College Algebra, 2013a

More documents

Recommendations

Info

602 Systems of Equations and Matrices to formalize and generalize the foregoing discussion. We begin with the formal definition of an invertible matrix. Definition 8.11. An n × n matrix A is said to be invertible if there exists a matrix A −1 ,read ‘A inverse’, such that A −1 A = AA −1 = I n . Note that, as a consequence of our definition, invertible matrices are square, and as such, the conditions in Definition 8.11 force the matrix A −1 to be same dimensions as A, thatis,n × n. Since not all matrices are square, not all matrices are invertible. However, just because a matrix is square doesn’t guarantee it is invertible. (See the exercises.) Our first result summarizes some of the important characteristics of invertible matrices and their inverses. Theorem 8.6. Suppose A is an n × n matrix. 1. If A is invertible then A −1 is unique. 2. A is invertible if and only if AX = B has a unique solution for every n × r matrix B. The proofs of the properties in Theorem 8.6 rely on a healthy mix of definition and matrix arithmetic. To establish the first property, we assume that A is invertible and suppose the matrices B and C act as inverses for A. Thatis,BA = AB = I n and CA = AC = I n . We need to show that B and C are, in fact, the same matrix. To see this, we note that B = I n B =(CA)B = C(AB) =CI n = C. Hence, any two matrices that act like A −1 are, in fact, the same matrix. 4 To prove the second property of Theorem 8.6, wenotethatifA is invertible then the discussion on page 599 shows the solution to AX = B to be X = A −1 B, and since A −1 is unique, so is A −1 B. Conversely, if AX = B has a unique solution for every n × r matrix B, then, in particular, there is a unique solution X 0 to the equation AX = I n . The solution matrix X 0 is our candidate for A −1 . We have AX 0 = I n by definition, but we need to also show X 0 A = I n . To that end, we note that A (X 0 A)=(AX 0 ) A = I n A = A. In other words, the matrix X 0 A is a solution to the equation AX = A. Clearly, X = I n is also a solution to the equation AX = A, and since we are assuming every such equation as a unique solution, we must have X 0 A = I n . Hence, we have X 0 A = AX 0 = I n , so that X 0 = A −1 and A is invertible. The foregoing discussion justifies our quest to find A −1 using our super-sized augmented matrix approach [ ] Gauss Jordan Elimination A In −−−−−−−−−−−−−−−−→ [ I n A −1 ] We are, in essence, trying to find the unique solution to the equation AX = I n using row operations. What does all of this mean for a system of linear equations? Theorem 8.6 tells us that if we write the system in the form AX = B, then if the coefficient matrix A is invertible, there is only one solution to the system − that is, if A is invertible, the system is consistent and independent. 5 We also know that the process by which we find A −1 is determined completely by A, and not by the 4 If this proof sounds familiar, it should. See the discussion following Theorem 5.2 on page 380. 5 It can be shown that a matrix is invertible if and only if when it serves as a coefficient matrix for a system of equations, the system is always consistent independent. It amounts to the second property in Theorem 8.6 where the matrices B are restricted to being n × 1 matrices. We note that, owing to how matrix multiplication is defined, being able to find unique solutions to AX = B for n × 1 matrices B gives you the same statement about solving such equations for n × r matrices − since we can find a unique solution to them one column at a time.
8.4 Systems of Linear Equations: Matrix Inverses 603 constants in B. This answers the question as to why we would bother doing row operations on a super-sized augmented matrix to find A −1 instead of an ordinary augmented matrix to solve a system; by finding A −1 we have done all of the row operations we ever need to do, once and for all, since we can quickly solve any equation AX = B using one multiplication, A −1 B. ⎡ Example 8.4.1. Let A = ⎣ 3 1 2 0 −1 5 2 1 4 ⎤ ⎦ 1. Use row operations to find A −1 . Check your answer by finding A −1 A and AA −1 . 2. Use A −1 to solve the following systems of equations (a) ⎧ ⎨ ⎩ 3x + y +2z = 26 −y +5z = 39 2x + y +4z = 117 (b) ⎧ ⎨ ⎩ 3x + y +2z = 4 −y +5z = 2 2x + y +4z = 5 (c) ⎧ ⎨ ⎩ 3x + y +2z = 1 −y +5z = 0 2x + y +4z = 0 Solution. 1. We begin with a super-sized augmented matrix and proceed with Gauss-Jordan elimination. ⎡ ⎣ ⎡ ⎣ ⎡ ⎣ ⎡ ⎣ ⎡ ⎣ ⎡ ⎣ 3 1 2 1 0 0 0 −1 5 0 1 0 2 1 4 0 0 1 1 1 3 2 3 1 3 0 0 0 −1 5 0 1 0 2 1 4 0 0 1 ⎤ ⎦ ⎤ ⎦ 1 2 1 1 3 3 3 0 0 0 −1 5 0 1 0 0 1 3 8 3 − 2 3 0 1 Replace R1 −−−−−−−→ with 1 3 R1 ⎡ Replace R3 with −−−−−−−−−−→ −2R1+R3 ⎤ ⎦ 1 2 1 1 3 3 3 0 0 0 1 −5 0 −1 0 0 1 3 8 3 − 2 3 0 1 1 2 1 1 3 3 3 0 0 0 1 −5 0 −1 0 0 0 13 3 − 2 3 1 3 1 1 2 1 1 3 3 3 0 0 0 1 −5 0 −1 0 0 0 1 − 2 13 1 13 3 13 Replace R2 −−−−−−−→ with (−1)R2 ⎤ ⎦ ⎤ ⎦ ⎤ ⎦ 2 1 3 3 0 0 0 −1 5 0 1 0 1 1 3 ⎤ ⎣ ⎦ 2 1 4 0 0 1 ⎡ 1 2 1 1 3 3 3 0 0 ⎣ 0 −1 5 0 1 0 1 0 3 ⎡ ⎣ Replace R3 with −−−−−−−−−−→ − 1 3 R2+R3 Replace R3 −−−−−−−→ with 3 13 R3 1 1 3 8 3 − 2 3 0 1 ⎤ ⎦ 2 1 3 3 0 0 0 1 −5 0 −1 0 1 0 3 ⎡ ⎣ ⎡ ⎣ Replace R1 with − 2 3 R3+R1 −−−−−−−−−−−−→ Replace R2 with 5R3+R2 1 1 3 8 3 − 2 3 0 1 ⎤ ⎦ 2 1 3 3 0 0 0 1 −5 0 −1 0 0 0 13 3 − 2 3 1 1 3 1 3 1 2 1 3 3 0 0 0 1 −5 0 −1 0 0 0 1 − 2 13 ⎡ ⎢ ⎣ 1 13 3 13 ⎤ ⎦ ⎤ ⎦ 1 17 1 3 0 39 − 2 39 − 2 13 0 1 0 − 10 13 − 8 13 0 0 1 − 2 13 1 13 15 13 3 13 ⎤ ⎥ ⎦
Page 1 and 2:
College Algebra Version ⌊π⌋ Co
Page 3 and 4:
Table of Contents Preface ix 1 Rela
Page 5 and 6:
Table of Contents v 4.3.3 Answers .
Page 7 and 8:
Preface Thank you for your interest
Page 9 and 10:
xi text and we have gone to great l
Page 11 and 12:
Chapter 1 Relations and Functions 1
Page 13 and 14:
1.1 Sets of Real Numbers and the Ca
Page 15 and 16:
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
1.2 Relations 21 Solution. 1. To gr
Page 33 and 34:
1.2 Relations 23 lines y = 1 and y
Page 35 and 36:
1.2 Relations 25 ‘connecting the
Page 37 and 38:
1.2 Relations 27 (x − 2) 2 + y 2
Page 39 and 40:
1.2 Relations 29 1.2.2 Exercises In
Page 41 and 42:
1.2 Relations 31 29. 5 4 3 y 30. 2
Page 43 and 44:
1.2 Relations 33 1.2.3 Answers 1. y
Page 45 and 46:
1.2 Relations 35 13. y 14. y 3 2
Page 47 and 48:
1.2 Relations 37 41. y = x 2 +1 The
Page 49 and 50:
1.2 Relations 39 45. y = √ x −
Page 51 and 52:
1.2 Relations 41 49. (x +2) 2 + y 2
Page 53 and 54:
