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Calculate " }{XPPEDIT 480 0 "int(9* x^2*sqrt(1+x^3),x = 0 .. 2);" "6#-%$intG6$*(\"\"*\"\"\"*$%\"xG\"\"#F(- %%sqrtG6#,&F(F(*$F*\"\"$F(F(/F*;\"\"!F+" }{TEXT 479 1 "." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 485 4 "a) \+ " }{XPPEDIT 265 0 "20;" "6#\"#?" }{TEXT 486 12 " b) " } {XPPEDIT 266 0 "24;" "6#\"#C" }{TEXT 487 12 " c) " }{XPPEDIT 267 0 "28;" "6#\"#G" }{TEXT 488 11 " d) " }{XPPEDIT 268 0 "32; " "6#\"#K" }{TEXT 489 10 " e) " }{TEXT 484 1 " " }{XPPEDIT 269 0 "36;" "6#\"#O" }{TEXT 483 19 " " }}{PARA 0 "" 0 " " {TEXT 482 4 "f) " }{XPPEDIT 270 0 "40;" "6#\"#S" }{TEXT 490 13 " \+ g) " }{XPPEDIT 271 0 "44;" "6#\"#W" }{TEXT 491 12 " h) \+ " }{XPPEDIT 273 0 "48;" "6#\"#[" }{TEXT 492 11 " i) " } {XPPEDIT 274 0 "52;" "6#\"#_" }{TEXT 493 12 " j) " }{XPPEDIT 275 0 "56;" "6#\"#c" }{TEXT 494 3 " " }}{PARA 3 "" 0 "" {TEXT 481 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 412 12 "Solution: i" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "J := Int(9*x^2*sqrt(1+x^3),x = 0 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$,$*(\"\"*\"\"\")%\"xG\"\"#F+, &F+F+*$)F-\"\"$F+F+#F+F.F+/F-;\"\"!F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "K := changevar(1+x^3 = u, J, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%$IntG6$,$*&\"\"$\"\"\"%\"uG#F+\"\"#F+/F,;F+\"\" *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(K);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"#_" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "antiderivative := int(integrand(K), u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/antiderivativeG,$*&\"\"#\"\"\")%\"uG#\"\"$F'F(F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "simplify( subs( u = 9, anti derivative) - subs( u = 1, antiderivative) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#_" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 262 3 "2. " } {TEXT 283 5 "Let " }{XPPEDIT 19 1 "F(x) = Int(sqrt(t^3-2),t = 2 .. x) ;" "6#/-%\"FG6#%\"xG-%$IntG6$-%%sqrtG6#,&*$%\"tG\"\"$\"\"\"\"\"#!\"\"/ F0;F3F'" }{TEXT 559 60 ". Calculate the derivative D( F )( 3 ) of F at 3. " }{TEXT 284 1 " " }}{PARA 3 "" 0 "" {TEXT 282 3 "a) " } {XPPEDIT 331 0 "3;" "6#\"\"$" }{TEXT 326 20 " b) " } {XPPEDIT 332 0 "4;" "6#\"\"%" }{TEXT 327 22 " c) " } {XPPEDIT 333 0 "5;" "6#\"\"&" }{TEXT 328 18 " d) " } {XPPEDIT 334 0 "6;" "6#\"\"'" }{TEXT 329 21 " e) " } {XPPEDIT 335 0 "7;" "6#\"\"(" }{TEXT 330 14 " \nf) " } {XPPEDIT 340 0 "3-sqrt(3);" "6#,&\"\"$\"\"\"-%%sqrtG6#F$!\"\"" }{TEXT 336 10 " g) " }{XPPEDIT 341 0 "4-sqrt(5);" "6#,&\"\"%\"\"\"-%%sq rtG6#\"\"&!\"\"" }{TEXT 337 9 " h) " }{XPPEDIT 342 0 "5-sqrt(6); " "6#,&\"\"&\"\"\"-%%sqrtG6#\"\"'!\"\"" }{TEXT 338 9 " i) " } {XPPEDIT 343 0 "6-sqrt(7);" "6#,&\"\"'\"\"\"-%%sqrtG6#\"\"(!\"\"" } {TEXT 339 11 " j) " }{XPPEDIT 345 0 "7-2*sqrt(2);" "6#,&\"\"(\" \"\"*&\"\"#F%-%%sqrtG6#F'F%!\"\"" }{TEXT 344 3 " " }{TEXT 346 6 " \+ \n" }{TEXT 413 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 1 "\n" } {TEXT 414 13 "Solution: c\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "F := (x) -> Int(sqrt(t^3-2),t = 2 .. x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$IntG6$-%%sq rtG6#,&*$)%\"tG\"\"$\"\"\"F7\"\"#!\"\"/F5;F89$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(F)(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#*$\"#D#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 257 19 "3. Calculat e " }{XPPEDIT 437 1 "Int((x+4)/(x+1)/(x+2),x = 0 .. 1);" "6#-%$In tG6$*(,&%\"xG\"\"\"\"\"%F)F),&F(F)F)F)!\"\",&F(F)\"\"#F)F,/F(;\"\"!F) " }{TEXT 436 4 " . " }{TEXT 348 2 "\n\n" }{TEXT 347 4 "a) " } {XPPEDIT 357 0 "ln(4/3);" "6#-%#lnG6#*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT 349 12 " b) " }{XPPEDIT 358 0 "ln(8/3);" "6#-%#lnG6#*&\"\")\" \"\"\"\"$!\"\"" }{TEXT 350 13 " c) " }{XPPEDIT 359 0 "ln(16/3 );" "6#-%#lnG6#*&\"#;\"\"\"\"\"$!\"\"" }{TEXT 351 11 " d) " } {XPPEDIT 360 0 "ln(25/3);" "6#-%#lnG6#*&\"#D\"\"\"\"\"$!\"\"" }{TEXT 352 10 " e) " }{TEXT 295 1 " " }{XPPEDIT 361 0 "ln(28/3);" "6#-% #lnG6#*&\"#G\"\"\"\"\"$!\"\"" }{TEXT 287 19 " " }} {PARA 0 "" 0 "" {TEXT 286 4 "f) " }{XPPEDIT 362 0 "ln(25/9);" "6#-%#l nG6#*&\"#D\"\"\"\"\"*!\"\"" }{TEXT 353 12 " g) " }{XPPEDIT 363 0 "ln(32/9);" "6#-%#lnG6#*&\"#K\"\"\"\"\"*!\"\"" }{TEXT 354 11 " \+ h) " }{XPPEDIT 364 0 "ln(35/6);" "6#-%#lnG6#*&\"#N\"\"\"\"\"'!\" \"" }{TEXT 355 10 " i) " }{XPPEDIT 365 0 "ln(35/9);" "6#-%#lnG6# *&\"#N\"\"\"\"\"*!\"\"" }{TEXT 356 11 " j) " }{XPPEDIT 366 0 "l n(36/5);" "6#-%#lnG6#*&\"#O\"\"\"\"\"&!\"\"" }{TEXT 367 3 " " }} {PARA 3 "" 0 "" {TEXT 285 1 " " }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 415 13 "Solution: g\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "J := Int((x+4)/(x+1)/(x+2),x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*(,&%\"x G\"\"\"\"\"%F+F+,&F*F+F+F+!\"\",&F*F+\"\"#F+F./F*;\"\"!F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "processedIntegrand := convert( inte grand(J), parfrac, x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%3process edIntegrandG,&*&\"\"$\"\"\",&%\"xGF(F(F(!\"\"F(*&\"\"#F(,&F*F(F-F(F+F+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "antiderivative := int(p rocessedIntegrand, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/antideriv ativeG,&*&\"\"$\"\"\"-%#lnG6#,&%\"xGF(F(F(F(F(*&\"\"#F(-F*6#,&F-F(F/F( F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "answer := simpli fy(subs(x = 1, antiderivative) - subs(x = 0, antiderivative));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG,&*&\"\"&\"\"\"-%#lnG6#\"\"# F(F(*&F,F(-F*6#\"\"$F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "combine(answer, ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6## \"#K\"\"*" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 321 15 "4. Calculate " }{XPPEDIT 577 1 "int((5*x^2+6)/(2+x^2)/x,x = 1 .. 2);" "6#-%$intG6$*(, &*&\"\"&\"\"\"*$%\"xG\"\"#F*F*\"\"'F*F*,&F-F**$F,F-F*!\"\"F,F1/F,;F*F- " }{TEXT 576 7 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 323 3 "a) " }{TEXT 439 2 " " }{XPPEDIT 19 1 "Pi/4+ln(2);" "6#,&*&%#PiG\"\"\"\"\"%!\"\"F&-%#lnG6#\"\"#F&" }{TEXT 440 8 " b) \+ " }{XPPEDIT 19 1 "Pi/4+2*ln(2);" "6#,&*&%#PiG\"\"\"\"\"%!\"\"F&*&\"\"# F&-%#lnG6#F*F&F&" }{TEXT 441 10 " c) " }{XPPEDIT 19 1 "Pi/4+ln(3 );" "6#,&*&%#PiG\"\"\"\"\"%!\"\"F&-%#lnG6#\"\"$F&" }{TEXT 442 10 " \+ d) " }{XPPEDIT 19 1 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT 443 9 " \+ e)" }{TEXT 324 1 " " }{XPPEDIT 19 1 "2*ln(2);" "6#*&\"\"#\"\"\"-%#lnG 6#F$F%" }{TEXT 444 1 " " }{TEXT 445 11 " " }}{PARA 0 "" 0 " " {TEXT 322 5 "f) " }{XPPEDIT 19 1 "4*ln(2);" "6#*&\"\"%\"\"\"-%#lnG 6#\"\"#F%" }{TEXT 438 14 " g) " }{XPPEDIT 19 1 "8*ln(2);" "6 #*&\"\")\"\"\"-%#lnG6#\"\"#F%" }{TEXT 446 11 " " }{TEXT 447 5 " h) " }{XPPEDIT 19 1 "ln(3);" "6#-%#lnG6#\"\"$" }{TEXT 448 19 " \+ i) " }{XPPEDIT 19 1 "2*ln(3);" "6#*&\"\"#\"\"\"-%#lnG6#\" \"$F%" }{TEXT 449 10 " j) " }{XPPEDIT 19 1 "ln(6);" "6#-%#lnG6# \"\"'" }{TEXT 450 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 416 13 "Solution: \+ f\n" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "J := \+ Int( (5*x^2+6)/(2+x^2)/x, x = 1 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*(,&*&\"\"&\"\"\")%\"xG\"\"#F,F,\"\"'F,F,,&F/F,*$ F-F,F,!\"\"F.F3/F.;F,F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " processedIntegrand := convert( integrand(J), parfrac, x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%3processedIntegrandG,&*(\"\"#\"\"\"%\"xGF( ,&F'F(*$)F)F'F(F(!\"\"F(*&\"\"$F(F)F-F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "antiderivative := int(processedIntegrand, x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%/antiderivativeG,&-%#lnG6#,&\"\"#\" \"\"*$)%\"xGF*F+F+F+*&\"\"$F+-F'6#F.F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "answer := simplify(subs(x = 2, antiderivative) - subs (x = 1, antiderivative));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answer G,$*&\"\"%\"\"\"-%#lnG6#\"\"#F(F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 299 15 "5. Calculate " }{XPPEDIT 579 1 "Int(x^2/sqrt(1-x^2),x = 0 .. 1);" "6#-%$IntG6$*&%\"xG\"\"#-%%sqrtG6#,&\"\"\"F-*$F'F(!\"\"F//F';\" \"!F-" }{TEXT 578 1 "." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 302 4 " a) " }{XPPEDIT 311 0 "Pi;" "6#%#PiG" }{TEXT 305 16 " b) \+ " }{XPPEDIT 312 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 306 15 " \+ c) " }{TEXT 457 1 " " }{XPPEDIT 256 0 "Pi/3;" "6#*&%#PiG\" \"\"\"\"$!\"\"" }{TEXT 458 14 " d) " }{XPPEDIT 313 0 "Pi/4; " "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 310 14 " e) " } {XPPEDIT 314 0 "Pi/6;" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 304 2 " " } {TEXT 303 20 " \n" }{TEXT 300 5 "f) " }{XPPEDIT 315 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT 309 13 " g) " } {XPPEDIT 316 0 "3*Pi/2;" "6#*(\"\"$\"\"\"%#PiGF%\"\"#!