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AND

APPLICATIONS

BY

ARTHUR SULLIVAN GALE, PH.D.

AND

CHARLES WILLIAM WATKEYS, A.M.

PROFESSORS OF MATHEMATICS
IN THE UNIVERSITY OF ROCHESTER

NEW YORK
HENRY HOLT AND COMPANY

COPYRIGHT, 1920

BY
HENRY HOLT AND COMPANY

March, 1923

PRINTED IN U.S. A.

Publishers 5-17-1924

63-18-245.me

PREFACE

This book presents a coherent year's work in mathematics for college freshmen, consisting of a study of the elementary functions, algebraic and transcendental, and their applications to problems arising in various fields of knowledge. The treatment is confined to functions of one variable, with incidental exceptions, and complex values of the independent and dependent variables are excluded. The subject matter includes the essentials of plane trigonometry and topics from advanced algebra, analytic geometry, and calculus.

The text is the result of experiments beginning in 1907-8. It has been used in the classroom since 1913-14, and each year extensive revisions have been made. Hence the content of the course, the order of topics, and the manner of presentation are based upon the experience of several years.

The unity of the course is gained by an explicit analysis of the functions studied, which enables the student to comprehend the purpose of the course as a whole and the nature of the investigation of properties of functions of a given type. This analysis consists of three parts:

First. Relations between a given function and its graph (see pages 42 and 274). Most of these relations are considered in the first chapter so that at the start the student is made aware of a number of questions which will be investigated in studying a particular type of functions.

Second. Relations between pairs of functions and their graphs (see page 152). These geometric transformations are introduced in connection with simple algebraic functions so that they are familiar tools by the time they are needed for the study of transcendental functions.

Third. Analogous properties of functions which have no immediate graphical interpretation. Several properties of x* are grouped together on page 153 in order to indicate further questions which should be investigated in studying transcendental functions.

Emphasis is also placed on characteristic properties which distinguish one class of functions from another.

A very large group of freshmen taking mathematics do not continue the study of this subject in the following years, and the needs of these students have received primary consideration. For the general student, the interpretation of a graph, the fundamental concepts of the calculus, and the usefulness of mathematics are of fundamental importance. Fortunately these matters are also important for the student of mathematics, and experience with the text has shown that it

possible in the second year satisfactorily to complete the usual first courses in analytic geometry and calculus.

To show the usefulness of mathematics, a wide range of problems which deal with matters of interest to the student have been introduced, although exercises which require considerable instruction in other subjects have been avoided. Applications to the solution of problems in mechanics, physics, chemistry, economics, and other subjects have been scattered throughout the course. The analysis

The analysis of a problem in a field other than mathematics is usually more difficult for a freshman than the solution after the conditions and requirements have been stated in mathematical language. But from the broad standpoint in which mathematics appears as part of an educational system, the training in such analysis is as important as the development of the mathematical processes to which the analysis leads. The obligation resting upon the teacher of mathematics to develop this power of analysis is increased by the proneness of other teachers to tread very lightly on the mathematical aspects of their own subjects, and it is quite possible that this inclination on the part of others is partly due to the failure of mathematicians to emphasize the applications sufficiently. Simple applied problems may furnish drill in mathematical technique, with added interest, and with but slightly increased difficulty. Simultaneously, they afford some training in analysis.

Attention is called to the following features:

1. The chapter on the theory of measurements gives an outline of statistical methods which are used in many fields such as economics, biology, physics, education, etc. It is, perhaps, not too much to say that the average college graduate will find more use for this topic than for any included in the traditional freshman course. The treatment given here is intermediate between the books on statistics which presuppose very little mathematical theory and those which the mathematical prerequisites render unsuitable for the average college man,

2. Emphasis is placed upon the determination of a function from a given table of empirical data. In problems of this sort, which illustrate an important method of discovery in science, there is an element of general culture which is too often nege lected in elementary courses in the various sciences on account of the mathematics involved.

3. Graphical methods of analyzing problems are used freely.

4. Graphical methods are used in problems which can be solved by straight lines, and an algebraic solution is then obtained by finding the equation of certain lines used in the graphical solution.

5. Use is made of the graphs of the trigonometric functions in tying together and affording the means of recalling many properties of these functions. This use of the graphs is merely a part of the general point of view of the course.

6. Trigonometric analysis, the most abstract topic included, is postponed until late in the course.

7. The introduction of a considerable amount of the elementary portions of the calculus gives the general student a knowledge of the importance and utility of the fundamental ideas of derivative and integral.

8. The average rate of change of a function is introduced at the start, and it is used in studying the linear function,

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