Methods Of Solving Problems In High School Mathematics | A. G. Tsypkin, A. I. Pinsky , V. I. Blagodatskikh (Editor)

First published 1986

Revised from the 1983 Russian edition

From the Editor

In this educational aid, intended for high-school students, an attempt has been made to classify the problems encountered in high-school mathematics by their solution methods.

It was rather difficult to attain the aim the authors set for them­ selves. On one hand, a detailed classification of problems by methods of solution would require the consideration of a large number of con­ crete problems and, on the other hand, a schematic classification would not yield a useful aid for solving different kinds of problems. There­ fore, alongside a large number of worked problems, the book includes many problems (about 2500) for the reader to solve.

In addition to the traditional problems from the course of high- school mathematics, the book includes methods for solving simple differential and integral calculus problems as well as problems which require the use of coordinates and vector algebra. These sections only include problems whose solutions require knowledge that is beyond the scope of high-school mathematics.

Some problems in the book can only be solved by a combined application of the knowledge from the traditional and new divisions of mathematics. These include, for instance, problems connected with the calculation of limits, derivatives and antiderivatives of functions which must first be simplified by means of identity transformations.

The authors consider all the most frequent methods of solving problems from the high-school course of mathematics. The fact that many problems are not followed by their solutions makes it possible to use the book for preparing for the entrance examinations to higher

educational establishments.

From the Authors

Our aim when writing this book was to help students to systema­ tize their knowledge of problem solving. The structure of the book Follows from this aim, hence each section begins with some theory (definitions, principal theorems and formulas) which must be known in order to cope with the subsequent problems without resorting to textbooks. Next the method of solving a specific kind of problem is indicated, followed by an example showing how to use the method. The remaining problems are left for the reader to solve.

We feel that this form of presentation is the best way of helping the student actively master the methods of problem solving. In a num­ ber of cases, the solution presented is not the briefest or most elegant. This is due, first of all, to our wish to give the most visual application to the method being suggested, rather than to demonstrate non- standard approaches.

We arranged the problems not followed by answers or hints in order of increasing difficulty, fully realizing that every reader may want to change the sequence of the problems according to his knowl­edge or inclination. Plane geometry and solid geometry, the tradi­tional divisions of school mathematics, are, in the main, represented by problems on calculations.

The material on solid geometry was not structured strictly in the way we explained above since, as distinct from problems in plane geometry, for which the solution methods can be neatly classified, any non-trivia1 problem in solid geometry may require the use of several different methods. Thus the problems considered in Chapter 12 are followed by quite detailed solutions, in which the techniques which reduce the original problem to simpler ones are emphasized.

Chapters 6-9 contain problems in mathematical analysis. Many of these should be solved using traditional high-school mathematics. Chapter 13 includes some very difficult problems in geometry whose solution can be considerably simplified with the use of vectors and the method of coordinates.

Since the theory of relativity and the related divisions of mathe­matics have drawn considerable attention lately, we deemed it neces­sary to include combinatorics and elements of relativity theory. We nave taken into account that this material is absolutely new to the majority of readers and so the theory part of the chapter is somewhat larger than those in the other chapters.

A double numeration system was adopted for both the problems and examples. The first digit indicates the number of the section and the other digit, the number of the problem or example within the sec­tion. An asterisk indicates a difficult problem and two asterisks mean that the problem is followed by a complete solution (given in “Answers and Hints” at the end of the book).

In conclusion, we wish to express our gratitude to all those who helped us to improve the structure and content of the book by their advice and remarks, especially to N. V. Reveruk who took part in writing Chapter 13 and who prepared some of the problems for it. Our thanks also go to K. K. Andreev who thoroughly studied the chapters on mathematical analysis and made valuable remarks. The authors are especially grateful to V. I. Blagodatskikh whose cordial attention was felt at every stage of writing this book.