Sixth Form Pure Mathematics

Volume 1

2nd Edition - January 1, 1968
Authors: C. Plumpton, W. A. Tomkys
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 1 - 3 7 1 9 - 3

Sixth Form Pure Mathematics, Volume 1, Second Edition, is the first of a series of volumes on Pure Mathematics and Theoretical Mechanics for Sixth Form students whose aim is… Read more

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Sixth Form Pure Mathematics, Volume 1, Second Edition, is the first of a series of volumes on Pure Mathematics and Theoretical Mechanics for Sixth Form students whose aim is entrance into British and Commonwealth Universities or Technical Colleges. A knowledge of Pure Mathematics up to G.C.E. O-level is assumed and the subject is developed by a concentric treatment in which each new topic is used to illustrate ideas already treated. The major topics of Algebra, Calculus, Coordinate Geometry, and Trigonometry are developed together. This volume covers most of the Pure Mathematics required for the single subject Mathematics at Advanced Level. Early and rapid progress in calculus is made at the beginning of this volume in order to facilitate the student's progress along the most satisfactory lines in Pure Mathematics, in Theoretical Mechanics and in Physics. The worked examples are an essential feature of this book and they are followed by routine exercises within the text of each chapter, associated closely with the work on which they are dependent. The exercises at the end of each chapter collectively embody all the topics of that chapter and, where possible, the preceding chapters also.

Preface to the Second Edition

Chapter I Introduction to the Calculus

1.1 Coordinates and Loci

1.2 The Idea of a Limit

1.3 The Gradient of a Curve

1.4 Differentiation

1.5 Tangents and Normals

1.6 Rates of Change

1.7 Differentiation of a Function of a Function

1.8 Maxima and Minima

1.9 Second Derivative

1.10 Parameters

Chapter II Methods of Coordinate Geometry

2.1 The Straight Line

2.2 The Division of a Line

2.3 The Equation of a Circle

2.4 The Intersection of Lines and Circles

2.5 The Parabola x=at2, y=2at, a>0

2.6 The Rectangular Hyperbola x=ct, y=c/t, c>0

2.7 The Semi-Cubical Parabola x=at2, y=at3, a>0

Chapter III Methods of the Calculus

3.1 Integration as The Reverse of Differentiation

3.2 The Constant of Integration

3.3 The Area under a Curve. Definite Integrals

3.4 Volumes of Revolution

3.5 Differentiation of Products and Quotients

3.6 Tangents to Conic Sections

Chapter IV The Circular Functions

4.1. Definition of An Angle

4.2. The Circular Functions

4.3. General Solutions of Trigonometric Equations

4.4. Circular Functions of 30°, 60°, 45°

4.5. Relations between The Circular Functions

4.6. Circular Measure

4.7. Vectors

4.8. The Addition Theorems

4.9. Double and Half Angles

4.10. The Addition of Sine Waves

4.11. The Sum-Product Transformations

Chapter V The Circular Functions in Calculus and Coordinate Geometry

5.1. The Derivatives of in x and cos x

5.2. Integral Forms

5.3. Differentiation and Integration of Other Circular Functions

5.4. Small Increments

5.5. The Angle between Two Straight Lines

5.6. The Sign of Ax+By+C

5.7. The Perpendicular Form of the Equation of a Straight Line

5.8. Tangents to Circles

5.9. The Ellipse x=a cos 0, y=b sin 0

Chapter VI The Quadratic Function and the Quadratic Equation

6.1. The Quadratic Function ax2+bx+c

6.2. The Function

6.3. The Quadratic Equation ax2+bx+c=0

6.4. Some Applications to Coordinate Geometry

6.5. The Cubic Function f(x) = ax3+bx3+cx+d

6.6. Co-Normal Points

6.7. The Hyperbola

Chapter VII Numerical Trigonometry

7.1. The Solution of Triangles

7.2. Trigonometry in Three Dimensions

7.3. The In-Center and e-Centers of a Triangle

7.4. The Orthocenter and the Altitudes

7.5. The Centroid and the Medians

Chapter VIII Finite Series

8.1. Definition and Notation

8.2. Arithmetical Progressions

8.3. Geometrical Progressions

8.4. Permutations and Combinations

8.5. Mathematical Induction

8.6. The Binomial Theorem

8.7. Some Other Finite Series

8.8. The Method of Differences

8.9. Finite Power Series

Chapter IX Infinite Series. Maclaurin's Expansion. The Binomial, Exponential and Logarithm Functions

9.1. Successive Approximations

9.2. Maclaurin's Expansion

9.3. The Binomial Series

9.4. The Exponential Function

9.5. The Expansion of Ex

9.6. Logarithms to Any Base

9.7. Natural Logarithms

9.8. Logarithmic Differentiation

9.9. The Logarithm Series

Chapter X Partial Fractions and their Applications. Some Further Methods of Integration

10.1. Partial Fractions

10.2. Application of Partial Fractions to Series Expansions

10.3. Application of Partial Fractions to the Summation of Series

10.4. Application of Partial Fractions to Integration

10.5. Integration by Substitution

10.6. Integration by Parts

Answers to the Exercises

Index