1.3 Introduction to Functions 43 1.
Page 55 and 56:
1.3 Introduction to Functions 45 So
Page 57 and 58:
1.3 Introduction to Functions 47 Al
Page 59 and 60:
Page 61 and 62:
1.3 Introduction to Functions 51 23
Page 63 and 64:
Page 65 and 66:
1.4 Function Notation 55 1.4 Functi
Page 67 and 68:
1.4 Function Notation 57 (c) To fin
Page 69 and 70:
1.4 Function Notation 59 1 − 4x x
Page 71 and 72:
1.4 Function Notation 61 these limi
Page 73 and 74:
1.4 Function Notation 63 1.4.2 Exer
Page 75 and 76:
1.4 Function Notation 65 ⎧ ⎪⎨
Page 77 and 78:
1.4 Function Notation 67 (b) How mu
Page 79 and 80:
1.4 Function Notation 69 1.4.3 Answ
Page 81 and 82:
1.4 Function Notation 71 18. For f(
Page 83 and 84:
1.4 Function Notation 73 26. For f(
Page 85 and 86:
1.4 Function Notation 75 69. C(0) =
Page 87 and 88:
1.5 Function Arithmetic 77 ( ) 4. F
Page 89 and 90:
1.5 Function Arithmetic 79 Next, we
Page 91 and 92:
1.5 Function Arithmetic 81 r(x + h)
Page 93 and 94:
1.5 Function Arithmetic 83 Solution
Page 95 and 96:
1.5 Function Arithmetic 85 27. f(x)
Page 97 and 98:
1.5 Function Arithmetic 87 1.5.2 An
Page 99 and 100:
1.5 Function Arithmetic 89 13. For
Page 101 and 102:
1.5 Function Arithmetic 91 37. 39.
Page 103 and 104:
1.6 Graphs of Functions 93 1.6 Grap
Page 105 and 106:
1.6 Graphs of Functions 95 In the p
Page 107 and 108:
1.6 Graphs of Functions 97 The calc
Page 109 and 110:
1.6 Graphs of Functions 99 The calc
Page 111 and 112:
1.6 Graphs of Functions 101 on [−
Page 113 and 114:
1.6 Graphs of Functions 103 1. Find
Page 115 and 116:
1.6 Graphs of Functions 105 Example
Page 117 and 118:
1.6 Graphs of Functions 107 1.6.2 E
Page 119 and 120:
1.6 Graphs of Functions 109 In Exer
Page 121 and 122:
1.6 Graphs of Functions 111 For Exe
Page 123 and 124:
1.6 Graphs of Functions 113 y y 4 5
Page 125 and 126:
1.6 Graphs of Functions 115 6. f(x)
Page 127 and 128:
1.6 Graphs of Functions 117 17. y 1
Page 129 and 130:
1.6 Graphs of Functions 119 90. h i
Page 131 and 132:
1.7 Transformations 121 7 y 7 y (5,
Page 133 and 134:
1.7 Transformations 123 2 to all of
Page 135 and 136:
1.7 Transformations 125 2 1 (0, 0)
Page 137 and 138:
1.7 Transformations 127 Solution. 1
Page 139 and 140:
1.7 Transformations 129 y y 3 2 1 (
Page 141 and 142:
1.7 Transformations 131 A few remar
Page 143 and 144:
1.7 Transformations 133 6 y 6 y (4,
Page 145 and 146:
1.7 Transformations 135 Finally, to
Page 147 and 148:
1.7 Transformations 137 (a, f(a)) a
Page 149 and 150:
1.7 Transformations 139 shift right
Page 151 and 152:
1.7 Transformations 141 Thecomplete
Page 153 and 154:
1.7 Transformations 143 64. The gra
Page 155 and 156:
1.7 Transformations 145 25. y =2−
Page 157 and 158:
1.7 Transformations 147 38. g(x) =f
Page 159 and 160:
1.7 Transformations 149 50. y = S 1
Page 161 and 162:
Chapter 2 Linear and Quadratic Func
Page 163 and 164:
2.1 Linear Functions 153 P 2 5. m =
Page 165 and 166:
2.1 Linear Functions 155 Equation 2
Page 167 and 168:
2.1 Linear Functions 157 y 4 3 2 y
Page 169 and 170:
2.1 Linear Functions 159 4. We find
Page 171 and 172:
2.1 Linear Functions 161 average sp
Page 173 and 174:
2.1 Linear Functions 163 2.1.1 Exer
Page 175 and 176:
2.1 Linear Functions 165 40. A rest
Page 177 and 178:
2.1 Linear Functions 167 61. y = 2
Page 179 and 180:
2.1 Linear Functions 169 2.1.2 Answ
Page 181 and 182:
2.1 Linear Functions 171 35. (a) F
Page 183 and 184:
2.2 Absolute Value Functions 173 2.
Page 185 and 186:
Page 187 and 188:
2.2 Absolute Value Functions 177 By
Page 189 and 190:
2.2 Absolute Value Functions 179 Ex
Page 191 and 192:
2.2 Absolute Value Functions 181 wh
Page 193 and 194:
Page 195 and 196:
2.2 Absolute Value Functions 185 24
Page 197 and 198:
2.2 Absolute Value Functions 187 31
Page 199 and 200:
2.3 Quadratic Functions 189 y y (
Page 201 and 202:
2.3 Quadratic Functions 191 shows t
Page 203 and 204:
2.3 Quadratic Functions 193 From g(
Page 205 and 206:
2.3 Quadratic Functions 195 ( a x +
Page 207 and 208:
2.3 Quadratic Functions 197 To find
Page 209 and 210:
2.3 Quadratic Functions 199 Neverth
Page 211 and 212:
2.3 Quadratic Functions 201 15. The
Page 213 and 214:
2.3 Quadratic Functions 203 2.3.2 A
Page 215 and 216:
2.3 Quadratic Functions 205 8. f(x)
Page 217 and 218:
2.3 Quadratic Functions 207 25. (a)
Page 219 and 220:
2.4 Inequalities with Absolute Valu
Page 221 and 222:
Page 223 and 224:
Page 225 and 226:
Page 227 and 228:
Page 229 and 230:
Page 231 and 232:
Page 233 and 234:
Page 235 and 236:
2.5 Regression 225 2.5 Regression W
Page 237 and 238:
2.5 Regression 227 2. Find the leas
Page 239 and 240:
2.5 Regression 229 The coefficient
Page 241 and 242:
2.5 Regression 231 4. The chart bel
Page 243 and 244:
2.5 Regression 233 2.5.2 Answers 1.
Page 245 and 246:
Chapter 3 Polynomial Functions 3.1
Page 247 and 248:
3.1 Graphs of Polynomials 237 of th
Page 249 and 250:
3.1 Graphs of Polynomials 239 In or
Page 251 and 252:
3.1 Graphs of Polynomials 241 Despi
Page 253 and 254:
3.1 Graphs of Polynomials 243 at th
Page 255 and 256:
3.1 Graphs of Polynomials 245 Theor
Page 257 and 258:
3.1 Graphs of Polynomials 247 In Ex
Page 259 and 260:
3.1 Graphs of Polynomials 249 37. T
Page 261 and 262:
3.1 Graphs of Polynomials 251 9. f(
Page 263 and 264:
3.1 Graphs of Polynomials 253 21. g
Page 265 and 266:
3.1 Graphs of Polynomials 255 33. (
Page 267 and 268:
3.2 The Factor Theorem and the Rema
Page 269 and 270:
Page 271 and 272:
Page 273 and 274:
Page 275 and 276:
Page 277 and 278:
Page 279 and 280:
3.3 Real Zeros of Polynomials 269 3
Page 281 and 282:
Page 283 and 284:
Page 285 and 286:
3.3 Real Zeros of Polynomials 275 f
Page 287 and 288:
3.3 Real Zeros of Polynomials 277 T
Page 289 and 290:
3.3 Real Zeros of Polynomials 279 (
Page 291 and 292:
3.3 Real Zeros of Polynomials 281 I
Page 293 and 294:
Page 295 and 296:
Page 297 and 298:
3.4 Complex Zeros and the Fundament
Page 299 and 300:
Page 301 and 302:
Page 303 and 304:
Page 305 and 306:
Page 307 and 308:
Page 309 and 310:
Page 311 and 312:
Chapter 4 Rational Functions 4.1 In
Page 313 and 314:
4.1 Introduction to Rational Functi
Page 315 and 316:
Page 317 and 318:
Page 319 and 320:
Page 321 and 322:
Page 323 and 324:
Page 325 and 326:
Page 327 and 328:
Page 329 and 330:
Page 331 and 332:
4.2 Graphs of Rational Functions 32
Page 333 and 334:
Page 335 and 336:
Page 337 and 338:
Page 339 and 340:
Page 341 and 342:
Page 343 and 344:
Page 345 and 346:
Page 347 and 348:
Page 349 and 350:
Page 351 and 352:
Page 353 and 354:
