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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 293 37 " Math 132 \n Fall 2007 Exam II" }}{PARA 19 "" 0 "" {TEXT -1 15 "22 October 2007" }} {PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 617 17 "Integra l Formula:" }{TEXT -1 9 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%$IntG6$*$)-%$secG6#%\"tG\"\"$\"\"\"F,,(*&#F.\"\"#F.*&F)F.-%$tanGF+F .F.F.*&F1F.-%#lnG6#-%$absG6#,&F)F.F4F.F.F.%\"CGF." }}{PARA 0 "" 0 "" {TEXT 256 3 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 258 1 "1" }{TEXT 264 2 ". " }{TEXT 437 13 "Suppose that " }{XPPEDIT 438 0 "f(x) = 2^(sq rt(x));" "6#/-%\"fG6#%\"xG)\"\"#-%%sqrtG6#F'" }{TEXT 436 13 ". Calcul ate " }{XPPEDIT 646 0 "D(f)(9);" "6#--%\"DG6#%\"fG6#\"\"*" }{TEXT 645 1 " " }{TEXT 647 1 "," }{TEXT -1 2 " " }{TEXT 601 19 "the derivative \+ of " }{XPPEDIT 649 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 648 24 " eval uated at x = 9.) " }{TEXT -1 1 " " }}{PARA 261 "" 0 "" {TEXT -1 3 "a) \+ " }{XPPEDIT 18 0 "2/3;" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 19 " ln( 2) b) " }{XPPEDIT 18 0 "2/9;" "6#*&\"\"#\"\"\"\"\"*!\"\"" } {TEXT -1 17 " ln(2) c) " }{XPPEDIT 18 0 "4/3;" "6#*&\"\"%\"\"\" \"\"$!\"\"" }{TEXT -1 19 " ln(2) d) " }{XPPEDIT 18 0 "4/9;" " 6#*&\"\"%\"\"\"\"\"*!\"\"" }{TEXT -1 20 " ln(2) e) " } {XPPEDIT 18 0 "4/27;" "6#*&\"\"%\"\"\"\"#F!\"\"" }{TEXT -1 12 " ln(2) \+ \nf) " }{XPPEDIT 18 0 "2/(3*ln(2));" "6#*&\"\"#\"\"\"*&\"\"$F%-%#lnG6 #F$F%!\"\"" }{TEXT -1 5 " " }{TEXT 294 5 " g) " }{TEXT -1 1 " " } {XPPEDIT 18 0 "2/(9*ln(2));" "6#*&\"\"#\"\"\"*&\"\"*F%-%#lnG6#F$F%!\" \"" }{TEXT 439 10 " h) " }{XPPEDIT 18 0 "4/(3*ln(2));" "6#*&\"\" %\"\"\"*&\"\"$F%-%#lnG6#\"\"#F%!\"\"" }{TEXT 650 13 " i) " } {XPPEDIT 18 0 "4/(9*ln(20));" "6#*&\"\"%\"\"\"*&\"\"*F%-%#lnG6#\"#?F%! \"\"" }{TEXT 440 12 " j) " }{XPPEDIT 18 0 "4/(27*ln(2));" "6#* &\"\"%\"\"\"*&\"#FF%-%#lnG6#\"\"#F%!\"\"" }{TEXT 441 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 651 15 "Solution ( c )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := x -> 2^ (sqrt(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)op eratorG%&arrowGF()\"\"#-%%sqrtG6#9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "D(f)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\" \"\"\"\"#F&*()F'*$%\"xGF%F&F+#!\"\"F'-%#lnG6#F'F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"%\"\"$\"\"\"-%#lnG6#\"\"#F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 335 16 "2. Calculate " }{XPPEDIT 465 0 "Int(log[3](x) ,x = 1 .. 3);" "6#-%$IntG6$-&%$logG6#\"\"$6#%\"xG/F,;\"\"\"F*" }{TEXT 464 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 346 5 "a) " }{XPPEDIT 467 0 "2-1/ln(3);" "6#,&\"\"#\"\"\"*&F%F%-%#lnG6# \"\"$!\"\"F+" }{TEXT 466 12 " b) " }{XPPEDIT 261 0 "2+1/ln(3); " "6#,&\"\"#\"\"\"*&F%F%-%#lnG6#\"\"$!\"\"F%" }{TEXT 347 13 " \+ c) " }{XPPEDIT 262 0 "2-ln(3);" "6#,&\"\"#\"\"\"-%#lnG6#\"\"$!\"\"" } {TEXT 348 11 " d) " }{XPPEDIT 469 0 "2+ln(3);" "6#,&\"\"#\"\"\" -%#lnG6#\"\"$F%" }{TEXT 468 12 " e) " }{XPPEDIT 263 0 "2*ln(3) -1;" "6#,&*&\"\"#\"\"\"-%#lnG6#\"\"$F&F&F&!\"\"" }{TEXT 349 9 " \nf) \+ " }{XPPEDIT 264 0 "2*ln(3)+1;" "6#,&*&\"\"#\"\"\"-%#lnG6#\"\"$F&F&F &F&" }{TEXT 350 10 " g) " }{TEXT 357 1 " " }{XPPEDIT 265 0 "3-2/ ln(3);" "6#,&\"\"$\"\"\"*&\"\"#F%-%#lnG6#F$!\"\"F+" }{TEXT 351 12 " \+ h) " }{XPPEDIT 266 0 "3+2/ln(3);" "6#,&\"\"$\"\"\"*&\"\"#F%-%#ln G6#F$!\"\"F%" }{TEXT 352 10 " i) " }{TEXT 355 1 " " }{XPPEDIT 267 0 "1-2/ln(3);" "6#,&\"\"\"F$*&\"\"#F$-%#lnG6#\"\"$!\"\"F+" }{TEXT 353 2 " " }{TEXT 356 10 " j) " }{XPPEDIT 268 0 "1+2/ln(3);" "6# ,&\"\"\"F$*&\"\"#F$-%#lnG6#\"\"$!\"\"F$" }{TEXT 345 1 "\n" }{TEXT 354 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 654 15 "S olution ( g )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Int('log[3](x)',x)= Int(log[ 3](x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-&%$logG6#\"\"$ 6#%\"xGF--F%6$*&-%#lnGF,\"\"\"-F2F*!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 98 "Int(ln(x)/ln(3),x)=student[intparts](Int(ln(x)/ln(3 ),x),ln(x)); #Integration by parts with u=ln(x)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%#lnG6#%\"xG\"\"\"-F)6#\"\"$!\"\"F+,&*(F-F 0F+F,F(F,F,-F%6$*&F,F,F-F0F+F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Int(ln(x)/ln(3),x)= value(student[intparts](Int(ln(x)/ln(3),x) ,ln(x)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%#lnG6#%\"xG \"\"\"-F)6#\"\"$!\"\"F+,&*(F-F0F+F,F(F,F,*&F-F0F+F,F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "Answer := Int('log[3](x)',x = 1..3 ) = subs(x=3, 1/ln(3)*x*ln(x)-1/ln(3)*x) - subs(x=1,1/ln(3)*x*ln(x)-1/ ln(3)*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AnswerG/-%$IntG6$-&%$l ogG6#\"\"$6#%\"xG/F/;\"\"\"F-,(F-F2*&\"\"#F2-%#lnGF,!\"\"F8*&F6F8-F76# F2F2F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Answer := value(r hs(Answer));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AnswerG,&\"\"$\"\" \"*&\"\"#F'-%#lnG6#F&!\"\"F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 260 "" 0 "" {TEXT 262 3 "3. " }{TEXT 296 1 " " }{TEXT 297 15 " Suppose that " }{XPPEDIT 18 0 "f(x) = x^sqrt(x);" "6#/-%\"fG 6#%\"xG)F'-%%sqrtG6#F'" }{TEXT 442 23 ". Calculate D(f)(4). " } {TEXT -1 1 " " }{TEXT 613 56 "(D(f)(4) is the derivative of f(x) evalu ated at x = 4.) " }}{PARA 3 "" 0 "" {TEXT 295 3 "a) " }{XPPEDIT 448 0 "1+ln(2);" "6#,&\"\"\"F$-%#lnG6#\"\"#F$" }{TEXT 443 12 " b) " }{XPPEDIT 449 0 "2*(2+ln(2));" "6#*&\"\"#\"\"\",&F$F%-%#lnG6#F$F%F%" } {TEXT 444 10 " c) " }{XPPEDIT 450 0 "4*(2+ln(2));" "6#*&\"\"%\" \"\",&\"\"#F%-%#lnG6#F'F%F%" }{TEXT 445 15 " d) " } {XPPEDIT 451 0 "2*(1+ln(2));" "6#*&\"\"#\"\"\",&F%F%-%#lnG6#F$F%F%" } {TEXT 446 13 " e) " }{XPPEDIT 452 0 "8+ln(2);" "6#,&\"\")\"\" \"-%#lnG6#\"\"#F%" }{TEXT 447 14 " \nf) " }{XPPEDIT 457 0 "2+ ln(2);" "6#,&\"\"#\"\"\"-%#lnG6#F$F%" }{TEXT 453 11 " g) " } {XPPEDIT 458 0 "4+ln(2);" "6#,&\"\"%\"\"\"-%#lnG6#\"\"#F%" }{TEXT 454 17 " h) " }{XPPEDIT 459 0 "4*(1+ln(2));" "6#*&\"\"%\"\"\" ,&F%F%-%#lnG6#\"\"#F%F%" }{TEXT 455 16 " i) " }{XPPEDIT 460 0 "2*(4+ln(2));" "6#*&\"\"#\"\"\",&\"\"%F%-%#lnG6#F$F%F%" }{TEXT 456 13 " j) " }{XPPEDIT 462 0 "8*(1+ln(2));" "6#*&\"\")\"\"\" ,&F%F%-%#lnG6#\"\"#F%F%" }{TEXT 461 3 " " }{TEXT 463 6 " \n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 655 15 "Solution ( j )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x -> x^sqrt(x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF()9$-%%s qrtG6#F-F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"xG*$F%#\"\"\"\"\"#F(,&*&F'F(*& F%#!\"\"F)-%#lnG6#F%F(F(F(*&F(F(*$F%#F(F)F.F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "D(f)(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&* &\"\"%\"\"\"-%#lnG6#F%F&F&\"\")F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 0 {PARA 3 "" 0 "" {TEXT 257 27 "4. A radioactive substance" } {TEXT 470 1 " " }{TEXT 471 24 "has mass 120g at time " }{XPPEDIT 474 0 "t = 4;" "6#/%\"tG\"\"%" }{TEXT 472 26 " and mass 90g at tim e " }{XPPEDIT 475 0 "t = 6;" "6#/%\"tG\"\"'" }{TEXT 473 25 ". What is the mass at " }{XPPEDIT 702 0 "t = 12;" "6#/%\"tG\"#7" }{TEXT 701 8 " ? \na) " }{XPPEDIT 484 0 "1979/32;" "6#*&\"%z>\"\"\"\"#K!