;Deductive methods;22
6.10;Order;23
7;CHAPTER 2.
SETS;28
7.1;Definition of a set;28
7.2;Equal sets;29
7.3;Subsets;29
7.4;Operations on sets;30
7.5;The algebra of sets;31
7.6;Union and intersection tables;32
7.7;De Morgan's rule;34
7.8;Counting sets;36
8;CHAPTER 3.
VECTORS AND CONGRUENCES;41
8.1;Directed lengths;41
8.2;Ordered pairs;42
8.3;Coordinates;42
8.4;Vectors;43
8.5;Subtraction of vectors;45
8.6;Scalar multiplier;46
8.7;Mid-points;47
8.8;The section formula;48
8.9;Homothetic figures;49
8.10;Three dimensions;51
8.11;Congruences;54
9;CHAPTER 4. FUNCTIONS;60
9.1;Relations;60
9.2;Mapping;62
9.3;Functions;64
9.4;Inverse functions;65
9.5;Locus and graph;66
10;CHAPTER 5. EXPONENTIAL AND LOGARITHMIC FUNCTION;74
10.1;Notation;74
10.2;The exponential function. The laws of indices;74
10.3;The function ax, a > 0;76
10.4;Logarithms—an inverse function;79
10.5;The laws of logarithms;80
10.6;Logarithms to base 10;81
10.7;Logarithms to any base;83
10.8;Napierian logarithms;84
10.9;The slide rule;85
10.10;Graphical solution of equations;88
11;CHAPTER 6. THE STRAIGHT LINE;92
11.1;Projections—the distance formula;92
11.2;Gradient;93
11.3;Positive and negative gradients;95
11.4;Equation of the straight line joining two points;96
11.5;The equation y = mx + c;97
11.6;The intercept form;97
11.7;Perpendicular lines;98
11.8;Distance of a point from a line;100
11.9;Intersection of two lines;101
12;CHAPTER 7.
THE LINEAR FUNCTION;106
12.1;Polynomials;106
12.2;The linear polynomial ax + b;107
12.3;Identity;107
12.4;Ordered triples in 3-space;109
12.5;Parametric equations of a line in 3-space;110
12.6;Linear polynomial in two variables;112
12.7;The sign of ax + by + c;113
12.8;Linear programming;114
13;CHAPTER 8.
THE QUADRATIC FUNCTION;118
13.1;Graph of a quadratic function;118
13.2;The sign of (x — a) (x — b);119
13.3;Graphical solution of inequalities;120
13.4;The quadratic equation—formula;125
13.5;A theorem concerning identities;125
13.6;Roots and coefficients of a quadratic equation;126
13.7;Complex numbers;131
13.8;The algebra of complex numbers;134
14;CHAPTER 9.
SEQUENCES, SERIES, LIMITS;137
14.1;Sequences;137
14.2;The arithmetic sequence (A.S.);138
14.3;The arithmetic mean;139
14.4;The geometric sequence (G.S.);141
14.5;The geometric mean;141
14.6;The geometric series;145
14.7;Limit of a sequence;146
14.8;Limit of a function;147
14.9;Formal definition of lim/(x);148
15;CHAPTER 10. MATHEMATICAL INDUCTION AND APPLICATIONS;150
15.1;The principle of mathematical induction;150
15.2;Proof of the principle;151
15.3;A warning;154
15.4;Summation of series;156
15.5;The sigma notation;157
15.6;The difference method;158
15.7;The limit sum of a series;161
16;CHAPTER 11. DIFFERENTIATION;164
16.1;Gradient of a chord;164
16.2;Gradient of a tangent;165
16.3;Gradient of a point on the curve y = x2;166
16.4;Gradient formula;168
16.5;Differentiation of powers of x;170
16.6;The derivative of xn;174
16.7;The chain rule for differentiation;175
16.8;Implicit differentiation;176
17;CHAPTER 12. APPLICATIONS OF DIFFERENTIATION AND THE INVERSE PROCESS;179
17.1;Rate of change;179
17.2;Stationary values of a function;181
17.3;Differential equations;187
17.4;Velocity and acceleration;190
18;CHAPTER 13.
FURTHER DIFFERENTIATION AND APPLICATIONS;194
18.1;Differentiation of a product;194
18.2;Differentiation of a quotient;195
18.3;Implicit differentiation;197
18.4;The second derivative;199
18.5;Acceleration;205
18.6;The Leibniz notation;207
19;CHAPTER 14. INTEGRATION AND APPLICATIONS;210
19.1;An important limit;210
19.2;An example of an integral;211
19.3;Area by integration;212
19.4;Area beneath a curve;213
19.5;A function defined as a definite integral;213
19.6;The fundamental theorem;214
19.7;Square bracket notation;216
19.8;Applications of integration;217
19.9;Volume of a solid of revolution;221
19.10;Mean values;224
19.11;Centre of mass;226
19.12;Numerical integration;228
19.13;The trapezium rule;229
19.14;Simpson's rule;230
20;ANSWERS;236
21;INDEX;246
22;MODERN MATHEMATICS IN SECONDARY SCHOOLS;248
23;EXERCISES IN MODERN MATHEMATICS;249