Calculus- Early Transcendentals, 2021a

More documents

Recommendations

Info

282 Techniques of Integration So all we need to do is find A and B so that 7x − 6 =(A + B)x + 3A − 2B, which is to say, we need 7 = A + B and −6 = 3A − 2B. This is a problem you’ve seen before: Solve a system of two equations in two unknowns. There are many ways to proceed; here’s one: If 7 = A + B then B = 7 − A and so −6 = 3A − 2B = 3A − 2(7 − A)=3A − 14 + 2A = 5A − 14. This is easy to solve for A: A = 8/5, and then B = 7 − A = 7 − 8/5 = 27/5. Thus ∫ ∫ 7x − 6 8 (x − 2)(x + 3) dx = 1 5 x − 2 + 27 1 5 x + 3 dx = 8 5 The answer to the original problem is now ∫ x 3 ∫ ∫ (x − 2)(x + 3) dx = x − 1dx+ 27 ln|x − 2| + ln|x + 3| +C. 5 7x − 6 (x − 2)(x + 3) dx = x2 2 − x + 8 27 ln|x − 2| + 5 5 ln|x + 3| +C. ♣ Now suppose that x 2 + bx + c doesn’t factor. Again we can use long division to ensure that the numerator has degree less than 2, then we complete the square. Example 7.37: Denominator Does Not Factor ∫ x + 1 Evaluate x 2 + 4x + 8 dx. Solution. The quadratic denominator does not factor. We could complete the square and use a trigonometric substitution, but it is simpler to rearrange the integrand: ∫ ∫ x + 1 x 2 + 4x + 8 dx = ∫ x + 2 x 2 + 4x + 8 dx− 1 x 2 + 4x + 8 dx. The first integral is an easy substitution problem, using u = x 2 + 4x + 8: ∫ x + 2 x 2 + 4x + 8 dx = 1 ∫ du 2 u = 1 2 ln|x2 + 4x + 8|. For the second integral we complete the square: ( (x ) + 2 2 x 2 + 4x + 8 =(x + 2) 2 + 4 = 4 + 1) , 2 making the integral Using u = x + 2 2 we get ∫ 1 4 ( x+2 2 ∫ 1 4 1 ) 2 dx = 1 ∫ + 1 4 ( x+2 2 1 ) 2 dx. + 1 2 u 2 + 1 dx = 1 ( ) x + 2 2 arctan . 2
7.5. Rational Functions 283 The final answer is now ∫ x + 1 x 2 + 4x + 8 dx = 1 2 ln|x2 + 4x + 8|− 1 ( ) x + 2 2 arctan +C. 2 ♣ Exercises for 7.5 Exercise 7.5.1 Exercise 7.5.2 Exercise 7.5.3 Exercise 7.5.4 Exercise 7.5.5 ∫ ∫ ∫ ∫ ∫ 1 4 − x 2 dx Exercise 7.5.6 x 4 ∫ 4 − x 2 dx Exercise 7.5.7 1 ∫ x 2 + 10x + 25 dx Exercise 7.5.8 x 2 ∫ 4 − x 2 dx Exercise 7.5.9 x 4 4 + x 2 dx Exercise 7.5.10 ∫ ∫ 1 x 2 + 10x + 29 dx x 3 4 + x 2 dx 1 x 2 + 10x + 21 dx 1 2x 2 − x − 3 dx 1 x 2 + 3x dx Evaluate the indefinite integral. ∫ 7x + 7 Exercise 7.5.11 x 2 + 3x − 10 dx Exercise 7.5.18 ∫ 7x − 2 Exercise 7.5.12 x 2 + x dx Exercise 7.5.19 ∫ −4 Exercise 7.5.13 3x 2 − 12 dx Exercise 7.5.20 ∫ x + 7 Exercise 7.5.14 (x + 5) 2 dx Exercise 7.5.21 ∫ −3x − 20 Exercise 7.5.15 (x + 8) 2 dx Exercise 7.5.22 ∫ 9x 2 + 11x + 7 Exercise 7.5.16 x(x + 1) 2 dx Exercise 7.5.23 ∫ −12x 2 − x + 33 Exercise 7.5.17 (x − 1)(x + 3)(3 − 2x) dx Exercise 7.5.24 ∫ 94x 2 − 10x (7x + 3)(5x − 1)(3x − 1) dx ∫ x 2 + x + 1 x 2 + x − 2 dx ∫ x 3 x 2 − x − 20 dx ∫ 2x 2 − 4x + 6 x 2 − 2x + 3 dx ∫ ∫ ∫ 1 x 3 + 2x 2 + 3x dx x 2 + x + 5 x 2 + 4x + 10 dx 12x 2 + 21x + 3 (x + 1)(3x 2 + 5x − 1) dx
Page 1:
with Open Texts Calculus Early Tran
Page 5:
Calculus: Early Transcendentals an
Page 9 and 10:
Calculus: Early Transcendentals an
Page 11 and 12:
Table of Contents Table of Contents
Page 13 and 14:
v 5.4.3 Taylor Polynomials . . . .
Page 15 and 16:
vii 13.2 Limits and Continuity . .
Page 17:
Introduction The emphasis in this c
Page 20 and 21:
4 Review In the table, the set of r
Page 22 and 23:
6 Review 1.1.3 The Quadratic Formul
Page 24 and 25:
8 Review 1.1.4 Inequalities, Interv
Page 26 and 27:
10 Review Inequality Rules Add/subt
Page 28 and 29:
12 Review Guidelines for Solving Ra
Page 30 and 31:
14 Review Looking where the
Page 32 and 33:
16 Review Example 1.17: Absolute Va
Page 34 and 35:
18 Review (a) |x|≥2 (b) |x − 3|
Page 36 and 37:
20 Review The most familiar form of
Page 38 and 39:
22 Review Example 1.25: Equa
Page 40 and 41:
24 Review |Δy|, as shown in figure
Page 42 and 43:
26 Review • (h,k) is the vert
Page 44 and 45:
28 Review Determining the Type of C
Page 46 and 47:
30 Review To determine a, we substi
Page 48 and 49:
32 Review (a) A(2,0),B(4,3) (b) A(
Page 50 and 51:
34 Review • Secant (abbreviated b
Page 52 and 53:
36 Review Reading from the unit cir
Page 54 and 55:
38 Review Notice that we can now fi
Page 56 and 57:
40 Review 1.3.4 Graphs of Trigonome
Page 58 and 59:
42 Review Exercises for 1.3 Exercis
Page 60 and 61:
44 Review Exercise 1.4.7 Simplify t
Page 62 and 63:
46 Functions Example 2.2: Domain of
Page 64 and 65:
48 Functions Example 2.5: Domain Fi
Page 66 and 67:
50 Functions For horizontal and ver
Page 68 and 69:
52 Functions Solution. The domain o
Page 70 and 71:
54 Functions After the positive int
Page 72 and 73:
56 Functions Example 2.11: Determin
Page 74 and 75:
58 Functions Example 2.16: Finding
Page 76 and 77:
60 Functions Since the function f (
Page 78 and 79:
62 Functions Example 2.21: Solve Lo
Page 80 and 81:
64 Functions We can do something si
Page 82 and 83:
66 Functions Cancellation Rules sin
Page 84 and 85:
68 Functions (a) sin −1 ( √ 3/2
Page 86 and 87:
70 Functions Example 2.32: Computin
Page 88 and 89:
72 Functions (a) f (x) (b) − f (x
Page 91 and 92:
Chapter 3 Limits 3.1 The Limit The
Page 93 and 94:
3.2. Precise Definition of a Limit
Page 95 and 96:
3.2. Precise Definition of a Limit
Page 97 and 98:
3.3. Computing Limits: Graphically
Page 99 and 100:
3.4. Computing Limits: Algebraicall
Page 101 and 102:
3.4. Computing Limits: Algebraicall
Page 103 and 104:
3.5. Infinite Limits and Limits at
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
5 + x −1 (n) lim x→∞ 1 + 2x
Page 115 and 116:
3.6. A Trigonometric Limit 99 Figur
Page 117 and 118:
3.6. A Trigonometric Limit 101 Solu
Page 119 and 120:
3.7. Continuity 103 § § ¥
Page 121 and 122:
3.7. Continuity 105 Definition 3.43
Page 123 and 124:
3.7. Continuity 107 Definition 3.45
Page 125 and 126:
3.7. Continuity 109 Solution. We wi
Page 127 and 128:
3.7. Continuity 111 Therefore, our
Page 129:
3.7. Continuity 113 Exercise 3.7.3
Page 132 and 133:
116 Derivatives doesn’t meet the
Page 134 and 135:
118 Derivatives of the slope of the
Page 136 and 137:
120 Derivatives algebra to find a s
Page 138 and 139:
122 Derivatives we often use f and
Page 140 and 141:
124 Derivatives you are required to
Page 142 and 143:
126 Derivatives We can summarize
Page 144 and 145:
128 Derivatives 4.2.3 Velocities Su
Page 146 and 147:
130 Derivatives Exercise 4.2.5 Find
Page 148 and 149:
132 Derivatives Proof. For convenie
Page 150 and 151:
134 Derivatives Exercises for Secti
Page 152 and 153:
136 Derivatives since sinx cosx −
Page 154 and 155:
138 Derivatives In practice, of cou
Page 156 and 157:
140 Derivatives Exercises for Secti
Page 158 and 159:
142 Derivatives Exercise 4.5.39 Fin
Page 160 and 161:
144 Derivatives . . . . Figure 4.4:
Page 162 and 163:
146 Derivatives = ( d dx x2 ln2) e
Page 164 and 165:
148 Derivatives Example 4.38: Deriv
Page 166 and 167:
150 Derivatives Example 4.42: Equat
Page 168 and 169:
152 Derivatives Solution. We take l
Page 170 and 171:
154 Derivatives Exercise 4.7.8 Find
Page 172 and 173:
156 Derivatives 4.8.1 Derivatives o
Page 174 and 175:
158 Derivatives Exercise 4.8.4 The
Page 177 and 178:
Chapter 5 Applications of Derivativ
Page 179 and 180:
5.1. Related Rates 163 Solution. He
Page 181 and 182:
5.1. Related Rates 165 Example 5.6:
Page 183 and 184:
5.2. Extrema of a Function 167 Exer
Page 185 and 186:
5.2. Extrema of a Function 169 poin
Page 187 and 188:
5.2. Extrema of a Function 171 Exer
Page 189 and 190:
5.2. Extrema of a Function 173 3. E
Page 191 and 192:
5.2. Extrema of a Function 175 A si
Page 193 and 194:
5.3. The Mean Value Theorem 177 2.
Page 195 and 196:
5.3. The Mean Value Theorem 179
Page 197 and 198:
5.3. The Mean Value Theorem 181 Exe
Page 199 and 200:
5.4. Linear and Higher Order Approx
Page 201 and 202:
Page 203 and 204:
Page 205 and 206:
Page 207 and 208:
5.5. L’Hôpital’s Rule 191 "bou
Page 209 and 210:
5.5. L’Hôpital’s Rule 193 Exam
Page 211 and 212:
5.5. L’Hôpital’s Rule 195 Exer
Page 213 and 214:
5.6. Curve Sketching 197 consistent
Page 215 and 216:
5.6. Curve Sketching 199 Solution.
Page 217 and 218:
5.6. Curve Sketching 201 the concav
Page 219 and 220:
5.6. Curve Sketching 203 If there a
Page 221 and 222:
5.6. Curve Sketching 205 Note that
Page 223 and 224:
5.7. Optimization Problems 207 Guid
Page 225 and 226:
5.7. Optimization Problems 209 is t
Page 227 and 228:
5.7. Optimization Problems 211 (Alt
Page 229 and 230:
5.7. Optimization Problems 213 Exer
Page 231 and 232:
Chapter 6 Integration 6.1 Displacem
Page 233 and 234:
6.1. Displacement and Area 217 Exam
Page 235 and 236:
6.1. Displacement and Area 219 draw
Page 237 and 238:
6.1. Displacement and Area 221 It i
Page 239 and 240:
6.1. Displacement and Area 223
Page 241 and 242:
6.1. Displacement and Area 225 We d
Page 243 and 244:
6.1. Displacement and Area 227 x i
Page 245 and 246:
6.1. Displacement and Area 229 Both
Page 247 and 248: 6.1. Displacement and Area 231 does
Page 249 and 250: 6.2. The Fundamental Theorem of Cal
Page 259 and 260: 6.3. Indefinite Integrals 243 Exerc
Page 261 and 262: 6.3. Indefinite Integrals 245 Solut
Page 263: 6.3. Indefinite Integrals 247 Exerc
Page 266 and 267: 250 Techniques of Integration This
Page 268 and 269: 252 Techniques of Integration ♣ E
Page 270 and 271: 254 Techniques of Integration u, th
Page 272 and 273: 256 Techniques of Integration 7.2 P
Page 274 and 275: 258 Techniques of Integration = u7
Page 276 and 277: 260 Techniques of Integration and
Page 278 and 279: 262 Techniques of Integration ∫ N
Page 280 and 281: 264 Techniques of Integration To in
Page 282 and 283: 266 Techniques of Integration Exerc
Page 284 and 285: 268 Techniques of Integration Expre
Page 286 and 287: 270 Techniques of Integration The t
Page 288 and 289: 272 Techniques of Integration Examp
Page 292 and 293: 276 Techniques of Integration = sec
Page 294 and 295: 278 Techniques of Integration Find
Page 296 and 297: 280 Techniques of Integration Solut
Page 308 and 309: 292 Techniques of Integration There
Page 310 and 311: 294 Techniques of Integration Now u
Page 312 and 313: 296 Techniques of Integration The f
Page 314 and 315: 298 Techniques of Integration (e)
Page 318 and 319: 302 Applications of Integration For
Page 320 and 321: 304 Applications of Integration Sup
Page 322 and 323: 306 Applications of Integration Gui
Page 324 and 325: 308 Applications of Integration The
Page 326 and 327: 310 Applications of Integration . y
Page 328 and 329: 312 Applications of Integration . F
Page 330 and 331: 314 Applications of Integration so
Page 332 and 333: 316 Applications of Integration Exe
Page 334 and 335: 318 Applications of Integration In
Page 336 and 337: 320 Applications of Integration Sol
Page 338 and 339: 322 Applications of Integration mag
Page 340 and 341: 324 Applications of Integration . 1
Page 342 and 343: 326 Applications of Integration and
Page 344 and 345: 328 Applications of Integration Exe
Page 346 and 347: 330 Applications of Integration Unf
Page 348 and 349:
332 Applications of Integration Fig
Page 350 and 351:
334 Applications of Integration is
Page 353 and 354:
Chapter 9 Sequences and Series Cons
Page 355 and 356:
9.1. Sequences 339 log 2 (1 + ε) >
Page 357 and 358:
9.1. Sequences 341 Example 9.8: Con
Page 359 and 360:
9.1. Sequences 343 below it is boun
Page 361 and 362:
9.2. Series 345 If {kx n } ∞ n=0
Page 363 and 364:
9.2. Series 347 for some number s.
Page 365 and 366:
9.3. The Integral Test 349 If all o
Page 367 and 368:
9.3. The Integral Test 351 Proof. W
Page 369 and 370:
9.4. Alternating Series 353 Exercis
Page 371 and 372:
9.5. Comparison Tests 355 Exercises
Page 373 and 374:
9.5. Comparison Tests 357 Solution.
Page 375 and 376:
9.7. The Ratio and Root Tests 359 E
Page 377 and 378:
9.7. The Ratio and Root Tests 361 P
Page 379 and 380:
9.8. Power Series 363 Definition 9.
Page 381 and 382:
Exercise 9.8.2 Find the radius of c
Page 383 and 384:
f ′′ (x)= f ′′′ (x)= ∞
Page 385 and 386:
9.10. Taylor Series 369 Solution. T
Page 387 and 388:
9.11. Taylor’s Theorem 371 a posi
Page 389 and 390:
9.11. Taylor’s Theorem 373 Note t
Page 391 and 392:
Chapter 10 Differential Equations M
Page 393 and 394:
10.1. First Order Differential Equa
Page 395 and 396:
10.1. First Order Differential Equa
Page 397 and 398:
10.2. First Order Homogeneous Linea
Page 399 and 400:
10.3. First Order Linear Equations
Page 401 and 402:
10.4. Approximation 385 Exercises f
Page 403 and 404:
10.4. Approximation 387 y .. 0.5 .
Page 405 and 406:
10.5. Second Order Homogeneous Equa
Page 407 and 408:
10.5. Second Order Homogeneous Equa
Page 409 and 410:
10.6. Second Order Linear Equations
Page 411 and 412:
Page 413 and 414:
Page 415 and 416:
Page 417 and 418:
Chapter 11 Polar Coordinates, Param
Page 419 and 420:
11.1. Polar Coordinates 403 Solutio
Page 421 and 422:
11.2. Slopes in Polar Coordinates 4
Page 423 and 424:
11.3. Areas in Polar Coordinates 40
Page 425 and 426:
11.3. Areas in Polar Coordinates 40
Page 427 and 428:
11.4. Parametric Equations 411 Figu
Page 429 and 430:
11.5 Calculus with Parametric Equat
Page 431 and 432:
11.6. Conics in Polar Coordinates 4
Page 433 and 434:
11.6. Conics in Polar Coordinates 4
Page 435 and 436:
Chapter 12 Three Dimensions 12.1 Th
Page 437 and 438:
12.1. The Coordinate System 421 Now
Page 439 and 440:
12.2. Vectors 423 12.2 Vectors A ve
Page 441 and 442:
12.2. Vectors 425 3.54 0.77 . . . .
Page 443 and 444:
12.3. The Dot Product 427 Exercise
Page 445 and 446:
12.3. The Dot Product 429 Solution.
Page 447 and 448:
12.3. The Dot Product 431 v .. v ..
Page 449 and 450:
12.4. The Cross Product 433 Exercis
Page 451 and 452:
12.4. The Cross Product 435 We know
Page 453 and 454:
12.5. Lines and Planes 437 Exercise
Page 455 and 456:
12.5. Lines and Planes 439 Solution
Page 457 and 458:
12.5. Lines and Planes 441 Example
Page 459 and 460:
12.5. Lines and Planes 443 Exercise
Page 461 and 462:
12.6. Other Coordinate Systems 445
Page 463 and 464:
Page 465:
Page 468 and 469:
452 Partial Differentiation the lin
Page 470 and 471:
454 Partial Differentiation Exercis
Page 472 and 473:
456 Partial Differentiation Fortuna
Page 474 and 475:
Page 476 and 477:
460 Partial Differentiation paralle
Page 478 and 479:
462 Partial Differentiation 3 z . 2
Page 480 and 481:
Page 482 and 483:
466 Partial Differentiation lim ε
Page 484 and 485:
468 Partial Differentiation 13.5 Di
Page 486 and 487:
470 Partial Differentiation Example
Page 488 and 489:
Page 490 and 491:
Page 492 and 493:
476 Partial Differentiation so we g
Page 494 and 495:
478 Partial Differentiation 0.0 0.2
Page 496 and 497:
480 Partial Differentiation and the
Page 498 and 499:
482 Partial Differentiation xz = 2y
Page 500 and 501:
484 Partial Differentiation x φ .
Page 502 and 503:
486 Multiple Integration point mult
Page 504 and 505:
488 Multiple Integration Figure 14.
Page 506 and 507:
490 Multiple Integration 1 0 0 1 Fi
Page 508 and 509:
492 Multiple Integration Exercise 1
Page 510 and 511:
Page 512 and 513:
496 Multiple Integration This examp
Page 514 and 515:
498 Multiple Integration Exercise 1
Page 516 and 517:
500 Multiple Integration Finally, M
Page 518 and 519:
Page 520 and 521:
504 Multiple Integration Example 14
Page 522 and 523:
506 Multiple Integration ∫ 1 = 1
Page 524 and 525:
508 Multiple Integration Solution.
Page 526 and 527:
510 Multiple Integration ∫ ∫
Page 528 and 529:
.. 512 Multiple Integration As befo
Page 530 and 531:
514 Multiple Integration y . ......
Page 532 and 533:
516 Multiple Integration ∫∫ Exe
Page 534 and 535:
518 Vector Functions separate funct
Page 536 and 537:
520 Vector Functions = 〈 f ′ (t
Page 538 and 539:
522 Vector Functions Figure 15.6:
Page 540 and 541:
524 Vector Functions 2.0 1.5 1.0 0.
Page 542 and 543:
526 Vector Functions Exercise 15.2.
Page 544 and 545:
528 Vector Functions (This integral
Page 546 and 547:
530 Vector Functions To remove the
Page 548 and 549:
532 Vector Functions Graphing this
Page 550 and 551:
534 Vector Functions Example 15.20
Page 552 and 553:
536 Vector Calculus which is the re
Page 554 and 555:
538 Vector Calculus Since f x = 2e
Page 556 and 557:
540 Vector Calculus Definition 16.4
Page 558 and 559:
542 Vector Calculus Example 16.8: W
Page 560 and 561:
544 Vector Calculus Proof. We write
Page 562 and 563:
546 Vector Calculus Exercise 16.3.2
Page 564 and 565:
548 Vector Calculus In this case, n
Page 566 and 567:
550 Vector Calculus We can now rewr
Page 568 and 569:
552 Vector Calculus 16.5 The Diverg
Page 570 and 571:
554 Vector Calculus The remaining f
Page 572 and 573:
556 Vector Calculus Suppose we inst
Page 574 and 575:
558 Vector Calculus circle of radiu
Page 576 and 577:
560 Vector Calculus Exercise 16.6.7
Page 578 and 579:
562 Vector Calculus we might want t
Page 580 and 581:
564 Vector Calculus where e is an e
Page 582 and 583:
566 Vector Calculus ∫ b = ∫ = a
Page 585 and 586:
Selected Exercise Answers 1.1.1 1.
Page 587 and 588:
571 2.8.4 1. [2,3) ∪ (3,∞) 2. (
Page 589 and 590:
573 4.5.18 −3(4 − x) 2 4.5.19 6
Page 591 and 592:
575 5.1.15 500/ √ 3 − 200 ≈ 8
Page 593 and 594:
577 5.6.21 min at x = 1 5.6.22 none
Page 595 and 596:
579 7.1.2 x 5 /5 + 2x 3 /3 + x +C 7
Page 597 and 598:
581 7.4.19 1/2e x2 +C 7.4.20 x 3 e
Page 599 and 600:
583 7.7.20 diverges 7.7.21 diverges
Page 601 and 602:
585 8.6.5 ¯x = 45/28, ȳ = 93/70 8
Page 603 and 604:
587 9.8.2 R = e 10.1.2 y = arctant
Page 605 and 606:
589 10.6.8 Ae t + Be 3t +(1/2)te 3t
Page 607 and 608:
591 12.5.2 4(x + 1)+5(y − 2) −
Page 609 and 610:
593 13.6.5 f x = 3cos(3x)cos(2y), f
Page 611 and 612:
595 14.3.6 ¯x = 6/5, ȳ = 12/5 14.
Page 613 and 614:
597 15.5.8 〈−3sint,2cost + 1/10
Page 615 and 616:
Index absolute extrema, 172, 206 ab
Page 617 and 618:
INDEX 601 at infinity, 87 indetermi
Page 619 and 620:
Table of Integrals Table of Integra
Page 621 and 622:
∫ 41. sin n ucos m udu= − sinn
Page 623 and 624:
Integrals Involving u 2 − a 2 , a
Page 625 and 626:
∫ du 108. u √ a + bu = √ 1
show all

Calculus- Early Transcendentals, 2021a

Create successful ePaper yourself

Delete template?

Save as template?