Cracking the GRE Math Test, 2nd Edition

by LEDUC, STEVE

• ISBN13: 9780375762673
• ISBN10: 0375762671
• Format: Trade Paper
• Copyright: 2002-09-01
• Publisher: Princeton Review

Summary

The Princeton Review realizes that acing the GRE Math Test is very different from getting straight A's in school. We don't try to teach you everything there is to know about mathonly the techniques you'll need to score higher on the test. There's a big difference. In Cracking the GRE Math Test, we'll teach you how to think like the test writers and Eliminate answer choices that look right but are planted to fool you Raise your score by focusing on the material most likely to appear on the test Test your knowledge with review questions for each math topic covered This book includes one full-length practice GRE Math Test. All of our practice questions are like the ones you'll see on the actual GRE Math Test, and we fully explain every solution.

Author Biography

Steve Leduc earned his Sc.B. in Theoretical Mathematics from MIT and his M.A. in Mathematics from UCSD, and has been teaching at the university level since the age of 19. After his graduate studies, Steve co-founded Hyperlearning, Inc., an educational services company that provided supplemental courses in undergraduate math and science for students from the University of California, where he lectured seventeen different courses in mathematics and physics. In addition to The Princeton Review's Cracking the GRE Math, he has published three other math books, Differential Equations in 1995, Linear Algebra in 1996, and The Princeton Review's Cracking the Virginia SOL Algebra II in 2001, as well as two physics books, The Princeton Review's Cracking the AP Physics B and C in 1999 and Cracking the SAT II Physics in 2000. Through Hyperlearning, Steve has directed the creation and administration of the most successful preparation course for the medical school entrance exam (the MCAT) in California, where he has taught mathematics and physics to thousands of undergraduates. Hyperlearning merged with The Princeton Review in 1996, and Steve now holds the position of National Director of MCAT Research, Production and Development for The Princeton Review. He currently owns two-to-the-eleventh-power CDs and has seen Monty Python and The Holy Grail, Star Trek II: The Wrath of Khan, and Blade Runner about two-to-the-eleventh-power times. -Paul Kanarek.