\"\"" } {TEXT 476 16 " b) " }{XPPEDIT 485 0 "1985/32;" "6#*&\"%&)> \"\"\"\"#K!\"\"" }{TEXT 477 17 " c) " }{XPPEDIT 486 0 "19 91/32;" "6#*&\"%\"*>\"\"\"\"#K!\"\"" }{TEXT 478 13 " d) " } {XPPEDIT 487 0 "1997/32;" "6#*&\"%(*>\"\"\"\"#K!\"\"" }{TEXT 479 15 " \+ e) " }{TEXT 317 1 " " }{XPPEDIT 488 0 "1203/21;" "6#*&\"%.7 \"\"\"\"#@!\"\"" }{TEXT 301 19 " " }}{PARA 0 "" 0 " " {TEXT 299 4 "f) " }{XPPEDIT 489 0 "1209/32;" "6#*&\"%47\"\"\"\"#K! \"\"" }{TEXT 480 18 " g) " }{XPPEDIT 490 0 "1215/32;" "6 #*&\"%:7\"\"\"\"#K!\"\"" }{TEXT 481 10 " " }{TEXT 491 1 " " } {TEXT 300 5 "h) " }{XPPEDIT 492 0 "1221/32;" "6#*&\"%@7\"\"\"\"#K!\" \"" }{TEXT 482 13 " i) " }{XPPEDIT 493 0 "1227/32;" "6#*&\"%F 7\"\"\"\"#K!\"\"" }{TEXT 483 16 " j) " }{XPPEDIT 494 0 "12 33/32;" "6#*&\"%L7\"\"\"\"#K!\"\"" }{TEXT 495 3 " " }}{PARA 3 "" 0 " " {TEXT 298 1 " " }}{PARA 0 "" 0 "" {TEXT 703 15 "Solution ( g )" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "m:= t -> A*exp(-la mbda*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mGf*6#%\"tG6\"6$%)oper atorG%&arrowGF(*&%\"AG\"\"\"-%$expG6#,$*&%'lambdaGF.9$F.!\"\"F.F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eqn1 := m(4) = 120;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/*&%\"AG\"\"\"-%$expG6#,$*&\" \"%F(%'lambdaGF(!\"\"F(\"$?\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "eqn2 := m(6) = 90;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/* &%\"AG\"\"\"-%$expG6#,$*&\"\"'F(%'lambdaGF(!\"\"F(\"#!*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqn3 := lhs(eqn1)/lhs(eqn2) = rhs(e qn1)/rhs(eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/*&-%$expG6 #,$*&\"\"%\"\"\"%'lambdaGF-!\"\"F--F(6#,$*&\"\"'F-F.F-F/F/#F,\"\"$" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eqn4 := combine(lhs(eqn3), \+ exp) = rhs(eqn3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/-%$expG6 #,$*&\"\"#\"\"\"%'lambdaGF,F,#\"\"%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "eqn5 := lambda = solve(eqn4, lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/%'lambdaG,$*&#\"\"\"\"\"#F*-%#lnG6##\"\"% \"\"$F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "eqn6 := subs(e qn5, eqn1); #Substitute this value of lambda into eqn1 to find A" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/*&%\"AG\"\"\"-%$expG6#,$*&\" \"#F(-%#lnG6##\"\"%\"\"$F(!\"\"F(\"$?\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn7 := A = solve(eqn6, A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn7G/%\"AG#\"$S'\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Answer := `m(12) = `*m(12) = subs( \{eqn5, eqn7, t = \+ 12\}, m(t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AnswerG/*(%)m(12)~ =~G\"\"\"%\"AGF(-%$expG6#,$*&\"#7F(%'lambdaGF(!\"\"F(,$*&#\"$S'\"\"$F( -F+6#,$*&\"\"'F(-%#lnG6##\"\"%F6F(F1F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "answer := simplify(Answer);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG#\"%:7\"#K" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 400 134 "5. The mass of a microbe colony splashing about in a nutrient bro th triples every 12 hours. \n What is the colony's doubling time? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 402 4 "a) \+ " }{XPPEDIT 678 0 "8;" "6#\"\")" }{TEXT 631 23 " b) \+ " }{XPPEDIT 679 0 "8*ln(2);" "6#*&\"\")\"\"\"-%#lnG6#\"\"#F%" }{TEXT 630 13 " c) " }{XPPEDIT 680 0 "8*ln(3);" "6#*&\"\")\"\"\"-%#l nG6#\"\"$F%" }{TEXT 629 15 " d) " }{XPPEDIT 681 0 "8*ln(2)/ ln(3);" "6#*(\"\")\"\"\"-%#lnG6#\"\"#F%-F'6#\"\"$!\"\"" }{TEXT 627 16 " e) " }{XPPEDIT 682 0 "12*ln(2)/ln(3);" "6#*(\"#7\"\"\"-% #lnG6#\"\"#F%-F'6#\"\"$!\"\"" }{TEXT 405 2 " " }{TEXT 404 3 " " } {TEXT 403 19 " " }}{PARA 0 "" 0 "" {TEXT 401 4 "f) \+ " }{XPPEDIT 683 0 "12*ln(3/2);" "6#*&\"#7\"\"\"-%#lnG6#*&\"\"$F%\"\"#! \"\"F%" }{TEXT 632 10 " g) " }{XPPEDIT 684 0 "3/2*ln(12);" "6#*( \"\"$\"\"\"\"\"#!\"\"-%#lnG6#\"#7F%" }{TEXT 633 10 " h) " } {XPPEDIT 685 0 "3*ln(2);" "6#*&\"\"$\"\"\"-%#lnG6#\"\"#F%" }{TEXT 634 15 " i) " }{XPPEDIT 686 0 "2*ln(3);" "6#*&\"\"#\"\"\"-%#lnG 6#\"\"$F%" }{TEXT 635 18 " j) " }{XPPEDIT 687 0 "2/3*ln( 12);" "6#*(\"\"#\"\"\"\"\"$!\"\"-%#lnG6#\"#7F%" }{TEXT 636 2 " " } {TEXT 688 2 " " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 628 15 "Solution ( e )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "T;" "6#%\"TG" }{TEXT -1 33 " \+ be the doubling time and let " }{XPPEDIT 18 0 "m(t) = A*2^(t/T);" "6 #/-%\"mG6#%\"tG*&%\"AG\"\"\")\"\"#*&F'F*%\"TG!\"\"F*" }{TEXT -1 39 " \+ be the colony's population at time " }{XPPEDIT 18 0 "t;" "6#%\"tG" } {TEXT -1 4 " . \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "m := t \+ -> A*2^(t/T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mGf*6#%\"tG6\"6$% )operatorG%&arrowGF(*&%\"AG\"\"\")\"\"#*&9$F.%\"TG!\"\"F.F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eqn := m(12) = 3*m(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/ *&%\"AG\"\"\")\"\"#,$*&\"#7F(%\"TG!\"\"F(F(,$*&\"\"$F(F'F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(eqn, T);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*(\"#7\"\"\"-%#lnG6#\"\"#F&-F(6#\"\"$!\"\"F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 377 20 "6. Suppose that " }{XPPEDIT 501 0 "u(t);" "6#-%\"uG6#%\"tG" }{TEXT 500 56 " is the unique solutio n of the initial value problem " }{XPPEDIT 497 0 "diff(u(t),t) = M-5 *u(t),u(0) = 1;" "6$/-%%diffG6$-%\"uG6#%\"tGF*,&%\"MG\"\"\"*&\"\"&F--F (6#F*F-!\"\"/-F(6#\"\"!F-" }{TEXT 496 9 " where " }{XPPEDIT 699 0 "M ;" "6#%\"MG" }{TEXT 698 27 " is a constant.\n If " }{XPPEDIT 499 0 "limit(u(t) = 12,t = infinity);" "6#-%&limitG6$/-%\"uG6#%\"tG\"# 7/F*%)infinityG" }{TEXT 498 17 " then what is " }{XPPEDIT 697 0 "M; " "6#%\"MG" }{TEXT 696 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 380 1 "\n" }{TEXT 381 4 "a) " }{XPPEDIT 390 0 "3;" "6 #\"\"$" }{TEXT 384 18 " b) " }{XPPEDIT 391 0 "4;" "6#\" \"%" }{TEXT 385 33 " c) 5 d) " }{XPPEDIT 392 0 "10;" "6#\"#5" }{TEXT 389 15 " e) " }{XPPEDIT 393 0 "12; " "6#\"#7" }{TEXT 383 2 " " }{TEXT 382 20 " \n" } {TEXT 378 5 "f) " }{XPPEDIT 394 0 "15;" "6#\"#:" }{TEXT 388 15 " \+ g) " }{XPPEDIT 395 0 "20;" "6#\"#?" }{TEXT 387 10 " \+ " }{TEXT 396 1 " " }{TEXT 379 20 "h) 30 i) " }{XPPEDIT 397 0 "60;" "6#\"#g" }{TEXT 386 15 " j) " }{XPPEDIT 398 0 "120; " "6#\"$?\"" }{TEXT 399 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 700 15 "Solution ( i )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqn1 := Int(1/(M-5*u),u) = Int(1,t)+C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% eqn1G/-%$IntG6$*&\"\"\"F*,&%\"MGF**&\"\"&F*%\"uGF*!\"\"F0F/,&-F'6$F*% \"tGF*%\"CGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn2 := ma p(value, eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,$*&#\"\" \"\"\"&F)-%#lnG6#,&%\"MGF)*&F*F)%\"uGF)!