\"\"" }{TEXT 308 7 " " }{TEXT 317 1 " " }{TEXT 301 3 "h) " }{XPPEDIT 257 0 "2 *Pi/3;" "6#*(\"\"#\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 456 13 " i) \+ " }{XPPEDIT 318 0 "3*Pi/4;" "6#*(\"\"$\"\"\"%#PiGF%\"\"%!\"\"" } {TEXT 307 12 " j) " }{XPPEDIT 319 0 "5*Pi/6;" "6#*(\"\"&\"\"\" %#PiGF%\"\"'!\"\"" }{TEXT 320 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 417 13 "Solution: d\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "J := Int(x^2/sqrt(1-x^2),x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&%\"xG\" \"#,&\"\"\"F,*$)F)F*F,!\"\"#F/F*/F);\"\"!F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "K := student[changevar](x=sin(t), J, t); \n#Note: \+ denominator of K is cos(t)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-% $IntG6$*(-%$sinG6#%\"tG\"\"#,&\"\"\"F/*$)F)F-F/!\"\"#F2F--%$cosGF+F//F ,;\"\"!,$*&F-F2%#PiGF/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "int(sin(t)^2, t = 0 .. Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*& \"\"%!\"\"%#PiG\"\"\"F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 291 4 "6. \+ " }{TEXT 451 187 "A radioactive substance has a half-life equal to ln ( 8 ) years. If m(t) is the mass of the substance at time t, measu red in years, then what is the logarithmic derivative of m(t) ? " } {TEXT 452 1 "\n" }{TEXT 290 0 "" }}{PARA 3 "" 0 "" {TEXT 289 4 "a) " }{XPPEDIT 257 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 580 17 " \+ b) " }{XPPEDIT 374 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 368 15 " c) " }{XPPEDIT 375 0 "1/sqrt(2);" "6#*&\"\"\"F$-%% sqrtG6#\"\"#!\"\"" }{TEXT 369 15 " d) " }{XPPEDIT 376 0 "1/ sqrt(3);" "6#*&\"\"\"F$-%%sqrtG6#\"\"$!\"\"" }{TEXT 370 15 " \+ e) " }{XPPEDIT 377 0 "1/(2*sqrt(2));" "6#*&\"\"\"F$*&\"\"#F$-%%sqrtG 6#F&F$!\"\"" }{TEXT 371 8 " \nf) " }{XPPEDIT 256 0 "-1/2;" "6#,$*& \"\"\"F%\"\"#!\"\"F'" }{TEXT 455 12 " g) " }{XPPEDIT 257 0 "-1 /3;" "6#,$*&\"\"\"F%\"\"$!\"\"F'" }{TEXT 454 13 " h) " } {XPPEDIT 378 0 "-1/sqrt(2);" "6#,$*&\"\"\"F%-%%sqrtG6#\"\"#!\"\"F*" } {TEXT 372 11 " i) " }{XPPEDIT 379 0 "-1/sqrt(3);" "6#,$*&\"\"\" F%-%%sqrtG6#\"\"$!\"\"F*" }{TEXT 373 11 " j) " }{XPPEDIT 257 0 "-1/(2*sqrt(2));" "6#,$*&\"\"\"F%*&\"\"#F%-%%sqrtG6#F'F%!\"\"F+" } {TEXT 453 3 " " }{TEXT 288 1 "\n" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 433 11 "Solution: g" }}{PARA 0 "" 0 "" {TEXT 434 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "k := ln(2)/ln(8); #Solve f or decay constant in terms of half-life" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG*&-%#lnG6#\"\"#\"\"\"-F'6#\"\")!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "simplify( diff( ln(A*exp(-k*t) ) , t) );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "#Note: -ln(2)/ln(8) = -ln(2)/ln(2^3) = -ln(2)/(3*l n(2)) = -1/3" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 296 13 "7. Calculate \+ " }{XPPEDIT 582 1 "Int(1/(1+x^2),x = 1 .. sqrt(3));" "6#-%$IntG6$*&\" \"\"F',&F'F'*$%\"xG\"\"#F'!\"\"/F*;F'-%%sqrtG6#\"\"$" }{TEXT 581 1 ". " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 585 4 "a) " }{XPPEDIT 268 0 "Pi;" "6#%#PiG" }{TEXT 588 16 " b) " }{XPPEDIT 269 0 "Pi /2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 589 15 " c) " } {TEXT 597 1 " " }{XPPEDIT 256 0 "Pi/3;" "6#*&%#PiG\"\"\"\"\"$!\"\"" } {TEXT 598 14 " d) " }{XPPEDIT 270 0 "Pi/4;" "6#*&%#PiG\"\"\" \"\"%!\"\"" }{TEXT 593 14 " e) " }{XPPEDIT 271 0 "Pi/6;" "6# *&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 587 2 " " }{TEXT 586 20 " \+ \n" }{TEXT 583 5 "f) " }{XPPEDIT 272 0 "5*Pi/6;" "6#*(\"\"&\" \"\"%#PiGF%\"\"'!\"\"" }{TEXT 592 13 " g) " }{XPPEDIT 273 0 " 5*Pi/12;" "6#*(\"\"&\"\"\"%#PiGF%\"#7!\"\"" }{TEXT 591 7 " " } {TEXT 594 1 " " }{TEXT 584 3 "h) " }{XPPEDIT 257 0 "2*Pi/3;" "6#*(\"\" #\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 596 13 " i) " }{XPPEDIT 275 0 "3*Pi/4;" "6#*(\"\"$\"\"\"%#PiGF%\"\"%!\"\"" }{TEXT 590 12 " \+ j) " }{XPPEDIT 276 0 "Pi/12;" "6#*&%#PiG\"\"\"\"#7!\"\"" }{TEXT 595 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 431 11 "Solution: j" }}{PARA 0 "" 0 "" {TEXT 432 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "antiderivative := \+ int(1/(1+x^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/antiderivative G-%'arctanG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "simpl ify(subs(x = sqrt(3), antiderivative) - subs(x = 1, antiderivative)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#7!\"\"%#PiG\"\"\"F(" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 293 3 "8. " }{TEXT 676 3 "Let" }{TEXT 679 2 " " }{XPPEDIT 260 1 "f(x) = (1+2*x)^x;" "6#/-%\" fG6#%\"xG),&\"\"\"F**&\"\"#F*F'F*F*F'" }{TEXT 678 2 ". " }{TEXT 677 2 " " }{TEXT 680 59 "What is D( f )(1) ? (The derivative of f(x) a t x = 1)" }{TEXT 681 2 " ?" }}{PARA 0 "" 0 "" {TEXT 292 17 " \+ " }}{PARA 0 "" 0 "" {TEXT 297 5 "a) " }{XPPEDIT 389 0 "2*ln( 2)+1;" "6#,&*&\"\"#\"\"\"-%#lnG6#F%F&F&F&F&" }{TEXT 384 10 " b) \+ " }{XPPEDIT 390 0 "3*ln(3)+2;" "6#,&*&\"\"$\"\"\"-%#lnG6#F%F&F&\"\"#F& " }{TEXT 385 10 " c) " }{XPPEDIT 391 0 "ln(2)+2;" "6#,&-%#lnG6# \"\"#\"\"\"F'F(" }{TEXT 386 10 " d) " }{XPPEDIT 392 0 "ln(3)+3; " "6#,&-%#lnG6#\"\"$\"\"\"F'F(" }{TEXT 387 13 " e) " } {XPPEDIT 393 0 "ln(2)+3;" "6#,&-%#lnG6#\"\"#\"\"\"\"\"$F(" }{TEXT 388 8 " \nf) " }{XPPEDIT 394 0 "ln(3)+2;" "6#,&-%#lnG6#\"\"$\"\"\"\"\"# F(" }{TEXT 383 14 " g) " }{XPPEDIT 395 0 "ln(2)+1;" "6#,&-%# lnG6#\"\"#\"\"\"F(F(" }{TEXT 382 12 " h) " }{XPPEDIT 396 0 "ln (3)+1;" "6#,&-%#lnG6#\"\"$\"\"\"F(F(" }{TEXT 381 11 " i) " } {XPPEDIT 397 0 "3*ln(2)+1;" "6#,&*&\"\"$\"\"\"-%#lnG6#\"\"#F&F&F&F&" } {TEXT 380 11 " j) " }{XPPEDIT 398 0 "3*ln(3)+1;" "6#,&*&\"\"$\" \"\"-%#lnG6#F%F&F&F&F&" }{TEXT 298 1 "\n" }{TEXT 399 0 "" }}{PAGEBK } {PARA 0 "" 0 "" {TEXT 429 11 "Solution: b" }}{PARA 0 "" 0 "" {TEXT 430 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> (1+2*x)^x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG 6\"6$%)operatorG%&arrowGF(),&\"\"\"F.*&\"\"#F.9$F.F.F1F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"$\"\"\"-%#lnG6#F%F&F&\"\"#F&" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 711 2 "9." }{TEXT -1 1 " " }{TEXT 712 39 "Consid er the following three statements" }{TEXT 713 16 " about a series " } {TEXT 715 1 " " }{XPPEDIT 261 1 "sum(a[n],n = 1 .. infinity);" "6#-%$s umG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT 714 54 " with positiv e terms:\nI: The series diverges because" }{TEXT 717 3 " " } {XPPEDIT 264 1 "limit(a[n],n = infinity) = 1;" "6#/-%&limitG6$&%\"aG6# %\"nG/F*%)infinityG\"\"\"" }{TEXT 716 34 ".\n\nII: The series diverges because" }{TEXT 719 3 " " }{XPPEDIT 257 1 "limit(a[n]/b[n],n = infi nity) = 1;" "6#/-%&limitG6$*&&%\"aG6#%\"nG\"\"\"&%\"bG6#F+!\"\"/F+%)in finityGF," }{TEXT 718 6 " and " }{TEXT 720 1 " " }{XPPEDIT 256 1 "sum (b[n],n = 1 .. infinity);" "6#-%$sumG6$&%\"bG6#%\"nG/F);\"\"\"%)infini tyG" }{TEXT 721 45 " diverges.\n\nIII: The series diverges because" } {TEXT 739 3 " " }{XPPEDIT 257 1 "limit(a[n]^(1/n),n = infinity) = 1; " "6#/-%&limitG6$)&%\"aG6#%\"nG*&\"\"\"F-F+!\"\"/F+%)infinityGF-" } {TEXT 738 97 ".\n\nFor each statement, determine whether the reasoning is correct ( $ ) or incorrect ( @ ). \n " }}{PARA 261 "" 0 "" {TEXT -1 6 "a) I: " }{TEXT 722 1 "$" }{TEXT -1 7 ", II: " }{TEXT 723 10 "$, III: $" }{TEXT -1 18 " \nb) I: " }{TEXT 724 1 "$" } {TEXT -1 7 ", II: " }{TEXT 725 9 "$, III:@" }{TEXT -1 12 " \nc) \+ I: " }{TEXT 726 1 "$" }{TEXT -1 7 ", II: " }{TEXT 729 10 "@, III: $ " }{TEXT -1 13 " \nd) I: " }{TEXT 727 1 "$" }{TEXT -1 7 ", II: \+ " }{TEXT 730 10 "@, III: @" }{TEXT -1 15 " \ne) I:" }{TEXT 732 1 "@" }{TEXT -1 6 ", II: " }{TEXT 731 10 "$, III: $" }{TEXT -1 23 " \nf) I: " }{TEXT 733 1 "@" }{TEXT -1 7 ", II: " } {TEXT 728 10 "$, III: @" }{TEXT -1 14 " \ng) I: " }{TEXT 737 1 "@" }{TEXT -1 7 ", II: " }{TEXT 736 10 "@, III: $" }{TEXT -1 13 " \+ \nh) I: " }{TEXT 734 1 "@" }{TEXT -1 6 ", II: " }{TEXT 735 9 "@, II I: @" }{TEXT -1 52 " \ni) Wrong answer \n j) Bonus wrong an swer" }}{PARA 3 "" 0 "" {TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT 741 1 " \n" }{TEXT 740 11 "Solution: b" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 259 44 "10. Consider the following three statements" }{TEXT 459 16 " about a series " }{TEXT 683 1 " " }{XPPEDIT 684 1 "sum(a[n],n = 1 .. infinity);" "6#-%$sumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" } {TEXT 682 55 " with positive terms:\nI: The series converges because " }{TEXT 686 3 " " }{XPPEDIT 687 1 "limit(a[n],n = infinity) = 0;" " 6#/-%&limitG6$&%\"aG6#%\"nG/F*%)infinityG\"\"!" }{TEXT 685 35 ".\n\nII : The series converges because" }{TEXT 689 3 " " }{XPPEDIT 259 1 "a[ n] < b[n];" "6#2&%\"aG6#%\"nG&%\"bG6#F'" }{TEXT 688 6 " and " }{TEXT 690 1 " " }{XPPEDIT 258 1 "sum(b[n],n = 1 .. infinity);" "6#-%$sumG6$& %\"bG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT 691 46 " diverges.