The rate of change is introduced informally in connection with the quadratic function, while formal treatment is postponed until later. The difficulties in grasping the concept of a derivative are thus separated, and time for thorough assimilation is afforded. 9. Integration is used to obtain the volumes of a pyramid, cone, and sphere. 10. Tables of squares, square roots etc., that is, tables of functions familiar to the student, are used in advance of other tables. The table of logarithms is introduced as a general tool, and it is not regarded as something to be used primarily with the trigonometric functions. Use is made of various tables throughout a large part of the course, so that the student acquires facility in their use and in the selection of the most suitable table for a given computation. 11. The use of the slide rule, and of logarithmic and semilogarithmic cross section paper, is explained in connection with the logarithmic function. 12. The functions en and bt, which occur frequently in the applications of mathematics, are treated at some length. 13. An effort has been made to render explicit the purpose of the various parts of the course. 14. Formal proofs of a number of theorems are omitted, and some are assumed without proof. The appeal to the intuition underlying most of these assumptions is justified by the belief that the logical presentation of these theorems requires a foundation too abstract for the general student, or too cumbersome for the purpose to be served. 15. The course includes more than a year's work so that teachers have an opportunity for choice of topics, and abundant material is provided for selected sections which progress more rapidly than the average. 16. The course increases in difficulty with a corresponding increase in interest and gain in power on the part of the student. 17. A very wide range of problems, varying in difficulty, makes it possible for the instructor to emphasize different aspects of the subject, to select exercises suitable for students of different abilities, and to assign different sets of exercises in different years. 18. Many of the exercises in the later chapters, while dealing with the newer topics, are constructed to afford review of principles presented earlier in the course, and to correlate various parts of the subject. We thank several friends for the inspiration of their interest and for their suggestions. We also thank the Trustees of The University of Rochester for making it possible for us to use the text in the classroom throughout our experimentation, in which we have participated equally. Combined courses are still to be regarded as in the experimental stage, but it is our conviction that they are fundamentally sound, and we shall feel well repaid if this volume contributes something of value to their development. ARTHUR SULLIVAN GALE CHARLES WILLIAM WATKEYS THE UNIVERSITY OF ROCHESTER May, 1920. CONTENTS PAGE XV FUNCTIONS, EQUATIONS, AND GRAPHS 1. Comparison of the Reasoning in Natural Science and in 2. Example of the Utility of Mathematics in Science. 3. Forms in which Data are Recorded..... 6. Determination of the Function which Expresses the Func- tional Relation between Two Variables 7. Graphical Representation. Directed Lines, 10. Discussion of the Table of Values... 11. Functions becoming Infinite. Asymptotes. 13. Average Rate of Change of a Function 41 17. Characteristic Property of a Straight Line. 18. Slope of a Straight Line.. 19. Graphical Solution of Problems Involving Functions ix 54 20. Graph of the Linear Function mx + b... 23. Equation of a Straight Line. 24. Application to the Solution of Problems. 26. Possible Errors in Arithmetic Calculations. Abridged Mul- 31. Translation of the Coördinate Axes 33. Rate of Change. Slope of Tangent Line. 34. Graph of the Quadratic Function ax+ bx + c. 37. Tables of Squares, Cubes, Square Roots, Cube Roots, and 39. Graph of 2", 0 <n<1. Graphs of Inverse Functions 40. Graph of x”, n < 0. Graphs of Reciprocal Functions . 45. The Linear Fractional Function 46. Integral Rational Functions. 133 133 134 136 137 140 144 |