\"\"F)F2,&%\"tGF)%\"CGF)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqn3 := map( z -> -5*z, eqn2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%#lnG6#,&%\"MG\"\"\"*& \"\"&F+%\"uGF+!\"\",&*&F-F+%\"tGF+F/*&F-F+%\"CGF+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn4 := map( z -> exp(z), eqn3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/,&%\"MG\"\"\"*&\"\"&F(%\"uGF( !\"\"-%$expG6#,&*&F*F(%\"tGF(F,*&F*F(%\"CGF(F," }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "eqn5 := u = solve(eqn4, u);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%eqn5G/%\"uG,&*&\"\"&!\"\"%\"MG\"\"\"F,*&#F,F)F,-%$ expG6#,&*&F)F,%\"tGF,F**&F)F,%\"CGF,F*F,F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 44 "eqn6 := limit(rhs(eqn5), t = infinity) = 12;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/,$*&\"\"&!\"\"%\"MG\"\"\"F+\" #7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "M = solve(eqn6, M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"MG\"#g" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 305 18 "7. Suppose that " }{XPPEDIT 504 0 "f(x) = arcsec(5* x);" "6#/-%\"fG6#%\"xG-%'arcsecG6#*&\"\"&\"\"\"F'F-" }{TEXT 502 16 ". \+ Calculate " }{XPPEDIT 505 0 "D(f)(-1/3);" "6#--%\"DG6#%\"fG6#,$*& \"\"\"F+\"\"$!\"\"F-" }{TEXT 503 3 ". " }{TEXT 520 31 "(The derivativ e of f(x) at " }{XPPEDIT 258 0 "x = -1/3;" "6#/%\"xG,$*&\"\"\"F'\" \"$!\"\"F)" }{TEXT 521 3 " )." }{TEXT -1 1 " " }{TEXT 304 0 "" }} {PARA 3 "" 0 "" {TEXT 303 4 "a) " }{XPPEDIT 693 0 "-9/4;" "6#,$*&\"\" *\"\"\"\"\"%!\"\"F(" }{TEXT 692 15 " b) " }{XPPEDIT 513 0 " -5/4;" "6#,$*&\"\"&\"\"\"\"\"%!\"\"F(" }{TEXT 506 15 " c) \+ " }{XPPEDIT 514 0 "-4/3;" "6#,$*&\"\"%\"\"\"\"\"$!\"\"F(" }{TEXT 507 17 " d) " }{XPPEDIT 515 0 "-5/3;" "6#,$*&\"\"&\"\"\"\"\"$ !\"\"F(" }{TEXT 508 15 " e) " }{XPPEDIT 516 0 "-9/20;" "6#, $*&\"\"*\"\"\"\"#?!\"\"F(" }{TEXT 509 11 " \nf) " }{XPPEDIT 517 0 "9/4;" "6#*&\"\"*\"\"\"\"\"%!\"\"" }{TEXT 510 19 " g) \+ " }{XPPEDIT 257 0 "5/4;" "6#*&\"\"&\"\"\"\"\"%!\"\"" }{TEXT 695 18 " \+ h) " }{XPPEDIT 518 0 "4/3;" "6#*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT 511 22 " i) " }{XPPEDIT 519 0 "5/3;" "6#*&\" \"&\"\"\"\"\"$!\"\"" }{TEXT 512 19 " j) " }{XPPEDIT 694 0 "9/20;" "6#*&\"\"*\"\"\"\"#?!\"\"" }{TEXT 302 1 "\n" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 691 15 "Solution ( f )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := x -> arcsec(5*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%'arcsecG6#,$ *&\"\"&\"\"\"9$F2F2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&)%\"xG\"\"#F$ ,&\"#DF$*&F$F$*$F&F$!\"\"F-#F$F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "D(f)(-1/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\" \"*\"\"\"\"#;!\"\"F'#F&\"\"#F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 320 17 "8. Suppose that " }{XPPEDIT 257 0 "f(x) = 120*arctan(sqrt(x));" "6#/-%\"fG6#%\"xG*&\"$?\"\"\"\"-%' arctanG6#-%%sqrtG6#F'F*" }{TEXT 321 15 ". What is " }{XPPEDIT 324 0 "D(f)(4);" "6#--%\"DG6#%\"fG6#\"\"%" }{TEXT 322 35 "? (The der ivative of f(x) at " }{XPPEDIT 325 0 "x = 4;" "6#/%\"xG\"\"%" } {TEXT 323 3 " )." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 327 4 "a) " }{XPPEDIT 524 0 "2;" "6#\"\"#" } {TEXT 523 16 " b) " }{XPPEDIT 336 0 "3;" "6#\"\"$" }{TEXT 328 16 " c) " }{XPPEDIT 337 0 "4;" "6#\"\"%" }{TEXT 329 17 " d) " }{XPPEDIT 525 0 "5;" "6#\"\"&" }{TEXT 522 17 " \+ e) " }{XPPEDIT 338 0 "6;" "6#\"\"'" }{TEXT 330 8 " \nf) \+ " }{XPPEDIT 339 0 "8;" "6#\"\")" }{TEXT 331 15 " g) " } {XPPEDIT 340 0 "10;" "6#\"#5" }{TEXT 332 14 " h) " } {XPPEDIT 341 0 "12;" "6#\"#7" }{TEXT 333 15 " i) " } {XPPEDIT 342 0 "15;" "6#\"#:" }{TEXT 334 17 " j) " } {XPPEDIT 343 0 "20;" "6#\"#?" }{TEXT 326 1 "\n" }{TEXT 344 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 637 15 "Solution ( e )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := x -> 120*arctan(sqrt(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,$*&\"$?\"\"\"\"-%'arctanG6#-%%sqrtG6#9$F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(f)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#g\"\"\"%\"xG#!\"\"\"\"#,&F&F&F'F&F)F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "simplify(D(f)(4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 260 1 " " }{TEXT 319 1 "9" }{TEXT -1 2 " . " }{TEXT 318 11 "Calculate " }{XPPEDIT 527 0 "Int(x*exp(2*x),x = 0 \+ .. 1/2);" "6#-%$IntG6$*&%\"xG\"\"\"-%$expG6#*&\"\"#F(F'F(F(/F';\"\"!*& F(F(F-!\"\"" }{TEXT 526 1 "." }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT 358 4 "a) " }{XPPEDIT 535 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT 528 20 " b) " }{XPPEDIT 536 0 "1/2;" "6#*&\"\"\"F$\"\" #!\"\"" }{TEXT 529 18 " c) " }{XPPEDIT 537 0 "3/4;" "6#* &\"\"$\"\"\"\"\"%!\"\"" }{TEXT 530 42 " d) 1 e) 2 \nf) " }{XPPEDIT 538 0 "exp(1)/4;" "6#*&-%$expG6#\"\"\"F'\"\"%! \"\"" }{TEXT 531 20 " g) " }{XPPEDIT 539 0 "exp(1)/2; " "6#*&-%$expG6#\"\"\"F'\"\"#!\"\"" }{TEXT 532 17 " h) " }{XPPEDIT 540 0 "3*exp(1)/4;" "6#*(\"\"$\"\"\"-%$expG6#F%F%\"\"%!\"\" " }{TEXT 533 15 " i) " }{XPPEDIT 541 0 "exp(1);" "6#-%$expG 6#\"\"\"" }{TEXT 534 16 " j) " }{XPPEDIT 542 0 "2*exp(1); " "6#*&\"\"#\"\"\"-%$expG6#F%F%" }{TEXT 359 1 "\n" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 653 15 "Solution ( a )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "J := Int(x*exp(2*x),x = 0 .. 1/2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&%\"xG\"\"\"-%$exp G6#,$*&\"\"#F*F)F*F*F*/F);\"\"!#F*F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "K := student[intparts](J, x); #Integration by parts \+ with u = x" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG,&*&#\"\"\"\"\"%F( -%$expG6#F(F(F(-%$IntG6$,$*&#F(\"\"#F(-F+6#,$*&F3F(%\"xGF(F(F(F(/F8;\" \"!F2!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(K);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 315 17 "10. Calculate " }{XPPEDIT 361 0 "int(x^2*sin(x),x = 0 .. Pi);" "6# -%$intG6$*&%\"xG\"\"#-%$sinG6#F'\"\"\"/F';\"\"!%#PiG" }{TEXT 360 2 " . " }}{PARA 0 "" 0 "" {TEXT 314 17 " " }}{PARA 0 "" 0 " " {TEXT 362 5 "a) " }{XPPEDIT 552 0 "Pi^2;" "6#*$%#PiG\"\"#" }{TEXT 547 16 " b) " }{XPPEDIT 553 0 "Pi^2-1;" "6#,&*$%#PiG\"\"# \"\"\"F'!\"\"" }{TEXT 548 14 " c) " }{XPPEDIT 554 0 "Pi^2-2; " "6#,&*$%#PiG\"\"#\"\"\"F&!\"\"" }{TEXT 549 12 " d) " } {XPPEDIT 555 0 "Pi^2-3;" "6#,&*$%#PiG\"\"#\"\"\"\"\"$!\"\"" }{TEXT 550 13 " e) " }{XPPEDIT 556 0 "Pi^2-4;" "6#,&*$%#PiG\"\"#\"\" \"\"\"%!\"\"" }{TEXT 551 8 " \nf) " }{XPPEDIT 557 0 "Pi^2+5;" "6#,& *$%#PiG\"\"#\"\"\"\"\"&F'" }{TEXT 546 10 " g) " }{XPPEDIT 558 0 "Pi^2+4;" "6#,&*$%#PiG\"\"#\"\"\"\"\"%F'" }{TEXT 545 12 " h) \+ " }{XPPEDIT 559 0 "Pi^2+3;" "6#,&*$%#PiG\"\"#\"\"\"\"\"$F'" }{TEXT 544 14 " i) " }{XPPEDIT 560 0 "Pi^2+2;" "6#,&*$%#PiG\"\"#\" \"\"F&F'" }{TEXT 543 14 " j) " }{XPPEDIT 561 0 "pi^2+1;" "6# ,&*$%#piG\"\"#\"\"\"F'F'" }{TEXT 363 1 "\n" }{TEXT 562 0 "" }}{PARA 0 "" 0 "" {TEXT 652 15 "Solution ( e )" }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "J := Int(x^2*sin(x),x = 0 .. Pi);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&)%\"xG\"\"#\"\"\"-%$sinG6#F*F,/ F*;\"\"!