# MTG Problems in Calculus of one Variable

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"MTGâ€™s ORIGINAL MASPTERPIECE is a series of collection of books that started their journey as best sellers and continue as a chart-topper generation after generation. Even today these books are considered as a masterpiece among the teachers and students fraternity which is passionate about the subject.
The USP of MTGâ€™s ORIGINAL MASTERPIECE Series lies in the fact that the work has been reproduced from the Original artifact and remains as true to the original work as possible.

This textbook on mathematical analysis is based on many yearsâ€™ 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.
The best part is that 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. Each chapter closes with a separate section of supplementary problems and questions aimed at reviewing and extending the material of the chapter. These sections should prove of interest to the inquiring student, and possibly also to lecturers in selecting material for class work or seminars.
The full solutions are provided to help students reaching the right answer systematically.
The student will find the book most useful if he uses it actively, that is to say, if he studies the relevant theoretical material carefully before going on to the worked-out solutions, and finally reinforces the newly-acquired knowledge by solving the problems given for independent work. The best results will be obtained when the student, having mastered the theoretical part, immediately attacks the unsolved problems without referring to the text solutions unless in difficulty.
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"Content 1. Introduction to Mathematical Analysis Â§ 1.1. Real Numbers. The Absolute Value of a Real Number Â§ 1.2. Function. Domain of Definition Â§ 1.3. Investigation of Functions Â§ 1.4. Inverse Functions Â§ 1.5. Graphical Representation of Functions Â§ 1.6. Number Sequences. Limit of a Sequence Â§ 1.7. Evaluation of Limits of Sequences Â§ 1.8. Testing Sequences for Convergence Â§ 1.9. The Limit of a Function Â§ 1.10. Calculation of Limits of Functions Â§ 1.11. Infinitesimal and Infinite Functions. Their Definition and Comparison Â§ 1.12. Equivalent Infinitesimals. Application to Finding Limits Â§ 1.13. One-Sided Limits Â§ 1.14. Continuity of a Function. Points of Discontinuity and Their Classification Â§ 1.15. Arithmetical Operations on Continuous Functions. Continuity of a Composite Function Â§ 1.16. The Properties of a Function Continuous on a Closed Interval. Continuity of an Inverse Function Â§ 1.17. Additional Problems 2. Differentiation of Functions Â§ 2.1. Definition of the Derivative Â§ 2.2. Differentiation of Explicit Functions CHAPTERS Â§ 2.3. Successive Differentiation of Explicit Functions.Leibniz Formula Â§ 2.4. Differentiation of Inverse, Implicit and Parametrically Represented Functions Â§ 2.5. Applications of the Derivative Â§ 2.6. The Differential of a Function. Application to Approximate Computations Â§ 2.7. Additional Problems 3. Application of Differential Calculus to Investigation of Functions Â§ 3.1. Basic Theorems on Differentiable Functions Â§ 3.2. Evaluation of Indeterminate Forms.Lâ€™Hospitalâ€™s Rule Â§ 3.3. Taylorâ€™s Formula. Application to Approximate Calculations Â§ 3.4. Application of Taylorâ€™s Formula to Evaluation of Limits Â§ 3.5. Testing a Function for Monotonicity Â§ 3.6. Maxima and Minima of a Function Â§ 3.7. Finding the Greatest and the Least Values of a Function Â§ 3.8. Solving Problems in Geometry and Physics Â§ 3.9. Convexity and Concavity of a Curve. Points of Inflection Â§ 3.10. Asymptotes Â§ 3.11. General Plan for Investigating Functions and Sketching Graphs Â§ 3.12. Approximate Solution of Algebraic and Transcendental Equations Â§ 3.13. Additional Problems 4. Indefinite Integrals. Basic Methods of Integration Â§ 4.1. Direct lntegration and the Method of Expansion Â§ 4.2. Integration by Substitution Â§ 4.3. Integration by Parts Â§ 4.4. Reduction Formulas CHAPTERS 5. Basic Classes of Integrable Functions Â§ 5.1. Integration of Rational Functions Â§ 5.2. Integration of Certain Irrational Expressions Â§ 5.3. Eulerâ€™s Substitutions Â§ 5.4. Other Methods of Integrating Irrational Expressions Â§ 5.5. Integration of a Binomial Differential Â§ 5.6. Integration of Trigonometric and Hyperbolic Functions Â§ 5.7. Integration of Certain Irrational Functions with the Aid of Trigonometric or Hyperbolic Substitutions Â§ 5.8. Integration of Other Transcendental Functions Â§ 5.9. Methods of Integration (List of Basic Forms of Integrals) 6. The Definite Integral Â§ 6.1. Statement of the Problem. The Lower and Upper Integral Sums Â§ 6.2. Evaluating Definite Integrals by the Newton-Leibniz Formula Â§ 6.3. Estimating an Integral. The Definite Integral as a Function of Its Limits Â§ 6.4. Changing the Variable in a Definite Integral Â§ 6.5. Simplification of Integrals Based on the Properties of Symmetry of Integrands Â§ 6.6. Integration by Parts. Reduction Formulas Â§ 6.7. Approximating Definite Integrals Â§ 6.8. Additional Problems 7. Applications of the Definite Integral Â§ 7.1. Computing the Limits of Sums with the Aid of Definite Integrals Â§ 7.2. Finding Average Values of a Function Â§ 7.3. Computing Areas in Rectangular Coordinates CHAPTERS Â§ 7.4. Computing Areas with Parametrically Represented Boundaries Â§ 7.5. The Area of a Curvilinear Sector in Polar Coordinates Â§ 7.6. Computing the Volume of a Solid Â§ 7.7. The Arc Length of a Plane Curve in Rectangular Coordinates Â§ 7.8. The Arc Length of a Curve Represented Parametrically Â§ 7.9. The Arc Length of a Curve in Polar Coordinates Â§ 7.10. Area of Surface of Revolution Â§ 7.11. Geometrical Applications of the Definite Integral. Â§ 7.12. Computing Pressure, Work and Other Physical Quantities by the Definite Integrals Â§ 7.13. Computing Static Moments and Moments of Inertia. Determining Coordinates of the Centre of Gravity Â§ 7.14. Additional Problems 8. Improper Integrals Â§ 8.1. Improper Integrals with Infinite Limits Â§ 8.2. Improper Integrals of Unbounded Functions Â§ 8.3. Geometric and Physical Applications of Improper Integrals Â§ 8.4. Additional Problems Answers and Hints "

## Details

"Content 1. Introduction to Mathematical Analysis Â§ 1.1. Real Numbers. The Absolute Value of a Real Number Â§ 1.2. Function. Domain of Definition Â§ 1.3. Investigation of Functions Â§ 1.4. Inverse Functions Â§ 1.5. Graphical Representation of Functions Â§ 1.6. Number Sequences. Limit of a Sequence Â§ 1.7. Evaluation of Limits of Sequences Â§ 1.8. Testing Sequences for Convergence Â§ 1.9. The Limit of a Function Â§ 1.10. Calculation of Limits of Functions Â§ 1.11. Infinitesimal and Infinite Functions. Their Definition and Comparison Â§ 1.12. Equivalent Infinitesimals. Application to Finding Limits Â§ 1.13. One-Sided Limits Â§ 1.14. Continuity of a Function. Points of Discontinuity and Their Classification Â§ 1.15. Arithmetical Operations on Continuous Functions. Continuity of a Composite Function Â§ 1.16. The Properties of a Function Continuous on a Closed Interval. Continuity of an Inverse Function Â§ 1.17. Additional Problems 2. Differentiation of Functions Â§ 2.1. Definition of the Derivative Â§ 2.2. Differentiation of Explicit Functions CHAPTERS Â§ 2.3. Successive Differentiation of Explicit Functions.Leibniz Formula Â§ 2.4. Differentiation of Inverse, Implicit and Parametrically Represented Functions Â§ 2.5. Applications of the Derivative Â§ 2.6. The Differential of a Function. Application to Approximate Computations Â§ 2.7. Additional Problems 3. Application of Differential Calculus to Investigation of Functions Â§ 3.1. Basic Theorems on Differentiable Functions Â§ 3.2. Evaluation of Indeterminate Forms.Lâ€™Hospitalâ€™s Rule Â§ 3.3. Taylorâ€™s Formula. Application to Approximate Calculations Â§ 3.4. Application of Taylorâ€™s Formula to Evaluation of Limits Â§ 3.5. Testing a Function for Monotonicity Â§ 3.6. Maxima and Minima of a Function Â§ 3.7. Finding the Greatest and the Least Values of a Function Â§ 3.8. Solving Problems in Geometry and Physics Â§ 3.9. Convexity and Concavity of a Curve. Points of Inflection Â§ 3.10. Asymptotes Â§ 3.11. General Plan for Investigating Functions and Sketching Graphs Â§ 3.12. Approximate Solution of Algebraic and Transcendental Equations Â§ 3.13. Additional Problems 4. Indefinite Integrals. Basic Methods of Integration Â§ 4.1. Direct lntegration and the Method of Expansion Â§ 4.2. Integration by Substitution Â§ 4.3. Integration by Parts Â§ 4.4. Reduction Formulas CHAPTERS 5. Basic Classes of Integrable Functions Â§ 5.1. Integration of Rational Functions Â§ 5.2. Integration of Certain Irrational Expressions Â§ 5.3. Eulerâ€™s Substitutions Â§ 5.4. Other Methods of Integrating Irrational Expressions Â§ 5.5. Integration of a Binomial Differential Â§ 5.6. Integration of Trigonometric and Hyperbolic Functions Â§ 5.7. Integration of Certain Irrational Functions with the Aid of Trigonometric or Hyperbolic Substitutions Â§ 5.8. Integration of Other Transcendental Functions Â§ 5.9. Methods of Integration (List of Basic Forms of Integrals) 6. The Definite Integral Â§ 6.1. Statement of the Problem. The Lower and Upper Integral Sums Â§ 6.2. Evaluating Definite Integrals by the Newton-Leibniz Formula Â§ 6.3. Estimating an Integral. The Definite Integral as a Function of Its Limits Â§ 6.4. Changing the Variable in a Definite Integral Â§ 6.5. Simplification of Integrals Based on the Properties of Symmetry of Integrands Â§ 6.6. Integration by Parts. Reduction Formulas Â§ 6.7. Approximating Definite Integrals Â§ 6.8. Additional Problems 7. Applications of the Definite Integral Â§ 7.1. Computing the Limits of Sums with the Aid of Definite Integrals Â§ 7.2. Finding Average Values of a Function Â§ 7.3. Computing Areas in Rectangular Coordinates CHAPTERS Â§ 7.4. Computing Areas with Parametrically Represented Boundaries Â§ 7.5. The Area of a Curvilinear Sector in Polar Coordinates Â§ 7.6. Computing the Volume of a Solid Â§ 7.7. The Arc Length of a Plane Curve in Rectangular Coordinates Â§ 7.8. The Arc Length of a Curve Represented Parametrically Â§ 7.9. The Arc Length of a Curve in Polar Coordinates Â§ 7.10. Area of Surface of Revolution Â§ 7.11. Geometrical Applications of the Definite Integral. Â§ 7.12. Computing Pressure, Work and Other Physical Quantities by the Definite Integrals Â§ 7.13. Computing Static Moments and Moments of Inertia. Determining Coordinates of the Centre of Gravity Â§ 7.14. Additional Problems 8. Improper Integrals Â§ 8.1. Improper Integrals with Infinite Limits Â§ 8.2. Improper Integrals of Unbounded Functions Â§ 8.3. Geometric and Physical Applications of Improper Integrals Â§ 8.4. Additional Problems Answers and Hints "