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Mathematics for Economist
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ISBN-10 : 0393957330
ISBN-13 : 9780393957334
Mathematics for Economist [Hardcover] 중고
저자 Carl P. Simon | 출판사 W.W.Norton
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Mathematics for Economists, a new text for advanced undergraduate and beginning graduate students in economics, is a thoroughly modern treatment of the mathematics that underlies economic theory. An abundance of applications to current economic analysis, illustrative diagrams, thought-provoking exercises, careful proofs, and a flexible organization-these are the advantages that Mathematics for Economists brings to today's classroom.

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목차

Preface xxi
PART I Introduction 3 (104)
1 Introduction 3 (7)
1.1 MATHEMATICS IN ECONOMIC THEORY 3 (2)
1.2 MODELS OF CONSUMER CHOICE 5 (5)
Two-Dimensional Model of Consumer Choice 5 (4)
Multidimensional Model of Consumer Choice 9 (1)
2 One-Variable Calculus: Foundations 10 (29)
2.1 FUNCTIONS ON R^1 10 (6)
Vocabulary of Functions 10 (1)
Polynomials 11 (1)
Graphs 12 (1)
Increasing and Decreasing Functions 12 (2)
Domain 14 (1)
Interval Notation 15 (1)
2.2 LINEAR FUNCTIONS 16 (6)
The Slope of a Line in the Plane 16 (3)
The Equation of a Line 19 (1)
Polynomials of Degree One Have Linear 19 (1)
Graphs
Interpreting the Slope of a Linear 20 (2)
Function
2.3 THE SLOPE OF NONLINEAR FUNCTIONS 22 (3)
2.4 COMPUTING DERIVATIVES 25 (4)
Rules for Computing Derivatives 27 (2)
2.5 DIFFERENTIABILITY AND CONTINUITY 29 (4)
A Nondifferentiable Function 30 (1)
Continuous Functions 31 (1)
Continuously Differentiable Functions 32 (1)
2.6 HIGHER-ORDER DERIVATIVES 33 (1)
2.7 APPROXIMATION BY DIFFERENTIALS 34 (5)
3 One-Variable Calculus: Applications 39 (31)
3.1 USING THE FIRST DERIVATIVE FOR GRAPHING 39 (4)
Positive Derivative Implies Increasing 39 (2)
Function
Using First Derivatives to Sketch Graphs 41 (2)
3.2 SECOND DERIVATIVES AND CONVEXITY 43 (4)
3.3 GRAPHING RATIONAL FUNCTIONS 47 (1)
Hints for Graphing 48 (1)
3.4 TAILS AND HORIZONTAL ASYMPTOTES 48 (3)
Tails of Polynomials 48 (1)
Horizontal Asymptotes of Rational 49 (2)
Functions
3.5 MAXIMA AND MINIMA 51 (7)
Local Maxima and Minima on the Boundary 51 (2)
and in the Interior
Second Order Conditions 53 (2)
Global Maxima and Minima 55 (1)
Functions with Only One Critical Point 55 (1)
Functions with Nowhere-Zero Second 56 (1)
Derivatives
Functions with No Global Max or Min 56 (1)
Functions Whose Domains Are Closed Finite 56 (2)
Intervals
3.6 APPLICATIONS TO ECONOMICS 58 (12)
Production Functions 58 (1)
Cost Functions 59 (3)
Revenue and Profit Functions 62 (2)
Demand Functions and Elasticity 64 (6)
4 One-Variable Calculus: Chain Rule 70 (12)
4.1 COMPOSITE FUNCTIONS AND THE CHAIN RULE 70 (5)
Composite Functions 70 (2)
Differentiating Composite Functions: The 72 (3)
Chain Rule
4.2 INVERSE FUNCTIONS AND THEIR DERIVATIVES 75 (7)
Definition and Examples of the Inverse of 75 (4)
a Function
The Derivative of the Inverse Function 79 (1)
The Derivative of x^m/n 80 (2)
5 Exponents and Logarithms 82 (25)
5.1 EXPONENTIAL FUNCTIONS 82 (3)
5.2 THE NUMBER e 85 (3)
5.3 LOGARITHMS 88 (3)
Base 10 Logarithms 88 (2)
Base e Logarithms 90 (1)
5.4 PROPERTIES OF EXP AND LOG 91 (2)
5.5 DERIVATIVES OF EXP AND LOG 93 (4)
5.6 APPLICATIONS 97 (10)
Present Value 97 (1)
Annuities 98 (1)
Optimal Holding Time 99 (1)
Logarithmic Derivative 100(7)

PART II Linear Algebra 107(146)
6 Introduction to Linear Algebra 107(15)
6.1 LINEAR SYSTEMS 107(1)
6.2 EXAMPLES OF LINEAR MODELS 108(14)
Example 1: Tax Benefits of Charitable 108(2)
Contributions
Example 2: Linear Models of Production 110(3)
Example 3: Markov Models of Employment 113(2)
Example 4: IS-LM Analysis 115(2)
Example 5: Investment and Arbitrage 117(5)
7 Systems of Linear Equations 122(31)
7.1 GAUSSIAN AND GAUSS-JORDAN ELIMINATION 122(7)
Substitution 123(2)
Elimination of Variables 125(4)
7.2 ELEMENTARY ROW OPERATIONS 129(5)
7.3 SYSTEMS WITH MANY OR NO SOLUTIONS 134(8)
7.4 RANK--THE FUNDAMENTAL CRITERION 142(8)
Application to Portfolio Theory 147(3)
7.5 THE LINEAR IMPLICIT FUNCTION THEOREM 150(3)
8 Matrix Algebra 153(35)
8.1 MATRIX ALGEBRA 153(7)
Addition 153(1)
Subtraction 154(1)
Scalar Multiplication 155(1)
Matrix Multiplication 155(1)
Laws of Matrix Algebra 156(1)
Transpose 157(1)
Systems of Equations in Matrix Form 158(2)
8.2 SPECIAL KINDS OF MATRICES 160(2)
8.3 ELEMENTARY MATRICES 162(3)
8.4 ALGEBRA OF SQUARE MATRICES 165(9)
8.5 INPUT-OUTPUT MATRICES 174(6)
Proof of Theorem 8.13 178(2)
8.6 PARTITIONED MATRICES (optional) 180(3)
8.7 DECOMPOSING MATRICES (optional) 183(5)
Mathematical Induction 185(1)
Including Row Interchanges 185(3)
9 Determinants: An Overview 188(11)
9.1 THE DETERMINANT OF A MATRIX 189(5)
Defining the Determinant 189(2)
Computing the Determinant 191(1)
Main Property of the Determinant 192(2)
9.2 USES OF THE DETERMINANT 194(3)
9.3 IS-LM ANALYSIS VIA CRAMER'S RULE 197(2)
10 Euclidean Spaces 199(38)
10.1 POINTS AND VECTORS IN EUCLIDEAN SPACE 199(3)
The Real Line 199(1)
The Plane 199(2)
Three Dimensions and More 201(1)
10.2 VECTORS 202(3)
10.3 THE ALGEBRA OF VECTORS 205(4)
Addition and Subtraction 205(2)
Scalar Multiplication 207(2)
10.4 LENGTH AND INNER PRODUCT IN R^n 209(13)
Length and Distance 209(4)
The Inner Product 213(9)
10.5 LINES 222(4)
10.6 PLANES 226(6)
Parametric Equations 226(2)
Nonparametric Equations 228(2)
Hyperplanes 230(2)
10.7 ECONOMIC APPLICATIONS 232(5)
Budget Sets in Commodity Space 232(1)
Input Space 233(1)
Probability Simplex 233(1)
The Investment Model 234(1)
IS-LM Analysis 234(3)
11 Linear Independence 237(16)
11.1 LINEAR INDEPENDENCE 237(7)
Definition 238(3)
Checking Linear Independence 241(3)
11.2 SPANNING SETS 244(3)
11.3 BASIS AND DIMENSION IN R^n 247(2)
Dimension 249(1)
11.4 EPILOGUE 249(4)

PART III Calculus of Several Variables 253(122)
12 Limits and Open Sets 253(20)
12.1 SEQUENCES OF REAL NUMBERS 253(7)
Definition 253(1)
Limit of a Sequence 254(2)
Algebraic Properties of Limits 256(4)
12.2 SEQUENCES IN R^m 260(4)
12.3 OPEN SETS 264(3)
Interior of a Set 267(1)
12.4 CLOSED SETS 267(3)
Closure of a Set 268(1)
Boundary of a Set 269(1)
12.5 COMPACT SETS 270(2)
12.6 EPILOGUE 272(1)
13 Functions of Several Variables 273(27)
13.1 FUNCTIONS BETWEEN EUCLIDEAN SPACES 273(4)
Functions from R^n to R 274(1)
Functions from R^k to R^m 275(2)
13.2 GEOMETRIC REPRESENTATION OF FUNCTIONS 277(10)
Graphs of Functions of Two Variables 277(3)
Level Curves 280(1)
Drawing Graphs from Level Sets 281(1)
Planar Level Sets in Economics 282(1)
Representing Functions from R^k to R^1 283(2)
for k is greater than 2
Images of Functions from R^1 to R^m 285(2)
13.3 SPECIAL KINDS OF FUNCTIONS 287(6)
Linear Functions on R^k 287(2)
Quadratic Forms 289(1)
Matrix Representation of Quadratic Forms 290(1)
Polynomials 291(2)
13.4 CONTINUOUS FUNCTIONS 293(2)
13.5 VOCABULARY OF FUNCTIONS 295(5)
Onto Functions and One-to-One Functions 297(1)
Inverse Functions 297(1)
Composition of Functions 298(2)
14 Calculus of Several Variables 300(34)
14.1 DEFINITIONS AND EXAMPLES 300(2)
14.2 ECONOMIC INTERPRETATION 302(3)
Marginal Products 302(2)
Elasticity 304(1)
14.3 GEOMETRIC INTERPRETATION 305(2)
14.4 THE TOTAL DERIVATIVE 307(6)
Geometric Interpretation 308(2)
Linear Approximation 310(1)
Functions of More than Two Variables 311(2)
14.5 THE CHAIN RULE 313(6)
Curves 313(1)
Tangent Vector to a Curve 314(2)
Differentiating along a Curve: The Chain 316(3)
Rule
14.6 DIRECTIONAL DERIVATIVES AND GRADIENTS 319(4)
Directional Derivatives 319(1)
The Gradient Vector 320(3)
14.7 EXPLICIT FUNCTIONS FROM R^n TO R^m 323(5)
Approximation by Differentials 324(2)
The Chain Rule 326(2)
14.8 HIGHER-ORDER DERIVATIVES 328(5)
Continuously Differentiable Functions 328(1)
Second Order Derivatives and Hessians 329(1)
Young's Theorem 330(1)
Higher-Order Derivatives 331(1)
An Economic Application 331(2)
14.9 Epilogue 333(1)
15 Implicit Functions and Their Derivatives 334(41)
15.1 IMPLICIT FUNCTIONS 334(8)
Examples 334(3)
The Implicit Function Theorem for R^2 337(4)
Several Exogenous Variables in an 341(1)
Implicit Function
15.2 LEVEL CURVES AND THEIR TANGENTS 342(8)
Geometric Interpretation of the Implicit 342(2)
Function Theorem
Proof Sketch 344(1)
Relationship to the Gradient 345(2)
Tangent to the Level Set Using 347(1)
Differentials
Level Sets of Functions of Several 348(2)
Variables
15.3 SYSTEMS OF IMPLICIT FUNCTIONS 350(10)
Linear Systems 351(2)
Nonlinear Systems 353(7)
15.4 APPLICATION: COMPARATIVE STATICS 360(4)
15.5 THE INVERSE FUNCTION THEOREM 364(4)
(optional)
15.6 APPLICATION: SIMPSON'S PARADOX 368(7)

PART IV Optimization 375(204)
16 Quadratic Forms and Definite Matrices 375(21)
16.1 QUADRATIC FORMS 375(1)
16.2 DEFINITENESS OF QUADRATIC FORMS 376(10)
Definite Symmetric Matrices 379(1)
Application: Second Order Conditions and 379(1)
Convexity
Application: Conic Sections 380(1)
Principal Minors of a Matrix 381(2)
The Definiteness of Diagonal Matrices 383(1)
The Definiteness of 2 x 2 Matrices 384(2)
16.3 LINEAR CONSTRAINTS AND BORDERED 386(7)
MATRICES
Definiteness and Optimality 386(4)
One Constraint 390(1)
Other Approaches 391(2)
16.4 APPENDIX 393(3)
17 Unconstrained Optimization 396(15)
17.1 DEFINITIONS 396(1)
17.2 FIRST ORDER CONDITIONS 397(1)
17.3 SECOND ORDER CONDITIONS 398(4)
Sufficient Conditions 398(3)
Necessary Conditions 401(1)
17.4 GLOBAL MAXIMA AND MINIMA 402(2)
Global Maxima of Concave Functions 403(1)
17.5 ECONOMIC APPLICATIONS 404(7)
Profit-Maximizing Firm 405(1)
Discriminating Monopolist 405(2)
Least Squares Analysis 407(4)
18 Constrained Optimization I: First Order 411(37)
Conditions
18.1 EXAMPLES 412(1)
18.2 EQUALITY CONSTRAINTS 413(11)
Two Variables and One Equality Constraint 413(7)
Several Equality Constraints 420(4)
18.3 INEQUALITY CONSTRAINTS 424(10)
One Inequality Constraint 424(6)
Several Inequality Constraints 430(4)
18.4 MIXED CONSTRAINTS 434(2)
18.5 CONSTRAINED MINIMIZATION PROBLEMS 436(3)
18.6 KUHN-TUCKER FORMULATION 439(3)
18.7 EXAMPLES AND APPLICATIONS 442(6)
Application: A Sales-Maximizing Firm with 442(1)
Advertising
Application: The Averch-Johnson Effect 443(2)
One More Worked Example 445(3)
19 Constrained Optimization II 448(35)
19.1 THE MEANING OF THE MULTIPLIER 448(5)
One Equality Constraint 449(1)
Several Equality Constraints 450(1)
Inequality Constraints 451(1)
Interpreting the Multiplier 452(1)
19.2 ENVELOPE THEOREMS 453(4)
Unconstrained Problems 453(2)
Constrained Problems 455(2)
19.3 SECOND ORDER CONDITIONS 457(12)
Constrained Maximization Problems 459(4)
Minimization Problems 463(3)
Inequality Constraints 466(1)
Alternative Approaches to the Bordered 467(1)
Hessian Condition
Necessary Second Order Conditions 468(1)
19.4 SMOOTH DEPENDENCE ON THE PARAMETERS 469(3)
19.5 CONSTRAINT QUALIFICATIONS 472(6)
19.6 PROOFS OF FIRST ORDER CONDITIONS 478(5)
Proof of Theorems 18.1 and 18.2: Equality 478(2)
Constraints
Proof of Theorems 18.3 and 18.4: 480(3)
Inequality Constraints
20 Homogeneous and Homothetic Functions 483(22)
20.1 HOMOGENEOUS FUNCTIONS 483(10)
Definition and Examples 483(2)
Homogeneous Functions in Economics 485(2)
Properties of Homogeneous Functions 487(4)
A Calculus Criterion for Homogeneity 491(1)
Economic Applications of Euler's Theorem 492(1)
20.2 HOMOGENIZING A FUNCTION 493(3)
Economic Applications of Homogenization 495(1)
20.3 CARDINAL VERSUS ORDINAL UTILITY 496(4)
20.4 HOMOTHETIC FUNCTIONS 500(5)
Motivation and Definition 500(1)
Characterizing Homothetic Functions 501(4)
21 Concave and Quasiconcave Functions 505(39)
21.1 CONCAVE AND CONVEX FUNCTIONS 505(12)
Calculus Criteria for Concavity 509(8)
21.2 PROPERTIES OF CONCAVE FUNCTIONS 517(5)
Concave Functions in Economics 521(1)
21.3 QUASICONCAVE AND QUASICONVEX FUNCTIONS 522(5)
Calculus Criteria 525(2)
21.4 PSEUDOCONCAVE FUNCTIONS 527(5)
21.5 CONCAVE PROGRAMMING 532(5)
Unconstrained Problems 532(1)
Constrained Problems 532(2)
Saddle Point Approach 534(3)
21.6 APPENDIX 537(7)
Proof of the Sufficiency Test of Theorem 537(1)
21.14
Proof of Theorem 21.15 538(2)
Proof of Theorem 21.17 540(1)
Proof of Theorem 21.20 541(3)
22 Economic Applications 544(35)
22.1 UTILITY AND DEMAND 544(13)
Utility Maximization 544(3)
The Demand Function 547(4)
The Indirect Utility Function 551(1)
The Expenditure and Compensated Demand 552(3)
Functions
The Slutsky Equation 555(2)
22.2 ECONOMIC APPLICATION: PROFIT AND COST 557(8)
The Profit-Maximizing Firm 557(3)
The Cost Function 560(5)
22.3 PARETO OPTIMA 565(4)
Necessary Conditions for a Pareto Optimum 566(1)
Sufficient Conditions for a Pareto Optimum 567(2)
22.4 THE FUNDAMENTAL WELFARE THEOREMS 569(10)
Competitive Equilibrium 572(1)
Fundamental Theorems of Welfare Economics 573(6)

PART V Eigenvalues and Dynamics 579(140)
23 Eigenvalues and Eigenvectors 579(54)
23.1 DEFINITIONS AND EXAMPLES 579(6)
23.2 SOLVING LINEAR DIFFERENCE EQUATIONS 585(12)
One-Dimensional Equations 585(1)
Two-Dimensional Systems: An Example 586(1)
Conic Sections 587(1)
The Leslie Population Model 588(2)
Abstract Two-Dimensional Systems 590(1)
k-Dimensional Systems 591(3)
An Alternative Approach: The Powers of a 594(2)
Matrix
Stability of Equilibria 596(1)
23.3 PROPERTIES OF EIGENVALUES 597(4)
Trace as Sum of the Eigenvalues 599(2)
23.4 REPEATED EIGENVALUES 601(8)
2 x 2 Nondiagonalizable Matrices 601(3)
3 x 3 Nondiagonalizable Matrices 604(2)
Solving Nondiagonalizable Difference 606(3)
Equations
23.5 COMPLEX EIGENVALUES AND EIGENVECTORS 609(6)
Diagonalizing Matrices with Complex 609(2)
Eigenvalues
Linear Difference Equations with Complex 611(3)
Eigenvalues
Higher Dimensions 614(1)
23.6 MARKOV PROCESSES 615(5)
23.7 SYMMETRIC MATRICES 620(6)
23.8 DEFINITENESS OF QUADRATIC FORMS 626(3)
23.9 APPENDIX 629(4)
Proof of Theorem 23.5 629(1)
Proof of Theorem 23.9 630(3)
24 Ordinary Differential Equations: Scalar 633(41)
Equations
24.1 DEFINITION AND EXAMPLES 633(6)
24.2 EXPLICIT SOLUTIONS 639(8)
Linear First Order Equations 639(2)
Separable Equations 641(6)
24.3 LINEAR SECOND ORDER EQUATIONS 647(10)
Introduction 647(1)
Real and Unequal Roots of the 648(2)
Characteristic Equation
Real and Equal Roots of the 650(1)
Characteristic Equation
Complex Roots of the Characteristic 651(2)
Equation
The Motion of a Spring 653(1)
Nonhomogeneous Second Order Equations 654(3)
24.4 EXISTENCE OF SOLUTIONS 657(9)
The Fundamental Existence and Uniqueness 657(2)
Theorem
Direction Fields Direction Fields 659(7)
24.5 PHASE PORTRAITS AND EQUILIBRIA ON R^1 666(4)
Drawing Phase Portraits 666(2)
Stability of Equilibria on the Line 668(2)
24.6 APPENDIX: APPLICATIONS 670(4)
Indirect Money Metric Utility Functions 671(1)
Converse of Euler's Theorem 672(2)
25 Ordinary Differential Equations: Systems 674(45)
of Equations
25.1 PLANAR SYSTEMS: AN INTRODUCTION 674(4)
Coupled Systems of Differential Equations 674(2)
Vocabulary 676(1)
Existence and Uniqueness 677(1)
25.2 LINEAR SYSTEMS VIA EIGENVALUES 678(4)
Distinct Real Eigenvalues 678(2)
Complex Eigenvalues 680(1)
Multiple Real Eigenvalues 681(1)
25.3 SOLVING LINEAR SYSTEMS BY SUBSTITUTION 682(2)
25.4 STEADY STATES AND THEIR STABILITY 684(5)
Stability of Linear Systems via 686(1)
Eigenvalues
Stability of Nonlinear Systems 687(2)
25.5 PHASE PORTRAITS OF PLANAR SYSTEMS 689(14)
Vector Fields 689(3)
Phase Portraits: Linear Systems 692(2)
Phase Portraits: Nonlinear Systems 694(9)
25.6 FIRST INTEGRALS 703(8)
The Predator-Prey System 705(2)
Conservative Mechanical Systems 707(4)
25.7 LIAPUNOV FUNCTIONS 711(4)
25.8 APPENDIX: LINEARIZATION 715(4)

PART VI Advanced Linear Algebra 719(84)
26 Determinants: The Details 719(31)
26.1 DEFINITIONS OF THE DETERMINANT 719(7)
26.2 PROPERTIES OF THE DETERMINANT 726(9)
26.3 USING DETERMINANTS 735(4)
The Adjoint Matrix 736(3)
26.4 ECONOMIC APPLICATIONS 739(4)
Supply and Demand 739(4)
26.5 APPENDIX 743(7)
Proof of Theorem 26.1 743(3)
Proof of Theorem 26.9 746(1)
Other Approaches to the Determinant 747(3)
27 Subspaces Attached to a Matrix 750(29)
27.1 VECTOR SPACES AND SUBSPACES 750(5)
R^n as a Vector Space 750(1)
Subspaces of R^n 751(4)
27.2 BASIS AND DIMENSION OF A PROPER 755(2)
SUBSPACE
27.3 ROW SPACE 757(3)
27.4 COLUMN SPACE 760(5)
Dimension of the Column Space of A 760(3)
The Role of the Column Space 763(2)
27.5 NULLSPACE 765(6)
Affine Subspaces 765(2)
Fundamental Theorem of Linear Algebra 767(3)
Conclusion 770(1)
27.6 ABSTRACT VECTOR SPACES 771(3)
27.7 APPENDIX 774(5)
Proof of Theorem 27.5 774(1)
Proof of Theorem 27.10 775(4)
28 Applications of Linear Independence 779(24)
28.1 GEOMETRY OF SYSTEMS OF EQUATIONS 779(4)
Two Equations in Two Unknowns 779(1)
Two Equations in Three Unknowns 780(2)
Three Equations in Three Unknowns 782(1)
28.2 PORTFOLIO ANALYSIS 783(1)
28.3 VOTING PARADOXES 784(7)
Three Alternatives 785(3)
Four Alternatives 788(1)
Consequences of the Existence of Cycles 789(1)
Other Voting Paradoxes 790(1)
Rankings of the Quality of Firms 790(1)
28.4 ACTIVITY ANALYSIS: FEASIBILITY 791(5)
Activity Analysis 791(2)
Simple Linear Models and Productive 793(3)
Matrices
28.5 ACTIVITY ANALYSIS: EFFICIENCY 796(7)
Leontief Models 796(7)

PART VII Advanced Analysis 803(44)
29 Limits and Compact Sets 803(19)
29.1 CAUCHY SEQUENCES 803(4)
29.2 COMPACT SETS 807(2)
29.3 CONNECTED SETS 809(2)
29.4 ALTERNATIVE NORMS 811(5)
Three Norms on R^n 811(2)
Equivalent Norms 813(2)
Norms on Function Spaces 815(1)
29.5 APPENDIX 816(6)
Finite Covering Property 816(1)
Heine-Borel Theorem 817(3)
Summary 820(2)
30 Calculus of Several Variables II 822(25)
30.1 WEIERSTRASS'S AND MEAN VALUE THEOREMS 822(5)
Existence of Global Maxima on Compact Sets 822(2)
Rolle's Theorem and the Mean Value Theorem 824(3)
30.2 TAYLOR POLYNOMIALS ON R^1 827(5)
Functions of One Variable 827(5)
30.3 TAYLOR POLYNOMIALS IN R^n 832(4)
30.4 SECOND ORDER OPTIMIZATION CONDITIONS 836(5)
Second Order Sufficient Conditions for 836(3)
Optimization
Indefinite Hessian 839(1)
Second Order Necessary Conditions for 840(1)
Optimization
30.5 CONSTRAINED OPTIMIZATION 841(6)

PART VIII Appendices 847(74)
A1 Sets, Numbers, and Proofs 847(12)
A1.1 SETS 847(1)
Vocabulary of Sets 847(1)
Operations with Sets 847(1)
A1.2 NUMBERS 848(3)
Vocabulary 848(1)
Properties of Addition and Multiplication 849(1)
Least Upper Bound Property 850(1)
A1.3 PROOFS 851(8)
Direct Proofs 851(2)
Converse and Contrapositive 853(1)
Indirect Proofs 854(1)
Mathematical Induction 855(4)
A2 Trigonometric Functions 859(17)
A2.1 DEFINITIONS OF THE TRIG FUNCTIONS 859(4)
A2.2 GRAPHING TRIG FUNCTIONS 863(2)
A2.3 THE PYTHAGOREAN THEOREM 865(1)
A2.4 EVALUATING TRIGONOMETRIC FUNCTIONS 866(2)
A2.5 MULTIANGLE FORMULAS 868(1)
A2.6 FUNCTIONS OF REAL NUMBERS 868(2)
A2.7 CALCULUS WITH TRIG FUNCTIONS 870(2)
A2.8 TAYLOR SERIES 872(1)
A2.9 PROOF OF THEOREM A2.3 873(3)
A3 Complex Numbers 876(11)
A3.1 BACKGROUND 876(2)
Definitions 877(1)
Arithmetic Operations 877(1)
A3.2 SOLUTIONS OF POLYNOMIAL EQUATIONS 878(1)
A3.3 GEOMETRIC REPRESENTATION 879(3)
A3.4 COMPLEX NUMBERS AS EXPONENTS 882(2)
A3.5 DIFFERENCE EQUATIONS 884(3)
A4 Integral Calculus 887(7)
A4.1 ANTIDERIVATIVES 887(2)
Integration by Parts 888(1)
A4.2 THE FUNDAMENTAL THEOREM OF CALCULUS 889(1)
A4.3 APPLICATIONS 890(4)
Area under a Graph 890(1)
Consumer Surplus 891(1)
Present Value of a Flow 892(2)
A5 Introduction to Probability 894(5)
A5.1 PROBABILITY OF AN EVENT 894(1)
A5.2 EXPECTATION AND VARIANCE 895(1)
A5.3 CONTINUOUS RANDOM VARIABLES 896(3)
A6 Selected Answers 899(22)

Index 921

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