\n\nIII: \+ The series converges because" }{TEXT 710 3 " " }{XPPEDIT 259 1 "limi t(a[n]/a[n+1],n = infinity) < 1;" "6#2-%&limitG6$*&&%\"aG6#%\"nG\"\"\" &F)6#,&F+F,F,F,!\"\"/F+%)infinityGF," }{TEXT 709 97 ".\n\nFor each sta tement, determine whether the reasoning is correct ( $ ) or incorrect ( @ ). \n " }}{PARA 261 "" 0 "" {TEXT -1 6 "a) I: " }{TEXT 693 1 "$ " }{TEXT -1 7 ", II: " }{TEXT 694 10 "$, III: $" }{TEXT -1 18 " \+ \nb) I: " }{TEXT 695 1 "$" }{TEXT -1 7 ", II: " }{TEXT 696 9 "$ , III:@" }{TEXT -1 12 " \nc) I: " }{TEXT 697 1 "$" }{TEXT -1 7 ", II: " }{TEXT 700 10 "@, III: $" }{TEXT -1 13 " \nd) I: " } {TEXT 698 1 "$" }{TEXT -1 7 ", II: " }{TEXT 701 10 "@, III: @" } {TEXT -1 15 " \ne) I:" }{TEXT 703 1 "@" }{TEXT -1 6 ", II: " } {TEXT 702 10 "$, III: $" }{TEXT -1 23 " \nf) I: " } {TEXT 704 1 "@" }{TEXT -1 7 ", II: " }{TEXT 699 10 "$, III: @" } {TEXT -1 14 " \ng) I: " }{TEXT 708 1 "@" }{TEXT -1 7 ", II: " } {TEXT 707 10 "@, III: $" }{TEXT -1 13 " \nh) I: " }{TEXT 705 1 " @" }{TEXT -1 6 ", II: " }{TEXT 706 9 "@, III: @" }{TEXT -1 52 " \+ \ni) Wrong answer \n j) Bonus wrong answer" }}{PARA 0 "" 0 "" {TEXT 692 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 " " 0 "" {TEXT 427 11 "Solution: h" }}{PARA 0 "" 0 "" {TEXT 428 1 " " } {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 477 5 "11. " }{TEXT 655 41 "Consider the three series \n I: " }{XPPEDIT 257 1 "S um(5^n*n/n!,n = 1 .. infinity);" "6#-%$SumG6$*()\"\"&%\"nG\"\"\"F)F*-% *factorialG6#F)!\"\"/F);F*%)infinityG" }{TEXT 656 12 " , II: " } {XPPEDIT 256 1 "Sum(1/n/ln(n),n = 3 .. infinity);" "6#-%$SumG6$*(\"\" \"F'%\"nG!\"\"-%#lnG6#F(F)/F(;\"\"$%)infinityG" }{TEXT 657 18 " , a nd III: " }{XPPEDIT 256 1 "Sum(n^(1/3)/(1+n^(13/9)),n = 0 .. infini ty);" "6#-%$SumG6$*&)%\"nG*&\"\"\"F*\"\"$!\"\"F*,&F*F*)F(*&\"#8F*\"\"* F,F*F,/F(;\"\"!%)infinityG" }{TEXT 661 2 " \n" }{TEXT 658 143 "and the statements\n( $ ) The series converges \n( @ ) The series diverges\n \nFor each series, decide which of statements ($), (@) is correct. \+ " }}{PARA 261 "" 0 "" {TEXT -1 6 "a) I: " }{TEXT 659 1 "$" }{TEXT -1 7 ", II: " }{TEXT 660 10 "$, III: $" }{TEXT -1 18 " \nb) I : " }{TEXT 662 1 "$" }{TEXT -1 7 ", II: " }{TEXT 663 9 "$, III:@" } {TEXT -1 12 " \nc) I: " }{TEXT 664 1 "$" }{TEXT -1 7 ", II: " } {TEXT 667 10 "@, III: $" }{TEXT -1 13 " \nd) I: " }{TEXT 665 1 " $" }{TEXT -1 7 ", II: " }{TEXT 668 10 "@, III: @" }{TEXT -1 15 " \+ \ne) I:" }{TEXT 670 1 "@" }{TEXT -1 6 ", II: " }{TEXT 669 10 "$, \+ III: $" }{TEXT -1 23 " \nf) I: " }{TEXT 671 1 "@" } {TEXT -1 7 ", II: " }{TEXT 666 10 "$, III: @" }{TEXT -1 14 " \+ \ng) I: " }{TEXT 675 1 "@" }{TEXT -1 7 ", II: " }{TEXT 674 10 "@, II I: $" }{TEXT -1 13 " \nh) I: " }{TEXT 672 1 "@" }{TEXT -1 6 ", II : " }{TEXT 673 9 "@, III: @" }{TEXT -1 52 " \ni) Wrong answer \+ \n j) Bonus wrong answer" }}{PARA 3 "" 0 "" {TEXT 654 1 "\n" } {TEXT 478 11 "Solution: c" }{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 294 5 "12. " }{TEXT 632 39 "Consider the two se ries \n I: " }{XPPEDIT 259 1 "Sum((-1)^n/ln(n),n = 3 .. infi nity);" "6#-%$SumG6$*&),$\"\"\"!\"\"%\"nGF)-%#lnG6#F+F*/F+;\"\"$%)infi nityG" }{TEXT 633 15 " and II: " }{XPPEDIT 256 1 "Sum((-1)^n*n^3 /(2^n),n = 0 .. infinity);" "6#-%$SumG6$*(),$\"\"\"!\"\"%\"nGF)*$F+\" \"$F))\"\"#F+F*/F+;\"\"!%)infinityG" }{TEXT 634 2 " \n" }{TEXT 635 198 "and the statements\n( $ ) The series converges absolutely\n( & ) The series converges conditionally\n( @ ) The series diverges\n\nFor \+ each series, decide which of statements ($), (&), (@) is correct. " }}{PARA 261 "" 0 "" {TEXT -1 6 "a) I: " }{TEXT 636 1 "$" }{TEXT -1 7 " , II: " }{TEXT 637 1 "$" }{TEXT -1 16 " b) I: " }{TEXT 638 1 "$" }{TEXT -1 7 ", II: " }{TEXT 653 1 "&" }{TEXT -1 10 " c) I: \+ " }{TEXT 639 1 "$" }{TEXT -1 7 ", II: " }{TEXT 640 1 "@" }{TEXT -1 12 " d) I: " }{TEXT 642 1 "&" }{TEXT -1 7 ", II: " }{TEXT 641 1 "$" }{TEXT -1 14 " e) I: " }{TEXT 643 1 "&" }{TEXT -1 7 ", II: " }{TEXT 644 1 "&" }{TEXT -1 21 " \nf) I: " }{TEXT 645 1 "&" }{TEXT -1 7 ", II: " }{TEXT 646 1 "@" }{TEXT -1 12 " g) I: " }{TEXT 648 1 "@" }{TEXT -1 7 " , II: " }{TEXT 647 1 "$" }{TEXT -1 12 " h) I: " }{TEXT 649 1 "@" }{TEXT -1 7 " , II: " }{TEXT 650 1 "&" }{TEXT -1 12 " i) I: " }{TEXT 651 1 "@" }{TEXT -1 7 " , II: \+ " }{TEXT 652 1 "@" }{TEXT -1 22 " j) Wrong answer" }}{PAGEBK } {PARA 0 "" 0 "" {TEXT 426 12 "Solution: d " }{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 276 43 "13. Consider the two series \n \+ I: " }{XPPEDIT 611 1 "Sum(10^n/(10^n+n^2),n = 0 .. infinity);" "6#-%$ SumG6$*&)\"#5%\"nG\"\"\",&)F(F)F**$F)\"\"#F*!\"\"/F);\"\"!%)infinityG " }{TEXT 610 15 " and II: " }{XPPEDIT 256 1 "Sum(n^3/(3^n),n = 0 .. infinity);" "6#-%$SumG6$*&%\"nG\"\"$)F(F'!\"\"/F';\"\"!%)infinityG " }{TEXT 612 2 " \n" }{TEXT 613 262 "and the statements\n( $ ) The te st establishes convergence\n( & ) The test establishes divergence\n( @ ) The test is not conclusive.\n\nApply the Ratio Test to series I and the Root Test to series II. For each, decide\nwhich of statements ($) , (&), (@) is correct. " }}{PARA 261 "" 0 "" {TEXT -1 6 "a) I: " } {TEXT 614 1 "$" }{TEXT -1 7 ", II: " }{TEXT 615 1 "$" }{TEXT -1 16 " \+ b) I: " }{TEXT 616 1 "$" }{TEXT -1 7 ", II: " }{TEXT 631 1 " &" }{TEXT -1 10 " c) I: " }{TEXT 617 1 "$" }{TEXT -1 7 ", II: " } {TEXT 618 1 "@" }{TEXT -1 12 " d) I: " }{TEXT 620 1 "&" }{TEXT -1 7 ", II: " }{TEXT 619 1 "$" }{TEXT -1 14 " e) I: " }{TEXT 621 1 "&" }{TEXT -1 7 ", II: " }{TEXT 622 1 "&" }{TEXT -1 21 " \+ \nf) I: " }{TEXT 623 1 "&" }{TEXT -1 7 ", II: " }{TEXT 624 1 " @" }{TEXT -1 12 " g) I: " }{TEXT 626 1 "@" }{TEXT -1 7 " , II: " }{TEXT 625 1 "$" }{TEXT -1 12 " h) I: " }{TEXT 627 1 "@" }{TEXT -1 7 " , II: " }{TEXT 628 1 "&" }{TEXT -1 12 " i) I: " }{TEXT 629 1 "@" }{TEXT -1 7 " , II: " }{TEXT 630 1 "@" }{TEXT -1 22 " j ) Wrong answer" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 461 12 "Solution: g " } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "A := int(Pi*(1/Pi)*y*4,y = 1 .. 2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"AG\"\"'" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 11 " 14. Let " }{XPPEDIT 606 1 "f(x) = Sum((n+3)*x^n/(n^2*(n+1)) ,n = 1 .. infinity);" "6#/-%\"fG6#%\"xG-%$SumG6$*(,&%\"nG\"\"\"\"\"$F. F.)F'F-F.*&F-\"\"#,&F-F.F.F.F.!\"\"/F-;F.%)infinityG" }{TEXT 599 5 " . " }{TEXT 460 10 "What is " }{XPPEDIT 608 1 "f*`'''`;" "6#*&%\"fG \"\"\"%$'''GF%" }{TEXT 607 6 " ( 0 )" }{TEXT 609 2 " ?" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 325 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 600 69 "a) 1 b) 2 c) 3 d) 6 \+ e) 12 " }{TEXT 603 3 " " }{TEXT 602 18 " " } }{PARA 0 "" 0 "" {TEXT 601 61 "f) 1/3 g) 1/2 h) 2/3 \+ i) 4/3 j)" }{TEXT -1 1 " " }{TEXT 605 1 " " }{TEXT 604 4 "1/6 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 424 11 "Solution: a " }}{PARA 0 "" 0 "" {TEXT 425 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "a := n -> (n+3)/(n^2*(n+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*(,&9$\" \"\"\"\"$F/F/F.!\"#,&F.F/F/F/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "simplify(3!*a(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "p := x -> sum((n +3)*x^n/(n^2*(n+1)),n = 1 .. 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$sumG6$**,&%\"nG\"\"\"\"\"$F 2F2)9$F1F2F1!\"#,&F1F2F2F2!\"\"/F1;F2\"#5F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "(D@@3)(p)(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 560 5 "15. " }{TEXT 562 6 "Let " }{XPPEDIT 288 1 "T(x) = a[0]+a[1]*x+a[2]*x^2+a[3]*x^3+a [4]*x^4+a[5]*x^5;" "6#/-%\"TG6#%\"xG,.&%\"aG6#\"\"!\"\"\"*&&F*6#F-F-F' F-F-*&&F*6#\"\"#F-*$F'F4F-F-*&&F*6#\"\"$F-*$F'F9F-F-*&&F*6#\"\"%F-*$F' F>F-F-*&&F*6#\"\"&F-*$F'FCF-F-" }{TEXT 561 2 " " }{TEXT 563 44 " be t he degree 5 Taylor polynomial \nof " }{XPPEDIT 19 1 "f(x) = x^2*e xp(-2*x);" "6#/-%\"fG6#%\"xG*&F'\"\"#-%$expG6#,$*&F)\"\"\"F'F/!\"\"F/ " }{TEXT 565 32 " centered about 0. What is " }{XPPEDIT 290 1 "T( 1);" "6#-%\"TG6#\"\"\"" }{TEXT 564 4 " ?\n " }}{PARA 0 "" 0 "" {TEXT 277 85 "a) 1/5 b) 2/5 c) 1/3 d) \+ 2/3 e) 4/15 " }{TEXT 411 3 " " }{TEXT 280 18 " \+ " }}{PARA 0 "" 0 "" {TEXT 278 4 "f) " }{TEXT 570 1 "-" }{TEXT 571 18 " 1/5 g) " }{TEXT 568 1 "-" }{TEXT 569 12 " 2/5 \+ " }{TEXT -1 2 " " }{TEXT 279 3 "h) " }{TEXT 566 1 "-" }{TEXT 567 18 " 1/3 i) " }{TEXT 572 1 "-" }{TEXT 573 14 " 2/3 j) " }{TEXT -1 1 " " }{TEXT 574 1 "-" }{TEXT 575 5 " 4/15" }{TEXT 464 1 " " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 422 12 "Solution: h " } }{PARA 0 "" 0 "" {TEXT 423 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "S := series(x^2*exp(-2*x), x = 0, 6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG+-%\"xG\"\"\"\"\"#!\"#\"\"$F(\"\"%#!\" %F*\"\"&-%\"OG6#F'\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " p := unapply(convert(S,polynom),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,**$)9$\"\"#\"\"\"F1*&F0F1)F /\"\"$F1!\"\"*&F0F1)F/\"\"%F1F1*&#F8F4F1*$)F/\"\"&F1F1F5F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"$" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 267 5 "1 6. " }{TEXT 466 42 "Calculate the interval of convergence of " } {XPPEDIT 19 1 "Sum((-1)^n*n*(x+2)^n/(n^2+1)/(3^n),n = 0 .. infinity); " "6#-%$SumG6$*,),$\"\"\"!\"\"%\"nGF)F+F)),&%\"xGF)\"\"#F)F+F),&*$F+F/ F)F)F)F*)\"\"$F+F*/F+;\"\"!%)infinityG" }{TEXT 465 1 "." }}{PARA 0 "" 0 "" {TEXT 467 14 "a) ( - 5 , 1 " }{TEXT 535 1 "]" }{TEXT 536 14 " \+ b) " }{TEXT 541 1 "[" }{TEXT 542 1 " " }{TEXT 543 9 " - 5 , 1 \+ " }{TEXT 539 1 "]" }{TEXT 540 24 " c) ( - 1 , 5 " }{TEXT 544 1 "]" }{TEXT 545 13 " d) " }{TEXT 550 1 "[" }{TEXT 551 1 " " }{TEXT 552 9 " - 1 , 5 " }{TEXT 548 1 "]" }{TEXT 549 12 " \+ e) " }{TEXT 553 2 " [" }{TEXT 554 9 " - 2 , 2 " }{TEXT 555 1 "]" } {TEXT 556 1 " " }{TEXT 471 1 " " }{TEXT 470 18 " " }} {PARA 0 "" 0 "" {TEXT 468 4 "f) " }{TEXT 537 1 "[" }{TEXT 538 45 " - \+ 5 , 1 ) g) ( - 5 , 1 ) " }{TEXT -1 1 " " }{TEXT 469 4 "h) " }{TEXT 546 1 "[" }{TEXT 547 54 " - 1 , 5 ) i) \+ ( - 1 , 5 ) j) ( " }{XPPEDIT 558 1 "-infinity*` `,infinity ;" "6$,$*&%)infinityG\"\"\"%\"~GF&!\"\"F%" }{TEXT 557 3 " )" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 421 14 "Solut ion: a " }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a := n -> n/(n^2+1)/(3^n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" aGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*(9$\"\"\",&*$)F-\"\"#F.F.F.F.!\" \")\"\"$F-F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "R := \+ limit(a(n)/a(n+1), n = infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"RG\"\"$" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 269 4 "17. " }{TEXT 275 1 " " }{TEXT 525 28 "What is the coefficient of " }{XPPEDIT 527 0 "(x -4)^3;" "6#*$,&%\"xG\"\"\"\"\"%!\"\"\"\"$" }{TEXT 526 27 " in the Tay lor series of " }{XPPEDIT 529 1 "sqrt(x);" "6#-%%sqrtG6#%\"xG" } {TEXT 528 20 " with base point 4?" }}{PARA 0 "" 0 "" {TEXT 403 6 "a) \+ " }{XPPEDIT 409 0 "1/32;" "6#*&\"\"\"F$\"#K!\"\"" }{TEXT 408 15 " \+ b) " }{XPPEDIT 257 0 "3/64;" "6#*&\"\"$\"\"\"\"#k!\"\"" } {TEXT 404 20 " c) " }{XPPEDIT 258 0 "1/128;" "6#*&\"\" \"F$\"$G\"!\"\"" }{TEXT 405 17 " d) " }{XPPEDIT 256 0 "3/ 256;" "6#*&\"\"$\"\"\"\"$c#!\"\"" }{TEXT 407 19 " e) " }{XPPEDIT 259 0 "1/512;" "6#*&\"\"\"F$\"$7&!\"\"" }{TEXT 406 12 " \+ \nf) " }{XPPEDIT 263 0 "-1/32;" "6#,$*&\"\"\"F%\"#K!\"\"F'" }{TEXT 534 11 " g) " }{XPPEDIT 257 0 "-3/64;" "6#,$*&\"\"$\"\"\"\"#k! \"\"F(" }{TEXT 530 17 " h) " }{XPPEDIT 258 0 "-1/128;" "6 #,$*&\"\"\"F%\"$G\"!\"\"F'" }{TEXT 531 14 " i) " }{XPPEDIT 256 0 "-3/256;" "6#,$*&\"\"$\"\"\"\"$c#!\"\"F(" }{TEXT 533 16 " \+ j) " }{XPPEDIT 259 0 "-1/512;" "6#,$*&\"\"\"F%\"$7&!\"\"F'" } {TEXT 532 3 " \n" }{TEXT 410 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 435 14 "Solution: e \n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "simplify(subs(x=4,diff(s qrt(x),x$3))/3!);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"$7&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "series(sqrt(x), x=4, 5); #al ternative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/,&%\"xG\"\"\"\"\"%!\"\" \"\"#\"\"!#F&F'F&#F(\"#kF)#F&\"$7&\"\"$#!\"&\"&%Q;F'-%\"OG6#F&\"\"&" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 271 4 "18. " }{TEXT 400 1 " " }{TEXT 474 25 "The Maclaurin se ries of " }{XPPEDIT 475 1 "sin(x^2);" "6#-%$sinG6#*$%\"xG\"\"#" } {TEXT 472 27 " is used to approximate " }{XPPEDIT 476 1 "int(42*sin (x^2),x = 0 .. `0.1`);" "6#-%$intG6$*&\"#U\"\"\"-%$sinG6#*$%\"xG\"\"#F (/F-;\"\"!%$0.1G" }{TEXT 473 1 " " }{TEXT 521 26 "with an error \nless than " }{XPPEDIT 524 0 "10^(-6);" "6#)\"#5,$\"\"'!\"\"" }{TEXT 523 154 ". The calculation uses only as many terms as are deemed necessar y for the required accuracy by the Alternating Series Test. What is th e approximation? \n " }{TEXT 522 2 " " }{TEXT 402 167 " \na) 0.012 \+ b) 0.013 c) 0.014 d) 0.015 e) 0.016 \+ \nf) 0.017 g) 0.018 h) 0.019 i) 0.020 \+ j) 0.021 " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 401 0 "" } {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 418 14 "Solution: c \n " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "series(42*sin(x^2),x=0,16);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#+-%\"xG\"#U\"\"#!\"(\"\"'#\"\"(\"#?\"#5#!\"\"\"$?\"\" #9-%\"OG6#\"\"\"\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "p : = convert( %, polynom );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,**& \"#U\"\"\")%\"xG\"\"#F(F(*&\"\"(F()F*\"\"'F(!\"\"*(F-F(\"#?F0F*\"#5F(* &\"$?\"F0F*\"#9F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "int(su bs(x=t, p), t = 0 .. x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"#9\" \"\")%\"xG\"\"$F&F&*$)F(\"\"(F&!\"\"*(F,F&\"$?#F-F(\"#6F&*&\"%+=F-F(\" #:F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(x=0.1, 14*x^3) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#9!\"$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "evalf(Int(42*sin(x^2),x = 0 .. 0.1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++***R\"!#6" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 272 33 "1 9. What is the coefficient of " }{XPPEDIT 258 0 "x^3;" "6#*$%\"xG\" \"$" }{TEXT 519 31 " in the Maclaurin series of " }{XPPEDIT 18 0 "x /(3+2*x);" "6#*&%\"xG\"\"\",&\"\"$F%*&\"\"#F%F$F%F%!\"\"" }{TEXT 520 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT 508 3 "a) " }{XPPEDIT 18 0 "-5/27;" "6#,$*&\"\"&\"\"\"\"#F!\"\"F(" }{TEXT 509 13 " b) " }{XPPEDIT 18 0 "-14/81;" "6#,$*&\"#9\"\"\"\"#\" )!\"\"F(" }{TEXT 510 11 " c) " }{XPPEDIT 19 1 "-4/27;" "6#,$*& \"\"%\"\"\"\"#F!\"\"F(" }{TEXT 511 12 " d) " }{XPPEDIT 18 0 "- 10/81;" "6#,$*&\"#5\"\"\"\"#\")!\"\"F(" }{TEXT 512 12 " e) " } {XPPEDIT 18 0 "-1/9;" "6#,$*&\"\"\"F%\"\"*!\"\"F'" }{TEXT 513 3 " \n " }{TEXT 514 4 "f) " }{XPPEDIT 18 0 "5/27;" "6#*&\"\"&\"\"\"\"#F!\"\" " }{TEXT 518 16 " g) " }{XPPEDIT 18 0 "14/81;" "6#*&\"#9\" \"\"\"#\")!\"\"" }{TEXT 517 15 " h) " }{XPPEDIT 18 0 "4/27; " "6#*&\"\"%\"\"\"\"#F!\"\"" }{TEXT 516 18 " i) " } {XPPEDIT 18 0 "10/81;" "6#*&\"#5\"\"\"\"#\")!\"\"" }{TEXT 515 18 " \+ j) " }{XPPEDIT 18 0 "1/9;" "6#*&\"\"\"F$\"\"*!\"\"" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 419 14 "Solution: h \n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "series(x/(3+2*x),x=0,8);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG#\"\"\"\"\"$F&#!\"#\"\"*\"\"## \"\"%\"#FF'#!\")\"#\")F-#\"#;\"$V#\"\"&#!#K\"$H(\"\"'#\"#k\"%(=#\"\"(- %\"OG6#F&\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "convert( \+ %, polynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&\"\"$!\"\"%\"xG\" \"\"F(*(\"\"#F(\"\"*F&F'F*F&*(\"\"%F(\"#FF&F'F%F(*(\"\")F(\"#\")F&F'F- F&*(\"#;F(\"$V#F&F'\"\"&F(*(\"#KF(\"$H(F&F'\"\"'F&*(\"#kF(\"%(=#F&F'\" \"(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "coeff(%,x^3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"%\"#F" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT 274 32 "20. What is the coefficient of " }{XPPEDIT 496 0 "x^3 ;" "6#*$%\"xG\"\"$" }{TEXT 495 32 " in the Maclaurin series of " } {XPPEDIT 498 0 "1/((1+x)^(1/3));" "6#*&\"\"\"F$),&F$F$%\"xGF$*&F$F$\" \"$!\"\"F*" }{TEXT 497 3 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 462 0 "" }}{PARA 260 "" 0 "" {TEXT 273 3 "a) " }{XPPEDIT 18 0 "-5/27; " "6#,$*&\"\"&\"\"\"\"#F!\"\"F(" }{TEXT 499 13 " b) " } {XPPEDIT 18 0 "-14/81;" "6#,$*&\"#9\"\"\"\"#\")!\"\"F(" }{TEXT 500 11 " c) " }{XPPEDIT 19 1 "-4/27;" "6#,$*&\"\"%\"\"\"\"#F!\"\"F(" } {TEXT 501 12 " d) " }{XPPEDIT 18 0 "-10/81;" "6#,$*&\"#5\"\"\" \"#\")!\"\"F(" }{TEXT 502 12 " e) " }{XPPEDIT 18 0 "-1/9;" "6# ,$*&\"\"\"F%\"\"*!\"\"F'" }{TEXT 503 2 " " }}{PARA 260 "" 0 "" {TEXT 463 4 "f) " }{XPPEDIT 18 0 "5/27;" "6#*&\"\"&\"\"\"\"#F!\"\"" }{TEXT 507 16 " g) " }{XPPEDIT 18 0 "14/81;" "6#*&\"#9\"\"\"\"#\" )!\"\"" }{TEXT 506 15 " h) " }{XPPEDIT 18 0 "4/27;" "6#*&\" \"%\"\"\"\"#F!\"\"" }{TEXT 505 18 " i) " }{XPPEDIT 18 0 "10/81;" "6#*&\"#5\"\"\"\"#\")!\"\"" }{TEXT 504 18 " j) \+ " }{XPPEDIT 18 0 "1/9;" "6#*&\"\"\"F$\"\"*!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 420 14 "Solution: b \n " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "binomial(-1/3,3); #From Newton's Binomial Seri es \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!#9\"#\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "series(1/(1+x)^(1/3), x = 0, 5); #direct \+ calculation" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"xG\"\"\"\"\"!#!\" \"\"\"$F%#\"\"#\"\"*F+#!#9\"#\")F)#\"#N\"$V#\"\"%-%\"OG6#F%\"\"&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "product(-1/3-k,k=0..2)/3!; # calculation of binomial coefficient" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 ##!#9\"#\")" }}}}}{MARK "17 0 2" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }