Mathematics for Economists

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

ISBN: 9780393957334 | 0393957330
Cover: Hardcover
Copyright: 4/17/1994

Rent

(Recommended)

$76.38

Term

Due

Price

Semester

Mar 23

$84.87

Quarter

Jan 2

$80.63

Short Term

Dec 12

$76.38

*This item is part of an exclusive publisher rental program and requires an additional convenience fee. This fee will be reflected in the shopping bag.
Buy New

Usually Ships in 3-5 Business Days

$135.82

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.

Preface

xxi

PART I Introduction

(104)

1 Introduction

(7)

1.1 MATHEMATICS IN ECONOMIC THEORY

(2)

1.2 MODELS OF CONSUMER CHOICE

(5)

Two-Dimensional Model of Consumer Choice

(4)

Multidimensional Model of Consumer Choice

(1)

2 One-Variable Calculus: Foundations

(29)

2.1 FUNCTIONS ON R^1

(6)

Vocabulary of Functions

(1)

Polynomials

(1)

Graphs

(1)

Increasing and Decreasing Functions

(2)

Domain

(1)

Interval Notation

(1)

2.2 LINEAR FUNCTIONS

(6)

The Slope of a Line in the Plane

(3)

The Equation of a Line

(1)

Polynomials of Degree One Have Linear Graphs

(1)

Interpreting the Slope of a Linear Function

(2)

2.3 THE SLOPE OF NONLINEAR FUNCTIONS

(3)

2.4 COMPUTING DERIVATIVES

(4)

Rules for Computing Derivatives

(2)

2.5 DIFFERENTIABILITY AND CONTINUITY

(4)

A Nondifferentiable Function

(1)

Continuous Functions

(1)

Continuously Differentiable Functions

(1)

2.6 HIGHER-ORDER DERIVATIVES

(1)

2.7 APPROXIMATION BY DIFFERENTIALS

(5)

3 One-Variable Calculus: Applications

(31)

3.1 USING THE FIRST DERIVATIVE FOR GRAPHING

(4)

Positive Derivative Implies Increasing Function

(2)

Using First Derivatives to Sketch Graphs

(2)

3.2 SECOND DERIVATIVES AND CONVEXITY

(4)

3.3 GRAPHING RATIONAL FUNCTIONS

(1)

Hints for Graphing

(1)

3.4 TAILS AND HORIZONTAL ASYMPTOTES

(3)

Tails of Polynomials

(1)

Horizontal Asymptotes of Rational Functions

(2)

3.5 MAXIMA AND MINIMA

(7)

Local Maxima and Minima on the Boundary and in the Interior

(2)

Second Order Conditions

(2)

Global Maxima and Minima

(1)

Functions with Only One Critical Point

(1)

Functions with Nowhere-Zero Second Derivatives

(1)

Functions with No Global Max or Min

(1)

Functions Whose Domains Are Closed Finite Intervals

(2)

3.6 APPLICATIONS TO ECONOMICS

(12)

Production Functions

(1)

Cost Functions

(3)

Revenue and Profit Functions

(2)

Demand Functions and Elasticity

(6)

4 One-Variable Calculus: Chain Rule

(12)

4.1 COMPOSITE FUNCTIONS AND THE CHAIN RULE

(5)

Composite Functions

(2)

Differentiating Composite Functions: The Chain Rule

(3)

4.2 INVERSE FUNCTIONS AND THEIR DERIVATIVES

(7)

Definition and Examples of the Inverse of a Function

(4)

The Derivative of the Inverse Function

(1)

The Derivative of x^m/n

(2)

5 Exponents and Logarithms

(25)

5.1 EXPONENTIAL FUNCTIONS

(3)

5.2 THE NUMBER e

(3)

5.3 LOGARITHMS

(3)

Base 10 Logarithms

(2)

Base e Logarithms

(1)

5.4 PROPERTIES OF EXP AND LOG

(2)

5.5 DERIVATIVES OF EXP AND LOG

(4)

5.6 APPLICATIONS

(10)

Present Value

(1)

Annuities

(1)

Optimal Holding Time

(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

Please wait while the item is added to your bag...

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Mathematics for Economists

Summary

Table of Contents

Supplemental Materials

FREE SHIPPING

Rent More, Save More! Use code: BBRENTAL

5% off 1 book, 7% off 2 books, 10% off 3+ books

Mathematics for Economists

Summary

Table of Contents

Supplemental Materials

Digital License