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