You are reading a page from An Elementary Treatise on Actuarial Mathematics by Harry Freeman (1932)
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CONTENTS
Introduction
Author's Preface
ELEMENTARY TRIGONOMETRY
CHAPTER I
Definitions    I
Negative Angles    4
Relations between the Ratios    4
Identities .    .    5
Magnitude of Angles. Degrees .    6
Magnitude of Angles. Radians .    6
Periodicity of the Trigonometrical Ratios    7
Ratios of (27r ± a)    9
Inverse Functions    IO
Projection .    II
Addition Theorems .    12

Sum and Difference Formulae . Double Angles and Half Angles
Examples I
FINITE DIFFERENCES
CHAPTER II
DEFINITIONS AND FUNDAMENTAL FORMULAE
Definitions
22
Difference Table
23
Symbolic Notation
24
Symbols of Operation
27
Separation of Symbols
36
Factorial Notation    .
38
Detached Coefficients
40
Examples 2    .
41

PAGE
xi

V1    CONTENTS
CHAPTER III
INTERPOLATION FOR EQUAL
INTERVALS

 
PAGE
Definition of Interpolation
44
Newton's Formula    .
47
Applications of Newton's Formula
48
Change of Origin    .
49
Subdivision of Intervals
51
Examples 3    .
53
CHAPTER IV
INTERPOLATION FOR UNEQUAL INTERVALS
Divided Differences .    .
57
Newton's Divided Difference Formula
58
Lagrange's Interpolation Formula    .
62
Examples 4    .
65
CHAPTER V
CENTRAL DIFFERENCES
Gauss's Formula
68
Stirling's Formula
70
Bessel's Formula
71
Everett's Formula
72
Sheppard's Rules
73
Relative Accuracy of the Formulae
76
Examples 5    .
8o
CHAPTER VI
INVERSE INTERPOLATION
Underlying Principles
84
Successive Approximation
89
Elimination of Third Differences
90
Examples 6    .
95
CHAPTER VII
SUMMATION
Definitions
97
Methods of Summation
98
Summation by Parts .
I02




CONTENTS
Vii PAGE
104 105 106 108
Relation between the Operators E and A Other Uses of the Symbol
Application of the Relation between and A Series in General
Examples 7
CHAPTER VIII
MISCELLANEOUS THEOREMS
Differences of Zero    .
114
The Compound Function uxv,. .
116
Functions of two Variables
I18
Central Difference Formulae: Fraser's Diagrams
123
Further Applications of the Calculus of Operations
126
"Summation n"
128
Examples 8
131

FUNCTIONS AND LIMITS
CHAPTER IX
Algebraic Functions . Transcendental Functions . Rates
Continuous Functions Limits    .
Limit of a Sequence .
The Function {(x + h)n — xn}/h
The Function (1 + 1/n)n . Asymptotes    .
Examples 9
134 135 135
139 140
143 144 145
147 148
DIFFERENTIAL CALCULUS
CHAPTER X
DEFINITIONS; STANDARD FORMS; SUCCESSIVE
DIFFERENTIATION
Definitions    .
Geometrical Interpretation
Standard Forms: Algebraic
Standard Forms: Trigonometrical
150 151 156 158

VIII    CONTENTS  
 
PAGE
Miscellaneous Examples of Differentiation
159
Successive Differentiation .
162
Leibnitz's Theorem .
164
Examples lo
167
CHAPTER XI
EXPANSIONS
Rolle's Theorem
174
Mean Value Theorem
176
Taylor's Theorem
177
Examples on the above Theorems    -
18o
The Series x/(ex — 1)
183
Differentiation of a Known Series
185
Trigonometrical Series
186
Examples I1
187
CHAPTER XII
MAXIMA AND MINIMA
Maxima and Minima
190
Examples on Maxima and Minima
194
Points of Inflexion
196
Miscellaneous Applications
199
Examples 12
202
CHAPT ER XIII
MISCELLANEOUS THEOREMS
Indeterminate Forms
206
Partial Differentiation
209
Euler's Theorem
212
Relation between the Operators d/dx and A
213
Osculatory Interpolation
216
Examples 13
220
I
INTEGRAL CALCULUS
CHAPTER XIV
DEFINITIONS AND STANDARD FORMS
Definitions
223
Geometrical Interpretation of an Integral .
225
Standard Forms
228
Examples 14
233



CONTENTS
CHAPTER XV
MORE DIFFICULT INTEGRALS; INTEGRATION
BY PARTS
  PAGE
Method of Substitution    . 235
Further Examples of Substitution 239
Integrals involving Simple Irrational Expressions 246
Integration by Parts . 249
Reduction Formulae . 252
Examples 15    . 256

CHAPTER XVI
DEFINITE INTEGRALS; AREAS; MISCELLANEOUS
THEOREMS
Definite Integrals    . 262
Product of two Functions . 267
The Functions x"e-x and x't-1 (1 — x)'n-1 268
Areas of Curves 271
Differentiation under the Integral Sign 274
Double Integrals 277
Examples 16 z8o
CHAPTER XVII
APPROXIMATE INTEGRATION
Simpson's Rule
286
Change of Unit
289
Change of Origin
290
The "three-eighths" Rule
293
Weddle's Rule .
294
Hardy's Formulae
295
Practical Applications of the Formulae
296
The Euler-Maclaurin Expansion
299
Lubbock's Formula .
302
Woolhouse's Formula
304
Other Formulae for Approximate Integration
306
Examples 17    .
309



x    CONTENTS PROBABILITY
CHAPTER XVIII
 
PAGE
Numerical Definitions of Probability.
314
The Addition Rule    .
316
Illustrative Examples
317
The Multiplication Rule
319
Illustrative Examples
321
Most Probable Value
328
Expectation and Probable Value
331
The Method of Induction.
333
Miscellaneous Examples on Probability
336
Examples 18    .
341

CHAPTER X1X
MEAN VALUE. THE APPLICATION OF THE CALCULUS TO THE SOLUTION OF QUESTIONS IN PROBABILITY
Mean Value    .
354
Application of the Calculus to Mean Value Problems
355
The Use of Double Integrals
357
Application of the Calculus to Probability .
36o
Geometrical Solutions
364
Examples 19
368
Miscellaneous Examples .
371
Answers to the Examples
383
Index
397