%#PiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "K := stude nt[intparts](J, x^2); #Integration by parts with u = x^2" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"KG,&*$)%#PiG\"\"#\"\"\"F*-%$IntG6$,$*(F)F*% \"xGF*-%$cosG6#F0F*!\"\"/F0;\"\"!F(F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "L := Pi^2 - student[intparts](Int(-2*x*cos(x),x = 0 . . Pi), x); #Integration by parts with u = x" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,&*$)%#PiG\"\"#\"\"\"F*-%$IntG6$,$*&F)F*-%$sinG6# %\"xGF*!\"\"/F3;\"\"!F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%#PiG\"\"#\"\"\"F( \"\"%!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 1 " " }{TEXT 259 19 "11. Calculate " }{XPPEDIT 258 0 " 25*Int(x^4*ln(x),x = 1 .. exp(1));" "6#*&\"#D\"\"\"-%$IntG6$*&%\"xG\" \"%-%#lnG6#F*F%/F*;F%-%$expG6#F%F%" }{TEXT 364 2 " ." }}{PARA 0 "" 0 " " {TEXT 269 4 "a) " }{XPPEDIT 366 1 "exp(5)-1;" "6#,&-%$expG6#\"\"&\" \"\"F(!\"\"" }{TEXT 306 13 " b) " }{XPPEDIT 367 1 "2*exp(5)-1 ;" "6#,&*&\"\"#\"\"\"-%$expG6#\"\"&F&F&F&!\"\"" }{TEXT 307 13 " \+ c) " }{XPPEDIT 368 1 "3*exp(5)-1;" "6#,&*&\"\"$\"\"\"-%$expG6#\"\"& F&F&F&!\"\"" }{TEXT 308 14 " d) " }{XPPEDIT 369 1 "4*exp(5)- 1;" "6#,&*&\"\"%\"\"\"-%$expG6#\"\"&F&F&F&!\"\"" }{TEXT 309 12 " \+ e) " }{XPPEDIT 370 1 "5*exp(5)-1;" "6#,&*&\"\"&\"\"\"-%$expG6#F%F&F &F&!\"\"" }{TEXT 310 11 " \n f) " }{XPPEDIT 371 1 "exp(5)+1;" "6# ,&-%$expG6#\"\"&\"\"\"F(F(" }{TEXT 311 13 " g) " }{XPPEDIT 372 1 "2*exp(5)+1;" "6#,&*&\"\"#\"\"\"-%$expG6#\"\"&F&F&F&F&" }{TEXT 312 13 " h) " }{XPPEDIT 373 1 "3*exp(5)+1;" "6#,&*&\"\"$\"\" \"-%$expG6#\"\"&F&F&F&F&" }{TEXT 313 14 " i) " }{XPPEDIT 374 0 "4*exp(5)+1;" "6#,&*&\"\"%\"\"\"-%$expG6#\"\"&F&F&F&F&" }{TEXT 365 14 " j) " }{XPPEDIT 375 0 "5*exp(5)+1;" "6#,&*&\"\"&\"\" \"-%$expG6#F%F&F&F&F&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 644 15 "Solution ( i )" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "25*int(x^4*ln(x),x = 1 .. exp(1)); #Maple unaided " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"%\"\"\"-%$expG6#\"\"&F&F&F &F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "J := Int(25*x^4*ln(x ),x = 1 .. exp(1)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6 $,$*(\"#D\"\"\")%\"xG\"\"%F+-%#lnG6#F-F+F+/F-;F+-%$expG6#F+" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "K := student[intparts](J, ln (x)); #Telling Maple to use integration by parts with u = ln(x)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG,&*&\"\"&\"\"\")-%$expG6#F(F'F(F (-%$IntG6$,$*&F'F()%\"xG\"\"%F(F(/F3;F(F*!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "value(K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& *&\"\"%\"\"\")-%$expG6#F&\"\"&F&F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 316 17 "12. Calcul ate " }{XPPEDIT 256 0 "int((5*x+2)/(x^2+x),x = 1 .. 2);" "6#-%$intG 6$*&,&*&\"\"&\"\"\"%\"xGF*F*\"\"#F*F*,&*$F+F,F*F+F*!\"\"/F+;F*F," } {TEXT 376 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 18 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT -1 14 " b) " }{XPPEDIT 18 0 "ln(3);" "6#-%#lnG6#\"\"$" }{TEXT -1 15 " c) " }{XPPEDIT 18 0 "2*ln(2);" "6#*&\"\"#\"\"\"-%#l nG6#F$F%" }{TEXT -1 12 " d) " }{XPPEDIT 18 0 "2*ln(3);" "6#*& \"\"#\"\"\"-%#lnG6#\"\"$F%" }{TEXT -1 13 " e) " }{XPPEDIT 18 0 "3*ln(2);" "6#*&\"\"$\"\"\"-%#lnG6#\"\"#F%" }{TEXT -1 20 " \+ \nf) " }{XPPEDIT 18 0 "ln(6);" "6#-%#lnG6#\"\"'" }{TEXT -1 14 " \+ g) " }{XPPEDIT 18 0 "ln(2/3);" "6#-%#lnG6#*&\"\"#\"\"\"\"\"$ !\"\"" }{TEXT -1 13 " h) " }{XPPEDIT 18 0 "ln(3/2);" "6#-%#ln G6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 13 " i) " }{XPPEDIT 18 0 "ln(9/2);" "6#-%#lnG6#*&\"\"*\"\"\"\"\"#!\"\"" }{TEXT -1 14 " \+ j) " }{XPPEDIT 18 0 "ln(27/2);" "6#-%#lnG6#*&\"#F\"\"\"\"\"#!\"\" " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 638 15 "Solution ( j )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "eqnForIntegrand := (5*x+2)/(x^2+x) = convert( (5*x+2) /(x^2+x) , parfrac, x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0eqnForI ntegrandG/*&,&*&\"\"&\"\"\"%\"xGF*F*\"\"#F*F*,&*$)F+F,F*F*F+F*!\"\",&* &F,F*F+F0F**&\"\"$F*,&F+F*F*F*F0F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "eqnForAntiderivative := Int( lhs(eqnForIntegrand), x) = int( rhs(eqnForIntegrand), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %5eqnForAntiderivativeG/-%$IntG6$*&,&*&\"\"&\"\"\"%\"xGF-F-\"\"#F-F-,& *$)F.F/F-F-F.F-!\"\"F.,&*&F/F--%#lnG6#F.F-F-*&\"\"$F--F76#,&F.F-F-F-F- F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "Answer := simplify( s ubs(x=2, rhs(eqnForAntiderivative)) - subs(x=1, rhs(eqnForAntiderivati ve))) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AnswerG,&-%#lnG6#\"\"#! \"\"*&\"\"$\"\"\"-F'6#F,F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Answer := combine(Answer, ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%'AnswerG,$-%#lnG6##\"\"#\"#F!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 287 29 "13. Find an ordered triple (" }{XPPEDIT 642 0 "alpha,beta,gam ma;" "6%%&alphaG%%betaG%&gammaG" }{TEXT 640 25 ") of positive integer s " }{XPPEDIT 641 0 "alpha,beta,gamma;" "6%%&alphaG%%betaG%&gammaG" } {TEXT 639 53 " such that \n " }{XPPEDIT 407 0 "Int((x^2+3*x-4)/x/((x+2)^2),x = 1 .. 2) = alpha*ln(be ta)-beta*ln(alpha)+1/gamma;" "6#/-%$IntG6$*(,(*$%\"xG\"\"#\"\"\"*&\"\" $F,F*F,F,\"\"%!\"\"F,F*F0*$,&F*F,F+F,F+F0/F*;F,F+,(*&%&alphaGF,-%#lnG6 #%%betaGF,F,*&F;F,-F96#F7F,F0*&F,F,%&gammaGF0F," }{TEXT 406 3 " . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 5 "a) ( " } {XPPEDIT 18 0 "2, 3, 4;" "6%\"\"#\"\"$\"\"%" }{TEXT -1 16 " ) b ) ( " }{XPPEDIT 18 0 "3,2,4;" "6%\"\"$\"\"#\"\"%" }{TEXT -1 16 " ) \+ c) (" }{XPPEDIT 18 0 "4,3,2;" "6%\"\"%\"\"$\"\"#" }{TEXT -1 15 ") d) (" }{XPPEDIT 18 0 "4,2,3;" "6%\"\"%\"\"#\"\"$" }{TEXT -1 14 ") e) ( " }{XPPEDIT 18 0 "3, 4, 2;" "6%\"\"$\"\"%\"\"#" } {TEXT -1 22 " ) \nf) (" }{XPPEDIT 18 0 "3,2,3;" "6%\"\"$ \"\"#F#" }{TEXT -1 17 ") g) ( " }{XPPEDIT 18 0 "2,3,2;" "6% \"\"#\"\"$F#" }{TEXT -1 16 ") h) (" }{XPPEDIT 18 0 "4,2,2;" "6%\"\"%\"\"#F$" }{TEXT -1 16 ") i) (" }{XPPEDIT 18 0 "4,3,3 ;" "6%\"\"%\"\"$F$" }{TEXT -1 14 ") j) (" }{XPPEDIT 18 0 "3,2, 2;" "6%\"\"$\"\"#F$" }{TEXT -1 2 ") " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 643 15 "Soluti on ( b )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "rationalFunction := (x^2+3*x-4)/x/( (x+2)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1rationalFunctionG*(,(* $)%\"xG\"\"#\"\"\"F+*&\"\"$F+F)F+F+\"\"%!\"\"F+F)F/,&F)F+F*F+!\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqn1 := rationalFunction = A /x+B/(x+2)+C/(x+2)^2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/*(, (*$)%\"xG\"\"#\"\"\"F,*&\"\"$F,F*F,F,\"\"%!