Ginzburg A. Calculus: Problems and Solutions

Dover Publications, 2011. — 480 p. — (Dover Books on Mathematics). — ISBN10: 0486432777, ISBN13: 978-0486432779This text helps students improve their understanding and problem-solving skills in analysis, analytic geometry, and higher algebra. Over 1,200 problems, with hints and complete solutions. Topics include sequences, functions of a single variable, limit of a function, differential calculus for functions of a single variable, the differential, indefinite and definite integrals, more. 1963 edition.Sequences
Basic definitions and theorems
Examples and exercises on general notions
Representation of a number by sequences
Evaluation of N(є)
Sequences given in the form n_n+1 = f(u_n)
Methods for the evaluation of limits
Functions of a Single Variable
Definition and notation
The elementary functions
Domain of definition
Even and odd functions
Rational functions
Logarithmic functions
Trigonometric functions
Hyperbolic functions
Inverse functions
The inverse trigonometric functions
The inverse hyperbolic functions
Composite functions
Periodic functions
Limit of a Function
Definitions and general exercises
Evaluation of limits
Continuity
Differential Calculus for Functions of a Single Variable
The notion of derivative and its physical and geometric interpretation
Evaluating derivatives
Evaluating derivatives of explicit functions
Differentiation of implicit functions
Parametric differentiation
Special cases in calculating derivatives
Higher derivatives
Calculation of y⁽ⁿ⁾
Graphical differentiation
Various examples
Fundamental Theorems of the Differential Calculus
The theorems of Rolle, Lagrange, and Cauchy
Taylor’s and Maclaurin’s formulas
Indeterminate forms: L’Hôpital’s rule
Applications of Differential Calculus
Rate of change
Locating intervals in which a function increases or decreases
Minima and MAXIMA
Concavity: points of inflection
Asymptotes
Curve tracing
Graphs in polar coordinates
Parametric equations
Tangent and normal
The order of contact
Osculating circle, radius of curvature
Evolute and involute
Solution of equations by Newton’s approximation method
The Differential
Definition of the differential
The invariance of the form of the differential
The differential as the principal part of the increment of the function: application to approximate calculations
Higher order differentials
The Indefinite Integral
Definition and basic properties
Immediate integrals
The method of substitution
Integration by parts
Integrals of rational functions
Irrational integrals
Trigonometric integrals
Integrals of exponential and hyperbolic functions
Miscellaneous integrals
The Definite Integral
Definition
Basic properties of the definite integral
Evaluation of the definite integral from its definition
Estimation of definite integrals
The mean value theorem of integral calculus
Integrals with variable limits
Evaluation of definite integrals
Changing the variable of integration
Approximate integration
Improper integrals
Miscellaneous problems
Applications of the Definite Integral
Computation of plane areas
Computation of arc length
Computation of volumes
Area of a surface of revolution
Moment of mass: centroids
Pappus7 theorems
Moment of inertia
Physics problems
Infinite Series
The general notion of a number series
Convergence of series with positive terms
Convergence of series with positive and negative terms
Arithmetic operations on series
Series of functions
Power series: radius of convergence
Taylor’s and Maclaurin’s series: operations on power series
Applications of Taylor’s and Maclaurin’s expansions
Various Problems
Solutions, hints, answers

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