Cracking the GRE Math Test

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Overview

The GRE subject tests are among the most difficult standardized exams. Rather than testing general problem-solving skills, they require highly specialized knowledge.

The experts at The Princeton Review have thoroughly research each subject test to provide students with the most thorough, up-to-date information available. Students don't need to relearn the entire histories of their fields--just what they need to know to earn high scores on the ...

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More About This Book

Overview

The GRE subject tests are among the most difficult standardized exams. Rather than testing general problem-solving skills, they require highly specialized knowledge.

The experts at The Princeton Review have thoroughly research each subject test to provide students with the most thorough, up-to-date information available. Students don't need to relearn the entire histories of their fields--just what they need to know to earn high scores on the exams.

Each guide includes one full-length practice exam, complete with comprehensive explanations for every solution.

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Product Details

  • ISBN-13: 9780375762673
  • Publisher: Random House Information Group
  • Publication date: 9/24/2002
  • Series: Graduate Test Prep Ser.
  • Edition description: 2ND
  • Edition number: 2
  • Pages: 464
  • Product dimensions: 7.96 (w) x 11.00 (h) x 1.06 (d)

Meet the Author

The Princeton Review is the fastest growing test-preparation company in the country, with over 60 franchise offices in the nation. Each year, we help more than 2 million students prepare for college, grad school, professional licensing exams, and successful careers.
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Table of Contents

Preface xv
1 Precalculus 1
Functions 1
Composition of Functions 2
Inverse Functions 3
Graphs in the x-y Plane 5
Analytic Geometry 7
Lines 7
Parabolas 7
Circles 9
Ellipses 10
Hyperbolas 11
Polynomial Equations 13
The Division Algorithm, Remainder Theorem, and Factor Theorem 13
The Fundamental Theorem of Algebra and Roots of Polynomial Equations 14
The Root Location Theorem 14
The Rational Roots Theorem 14
The Conjugate Radical Roots Theorem 14
The Complex Conjugate Roots Theorem 15
Sum and Product of the Roots 15
Logarithms 16
Trigonometry 18
Trig Functions of Acute Angles 18
Trig Functions of Arbitrary Angles 19
Trig Functions of Real Numbers 19
Trig Identities and Formulas 20
Fundamental Identities 20
Opposite-Angle Identities 21
Pythagorean Identities 21
Addition and Subtraction Formulas 21
Double-Angle Formulas 21
Complementary-Angle (Reduction) Formulas 21
Half-Angle Formulas 21
Periodicity of the Trig Functions 23
Graphs of the Trig Functions 23
The Inverse Trig Functions 24
Chapter 1 Review Questions 26
2 Calculus I 31
Limits of Sequences 31
Limits of Functions 34
Limits of Functions as x to [plus or minus infinity] 36
Continuous Functions 37
Theorems Concerning Continuous Functions 40
The Derivative 42
Linear Approximations Using Differentials 46
Implicit Differentiation 47
Higher-Order Derivatives 47
Curve Sketching 48
Properties of the First Derivative 48
Properties of the Second Derivative 48
Theorems Concerning Differentiable Functions 50
Max/Min Problems 51
Related Rates 53
Indefinite Integration (Antidifferentiation) 55
Techniques of Integration 55
Integration by Substitution 55
Integration by Parts 57
Trig Substitutions 58
The Method of Partial Fractions 59
Definite Integration 60
The Fundamental Theorem of Calculus 62
The Average Value of a Function 64
Finding the Area Between Two Curves 65
Polar Coordinates 67
Volumes of Solids of Revolution 69
Arc Length 71
The Natural Exponential and Logarithm Functions 72
L'Hopital's Rule 75
Improper Integrals 77
Infinite Series 80
Alternating Series 84
Power Series 84
Functions Defined by Power Series 86
Taylor Series 87
Taylor Polynomials 89
Chapter 2 Review Questions 91
3 Calculus II 101
Analytic Geometry of R[superscript 3] 101
The Dot Product 103
The Cross Product 105
The Triple Scalar Product 106
Lines in 3-Space 108
Planes in 3-Space 109
Cylinders 111
Surfaces of Revolution 113
Level Curves and Level Surfaces 115
Cylindrical Coordinates 117
Spherical Coordinates 117
Partial Derivatives 118
Geometric Interpretation of f[subscript x] and f[subscript y] 119
Higher-Order Partial Derivatives 120
The Tangent Plane to a Surface 122
Linear Approximations 123
The Chain Rule for Partial Derivatives 124
Directional Derivatives and the Gradient 128
Max/Min Problems 131
Max/Min Problems with a Constraint 133
The Lagrange Multiplier Method 134
Line Integrals 135
Line Integrals with Respect to Arc Length 136
The Line Integral of a Vector Field 139
The Fundamental Theorem of Calculus for Line Integrals 143
Double Integrals 146
Double Integrals in Polar Coordinates 150
Green's Theorem 152
Path Independence and Gradient Fields 154
Chapter 3 Review Questions 157
4 Differential Equations 165
Separable Equations 167
Homogeneous Equations 168
Exact Equations 168
Nonexact Equations and Integrating Factors 170
First-Order Linear Equations 171
Higher-Order Linear Equations with Constant Coefficients 173
Chapter 4 Review Questions 176
5 Linear Algebra 181
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[superscript -1] 193
Vector Spaces 195
The Nullspace 196
Linear Combinations 196
The Rank, Column Space, and Row Space of a Matrix 198
Other Vector Spaces 200
Determinants 200
Laplace Expansions 204
The Adjugate Matrix 205
Cramer's Rule 206
Linear Transformations 207
Standard Matrix Representative 208
The Rank Plus Nullity Theorem 208
A Note on Inverses and Compositions 209
Eigenvalues and Eigenvectors 209
Eigenspaces 211
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
Primes 221
The Greatest Common Divisor and the Least Common Multiple 221
The Euclidean Algorithm 222
The Diophantine Equation ax + by = c 223
Congruences 225
The Congruence Equation ax [identical with] b (mod n) 226
Part B Abstract Algebra 227
Binary Structures and the Definition of a Group 227
Examples of Groups 228
Cyclic Groups 231
Subgroups 232
Cyclic Subgroups 233
Generators and Relations 234
Some Theorems Concerning Subgroups 234
The Concept of Isomorphism 235
The Classification of Finite Abelian Groups 237
Group Homomorphisms 241
Rings 245
Ring Homomorphisms 248
Integral Domains 252
Fields 253
Chapter 6 Review Questions 256
7 Additional Topics 261
Set Theory 261
Subsets and Complements 262
Union and Intersection 262
Cartesian Products 263
Intervals of the Real Line 263
Venn Diagrams 265
Cardinality 266
Combinatorics 268
Permutations and Combinations 269
With Repetitions Allowed 271
The Pigeonhole Principle 272
Probability and Statistics 272
Probability Spaces 274
Bernoulli Trials 277
Random Variables 278
Expectation, Variance, and Standard Deviation 280
The Normal Distribution 281
The Normal Approximation to the Binomial Distribution 283
Point-Set Topology 284
The Subspace Topology 285
The Interior, Exterior, Boundary, Limit Points, and Closure of a Set 285
Basis for a Topology 287
The Product Topology 288
Connectedness 289
Compactness 289
Metric Spaces 290
Continuous Functions 291
Open Maps and Homeomorphisms 293
Real Analysis 294
The Completeness of the Real Numbers 294
Lebesgue Measure 295
Lebesgue Measurable Functions 296
Lebesgue Integrable Functions 297
Complex Variables 299
The Polar Form 300
The Exponential Form 301
Complex Roots 301
Complex Logarithms 303
Complex Powers 303
The Trigonometric Functions 304
The Hyperbolic Functions 305
The Derivative of a Function of a Complex Variable 306
The Cauchy-Riemann Equations 307
Analytic Functions 309
Complex Line Integrals 310
Theorems Concerning Analytic Functions 311
Taylor Series for Functions of a Complex Variable 312
Singularities, Poles, and Laurent Series 312
The Residue Theorem 315
8 Solutions to the Chapter Review Questions 319
9 Practice Test 383
10 Practice Test Answers and Explanations 415
About the Author 439
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  • Anonymous

    Posted August 8, 2006

    big mistakes

    There are 4 mistakes in page 21. They are Opposite-Angle Identities of sin and cos and Addidtion and subtraction formulas of cos(alpha+beta) and cos(alpha-beta). The book give the totally opposite answers and I am afraid whether I should trust other parts in this book.I am very disappointed.

    Was this review helpful? Yes  NoThank you for your feedback.   Report this reviewThank you, this review has been flagged.
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