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{TEXT 256 3 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 258 14 "1. Calculate " }{XPPEDIT 661 0 "int(x/(5+2*x+ x^2),x = -1 .. 1);" "6#-%$intG6$*&%\"xG\"\"\",(\"\"&F(*&\"\"#F(F'F(F(* $F'F,F(!\"\"/F';,$F(F.F(" }{TEXT 660 1 "." }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 667 4 "a) " }{XPPEDIT 270 0 "ln(2)-Pi/2;" "6#,&-%#lnG6#\"\"#\"\"\"*&%#PiGF(F'!\"\"F+" } {TEXT 668 10 " b) " }{XPPEDIT 271 0 "ln(2)/4-Pi/6;" "6#,&*&-%#ln G6#\"\"#\"\"\"\"\"%!\"\"F)*&%#PiGF)\"\"'F+F+" }{TEXT 669 13 " c ) " }{XPPEDIT 272 0 "ln(2)/2-Pi/8;" "6#,&*&-%#lnG6#\"\"#\"\"\"F(!\" \"F)*&%#PiGF)\"\")F*F*" }{TEXT 670 11 " d) " }{XPPEDIT 273 0 "l n(2)/2-3*Pi/4;" "6#,&*&-%#lnG6#\"\"#\"\"\"F(!\"\"F)*(\"\"$F)%#PiGF)\" \"%F*F*" }{TEXT 671 11 " e) " }{TEXT 666 1 " " }{XPPEDIT 274 0 "ln(2)-4/3*Pi;" "6#,&-%#lnG6#\"\"#\"\"\"*(\"\"%F(\"\"$!\"\"%#PiGF(F," }{TEXT 665 19 " " }}{PARA 0 "" 0 "" {TEXT 663 4 "f) \+ " }{XPPEDIT 275 0 "ln(2)/2-Pi/4;" "6#,&*&-%#lnG6#\"\"#\"\"\"F(!\"\"F) *&%#PiGF)\"\"%F*F*" }{TEXT 672 11 " g) " }{XPPEDIT 276 0 "3*ln( 2)-Pi/3;" "6#,&*&\"\"$\"\"\"-%#lnG6#\"\"#F&F&*&%#PiGF&F%!\"\"F-" } {TEXT 673 4 " " }{TEXT 676 1 " " }{TEXT 664 5 "h) " }{XPPEDIT 278 0 "2*ln(2)-Pi/4;" "6#,&*&\"\"#\"\"\"-%#lnG6#F%F&F&*&%#PiGF&\"\"%! \"\"F-" }{TEXT 674 9 " i) " }{XPPEDIT 279 0 "2*ln(2)-3/4*Pi;" "6# ,&*&\"\"#\"\"\"-%#lnG6#F%F&F&*(\"\"$F&\"\"%!\"\"%#PiGF&F-" }{TEXT 675 10 " j) " }{XPPEDIT 280 0 "2*ln(2)-4/3*Pi;" "6#,&*&\"\"#\"\"\"-% #lnG6#F%F&F&*(\"\"%F&\"\"$!\"\"%#PiGF&F-" }{TEXT 677 3 " " }}{PARA 3 "" 0 "" {TEXT 662 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK } {PARA 0 "" 0 "" {TEXT 508 12 "Solution: c" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "J := Int(x/(5+2*x+x^2),x = \+ -1 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&%\"xG\" \"\",(\"\"&F**&\"\"#F*F)F*F**$)F)F.F*F*!\"\"/F);F1F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "A := numer(integrand(J)); B := denom(int egrand(J));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,(\"\"&\"\"\"*&\"\"#F'%\"xGF'F'*$)F*F) F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "C := completesquare (B,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG,&*$),&%\"xG\"\"\"F*F* \"\"#F*F*\"\"%F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn := \+ x + 1 = 2*tan(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&%\"xG\" \"\"F(F(,$*&\"\"#F(-%$tanG6#%\"tGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "J1 := changevar(eqn, J, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J1G-%$IntG6$,$**\"\"#\"\"\",&F+!\"\"*&F*F+-%$tanG6#% \"tGF+F+F+,&F+F+*$)F/F*F+F+F+,(\"\"$F+*&\"\"%F+F/F+F+*$)F,F*F+F+F-F+/F 2;\"\"!,$*&F9F-%#PiGF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "J2 := map(simplify, J1 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J2G-% $IntG6$,&#\"\"\"\"\"#!\"\"-%$tanG6#%\"tGF*/F0;\"\"!,$*&\"\"%F,%#PiGF*F *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "antiderivative := int( integrand(J2), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/antiderivativ eG,&*&\"\"#!\"\"%\"tG\"\"\"F(-%#lnG6#-%$cosG6#F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "simplify(subs(t=Pi/4, antiderivative) - sub s(t=0, antiderivative));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\")! \"\"%#PiG\"\"\"F&*&#F(\"\"#F(-%#lnG6#F+F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "int(x/(5+2*x+x^2),x = -1 .. 1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&-%#lnG6#F'F&F&*&\"\")!\"\"%#PiGF&F- " }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 262 3 "2. " }{TEXT 288 1 " " } {TEXT 289 57 " The triangular region in the first quadrant bounded by \+ " }{XPPEDIT 19 1 "y = 2*x;" "6#/%\"yG*&\"\"#\"\"\"%\"xGF'" }{TEXT 640 2 ", " }{XPPEDIT 19 1 "y = x;" "6#/%\"yG%\"xG" }{TEXT 641 6 ", and " }{XPPEDIT 19 1 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT 642 97 " is rotate d about the line \ny = -1. What is the volume of the solid of revolut ion that results?" }}{PARA 3 "" 0 "" {TEXT 287 3 "a) " }{XPPEDIT 391 0 "Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 386 13 " b) " } {XPPEDIT 392 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 387 14 " \+ c) " }{XPPEDIT 393 0 "3*Pi/4;" "6#*(\"\"$\"\"\"%#PiGF%\"\"%!\" \"" }{TEXT 388 12 " d) " }{XPPEDIT 394 0 "Pi;" "6#%#PiG" } {TEXT 389 15 " e) " }{XPPEDIT 395 0 "4*Pi/3;" "6#*(\"\"%\" \"\"%#PiGF%\"\"$!