4.3 Rational Inequalities and Appli
Page 355 and 356:
Page 357 and 358:
Page 359 and 360:
Page 361 and 362:
Page 363 and 364:
Page 365 and 366:
Page 367 and 368:
Page 369 and 370:
Chapter 5 Further Topics in Functio
Page 371 and 372:
5.1 Function Composition 361 2. As
Page 373 and 374:
5.1 Function Composition 363 (+)
Page 375 and 376:
5.1 Function Composition 365 9. The
Page 377 and 378:
5.1 Function Composition 367 Theore
Page 379 and 380:
5.1 Function Composition 369 5.1.1
Page 381 and 382:
5.1 Function Composition 371 50. (g
Page 383 and 384:
5.1 Function Composition 373 7. For
Page 385 and 386:
5.1 Function Composition 375 20. Fo
Page 387 and 388:
5.1 Function Composition 377 56. (g
Page 389 and 390:
5.2 Inverse Functions 379 The main
Page 391 and 392:
5.2 Inverse Functions 381 y = f −
Page 393 and 394:
5.2 Inverse Functions 383 (b) We ca
Page 395 and 396:
5.2 Inverse Functions 385 y = f(x)
Page 397 and 398:
5.2 Inverse Functions 387 = = 2x 2x
Page 399 and 400:
5.2 Inverse Functions 389 We get g
Page 401 and 402:
5.2 Inverse Functions 391 Using wha
Page 403 and 404:
5.2 Inverse Functions 393 1. Explai
Page 405 and 406:
5.2 Inverse Functions 395 26. Show
Page 407 and 408:
5.3 Other Algebraic Functions 397 5
Page 409 and 410:
5.3 Other Algebraic Functions 399 I
Page 411 and 412:
5.3 Other Algebraic Functions 401 b
Page 413 and 414:
Page 415 and 416:
5.3 Other Algebraic Functions 405
Page 417 and 418:
Page 419 and 420:
Page 421 and 422:
Page 423 and 424:
Page 425 and 426:
5.3 Other Algebraic Functions 415 (
Page 427 and 428:
Chapter 6 Exponential and Logarithm
Page 429 and 430:
6.1 Introduction to Exponential and
Page 431 and 432:
Page 433 and 434:
Page 435 and 436:
Page 437 and 438:
Page 439 and 440:
Page 441 and 442:
Page 443 and 444:
Page 445 and 446:
Page 447 and 448:
6.2 Properties of Logarithms 437 6.
Page 449 and 450:
6.2 Properties of Logarithms 439 lo
Page 451 and 452:
6.2 Properties of Logarithms 441 ex
Page 453 and 454:
6.2 Properties of Logarithms 443 as
Page 455 and 456:
Page 457 and 458:
Page 459 and 460:
6.3 Exponential Equations and Inequ
Page 461 and 462:
Page 463 and 464:
Page 465 and 466:
Page 467 and 468:
Page 469 and 470:
6.4 Logarithmic Equations and Inequ
Page 471 and 472:
Page 473 and 474:
Page 475 and 476:
Page 477 and 478:
Page 479 and 480:
6.5 Applications of Exponential and
Page 481 and 482:
Page 483 and 484:
Page 485 and 486:
Page 487 and 488:
Page 489 and 490:
Page 491 and 492:
Page 493 and 494:
Page 495 and 496:
Page 497 and 498:
Page 499 and 500:
Page 501 and 502:
Page 503 and 504:
Page 505 and 506:
Chapter 7 Hooked on Conics 7.1 Intr
Page 507 and 508:
7.1 Introduction to Conics 497 If t
Page 509 and 510:
7.2 Circles 499 4 y 3 2 1 −4 −3
Page 511 and 512:
7.2 Circles 501 We close this secti
Page 513 and 514:
7.2 Circles 503 7.2.2 Answers 1. (x
Page 515 and 516:
7.3 Parabolas 505 7.3 Parabolas We
Page 517 and 518:
7.3 Parabolas 507 The distance from
Page 519 and 520:
7.3 Parabolas 509 Example 7.3.3. Gr
Page 521 and 522:
7.3 Parabolas 511 Every cross secti
Page 523 and 524:
7.3 Parabolas 513 7.3.2 Answers 1.
Page 525 and 526:
7.3 Parabolas 515 8. ( y + 2) 3 2 (
Page 527 and 528:
7.4 Ellipses 517 Minor Axis Major A
Page 529 and 530:
7.4 Ellipses 519 This equation is f
Page 531 and 532:
7.4 Ellipses 521 As with circles an
Page 533 and 534:
7.4 Ellipses 523 From this sketch,
Page 535 and 536:
7.4 Ellipses 525 7.4.1 Exercises In
Page 537 and 538:
7.4 Ellipses 527 7.4.2 Answers x 2
Page 539 and 540:
7.4 Ellipses 529 (x − 4) 2 (y −
Page 541 and 542:
7.5 Hyperbolas 531 7.5 Hyperbolas I
Page 543 and 544:
7.5 Hyperbolas 533 endpoints of the
Page 545 and 546:
7.5 Hyperbolas 535 from the center
Page 547 and 548:
7.5 Hyperbolas 537 As with the othe
Page 549 and 550:
7.5 Hyperbolas 539 y 6 5 4 3 2 Jeff
Page 551 and 552:
7.5 Hyperbolas 541 7.5.1 Exercises
Page 553 and 554:
7.5 Hyperbolas 543 is a hyperbola.
Page 555 and 556:
7.5 Hyperbolas 545 (y − 3) 2 (x
Page 557 and 558:
7.5 Hyperbolas 547 19. (x − 1) 2
Page 559 and 560:
Chapter 8 Systems of Equations and
Page 561 and 562: 8.1 Systems of Linear Equations: Ga
Page 577 and 578: 8.2 Systems of Linear Equations: Au
Page 589 and 590: 8.3 Matrix Arithmetic 579 ⎡ ⎤
Page 591 and 592: 8.3 Matrix Arithmetic 581 As did ma
Page 593 and 594: 8.3 Matrix Arithmetic 583 While the
Page 595 and 596: 8.3 Matrix Arithmetic 585 Using Ri
Page 597 and 598: 8.3 Matrix Arithmetic 587 C 2 − 5
Page 599 and 600: 8.3 Matrix Arithmetic 589 ( √2 2
Page 601 and 602: 8.3 Matrix Arithmetic 591 8.3.1 Exe
Page 603 and 604: 8.3 Matrix Arithmetic 593 28. Now l
Page 605 and 606: 8.3 Matrix Arithmetic 595 8.3.2 Ans
Page 607 and 608: 8.3 Matrix Arithmetic 597 [ ] −9
Page 609 and 610: 8.4 Systems of Linear Equations: Ma
Page 611: 8.4 Systems of Linear Equations: Ma
Page 625 and 626: 8.5 Determinants and Cramer’s Rul
Page 639 and 640: 8.6 Partial Fraction Decomposition
Page 647 and 648: 8.7 Systems of Non-Linear Equations
Page 657 and 658: 20. Solve the following system ⎧
Page 661 and 662: Chapter 9 Sequences and the Binomia
Page 663 and 664:
9.1 Sequences 653 3. From {2n − 1
Page 665 and 666:
9.1 Sequences 655 Solution. A good
Page 667 and 668:
9.1 Sequences 657 Looking at the de
Page 669 and 670:
9.1 Sequences 659 28. 0.9, 0.99, 0.
Page 671 and 672:
9.2 Summation Notation 661 9.2 Summ
Page 673 and 674:
9.2 Summation Notation 663 5∑ (
Page 675 and 676:
9.2 Summation Notation 665 S = n (2
Page 677 and 678:
9.2 Summation Notation 667 payment
Page 679 and 680:
9.2 Summation Notation 669 n∑ k=1
Page 681 and 682:
9.2 Summation Notation 671 In Exerc
Page 683 and 684:
9.3 Mathematical Induction 673 9.3
Page 685 and 686:
9.3 Mathematical Induction 675 This
Page 687 and 688:
9.3 Mathematical Induction 677 it i
Page 689 and 690:
9.3 Mathematical Induction 679 9.3.
Page 691 and 692:
9.4 The Binomial Theorem 681 9.4 Th
Page 693 and 694:
9.4 The Binomial Theorem 683 songs
Page 695 and 696:
9.4 The Binomial Theorem 685 (a + b
Page 697 and 698:
9.4 The Binomial Theorem 687 ∑k
Page 699 and 700:
9.4 The Binomial Theorem 689 ( 0 0)
Page 701 and 702:
9.4 The Binomial Theorem 691 9.4.1
Page 703 and 704:
Index n th root of a complex number
Page 705 and 706:
Index 1071 central angle, 701 chang
Page 707 and 708:
Index 1073 reflective property, 523
Page 709 and 710:
Index 1075 matrix, multiplicative,
Page 711 and 712:
Index 1077 midpoint definition of,
Page 713 and 714:
Index 1079 instantaneous, 161, 472
Page 715 and 716:
Index 1081 inconsistent, 553 indepe
show all

College Algebra, 2013a

Create successful ePaper yourself

Delete template?

Save as template?