\"\"F,F*F0,&F*F,F+F,!\"#,(* &%\"AGF,F*F0F,*&%\"BGF,F1F0F,*&%\"CGF,F1F2F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eqn2 := rationalFunction = normal(rhs(eqn1));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/*(,(*$)%\"xG\"\"#\"\"\"F,*& \"\"$F,F*F,F,\"\"%!\"\"F,F*F0,&F*F,F+F,!\"#*(,.*&%\"AGF,F)F,F,*(F/F,F6 F,F*F,F,*&F/F,F6F,F,*&%\"BGF,F)F,F,*(F+F,F:F,F*F,F,*&%\"CGF,F*F,F,F,F* F0F1F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eqn3 := numer(lhs (eqn2)) = numer(rhs(eqn2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3 G/,(*$)%\"xG\"\"#\"\"\"F+*&\"\"$F+F)F+F+\"\"%!\"\",.*&%\"AGF+F(F+F+*(F .F+F2F+F)F+F+*&F.F+F2F+F+*&%\"BGF+F(F+F+*(F*F+F6F+F)F+F+*&%\"CGF+F)F+F +" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn4 := subs(x=0, eqn3 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/!\"%,$*&\"\"%\"\"\"%\"A GF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn5 := A = solve( eqn4, A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/%\"AG!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn6 := subs(x=-2, eqn3);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/!\"',$*&\"\"#\"\"\"%\"CGF*! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn7 := C = solve(e qn6, C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn7G/%\"CG\"\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn8 := subs(\{eqn5, eqn7\}, eqn3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn8G/,(*$)%\"xG\"\"#\" \"\"F+*&\"\"$F+F)F+F+\"\"%!\"\",,F'F/F)F/F.F/*&%\"BGF+F(F+F+*(F*F+F2F+ F)F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eqn9 := lhs(eqn8) = collect(rhs(eqn8), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn9G/ ,(*$)%\"xG\"\"#\"\"\"F+*&\"\"$F+F)F+F+\"\"%!\"\",(*&,&%\"BGF+F+F/F+F(F +F+*&,&*&F*F+F3F+F+F+F/F+F)F+F+F.F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "By comparing the coefficients of " } {XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 27 " on each side, we \+ see that " }{XPPEDIT 18 0 "B-1 = 1;" "6#/,&%\"BG\"\"\"F&!\"\"F&" } {TEXT -1 6 " or " }{XPPEDIT 18 0 "B = 2;" "6#/%\"BG\"\"#" }{TEXT -1 36 ". (Notice that with this value of " }{XPPEDIT 18 0 "B;" "6#%\"BG " }{TEXT -1 22 ", the coefficients of " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 17 ", namely 3 and " }{XPPEDIT 18 0 "2*B-1;" "6#,&*&\"\"# \"\"\"%\"BGF&F&F&!\"\"" }{TEXT -1 24 ", also equal each other." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eqn10 := subs(\{A = -1, B = \+ 2, C = 3\}, rationalFunction = A/x+B/(x+2)+C/(x+2)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqn10G/*(,(*$)%\"xG\"\"#\"\"\"F,*&\"\"$F,F*F,F, \"\"%!\"\"F,F*F0,&F*F,F+F,!\"#,(*&F+F,F1F0F,*&F.F,F1F2F,*&F,F,F*F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Int(rhs(eqn10), x = 1 .. \+ 2) = int(rhs(eqn10), x = 1 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%$IntG6$,(*&\"\"#\"\"\",&%\"xGF*F)F*!\"\"F**&\"\"$F*F+!\"#F**&F*F*F,F -F-/F,;F*F),(#F*\"\"%F**&F/F*-%#lnG6#F)F*F**&F)F*-F96#F/F*F-" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 261 1 " " }{TEXT 270 17 "14. Calculate \+ " }{XPPEDIT 564 0 "Int(sin(x)^3*cos(x)^2,x = 0 .. Pi);" "6#-%$IntG6$ *&-%$sinG6#%\"xG\"\"$-%$cosG6#F*\"\"#/F*;\"\"!%#PiG" }{TEXT 563 1 "." }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 408 80 "a) 1/1 5 b) 2/15 c) 1/5 d) 4/15 e) 1 /3 " }{TEXT 411 1 " " }{TEXT 412 18 " " }}{PARA 0 " " 0 "" {TEXT 409 39 "f) 2/5 g) 7/15 " }{TEXT -1 1 " " }{TEXT 410 42 "h) 8/15 i) 3/5 j) 2/3 " } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 656 15 "Solution ( d )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "J := Int(sin(x)^3*cos(x)^2,x = 0 .. Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"%\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "J1 := Int((1-cos(x)^2)*cos(x)^2*sin(x),x = 0 .. Pi); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J1G-%$IntG6$*(,&\"\"\"F**$)-%$c osG6#%\"xG\"\"#F*!\"\"F*F,F*-%$sinGF/F*/F0;\"\"!%#PiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "K := student[changevar](u = cos(x), J1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%$IntG6$*&,&\"\"\"F **$)%\"uG\"\"#F*!\"\"F*F,F*/F-;F/F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"%\"# :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 0 {PARA 260 "" 0 "" {TEXT -1 38 "15. Use the reduction formula \n " }{XPPEDIT 18 0 "Int(x^n*exp(-x^2),x = 0 .. 1) = (n-1)/2*In t(x^(n-2)*exp(-x^2),x = 0 .. 1)-1/2/exp(1);" "6#/-%$IntG6$*&)%\"xG%\"n G\"\"\"-%$expG6#,$*$F)\"\"#!\"\"F+/F);\"\"!F+,&*(,&F*F+F+F2F+F1F2-F%6$ *&)F),&F*F+F1F2F+-F-6#,$*$F)F1F2F+/F);F5F+F+F+*(F+F+F1F2-F-6#F+F2F2" } {TEXT -1 25 "\n\n and the approximation " }{XPPEDIT 18 0 "Int(exp(-x^2 ),x = 0 .. 1);" "6#-%$IntG6$-%$expG6#,$*$%\"xG\"\"#!\"\"/F+;\"\"!\"\" \"" }{TEXT -1 2 " " }{OLE 1 4612 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>: B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::fyyyyyq:nY^:f:v:nYnyyyYyA::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^ H_mvJ::::::::gj:;JZC:bKB::>`hgyP_mO>s;V:>b@Z:j:VBYmp>HYL kNG>::::::::N:F:YLpfF]KRnC=MtFGgml>:;::::::JGNZ; F:V:;B:yayA:<::::::?:CR;:;::::::wJ;@jyyyyyI:;Z::: :::jEB:EJ:>:F:;Jyk;B:CB:;jj>J?>:Q:S:UJ:n;v;;JBB:]:_J:V<^:;`:Z@w:>d=>:[G;nsHJfu:;B::yay=:::::::::::::::::::::::::::::::::::::: :::::::::::J:B::::::::vYxI:;Z::::::::jysy::::::::::::::::::::::::::::: ::>:jy;::B:>::xIyA::>:<:yay=:J:B:::vYxI::B:>:xIyAZ:J:ryvY:<:;Zymy;::;Z :::jysy::>:<:yay=::<:;:Zymy;::B:Zy=::::::::::::::::::::::::::::::::::: ::::::::::::::J:B::::::::::::::j:>:;::::::::::F:Y:<::::::::=J:DZ:B:::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::5:" }{TEXT -1 3 " \+ " }{XPPEDIT 18 0 "3/4;" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 21 " \+ to approximate " }{XPPEDIT 18 0 "Int(x^4*exp(-x^2),x = 0 .. 1)" "6# -%$IntG6$*&%\"xG\"\"%-%$expG6#,$*$F'\"\"#!\"\"\"\"\"/F';\"\"!F0" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 289 4 "a) " }{XPPEDIT 256 0 "1/2-1/4/exp(1);" "6#,&*&\"\"\"F%\" \"#!\"\"F%*(F%F%\"\"%F'-%$expG6#F%F'F'" }{TEXT 669 1 " " }{TEXT 670 12 " b) " }{XPPEDIT 256 0 "1/2-3/4/exp(1);" "6#,&*&\"\"\"F%\" \"#!\"\"F%*(\"\"$F%\"\"%F'-%$expG6#F%F'F'" }{TEXT 671 1 " " }{TEXT 672 13 " c) " }{XPPEDIT 256 0 "3/4-1/2/exp(1);" "6#,&*&\"\"$ \"\"\"\"\"%!\"\"F&*(F&F&\"\"#F(-%$expG6#F&F(F(" }{TEXT 667 1 " " } {TEXT 668 10 " d) " }{XPPEDIT 257 0 "3/4-3/2/exp(1);" "6#,&*&\" \"$\"\"\"\"\"%!\"\"F&*(F%F&\"\"#F(-%$expG6#F&F(F(" }{TEXT 665 1 " " } {TEXT 666 9 " e)" }{TEXT 600 2 " " }{XPPEDIT 257 0 "9/16-3/4/ex p(1);" "6#,&*&\"\"*\"\"\"\"#;!\"\"F&*(\"\"$F&\"\"%F(-%$expG6#F&F(F(" } {TEXT 663 2 " " }{TEXT 664 1 " " }{TEXT 292 18 " " } }{PARA 0 "" 0 "" {TEXT 290 4 "f) " }{XPPEDIT 659 0 "9/16-5/4/exp(1); " "6#,&*&\"\"*\"\"\"\"#;!\"\"F&*(\"\"&F&\"\"%F(-%$expG6#F&F(F(" } {TEXT 658 12 " g) " }{XPPEDIT 257 0 "5/8-5/4/exp(1);" "6#,&*& \"\"&\"\"\"\"\")!\"\"F&*(F%F&\"\"%F(-%$expG6#F&F(F(" }{TEXT 661 9 " \+ " }{TEXT -1 1 " " }{TEXT 291 4 "h) " }{XPPEDIT 256 0 "5/8-3/4/e xp(1);" "6#,&*&\"\"&\"\"\"\"\")!