Table of Contents

Preface	p. xv
Precalculus	p. 1
Functions	p. 1
Composition of Functions	p. 2
Inverse Functions	p. 3
Graphs in the x-y Plane	p. 5
Analytic Geometry	p. 7
Lines	p. 7
Parabolas	p. 7
Circles	p. 9
Ellipses	p. 10
Hyperbolas	p. 11
Polynomial Equations	p. 13
The Division Algorithm, Remainder Theorem, and Factor Theorem	p. 13
The Fundamental Theorem of Algebra and Roots of Polynomial Equations	p. 14
The Root Location Theorem	p. 14
The Rational Roots Theorem	p. 14
The Conjugate Radical Roots Theorem	p. 14
The Complex Conjugate Roots Theorem	p. 15
Sum and Product of the Roots	p. 15
Logarithms	p. 16
Trigonometry	p. 18
Trig Functions of Acute Angles	p. 18
Trig Functions of Arbitrary Angles	p. 19
Trig Functions of Real Numbers	p. 19
Trig Identities and Formulas	p. 20
Periodicity of the Trig Functions	p. 22
Graphs of the Trig Functions	p. 23
The Inverse Trig Functions	p. 24
Chapter 1 Review Questions	p. 26
Calculus I	p. 31
Limits of Sequences	p. 31
Limits of Functions	p. 34
Limits of Functions as x . ± 8	p. 36
Continuous Functions	p. 37
Theorems Concerning Continuous Functions	p. 39
The Derivative	p. 41
Linear Approximations Using Differentials	p. 45
Implicit Differentiation	p. 46
Higher-Order Derivatives	p. 46
Curve Sketching	p. 47
Theorems Concerning Differentiable Functions	p. 49
Max/Min Problems	p. 50
Related Rates	p. 52
Indefinite Integration (Antidifferentiation)	p. 54
Techniques of Integration	p. 54
Integration by Substitution	p. 54
Integration by Parts	p. 56
Trig Substitutions	p. 57
The Method of Partial Fractions	p. 58
Definite Integration	p. 59
The Fundamental Theorem of Calculus	p. 61
The Average Value of a Function	p. 63
Finding the Area Between Two Curves	p. 64
Polar Coordinates	p. 66
Volumes of Solids of Revolution	p. 68
Arc Length	p. 70
The Natural Exponential and Logarithm Functions	p. 71
L'Hôpital's Rule	p. 74
Improper Integrals	p. 76
Infinite Series	p. 79
Alternating Series	p. 83
Power Series	p. 83
Functions Defined by Power Series	p. 85
Taylor Series	p. 86
Taylor Polynomials	p. 88
Chapter 2 Review Questions	p. 90
Calculus II	p. 101
Analytic Geometry of R3	p. 101
The Dot Product	p. 103
The Cross Product	p. 105
The Triple Scalar Product	p. 106
Lines in 3-Space	p. 108
Planes in 3-Space	p. 109
Cylinders	p. 111
Surfaces of Revolution	p. 113
Level Curves and Level Surfaces	p. 115
Cylindrical Coordinates	p. 117
Spherical Coordinates	p. 117
Partial Derivatives	p. 118
Geometric Interpretation of fx and fy	p. 119
Higher-Order Partial Derivatives	p. 120
The Tangent Plane to a Surface	p. 122
Linear Approximations	p. 123
The Chain Rule for Partial Derivatives	p. 124
Directional Derivatives and the Gradient	p. 128
Max/Min Problems	p. 130
Max/Min Problems with a Constraint	p. 133
The Lagrange Multiplier Method	p. 134
Line Integrals	p. 135
Line Integrals with Respect to Arc Length	p. 136
The Line Integral of a Vector Field	p. 139
The Fundamental Theorem of Calculus for Line Integrals	p. 143
Double Integrals	p. 146
Double Integrals in Polar Coordinates	p. 150
Green's Theorem	p. 152
Path Independence and Gradient Fields	p. 154
Chapter 3 Review Questions	p. 157
Differential Equations	p. 165
Separable Equations	p. 167
Homogeneous Equations	p. 168
Exact Equations	p. 168
Non-Exact Equations and Integrating Factors	p. 170
First-Order Linear Equations	p. 171
Higher-Order Linear Equations with Constant Coefficients	p. 173
Chapter 4 Review Questions	p. 176
Linear Algebra	p. 181
Solutions of Linear Systems	p. 181
Matrices and Matrix Algebra	p. 183
Matrix Operations	p. 183
Identity Matrices and Inverses	p. 186
Gaussian Elimination	p. 188
Solving Matrix Equations Using A-1	p. 193
Vector Spaces	p. 195
The Nullspace	p. 196
Linear Combinations	p. 196
The Rank, Column Space, and Row Space of a Matrix	p. 198
Other Vector Spaces	p. 200
Determinants	p. 200
Laplace Expansions	p. 204
The Adjugate Matrix	p. 205
Cramer's Rule	p. 206
Linear Transformations	p. 207
Standard Matrix Representative	p. 208
The Rank Plus Nullity Theorem	p. 208
A Note on Inverses and Compositions	p. 209
Eigenvalues and Eigenvectors	p. 209
Eigenspaces	p. 211
The Cayley-Hamilton Theorem	p. 213
Chapter 5 Review Questions	p. 214
Number Theory and Abstract Algebra	p. 219
Part A: Number Theory	p. 220
Divisibility	p. 220
The Division Algorithm	p. 221
Primes	p. 221
The Greatest Common Divisor and the Least Common Multiple	p. 221
The Euclidean Algorithm	p. 222
The Diophantine Equation ax + by = c	p. 223
Congruences	p. 225
The Congruence Equation ax = b (mod n)	p. 226
Part B: Abstract Algebra	p. 227
Binary Structures and the Definition of a Group	p. 227
Examples of Groups	p. 228
Cyclic Groups	p. 231
Subgroups	p. 232
Cyclic Subgroups	p. 233
Generators and Relations	p. 234
Some Theorems Concerning Subgroups	p. 234
The Concept of Isomorphism	p. 235
The Classification of Finite Abelian Groups	p. 237
Group Homomorphisms	p. 241
Rings	p. 245
Ring Homomorphisms	p. 248
Integral Domains	p. 252
Fields	p. 253
Chapter 6 Review Questions	p. 256
Additional Topics	p. 261
Set Theory	p. 261
Subsets and Complements	p. 262
Union and Intersection	p. 262
Cartesian Products	p. 263
Intervals of the Real Line	p. 263
Venn Diagrams	p. 265
Cardinality	p. 266
Combinatorics	p. 268
Permutations and Combinations	p. 269
With Repetitions Allowed	p. 270
The Pigeonhole Principle	p. 272
Probability and Statistics	p. 272
Probability Spaces	p. 274
Bernoulli Trials	p. 276
Random Variables	p. 278
Expectation, Variance, and Standard Deviation	p. 279
The Normal Distribution	p. 281
The Normal Approximation to the Binomial Distribution	p. 283
Point-Set Topology	p. 284
The Subspace Topology	p. 285
The Interior, Exterior, Boundary, Limit Points, and Closure of a Set	p. 285
Basis for a Topology	p. 287
The Product Topology	p. 288
Connectedness	p. 289
Compactness	p. 289
Metric Spaces	p. 290
Continuous Functions	p. 291
Open Maps and Homeomorphisms	p. 293
Real Analysis	p. 294
The Completeness of the Real Numbers	p. 294
Lebesgue Measure	p. 295
Lebesgue Measurable Functions	p. 296
Lebesgue Integrable Functions	p. 297
Complex Variables	p. 299
The Polar Form	p. 300
The Exponential Form	p. 301
Complex Roots	p. 301
Complex Logarithms	p. 303
Complex Powers	p. 303
The Trigonometric Functions	p. 304
The Hyperbolic Functions	p. 305
The Derivative of a Function of a Complex Variable	p. 306
The Cauchy-Riemann Equations	p. 307
Analytic Functions	p. 309
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