\"\"" }{TEXT 390 14 " \nf) " }{XPPEDIT 400 0 "3*Pi/2;" "6#*(\"\"$\"\"\"%#PiGF%\"\"#!\"\"" }{TEXT 396 11 " g ) " }{XPPEDIT 401 0 "5*Pi/3;" "6#*(\"\"&\"\"\"%#PiGF%\"\"$!\"\"" } {TEXT 397 10 " h) " }{XPPEDIT 402 0 "2*Pi;" "6#*&\"\"#\"\"\"%#Pi GF%" }{TEXT 398 14 " i) " }{XPPEDIT 403 0 "3*Pi;" "6#*&\"\"$ \"\"\"%#PiGF%" }{TEXT 399 12 " j) " }{XPPEDIT 405 0 "4*Pi;" "6 #*&\"\"%\"\"\"%#PiGF%" }{TEXT 404 3 " " }{TEXT 406 6 " \n" } {TEXT 509 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 510 13 "Solution: h\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "volume := Pi*int((1+2*x)^2-(1+x)^2, x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'volumeG,$*&\"\"#\"\"\"%#PiGF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 257 36 "3. Calculate the arc length of " }{XPPEDIT 535 1 "y = x^2/2-ln(x)/4;" "6#/%\"yG,&*&%\"xG\"\"#F(!\"\"\"\"\"*&-%#lnG6#F 'F*\"\"%F)F)" }{TEXT 534 9 " for " }{XPPEDIT 536 1 "1 <= x;" "6#1 \"\"\"%\"xG" }{XPPEDIT 537 1 "`` <= 2;" "6#1%!G\"\"#" }{TEXT 409 1 ". " }{TEXT 408 3 " \n\n" }{TEXT 407 4 "a) " }{XPPEDIT 418 0 "1/2+ln(2)/ 3;" "6#,&*&\"\"\"F%\"\"#!\"\"F%*&-%#lnG6#F&F%\"\"$F'F%" }{TEXT 410 10 " b) " }{XPPEDIT 419 0 "2/3+ln(2)/4;" "6#,&*&\"\"#\"\"\"\"\"$!\" \"F&*&-%#lnG6#F%F&\"\"%F(F&" }{TEXT 411 13 " c) " }{XPPEDIT 420 0 "3/2+ln(2)/4;" "6#,&*&\"\"$\"\"\"\"\"#!\"\"F&*&-%#lnG6#F'F&\"\"% F(F&" }{TEXT 412 11 " d) " }{XPPEDIT 421 0 "3/4+ln(2)/2;" "6#,& *&\"\"$\"\"\"\"\"%!\"\"F&*&-%#lnG6#\"\"#F&F-F(F&" }{TEXT 413 11 " \+ e) " }{TEXT 309 1 " " }{XPPEDIT 422 0 "4/3+ln(2)/2;" "6#,&*&\"\"%\" \"\"\"\"$!\"\"F&*&-%#lnG6#\"\"#F&F-F(F&" }{TEXT 293 19 " \+ " }}{PARA 0 "" 0 "" {TEXT 291 4 "f) " }{XPPEDIT 423 0 "1/2-ln(2) /3;" "6#,&*&\"\"\"F%\"\"#!\"\"F%*&-%#lnG6#F&F%\"\"$F'F'" }{TEXT 414 12 " g) " }{XPPEDIT 424 0 "2/3-ln(2)/4;" "6#,&*&\"\"#\"\"\"\" \"$!\"\"F&*&-%#lnG6#F%F&\"\"%F(F(" }{TEXT 415 6 " " }{TEXT 425 1 " " }{TEXT 292 5 "h) " }{XPPEDIT 426 0 "3/2-ln(2)/4;" "6#,&*&\"\"$\" \"\"\"\"#!\"\"F&*&-%#lnG6#F'F&\"\"%F(F(" }{TEXT 416 11 " i) " } {XPPEDIT 427 0 "3/4-ln(2)/2;" "6#,&*&\"\"$\"\"\"\"\"%!\"\"F&*&-%#lnG6# \"\"#F&F-F(F(" }{TEXT 417 12 " j) " }{XPPEDIT 428 0 "4/3-ln(2) /2;" "6#,&*&\"\"%\"\"\"\"\"$!\"\"F&*&-%#lnG6#\"\"#F&F-F(F(" }{TEXT 429 3 " " }}{PARA 3 "" 0 "" {TEXT 290 1 " " }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 511 13 "Solution: c\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := x -> x^ 2/2-ln(x)/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%) operatorG%&arrowGF(,&*&#\"\"\"\"\"#F/*$)9$F0F/F/F/*&#F/\"\"%F/-%#lnG6# F3F/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "toBeInte grated := simplify(sqrt(1+diff(f(x),x)^2)) assuming x > 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/toBeIntegratedG,$*(\"\"%!\"\",&\"\"\"F**& F'F*)%\"xG\"\"#F*F*F*F-F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "int(toBeIntegrated, x=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& #\"\"$\"\"#\"\"\"*&#F'\"\"%F'-%#lnG6#F&F'F'" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 368 17 "4. The graph of " }{XPPEDIT 554 1 "y = x^3/3;" "6#/% \"yG*&%\"xG\"\"$F'!\"\"" }{TEXT 551 3 " " }{TEXT 552 7 " for " } {XPPEDIT 257 1 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 258 1 "`` <= 1;" "6 #1%!G\"\"\"" }{TEXT 553 87 " is rotated about the x-axis. What is t he surface area of the resulting figure? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 370 3 "a) " }{TEXT 539 2 " " } {XPPEDIT 19 1 "Pi*(sqrt(2)-1)/3;" "6#*(%#PiG\"\"\",&-%%sqrtG6#\"\"#F%F %!\"\"F%\"\"$F+" }{TEXT 540 9 " b) " }{XPPEDIT 19 1 "Pi*(sqrt(2)- 1)/6;" "6#*(%#PiG\"\"\",&-%%sqrtG6#\"\"#F%F%!\"\"F%\"\"'F+" }{TEXT 541 9 " c) " }{XPPEDIT 19 1 "Pi*(sqrt(2)-1)/9;" "6#*(%#PiG\"\"\", &-%%sqrtG6#\"\"#F%F%!\"\"F%\"\"*F+" }{TEXT 542 10 " d) " } {XPPEDIT 19 1 "Pi*(2*sqrt(2)-1)/3;" "6#*(%#PiG\"\"\",&*&\"\"#F%-%%sqrt G6#F(F%F%F%!\"\"F%\"\"$F," }{TEXT 543 7 " e)" }{TEXT 371 2 " " } {XPPEDIT 19 1 "Pi*(2*sqrt(2)-1)/6;" "6#*(%#PiG\"\"\",&*&\"\"#F%-%%sqrt G6#F(F%F%F%!\"\"F%\"\"'F," }{TEXT 544 1 " " }{TEXT 545 11 " \+ " }}{PARA 0 "" 0 "" {TEXT 369 5 "f) " }{XPPEDIT 19 1 "Pi*(2*sqrt(2)- 1)/9;" "6#*(%#PiG\"\"\",&*&\"\"#F%-%%sqrtG6#F(F%F%F%!\"\"F%\"\"*F," } {TEXT 538 6 " g) " }{XPPEDIT 19 1 "Pi*(3*sqrt(2)-1)/3;" "6#*(%#PiG\" \"\",&*&\"\"$F%-%%sqrtG6#\"\"#F%F%F%!\"\"F%F(F-" }{TEXT 546 2 " " } {TEXT 547 4 " h) " }{XPPEDIT 19 1 "Pi*(3*sqrt(2)-1)/6;" "6#*(%#PiG\"\" \",&*&\"\"$F%-%%sqrtG6#\"\"#F%F%F%!