\"\"F&*(\"\"$F&\"\"%F(-%$expG6#F&F(F( " }{TEXT 662 12 " i) " }{TEXT 675 1 " " }{XPPEDIT 256 0 "3/4-1 /exp(1);" "6#,&*&\"\"$\"\"\"\"\"%!\"\"F&*&F&F&-%$expG6#F&F(F(" }{TEXT 673 1 " " }{TEXT 674 1 " " }{TEXT 676 11 " j)" }{TEXT -1 2 " \+ " }{XPPEDIT 257 0 "9/16-1/exp(1);" "6#,&*&\"\"*\"\"\"\"#;!\"\"F&*&F&F& -%$expG6#F&F(F(" }{TEXT 677 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 660 15 "Solution ( f )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "J := n \+ -> Int(x^n*exp(-x^2),x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"JGf*6#%\"nG6\"6$%)operatorG%&arrowGF(-%$IntG6$*&)%\"xG9$\"\"\"-%$ex pG6#,$*$)F1\"\"#F3!\"\"F3/F1;\"\"!F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "Reduction := n-> J(n) = (n-1)/2*J(n-2)-1/2/exp(1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*ReductionGf*6#%\"nG6\"6$%)opera torG%&arrowGF(/-%\"JG6#9$,&*&#\"\"\"\"\"#F4*&,&F0F4F4!\"\"F4-F.6#,&F0F 4F5F8F4F4F4*&#F4F5F4*&F4F4-%$expG6#F4F8F4F8F(F(F(" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "eqn := Reduction(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%$IntG6$*&)%\"xG\"\"%\"\"\"-%$expG6#,$*$)F+\" \"#F-!\"\"F-/F+;\"\"!F-,&*&#\"\"$F4F--F'6$*&F3F-F.F-F6F-F-*&#F-F4F-*&F -F--F/6#F-F5F-F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Reducti on(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)%\"xG\"\"#\"\" \"-%$expG6#,$*$F(F+!\"\"F+/F);\"\"!F+,&*&#F+F*F+-F%6$F,F2F+F+*&#F+F*F+ *&F+F+-F-6#F+F1F+F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn2 := subs(Reduction(2), eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn 2G/-%$IntG6$*&)%\"xG\"\"%\"\"\"-%$expG6#,$*$)F+\"\"#F-!\"\"F-/F+;\"\"! F-,&*&#\"\"$F,F--F'6$F.F6F-F-*&#\"\"&F,F-*&F-F--F/6#F-F5F-F5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "approximation := Int(exp(-x ^2),x = 0 .. 1) = 3/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.approxima tionG/-%$IntG6$-%$expG6#,$*$)%\"xG\"\"#\"\"\"!\"\"/F/;\"\"!F1#\"\"$\" \"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eqn3 := subs(approxi mation, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%$IntG6$*& )%\"xG\"\"%\"\"\"-%$expG6#,$*$)F+\"\"#F-!\"\"F-/F+;\"\"!F-,&#\"\"*\"#; F-*&#\"\"&F,F-*&F-F--F/6#F-F5F-F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 272 14 "16. Calculate " }{XPPEDIT 580 0 "int(sqrt(1-x^2),x = 1/sqrt(2) .. 1);" "6#-%$intG6$-%%sqrtG6#,&\"\"\"F **$%\"xG\"\"#!\"\"/F,;*&F*F*-F'6#F-F.F*" }{TEXT 579 1 "." }}{PARA 3 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 657 15 "Solution ( a )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 288 3 "a) " }{XPPEDIT 256 0 "(Pi-2)/8;" "6#*&,&%#PiG\"\"\"\"\"#! \"\"F&\"\")F(" }{TEXT 581 12 " b) " }{XPPEDIT 257 0 "(Pi-2)/4; " "6#*&,&%#PiG\"\"\"\"\"#!\"\"F&\"\"%F(" }{TEXT 582 14 " c) \+ " }{XPPEDIT 258 0 "(Pi-2)/2;" "6#*&,&%#PiG\"\"\"\"\"#!\"\"F&F'F(" } {TEXT 583 15 " d) " }{XPPEDIT 256 0 "(Pi-1)/8;" "6#*&,&%#Pi G\"\"\"F&!\"\"F&\"\")F'" }{TEXT 585 13 " e) " }{XPPEDIT 256 0 "(Pi-1)/4;" "6#*&,&%#PiG\"\"\"F&!\"\"F&\"\"%F'" }{TEXT 586 10 " \+ \nf) " }{XPPEDIT 261 0 "(Pi-1)/2;" "6#*&,&%#PiG\"\"\"F&!\"\"F&\"\"#F' " }{TEXT 584 15 " g) " }{XPPEDIT 256 0 "(4-Pi)/8;" "6#*&,& \"\"%\"\"\"%#PiG!\"\"F&\"\")F(" }{TEXT 587 13 " h) " } {XPPEDIT 256 0 "(4-Pi)/4;" "6#*&,&\"\"%\"\"\"%#PiG!\"\"F&F%F(" }{TEXT 588 13 " i) " }{XPPEDIT 256 0 "(4-Pi)/2;" "6#*&,&\"\"%\"\"\"% #PiG!\"\"F&\"\"#F(" }{TEXT 589 14 " j) " }{XPPEDIT 265 0 "(8 -Pi)/8;" "6#*&,&\"\")\"\"\"%#PiG!\"\"F&F%F(" }{TEXT 590 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "J := Int((1-x^2)^(1/2),x = 1/sqrt(2 )..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*$,&\"\"\"F** $)%\"xG\"\"#F*!\"\"#F*F./F-;,$*&F.F/F.F0F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "J1 := student[changevar](x = sin(t), J, t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J1G-%$IntG6$*&,&\"\"\"F**$)-%$sinG6 #%\"tG\"\"#F*!\"\"#F*F1-%$cosGF/F*/F0;,$*&\"\"%F2%#PiGF*F*,$*&F1F2F;F* F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "J1 := Int(cos(t)^2, t = Pi/4 .. Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J1G-%$IntG6$*$ )-%$cosG6#%\"tG\"\"#\"\"\"/F-;,$*&\"\"%!\"\"%#PiGF/F/,$*&F.F5F6F/F/" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "J1 := Int((1+cos(2*t))/2, \+ t = Pi/4 .. Pi/2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J1G-%$IntG6$ ,&#\"\"\"\"\"#F**&F)F*-%$cosG6#,$*&F+F*%\"tGF*F*F*F*/F2;,$*&\"\"%!\"\" %#PiGF*F*,$*&F+F8F9F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " value(J1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\")!\"\"%#PiG\"\" \"F(#F(\"\"%F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 274 4 "17. " }{TEXT 286 12 " Calculate " }{XPPEDIT 256 0 "int(x^2/sqrt(1+x^2),x = 0 .. 1) ;" "6#-%$intG6$*&%\"xG\"\"#-%%sqrtG6#,&\"\"\"F-*$F'F(F-!\"\"/F';\"\"!F -" }{TEXT 591 1 "." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 602 4 "a) " }{XPPEDIT 278 0 "1-ln(1+sqrt(2))/2; " "6#,&\"\"\"F$*&-%#lnG6#,&F$F$-%%sqrtG6#\"\"#F$F$F-!\"\"F." }{TEXT 611 4 " " }}{PARA 0 "" 0 "" {TEXT 624 4 "b) " }{XPPEDIT 257 0 "1+l n(1+sqrt(2))/2;" "6#,&\"\"\"F$*&-%#lnG6#,&F$F$-%%sqrtG6#\"\"#F$F$F-!\" \"F$" }{TEXT 603 22 " " }}{PARA 0 "" 0 "" {TEXT 623 4 "c) " }{XPPEDIT 258 0 "sqrt(2)+ln(1+sqrt(2));" "6#,&-%%sqrtG6# \"\"#\"\"\"-%#lnG6#,&F(F(-F%6#F'F(F(" }{TEXT 604 15 " " }}{PARA 0 "" 0 "" {TEXT 622 4 "d) " }{XPPEDIT 256 0 "sqrt(2)-ln(1+sqr t(2));" "6#,&-%%sqrtG6#\"\"#\"\"\"-%#lnG6#,&F(F(-F%6#F'F(!\"\"" } {TEXT 609 18 " " }}{PARA 0 "" 0 "" {TEXT 621 4 "e) \+ " }{XPPEDIT 259 0 "1-ln(1+sqrt(2));" "6#,&\"\"\"F$-%#lnG6#,&F$F$-%%sqr tG6#\"\"#F$!\"\"" }{TEXT 605 13 " \nf) " }{XPPEDIT 260 0 "1+ln (1+sqrt(2));" "6#,&\"\"\"F$-%#lnG6#,&F$F$-%%sqrtG6#\"\"#F$F$" }{TEXT 606 3 " " }}{PARA 0 "" 0 "" {TEXT 618 4 "g) " }{XPPEDIT 261 0 "1/2- ln(1+1/sqrt(2));" "6#,&*&\"\"\"F%\"\"#!\"\"F%-%#lnG6#,&F%F%*&F%F%-%%sq rtG6#F&F'F%F'" }{TEXT 607 4 " " }}{PARA 0 "" 0 "" {TEXT 619 4 "h) \+ " }{XPPEDIT 262 0 "1/2+ln(1+1/sqrt(2));" "6#,&*&\"\"\"F%\"\"#!\"\"F%-% #lnG6#,&F%F%*&F%F%-%%sqrtG6#F&F'F%F%" }{TEXT 610 6 " " }}{PARA 0 "" 0 "" {TEXT 625 4 "i) " }{XPPEDIT 263 0 "sqrt(2)/2-ln(1+sqrt(2))/2; " "6#,&*&-%%sqrtG6#\"\"#\"\"\"F(!\"\"F)*&-%#lnG6#,&F)F)-F&6#F(F)F)F(F* F*" }{TEXT 608 6 " " }}{PARA 0 "" 0 "" {TEXT 620 4 "j) " } {XPPEDIT 264 0 "sqrt(2)/2+ln(1+sqrt(2))/2;" "6#,&*&-%%sqrtG6#\"\"#\"\" \"F(!\"\"F)*&-%#lnG6#,&F)F)-F&6#F(F)F)F(F*F)" }{TEXT 612 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 626 15 "Solution ( i )" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 704 64 "Method 1 (Using the form ula given at the beginning of the exam.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "J := Int(x^2/sqrt(1+x ^2),x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&% \"xG\"\"#,&\"\"\"F,*$)F)F*F,F,#!\"\"F*/F);\"\"!F," }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 40 "K := student[changevar](x=tan(t), J, t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%$IntG6$*&)-%$tanG6#%\"tG\"\"# \"\"\",&F/F/*$F)F/F/#F/F./F-;\"\"!,$*&\"\"%!