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261 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 280 40 " Math 132 \n Fall 2005 Final Exam" }}{PARA 0 "" 0 "" {TEXT 256 3 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 258 15 "1. Calculate " }{XPPEDIT 479 0 "int(x* sin(x^2),x = 0 .. sqrt(Pi));" "6#-%$intG6$*&%\"xG\"\"\"-%$sinG6#*$F'\" \"#F(/F';\"\"!-%%sqrtG6#%#PiG" }{TEXT 478 1 "." }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 484 4 "a) " } {XPPEDIT 265 0 "1;" "6#\"\"\"" }{TEXT 485 14 " b) " } {XPPEDIT 266 0 "2;" "6#\"\"#" }{TEXT 486 12 " c) " }{XPPEDIT 267 0 "3;" "6#\"\"$" }{TEXT 487 12 " d) " }{XPPEDIT 268 0 "4; " "6#\"\"%" }{TEXT 488 14 " e) " }{TEXT 483 1 " " }{XPPEDIT 269 0 "5;" "6#\"\"&" }{TEXT 482 19 " " }}{PARA 0 "" 0 "" {TEXT 481 4 "f) " }{XPPEDIT 270 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\" " }{TEXT 489 13 " g) " }{XPPEDIT 271 0 "3/2;" "6#*&\"\"$\"\" \"\"\"#!\"\"" }{TEXT 490 12 " h) " }{XPPEDIT 273 0 "5/2;" "6#* &\"\"&\"\"\"\"\"#!\"\"" }{TEXT 491 11 " i) " }{XPPEDIT 274 0 "s qrt(2)/2;" "6#*&-%%sqrtG6#\"\"#\"\"\"F'!\"\"" }{TEXT 492 12 " j) " }{XPPEDIT 275 0 "2*sqrt(2);" "6#*&\"\"#\"\"\"-%%sqrtG6#F$F%" } {TEXT 493 3 " " }}{PARA 3 "" 0 "" {TEXT 480 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 411 12 "Solution: a" } }{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(x*sin(x^2),x = 0 .. sqrt(Pi));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"JG-%$IntG6$*&%\"xG\"\"\"-%$sinG6#*$)F)\"\"#F*F*/F);\"\"!*$%#PiG# F*F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "K := changevar(x^2 \+ = u, J, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%$IntG6$,$*&#\" \"\"\"\"#F+-%$sinG6#%\"uGF+F+/F0;\"\"!%#PiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\" " }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 262 3 "2. " }{TEXT 282 5 "Let \+ " }{XPPEDIT 19 1 "F(x) = Int((8+t^5)/(1+t^2),t = x .. 10);" "6#/-%\"FG 6#%\"xG-%$IntG6$*&,&\"\")\"\"\"*$%\"tG\"\"&F.F.,&F.F.*$F0\"\"#F.!\"\"/ F0;F'\"#5" }{TEXT 558 60 ". Calculate the derivative D( F )( 2 ) \+ of F at 2. " }{TEXT 283 1 " " }}{PARA 3 "" 0 "" {TEXT 281 3 "a) \+ " }{XPPEDIT 330 0 "4;" "6#\"\"%" }{TEXT 325 17 " b) " } {XPPEDIT 331 0 "5;" "6#\"\"&" }{TEXT 326 18 " c) " } {XPPEDIT 332 0 "6;" "6#\"\"'" }{TEXT 327 17 " d) " } {XPPEDIT 333 0 "7;" "6#\"\"(" }{TEXT 328 18 " e) " } {XPPEDIT 334 0 "8;" "6#\"\")" }{TEXT 329 14 " \nf) " } {XPPEDIT 339 0 "-4;" "6#,$\"\"%!\"\"" }{TEXT 335 15 " g) " }{XPPEDIT 340 0 "-5;" "6#,$\"\"&!\"\"" }{TEXT 336 13 " h) " } {XPPEDIT 341 0 "-6;" "6#,$\"\"'!\"\"" }{TEXT 337 16 " i) \+ " }{XPPEDIT 342 0 "-7;" "6#,$\"\"(!\"\"" }{TEXT 338 15 " j) " }{XPPEDIT 344 0 "-8;" "6#,$\"\")!\"\"" }{TEXT 343 3 " " }{TEXT 345 6 " \n" }{TEXT 412 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 413 13 "Solution: j\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "F := (x) -> - Int((8+t^5)/(1+t^2), t = 10 .. x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,$-%$IntG6$*&,&\"\")\"\"\"*$)%\"tG\"\"&F3F3F3,&F3F3*$)F6\"\"#F3F3! \"\"/F6;\"#59$F " 0 "" {MPLTEXT 1 0 8 "D( F)(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\")" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "subs( t = 2, integrand(F(x)) );" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 257 19 "3. C alculate " }{XPPEDIT 436 1 "Int((5*x+12)/(x+2)/(x+3),x = 0 .. 1); " "6#-%$IntG6$*(,&*&\"\"&\"\"\"%\"xGF*F*\"#7F*F*,&F+F*\"\"#F*!\"\",&F+ F*\"\"$F*F//F+;\"\"!F*" }{TEXT 435 4 " . " }{TEXT 347 2 "\n\n" } {TEXT 346 4 "a) " }{XPPEDIT 356 0 "ln(4/3);" "6#-%#lnG6#*&\"\"%\"\"\" \"\"$!\"\"" }{TEXT 348 12 " b) " }{XPPEDIT 357 0 "ln(8/3);" "6 #-%#lnG6#*&\"\")\"\"\"\"\"$!\"\"" }{TEXT 349 13 " c) " } {XPPEDIT 358 0 "ln(16/3);" "6#-%#lnG6#*&\"#;\"\"\"\"\"$!\"\"" }{TEXT 350 11 " d) " }{XPPEDIT 359 0 "ln(25/3);" "6#-%#lnG6#*&\"#D\"\" \"\"\"$!\"\"" }{TEXT 351 10 " e) " }{TEXT 294 1 " " }{XPPEDIT 360 0 "ln(28/3);" "6#-%#lnG6#*&\"#G\"\"\"\"\"$!\"\"" }{TEXT 286 19 " \+ " }}{PARA 0 "" 0 "" {TEXT 285 4 "f) " }{XPPEDIT 361 0 "ln(25/9);" "6#-%#lnG6#*&\"#D\"\"\"\"\"*!\"\"" }{TEXT 352 12 " \+ g) " }{XPPEDIT 362 0 "ln(32/9);" "6#-%#lnG6#*&\"#K\"\"\"\"\"*!\"\" " }{TEXT 353 11 " h) " }{XPPEDIT 363 0 "ln(35/6);" "6#-%#lnG6#* &\"#N\"\"\"\"\"'!