\"\"F%\"\"'F-" }{TEXT 548 8 " i ) " }{XPPEDIT 19 1 "Pi*(3*sqrt(2)-1)/9;" "6#*(%#PiG\"\"\",&*&\"\"$F%-% %sqrtG6#\"\"#F%F%F%!\"\"F%\"\"*F-" }{TEXT 549 10 " j) " } {XPPEDIT 19 1 "Pi*(4*sqrt(2)-1)/3;" "6#*(%#PiG\"\"\",&*&\"\"%F%-%%sqrt G6#\"\"#F%F%F%!\"\"F%\"\"$F-" }{TEXT 550 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 512 13 "Solution: f\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := x -> x ^3/3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operato rG%&arrowGF(,$*&#\"\"\"\"\"$F/*$)9$F0F/F/F/F(F(F(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 55 "answer := 2*Pi*int(f(x)*sqrt(1+diff(f(x),x)^ 2),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG,$*(\"\"#\"\" \"%#PiGF(,&*&\"\"*!\"\"F'#F(F'F(#F(\"#=F-F(F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 345 52 "5. Let R be the region bounded by the graph of \+ " }{XPPEDIT 558 1 "y = sqrt(x);" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT 556 38 ", the x-axis, and the vertical line " }{XPPEDIT 557 1 "x = 4 ;" "6#/%\"xG\"\"%" }{TEXT 555 56 ". What is the x-coordinate of the c enter of mass of R?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 348 1 "\n" }{TEXT 349 4 "a) " }{XPPEDIT 358 0 "3;" "6#\"\"$" }{TEXT 352 16 " b) " }{XPPEDIT 359 0 "7/4;" "6#*&\"\"(\"\" \"\"\"%!\"\"" }{TEXT 353 15 " c) " }{TEXT 565 1 " " } {XPPEDIT 256 0 "11/4;" "6#*&\"#6\"\"\"\"\"%!\"\"" }{TEXT 566 14 " \+ d) " }{XPPEDIT 360 0 "14/5;" "6#*&\"#9\"\"\"\"\"&!\"\"" }{TEXT 357 12 " e) " }{XPPEDIT 361 0 "10/3;" "6#*&\"#5\"\"\"\"\"$!\" \"" }{TEXT 351 2 " " }{TEXT 350 20 " \n" }{TEXT 346 5 "f) " }{XPPEDIT 362 0 "9/5;" "6#*&\"\"*\"\"\"\"\"&!\"\"" } {TEXT 356 15 " g) " }{XPPEDIT 363 0 "9/4;" "6#*&\"\"*\"\"\" \"\"%!\"\"" }{TEXT 355 10 " " }{TEXT 364 1 " " }{TEXT 347 3 " h) " }{XPPEDIT 257 0 "12/5;" "6#*&\"#7\"\"\"\"\"&!\"\"" }{TEXT 564 16 " i) " }{XPPEDIT 365 0 "5/3;" "6#*&\"\"&\"\"\"\"\"$!\"\"" }{TEXT 354 15 " j) " }{XPPEDIT 366 0 "8/3;" "6#*&\"\")\"\" \"\"\"$!\"\"" }{TEXT 367 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 513 13 "Solution: h\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "x_bar := int(x*sqrt(x),x=0.. 4)/int(sqrt(x),x=0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&x_barG# \"#7\"\"&" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 297 4 "6. " }{TEXT 559 92 "What is the y-coordinate of the center of mass of the region R o f the preceding question? " }{TEXT 560 1 "\n" }{TEXT 296 0 "" }}{PARA 3 "" 0 "" {TEXT 295 16 "a) 1 b) " }{XPPEDIT 436 0 "sqrt(2);" " 6#-%%sqrtG6#\"\"#" }{TEXT 430 13 " c) " }{XPPEDIT 437 0 "1/sq rt(2);" "6#*&\"\"\"F$-%%sqrtG6#\"\"#!\"\"" }{TEXT 431 11 " d) \+ " }{XPPEDIT 438 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT 432 11 " \+ e) " }{XPPEDIT 439 0 "2/sqrt(3);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"$!\" \"" }{TEXT 433 8 " \nf) " }{XPPEDIT 256 0 "2/5;" "6#*&\"\"#\"\"\"\" \"&!\"\"" }{TEXT 563 12 " g) " }{XPPEDIT 257 0 "3/5;" "6#*&\" \"$\"\"\"\"\"&!\"\"" }{TEXT 562 13 " h) " }{XPPEDIT 440 0 "2/ 3;" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 434 15 " i) " } {XPPEDIT 441 0 "3/4;" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 435 15 " \+ j) " }{XPPEDIT 257 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 561 3 " " }{TEXT 294 1 "\n" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 531 11 "Solution: i" }}{PARA 0 "" 0 "" {TEXT 532 1 " " }{TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "int(sqrt(x)^2,x=0..4)/2/int( sqrt(x),x=0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\"%" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT 311 17 "7. Suppose that " }{XPPEDIT 568 1 "f(x) = sqrt(x);" "6#/-%\"fG6#%\"xG-%%sqrtG6#F'" }{TEXT 567 12 " . For what " }{XPPEDIT 576 1 "c;" "6#%\"cG" }{TEXT 575 6 " is " } {XPPEDIT 574 1 "f(c);" "6#-%\"fG6#%\"cG" }{TEXT 573 33 " equal to the average value of " }{XPPEDIT 572 1 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 571 21 " over the interval " }{XPPEDIT 570 1 "[0, 9];" "6#7$\"\"!\" \"*" }{TEXT 569 2 " ?" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 313 4 "a) " }{XPPEDIT 444 0 "sqrt(5);" "6#- %%sqrtG6#\"\"&" }{TEXT 443 13 " b) " }{XPPEDIT 321 0 "3;" "6# \"\"$" }{TEXT 314 16 " c) " }{XPPEDIT 322 0 "4;" "6#\"\"% " }{TEXT 315 17 " d) " }{XPPEDIT 445 0 "5;" "6#\"\"&" } {TEXT 442 16 " e) " }{XPPEDIT 323 0 "3*sqrt(3);" "6#*&\"\" $\"\"\"-%%sqrtG6#F$F%" }{TEXT 316 7 " \nf) " }{XPPEDIT 324 0 "sqrt(6 );" "6#-%%sqrtG6#\"\"'" }{TEXT 317 13 " g) " }{XPPEDIT 325 0 "2*sqrt(5);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"&F%" }{TEXT 318 11 " \+ h) " }{XPPEDIT 326 0 "4*sqrt(3);" "6#*&\"\"%\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT 319 11 " i) " }{XPPEDIT 327 0 "2*sqrt(2);" "6#*&\"\"#\" \"\"-%%sqrtG6#F$F%" }{TEXT 320 11 " j) " }{XPPEDIT 328 0 "4*sqr t(2);" "6#*&\"\"%\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 312 1 "\n" }{TEXT 329 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 529 11 "Solution: c" }}{PARA 0 "" 0 "" {TEXT 530 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a :=0; b := 9; f : = x -> sqrt(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%%sqrtG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f_ave := int(f(x),x = a .. b)/(b-a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&f_aveG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(f(c)=f_ave,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" } }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 260 1 " " }{TEXT 310 1 "8" }{TEXT -1 1 "." }{TEXT 577 30 " For what constant c is " }{XPPEDIT 579 1 " f(x) = c*x*(2-x);" "6#/-%\"fG6#%\"xG*(%\"cG\"\"\"F'F*,&\"\"#F*F'!\"\"F *" }{TEXT 578 37 " a probability density function for" }{TEXT 582 2 " " }{XPPEDIT 259 1 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 260 1 "`` <= \+ 2;" "6#1%!G\"\"#" }{TEXT 581 1 "." }{TEXT 580 1 " " }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT 330 4 "a) " }{XPPEDIT 453 0 "1/4;" "6#*&\"\"\"F $\"\"%!\"\"" }{TEXT 446 15 " b) " }{XPPEDIT 454 0 "1/2;" "6 #*&\"\"\"F$\"\"#!\"\"" }{TEXT 447 15 " c) " }{XPPEDIT 455 0 "2/3;" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 448 38 " d) 1 \+ e) 2 \nf) " }{XPPEDIT 456 0 "4/3;" "6#*&\"\"%\"\"\"\"\"$!\" \"" }{TEXT 449 15 " g) " }{XPPEDIT 457 0 "1/3;" "6#*&\"\"\" F$\"\"$!\"\"" }{TEXT 450 15 " h) " }{XPPEDIT 458 0 "3/4;" " 6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 451 14 " i) " }{XPPEDIT 459 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT 452 15 " j) \+ " }{XPPEDIT 460 0 "3;" "6#\"\"$" }{TEXT 331 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 527 11 "Solution: h" }}{PARA 0 "" 0 "" {TEXT 528 1 " " } {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eqn := int(c *x*(2-x), x=0..2) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,$*( \"\"%\"\"\"\"\"$!\"\"%\"cGF)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "c = solve(eqn, c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%\"cG#\"\"$\"\"%" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 307 54 "9. A random variable assumes values in the interval " } {XPPEDIT 584 1 "[0, 1];" "6#7$\"\"!\"\"\"" }{TEXT 583 39 " and has pro bability density function " }{XPPEDIT 588 1 "f(x) = 4/(Pi*(1+x^2));" "6#/-%\"fG6#%\"xG*&\"\"%\"\"\"*&%#PiGF*,&F*F**$F'\"\"#F*F*!\"\"" } {TEXT 585 6 " for " }{TEXT 586 2 " " }{XPPEDIT 256 1 "0 <= x;" "6#1 \"\"!%\"xG" }{XPPEDIT 257 1 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT 587 27 ". What is the mean of X?" }}{PARA 0 "" 0 "" {TEXT 306 17 " \+ " }}{PARA 0 "" 0 "" {TEXT 332 5 "a) " }{XPPEDIT 470 0 "2/Pi; " "6#*&\"\"#\"\"\"%#PiG!\"\"" }{TEXT 465 14 " b) " } {XPPEDIT 471 0 "sqrt(2)/Pi;" "6#*&-%%sqrtG6#\"\"#\"\"\"%#PiG!\"\"" } {TEXT 466 16 " c) " }{XPPEDIT 472 0 "2*sqrt(2)/Pi;" "6#*( \"\"#\"\"\"-%%sqrtG6#F$F%%#PiG!\"\"" }{TEXT 467 16 " d) " }{XPPEDIT 473 0 "(1-ln(2))/Pi;" "6#*&,&\"\"\"F%-%#lnG6#\"\"#!\"\"F%%#P iGF*" }{TEXT 468 10 " e) " }{XPPEDIT 474 0 "ln(2)/2/Pi;" "6#*(-% #lnG6#\"\"#\"\"\"F'!\"\"%#PiGF)" }{TEXT 469 8 " \nf) " }{XPPEDIT 475 0 "ln(2)/Pi;" "6#*&-%#lnG6#\"\"#\"\"\"%#PiG!\"\"" }{TEXT 464 11 " \+ g) " }{XPPEDIT 476 0 "2*ln(2)/Pi;" "6#*(\"\"#\"\"\"-%#lnG6#F$F% %#PiG!\"\"" }{TEXT 463 8 " h) " }{XPPEDIT 477 0 "(3-ln(2))/Pi;" "6 #*&,&\"\"$\"\"\"-%#lnG6#\"\"#!\"\"F&%#PiGF+" }{TEXT 462 9 " i) " }{XPPEDIT 478 0 "(2*ln(2)-1)/Pi;" "6#*&,&*&\"\"#\"\"\"-%#lnG6#F&F'F'F' !\"\"F'%#PiGF+" }{TEXT 461 9 " j) " }{XPPEDIT 479 0 "exp(1)/Pi;" "6#*&-%$expG6#\"\"\"F'%#PiG!\"\"" }{TEXT 333 1 "\n" }{TEXT 480 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 525 11 "Solution: g" }}{PARA 0 "" 0 "" {TEXT 526 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "int(4/(1+x^2)/Pi,x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eqn := mu = Int( 4*x/(1+x^2)/Pi, x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqn G/%#muG-%$IntG6$,$**\"\"%\"\"\"%\"xGF-,&F-F-*$)F.