# Problems in Calculus of one Variable

160.00

The book consists of principal definition, theorems and formulas followed by solved examples and problems for practice. Answers are given in the end of the book.

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• 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
• 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
• 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
• 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
• 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