Problems in Calculus of One Variable is an exhaustive reference on Calculus for students studying a preliminary course on Calculus. The book is intended as an introduction to calculus, helping students in class-11 and class-12 understand the mathematical basis required for an additional or more advanced course. The book covers mathematical analysis, limits of a function, differential calculus, integral calculus, applications of differentiation and integration and improper integrals among other associated topics. The book is an indispensable resource for students aspiring for admission into one of the Indian Institutes of Technology for a Bachelor program in Engineering.
About I. A. Maron
I. A. Maron was a Russian mathematician. His works on Calculus are used throughout the world as introductory books on the subject.
Table of Contents
From the Author
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 Function
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 Definitions 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 Function, Continuity of a Composite
Function
1.16 The Properties of a Function, Continuous 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
2.3 Successive Differentiation of Explicit Functions, Leibnitz 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 Problems
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 Integration and the Methods of Expansion
4.2 Integration by Substitution
4.3 Integration by Parts
4.4 Reduction Formula
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 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 Other Transcendental Functions
5.8 Integration of Other Transcendental Functions
5.9 Methods of Integration (List of Basic Forms of Integrals)
6. The Definite Integrals
6.1 Statement of the Problem, The Lower and Upper Integral Sums
6.2 Evaluating Definite Integrals by the Newton-Leibnitz 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
Integrals
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
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
Answer and Hints
Attachment

Written by: I. A. Maron

Published by: Arihant

Date published: 03/03/2014

Edition: 4th

ISBN: 9351414876