\"\"#F-F-!\"\"%#PiGF3 F-/F.;\"\"!F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "map(value, eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#muG,$*(\"\"#\"\"\"-%#lnG6 #F'F(%#PiG!\"\"F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 259 5 "10. " }{TEXT 589 54 "A random variable X assumes values in t he interval " }{XPPEDIT 260 1 "[0, Pi];" "6#7$\"\"!%#PiG" }{TEXT 590 39 " and has probability density function " }{XPPEDIT 264 1 "f(x) = s in(x)/2;" "6#/-%\"fG6#%\"xG*&-%$sinG6#F'\"\"\"\"\"#!\"\"" }{TEXT 591 6 " for " }{TEXT 592 2 " " }{XPPEDIT 256 1 "0 <= x;" "6#1\"\"!%\"xG " }{XPPEDIT 257 1 "`` <= Pi;" "6#1%!G%#PiG" }{TEXT 593 29 ". What is the value of " }{TEXT 597 1 "P" }{TEXT 678 2 " (" }{TEXT 598 2 " \+ " }{XPPEDIT 595 1 "Pi/4 <= X;" "6#1*&%#PiG\"\"\"\"\"%!\"\"%\"XG" } {XPPEDIT 596 1 "`` <= Pi/3;" "6#1%!G*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 594 1 " " }{TEXT 599 1 ")" }{TEXT 600 7 " ? " }}{PARA 0 "" 0 "" {TEXT 269 4 "a) " }{XPPEDIT 335 1 "(sqrt(2)-1)/2;" "6#*&,&-%%sqrtG6# \"\"#\"\"\"F)!\"\"F)F(F*" }{TEXT 298 13 " b) " }{XPPEDIT 336 1 "(sqrt(2)-1)/4;" "6#*&,&-%%sqrtG6#\"\"#\"\"\"F)!\"\"F)\"\"%F*" } {TEXT 299 13 " c) " }{XPPEDIT 337 1 "(sqrt(3)-1)/2;" "6#*&,&- %%sqrtG6#\"\"$\"\"\"F)!\"\"F)\"\"#F*" }{TEXT 300 14 " d) " } {XPPEDIT 338 1 "(sqrt(3)-1)/4;" "6#*&,&-%%sqrtG6#\"\"$\"\"\"F)!\"\"F) \"\"%F*" }{TEXT 301 12 " e) " }{XPPEDIT 339 1 "(2-sqrt(2))/2; " "6#*&,&\"\"#\"\"\"-%%sqrtG6#F%!\"\"F&F%F*" }{TEXT 302 11 " \n f) " }{XPPEDIT 340 1 "(2*sqrt(2)-1)/2;" "6#*&,&*&\"\"#\"\"\"-%%sqrtG6#F &F'F'F'!\"\"F'F&F+" }{TEXT 303 10 " g) " }{XPPEDIT 341 1 "(2*sqr t(2)-1)/4;" "6#*&,&*&\"\"#\"\"\"-%%sqrtG6#F&F'F'F'!\"\"F'\"\"%F+" } {TEXT 304 10 " h) " }{XPPEDIT 342 1 "(2*sqrt(3)-1)/2;" "6#*&,&*& \"\"#\"\"\"-%%sqrtG6#\"\"$F'F'F'!\"\"F'F&F," }{TEXT 305 11 " i) " }{XPPEDIT 343 0 "(2*sqrt(3)-1)/4;" "6#*&,&*&\"\"#\"\"\"-%%sqrtG6#\" \"$F'F'F'!\"\"F'\"\"%F," }{TEXT 334 11 " j) " }{XPPEDIT 344 0 " (sqrt(3)-sqrt(2))/4;" "6#*&,&-%%sqrtG6#\"\"$\"\"\"-F&6#\"\"#!\"\"F)\" \"%F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 523 11 "Solution: b" }}{PARA 0 "" 0 "" {TEXT 524 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "int(sin(x)/2,x = P i/4 .. Pi/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"\"\"\"%!\"\"*& F&F'\"\"##F%F)F%" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 643 45 "11. A ran dom variable X has pdf given by " }{XPPEDIT 644 1 "f(x) = 3*x^2/124 ;" "6#/-%\"fG6#%\"xG*(\"\"$\"\"\"*$F'\"\"#F*\"$C\"!\"\"" }{TEXT 645 8 " for " }{XPPEDIT 646 1 "1 <= x;" "6#1\"\"\"%\"xG" }{XPPEDIT 647 1 "`` <= 5;" "6#1%!G\"\"&" }{TEXT 648 2 ". " }{TEXT -1 3 " " }{TEXT 649 5 "\nIf " }{XPPEDIT 650 1 "m;" "6#%\"mG" }{TEXT 651 19 " is the \+ median of " }{XPPEDIT 652 1 "X;" "6#%\"XG" }{TEXT 653 7 ", then " } {XPPEDIT 654 1 "m^3;" "6#*$%\"mG\"\"$" }{TEXT 655 11 " equals:\n\n" } {TEXT 659 11 "Solution: f" }}{PARA 261 "" 0 "" {TEXT -1 79 "a) 58 \+ b) 59 c) 60 d) 61 e) 62\nf) 63 \+ " }{TEXT 656 39 " g) 64 h) 65 i) " }{TEXT -1 2 "66" }{TEXT 657 18 " j) 67" }}{PAGEBK }{PARA 3 "" 0 "" {TEXT 658 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eqn := In t( 3*x^2/124, x = 1..m) = 1/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e qnG/-%$IntG6$,$*(\"\"$\"\"\"\"$C\"!\"\"%\"xG\"\"#F,/F/;F,%\"mG#F,F0" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn2 := map(value, eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,&*&\"$C\"!\"\"%\"mG\"\"$\" \"\"#F,F(F)#F,\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "m^3 \+ = solve(eqn2, m^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"mG\"\"$ \"\"\"\"#j" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 308 212 "12. \+ A spring is maintained 2m beyond equilibrium by a 60N force. If, start ing from equilibrium, 120J of work have been expended stretching a spr ing, then how many meters beyond equilibrium has it been stretched? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 14 " \+ b) " }{XPPEDIT 18 0 "sqrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT -1 15 " \+ c) " }{XPPEDIT 18 0 "2;" "6#\"\"#" }{TEXT -1 11 " d) " }{XPPEDIT 18 0 "sqrt(5);" "6#-%%sqrtG6#\"\"&" }{TEXT -1 12 " e ) " }{XPPEDIT 18 0 "sqrt(6);" "6#-%%sqrtG6#\"\"'" }{TEXT -1 20 " \+ \nf) " }{XPPEDIT 18 0 "sqrt(7);" "6#-%%sqrtG6#\"\"(" }{TEXT -1 14 " g) " }{XPPEDIT 18 0 "2*sqrt(2);" "6#*&\"\"#\"\"\"-%% sqrtG6#F$F%" }{TEXT -1 12 " h) " }{XPPEDIT 18 0 "3;" "6#\"\"$ " }{TEXT -1 13 " i) " }{XPPEDIT 18 0 "sqrt(10);" "6#-%%sqrtG6 #\"#5" }{TEXT -1 11 " j) " }{XPPEDIT 18 0 "2*sqrt(3);" "6#*&\" \"#\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 522 12 "Solution: g " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "HookesLaw := F = k *x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*HookesLawG/%\"FG*&%\"kG\"\" \"%\"xGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eqn := subs(\{ F=60, x=2\}, HookesLaw);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/\" #g,$*&\"\"#\"\"\"%\"kGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eqn2a := k = solve(eqn, k);\neqn2b := W = 120;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&eqn2aG/%\"kG\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&eqn2bG/%\"WG\"$?\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " eqn3 := W = int(k*x, x = 0..a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% eqn3G/%\"WG,$*(\"\"#!\"\"%\"kG\"\"\"%\"aGF)F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eqn4 := subs( \{eqn2a, eqn2b\}, eqn3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/\"$?\",$*&\"#:\"\"\")%\"aG\"\"#F*F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "a = solve(eqn4, a);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"aG6$,$*&\"\"#\"\"\"F(#F)F(!\"\",$* &F(F)F(F*F)" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 281 184 "13. The chemis try department has a cylindrical tank that has height 4m, base radius \+ 2m, and that is partially filled with a fluid that has been concocted \+ to have a weight density of " }{XPPEDIT 603 1 "1/Pi;" "6#*&\"\"\"F$% #PiG!\"\"" }{TEXT 602 1 " " }{XPPEDIT 604 1 "N/(m^3);" "6#*&%\"NG\"\" \"*$%\"mG\"\"$!\"\"" }{TEXT 601 151 ". If the initial depth of the fl uid is 3m and 1/3 of the initial volume is pumped to the top of the ta nk, then how many Joules of work have been done?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 3 "a) " }{XPPEDIT 18 0 "2; " "6#\"\"#" }{TEXT -1 13 " b) " }{XPPEDIT 18 0 "3;" "6#\"\"$ " }{TEXT -1 14 " c) " }{XPPEDIT 18 0 "4;" "6#\"\"%" }{TEXT -1 10 " d) " }{XPPEDIT 18 0 "6;" "6#\"\"'" }{TEXT -1 11 " \+ e) " }{XPPEDIT 18 0 "8;" "6#\"\")" }{TEXT -1 20 " \nf) \+ " }{XPPEDIT 18 0 "9;" "6#\"\"*" }{TEXT -1 13 " g) " } {XPPEDIT 18 0 "12;" "6#\"#7" }{TEXT -1 11 " h) " }{XPPEDIT 18 0 "16;" "6#\"#;" }{TEXT -1 10 " i) " }{XPPEDIT 18 0 "18;" "6#\"# =" }{TEXT -1 10 " j) " }{XPPEDIT 18 0 "24;" "6#\"#C" }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 606 12 "Solution: d " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A := int(Pi*(1/Pi)*y*4,y = 1 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG\"\"'" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 226 " 14. Forty feet of a uniform cable hang over the side of a building. \+ The cable weighs 6 lbs/ft. A 48 pound hook is attached to the dangling end of the cable. The cable is pulled up 10 feet before the hook snag s on a flag pole." }{TEXT -1 1 " " }{TEXT 605 58 "How many foot-pounds of work have been done to that point?" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 372 77 "a) 2500 b) 2520 c) 2540 d) 2560 e) 2580 " }{TEXT 375 1 " " }{TEXT 376 18 " \+ " }}{PARA 0 "" 0 "" {TEXT 373 35 "f) 2600 g) 2620 \+ " }{TEXT -1 1 " " }{TEXT 374 44 "h) 2640 i) 2660 \+ j) 2680 " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 520 11 "Solution: e" }}{PARA 0 "" 0 "" {TEXT 521 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "48*10+ 30*6*10+int(6*y,y = 0 .. 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%!e#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 15 "15. Calculate " }{XPPEDIT 19 1 "int(1/sqrt(5-x),x = 1 .. 5);" "6#-%$intG6$*&\"\"\"F'-%%sqrtG6#,&\"\"&F'%\"xG!\"\"F./F-;F' F," }{TEXT 618 1 "." }}{PARA 0 "" 0 "" {TEXT 282 74 "a) 2 \+ b) 3 c) 4 d) 5 e) " }{XPPEDIT 19 1 "2*sqrt(5)-1;" "6#,&*&\"\"#\"\"\"-%%sqrtG6#\"\"&F&F&F&!\"\"" } {TEXT 507 3 " " }{TEXT 285 18 " " }}{PARA 0 "" 0 " " {TEXT 283 33 "f) 9/2 g) 16/3 " }{TEXT -1 1 " " } {TEXT 284 33 "h) 17/4 i) 21/4 j)" }{TEXT -1 1 " " } {TEXT 619 2 " " }{XPPEDIT 19 1 "3*sqrt(5)-1;" "6#,&*&\"\"$\"\"\"-%%sq rtG6#\"\"&F&F&F&!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 518 12 "Solution: c " }}{PARA 0 "" 0 "" {TEXT 519 1 " " }{TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "int(1/sqrt(5-x),x = 1 .. 5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 272 5 "16. \+ " }{TEXT 621 10 "Calculate " }{XPPEDIT 622 1 "int(cos(x)/sqrt(sin(x)), x = 0 .. Pi/2);" "6#-%$intG6$*&-%$cosG6#%\"xG\"\"\"-%%sqrtG6#-%$sinG6# F*!\"\"/F*;\"\"!*&%#PiGF+\"\"#F2" }{TEXT 620 1 "." }}{PARA 0 "" 0 "" {TEXT 623 56 "a) 2 b) 3 c) 4 d) \+ " }{XPPEDIT 19 1 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT 629 17 " \+ e) " }{XPPEDIT 19 1 "(sqrt(3)-1)/2;" "6#*&,&-%%sqrtG6#\"\"$\"\" \"F)!\"\"F)\"\"#F*" }{TEXT 627 3 " " }{TEXT 626 18 " \+ " }}{PARA 0 "" 0 "" {TEXT 624 4 "f) " }{XPPEDIT 19 1 "sqrt(3)/2;" " 6#*&-%%sqrtG6#\"\"$\"\"\"\"\"#!\"\"" }{TEXT 630 12 " g) " } {XPPEDIT 19 1 "sqrt(2)/2;" "6#*&-%%sqrtG6#\"\"#\"\"\"F'!\"\"" }{TEXT 631 6 " " }{TEXT -1 1 " " }{TEXT 625 3 "h) " }{XPPEDIT 19 1 "sqrt (2)-1/2;" "6#,&-%%sqrtG6#\"\"#\"\"\"*&F(F(F'!\"\"F*" }{TEXT 632 9 " \+ i) " }{XPPEDIT 19 1 "1-sqrt(2)/2;" "6#,&\"\"\"F$*&-%%sqrtG6#\"\"#F$ F)!\"\"F*" }{TEXT 633 9 " j)" }{TEXT -1 1 " " }{TEXT 628 2 " " }{XPPEDIT 19 1 "(sqrt(2)-1)/2;" "6#*&,&-%%sqrtG6#\"\"#\"\"\"F)!\"\"F)F (F*" }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 517 14 "Solution: a " }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "int(cos(x)/sqrt(sin(x)), x = 0 .. P i/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}}{SECT 0 {PARA 3 " " 0 "" {TEXT 274 4 "17. " }{TEXT 280 12 " Calculate " }{XPPEDIT 256 0 "int(x/((1+x^2)^(3/2)),x = 0 .. infinity);" "6#-%$intG6$*&%\"xG\"\" \"),&F(F(*$F'\"\"#F(*&\"\"$F(F,!\"\"F//F';\"\"!%)infinityG" }{TEXT 493 1 "." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 494 4 "a) " } {XPPEDIT 504 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 502 11 " b ) " }{XPPEDIT 257 0 "2/3;" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 495 18 " c) " }{XPPEDIT 258 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\" " }{TEXT 496 15 " d) " }{XPPEDIT 256 0 "3/4;" "6#*&\"\"$\" \"\"\"\"%!\"\"" }{TEXT 500 21 " e) " }{XPPEDIT 259 0 "1;" "6#\"\"\"" }{TEXT 497 13 " \nf) " }{XPPEDIT 19 1 "4/3;" " 6#*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT 634 10 " g) " }{XPPEDIT 261 0 "2*sqrt(2);" "6#*&\"\"#\"\"\"-%%sqrtG6#F$F%" }{TEXT 498 12 " h ) " }{XPPEDIT 262 0 "sqrt(2)/2;" "6#*&-%%sqrtG6#\"\"#\"\"\"F'!\"\"" } {TEXT 501 14 " i) " }{XPPEDIT 263 0 "(sqrt(2)-1)/2;" "6#*&,& -%%sqrtG6#\"\"#\"\"\"F)!\"\"F)F(F*" }{TEXT 499 11 " j) " } {XPPEDIT 264 0 "2;" "6#\"\"#" }{TEXT 505 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 533 14 "Solution: e \n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "int(x/((1+x^ 2)^(3/2)),x = 0 .. infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 276 4 "18. " }{TEXT 481 1 " " }{TEXT 637 24 "It may be verified that " } {XPPEDIT 638 1 "Diff(ln(x)/x,x) = 1/(x^2)-ln(x)/(x^2);" "6#/-%%DiffG6$ *&-%#lnG6#%\"xG\"\"\"F+!\"\"F+,&*&F,F,*$F+\"\"#F-F,*&-F)6#F+F,*$F+F1F- F-" }{TEXT 635 32 ". Hence or otherwise, evaluate " }{XPPEDIT 639 1 " int((1-ln(x))/(x^2),x = 1 .. infinity);" "6#-%$intG6$*&,&\"\"\"F(-%#ln G6#%\"xG!\"\"F(*$F,\"\"#F-/F,;F(%)infinityG" }{TEXT 636 4 ". " } {TEXT 492 5 " \na) " }{XPPEDIT 256 0 "0;" "6#\"\"!" }{TEXT 483 21 " \+ b) " }{XPPEDIT 262 0 "-1;" "6#,$\"\"\"!\"\"" }{TEXT 484 16 " c) " }{XPPEDIT 270 0 "1;" "6#\"\"\"" }{TEXT 485 14 " d) " }{XPPEDIT 271 0 "exp(1);" "6#-%$expG6#\"\"\"" } {TEXT 486 14 " e) " }{XPPEDIT 272 0 "1/exp(1);" "6#*&\"\"\"F $-%$expG6#F$!\"\"" }{TEXT 487 10 " \nf) " }{XPPEDIT 273 0 "exp(1) -1;" "6#,&-%$expG6#\"\"\"F'F'!\"\"" }{TEXT 488 15 " g) " } {XPPEDIT 274 0 "(exp(1)-1)/2;" "6#*&,&-%$expG6#\"\"\"F(F(!\"\"F(\"\"#F )" }{TEXT 489 9 " h) " }{XPPEDIT 275 0 "2*exp(1);" "6#*&\"\"#\"\" \"-%$expG6#F%F%" }{TEXT 490 11 " i) " }{XPPEDIT 276 0 "2/exp(1) ;" "6#*&\"\"#\"\"\"-%$expG6#F%!\"\"" }{TEXT 491 16 " j) " }{XPPEDIT 503 0 "exp(1)/2;" "6#*&-%$expG6#\"\"\"F'\"\"#!\"\"" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 482 0 "" }{TEXT -1 0 "" }}{PAGEBK } {PARA 0 "" 0 "" {TEXT 514 14 "Solution: a \n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "e qn := Int((1-ln(x))/(x^2),x = 1 .. infinity) = Limit(subs(x=N, ln(x)/x ) - subs(x=1, ln(x)/x ), N = infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%$IntG6$*&,&\"\"\"F+-%#lnG6#%\"xG!\"\"F+F/!\"#/F/;F+%)i nfinityG-%&LimitG6$,&*&-F-6#%\"NGF+F " 0 "" {MPLTEXT 1 0 40 "eqn2 := lhs(eqn) = simplify( rhs(eq n) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%$IntG6$*&,&\"\"\"F +-%#lnG6#%\"xG!\"\"F+F/!\"#/F/;F+%)infinityG-%&LimitG6$*&-F-6#%\"NGF+F ;F0/F;F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "eqn3 := lhs(eqn ) = Limit(Diff(ln(N),N)/Diff(N,N), N = infinity); #L'Hopital" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%$IntG6$*&,&\"\"\"F+-%#lnG6#%\"xG! \"\"F+F/!\"#/F/;F+%)infinityG-%&LimitG6$*&-%%DiffG6$-F-6#%\"NGF>F+-F:6 $F>F>F0/F>F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eqn4 := lhs (eqn) = value( rhs( eqn3) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn 4G/-%$IntG6$*&,&\"\"\"F+-%#lnG6#%\"xG!\"\"F+F/!\"#/F/;F+%)infinityG\" \"!" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 277 41 "19. Calculate the fi fth partial sum of " }{XPPEDIT 19 1 "sum(ln((n+1)/n),n = 1 .. infinit y);" "6#-%$sumG6$-%#lnG6#*&,&%\"nG\"\"\"F,F,F,F+!\"\"/F+;F,%)infinityG " }{TEXT 617 1 "." }}{PARA 260 "" 0 "" {TEXT 506 5 "a) " }{XPPEDIT 18 0 "ln(5);" "6#-%#lnG6#\"\"&" }{TEXT 385 11 " b) " }{XPPEDIT 265 0 "ln(6);" "6#-%#lnG6#\"\"'" }{TEXT 377 16 " c) " } {XPPEDIT 266 0 "ln(6/5);" "6#-%#lnG6#*&\"\"'\"\"\"\"\"&!\"\"" }{TEXT 378 12 " d) " }{XPPEDIT 256 0 "ln(5/6);" "6#-%#lnG6#*&\"\"&\" \"\"\"\"'!\"\"" }{TEXT 383 15 " e) " }{XPPEDIT 267 0 "2*ln( 5);" "6#*&\"\"#\"\"\"-%#lnG6#\"\"&F%" }{TEXT 379 12 " \nf) " } {XPPEDIT 268 0 "2*ln(6);" "6#*&\"\"#\"\"\"-%#lnG6#\"\"'F%" }{TEXT 380 10 " g) " }{XPPEDIT 273 0 "5*ln(6/5);" "6#*&\"\"&\"\"\"-%#lnG6#* &\"\"'F%F$!\"\"F%" }{TEXT 381 10 " h) " }{XPPEDIT 274 0 "6*ln(6/ 5);" "6#*&\"\"'\"\"\"-%#lnG6#*&F$F%\"\"&!\"\"F%" }{TEXT 384 11 " \+ i) " }{XPPEDIT 275 0 "5*ln(5/6);" "6#*&\"\"&\"\"\"-%#lnG6#*&F$F%\"\" '!\"\"F%" }{TEXT 382 11 " j) " }{XPPEDIT 276 0 "6*ln(5/6);" "6# *&\"\"'\"\"\"-%#lnG6#*&\"\"&F%F$!\"\"F%" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 515 14 "Solution: b \n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "answer := sum( ln((n+1)/n), n = 1..5);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'answerG,,-%#lnG6#\"\"#\"\"\"-F'6##\"\"$F)F*-F '6##\"\"%F.F*-F'6##\"\"&F2F*-F'6##\"\"'F6F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "answer := simplify( answer );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG,&-%#lnG6#\"\"#\"\"\"-F'6#\"\"$F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "combine( answer , ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#\"\"'" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 279 12 "20. Consider" }}{PARA 3 "" 0 "" {TEXT 608 4 "I) " } {XPPEDIT 609 1 "sum(1/n,n = 1 .. infinity);" "6#-%$sumG6$*&\"\"\"F'%\" nG!\"\"/F(;F'%)infinityG" }{TEXT 610 9 " II) " }{XPPEDIT 611 1 "su m(sin(1/n),n = 1 .. infinity);" "6#-%$sumG6$-%$sinG6#*&\"\"\"F*%\"nG! \"\"/F+;F*%)infinityG" }{TEXT 612 9 " III) " }{XPPEDIT 613 1 "sum(c os(1/n),n = 1 .. infinity);" "6#-%$sumG6$-%$cosG6#*&\"\"\"F*%\"nG!\"\" /F+;F*%)infinityG" }{TEXT 614 8 " IV) " }{XPPEDIT 615 1 "sum((n-1)/ n,n = 1 .. infinity);" "6#-%$sumG6$*&,&%\"nG\"\"\"F)!\"\"F)F(F*/F(;F)% )infinityG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 607 72 "List all given series for which the Divergence Test yields a c onclusion." }}{PARA 260 "" 0 "" {TEXT 278 73 "a) I b) II \+ c) III d) IV e) I, II " }}{PARA 260 "" 0 " " {TEXT 616 70 "f) I, III g) I, IV h) II, III i) II, IV \+ j) III, IV" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 " " 0 "" {TEXT 516 14 "Solution: j \n" }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "a := n -> \+ 1/n; \nb := n -> sin(1/n); \nc := n -> cos(1/n);\nd := n -> (n-1)/n; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operatorG% &arrowGF(*&\"\"\"F-9$!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"bGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%$sinG6#*&\"\"\"F09$!\"\"F(F(F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cGf*6#%\"nG6\"6$%)operatorG%& arrowGF(-%$cosG6#*&\"\"\"F09$!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&,&9$\"\"\"F/!\"\"F/F. F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "testeq( limit(a (n), n = infinity) = 0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "testeq( limit(b(n), n = i nfinity) = 0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "testeq( limit(c(n), n = infinity) = 0 ); #Divergence Test applies" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&fa lseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "testeq( limit(d(n), n = infinity) = 0 ); #Divergence Test applies" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}}{MARK "5 0 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }