Basic identities. Equations. More calculus. Involute. Cycloid. Damped oscillations. Addition formulas. Sums and products, etc.
CHAPTER XII. DEFINITE INTEGRALS 419
Summation of "elements": length, surface of revolution, etc. Plotting a surface. Double integration. Partial derivatives. Simpson's rule.
CHAPTER XIII. PROGRESSIONS AND SERIES . . . . 441
A.P. and G.P. Investment problems. Maclaurin series. Binomial theorem. Further laws; least squares.
CHAPTER XIV. COMBINATIONS, PROBABILITY, AND STATISTICAL METHOD 468
Pn,r.; Cn,r. Chance. Normal Probability Curve. Other distributions; averages; measures of dispersion.
CHAPTER XV. COMPLEX NUMBERS . . . . . . 500
Definition. Geometric representation. Operations. Roots of unity.
EXERCISES FOR REVIEW 513
Some proofs. Some standard formulas. The idea of "infinity." Use of a protractor. Table of integrals. Some important constants. Numerical tables: constants; roots; logarithms, base e or 10; trigonometric functions, radians and degrees.
ABBREVIATIONS AND SYMBOLS. . . . . . . 542
(Introduction) A PRELIMINARY WORD TO STUDENTS
I. "What It is All About." In scientific work and in daily affairs, we frequently observe that some two things seem to be related - that any change in the one produces some corresponding change in the other. Often it is important to ascertain precisely how the one will change with the other.
To illustrate: the speed of a locomotive depends in part on the amount of fuel consumed. Just how will the speed vary with the consumption of fuel? The blood-pressure in a healthy person is different at different ages. Just how should the pressure vary with the age? How should the price of corn vary with the size of the crop? Or the cost of a reservoir with the capacity? Or the speed of development of a photograph with the temperature of the developer? And so on.
Mathematical Analysis makes a systematic study of many different modes of variation, discovers exact relations between varying quantities, and devises suitable methods of making any necessary calculations. It has played a leading part in the modern development of the exact sciences and is being used extensively in other fields - the biological and social sciences, psychology and medicine, engineering, and business administration.
The subject is a large one, and could be studied for many years without exhausting it. But the introduction given by the present course will provide mathematical tools adequate for many kinds of scientific work. Also - what is desirable as part of a liberal education - it will give some idea of the nature and power of modern mathematics, and its important place in modern life and thought.
II. Some Suggestions as to Methods of Study. No subject can be mastered by merely receiving instruction. One must study it actively for himself. Where possible, try to react on each new question, and to devise some plan of your own for dealing with it.
Before studying each new lesson think over the recent work. Recall it clearly. Then, after reading the assignment, reflect upon this, too; formulate briefly in your own words just what each new process is, what it does, and why it is valid. This will save you much time in working the exercises. Study with care the examples solved in the text, as these often cover elusive points. Rework such examples for yourself, with the book laid aside: then compare.
Now and then run rapidly over in your mind an outline of the course to date, in order to see each major topic in perspective. Re-read occasionally the "summaries" of preceding chapters. Make free use of the Index, pp. 543-546, and of reference material in the Appendix.
Practice quizzing yourself. That is, think of questions which might come up in class, and see whether you can answer them. (In the first few lessons some ideas for such questions are listed, to indicate what is intended by this suggestion.) If any point is not clear, make a note of it, and ask about it or look it up soon. Note carefully the exact meaning of each new technical term that is introduced. In short, "get into the game," actively.
Some effort may be required for a thorough mastery of the course, but the final achievement will be well worth it.
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