About the Book
This book is intended as an introduction to mathematical analysis. It includes material covering all sections of mathematical analysis taught at the secondary school level. The book examines derivatives of polynomials, trigonometric functions, exponential and logarithmic functions. The integral is presented as an operation inverse to differentiation, as the area under a graph, and as the limit of finite sums. At the end of book, exercises are provided for each section.
The emphasis is placed not on the rigour of exposition, but on computational technique.
For senior school students and for anyone beginning to study mathematical analysis.
Translated from the Russian and Typeset by Damitr Mazanav.
Released under Creative Commons by Share Alike 4.0
You can get the book here
Contents
Preface v
1 The Derivative 1
The Derivative and the Tangent . . . . . . . . . 2
Continuity of a Function . . . . . . . . . . . . . 13
2 Computing the Derivative of a Polynomial 15
3 Maximum and Minimum 23
Positivity and Negativity of the Derivative . . 24
Maximum and Minimum . . . . . . . . . . . . 26
Rolle’s Theorem . . . . . . . . . . . . . . . . . . 28
Lagrange’s Formula of Finite Increments of a
Function . . . . . . . . . . . . . . . . . . . 30
The Second Derivative . . . . . . . . . . . . . . 33
Distinguishing Between a Maximum and a
Minimum . . . . . . . . . . . . . . . . . . 34
4 Study Of Functions 36
Inflection Point . . . . . . . . . . . . . . . . . . 42
Exercises . . . . . . . . . . . . . . . . . . . . . . 46
5 Derivatives of Trigonometric Functions 49
Derivative of a Product and a Quotient . . . . 53
The Derivative of tan x function . . . . . . . . . 55
Derivative of a Composite Function . . . . . . 56
Inverse Function . . . . . . . . . . . . . . . . . . 58
Derivative of a Rational Power of x. . . . . . . 62
6 Indefinite Integral 65
7 Definite Integral 75
Definite Integral as the Limit of a Sequence of Finite Sums . . . . . . . . . . . . . . . . . 82
8 Postulate of Convergence 85
Postulate of Convergence . . . . . . . . . . . . 88
9 Newton’s Binomial and the Sum of a Geometric Progression 92
Newton’s Binomial . . . . . . . . . . . . . . . . 92
Sum of a Geometric Progression . . . . . . . . 96
10 The Function e^x 98
Study of the Function ωn(x). . . . . . . . . . . 99
Case |x| 1 . . . . . . . . . . . . . . . . . . . . 102
Fundamental Property of the Function exp(x) 105
The Number e and the Function exp x. . . . . 108
The Derivative of the Function exp x . . . . . . . . 110
11 The Function ln x 112
12 Expansion of the Function ex into a Series 114
13 Epilogue: On the Theory of Limits 117
The Theory of Limits . . . . . . . . . . . . . . . 118
Continuous Functions . . . . . . . . . . . . . . 122
14 Exercises 124
Section 1 . . . . . . . . . . . . . . . . . . . . . . 124
Section 2 . . . . . . . . . . . . . . . . . . . . . . 128
Section 3 . . . . . . . . . . . . . . . . . . . . . . 128
Section 4 . . . . . . . . . . . . . . . . . . . . . . 130
Section 5 . . . . . . . . . . . . . . . . . . . . . . 137
Section 6 . . . . . . . . . . . . . . . . . . . . . . 145
Section 7 . . . . . . . . . . . . . . . . . . . . . . 151
Section 8 . . . . . . . . . . . . . . . . . . . . . . 154
Section 10 . . . . . . . . . . . . . . . . . . . . . 155
Section 11 . . . . . . . . . . . . . . . . . . . . . 156
Section 13 . . . . . . . . . . . . . . . . . . . . . 159
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