Introduction to Calculus and Analysis: Volume I (Paperback, Softcover Repri)

1 Introduction.- 1.1 The Continuum of Numbers.- a. The System of Natural Numbers and Its Extension. Counting and Measuring, 1 b. Real Numbers and Nested Intervals, 7 c. Decimal Fractions. Bases Other Than Ten, 9 d. Definition of Neighborhood, 12 e. Inequalities, 12.- 1.2 The Concept of Function.- a. Mapping-Graph, 18 b. Definition of the Concept of Functions of a Continuous Variable. Domain and Range of a Function, 21 c. Graphical Representation. Monotonic Functions, 24 d. Continuity, 31 e. The Intermediate Value Theorem. Inverse Functions, 44.- 1.3 The Elementary Functions.- a. Rational Functions, 47 b. Algebraic Functions, 49 c. Trigonometric Functions, 49 d. The Exponential Function and the Logarithm, 51 e. Compound Functions, Symbolic Products, Inverse Functions, 52.- 1.4 Sequences.- 1.5 Mathematical Induction.- 1.6 The Limit of a Sequence.- a. $${a_n} = rac{1}{n}$$, 61 b. $${a_{2m}} = rac{1}{m}$$; 62 c. $${a_{2m - 1}} = rac{1}{{2m}}$$ 63 d. $${a_n} = \sqrt[n]{p}$$, 64 e.an=?n 65 f. Geometrical Illustration of the Limits of ?nand $$\sqrt[n]{p}$$ 65 g. The Geometric Series, 67 h. $${a_n} = \sqrt[n]{n}$$, 67 i. $${a_n} = \sqrt {n + 1} - \sqrt n $$, 69.- 1.7 Further Discussion of the Concept of Limit.- a. Definition of Convergence and Divergence, 70 b. Rational Operations with Limits, 71 c. Intrinsic Convergence Tests. Monotone Sequences, 73 d. Infinite Series and the Summation Symbol, 75 e. The Number e, 77 f. The Number ? as a Limit, 80.- 1.8 The Concept of Limit for Functions of a Continuous Variable.- a. Some Remarks about the Elementary Functions, 86.- Supplements.- S.1 Limits and the Number Concept.- a. The Rational Numbers, 89 b. Real Numbers Determined by Nested Sequences of Rational Intervals, 90 c. Order, Limits, and Arithmetic Operations for Real Numbers, 92 d. Completeness of the Number Continuum. Compactness of Closed Intervals. Convergence Criteria, 94 e. Least Upper Bound and Greatest Lower Bound, 97 f. Denumerability of the Rational Numbers, 98.- S.2 Theorems on Continuous Functions.- S.3 Polar Coordinates.- S.4 Remarks on Complex Numbers.- Problems.- 2 The Fundamental Ideas of the Integral and Differential Calculus.- 2.1 The Integral.- a. Introduction, 120 b. The Integral as an Area, 121 c. Analytic Definition of the Integral. Notations, 122.- 2.2 Elementary Examples of Integration.- a. Integration of Linear Function, 128 b. Integration of x2, 130 c. Integration of x? for Integers ? ? 1, 131 d. Integration of x? for Rational ? Other Than -1, 134 e. Integration of sin x and cos x, 135.- 2.3 Fundamental Rules of Integration.- a. Additivity, 136 b. Integral of a Sum of a Product with a Constant, 137 c. Estimating Integrals, 138, d. The Mean Value Theorem for Integrals, 139.- 2.4 The Integral as a Function of the Upper Limit (Indefinite Integral).- 2.5 Logarithm Defined by an Integral.- a. Definition of the Logarithm Function, 145 b. The Addition Theorem for Logarithms, 147.- 2.6 Exponential Function and Powers.- a. The Logarithm of the Number e, 149 b. The Inverse Function of the Logarithm. The Exponential Function, 150 c. The Exponential Function as Limit of Powers, 152 d. Definition of Arbitrary Powers of Positive Numbers, 152 e. Logarithms to Any Base, 153.- 2.7 The Integral of an Arbitrary Power of x.- 2.8 The Derivative.- a. The Derivative and the Tangent, 156 b. The Derivative as a Velocity, 162 c. Examples of Differentiation, 163 d. Some Fundamental Rules for Differentiation, 165 e. Differentiability and Continuity of Functions, 166 f. Higher Derivatives and Their Significance, 169 g. Derivative and Difference Quotient. Leibnitz's Notation, 171 h. The Mean Value Theorem of Differential Calculus, 173 i. Proof of the Theorem, 175 j. The Approximation of Functions by Linear Functions. Definition of Differentials, 179 k. Remarks on Applications to the Natural Sciences, 183.- 2.9 The Integral, the Primitive Function, and theFundamental Theorems of the Calculus.- a. The Derivative of the Integral, 184 b.