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. . . .4.2 Properties of Continuous Functions . . . .4.2.1 Local Properties . . . . . . . . . .........................................................................44464649525466707171727374747677797979818595103106106110126130146149149149154157157Contentsxv4.2.24.2.35Global Properties of Continuous Functions . . . . . . . .Problems and Exercises . . . . . . . . . . . . . . . . . .Differential Calculus . . . .
. . . . . . . . . . . . . . . . . . . . . .5.1 Differentiable Functions . . . . . . . . . . . . . . . . . . . . . .5.1.1 Statement of the Problem and Introductory Considerations5.1.2 Functions Differentiable at a Point . . . . . . . . . . . .5.1.3 The Tangent Line; Geometric Meaning of the Derivativeand Differential . .
. . . . . . . . . . . . . . . . . . . .5.1.4 The Role of the Coordinate System . . . . . . . . . . . .5.1.5 Some Examples . . . . . . . . . . . . . . . . . . . . . .5.1.6 Problems and Exercises . . . . . . . . . . . . . . . . . .5.2 The Basic Rules of Differentiation .
. . . . . . . . . . . . . . .5.2.1 Differentiation and the Arithmetic Operations . . . . . .5.2.2 Differentiation of a Composite Function (Chain Rule) . .5.2.3 Differentiation of an Inverse Function . . . . . . . . . . .5.2.4 Table of Derivatives of the Basic Elementary Functions .5.2.5 Differentiation of a Very Simple Implicit Function . . . .5.2.6 Higher-Order Derivatives . . .
. . . . . . . . . . . . . .5.2.7 Problems and Exercises . . . . . . . . . . . . . . . . . .5.3 The Basic Theorems of Differential Calculus . . . . . . . . . . .5.3.1 Fermat’s Lemma and Rolle’s Theorem . . . . . . . . . .5.3.2 The Theorems of Lagrange and Cauchy on FiniteIncrements . . . . . . . . . . . . . . .
. . . . . . . . . .5.3.3 Taylor’s Formula . . . . . . . . . . . . . . . . . . . . . .5.3.4 Problems and Exercises . . . . . . . . . . . . . . . . . .5.4 The Study of Functions Using the Methods of Differential Calculus5.4.1 Conditions for a Function to be Monotonic . .
. . . . . .5.4.2 Conditions for an Interior Extremum of a Function . . . .5.4.3 Conditions for a Function to be Convex . . . . . . . . . .5.4.4 L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . .5.4.5 Constructing the Graph of a Function . . . . . . . . . . .5.4.6 Problems and Exercises . . . . . . . . . .
. . . . . . . .5.5 Complex Numbers and the Connections Among the ElementaryFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.5.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . .5.5.2 Convergence in C and Series with Complex Terms . . . .5.5.3 Euler’s Formula and the Connections Amongthe Elementary Functions . . . . . . . . . . . . . . . . .5.5.4 Power Series Representation of a Function.
Analyticity .5.5.5 Algebraic Closedness of the Field C of Complex Numbers5.5.6 Problems and Exercises . . . . . . . . . . . . . . . . . .5.6 Some Examples of the Application of Differential Calculusin Problems of Natural Science . . .
. . . . . . . . . . . . . . .5.6.1 Motion of a Body of Variable Mass . . . . . . . . . . . .5.6.2 The Barometric Formula . . . . . . . . . . . . . . . . . .158167171171171175178181183188190190194197202202206210211211213217230234234235241248250259263263267271275280285287287289xvi6Contents5.6.3 Radioactive Decay, Chain Reactions, and Nuclear Reactors5.6.4 Falling Bodies in the Atmosphere . . . . . . . . . . . . .5.6.5 The Number e and the Function exp x Revisited . . .
. .5.6.6 Oscillations . . . . . . . . . . . . . . . . . . . . . . . .5.6.7 Problems and Exercises . . . . . . . . . . . . . . . . . .5.7 Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.7.1 The Primitive and the Indefinite Integral . . .
. . . . . .5.7.2 The Basic General Methods of Finding a Primitive . . . .5.7.3 Primitives of Rational Functions. . . . . . . . . . . . .5.7.4 Primitives of the Form R(cos x, sin x) dx . . . . . . . .5.7.5 Primitives of the Form R(x, y(x)) dx . . . . . . . . . .5.7.6 Problems and Exercises .
. . . . . . . . . . . . . . . . .291293295298302306306308314318321324Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.1 Definition of the Integral and Description of the Set of IntegrableFunctions . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .6.1.1 The Problem and Introductory Considerations . . . . . .6.1.2 Definition of the Riemann Integral . . . . . . . . . . . .6.1.3 The Set of Integrable Functions . . . . . . . . . . . . . .6.1.4 Problems and Exercises . . . . . . . . .
. . . . . . . . .6.2 Linearity, Additivity and Monotonicity of the Integral . . . . . .6.2.1 The Integral as a Linear Function on the Space R[a, b] .6.2.2 The Integral as an Additive Function of the Intervalof Integration . . . . . . . . . . . . . . .
. . . . . . . .6.2.3 Estimation of the Integral, Monotonicity of the Integral,and the Mean-Value Theorem . . . . . . . . . . . . . . .6.2.4 Problems and Exercises . . . . . . . . . . . . . . . . . .6.3 The Integral and the Derivative . . . . . . . . . . . . . . . . . .6.3.1 The Integral and the Primitive . . . . . . . . .
. . . . . .6.3.2 The Newton–Leibniz Formula . . . . . . . . . . . . . . .6.3.3 Integration by Parts in the Definite Integral and Taylor’sFormula . . . . . . . . . . . . . . . . . . . . . . . . . .6.3.4 Change of Variable in an Integral . . . . . . . . . . . . .6.3.5 Some Examples . .
. . . . . . . . . . . . . . . . . . . .6.3.6 Problems and Exercises . . . . . . . . . . . . . . . . . .6.4 Some Applications of Integration . . . . . . . . . . . . . . . . .6.4.1 Additive Interval Functions and the Integral . . . . . . .6.4.2 Arc Length . . . . . . . . . . . . . . . . . . . . . . . .
.6.4.3 The Area of a Curvilinear Trapezoid . . . . . . . . . . .6.4.4 Volume of a Solid of Revolution . . . . . . . . . . . . .6.4.5 Work and Energy . . . . . . . . . . . . . . . . . . . . .6.4.6 Problems and Exercises . . . . . . . . . . . . . . .
. . .6.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . .6.5.1 Definition, Examples, and Basic Properties of ImproperIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . .331331331333335347349349350352359360360363364366368372375375377383384385391393393Contentsxvii6.5.26.5.36.5.478Convergence of an Improper Integral . . . . . .
. . . . .Improper Integrals with More than One Singularity . . .Problems and Exercises . . . . . . . . . . . . . . . . . .Functions of Several Variables: Their Limits and Continuity . .7.1 The Space Rm and the Most Important Classes of Its Subsets7.1.1 The Set Rm and the Distance in It . . . . . . . . . . .7.1.2 Open and Closed Sets in Rm . . . . .
. . . . . . . .7.1.3 Compact Sets in Rm . . . . . . . . . . . . . . . . . .7.1.4 Problems and Exercises . . . . . . . . . . . . . . . .7.2 Limits and Continuity of Functions of Several Variables . . .7.2.1 The Limit of a Function . . . . . . . . . . . . . . . .7.2.2 Continuity of a Function of Several Variables andProperties of Continuous Functions . .
. . . . . . . .7.2.3 Problems and Exercises . . . . . . . . . . . . . . . .........398404406........409409409411413415416416. .. .421426The Differential Calculus of Functions of Several Variables . . . .8.1 The Linear Structure on Rm . . . . . . . . . . . . . . . . . . . .8.1.1 Rm as a Vector Space . . . . . . . .
. . . . . . . . . . .8.1.2 Linear Transformations L : Rm → Rn . . . . . . . . . . .8.1.3 The Norm in Rm . . . . . . . . . . . . . . . . . . . . . .8.1.4 The Euclidean Structure on Rm . . . . . . . . . . . . . .8.2 The Differential of a Function of Several Variables . .
. . . . . .8.2.1 Differentiability and the Differential of a Functionat a Point . . . . . . . . . . . . . . . . . . . . . . . . . .8.2.2 The Differential and Partial Derivatives of a Real-ValuedFunction . . . . . . . . . . . . . . . . . . . . . . . . . .8.2.3 Coordinate Representation of the Differentialof a Mapping.
The Jacobi Matrix . . . . . . . . . . . . .8.2.4 Continuity, Partial Derivatives, and Differentiabilityof a Function at a Point . . . . . . . . . . . . . . . . . .8.3 The Basic Laws of Differentiation . . . . . . . . . . . . . . . . .8.3.1 Linearity of the Operation of Differentiation . . . .
. . .8.3.2 Differentiation of a Composition of Mappings (Chain Rule)8.3.3 Differentiation of an Inverse Mapping . . . . . . . . . . .8.3.4 Problems and Exercises . . . . . . . . . . . . . . . . . .8.4 The Basic Facts of Differential Calculus of Real-ValuedFunctions of Several Variables . .
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