We now come to Problems in Calculus of One Variable (With Elements  of Theory) by Issac A. Maron

This textbook on mathematical analysis is based on many years’
experience of lecturing at a higher technical college. Its aim is to
train the students in active approach to mathematical exercises, as
is done at a seminar. Much attention is given to problems improving
the theoretical background. Therefore standard computational
exercises are supplemented by examples and problems explaining the  theory, promoting its deeper understanding and stimulating precise  mathematical thinking. Some counter examples explaining the need for  certain conditions in the formulation of basic theorems are also  included.

The book is designed along the following lines. Each section opens
with a concise theoretical introduction containing the principal
definitions, theorems and formulas. Then follows a detailed solution
of one or more typical problems. Finally, problems without solution
are given, which are similar to those solved but contain certain
peculiarities. Some of them are provided with hints.

These sections should prove of interest to the inquiring student,
and possibly also to lecturers in selecting material for classwork
or seminars.

This book was translated from the Russian by Leonid Levant. The book was published by first Mir Publishers in 1973. The book is still in print in Indian edition by CBSPD.

All credits to the original uploader.

DJVU | 4.2 MB | Pages: 453 | Cover | OCR
You can get the book here.

Table of Contents
From the Author 9
Chapter I
Introduction to Mathematical Analysis 11
1.1. Real Numbers. The Absolute Value of a Real Number 11
1.2. Function. Domain of Definition 15
1.3. Investigation of Functions 22
1.4. Inverse Functions 28
1.5. Graphical Representation of Functions 30
1.6. Number Sequences. Limit of a Sequence 41
1.7. Evaluation of Limits of Sequences 48
1.8. Testing Sequences for Convergence 50
1.9. The Limit of a Function 55
1.10. Calculation of Limits of Functions 60
1.11. Infinitesimal and Infinite Functions.Their Definition and Comparison 68
1.12. Equivalent Infinitesimals. Application to Finding Limits 71
1.13. One-Sided Limits 75
1.14. Continuity of a Function. Points of Discontinuity and Their Classification 77
1.15. Arithmetical Operations on Continuous Functions. Continuity of a
Composite Function 84
1.16. The Properties of a Function Continuous on a Closed Interval. Continuity
of an Inverse Function 87
1.17. Additional Problems 91

Chapter II
Differentiation of Functions 98

2.1. Definition of the Derivative 98
2.2. Differentiation of Explicit Functions 100
2.3. Successive Differentiation of Explicit Functions. Leibniz Formula 107
2.4. Differentiation of Inverse, Implicit and Parametrically Represented Functions 111
2.5. Applications of the Derivative 115
2.6. The Differential of a Function. Application to Approximate Computations 122
2.7. Additional Problems 126

Chapter III
Application of Differential Calculus to Investigation of Functions 131

3.1. Basic Theorems on Differentiable Functions 131
3.2. Evaluation of Indeterminate Forms. L’Hospital’s Rule 138
3.3. Taylor’s Formula. Application to Approximate Calculations 143
3.4. Application of Taylor’s Formula to Evaluation of Limits 147
3.5. Testing a Function for Monotonicity 148
3.6. Maxima and Minima of a Function 152
3.7. Finding the Greatest and the Least Values of a Function 159
3.8. Solving Problems in Geometry and Physics 162
3.9. Convexity and Concavity of a Curve. Points of Inflection 166
3.10. Asymptotes 170
3.11. General Plan for Investigating Functions and Sketching Graphs 174
3.12. Approximate Solution of Algebraic and Transcendental Equations 183
3.13. Additional Problems 190

Chapter IV
Indefinite Integrals. Basic Methods of Integration 190

4.1. Direct Integration and the Method of Expansion 195
4.2. Integration by Substitution 199
4.3. Integration by Parts 202
4.4. Reduction Formulas 211

Chapter V
Basic Classes of Integrable Functions 214

5.1. Integration of Rational Functions214
5.2. Integration of Certain Irrational Expressions219
5.3. Euler’s Substitutions222
5.4. Other Methods of Integrating IrrationalExpressions224
5.5. Integration of a Binomial Differential228
5.6. Integration of Trigonometric and Hyperbolic Functions230
5.7. Integration of Certain Irrational Function with the Aid of
Trigonometric or Hyperbolic Substitutions237
5.8. Integration of Other Transcendental Functions 240
5.9. Methods of Integration (List of Basic Forms of Integrals) 242

Chapter VI
The Definite Integral 247

6.1. Statement of the Problem. The Lower and Upper Integral Sums 247
6.2. Evaluating Definite Integrals by the Newton-Leibniz Formula 256
6.3. Estimating an Integral. The Definite Integral as a Function of
Its Limits. 262
6.4. Changing the Variable in a Definite Integral 275
6.5. Simplification of Integrals Based on the Properties of Symmetry of Integrands. 288
6.6. Integration by Parts. Reduction Formulas 294
6.7. Approximating Definite Integrals. 301
6.8. Additional Problems 307

Chapter VII
Applications of the Definite Integral 310

7.1. Computing the Limits of Sums with the Aid of Definite Integrals. 310
7.2. Finding Average Values of a Function 312
7.3. Computing Areas in Rectangular Coordinates 317
7.4. Computing Areas with Parametrically Represented Boundaries 327
7.5. The Area of a Curvilinear Sector in Polar Coordinates 331
7.6. Computing the Volume of a Solid 336
7.7. The Arc Length of a Plane Curve in Rectangular Coordinates 345
7.8. The Arc Length of a Curve Represented Parametrically 348
7.9. The Arc Length of a Curve in Polar Coordinates 351
7.10.Area of Surface of Revolution 354
7.11.Geometrical Applications of the Definite Integral 360
7.12.Computing Pressure, Work and Other Physical Quantities by the
Definite Integrals 367
7.13.Computing Static Moments and Moments of Inertia. Determining
Coordinates of the Centre of Gravity 372
7.14.Additional Problems 383

Chapter VIII
Improper Integrals 387

8.1. Improper Integrals with Infinite Limits 387
8.2. Improper Integrals of Unbounded Functions 397
8.3. Geometric and Physical Applications of Improper Integrals 409
8.4. Additional Problems 415

Answers and Hints 418