APPLIED CALCULUS
1 INTRODUCTION PART 1 HISTORICAL INTRODUCTION THE TWO BASIC CONCEPTS OF CALCULUS HISTORICAL BACKGROUND THE METHOD OF EXHAUSTION FOR THE AREA OF A PARABOLIC SEGMENT EXERCISES A CRITICAL ANALYSIS OF ARCHIMEDES’ METHOD THE APPROACH TO CALCULUS TO BE USED IN THIS BOOK PART 2 SOME BASIC CONCEPTS OF THE THEORY OF SETS INTRODUCTION TO SET THEORY NOTATIONS FOR DESIGNATING SETS SUBSETS UNIONS, INTERSECTIONS, COMPLEMENTS EXERCISES PART 3 A SET OF AXIOMS FOR THE REAL-NUMBER SYSTEM INTRODUCTION THE FIELD AXIOMS EXERCISES THE ORDER AXIOMS EXERCISES INTEGERS AND RATIONAL NUMBERS EXERCISES 1 12 THE DEFINITION OF THE INTEGRAL FOR STEP FUNCTIONS 1 13 PROPERTIES OF THE INTEGRAL OF A STEP FUNCTION 1 14 OTHER NOTATIONS FOR INTEGRALS EXERCISES 1 27 PROOFS OF THE BASIC PROPERTIES OF THE INTEGRAL EXERCISES INFORMA1 DESCRIPTION OF CONTINUITY THE DEFINITION OF THE LIMIT OF A FUNCTION HISTORICAL INTRODUCTION A PROBLEM INVOLVING VELOCITY THE DERIVATIVE OF A FUNCTION EXAMPLES OF DERIVATIVES THE ALGEBRA OF DERIVATIVES GEOMETRIC INTERPRETATION OF THE DERIVATIVE AS A SLOPE OTHER NOTATIONS FOR DERIVATIVES 4 10 THE CHAIN RULE FOR DIFFERENTIATING COMPOSITE FUNCTIONS 174 - 4 11 APPLICATIONS OF THE CHAIN RULE RELATED RATES AND IMPLICIT DIFFERENTIATION 176 CC 4 12 EXERCISES 4 13 APPLICATIONS OF DIFFERENTIATION TO EXTREME VALUES OF FUNCTIONS 4 14 THE MEAN-VALUE THEOREM FOR DERIVATIVES 4 15 EXERCISES 4 16 APPLICATIONS OF THE MEAN-VALUE THEOREM TO GEOMETRIC PROPERTIES OF FUNCTIONS 4 17 SECOND-DERIVATIVE TEST FOR EXTREMA 4 18 CURVE SKETCHING 4 19 EXERCISES 4 20 WORKED EXAMPLES OF EXTREMUM PROBLEMS “4 22 PARTIAL DERIVATIVES 5 THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION 5 1 THE DERIVATIVE OF AN INDEFINITE INTEGRAL THE FIRST FUNDAMENTAL THEOREM OF CALCULUS THE ZERO-DERIVATIVE THEOREM PRIMITIVE FUNCTIONS AND THE SECOND FUNDAMENTAL THEOREM OF CALCULUS PROPERTIES OF A FUNCTION DEDUCED FROM PROPERTIES OF ITS DERIVATIVE INTEGRATION BY SUBSTITUTION INTEGRATION BY PARTS 217 - 5 10 EXERCISES *5 11 MISCELLANEOUS REVIEW EXERCISES LOGARITHMS REFERRED TO ANY POSITIVE BASE B # EXERCISES 6 12 THE EXPONENTIAL FUNCTION 6 13 EXPONENTIALS EXPRESSED AS POWERS OF E 6 14 THE DEFINITION OF E” FOR ARBITRARY REAL X 6 15 THE DEFINITION OF A” FOR A > 0 AND X REAL 6 16 DIFFERENTIATION AND INTEGRATION FORMULAS INVOLVING EXPONENTIALS 6 17 EXERCISES 6 18 THE HYPERBOLIC FUNCTIONS 6 19 EXERCISES 6 20 DERIVATIVES OF INVERSE FUNCTIONS 6 21 INVERSES OF THE TRIGONOMETRIC FUNCTIONS 6 22 EXERCISES 6 23 INTEGRATION BY PARTIAL FRACTIONS 6 24 INTEGRALS WHICH CARI BE TRANSFORMED INTO INTEGRALS OF RATIONAL FUNCTIONS 6 25 EXERCISES 6 26 MISCELLANEOUS REVIEW EXERCISES EXERCISES 8 INTRODUCTION TO DIFFERENTIAL EQUATIONS INTRODUCTION TERMINOLOGY AND NOTATION A FIRST-ORDER DIFFERENTIAL EQUATION FOR THE EXPONENTIAL FUNCTION FIRST-ORDER LINEAR DIFFERENTIAL EQUATIONS 8 5 EXERCISES SOME PHYSICAL PROBLEMS LEADING TO FIRST-ORDER LINEAR DIFFERENTIAL EQUATIONS 8 7 EXERCISES LINEAR EQUATIONS OF SECOND ORDER WITH CONSTANT COEFFICIENTS EXISTENCE OF SOLUTIONS OF THE EQUATION Y” + ~JJ = 8 10 REDUCTION OF THE GENERAL EQUATION TO THE SPECIAL CASE Y” + ~JJ = 8 11 UNIQUENESS THEOREM FOR THE EQUATION Y” + BU = 8 12 COMPLETE SOLUTION OF THE EQUATION Y” + BU = 8 13 COMPLETE SOLUTION OF THE EQUATION Y” + AY’ + BR = 8 15 NONHOMOGENEOUS LINEAR EQUATIONS OF SECOND ORDER WITH CONSTANT COEFFI- CIENTS EXERCISES 8 27 SOME GEOMETRICAL AND PHYSICAL PROBLEMS LEADING TO FIRST-ORDER EQUATIONS 8 28 MISCELLANEOUS REVIEW EXERCISES POINTWISE CONVERGENCE OF SEQUENCES OF FUNCTIONS UNIFORM CONVERGENCE OF SEQUENCES OF FUNCTIONS 11 3 UNIFORM CONVERGENCE AND CONTINUITY 11 4 UNIFORM CONVERGENCE AND INTEGRATION A SUFFICIENT CONDITION FOR UNIFORM CONVERGENCE 11 6 POWER SERIES CIRCLE OF CONVERGENCE 11 7 EXERCISES PROPERTIES OF FUNCTIONS REPRESENTED BY REAL POWER SERIES THE TAYLOR’S SERIES GENERATED BY A FUNCTION 11 10 A SUFFICIENT CONDITION FOR CONVERGENCE OF A TAYLOR’S SERIES 11 11 POWER-SERIES EXPANSIONS FOR THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS *IL 12 BERNSTEIN’S THEOREM 11 13 EXERCISES 11 14 POWER SERIES AND DIFFERENTIAL EQUATIONS 11 15 THE BINOMIAL SERIES 11 16 EXERCISES GEOMETRIC INTERPRETATION FOR N < LINES IN N-SPACE SOME SIMPLE PROPERTIES OF STRAIGHT LINES LINES AND VECTOR-VALUED FUNCTIONS PLANES IN EUCLIDEAN N-SPACE PLANES AND VECTOR-VALUED FUNCTIONS THE CROSS PRODUCT 13 APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY EXERCISES 14 20 APPLICATIONS TO PLANETARY MOTION 14 2 1 MISCELLANEOUS REVIEW EXERCISES THE DEFINITION OF A LINEAR SPACE EXAMPLES OF LINEAR SPACES ELEMENTARY OF THE AXIOMS CONSEQUENCES SUBSPACES OF A LINEAR SPACE DEPENDENT AND INDEPENDENT SETS IN A LINEAR SPACE BASES AND DIMENSION 15 10 INNER PRODUCTS, EUCLIDEAN NORMS SPACES, 15 11 ORTHOGONALITY IN A EUCLIDEAN SPACE 15 12 EXERCISES 15 13 CONSTRUCTION OF ORTHOGONAL SETS THE GRAM-SCHMIDT PROCESS 15 14 ORTHOGONAL COMPLEMENTS PROJECTIONS 15 15 BEST APPROXIMATION OF ELEMENTS IN A EUCLIDEAN SPACE BY ELEMENTS IN A FINITE- DIMENSIONAL SUBSPACE 15 16 EXERCISES EXERCISES 16 17 SYSTEMS OF LINEAR EQUATIONS 16 18 COMPUTATION TECHNIQUES 16 19 INVERSES OF MATRICES 16 20 EXERCISES 16 21 MISCELLANEOUS EXERCISES ON MATRICES ANSWERS TO EXERCISES