Mathematics for Economists,9780393957334

Mathematics for Economists

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ISBN13: 9780393957334
ISBN10: 0393957330
Format: Hardcover
Pub. Date: 1994-04-17
Publisher(s): W. W. Norton & Company
List Price: $137.33

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Summary

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.

Table of Contents

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 Graphs
19(1)
Interpreting the Slope of a Linear Function
20(2)
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 Function
39(2)
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 Functions
49(2)
3.5 MAXIMA AND MINIMA
51(7)
Local Maxima and Minima on the Boundary and in the Interior
51(2)
Second Order Conditions
53(2)
Global Maxima and Minima
55(1)
Functions with Only One Critical Point
55(1)
Functions with Nowhere-Zero Second Derivatives
56(1)
Functions with No Global Max or Min
56(1)
Functions Whose Domains Are Closed Finite Intervals
56(2)
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 Chain Rule
72(3)
4.2 INVERSE FUNCTIONS AND THEIR DERIVATIVES
75(7)
Definition and Examples of the Inverse of a Function
75(4)
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 Contributions
108(2)
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 for k is greater than 2
283(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 Rule
316(3)
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 Implicit Function
341(1)
15.2 LEVEL CURVES AND THEIR TANGENTS
342(8)
Geometric Interpretation of the Implicit Function Theorem
342(2)
Proof Sketch
344(1)
Relationship to the Gradient
345(2)
Tangent to the Level Set Using Differentials
347(1)
Level Sets of Functions of Several Variables
348(2)
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 (optional)
364(4)
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 Convexity
379(1)
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 MATRICES
386(7)
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 Conditions
411(37)
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 Advertising
442(1)
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 Hessian Condition
467(1)
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 Constraints
478(2)
Proof of Theorems 18.3 and 18.4: Inequality Constraints
480(3)
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 21.14
537(1)
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 Functions
552(3)
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 Matrix
594(2)
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 Equations
606(3)
23.5 COMPLEX EIGENVALUES AND EIGENVECTORS
609(6)
Diagonalizing Matrices with Complex Eigenvalues
609(2)
Linear Difference Equations with Complex Eigenvalues
611(3)
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 Equations
633(41)
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 Characteristic Equation
648(2)
Real and Equal Roots of the Characteristic Equation
650(1)
Complex Roots of the Characteristic Equation
651(2)
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 Theorem
657(2)
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 of Equations
674(45)
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 Eigenvalues
686(1)
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 SUBSPACE
755(2)
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 Matrices
793(3)
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 Optimization
836(3)
Indefinite Hessian
839(1)
Second Order Necessary Conditions for Optimization
840(1)
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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