Calculus for the Practical Man

J. E. Thompson

Description

A step-by-step guide to calculus featuring practice questions and exercises to help people improve their understanding of the mathematical study of change.

First published in 1945, this edition of J. E. Thompson’s Calculus for the Practical Man is the ideal simple guide for those who are studying physics or mathematical courses at university, or for those who wish to brush up on the calculus they learnt while in higher education. Each chapter features illustrated examples of solved problems, and there are practice exercises for the reader to try at the end of each section.

The contents of this volume includes:

- Fundamental Ideas. Rates and Differentials

- Functions and Derivatives

- Differentials of Algebraic Functions

- Use of Rates and Differentials in Solving Problems

- Differentials of Trigonometric Functions

- Velocity, Acceleration and Derivatives

- Interpretation of Functions and Derivatives by Means of Graphs

- Maximum and Minimum Values

- Problems in Maxima and Minima

- Differentials of Logarithmic and Exponential Functions

Chapter I. Fundamental Ideas. Rates and Differentials, Chapter II. Functions and Derivatives, Chapter III. Differentials of Algebraic Functions, Chapter IV. Use of Rates and Differentials in Solving Problems, Chapter V. Differentials of Trigonometric Functions, Chapter VI. Velocity, Acceleration and Derivatives, Chapter VII. Interpretation of Functions and Derivatives by Means of Graphs, Chapter VIII. Maximum and Minimum Values, Chapter IX. Problems in Maxima and Minima, Chapter X. Differentials of Logarithmic and Exponential Functions, Chapter XI. Summary of Differential Formulas, Chapter XII. Reversing the Process of Differentiation, Chapter XIII. Integral Formulas, Chapter XIV. How to Use Integral Formulas, Chapter XV. Interpretation of Integrals by Means of Graphs, Chapter XVI. Graphical Applications of Integration, Chapter XVII. Use of Integrals in Solving Problems, Chapter XVIII. The Natural Law of Growth and The Number “e”

Sujets

• Sciences formelles
• Calcul
• Guides de référence et manuels mathématiques
• Références et manuels de mathématiques
• Calculus
• Mathematics
• Sciences et techniques

Informations

• Publié par: Read Books Ltd.
• Date de parution: 23 mars 2011
• Nombre de lectures: 1
• EAN13: 9781446547083
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

CALCULUS
For the Practical Man
by
J. E. THOMPSON, B.S. in E.E., A.M.
Associate Professor of Mathematics
School of Engineering
Pratt Institute
PRINTED IN THE UNITED STATES OF AMERICA
PREFACE
T HIS book on simplified calculus is one of a series designed by the author and publisher for the reader with an interest in the meaning and simpler technique of mathematical science, and for those who wish to obtain a practical mastery of some of the more usual and directly useful branches of the science without the aid of a teacher. Like the other books in the series it is the outgrowth of the author s experience with students such as those mentioned and the demand experienced by the publisher for books which may be read as well as studied.
One of the outstanding features of the book is the use of the method of rates instead of the method of limits. To the conventional teacher of mathematics, whose students work for a college degree and look toward the modern theory of functions, the author hastens to say that for their purposes the limit method is the only method which can profitably be used. To the readers contemplated in the preparation of this book, however, the notion of a limit and any method of calculation based upon it always seem artificial and not in any way connected with the familiar ideas of numbers, algebraic symbolism or natural phenomena. On the other hand, the method of rates seems a direct application of the principle which such a reader has often heard mentioned as the extension of arithmetic and algebra with which he must become acquainted before he can perform calculations which involve changing quantities. The familiarity of examples of changing quantities in every-day life also makes it a simple matter to introduce the terminology of the calculus; teachers and readers will recall the difficulty encountered in this connection in more formal treatments.
The scope and range of the book are evident from the table of contents. The topics usually found in books on the calculus but not appearing here are omitted in conformity with the plan of the book as stated in the first paragraph above. An attempt has been made to approach the several parts of the subject as naturally and directly as possible, to show as clearly as possible the unity and continuity of the subject as a whole, to show what the calculus is all about and how it is used, and to present the material in as simple, straightforward and informal a style as it will permit. It is hoped thus that the book will be of the greatest interest and usefulness to the readers mentioned above.
The first edition of this book was prepared before the other volumes of the series were written and the arrangement of the material in this volume was not the same as in the others. In this revised edition the arrangement has been changed somewhat so that it is now the same in all the volumes of the series. Some changes and additions have been made in the text, but the experience of readers has indicated that the text is in the main satisfactory, and beyond corrections and improvements in presentation these changes are few. The last section of the book (Article 109) is new, and in this edition a fairly complete table of the more useful integral formulas has been added. The greatest changes are in the exercises and problems. These have been increased considerably in number, some of the original exercises have been replaced by better ones, and answers have been provided for all. Some of the new problems bring in up-to-date illustrations and applications of the principles, and it is hoped that all will now be found more useful and satisfactory.
Many of the corrections and improvements in the text are the results of suggestions received from readers of the first edition, and it is hoped that readers of the new edition will call attention to errors or inaccuracies which may be found in the revised text.
J. E. T HOMPSON .
Brooklyn, N. Y.
October, 1945.
CONTENTS
CHAPTER I
FUNDAMENTAL IDEAS. RATES AND DIFFERENTIALS
1. R ATES
2. V ARYING R ATES
3. D IFFERENTIALS
4. M ATHEMATICAL E XPRESSION OF A S TEADY R ATE
5. D IFFERENTIAL OF A S UM OR D IFFERENCE OF V ARIABLES
6. D IFFERENTIAL OF A C ONSTANT AND OF A N EGATIVE V ARIABLE
7. D IFFERENTIAL OF THE P RODUCT OF A C ONSTANT AND A V ARIABLE
CHAPTER II
FUNCTIONS AND DERIVATIVES
8. M EANING OF A F UNCTION
9. C LASSIFICATION OF F UNCTIONS
10. D IFFERENTIAL OF A F UNCTION OF AN I NDEPENDENT V ARIABLE
11. T HE D ERIVATIVE OF A F UNCTION
CHAPTER III
DIFFERENTIALS OF ALGEBRAIC FUNCTIONS
12. I NTRODUCTION
13. D IFFERENTIAL OF THE S QUARE OF A V ARIABLE
14. D IFFERENTIAL OF THE S QUARE R OOT OF A V ARIABLE
15. D IFFERENTIAL OF THE P RODUCT OF T WO V ARIABLES
16. D IFFERENTIAL OF THE Q UOTIENT OF T WO V ARIABLES
17. D IFFERENTIAL OF A P OWER OF A V ARIABLE
18. F ORMULAS
19. I LLUSTRATIVE E XAMPLES
E XERCISES
CHAPTER IV
USE OF RATES AND DIFFERENTIALS IN SOLVING PROBLEMS
20. I NTRODUCTORY R EMARKS
21. I LLUSTRATIVE P ROBLEMS
22. P ROBLEMS FOR S OLUTION
CHAPTER V
DIFFERENTIALS OF TRIGONOMETRIC FUNCTIONS
23. A NGLE M EASURE AND A NGLE F UNCTIONS
24. D IFFERENTIALS OF THE S INE AND C OSINE OF AN A NGLE
25. D IFFERENTIALS OF THE T ANGENT AND C OTANGENT OF AN A NGLE
26. D IFFERENTIALS OF THE S ECANT AND C OSECANT OF AN A NGLE
27. I LLUSTRATIVE E XAMPLES I NVOLVING THE T RIGONOMETRIC D IFFERENTIALS
28. I LLUSTRATIVE P ROBLEMS
E XERCISES
CHAPTER VI
VELOCITY, ACCELERATION AND DERIVATIVES
29. R ATE , D ERIVATIVE AND V ELOCITY
30. A CCELERATION AND D ERIVATIVES
31. S ECOND AND H IGHER D ERIVATIVES OF F UNCTIONS
32. I LLUSTRATIVE E XAMPLES
E XERCISES
CHAPTER VII
INTERPRETATION OF FUNCTIONS AND DERIVATIVES BY MEANS OF GRAPHS
33. G RAPHS AND F UNCTIONS
34. D IFFERENTIALS OF C OORDINATES OF A C URVE
35. G RAPHICAL I NTERPRETATION OF THE D ERIVATIVE
36. I LLUSTRATIVE E XAMPLES
E XERCISES
CHAPTER VIII
MAXIMUM AND MINIMUM VALUES
37. M AXIMUM AND M INIMUM P OINTS ON A C URVE
38. M AXIMUM AND M INIMUM V ALUES OF F UNCTIONS
39. D ETERMINING AND D ISTINGUISHING M AXIMA AND M INIMA
40. I LLUSTRATIVE E XAMPLES
E XERCISES
CHAPTER IX
PROBLEMS IN MAXIMA AND MINIMA
41. I NTRODUCTORY R EMARKS
42. I LLUSTRATIVE P ROBLEM S OLUTIONS
43. P ROBLEMS FOR S OLUTION
CHAPTER X
DIFFERENTIALS OF LOGARITHMIC AND EXPONENTIAL FUNCTIONS
44. L OGARITHMIC AND E XPONENTIAL F UNCTIONS
45. D IFFERENTIAL OF THE L OGARITHM OF A V ARIABLE
46. N ATURAL L OGARITHMS AND T HEIR D IFFERENTIALS
47. D IFFERENTIAL OF THE E XPONENTIAL F UNCTION OF A V ARIABLE
48. I LLUSTRATIVE E XAMPLES
E XERCISES
CHAPTER XI
SUMMARY OF DIFFERENTIAL FORMULAS
49. R EMARKS
50. D IFFERENTIAL AND D ERIVATIVE F ORMULAS
CHAPTER XII
REVERSING THE PROCESS OF DIFFERENTIATION
51. A N EW T YPE OF P ROBLEM
52. I NTEGRATION AND I NTEGRALS
53. I NTEGRAL F ORMULAS
CHAPTER XIII
INTEGRAL FORMULAS
54. I NTEGRAL F ORMULAS O BTAINED D IRECTLY FROM D IFFERENTIALS
55. I NTEGRAL F ORMULAS D ERIVED I NDIRECTLY
56. S UMMARY OF I NTEGRALS
57. I LLUSTRATIVE E XAMPLES
E XERCISES
CHAPTER XIV
HOW TO USE INTEGRAL FORMULAS
58. T HE S TANDARD F ORMS
59. A LGEBRAIC S IMPLIFICATION
60. M ETHOD OF S UBSTITUTION
61. I NTEGRATION BY P ARTS
62. R EMARKS
E XERCISES
CHAPTER XV
INTERPRETATION OF INTEGRALS BY MEANS OF GRAPHS
63. M EANING OF THE I NTEGRAL S IGN
64. T HE A REA UNDER A C URVE
65. T HE D EFINITE I NTEGRAL
66. T HE C ONSTANT OF I NTEGRATION
67. T HE L ENGTH OF A C URVE
68. R EMARKS
69. I LLUSTRATIVE E XAMPLES OF D EFINITE I NTEGRALS
E XERCISES
CHAPTER XVI
GRAPHICAL APPLICATIONS OF INTEGRATION
A. Areas under Curves
70. I LLUSTRATIVE P ROBLEMS
71. P ROBLEMS FOR S OLUTION
B. Lengths of Curves
72. I LLUSTRATIVE P ROBLEMS
73. P ROBLEMS FOR S OLUTION
CHAPTER XVII
USE OF INTEGRALS IN SOLVING PROBLEMS
74. I NTRODUCTION
75. P OPULATION I NCREASE
76. T HE L AWS OF F ALLING B ODIES
77. P ATH AND R ANGE OF P ROJECTILES
78. L ENGTH OF B ELTING OR P APER IN A R OLL
79. C HARGE AND D ISCHARGE OF AN E LECTRIC C ONDENSER
80. R ISE AND F ALL OF E LECTRIC C URBENT IN A C OIL
81. D IFFERENTIAL E QUATIONS
82. E FFECTIVE V ALUE OF A LTERNATING C URRENT
83. A CCELERATION AGAINST A IR R ESISTANCE
84. T IME OF S WING OF A P ENDULUM
85. S URFACE OF A L IQUID IN A R OTATING V ESSEL
86. T HE P ARABOLA
87. W ORK D ONE BY E XPANDING G AS OR S TEAM
88. R EMARKS ON THE A PPLICATION OF I NTEGRATION
89. P ROBLEMS FOR S OLUTION
CHAPTER XVIII
THE NATURAL LAW OF GROWTH AND THE NUMBER e
A. The Law of Growth and e
90. I NTRODUCTION
91. T HE N ATURAL L AW OF G ROWTH
92. M ATHEMATICAL E XPRESSION OF THE L AW OF G ROWTH
93. T HE C OMPOUND I NTEREST F ORMULA
94. A N EW D EFINITION OF THE D ERIVATIVE
95. D ERIVATIVE OF L OG b x . F ORMULA FOR e
96. C ALCULATION OF e
97. C ALCULATION OF e x FOR A NY V ALUE OF x
98. D ERIVATIVE OF e x
B. The Number e and Other Numbers and Functions
99. I NTRODUCTORY R EMARKS
100. C ALCULATION OF T ABLES OF L OGARITHMS
101. H YPERBOLIC L OGARITHMS AND THE H YPERBOLA
102. T HE E QUILATERAL H YPERBOLA AND THE C IRCLE
103. H YPERBOLIC F UNCTIONS
104. H YPERBOLIC F UNCTIONS AND THE N UMBER e
105. T HE C IRCULAR F UNCTIONS AND e
106. C OMPUTATION OF T RIGONOMETRIC T ABLES
107. C ALCULATION OF
108. T HE M OST R EMARKABLE F ORMULA IN M ATHEMATICS
109. L OGARITHM OF A N EGATIVE N UMBER
A NSWERS TO E XERCISES AND P ROBLEMS
I NTEGRALS
I NDEX
INTRODUCTION
I N arithmetic we study numbers which retain always a fixed value (constants). The numbers studied in algebra may be constants or they may vary (variables), but in any particular problem the numbers remain constant while a calculation is being made , that is, throughout the consideration of that one problem,
There are, however, certain kinds of problems, not considered in algebra or ari