An Analytical Calculus: Volume 1: For School and University

Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
CONTENTS......Page 8
PREFACE......Page 10
INTRODUCTION......Page 12
1. Functions......Page 14
2. Discrete and continuous variables......Page 16
3. Notation......Page 17
4. Limits......Page 18
5. Continuity......Page 25
6. Rate of change......Page 27
7. Gradient......Page 29
8. The differential coefficient......Page 32
9. Notation......Page 34
10. Tangent and normal......Page 36
1. Some theorems on limits......Page 38
3. The differential coefficient of the product of two functions......Page 39
4. The differential coefficient of a 'function of a function'......Page 40
5. The differential coefficient of xn is nxn-  1......Page 43
6. The limit as x->0 of sin x/x is unity......Page 45
7. The differential coeffcient of sin x is cos x.......Page 46
8. The differential coefficient of cos x is - sin x......Page 47
9. Differential coefficients of higher order......Page 48
10. Some standard forms......Page 49
I. THE INVERSE SINE......Page 51
II. THE INVERSE COSINE......Page 53
III. THE INVERSE TANGENT......Page 54
12.* Differentials......Page 55
13.* The differentiation of determinants......Page 58
1. Illustration from dynamics; velocity, acceleration......Page 61
2. Maxima and minima illustrated......Page 62
3. The determination of maxima and minima......Page 64
4. Increasing and decreasing functions......Page 65
5. The second differential coefficient and concavity......Page 67
6, Points of inflexion......Page 68
7. Discrimination between maxima and minima......Page 69
8. The sketching of simple curves......Page 72
9. Rolle's theorem......Page 73
10. The mean value theorem......Page 74
11. The real roots of the equations ƒ(x) = 0,ƒ'(x) = 0......Page 76
12.* Mean value theorems for two functions......Page 79
13. Application to certain limits......Page 81
1. The area 'under' a curve......Page 90
2. The integral......Page 94
3. Notation......Page 96
4. The sign of the area......Page 97
5. Definite and indefinite integrals......Page 98
6. The evaluation of an integral......Page 99
8. The evaluation of a definite integral......Page 102
9. Some simple standard forms......Page 103
1. Substitution, or change of variable; indefinite integrals......Page 107
2. Substitution, or change of variable; definite integrals......Page 111
3. Integration by parts......Page 116
4.* Formulae of reduction......Page 119
1. Use in dynamics......Page 124
2. Area; Cartesian coordinates......Page 126
3. The area of a sector, in polar coordinates......Page 127
4. Centre of gravity, or centroid......Page 131
5. The moment of inertia of a lamina......Page 136
6. Volume of revolution......Page 138
7. The centre of gravity of a uniform solid of revolution......Page 140
8. First theorem of Pappus......Page 141
9. The moment of inertia of a uniform solid of revolution about its axis......Page 142
10.* The area of a surface of revolution......Page 143
11. Approximate integration......Page 147
12. Simpson's rule......Page 148
13. Mean values......Page 151
CHAPTER I......Page 166
CHAPTER II......Page 167
CHAPTER III......Page 170
CHAPTER V......Page 173
CHAPTER VI......Page 176
INDEX......Page 178