\"\"" }{TEXT 354 10 " i) " }{XPPEDIT 364 0 "ln( 35/9);" "6#-%#lnG6#*&\"#N\"\"\"\"\"*!\"\"" }{TEXT 355 11 " j) \+ " }{XPPEDIT 365 0 "ln(36/5);" "6#-%#lnG6#*&\"#O\"\"\"\"\"&!\"\"" } {TEXT 366 3 " " }}{PARA 3 "" 0 "" {TEXT 284 1 " " }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 414 13 "Solution: c\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "J := Int( (5*x+12)/(x+2)/(x+3),x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"JG-%$IntG6$*(,&*&\"\"&\"\"\"%\"xGF,F,\"#7F,F,,&F-F,\"\"#F,!\"\",&F- F,\"\"$F,F1/F-;\"\"!F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A := convert(integrand(J), parfrac, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,&*&\"\"#\"\"\",&%\"xGF(F'F(!\"\"F(*&\"\"$F(,&F*F(F-F(F+F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "B := int(A, x = 0 .. 1) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,&-%#lnG6#\"\"$!\"\"*&\"\"% \"\"\"-F'6#\"\"#F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "com bine( B , ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6##\"\"$\"#; !\"\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 320 15 "4. Calculate " } {XPPEDIT 566 1 "int((1+3*x^2+2*x)/(1+x)/(1+x^2),x = 0 .. 1);" "6#-%$in tG6$*(,(\"\"\"F(*&\"\"$F(*$%\"xG\"\"#F(F(*&F-F(F,F(F(F(,&F(F(F,F(!\"\" ,&F(F(*$F,F-F(F0/F,;\"\"!F(" }{TEXT 565 7 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 322 3 "a) " }{TEXT 438 2 " " } {XPPEDIT 19 1 "Pi/4+ln(2);" "6#,&*&%#PiG\"\"\"\"\"%!\"\"F&-%#lnG6#\"\" #F&" }{TEXT 439 8 " b) " }{XPPEDIT 19 1 "Pi/4+2*ln(2);" "6#,&*&%#P iG\"\"\"\"\"%!\"\"F&*&\"\"#F&-%#lnG6#F*F&F&" }{TEXT 440 10 " c) \+ " }{XPPEDIT 19 1 "Pi/4+ln(3);" "6#,&*&%#PiG\"\"\"\"\"%!\"\"F&-%#lnG6# \"\"$F&" }{TEXT 441 10 " d) " }{XPPEDIT 19 1 "ln(2);" "6#-%#lnG6 #\"\"#" }{TEXT 442 9 " e)" }{TEXT 323 1 " " }{XPPEDIT 19 1 "2*ln (2);" "6#*&\"\"#\"\"\"-%#lnG6#F$F%" }{TEXT 443 1 " " }{TEXT 444 11 " \+ " }}{PARA 0 "" 0 "" {TEXT 321 5 "f) " }{XPPEDIT 19 1 "4*ln( 2);" "6#*&\"\"%\"\"\"-%#lnG6#\"\"#F%" }{TEXT 437 14 " g) " } {XPPEDIT 19 1 "8*ln(2);" "6#*&\"\")\"\"\"-%#lnG6#\"\"#F%" }{TEXT 445 11 " " }{TEXT 446 5 " h) " }{XPPEDIT 19 1 "ln(3);" "6#-%#ln G6#\"\"$" }{TEXT 447 19 " i) " }{XPPEDIT 19 1 "2*ln(3); " "6#*&\"\"#\"\"\"-%#lnG6#\"\"$F%" }{TEXT 448 10 " j) " } {XPPEDIT 19 1 "ln(6);" "6#-%#lnG6#\"\"'" }{TEXT 449 2 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 415 13 "Solution: e\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "J := Int( (1+3*x^2+2*x)/(1+x)/(1+x^2),x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*(,(\"\"\"F**&\"\"$F*)%\"xG\"\"#F*F**&F/F*F.F*F*F *,&F*F*F.F*!\"\",&F*F**$F-F*F*F2/F.;\"\"!F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A := convert( integrand(J), parfrac, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,&*&\"\"\"F',&F'F'%\"xGF'!\"\"F'*(\"\" #F'F)F',&F'F'*$)F)F,F'F'F*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "int(A, x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"# \"\"\"-%#lnG6#F%F&F&" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 298 15 "5. Ca lculate " }{XPPEDIT 568 1 "Int(1/sqrt(16+x^2),x = 0 .. 3);" "6#-%$Int G6$*&\"\"\"F'-%%sqrtG6#,&\"#;F'*$%\"xG\"\"#F'!\"\"/F.;\"\"!\"\"$" } {TEXT 567 1 "." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 4 "a) " } {XPPEDIT 310 0 "Pi;" "6#%#PiG" }{TEXT 304 18 " b) " } {XPPEDIT 311 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 305 16 " \+ c) " }{TEXT 456 1 " " }{XPPEDIT 256 0 "Pi/3;" "6#*&%#PiG\"\" \"\"\"$!\"\"" }{TEXT 457 18 " d) " }{XPPEDIT 312 0 "Pi/4 ;" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 309 15 " e) " } {XPPEDIT 313 0 "Pi/6;" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 303 2 " " } {TEXT 302 20 " \n" }{TEXT 299 5 "f) " }{XPPEDIT 314 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT 308 12 " g) " } {XPPEDIT 315 0 "ln(3);" "6#-%#lnG6#\"\"$" }{TEXT 307 7 " " } {TEXT 316 1 " " }{TEXT 300 3 "h) " }{XPPEDIT 257 0 "2*ln(2);" "6#*&\" \"#\"\"\"-%#lnG6#F$F%" }{TEXT 455 13 " i) " }{XPPEDIT 317 0 " ln(5);" "6#-%#lnG6#\"\"&" }{TEXT 306 12 " j) " }{XPPEDIT 318 0 "2*ln(3);" "6#*&\"\"#\"\"\"-%#lnG6#\"\"$F%" }{TEXT 319 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 416 13 "Solution: f\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "J := Int(1/sqrt(16+x^2),x=0..