Calculus- Early Transcendentals, 2021a

More documents

Recommendations

Info

164 Applications of Derivatives Solution. We start out by asking: What is the geometric quantity whose rate of change we know, and what is the geometric quantity whose rate of change we’re being asked about? Note that the person pushing the swing is moving horizontally at a rate we know. In other words, the horizontal coordinate of P is increasing at 6 ft/sec. In the xy-plane let us make the convenient choice of putting the origin at the location of P at time t = 0, i.e., a distance 10 directly below the point of attachment. Then the rate we know is dx/dt, and in part (a) the rate we want is dy/dt (the rate at which P is rising). In part (b) the rate we want is ˙θ = dθ/dt, whereθ stands for the angle in radians through which the swing has swung from the vertical. (Actually, since we want our answer in deg/sec, at the end we must convert dθ/dt from rad/sec by multiplying by 180/π.) (a) From the diagram we see that we have a right triangle whose legs are x and 10 − y, and whose hypotenuse is 10. Hence x 2 +(10 − y) 2 = 100. Taking the derivative of both sides we obtain: 2xẋ + 2(10 − y)(0 − ẏ) =0. We now look at what we know after 1 second, namely x = 6 (because x started at 0 and has been increasing at the rate of 6 ft/sec for 1 sec), thus y = 2 (because we get 10 − y = 8 from the Pythagorean theorem applied to the triangle with hypotenuse 10 and leg 6), and ẋ = 6. Putting in these values gives us 2 · 6 · 6 − 2 · 8ẏ = 0, from which we can easily solve for ẏ: ẏ = 4.5 ft/sec. (b) Here our two variables are x and θ, so we want to use the same right triangle as in part (a), but this time relate θ to x. Since the hypotenuse is constant (equal to 10), the best way to do this is to use the sine: sinθ = x/10. Taking derivatives we obtain (cosθ) ˙θ = 0.1ẋ. At the instant in question (t = 1 sec), when we have a right triangle with sides 6–8–10, cosθ = 8/10 and ẋ = 6. Thus (8/10) ˙θ = 6/10, i.e., ˙θ = 6/8 = 3/4 rad/sec, or approximately 43 deg/sec. ♣ Q • θ . y x . •P Figure 5.3: Swing. We have seen that sometimes there are apparently more than two variables that change with time, but in reality there are just two, as the others can be expressed in terms of just two. However sometimes there really are several variables that change with time; as long as you know the rates of change of all but one of them you can find the rate of change of the remaining one. As in the case when there are just two variables, take the derivative of both sides of the equation relating all of the variables, and then substitute all of the known values and solve for the unknown rate.
5.1. Related Rates 165 Example 5.6: Distance Changing Rate A road running north to south crosses a road going east to west at the point P. Car A is driving north along the first road, and car B is driving east along the second road. At a particular time car A is 10 kilometers to the north of P and traveling at 80 km/hr, while car B is 15 kilometers to the east of P and traveling at 100 km/hr. How fast is the distance between the two cars changing? Solution. Let a(t) be the distance of car A north of P at time t, andb(t) the distance of car B east of P at time t, andletc(t) be the distance from car A to car B at time t. By the Pythagorean Theorem, c(t) 2 = a(t) 2 + b(t) 2 . Taking derivatives we get 2c(t)c ′ (t)=2a(t)a ′ (t)+2b(t)b ′ (t),so Substituting known values we get: ċ = ċ = aȧ + bḃ c = aȧ + bḃ √ a 2 + b 2 . 10 · 80 + 15 · 100 √ 10 2 + 15 2 = 460 √ 13 ≈ 127.6km/hr at the time of interest. ♣ . . (0,a(t))• c(t) . . . . . . . . . . . . • P • . (b(t),0) Figure 5.4: Cars moving apart. Notice how this problem differs from Example 5.2. In both cases we started with the Pythagorean Theorem and took derivatives on both sides. However, in Example 5.2 one of the sides was a constant (the altitude of the plane), and so the derivative of the square of that side of the triangle was simply zero. In this Example, on the other hand, all three sides of the right triangle are variables, even though we are interested in a specific value of each side of the triangle (namely, when the sides have lengths 10 and 15). Make sure that you understand at the start of the problem what are the variables and what are the constants. Exercises for Section 5.1 Exercise 5.1.1 Air is being pumped into a spherical balloon at a constant rate of 3 cm 3 /s. How fast is the radius of the balloon increasing when the radius reaches 5cm? Exercise 5.1.2 A cylindrical tank standing upright (with one circular base on the ground) has radius 20 cm. How fast does the water level in the tank drop when the water is being drained at 25 cm 3 /sec?
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: 5.1. Related Rates 163 Solution. He
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 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 251 and 252:
Page 253 and 254:
Page 255 and 256:
Page 257 and 258:
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 290 and 291:
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 298 and 299:
282 Techniques of Integration So al
Page 300 and 301:
Page 302 and 303:
Page 304 and 305:
Page 306 and 307:
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 316 and 317:
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?