"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 "
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