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&\"\"\"F)*$,&\"#;F)*$)%\"xG\"\"#F)F)#F)F0!\"\"/F /;\"\"!\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "K := studen t[changevar](x=4*tan(t), J, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"KG-%$IntG6$*&,&\"#;\"\"\"*&F*F+)-%$tanG6#%\"tG\"\"#F+F+#!\"\"F2,&\" \"%F+*&F6F+F-F+F+F+/F1;\"\"!-%'arctanG6##\"\"$F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "simplify(student[integrand](K)) assuming t>0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,&\"\"\"F%*$)-%$tanG6#%\"tG\"\"# F%F%#F%F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A := int(sec(t ), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%#lnG6#,&-%$secG6#%\" tG\"\"\"-%$tanGF+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "B := subs(t = arctan(3/4), A) - subs(t = 0, A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,&-%#lnG6#,&-%$secG6#-%'arctanG6##\"\"$\"\"%\"\" \"-%$tanGF,F3F3-F'6#,&-F+6#\"\"!F3-F5F:F3!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "simplify(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%#lnG6#\"\"#" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 290 4 "6. " } {TEXT 450 40 "What is the logarithmic derivative of " }{TEXT 730 1 " " }{XPPEDIT 731 1 "y(t) = 2^(k*t);" "6#/-%\"yG6#%\"tG)\"\"#*&%\"kG\" \"\"F'F," }{TEXT 729 3 " ? " }{TEXT 451 1 "\n" }{TEXT 289 1 " " }} {PARA 3 "" 0 "" {TEXT 288 4 "a) " }{XPPEDIT 257 0 "k;" "6#%\"kG" } {TEXT 569 21 " b) " }{XPPEDIT 373 0 "1/k;" "6#*&\"\" \"F$%\"kG!\"\"" }{TEXT 367 16 " c) " }{XPPEDIT 374 0 "k*t; " "6#*&%\"kG\"\"\"%\"tGF%" }{TEXT 368 18 " d) " } {XPPEDIT 375 0 "t/k;" "6#*&%\"tG\"\"\"%\"kG!\"\"" }{TEXT 369 28 " \+ e) " }{XPPEDIT 376 0 "k/ln(2);" "6#*&%\"kG\"\"\"-% #lnG6#\"\"#!\"\"" }{TEXT 370 8 " \nf) " }{XPPEDIT 256 0 "k*ln(2);" "6#*&%\"kG\"\"\"-%#lnG6#\"\"#F%" }{TEXT 454 10 " g) " }{XPPEDIT 257 0 "ln(k);" "6#-%#lnG6#%\"kG" }{TEXT 453 13 " h) " } {XPPEDIT 377 0 "2*ln(k);" "6#*&\"\"#\"\"\"-%#lnG6#%\"kGF%" }{TEXT 371 10 " i) " }{XPPEDIT 378 0 "ln(2)*2^(k*t);" "6#*&-%#lnG6#\"\"#\" \"\")F'*&%\"kGF(%\"tGF(F(" }{TEXT 372 11 " j) " }{XPPEDIT 257 0 "2^(k*t)/ln(2);" "6#*&)\"\"#*&%\"kG\"\"\"%\"tGF(F(-%#lnG6#F%!\"\"" } {TEXT 452 3 " " }{TEXT 287 1 "\n" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 432 11 "Solution: f" }}{PARA 0 "" 0 "" {TEXT 433 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "diff(ln(2^(k*t)),t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"kG\"\"\"-%#lnG6#\"\"#F%" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT 295 13 "7. Calculate " }{XPPEDIT 571 1 " Int(1/sqrt(1-x^2),x = 1/2 .. sqrt(3)/2);" "6#-%$IntG6$*&\"\"\"F'-%%sqr tG6#,&F'F'*$%\"xG\"\"#!\"\"F//F-;*&F'F'F.F/*&-F)6#\"\"$F'F.F/" }{TEXT 570 1 "." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 574 4 "a) " } {XPPEDIT 268 0 "Pi;" "6#%#PiG" }{TEXT 577 17 " b) " } {XPPEDIT 269 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 578 15 " \+ c) " }{TEXT 586 1 " " }{XPPEDIT 256 0 "Pi/3;" "6#*&%#PiG\"\"\" \"\"$!\"\"" }{TEXT 587 15 " d) " }{XPPEDIT 270 0 "Pi/4;" "6 #*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 582 15 " e) " }{XPPEDIT 271 0 "Pi/6;" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 576 2 " " }{TEXT 575 20 " \n" }{TEXT 572 5 "f) " }{XPPEDIT 272 0 "5 *Pi/6;" "6#*(\"\"&\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT 581 13 " g) \+ " }{XPPEDIT 273 0 "5*Pi/12;" "6#*(\"\"&\"\"\"%#PiGF%\"#7!\"\"" } {TEXT 580 7 " " }{TEXT 583 1 " " }{TEXT 573 3 "h) " }{XPPEDIT 257 0 "2*Pi/3;" "6#*(\"\"#\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 585 14 " \+ i) " }{XPPEDIT 275 0 "3*Pi/4;" "6#*(\"\"$\"\"\"%#PiGF%\"\"%!\" \"" }{TEXT 579 13 " j) " }{XPPEDIT 276 0 "Pi/12;" "6#*&%#PiG \"\"\"\"#7!\"\"" }{TEXT 584 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 430 11 "Solution: e" }}{PARA 0 "" 0 "" {TEXT 431 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A := int(1/sqrt(1-x^2), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"AG-%'arcsinG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "B := subs(x = sqrt(3)/2, A) - subs(x = 1/2, A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,&-%'arcsinG6#,$*&\"\"#!\"\"\"\"$#\"\"\"F+F/F/-F' 6#F.F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify( B );" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"'!\"\"%#PiG\"\"\"F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 1 " " }{TEXT 292 3 "8. " }{TEXT 665 3 "Let" }{TEXT 668 2 " \+ " }{XPPEDIT 260 1 "f(x) = x^(2*x);" "6#/-%\"fG6#%\"xG)F'*&\"\"#\"\"\" F'F+" }{TEXT 667 2 ". " }{TEXT 666 2 " " }{TEXT 669 63 "What is D( f )(1/2) ? (The derivative of f(x) at x = 1/2)" }{TEXT 670 2 " ?" }}{PARA 0 "" 0 "" {TEXT 291 17 " " }}{PARA 0 "" 0 "" {TEXT 296 5 "a) " }{XPPEDIT 388 0 "2*ln(2)+1;" "6#,&*&\"\"#\"\"\"-%# lnG6#F%F&F&F&F&" }{TEXT 383 12 " b) " }{XPPEDIT 389 0 "3*ln(3) +2;" "6#,&*&\"\"$\"\"\"-%#lnG6#F%F&F&\"\"#F&" }{TEXT 384 12 " c ) " }{XPPEDIT 390 0 "ln(2)+2;" "6#,&-%#lnG6#\"\"#\"\"\"F'F(" }{TEXT 385 13 " d) " }{XPPEDIT 391 0 "ln(3)+3;" "6#,&-%#lnG6#\"\"$\" \"\"F'F(" }{TEXT 386 17 " e) " }{XPPEDIT 392 0 "ln(2)+3; " "6#,&-%#lnG6#\"\"#\"\"\"\"\"$F(" }{TEXT 387 8 " \nf) " }{XPPEDIT 393 0 "-ln(3)+2;" "6#,&-%#lnG6#\"\"$!\"\"\"\"#\"\"\"" }{TEXT 382 14 " \+ g) " }{XPPEDIT 394 0 "-ln(2)+1;" "6#,&-%#lnG6#\"\"#!\"\"\"\" \"F)" }{TEXT 381 11 " h) " }{XPPEDIT 395 0 "-ln(3)+1;" "6#,&-%# lnG6#\"\"$!\"\"\"\"\"F)" }{TEXT 380 11 " i) " }{XPPEDIT 396 0 " -3*ln(2)+1;" "6#,&*&\"\"$\"\"\"-%#lnG6#\"\"#F&!\"\"F&F&" }{TEXT 379 11 " j) " }{XPPEDIT 397 0 "-3*ln(3)+1;" "6#,&*&\"\"$\"\"\"-%#ln G6#F%F&!\"\"F&F&" }{TEXT 297 1 "\n" }{TEXT 398 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 428 11 "Solution: g" }}{PARA 0 "" 0 "" {TEXT 429 1 " \+ " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := x -> x^(2*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF()9$,$*&\"\"#\"\"\"F-F1F1F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "eqn := ln('f(x)') = ln(f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%#lnG6#-%\"fG6#%\"xG-F'6#)F,,$*&\"\"#\"\" \"F,F3F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqn2 := Diff(f( x),x)/'f(x)' = diff(rhs( eqn ), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%eqn2G/*&-%%DiffG6$)%\"xG,$*&\"\"#\"\"\"F+F/F/F+F/-%\"fG6#F+!\"\",& *&F.F/-%#lnGF2F/F/F.F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "e qn3 := Diff(f(x),x) = solve(eqn2, Diff(f(x),x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%%DiffG6$)%\"xG,$*&\"\"#\"\"\"F*F.F.F*,&*(F-F .-%#lnG6#F*F.F)F.F.*&F-F.F)F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Answer := subs( x = 1/2, rhs(eqn3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AnswerG,&-%#lnG6##\"\"\"\"\"#F*F*F*" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 698 2 "9." }{TEXT -1 1 " " }{TEXT 699 39 "Consid er the following three statements" }{TEXT 700 16 " about a series " } {TEXT 702 1 " " }{XPPEDIT 261 1 "sum(a[n],n = 1 .. infinity);" "6#-%$s umG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" }{TEXT 701 55 " with positiv e terms:\nI: The series converges because" }{TEXT 704 3 " " } {XPPEDIT 264 1 "limit(a[n],n = infinity) = 0;" "6#/-%&limitG6$&%\"aG6# %\"nG/F*%)infinityG\"\"!" }{TEXT 703 35 ".\n\nII: The series converges because" }{TEXT 706 3 " " }{XPPEDIT 257 1 "limit(a[n]/b[n],n = infi nity) = 1;" "6#/-%&limitG6$*&&%\"aG6#%\"nG\"\"\"&%\"bG6#F+!\"\"/F+%)in finityGF," }{TEXT 705 6 " and " }{TEXT 707 1 " " }{XPPEDIT 256 1 "sum (b[n],n = 1 .. infinity);" "6#-%$sumG6$&%\"bG6#%\"nG/F);\"\"\"%)infini tyG" }{TEXT 708 47 " converges.\n\nIII: The series converges because " }{TEXT 726 3 " " }{XPPEDIT 257 1 "limit(a[n]^(1/n),n = infinity) = 1;" "6#/-%&limitG6$)&%\"aG6#%\"nG*&\"\"\"F-F+!\"\"/F+%)infinityGF-" } {TEXT 725 97 ".\n\nFor each statement, determine whether the reasoning is correct ( $ ) or incorrect ( @ ). \n " }}{PARA 261 "" 0 "" {TEXT -1 6 "a) I: " }{TEXT 709 1 "$" }{TEXT -1 7 ", II: " }{TEXT 710 10 "$, III: $" }{TEXT -1 18 " \nb) I: " }{TEXT 711 1 "$" } {TEXT -1 7 ", II: " }{TEXT 712 9 "$, III:@" }{TEXT -1 12 " \nc) \+ I: " }{TEXT 713 1 "$" }{TEXT -1 7 ", II: " }{TEXT 716 10 "@, III: $ " }{TEXT -1 13 " \nd) I: " }{TEXT 714 1 "$" }{TEXT -1 7 ", II: \+ " }{TEXT 717 10 "@, III: @" }{TEXT -1 15 " \ne) I:" }{TEXT 719 1 "@" }{TEXT -1 6 ", II: " }{TEXT 718 10 "$, III: $" }{TEXT -1 23 " \nf) I: " }{TEXT 720 1 "@" }{TEXT -1 7 ", II: " } {TEXT 715 10 "$, III: @" }{TEXT -1 14 " \ng) I: " }{TEXT 724 1 "@" }{TEXT -1 7 ", II: " }{TEXT 723 10 "@, III: $" }{TEXT -1 13 " \+ \nh) I: " }{TEXT 721 1 "@" }{TEXT -1 6 ", II: " }{TEXT 722 9 "@, II I: @" }{TEXT -1 52 " \ni) Wrong answer \n j) Bonus wrong an swer" }}{PARA 3 "" 0 "" {TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT 728 1 " \n" }{TEXT 727 12 "Solution: f " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 259 44 "10. Consider the following three statements" }{TEXT 458 16 " about a series " }{TEXT 672 1 " " }{XPPEDIT 673 1 "sum(a[n],n = 1 .. infinity);" "6#-%$sumG6$&%\"aG6#%\"nG/F);\"\"\"%)infinityG" } {TEXT 671 55 " with positive terms:\nI: The series converges because " }{TEXT 675 3 " " }{XPPEDIT 676 1 "a[n] < 1/n;" "6#2&%\"aG6#%\"nG*& \"\"\"F)F'!\"\"" }{TEXT 674 35 ".\n\nII: The series converges because " }{TEXT 677 3 " " }{XPPEDIT 259 1 "a[n] < (n/(1+2*n))^n;" "6#2&%\"a G6#%\"nG)*&F'\"\"\",&F*F**&\"\"#F*F'F*F*!\"\"F'" }{TEXT 678 36 ".\n\nI II: The series converges because" }{TEXT 697 3 " " }{XPPEDIT 259 1 " limit(a[n+1]/a[n],n = infinity) < 1;" "6#2-%&limitG6$*&&%\"aG6#,&%\"nG \"\"\"F-F-F-&F)6#F,!\"\"/F,%)infinityGF-" }{TEXT 696 97 ".\n\nFor each statement, determine whether the reasoning is correct ( $ ) or incor rect ( @ ). \n " }}{PARA 261 "" 0 "" {TEXT -1 6 "a) I: " }{TEXT 680 1 "$" }{TEXT -1 7 ", II: " }{TEXT 681 10 "$, III: $" }{TEXT -1 18 " \+ \nb) I: " }{TEXT 682 1 "$" }{TEXT -1 7 ", II: " }{TEXT 683 9 "$, III:@" }{TEXT -1 12 " \nc) I: " }{TEXT 684 1 "$" }{TEXT -1 7 ", II: " }{TEXT 687 10 "@, III: $" }{TEXT -1 13 " \nd) I: " } {TEXT 685 1 "$" }{TEXT -1 7 ", II: " }{TEXT 688 10 "@, III: @" } {TEXT -1 15 " \ne) I:" }{TEXT 690 1 "@" }{TEXT -1 6 ", II: " } {TEXT 689 10 "$, III: $" }{TEXT -1 23 " \nf) I: " } {TEXT 691 1 "@" }{TEXT -1 7 ", II: " }{TEXT 686 10 "$, III: @" } {TEXT -1 14 " \ng) I: " }{TEXT 695 1 "@" }{TEXT -1 7 ", II: " } {TEXT 694 10 "@, III: $" }{TEXT -1 13 " \nh) I: " }{TEXT 692 1 " @" }{TEXT -1 6 ", II: " }{TEXT 693 9 "@, III: @" }{TEXT -1 52 " \+ \ni) Wrong answer \n j) Bonus wrong answer" }}{PARA 0 "" 0 "" {TEXT 679 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 " " 0 "" {TEXT 426 11 "Solution: e" }}{PARA 0 "" 0 "" {TEXT 427 1 " " } {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 476 5 "11. " }{TEXT 644 41 "Consider the three series \n I: " }{XPPEDIT 257 1 "S um(n/(ln(n)^3),n = 2 .. infinity);" "6#-%$SumG6$*&%\"nG\"\"\"*$-%#lnG6 #F'\"\"$!\"\"/F';\"\"#%)infinityG" }{TEXT 645 15 " , II: " } {XPPEDIT 256 1 "Sum(7^n/n!,n = 0 .. infinity);" "6#-%$SumG6$*&)\"\"(% \"nG\"\"\"-%*factorialG6#F)!\"\"/F);\"\"!%)infinityG" }{TEXT 646 22 " \+ , and III: " }{XPPEDIT 256 1 "Sum(n^2/(2^n),n = 0 .. infinit y);" "6#-%$SumG6$*&%\"nG\"\"#)F(F'!\"\"/F';\"\"!%)infinityG" }{TEXT 650 2 " \n" }{TEXT 647 151 "and the statements\n\n( $ ) The series co nverges \n( @ ) The series diverges\n\nFor each series, decide which o f statements ($), (&), (@) is correct. \n" }}{PARA 261 "" 0 "" {TEXT -1 6 "a) I: " }{TEXT 648 1 "$" }{TEXT -1 7 ", II: " }{TEXT 649 10 "$, III: $" }{TEXT -1 18 " \nb) I: " }{TEXT 651 1 "$" } {TEXT -1 7 ", II: " }{TEXT 652 9 "$, III:@" }{TEXT -1 12 " \nc) \+ I: " }{TEXT 653 1 "$" }{TEXT -1 7 ", II: " }{TEXT 656 10 "@, III: $ " }{TEXT -1 13 " \nd) I: " }{TEXT 654 1 "$" }{TEXT -1 7 ", II: \+ " }{TEXT 657 10 "@, III: @" }{TEXT -1 15 " \ne) I:" }{TEXT 659 1 "@" }{TEXT -1 6 ", II: " }{TEXT 658 10 "$, III: $" }{TEXT -1 23 " \nf) I: " }{TEXT 660 1 "@" }{TEXT -1 7 ", II: " } {TEXT 655 10 "$, III: @" }{TEXT -1 14 " \ng) I: " }{TEXT 664 1 "@" }{TEXT -1 7 ", II: " }{TEXT 663 10 "@, III: $" }{TEXT -1 13 " \+ \nh) I: " }{TEXT 661 1 "@" }{TEXT -1 6 ", II: " }{TEXT 662 9 "@, II I: @" }{TEXT -1 52 " \ni) Wrong answer \n j) Bonus wrong an swer" }}{PARA 3 "" 0 "" {TEXT 643 1 "\n" }{TEXT 477 10 "Solution: " } {TEXT -1 1 "e" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 293 5 "12. " }{TEXT 621 39 "Consider the two series \n I: " } {XPPEDIT 259 1 "Sum((-1)^n*ln(n)/(n^2),n = 3 .. infinity);" "6#-%$SumG 6$*(),$\"\"\"!\"\"%\"nGF)-%#lnG6#F+F)*$F+\"\"#F*/F+;\"\"$%)infinityG" }{TEXT 622 15 " and II: " }{XPPEDIT 256 1 "Sum((-1)^n*n/(1+n^(3/ 2)),n = 0 .. infinity);" "6#-%$SumG6$*(),$\"\"\"!\"\"%\"nGF)F+F),&F)F) )F+*&\"\"$F)\"\"#F*F)F*/F+;\"\"!%)infinityG" }{TEXT 623 2 " \n" } {TEXT 624 198 "and the statements\n( $ ) The series converges absolut ely\n( & ) The series converges conditionally\n( @ ) The series diverg es\n\nFor each series, decide which of statements ($), (&), (@) is co rrect. " }}{PARA 261 "" 0 "" {TEXT -1 6 "a) I: " }{TEXT 625 1 "$" } {TEXT -1 7 ", II: " }{TEXT 626 1 "$" }{TEXT -1 16 " b) I: " }{TEXT 627 1 "$" }{TEXT -1 7 ", II: " }{TEXT 642 1 "&" }{TEXT -1 10 " c) I: " }{TEXT 628 1 "$" }{TEXT -1 7 ", II: " }{TEXT 629 1 "@" } {TEXT -1 12 " d) I: " }{TEXT 631 1 "&" }{TEXT -1 7 ", II: " } {TEXT 630 1 "$" }{TEXT -1 14 " e) I: " }{TEXT 632 1 "&" }{TEXT -1 7 ", II: " }{TEXT 633 1 "&" }{TEXT -1 21 " \nf) I: " }{TEXT 634 1 "&" }{TEXT -1 7 ", II: " }{TEXT 635 1 "@" }{TEXT -1 12 " g) I: " }{TEXT 637 1 "@" }{TEXT -1 7 " , II: " }{TEXT 636 1 "$" }{TEXT -1 12 " h) I: " }{TEXT 638 1 "@" }{TEXT -1 7 " , II: " } {TEXT 639 1 "&" }{TEXT -1 12 " i) I: " }{TEXT 640 1 "@" }{TEXT -1 7 " , II: " }{TEXT 641 1 "@" }{TEXT -1 22 " j) Wrong answer" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 425 12 "Solution: b " }{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 276 43 "13. Consider the two series \+ \n I: " }{XPPEDIT 600 1 "Sum(n!/(2*n)!,n = 0 .. infinity);" "6#-%$SumG6$*&-%*factorialG6#%\"nG\"\"\"-F(6#*&\"\"#F+F*F+!\"\"/F*;\" \"!%)infinityG" }{TEXT 599 15 " and II: " }{XPPEDIT 256 1 "Sum(1 /n,n = 0 .. infinity);" "6#-%$SumG6$*&\"\"\"F'%\"nG!\"\"/F(;\"\"!%)inf inityG" }{TEXT 601 2 " \n" }{TEXT 602 262 "and the statements\n( $ ) \+ The test establishes convergence\n( & ) The test establishes divergenc e\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 statemen ts ($), (&), (@) is correct. " }}{PARA 261 "" 0 "" {TEXT -1 6 "a) I: " }{TEXT 603 1 "$" }{TEXT -1 7 ", II: " }{TEXT 604 1 "$" }{TEXT -1 16 " b) I: " }{TEXT 605 1 "$" }{TEXT -1 7 ", II: " }{TEXT 620 1 "&" }{TEXT -1 10 " c) I: " }{TEXT 606 1 "$" }{TEXT -1 7 ", I I: " }{TEXT 607 1 "@" }{TEXT -1 12 " d) I: " }{TEXT 609 1 "&" } {TEXT -1 7 ", II: " }{TEXT 608 1 "$" }{TEXT -1 14 " e) I: " } {TEXT 610 1 "&" }{TEXT -1 7 ", II: " }{TEXT 611 1 "&" }{TEXT -1 21 " \+ \nf) I: " }{TEXT 612 1 "&" }{TEXT -1 7 ", II: " }{TEXT 613 1 "@" }{TEXT -1 12 " g) I: " }{TEXT 615 1 "@" }{TEXT -1 7 " , II: " }{TEXT 614 1 "$" }{TEXT -1 12 " h) I: " }{TEXT 616 1 "@" } {TEXT -1 7 " , II: " }{TEXT 617 1 "&" }{TEXT -1 12 " i) I: " } {TEXT 618 1 "@" }{TEXT -1 7 " , II: " }{TEXT 619 1 "@" }{TEXT -1 22 " \+ j) Wrong answer" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 460 12 "Solution: c \+ " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 " " 0 "" {TEXT 261 11 " 14. Let " }{XPPEDIT 595 1 "f(x) = sum((n-1)*(x /2)^n,n = 0 .. infinity);" "6#/-%\"fG6#%\"xG-%$sumG6$*&,&%\"nG\"\"\"F. !\"\"F.)*&F'F.\"\"#F/F-F./F-;\"\"!%)infinityG" }{TEXT 588 5 " . " } {TEXT 459 10 "What is " }{XPPEDIT 597 1 "f^` (4)`;" "6#)%\"fG%%~(4)G " }{TEXT 596 6 " ( 0 )" }{TEXT 598 2 " ?" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 324 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 589 69 "a) \+ 1 b) 3/2 c) 2 d) 5/2 e) 3 " } {TEXT 592 3 " " }{TEXT 591 18 " " }}{PARA 0 "" 0 " " {TEXT 590 65 "f) 7/2 g) 4 h) 9/2 i) 5 \+ j)" }{TEXT -1 1 " " }{TEXT 594 2 " 1" }{TEXT 593 4 "1/2 " } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 423 11 "Solution: h" }} {PARA 0 "" 0 "" {TEXT 424 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "a := n -> (n-1)/2^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*&,&9$\"\"\"F/ !\"\"F/)\"\"#F.F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " Answer := simplify(4!*a(4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Ans werG,$*&\"#C\"\"\"-%\"aG6#\"\"%F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "p := x -> sum((n-1)*(x/2)^n,n = 0 .. 10); \n#Any upp er summation index greater than 3 will work as well as 10" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$sum G6$*&,&%\"nG\"\"\"F2!\"\"F2),$*&#F2\"\"#F29$F2F2F1F2/F1;\"\"!\"#5F(F(F (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Direct_differentiation _answer := (D@@4)(p)(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"*\"\" #" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 559 5 "15. " }{TEXT 561 6 "Let " }{XPPEDIT 287 1 "T(x) = a[0]+a[1]*x+a[2]*x^2+a[3]*x^3;" "6#/-%\"T G6#%\"xG,*&%\"aG6#\"\"!\"\"\"*&&F*6#F-F-F'F-F-*&&F*6#\"\"#F-*$F'F4F-F- *&&F*6#\"\"$F-*$F'F9F-F-" }{TEXT 560 2 " " }{TEXT 562 44 " be the deg ree 3 Taylor polynomial \nof " }{XPPEDIT 19 1 "f(x) = x^2+sin(3*x );" "6#/-%\"fG6#%\"xG,&*$F'\"\"#\"\"\"-%$sinG6#*&\"\"$F+F'F+F+" } {TEXT 564 32 " centered about 0. What is " }{XPPEDIT 289 1 "T(1/2 );" "6#-%\"TG6#*&\"\"\"F'\"\"#!\"\"" }{TEXT 563 4 " ?\n " }}{PARA 0 " " 0 "" {TEXT 277 87 "a) 11/16 b) 3/4 c) 13/16 d) 7/8 e) 15/16 " }{TEXT 410 3 " " }{TEXT 279 18 " " }}{PARA 0 "" 0 "" {TEXT 278 80 "f) 17/16 \+ g) 9/8 h) 19/16 i) 5/4 j)" } {TEXT -1 3 " " }{TEXT 463 6 "21/16 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 " " 0 "" {TEXT 421 12 "Solution: h " }}{PARA 0 "" 0 "" {TEXT 422 1 " " } {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "S := series( x^2+sin(3*x), x = 0, 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG++% \"xG\"\"$\"\"\"F(\"\"##!\"*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$F/F/*$)F0\"\"#F/F/*&#\"\"*F3F/*$)F0F.F/F/!\"\"F(F(F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#>\"#;" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 267 5 "16. " }{TEXT 465 43 "Calculate the interval of convergence of " } {XPPEDIT 734 1 "Sum((x+1)^n/(n+1)/(2^n),n = 0 .. infinity);" "6#-%$Sum G6$*(),&%\"xG\"\"\"F*F*%\"nGF*,&F+F*F*F*!\"\")\"\"#F+F-/F+;\"\"!%)infi nityG" }{TEXT 464 2 " ." }}{PARA 0 "" 0 "" {TEXT 466 14 "a) ( - 3 , 1 " }{TEXT 534 1 "]" }{TEXT 535 14 " b) " }{TEXT 540 1 "[" } {TEXT 541 1 " " }{TEXT 542 9 " - 3 , 1 " }{TEXT 538 1 "]" }{TEXT 539 24 " c) ( - 1 , 3 " }{TEXT 543 1 "]" }{TEXT 544 13 " \+ d) " }{TEXT 549 1 "[" }{TEXT 550 1 " " }{TEXT 551 9 " - 1 , 3 " } {TEXT 547 1 "]" }{TEXT 548 12 " e) " }{TEXT 552 2 " [" }{TEXT 553 9 " - 2 , 2 " }{TEXT 554 1 "]" }{TEXT 555 1 " " }{TEXT 470 1 " " } {TEXT 469 18 " " }}{PARA 0 "" 0 "" {TEXT 467 4 "f) \+ " }{TEXT 536 1 "[" }{TEXT 537 45 " - 3 , 1 ) g) ( - 3 , 1 ) \+ " }{TEXT -1 1 " " }{TEXT 468 4 "h) " }{TEXT 545 1 "[" } {TEXT 546 54 " - 1 , 3 ) i) ( - 1 , 3 ) j) ( " }{XPPEDIT 557 1 "-infinity*` `,infinity;" "6$,$*&%)infinityG\"\"\"%\"~ GF&!\"\"F%" }{TEXT 556 3 " )" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 420 14 "Solution: f " }{TEXT -1 1 "\n " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 269 4 "17. " }{TEXT 275 1 " " } {TEXT 524 28 "What is the coefficient of " }{XPPEDIT 526 0 "(x-1)^3; " "6#*$,&%\"xG\"\"\"F&!\"\"\"\"$" }{TEXT 525 27 " in the Taylor serie s of " }{XPPEDIT 528 1 "x^(1/3);" "6#)%\"xG*&\"\"\"F&\"\"$!\"\"" } {TEXT 527 20 " with base point 1?" }}{PARA 0 "" 0 "" {TEXT 402 6 "a) \+ " }{XPPEDIT 408 0 "1/9;" "6#*&\"\"\"F$\"\"*!\"\"" }{TEXT 407 15 " \+ b) " }{XPPEDIT 257 0 "4/27;" "6#*&\"\"%\"\"\"\"#F!\"\"" } {TEXT 403 20 " c) " }{XPPEDIT 258 0 "5/81;" "6#*&\"\"& \"\"\"\"#\")!\"\"" }{TEXT 404 17 " d) " }{XPPEDIT 256 0 " 10/243;" "6#*&\"#5\"\"\"\"$V#!\"\"" }{TEXT 406 19 " e) \+ " }{XPPEDIT 259 0 "1/512;" "6#*&\"\"\"F$\"$7&!\"\"" }{TEXT 405 12 " \+ \nf) " }{XPPEDIT 263 0 "-1/9;" "6#,$*&\"\"\"F%\"\"*!\"\"F'" } {TEXT 533 11 " g) " }{XPPEDIT 257 0 "-4/27;" "6#,$*&\"\"%\"\"\" \"#F!\"\"F(" }{TEXT 529 17 " h) " }{XPPEDIT 258 0 "-5/81; " "6#,$*&\"\"&\"\"\"\"#\")!\"\"F(" }{TEXT 530 14 " i) " } {XPPEDIT 256 0 "-10/243;" "6#,$*&\"#5\"\"\"\"$V#!\"\"F(" }{TEXT 532 16 " j) " }{XPPEDIT 259 0 "-1/512;" "6#,$*&\"\"\"F%\"$7&! \"\"F'" }{TEXT 531 3 " \n" }{TEXT 409 0 "" }}{PAGEBK }{PARA 0 "" 0 " " {TEXT 434 14 "Solution: c \n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "simplify( su bs(x=1,diff(x^(1/3),x$3))/3! );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\" \"&\"#\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "series(x^(1/3) , x=1, 5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/,&%\"xG\"\"\"F&!\"\"F& \"\"!#F&\"\"$F&#F'\"\"*\"\"##\"\"&\"#\")F*#!#5\"$V#\"\"%-%\"OG6#F&F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 271 4 "18. " }{TEXT 399 1 " " }{TEXT 473 25 "The Maclaurin \+ series of " }{XPPEDIT 474 1 "1-exp(-x^2);" "6#,&\"\"\"F$-%$expG6#,$*$ %\"xG\"\"#!\"\"F," }{TEXT 471 27 " is used to approximate " } {XPPEDIT 475 1 "int(1-exp(-x^2),x = 0 .. ` 0.3`);" "6#-%$intG6$,&\"\" \"F'-%$expG6#,$*$%\"xG\"\"#!\"\"F//F-;\"\"!%%~0.3G" }{TEXT 472 1 " " } {TEXT 520 26 "with an error \nless than " }{XPPEDIT 523 0 "10^(-5);" "6#)\"#5,$\"\"&!\"\"" }{TEXT 522 154 ". The calculation uses only as \+ many terms as are deemed necessary for the required accuracy by the Al ternating Series Test. What is the approximation? \n " }{TEXT 521 2 " \+ " }{TEXT 401 193 " \na) 0.007957 b) 0.008057 c) 0.0 08157 d) 0.008257 e) 0.008357 \nf) 0.008457 \+ g) 0.008557 h) 0.008657 i) 0.008757 j) 0.008 857 " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 400 0 "" }{TEXT -1 0 "" } }{PAGEBK }{PARA 0 "" 0 "" {TEXT 417 14 "Solution: i \n" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "series(1-exp(-x^2),x=0,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ++%\"xG\"\"\"\"\"##!\"\"F&\"\"%#F%\"\"'F+-%\"OG6#F%\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "p := convert( %, polynom );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,(*$)%\"xG\"\"#\"\"\"F**&F)!\"\" F(\"\"%F,*&\"\"'F,F(F/F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "int(subs(x=t, p), t = 0 .. x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(* &\"\"$!\"\"%\"xGF%\"\"\"*&\"#5F&F'\"\"&F&*&\"#UF&F'\"\"(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(x=.3, 1/3*x^3-1/10*x^5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++d()!#7" }}}}{SECT 0 {PARA 260 " " 0 "" {TEXT 272 33 "19. What is the coefficient of " }{XPPEDIT 258 0 "x^3;" "6#*$%\"xG\"\"$" }{TEXT 518 31 " in the Maclaurin series of \+ " }{XPPEDIT 18 0 "25*x/(5-2*x);" "6#*(\"#D\"\"\"%\"xGF%,&\"\"&F%*&\" \"#F%F&F%!\"\"F+" }{TEXT 519 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 260 "" 0 "" {TEXT 507 6 "a) " }{XPPEDIT 18 0 "5/27;" "6#*&\" \"&\"\"\"\"#F!\"\"" }{TEXT 508 16 " b) " }{XPPEDIT 18 0 "3 /10;" "6#*&\"\"$\"\"\"\"#5!\"\"" }{TEXT 509 16 " c) " } {XPPEDIT 733 1 "4/5;" "6#*&\"\"%\"\"\"\"\"&!\"\"" }{TEXT 510 17 " \+ d) " }{XPPEDIT 18 0 "4/15;" "6#*&\"\"%\"\"\"\"#:!\"\"" } {TEXT 511 17 " e) " }{XPPEDIT 18 0 "5/9;" "6#*&\"\"&\"\" \"\"\"*!\"\"" }{TEXT 512 3 " \n" }{TEXT 513 4 "f) " }{XPPEDIT 18 0 " -5/27;" "6#,$*&\"\"&\"\"\"\"#F!\"\"F(" }{TEXT 517 13 " g) " } {XPPEDIT 18 0 "-3/10;" "6#,$*&\"\"$\"\"\"\"#5!\"\"F(" }{TEXT 516 13 " \+ h) " }{XPPEDIT 18 0 "-4/5;" "6#,$*&\"\"%\"\"\"\"\"&!\"\"F(" } {TEXT 515 14 " i) " }{XPPEDIT 18 0 "-4/15;" "6#,$*&\"\"%\"\" \"\"#:!\"\"F(" }{TEXT 514 15 " j) " }{XPPEDIT 18 0 "-5/9;" "6#,$*&\"\"&\"\"\"\"\"*!\"\"F(" }}{PARA 0 "" 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(25*x/(5-2*x),x=0,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #+3%\"xG\"\"&\"\"\"\"\"#F'#\"\"%F%\"\"$#\"\")\"#DF)#\"#;\"$D\"F%#\"#K \"$D'\"\"'#\"#k\"%DJ\"\"(-%\"OG6#F&F," }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 274 32 "20. What is the coefficient of " }{XPPEDIT 495 0 "x^4; " "6#*$%\"xG\"\"%" }{TEXT 494 32 " in the Maclaurin series of " } {XPPEDIT 497 0 "(1+x^2)^(1/2);" "6#),&\"\"\"F%*$%\"xG\"\"#F%*&F%F%F(! \"\"" }{TEXT 496 3 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 461 0 "" }}{PARA 260 "" 0 "" {TEXT 273 3 "a) " }{XPPEDIT 18 0 "-1/32;" "6#,$ *&\"\"\"F%\"#K!\"\"F'" }{TEXT 498 13 " b) " }{XPPEDIT 18 0 "- 1/16;" "6#,$*&\"\"\"F%\"#;!\"\"F'" }{TEXT 499 11 " c) " } {XPPEDIT 732 1 "-1/8;" "6#,$*&\"\"\"F%\"\")!\"\"F'" }{TEXT 500 12 " \+ d) " }{XPPEDIT 18 0 "-1/4;" "6#,$*&\"\"\"F%\"\"%!\"\"F'" }{TEXT 501 12 " e) " }{XPPEDIT 18 0 "-1/2;" "6#,$*&\"\"\"F%\"\"#!\"\" F'" }{TEXT 502 2 " " }}{PARA 260 "" 0 "" {TEXT 462 4 "f) " } {XPPEDIT 18 0 "1/32;" "6#*&\"\"\"F$\"#K!\"\"" }{TEXT 506 16 " \+ g) " }{XPPEDIT 18 0 "1/16;" "6#*&\"\"\"F$\"#;!\"\"" }{TEXT 505 16 " h) " }{XPPEDIT 18 0 "1/8;" "6#*&\"\"\"F$\"\")!\"\"" } {TEXT 504 17 " i) " }{XPPEDIT 18 0 "1/4;" "6#*&\"\"\"F$\" \"%!\"\"" }{TEXT 503 18 " j) " }{XPPEDIT 18 0 "1/2;" "6# *&\"\"\"F$\"\"#!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK } {PARA 0 "" 0 "" {TEXT 419 14 "Solution: c \n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "bi nomial(1/2,2); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "series((1+x^2)^(1/2), x = 0, 5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"xG\"\"\"\"\"!#F%\"\"#F(#! \"\"\"\")\"\"%-%\"OG6#F%\"\"'" }}}}}{MARK "16 5 0" 11 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }