实例介绍
【实例简介】
Cracking the GRE Mathematics Subject Test, 4th Edition
Princeton Reviev° 〔 racking the GRE Mathematics Subiect Test STEVEN A. LEDUC FOURTH EDITION RANDOM HOUSE, INC NEW YORK Princeton review. com The Princeton Review Editorial The princeton review Inc Rob Fronek, VP Test Prep Books, Publisher 2315 Broadway Seamus Mullarkey, Associate Publisher New York. NY 10024 Rebecco lessem. Senior Editor Email:editoriolsupport@review.com louro Boswell. Serior Editor n0p00 Copyright C 2010 by The Princeton Review, Inc Heather Brody, Editor All rights reserved. Published in the United States by Random House, Inc, New York, and in Canada by Production Services Rondom house of conad limited Toronto ott Horris Executive Director, Production Services Kim Howie, Senior Grophic Designer SBk:978037542972 Ryon ToZzi, Production Monoger SS:1541-4957 Production Editorial Editor: Rebecca lessem Meove Shelton Produchon Editor Production Editor Meave shelton jennifer graham Production editor Production Coordinator: Mary Kinzel Kristen 0'Toole Production Editor Printed in the United States of Americo on partially recycled Random House Publishing Team paper Russell. publishe 1098765 Nicole Benhabib, Publishing Manager Ellen L. Reed, Production Manager Fourth Edition Alison Stoltzfus, Associote Manuging Editor Elhom Shobahot, Publishing Assistant This work is dedicated to the memories of my grandparents, Eugenia and Joseph paul, and my great aunt, Carolyn Lamb Wilks I miss you ACKNOWLEDGMENTS I'd like to thank Evan Schnittman, John Katzman, Steve Quattrociocchi, Paul Maniscalco, Ian Stewart Suellen Glasser, Chris Volpe, and Kris Gamache for their confidence and support. Thanks to Christopher Anton for his help in updating this edition. Special thanks to Paul Kanarek for his support, friendship, and encouragement To the mathematics teachers in my life who provided instruction, encouragement, and guidance, I'd like to say thank you: Edna Nawrocki(Kendall Dean Elementary ) Elizabeth Crisafulli(Union Village Elementary ) Janice Lanik and Charles Brady(Halliwell Memorial); Robert Deroy, Malcolm Andrews, and especially Mary Provost(North Smithfield Junior-Senior High School, North Smithfield, RI); and to my professors at MIT and UCSD, including Frank Morgan, David Jerison, Linda Rothschild, and especially Arthur Mattuck and James Munkres. They not only taught me how to learn, they also taught me how to teach, and their excellence has always inspired me CONTENTS Preface PRECALCULUS 5 Functions 5 Composition of Functions 6 Inverse Functions 7 Graphs in the xy-Plane 9 Analytic Geometry 11 Lines I Parabolas 1 1 Circles 13 Ellipses 14 pergola 15 lynomial Equations 17 The Division Algorithm, Remainder Theorem, and Factor Theorem 17 The Fundamental Theorem of algebra and Roots of Polynomial Equations 18 The Root Location Theorem 18 The Rational Roots theorem 18 The Conjugate Radical Roots Theorem 18 The Complex Conjugate Roots Theorem 18 Sum and product of the roots 19 Logarithms 20 Trigonometry 22 Tria Functions of Acute Angles 22 ig Functions of Arbitrary Angles 23 Trig Functions of Real Numbers 24 Trig Identities and Formulas 24 Fundamental Identities 25 Opposite-Angle identities 25 thagorean identities 25 Addition and Subtraction Formulas 25 Double-Angle Formulas 25 Complementary Angle(Reduction)Formulas 25 Half-Angle Formulas 25 Periodicity of the Trig Functions 27 Graphs of the Trig Functions 27 The Inverse Trig Functions 28 Chapter 1 Review Questions 30 CALCULUS I 35 Limits of s 35 Limits of functions 38 Limits of functions as x→)± Continuous Functions 41 Theorems Concerning Continuous Functions 44 The Derivative 45 Linear Approximations Using differentials 50 Implicit Differenti 51 Higher-Order Derivatives 51 Curve Sketching 52 f the first 52 Properties of the Second Derivative 52 Theorems Concerning Differentiable Functions 54 Max/Min Problems 55 Related rates 57 Indefinite Integration (Antidifferentiation) 59 Techniques of Integration Integration by Substitution Integration by Parts 61 Trig Substitutions 62 The Method of Partial Fractions 63 Definite Integration 64 The Fundamental theorem of calculus 66 The Average Value of a Function 68 Finding the Area Between Two Curves 69 Polar coordinates 71 Volumes of solids of revolution 73 Arc Length 75 The Natural Exponential and Logarithm Functions 76 L'HOpital's Rule 79 Improper Integrals 81 Infinite Series 84 Alternating Series 88 Power Series 89 Functions Defined by Power Series 90 Taylor Series 92 Taylor Polynomials 93 Chapter 2 Review Questions 95 CALCULUS‖105 Analytic Geometry of R3 105 The Dot Product 107 The Cross Product 109 The Triple Scalar Product 110 Lines in 3-Space 112 P anes In 113 Cylinders 115 Surfaces of revolution 117 Level curves and Level surfaces 119 Cylindrical Coordinates 121 Spherical Coordinates 121 Partial derivatives 122 Geometric Interpretation of f and f 123 Higher-Order Partial Derivatives 124 The Tangent Plane to a Surface 126 Linear Approximations 127 The Chain rule for partial Derivatives 128 Directional Derivatives and the Gradient 132 Max/Min Problems 135 Max/Min Problems with a Constraint 137 The Lagrange Multip plier Method 138 Line Integrals 139 Line Integrals with Respect to Arc Length 140 The Line Integral of a Vector Field 143 The Fundamental Theorem of Calculus for Line Integrals 147 Double Integrals 150 Double integrals in Polar Coordinates 154 Green's Theorem 156 Path Independence and Gradient Fields 158 Chapter 3 Review Questions 160 DIFFERENTIAL EQUATIONS 167 Separable Equations 169 Homogeneous Equations 170 vi◆C0 NTENTS Exact Equations 170 Nonexact Equations and Integrating Factors 172 First-Order Linear Equations 174 Higher-Order Linear Equations with Constant Coefficients 175 Chapter 4 Review Questions 178 5 LINEAR ALGEBRA 18 Solutions of Linear Systems 181 Matrices and Matrix Algebra 183 Matrix Operations 183 Identity Matrices and Inverses 186 Gaussian Elimination 188 Solving Matrix Equations Using A-193 Vector s 195 The Nullspace 196 196 The Rank, Column Space, and Row Space of a Matrix 198 200 Determinants 200 Laplace e 204 The Adjugate Matrix 205 Cramer's Rule 206 Linear Transformations 207 Standard Matrix Representative 208 The Rank Plus Nullity Thec A Note on Inverses and Compositions 209 Eigenvalues and Eigenvectors 209 212 The Cayley-Hamilton Theorem 213 Chapter 5 Review Questions 214 6 NUMBER THEORY AND ABSTRACT ALGEBRA 219 Part A: Number Theory 220 Divisibility 220 The Division Algorithm 221 221 The Greatest Common Divisor and the Least Common Multiple 221 The Euclidean Algorithm 222 The Diophantine Equation ox+ by=c 223 Congruences 225 The Congruence Equation ax=b(mod n) 226 Part B: Abstract Algebra 227 Binary Structures and the Defi 228 Cyclic Groups 231 Subgroups 232 Cyclic Subgroups 23 Generators and Relations 233 Some Theorems Concerning Subgroups 234 The Concept of isomorphism 235 The Classification of Finite Abelian Groups 237 Group Homomorphisms 240 Rings 244 Ring ho Integral De Fields 251 Chapter 6 Review Questions 254 C0 NTENTS◆vi ADDIIONAL TOPICS 257 Logic 257 Set Theory 259 Subsets and Complements 259 Union and Intersection 260 Cartesian Products 260 Intervals of the Real Line 261 Venn di 263 Cardinality 263 Graph Theory 265 Algorithms 267 Combinatorics 267 Permutations and Combinations 268 With Re flowed 270 The Pigeonhole Principle 271 Probability and Statistics 271 P bility Spaces 273 Bernoulli trials 276 Random variables 277 Expectation, Variance, and Standard Deviation 279 The Normal Distribution 280 The Normal Approximation to the Binomial Distribution 282 Point-Set Topology 283 The Subspace Topology 284 The Interior, Exterior, Boundary, Limit Points, and Closure of a Set 284 Basis for a Topology 286 The Product Topology 287 Connectedness 288 Compactness 288 Metric Spaces 289 Continuous functions 290 Open Maps and Homeomorphisms 292 Real Analysis 293 f the Real nu 293 Leben gue Measure Lebesgue Measurable Functions 295 Lebesque Integrable Functions 296 Complex Variables 298 e polar Form The Exponential Form 300 Complex Roots 300 Complex Logarithms 302 302 The Trigonometric Functions 303 The Hyperbolic Ft 304 The Derivative of a Function of a Complex Variable 305 The Cauchy-Riemann Equations 306 Analytic Functions 308 Complex Line Integrals 309 Theorems Concerning Analytic Functions 310 Taylor Series for Functions of a Complex Variable 311 Singularities, Poles, and Laurent Series 311 The Residue Theorem 314 al Analysis 317 Chapter 7 Review Questions 319 8 SOLUTIONS TO THE CHAPTER REVIEW QUESTIONS 331 PRACTICE TEST 405 10 PRACTICE TEST ANSWERS AND EXPLANATIONS 421 About the author 445 vi◆〔 ONTENTS 【实例截图】
【核心代码】
Cracking the GRE Mathematics Subject Test, 4th Edition
Princeton Reviev° 〔 racking the GRE Mathematics Subiect Test STEVEN A. LEDUC FOURTH EDITION RANDOM HOUSE, INC NEW YORK Princeton review. com The Princeton Review Editorial The princeton review Inc Rob Fronek, VP Test Prep Books, Publisher 2315 Broadway Seamus Mullarkey, Associate Publisher New York. NY 10024 Rebecco lessem. Senior Editor Email:editoriolsupport@review.com louro Boswell. Serior Editor n0p00 Copyright C 2010 by The Princeton Review, Inc Heather Brody, Editor All rights reserved. Published in the United States by Random House, Inc, New York, and in Canada by Production Services Rondom house of conad limited Toronto ott Horris Executive Director, Production Services Kim Howie, Senior Grophic Designer SBk:978037542972 Ryon ToZzi, Production Monoger SS:1541-4957 Production Editorial Editor: Rebecca lessem Meove Shelton Produchon Editor Production Editor Meave shelton jennifer graham Production editor Production Coordinator: Mary Kinzel Kristen 0'Toole Production Editor Printed in the United States of Americo on partially recycled Random House Publishing Team paper Russell. publishe 1098765 Nicole Benhabib, Publishing Manager Ellen L. Reed, Production Manager Fourth Edition Alison Stoltzfus, Associote Manuging Editor Elhom Shobahot, Publishing Assistant This work is dedicated to the memories of my grandparents, Eugenia and Joseph paul, and my great aunt, Carolyn Lamb Wilks I miss you ACKNOWLEDGMENTS I'd like to thank Evan Schnittman, John Katzman, Steve Quattrociocchi, Paul Maniscalco, Ian Stewart Suellen Glasser, Chris Volpe, and Kris Gamache for their confidence and support. Thanks to Christopher Anton for his help in updating this edition. Special thanks to Paul Kanarek for his support, friendship, and encouragement To the mathematics teachers in my life who provided instruction, encouragement, and guidance, I'd like to say thank you: Edna Nawrocki(Kendall Dean Elementary ) Elizabeth Crisafulli(Union Village Elementary ) Janice Lanik and Charles Brady(Halliwell Memorial); Robert Deroy, Malcolm Andrews, and especially Mary Provost(North Smithfield Junior-Senior High School, North Smithfield, RI); and to my professors at MIT and UCSD, including Frank Morgan, David Jerison, Linda Rothschild, and especially Arthur Mattuck and James Munkres. They not only taught me how to learn, they also taught me how to teach, and their excellence has always inspired me CONTENTS Preface PRECALCULUS 5 Functions 5 Composition of Functions 6 Inverse Functions 7 Graphs in the xy-Plane 9 Analytic Geometry 11 Lines I Parabolas 1 1 Circles 13 Ellipses 14 pergola 15 lynomial Equations 17 The Division Algorithm, Remainder Theorem, and Factor Theorem 17 The Fundamental Theorem of algebra and Roots of Polynomial Equations 18 The Root Location Theorem 18 The Rational Roots theorem 18 The Conjugate Radical Roots Theorem 18 The Complex Conjugate Roots Theorem 18 Sum and product of the roots 19 Logarithms 20 Trigonometry 22 Tria Functions of Acute Angles 22 ig Functions of Arbitrary Angles 23 Trig Functions of Real Numbers 24 Trig Identities and Formulas 24 Fundamental Identities 25 Opposite-Angle identities 25 thagorean identities 25 Addition and Subtraction Formulas 25 Double-Angle Formulas 25 Complementary Angle(Reduction)Formulas 25 Half-Angle Formulas 25 Periodicity of the Trig Functions 27 Graphs of the Trig Functions 27 The Inverse Trig Functions 28 Chapter 1 Review Questions 30 CALCULUS I 35 Limits of s 35 Limits of functions 38 Limits of functions as x→)± Continuous Functions 41 Theorems Concerning Continuous Functions 44 The Derivative 45 Linear Approximations Using differentials 50 Implicit Differenti 51 Higher-Order Derivatives 51 Curve Sketching 52 f the first 52 Properties of the Second Derivative 52 Theorems Concerning Differentiable Functions 54 Max/Min Problems 55 Related rates 57 Indefinite Integration (Antidifferentiation) 59 Techniques of Integration Integration by Substitution Integration by Parts 61 Trig Substitutions 62 The Method of Partial Fractions 63 Definite Integration 64 The Fundamental theorem of calculus 66 The Average Value of a Function 68 Finding the Area Between Two Curves 69 Polar coordinates 71 Volumes of solids of revolution 73 Arc Length 75 The Natural Exponential and Logarithm Functions 76 L'HOpital's Rule 79 Improper Integrals 81 Infinite Series 84 Alternating Series 88 Power Series 89 Functions Defined by Power Series 90 Taylor Series 92 Taylor Polynomials 93 Chapter 2 Review Questions 95 CALCULUS‖105 Analytic Geometry of R3 105 The Dot Product 107 The Cross Product 109 The Triple Scalar Product 110 Lines in 3-Space 112 P anes In 113 Cylinders 115 Surfaces of revolution 117 Level curves and Level surfaces 119 Cylindrical Coordinates 121 Spherical Coordinates 121 Partial derivatives 122 Geometric Interpretation of f and f 123 Higher-Order Partial Derivatives 124 The Tangent Plane to a Surface 126 Linear Approximations 127 The Chain rule for partial Derivatives 128 Directional Derivatives and the Gradient 132 Max/Min Problems 135 Max/Min Problems with a Constraint 137 The Lagrange Multip plier Method 138 Line Integrals 139 Line Integrals with Respect to Arc Length 140 The Line Integral of a Vector Field 143 The Fundamental Theorem of Calculus for Line Integrals 147 Double Integrals 150 Double integrals in Polar Coordinates 154 Green's Theorem 156 Path Independence and Gradient Fields 158 Chapter 3 Review Questions 160 DIFFERENTIAL EQUATIONS 167 Separable Equations 169 Homogeneous Equations 170 vi◆C0 NTENTS Exact Equations 170 Nonexact Equations and Integrating Factors 172 First-Order Linear Equations 174 Higher-Order Linear Equations with Constant Coefficients 175 Chapter 4 Review Questions 178 5 LINEAR ALGEBRA 18 Solutions of Linear Systems 181 Matrices and Matrix Algebra 183 Matrix Operations 183 Identity Matrices and Inverses 186 Gaussian Elimination 188 Solving Matrix Equations Using A-193 Vector s 195 The Nullspace 196 196 The Rank, Column Space, and Row Space of a Matrix 198 200 Determinants 200 Laplace e 204 The Adjugate Matrix 205 Cramer's Rule 206 Linear Transformations 207 Standard Matrix Representative 208 The Rank Plus Nullity Thec A Note on Inverses and Compositions 209 Eigenvalues and Eigenvectors 209 212 The Cayley-Hamilton Theorem 213 Chapter 5 Review Questions 214 6 NUMBER THEORY AND ABSTRACT ALGEBRA 219 Part A: Number Theory 220 Divisibility 220 The Division Algorithm 221 221 The Greatest Common Divisor and the Least Common Multiple 221 The Euclidean Algorithm 222 The Diophantine Equation ox+ by=c 223 Congruences 225 The Congruence Equation ax=b(mod n) 226 Part B: Abstract Algebra 227 Binary Structures and the Defi 228 Cyclic Groups 231 Subgroups 232 Cyclic Subgroups 23 Generators and Relations 233 Some Theorems Concerning Subgroups 234 The Concept of isomorphism 235 The Classification of Finite Abelian Groups 237 Group Homomorphisms 240 Rings 244 Ring ho Integral De Fields 251 Chapter 6 Review Questions 254 C0 NTENTS◆vi ADDIIONAL TOPICS 257 Logic 257 Set Theory 259 Subsets and Complements 259 Union and Intersection 260 Cartesian Products 260 Intervals of the Real Line 261 Venn di 263 Cardinality 263 Graph Theory 265 Algorithms 267 Combinatorics 267 Permutations and Combinations 268 With Re flowed 270 The Pigeonhole Principle 271 Probability and Statistics 271 P bility Spaces 273 Bernoulli trials 276 Random variables 277 Expectation, Variance, and Standard Deviation 279 The Normal Distribution 280 The Normal Approximation to the Binomial Distribution 282 Point-Set Topology 283 The Subspace Topology 284 The Interior, Exterior, Boundary, Limit Points, and Closure of a Set 284 Basis for a Topology 286 The Product Topology 287 Connectedness 288 Compactness 288 Metric Spaces 289 Continuous functions 290 Open Maps and Homeomorphisms 292 Real Analysis 293 f the Real nu 293 Leben gue Measure Lebesgue Measurable Functions 295 Lebesque Integrable Functions 296 Complex Variables 298 e polar Form The Exponential Form 300 Complex Roots 300 Complex Logarithms 302 302 The Trigonometric Functions 303 The Hyperbolic Ft 304 The Derivative of a Function of a Complex Variable 305 The Cauchy-Riemann Equations 306 Analytic Functions 308 Complex Line Integrals 309 Theorems Concerning Analytic Functions 310 Taylor Series for Functions of a Complex Variable 311 Singularities, Poles, and Laurent Series 311 The Residue Theorem 314 al Analysis 317 Chapter 7 Review Questions 319 8 SOLUTIONS TO THE CHAPTER REVIEW QUESTIONS 331 PRACTICE TEST 405 10 PRACTICE TEST ANSWERS AND EXPLANATIONS 421 About the author 445 vi◆〔 ONTENTS 【实例截图】
【核心代码】
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