\"\"%#PiGF/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "L := subs(tan(t)^2 = sec(t)^2-1, K) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG-%$IntG6$*&,&*$)-%$secG6#% \"tG\"\"#\"\"\"F1F1!\"\"F1F*#F1F0/F/;\"\"!,$*&\"\"%F2%#PiGF1F1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Answer := Int(sec(t)^3, t = \+ 0 .. 1/4*Pi)-Int(sec(t), t = 0 .. 1/4*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AnswerG,&-%$IntG6$*$)-%$secG6#%\"tG\"\"$\"\"\"/F.;\" \"!,$*&\"\"%!\"\"%#PiGF0F0F0-F'6$F+F1F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "antiderivative := 1/2*sec(t)*tan(t)+1/2*ln(abs(sec(t) +tan(t)))-ln(abs(sec(t)+tan(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %/antiderivativeG,&*&#\"\"\"\"\"#F(*&-%$secG6#%\"tGF(-%$tanGF-F(F(F(*& #F(F)F(-%#lnG6#-%$absG6#,&F+F(F/F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 78 "prelimAnswer := subs(t = Pi/4, antiderivative) - su bs(t = 0, antiderivative): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Answer := simplify(prelimAnswer);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'AnswerG,&*&\"\"#!\"\"F'#\"\"\"F'F**&#F*F'F*-%#lnG6#,&F*F**$F'F)F *F*F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 705 84 "Method 2 (Without using the given integral formula, as pointed out by Michael Hamm.)" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 19 "We write integral " }{XPPEDIT 18 0 "K;" "6#%\"KG" }{TEXT -1 21 " above in terms of " }{XPPEDIT 18 0 "sin(t);" "6#-%$sinG6#%\"tG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "cos(t);" "6#-%$cosG6#%\"tG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "K1 := Int(sin(t)^2/cos(t)^3, t = 0 .. Pi/4); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#K1G-%$IntG6$*&-%$sinG6#%\"tG\"\"#-%$cosGF+!\"$/ F,;\"\"!,$*&\"\"%!\"\"%#PiG\"\"\"F9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "To motivate the next step, think of the integrand " }{XPPEDIT 18 0 "sin(t)^2/(cos(t)^3);" "6#*&-%$sinG6#%\" tG\"\"#*$)-%$cosG6#F'\"\"$\"\"\"!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dt;" "6#%#dtG" }{TEXT -1 6 " as " }{XPPEDIT 18 0 "sin(t)^2/(cos(t)^ 4);" "6#*&-%$sinG6#%\"tG\"\"#*$-%$cosG6#F'\"\"%!\"\"" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "cos(t)*dt;" "6#*&-%$cosG6#%\"tG\"\"\"%#dtGF(" } {TEXT -1 11 " or " }{XPPEDIT 18 0 "sin(t)^2/((1-sin(t)^2)^2);" "6#*&-%$sinG6#%\"tG\"\"#*$),&\"\"\"F,*$)-%$sinG6#%\"tGF(F,!\"\"F(F,F3 " }{TEXT -1 2 " " }{XPPEDIT 18 0 "cos(t)*dt;" "6#*&-%$cosG6#%\"tG\"\" \"%#dtGF(" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "K2 := student[changevar](u = sin(t), K1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#K2G-%$IntG6$*&%\"uG\"\"#,&\"\"\"F,*$)F)F*F,!\"\"!\"# /F);\"\"!,$*&F*F/F*#F,F*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "K3 := Int( convert(student[integrand](K2), parfrac, u), u = 0..sqr t(2)/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#K3G-%$IntG6$,**&\"\"\"F **&\"\"%F*),&%\"uGF*F*F*\"\"#F*!\"\"F**&F*F**&F,F*),&F/F*F*F1F0F*F1F** &F*F**&F,F*F5F*F1F**&F*F**&F,F*F.F*F1F1/F/;\"\"!,$*&F0F1F0#F*F0F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "J = value(K3);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"#,&\"\"\"F+*$)F(F)F+F+#!\" \"F)/F(;\"\"!F+,(*&#F+\"\"%F+-%#lnG6#,&F)F+*$F)#F+F)F/F+F+*&F)F/F)F \+ " 0 "" {MPLTEXT 1 0 31 "R := (4*x^2+2*x+4)/((x^2+1)^2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"RG*&,(*&\"\"%\"\"\")%\"xG\"\"#F)F)*&F,F)F+F) F)F(F)F),&*$F*F)F)F)F)!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "eqn := R = (A*x+B)/(x^2+1) + (C*x+D)/(x^2+1)^2;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$eqnG/*&,(*&\"\"%\"\"\")%\"xG\"\"#F*F**&F-F*F,F*F*F )F*F*,&*$F+F*F*F*F*!\"#,&*&,&*&%\"AGF*F,F*F*%\"BGF*F*F/!\"\"F**&,&*&% \"CGF*F,F*F*%\"DGF*F*F/F1F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn2 := R = normal(rhs(eqn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%eqn2G/*&,(*&\"\"%\"\"\")%\"xG\"\"#F*F**&F-F*F,F*F*F)F*F*,&*$F+F*F*F* F*!\"#*&,.*&%\"AGF*)F,\"\"$F*F**&F5F*F,F*F**&%\"BGF*F+F*F*F:F**&%\"CGF *F,F*F*%\"DGF*F*F/F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "eqn 3 := numer(lhs(eqn2)) = collect(numer(rhs(eqn2)), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,(*&\"\"%\"\"\")%\"xG\"\"#F)F)*&F,F)F+F)F )F(F),,*&%\"AGF))F+\"\"$F)F)*&%\"BGF)F*F)F)*&,&%\"CGF)F0F)F)F+F)F)%\"D GF)F4F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "By matching the coefficients of like powers on each side of eqn3 we obtain " }{XPPEDIT 18 0 "A = 0,B = 4;" "6$/%\"AG\"\"!/%\"BG\"\" %" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 35 "eqn4 := subs(\{A = 0, B = 4\}, eqn3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/,(*&\"\"%\"\"\")%\"xG\"\"#F)F )*&F,F)F+F)F)F(F),*F(F)*&F(F)F*F)F)*&%\"CGF)F+F)F)%\"DGF)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 80 "By matching the coefficients of like powers on each sid e of eqn3 we obtain " }{XPPEDIT 18 0 "C = 2,D = 0;" "6$/%\"CG\"\" #/%\"DG\"\"!" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "partialFractionDecomp := subs(\{A = 0, B = 4, C = 2, D = 0\}, rhs( eqn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%6partialFractionDecompG,&* &\"\"%\"\"\",&*$)%\"xG\"\"#F(F(F(F(!\"\"F(*(F-F(F,F(F)!\"#F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "antiDerivative := int(partia lFractionDecomp, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/antiDerivat iveG,&*&\"\"%\"\"\"-%'arctanG6#%\"xGF(F(*&F(F(,&*$)F,\"\"#F(F(F(F(!\" \"F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "preliminaryAnswer : = subs(x=1,antiDerivative) - subs(x=0,antiDerivative); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%2preliminaryAnswerG,(*&\"\"%\"\"\"-%'arctanG6# F(F(F(#F(\"\"#F(*&F'F(-F*6#\"\"!F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Answer := simplify(preliminaryAnswer);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'AnswerG,&#\"\"\"\"\"#F'%#PiGF'" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 707 45 "Method 1 (Using a Trigonometric Substitution)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "J := Int((4*x^2+2*x+4)/((x^2+1)^2),x = 0 .. 1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&,(*&\"\"%\"\"\")%\"xG \"\"#F,F,*&F/F,F.F,F,F+F,F,,&F,F,*$F-F,F,!\"#/F.;\"\"!F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "K := student[changevar](x=tan(t), J , t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%$IntG6$*&,(*&\"\"%\" \"\")-%$tanG6#%\"tG\"\"#F,F,*&F2F,F.F,F,F+F,F,,&F,F,*$F-F,F,!\"\"/F1; \"\"!,$*&F+F6%#PiGF,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "Notice that the sum of the first and third summands in the numerator is 4 times the denominator. Therefore" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "Int((4*x^2+2*x+4)/((x^2+1)^2),x = 0 .. 1) = I nt(4 + 2*tan(t)/(1+tan(t)^2), t = 0 .. Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,(*&\"\"%\"\"\")%\"xG\"\"#F+F+*&F.F+F-F+F+F *F+F+,&F+F+*$F,F+F+!\"#/F-;\"\"!F+-F%6$,&F*F+*(F.F+-%$tanG6#%\"tGF+,&F +F+*$)F:F.F+F+!\"\"F+/F=;F5,$*&F*FA%#PiGF+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Int((4*x^2+2 *x+4)/((x^2+1)^2),x = 0 .. 1) = Pi + Int(2*tan(t)/(1+tan(t)^2), t = 0 \+ .. Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,(*&\"\"%\"\" \")%\"xG\"\"#F+F+*&F.F+F-F+F+F*F+F+,&F+F+*$F,F+F+!\"#/F-;\"\"!F+,&%#Pi GF+-F%6$,$*(F.F+-%$tanG6#%\"tGF+,&F+F+*$)F " 0 "" {MPLTEXT 1 0 114 "Int((4*x^2+2*x+4)/(1+x^ 2)^2,x = 0 .. 1) = Pi+Int(2*simplify(convert(tan(t)/(1+tan(t)^2),sinco s)),t = 0 .. 1/4*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&, (*&\"\"%\"\"\")%\"xG\"\"#F+F+*&F.F+F-F+F+F*F+F+,&F+F+*$F,F+F+!\"#/F-; \"\"!F+,&%#PiGF+-F%6$,$*(F.F+-%$sinG6#%\"tGF+-%$cosGF>F+F+/F?;F5,$*&F* !\"\"F7F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Int((4*x^2+2 *x+4)/(1+x^2)^2,x = 0 .. 1) = Pi+Int(sin(2*t),t = 0 .. 1/4*Pi);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,(*&\"\"%\"\"\")%\"xG\"\"# F+F+*&F.F+F-F+F+F*F+F+,&F+F+*$F,F+F+!\"#/F-;\"\"!F+,&%#PiGF+-F%6$-%$si nG6#,$*&F.F+%\"tGF+F+/F?;F5,$*&F*!\"\"F7F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Int((4*x^2+2*x+4)/ (1+x^2)^2,x = 0 .. 1) = value(Pi+Int(sin(2*t),t = 0 .. 1/4*Pi));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,(*&\"\"%\"\"\")%\"xG\"\"# F+F+*&F.F+F-F+F+F*F+F+,&F+F+*$F,F+F+!\"#/F-;\"\"!F+,&%#PiGF+#F+F.F+" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 260 "" 0 "" {TEXT 277 55 "19. The region in the first qu adrant bounded above by " }{XPPEDIT 18 0 "y = 2-x;" "6#/%\"yG,&\"\"# \"\"\"%\"xG!\"\"" }{TEXT 598 17 " and below by " }{XPPEDIT 18 0 "y \+ = sqrt(x);" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT 599 88 " is rotated abou t the x-axis. What is the volume of the resulting solid of revolution ?" }}{PARA 260 "" 0 "" {TEXT 597 5 "a) " }{XPPEDIT 18 0 "2*Pi;" "6#* &\"\"#\"\"\"%#PiGF%" }{TEXT 421 12 " b) " }{XPPEDIT 265 0 "11* Pi/6;" "6#*(\"#6\"\"\"%#PiGF%\"\"'!\"\"" }{TEXT 413 14 " c) \+ " }{XPPEDIT 266 0 "5*Pi/3;" "6#*(\"\"&\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 414 12 " d) " }{XPPEDIT 256 0 "3*Pi/2;" "6#*(\"\"$\"\"\"%#PiGF %\"\"#!\"\"" }{TEXT 419 13 " e) " }{XPPEDIT 267 0 "4*Pi/3;" " 6#*(\"\"%\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 415 12 " \nf) " } {XPPEDIT 268 0 "5*Pi/2;" "6#*(\"\"&\"\"\"%#PiGF%\"\"#!\"\"" }{TEXT 416 14 " g) " }{XPPEDIT 273 0 "8*Pi/3;" "6#*(\"\")\"\"\"%#Pi GF%\"\"$!\"\"" }{TEXT 417 14 " h) " }{XPPEDIT 274 0 "9*Pi/4; " "6#*(\"\"*\"\"\"%#PiGF%\"\"%!\"\"" }{TEXT 420 15 " i) " } {XPPEDIT 275 0 "7*Pi/4;" "6#*(\"\"(\"\"\"%#PiGF%\"\"%!\"\"" }{TEXT 418 13 " j) " }{XPPEDIT 276 0 "5*Pi/4;" "6#*(\"\"&\"\"\"%#PiG F%\"\"%!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 689 15 "Solution ( b )" }{TEXT -1 0 "" }}{PARA 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}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "`Volume = `*Pi*Int( (2-x)^2- '(sqrt(x))^2',x=0..1) = Pi*int( (2-x) ^2- (sqrt(x))^2,x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Volum e~=~G\"\"\"%#PiGF&-%$IntG6$,&*$),&\"\"#F&%\"xG!\"\"F/F&F&*$)-%%sqrtG6# F0F/F&F1/F0;\"\"!F&F&,$*(\"#6F&\"\"'F1F'F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 280 57 "20. The region \+ above the x-axis and under the parabola " }{XPPEDIT 594 0 "y = 1-(x- 2)^2;" "6#/%\"yG,&\"\"\"F&*$,&%\"xGF&\"\"#!\"\"F*F+" }{TEXT 592 6 " \+ , " }{XPPEDIT 595 0 "1 < x;" "6#2\"\"\"%\"xG" }{XPPEDIT 596 0 "`` <= \+ 3;" "6#1%!G\"\"$" }{TEXT 593 89 " is rotated about the y-axis. What \+ is the volume of the resulting solid of revolution?\n" }}{PARA 260 "" 0 "" {TEXT 279 5 "a) " }{XPPEDIT 427 1 "5*Pi/3;" "6#*(\"\"&\"\"\"%#P iGF%\"\"$!\"\"" }{TEXT 422 13 " b) " }{XPPEDIT 428 1 "8*Pi/3; " "6#*(\"\")\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 281 15 " c) " } {XPPEDIT 429 1 "10*Pi/3;" "6#*(\"#5\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 282 12 " d) " }{XPPEDIT 430 1 "4*Pi;" "6#*&\"\"%\"\"\"%#PiGF% " }{TEXT 283 11 " e) " }{XPPEDIT 431 0 "9*Pi/2;" "6#*(\"\"*\"\" \"%#PiGF%\"\"#!\"\"" }{TEXT 423 10 " \nf) " }{XPPEDIT 432 1 "5*Pi ;" "6#*&\"\"&\"\"\"%#PiGF%" }{TEXT 424 13 " g) " }{XPPEDIT 18 0 "16*Pi/3;" "6#*(\"#;\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 284 14 " \+ h) " }{XPPEDIT 433 0 "21*Pi/4;" "6#*(\"#@\"\"\"%#PiGF%\"\"%!\"\" " }{TEXT 285 11 " i) " }{XPPEDIT 434 0 "6*Pi;" "6#*&\"\"'\"\"\" %#PiGF%" }{TEXT 425 13 " j) " }{XPPEDIT 435 1 "20*Pi/3;" "6#* (\"#?\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 426 2 " " }{TEXT 278 1 "\n" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 690 15 "Solution ( g )" }{TEXT -1 0 "" }}{PARA 0 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"6$,&\"\"#\"\"\"*$,&F%F%%\"yG!\"\"#F%F$F% ,&F$F%F&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "`Volume = `* Int(Pi*((2+(1-y)^(1/2))^2 - (2-(1-y)^(1/2))^2), y = 0..1) = int(Pi*((2 +(1-y)^(1/2))^2 - (2-(1-y)^(1/2))^2), y = 0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%*Volume~=~G\"\"\"-%$IntG6$*&%#PiGF&,&*$),&\"\"#F&*$ ,&F&F&%\"yG!\"\"#F&F0F&F0F&F&*$),&F0F&F1F4F0F&F4F&/F3;\"\"!F&F&,$*(\"# ;F&\"\"$F4F+F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 37 " \+ #20 Figure 5: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 21 "21. Code for Fi gures" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 708 58 "#19 Figure 2: Outl ine of Solid of Revolution with Washer" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 626 "outerS urface := plot3d([(2-y)*cos(t),y,(2-y)*sin(t)], y=0..1, t= 0..2*Pi, st yle=WIREFRAME,color=WHEAT):\ninnerSurface := plot3d([sqrt(y)*cos(t),y, sqrt(y)*sin(t)], y=0..1, t= 0..2*Pi, style=WIREFRAME, color=TAN):\nbas e := plot3d([rho*cos(t),0,rho*sin(t)], rho=0..2, t= 0..2*Pi, style=WIR EFRAME, color=WHEAT):\nc := 1/2:\nwasherFace := plot3d([rho*cos(t),c, rho*sin(t)], rho=sqrt(c)..2-c, t= 0..2*Pi, style=PATCH):\nwasherSide : = plot3d([(2-c)*cos(t),y,(2-c)*sin(t)], y=c-0.04..c, t= 0..2*Pi, style =PATCHNOGRID,color=COLOR(RGB,.4,.4,.4)):\nplots[display](outerSurface, innerSurface, base, washerFace, washerSide, orientation=[12,90]);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 711 58 "#20 Figure 5: Outline of Solid of Revo lution with Washer" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "r := 'r':\nh := 1/3:\nwashe rtop := plot3d([r*cos(t),r*sin(t),h], r = 2 - (1-h)^(1/2)..2 + (1-h)^( 1/2), t=0..2*Pi, style=PATCH, orientation = [45,50]):\nwashertop := pl ot3d([r*cos(t),r*sin(t),h], r = 2 - (1-h)^(1/2)..2 + (1-h)^(1/2), t=0. .2*Pi, style=PATCH, orientation = [45,50]):\nwasherside := plot3d([(2 \+ + (1-h)^(1/2))*cos(t),(2 + (1-h)^(1/2))*sin(t),z], z = h-0.1 .. h, t=0 ..2*Pi, style=PATCHNOGRID):\nplots[display](surface, washertop, washer side, orientation = [50,64]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "20 19 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }