We begin with the Little Mathematics Library series once again, after a long break.  We first see the book titled Differentiation Explained by V. G. Boltyansky (Boltyanskii).

The author in the Preface says:

High school students, especially those interested in mathematics, physics and engineering, often ask, ‘What is “higher” mathematics?’ Sometimes they discuss this and similar questions at mathematics clubs at schools.

In this book I have tried to explain, in a way a high school pupil would understand, certain concepts of higher mathematics  such as the derivative, differential equation, the number e, and natural logarithm (pupils are more apt to be aware of and interested in the latter two concepts). Wherever possible, I have tried to illustrate the concepts with problems taken from physics. In addition, I have tried to show that the concepts of “higher mathematics” are mathematical reflections of actual processes, that mathematics and life are connected, not separated, and that mathematics is a growing, not an unchanging, completed science. Not all proofs and arguments are presented with complete mathematical rigour. Some arguments are presented for illustration. This method seems to me more appropriate for a general book.

The book can be used by mathematics and physics clubs at school. Part of the material is taken from lectures the author gave at the request of the advisers of school mathematics clubs at the Moscow State University.

The book was translated from Russian by M. Samokhvalov and was first published by Mir in 1977.  All credits to the original uploader.

The Internet Archive Link

Contents

Author’s Preface 6
The Problem of a Free Falling Body 7
Statement of the Problem 7
Qualitative Solution of the Problem 9
Formula for the Velocity of a Falling Body The Number e  13
Differentiation 25
The Concept of the Derivative 25
The Differential Equation 27
Two Problems Leading to Differential Equations 28
Napierian Logarithms 33
Harmonic Oscillations 34
The Problem of Small Oscillations of a Pendulum 34
The Differential Equation of Harmonic Oscillations 42
The Oscillatory Circuit 45
Oscillations Produced by the Action of the Elastic Force of a Spring 47
Some Other Applications of the Concept of the Derivative 52
Maximum and Minimum Values 52
The Problem of Drawing a Tangent 58
Modelling 60
Afterword 62

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