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"Heading 1" -1 260 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 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "T imes" 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 333 36 " Math 132 \n Fall 2006 Exam I" }}{PARA 0 "" 0 "" {TEXT 256 3 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 258 1 "1" }{TEXT 264 2 ". " }{TEXT 334 16 " A Riemann su m " }{XPPEDIT 383 0 "sum(f(xi[j])*Delta*x,j = 1 .. N);" "6#-%$sumG6$* (-%\"fG6#&%#xiG6#%\"jG\"\"\"%&DeltaGF.%\"xGF./F-;F.%\"NG" }{TEXT 382 19 " for a function " }{XPPEDIT 380 0 "f;" "6#%\"fG" }{TEXT 378 18 " on an interval " }{XPPEDIT 381 0 "[a, b];" "6#7$%\"aG%\"bG" } {TEXT 379 18 " is said to be a " }{TEXT 431 17 "lower Riemann sum" } {TEXT 432 16 " if, for each " }{XPPEDIT 385 0 "j;" "6#%\"jG" }{TEXT 384 14 ", the point " }{XPPEDIT 387 0 "xi[j];" "6#&%#xiG6#%\"jG" } {TEXT 386 10 " in the " }{XPPEDIT 552 0 "j;" "6#%\"jG" }{XPPEDIT 18 0 "` `^th;" "6#)%\"~G%#thG" }{TEXT 553 34 " subinterval is chosen so that " }{XPPEDIT 256 0 "f(xi[j])" "6#-%\"fG6#&%#xiG6#%\"jG" }{TEXT 388 55 " is minimized. Calculate the lower Riemann sum for " } {XPPEDIT 391 0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT 390 6 " , " }{XPPEDIT 394 0 "[a, b] = [-3/2, 5/2];" "6#/7$%\"aG%\"bG7$,$* &\"\"$\"\"\"\"\"#!\"\"F-*&\"\"&F+F,F-" }{TEXT 393 11 " , and " } {XPPEDIT 392 0 "N = 4;" "6#/%\"NG\"\"%" }{TEXT 389 61 ". (Use a parti tion of [a,b] into equal length subintervals.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 94 "a) 7/42 b) 2 \+ c) 9/4 d) 5/2 e) 11/4 \nf) 3 \+ " }{TEXT 335 63 " g) 13/4 h) 7/2 i) 15/4 \+ j) 4 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 336 8 "Solution" }{TEXT 337 6 ": (e)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f := x -> x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$) 9$\"\"#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot (f(x),x=-3/2..5/2);" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!3++++++++:!#<$\"3+++++++]AF*7$$!3MLLL $Q6GT\"F*$\"3)oCw[+Og*>F*7$$!3bmm;M!\\pL\"F*$\"3i#=f*>FV(y\"F*7$$!3MLL L))Qj^7F*$\"3?>WU!R(em:F*7$$!3ALLL=Kvl6F*$\"3+CX0m0)*e8F*7$$!3wmm;C2G! 3\"F*$\"3/eg+Vk+n6F*7$$!3OLL$3yO5+\"F*$\"3.j6eOY2-5F*7$$!3i+++vE%)*=*! #=$\"3m;^7R3KX%)FN7$$!3)RLL$3WDT$)FN$\"3mFSX5DldpFN7$$!3'4++]d(Q&\\(FN $\"3ORk%**[$3=cFN7$$!3:mmmm&4`i'FN$\"3]%[;aos%*Q%FN7$$!3ALLL$QW*eeFN$ \"30@$*pGHsKMFN7$$!3w++++()>'*\\FN$\"3))p\"))\\9+i\\#FN7$$!3E++++0\"*H TFN$\"34D5!Q2;cq\"FN7$$!35++++83&H$FN$\"35p4Oxgv&3\"FN7$$!3\\LLL3k(p`# FN$\"3C,*RkH\\iV'!#>7$$!3Anmmmj^N;FN$\"3?OXj&y8\\n#Fcp7$$!3tzmmmYh=()F cp$\"3+M:eqTU,w!#?7$$\"3-+*****\\s]k\"F^q$\"39@fDINE1F!#B7$$\"3U9LLL`d F!)Fcp$\"3yZTBtl>WkF^q7$$\"3'3++]sgam\"FN$\"3aGv^E%fPx#Fcp7$$\"3G+++v \"ep[#FN$\"3'>L*>k4'\\='Fcp7$$\"3#QLLLe/TM$FN$\"3+,rUYNI=6FN7$$\"39LLL eDBJTFN$\"3uPL5X#3nq\"FN7$$\"3Immm;kD!)\\FN$\"3K]\\d(R&H![#FN7$$\"3Qjm m\"f`@'eFN$\"3;O!HKZ%[OMFN7$$\"3%z****\\nZ)HmFN$\"3w(Gq$>!)[&R%FN7$$\" 3ckmm;$y*euFN$\"3iY.&GvNOc&FN7$$\"3f)******R^bJ)FN$\"3h$>/'3&R[\"pFN7$ $\"3'e*****\\5a`\"*FN$\"35.NSv8ty$)FN7$$\"3'o****\\7RV'**FN$\"3Tq0!)>a !)G**FN7$$\"3k*****\\@fk3\"F*$\"3c:%eei$R!=\"F*7$$\"3/LLL`4Nn6F*$\"3aT #[#[#3FO\"F*7$$\"3#*******\\,s`7F*$\"3KAg^9U\"=d\"F*7$$\"3[mm;zM)>L\"F *$\"3[PHx))*zTx\"F*7$$\"3$*******pfa<9F*$\"3vSK1xlV4?F*7$$\"3#HLLeg`!) \\\"F*$\"3\"4D]fgkTC#F*7$$\"3w****\\#G2Ae\"F*$\"3-Mgz%))zL]#F*7$$\"3;L LL$)G[k;F*$\"3EY'4*oK]qFF*7$$\"3#)****\\7yh]F*$\"3k)*f\"RH2.o$F*7$$\"3F LL$e#pa-?F*$\"3FLX;!>%>5SF*7$$\"3!*******Rv&)z?F*$\"3V^[p'Q2eK%F*7$$\" 3ILLLGUYo@F*$\"3\\y7c4rB-ZF*7$$\"3_mmm1^rZAF*$\"3gVvt+KA_]F*7$$\"34++] sI@KBF*$\"3#4*)Rb\"y@RaF*7$$\"34++]2%)38CF*$\"3t')3Tic*H#eF*7$$\"3++++ ++++DF*$\"3+++++++]iF*-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fc[lFb[l-%+AXE SLABELSG6$Q\"x6\"Q!Fh[l-%%VIEWG6$;$!+++++:!\"*$\"+++++DF`\\l%(DEFAULTG " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "a := -3/2: b := 5/2: N := \+ 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Delta := (b-a)/N;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&DeltaG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "(f(-1/2)+f(0)+f(1/2)+f(3/2))*Delta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#6\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "int(x^2,x=-3/2..5/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#>\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 430 15 "2. Calculate " }{XPPEDIT 434 0 "int(sec(theta) ^2,theta = 0 .. Pi/3);" "6#-%$intG6$*$-%$secG6#%&thetaG\"\"#/F*;\"\"!* &%#PiG\"\"\"\"\"$!\"\"" }{TEXT 433 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 445 27 "a) \+ 1 b) " }{XPPEDIT 261 0 "sqrt(2);" "6#-%%sqrtG6#\"\" #" }{TEXT 446 18 " c) " }{XPPEDIT 262 0 "sqrt(3);" "6#-% %sqrtG6#\"\"$" }{TEXT 447 37 " d) 2 e) " } {XPPEDIT 263 0 "3*sqrt(2)/2;" "6#*(\"\"$\"\"\"-%%sqrtG6#\"\"#F%F)!\"\" " }{TEXT 448 9 " \nf) " }{XPPEDIT 264 0 "3*sqrt(3)/2;" "6#*(\"\"$ \"\"\"-%%sqrtG6#F$F%\"\"#!\"\"" }{TEXT 449 15 " g) " } {TEXT 456 1 " " }{XPPEDIT 265 0 "2*sqrt(2);" "6#*&\"\"#\"\"\"-%%sqrtG6 #F$F%" }{TEXT 450 12 " h) " }{XPPEDIT 266 0 "2*sqrt(3);" "6#*& \"\"#\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT 451 13 " i) " }{TEXT 454 1 " " }{XPPEDIT 267 0 "3*sqrt(2);" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"# F%" }{TEXT 452 2 " " }{TEXT 455 12 " j) " }{XPPEDIT 268 0 "3* sqrt(3);" "6#*&\"\"$\"\"\"-%%sqrtG6#F$F%" }{TEXT 444 1 "\n" }{TEXT 453 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 457 8 "Solution" }{TEXT 458 8 ": ( c )\n" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(sec(theta)^2,theta = 0 . . Pi/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"$#\"\"\"\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "subs(theta=Pi/3, tan(theta)) - subs(theta=0, tan(theta)); #alternative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$tanG6#,$*&\"\"$!\"\"%#PiG\"\"\"F,F,-F%6#\"\"!F*" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify( % );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"$#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT 262 3 "3. " }{TEXT 339 1 " " }{TEXT 340 21 " An antiderivat ive of" }{TEXT 395 2 " " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 396 3 " " }{TEXT 398 17 "is the function " }{XPPEDIT 18 0 "proc (x) option s operator, arrow; (3*x^2+4*x+2)/(x^2+x+1) end proc;" "6#f*6#%\"xG7\"6 $%)operatorG%&arrowG6\"*&,(*&\"\"$\"\"\"*$F%\"\"#F/F/*&\"\"%F/F%F/F/F1 F/F/,(*$F%F1F/F%F/F/F/!\"\"F*F*F*" }{TEXT 397 1 " " }{TEXT 554 1 "." } {TEXT 555 9 " If " }{XPPEDIT 18 0 "int(f(x),x = 0 .. b) = 1;" "6# /-%$intG6$-%\"fG6#%\"xG/F*;\"\"!%\"bG\"\"\"" }{TEXT 400 18 ", then w hat is " }{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT 399 1 "?" }}{PARA 3 "" 0 "" {TEXT 338 169 "a) 1 b) 4/3 c) 5/4 \+ d) 5/3 e) 3/2 \nf) 7/4 g) 7/3 \+ h) 9/4 i) 8/3 j) 5/2 " }{TEXT 341 7 " \+ \n" }}{PARA 3 "" 0 "" {TEXT 342 8 "Solution" }{TEXT 343 6 ": (a)\n" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "F := x -> (3*x^2+4*x+2)/(x^ 2+x+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)opera torG%&arrowGF(*&,(*&\"\"$\"\"\")9$\"\"#F0F0*&\"\"%F0F2F0F0F3F0F0,(*$F1 F0F0F2F0F0F0!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " eqn := F(b) - F(0) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&* &,(*&\"\"$\"\"\")%\"bG\"\"#F+F+*&\"\"%F+F-F+F+F.F+F+,(*$F,F+F+F-F+F+F+ !\"\"F+F.F3F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve(eqn, b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 257 17 "4. Calculate " }{XPPEDIT 402 0 "int(diff((7*x^2+9)/( x^2+3),x),x = 0 .. 3);" "6#-%$intG6$-%%diffG6$*&,&*&\"\"(\"\"\"*$%\"xG \"\"#F-F-\"\"*F-F-,&*$F/F0F-\"\"$F-!\"\"F//F/;\"\"!F4" }{TEXT 401 2 " \+ ." }{TEXT 348 2 "\n\n" }{TEXT 349 68 "a) 1 b) 2 \+ c) 3 d) 4 e) " }{TEXT 345 1 "5" }{TEXT 403 2 " \+ " }{TEXT 350 19 " " }}{PARA 0 "" 0 "" {TEXT 346 33 "f) 6 g) 7 " }{TEXT -1 1 " " }{TEXT 347 38 "h) \+ 8 i) 9 j) 10" }{TEXT -1 3 " " }}{PARA 3 "" 0 "" {TEXT 344 1 " " }}{PARA 3 "" 0 "" {TEXT 351 8 "Solution" }{TEXT 352 8 ": ( c )\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "F := x - > (7*x^2+9)/(x^2+3); #This is an antiderivative of the integrand " } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operatorG%&arro wGF(*&,&*&\"\"(\"\"\")9$\"\"#F0F0\"\"*F0F0,&*$F1F0F0\"\"$F0!\"\"F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "F(3) - F(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 517 16 "5. Suppose that " }{XPPEDIT 518 0 "int(f(x),x = 1 .. 4) = 8;" "6#/-%$intG6$-%\"fG6#%\"xG/F*;\"\"\"\"\" %\"\")" }{TEXT 519 7 " and " }{XPPEDIT 520 0 "int(`(`*2+f(x)*`)`,x = -1 .. 4) = 22;" "6#/-%$intG6$,&*&%\"(G\"\"\"\"\"#F*F**&-%\"fG6#%\"xGF *%\")GF*F*/F0;,$F*!\"\"\"\"%\"#A" }{TEXT 521 14 ". What is " } {XPPEDIT 522 0 "int(f(x),x = -1 .. 1);" "6#-%$intG6$-%\"fG6#%\"xG/F);, $\"\"\"!\"\"F-" }{TEXT 523 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 527 66 "a) 1 b) 2 c) 3 d) 4 e)" }{TEXT 530 2 " " }{TEXT 524 1 "5" }{TEXT 529 2 " " }{TEXT 528 19 " " }}{PARA 0 "" 0 "" {TEXT 525 33 "f) 6 g) 7 " }{TEXT -1 1 " " } {TEXT 526 38 "h) 8 i) 9 j) 10" }{TEXT -1 3 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT 531 8 "Solution" }{TEXT 532 8 ": ( d )\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 39 "eqn := int(2+f(x),x = -1 .. 4) = 22; " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%$intG6$,&\"\"#\"\"\"-%\"fG6# %\"xGF+/F/;!\"\"\"\"%\"#A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eqn2 := 10 + int(f(x),x = -1 .. 4) = 22;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,&\"#5\"\"\"-%$intG6$-%\"fG6#%\"xG/F/;!\"\"\" \"%F(\"#A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "eqn3 := 10 + i nt(f(x),x = -1 .. 1) + int(f(x),x = 1 .. 4) = 22;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,(\"#5\"\"\"-%$intG6$-%\"fG6#%\"xG/F/;!\"\"F(F (-F*6$F,/F/;F(\"\"%F(\"#A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "eqn3 := 10 + int(f(x),x = -1 .. 1) + 8 = 22;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,&\"#=\"\"\"-%$intG6$-%\"fG6#%\"xG/F/;!\"\"F(F (\"#A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "solve( eqn3, int(f (x),x = -1 .. 1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 478 17 "6. Suppose that " }{XPPEDIT 479 0 "f(x) = x^2+1;" "6#/-%\"fG6#%\"xG,&*$F'\"\"# \"\"\"F+F+" }{TEXT 480 72 " . The Mean Value Theorem for Integrals as serts that there is a point " }{XPPEDIT 481 0 "c;" "6#%\"cG" }{TEXT 482 38 " in the interval [1,7] such that " }{XPPEDIT 483 0 "f(c) \+ = f[ave];" "6#/-%\"fG6#%\"cG&F%6#%$aveG" }{TEXT 484 10 " where " } {XPPEDIT 485 0 "f[ave];" "6#&%\"fG6#%$aveG" }{TEXT 486 27 " is the av erage value of " }{XPPEDIT 487 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT 488 6 " for " }{XPPEDIT 489 0 "x;" "6#%\"xG" }{TEXT 490 34 " in the \+ interval [1,7]. What is " }{XPPEDIT 491 0 "c;" "6#%\"cG" }{TEXT 492 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 495 1 " \n" }{TEXT 496 4 "a) " }{XPPEDIT 505 0 "sqrt(5);" "6#-%%sqrtG6#\"\"& " }{TEXT 499 17 " b) " }{XPPEDIT 506 0 "sqrt(6);" "6#-%%s qrtG6#\"\"'" }{TEXT 500 33 " c) 3 d) " } {XPPEDIT 507 0 "sqrt(10);" "6#-%%sqrtG6#\"#5" }{TEXT 504 15 " \+ e) " }{XPPEDIT 508 0 "2*sqrt(3);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"\"$F% " }{TEXT 498 2 " " }{TEXT 497 20 " \n" }{TEXT 493 5 "f) " }{XPPEDIT 509 0 "sqrt(14);" "6#-%%sqrtG6#\"#9" }{TEXT 503 16 " g) " }{XPPEDIT 510 0 "sqrt(15);" "6#-%%sqrtG6#\"#:" } {TEXT 502 10 " " }{TEXT 511 1 " " }{TEXT 494 19 "h) 4 \+ i) " }{XPPEDIT 512 0 "sqrt(19);" "6#-%%sqrtG6#\"#>" }{TEXT 501 16 " j) " }{XPPEDIT 513 0 "sqrt(21);" "6#-%%sqrtG6#\"#@" } {TEXT 514 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 515 8 "Solution" }{TEXT 516 8 ": ( i )\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f := x -> x^2+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"#\"\"\"F1F1F1F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "m := int(f(x),x=1..7)/( 7-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(f(c) = m, c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*$\"#>#\"\"\"\"\"#,$F#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 356 19 "7. Calculate " }{XPPEDIT 406 0 "`d `/(d*x);" "6#*& %#d~G\"\"\"*&%\"dGF%%\"xGF%!\"\"" }{TEXT 405 1 " " }{XPPEDIT 257 0 "in t((7*t^2+9)/(t^2+3),t = -1 .. x);" "6#-%$intG6$*&,&*&\"\"(\"\"\"*$%\"t G\"\"#F*F*\"\"*F*F*,&*$F,F-F*\"\"$F*!\"\"/F,;,$F*F2%\"xG" }{TEXT 404 11 " at " }{XPPEDIT 408 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT 407 1 "." }}{PARA 3 "" 0 "" {TEXT 355 1 "\n" }}{PARA 3 "" 0 "" {TEXT 354 132 "a) 0 b) 1 c) 2 d) 3 e) 4 \+ \nf) 5 g) 6 h) 7 i) 8 j) 9" }{TEXT 353 1 "\n" }}{PARA 3 "" 0 "" {TEXT 357 8 "Solution" }{TEXT 358 8 ": ( e )\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "subs(t=1,(7* t^2+9)/(t^2+3)); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 413 17 "8. Suppose that " }{XPPEDIT 257 0 "F(x) = int(sqrt(1+t^2),t = 0 .. ` `*tan(x));" "6#/-%\"FG6#%\"xG-%$intG6$-%%sqrtG6#,&\"\"\"F/*$%\"tG\"\" #F//F1;\"\"!*&%\"~GF/-%$tanG6#F'F/" }{TEXT 414 15 ". What is " } {XPPEDIT 417 0 "D(F)(Pi/4);" "6#--%\"DG6#%\"FG6#*&%#PiG\"\"\"\"\"%!\" \"" }{TEXT 415 35 "? (The derivative of F(x) at " }{XPPEDIT 418 0 "x = Pi/4;" "6#/%\"xG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 416 3 " )." } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 420 26 "a) 1 b) " }{XPPEDIT 435 0 "sqrt(2);" " 6#-%%sqrtG6#\"\"#" }{TEXT 421 16 " c) " }{XPPEDIT 436 0 "s qrt(3);" "6#-%%sqrtG6#\"\"$" }{TEXT 422 37 " d) 2 \+ e) " }{XPPEDIT 437 0 "3*sqrt(2)/2;" "6#*(\"\"$\"\"\"-%%sqrtG6# \"\"#F%F)!\"\"" }{TEXT 423 9 " \nf) " }{XPPEDIT 438 0 "3*sqrt(3)/2 ;" "6#*(\"\"$\"\"\"-%%sqrtG6#F$F%\"\"#!\"\"" }{TEXT 424 14 " g ) " }{XPPEDIT 439 0 "2*sqrt(2);" "6#*&\"\"#\"\"\"-%%sqrtG6#F$F%" } {TEXT 425 11 " h) " }{XPPEDIT 440 0 "2*sqrt(3);" "6#*&\"\"#\"\" \"-%%sqrtG6#\"\"$F%" }{TEXT 426 14 " i) " }{XPPEDIT 441 0 "3 *sqrt(2);" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 427 14 " \+ j) " }{XPPEDIT 442 0 "3*sqrt(3);" "6#*&\"\"$\"\"\"-%%sqrtG6#F$F%" } {TEXT 419 1 "\n" }{TEXT 443 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT 428 8 "Solution" }{TEXT 429 7 ": ( g )" } {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "F := (x) - > int(sqrt(1+t^2),t = 0 .. tan(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"FGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$intG6$-%%sqrtG6#,&\"\"\"F 3*$)%\"tG\"\"#F3F3/F6;\"\"!-%$tanG6#9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "D(F)(Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*&\"\"#\"\"\"F%#F&F%F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 260 1 " " }{TEXT 412 1 "9" }{TEXT -1 2 ". " } {TEXT 409 14 "Suppose that " }{XPPEDIT 410 0 "F(x) = int(sqrt(8+t^2), t = ` `*x .. ` `*x^5);" "6#/-%\"FG6#%\"xG-%$intG6$-%%sqrtG6#,&\"\")\" \"\"*$%\"tG\"\"#F0/F2;*&%\"~GF0F'F0*&F7F0*$F'\"\"&F0" }{TEXT 411 60 ". What is D(F)(1)? (The derivative of F(x) at x = 1)." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 459 157 "a) 3 b) 4 \+ c) 5 d) 6 e) 7 \nf) 8 g) 9 h) 10 i) 11 j) 12" }{TEXT 460 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 3 "" 0 "" {TEXT 369 8 "Solution" }{TEXT 370 8 ": ( j )\n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "F := (x) -> int(sqrt(8+t^2),t = x .. x^5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$intG6$-%%sq rtG6#,&\"\")\"\"\"*$)%\"tG\"\"#F4F4/F7;9$*$)F;\"\"&F4F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "D(F)(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 372 17 "10. Calculate " }{XPPEDIT 462 0 "int(4*x^2*(x^3+1) ^3,x = 0 .. 1);" "6#-%$intG6$*(\"\"%\"\"\"*$%\"xG\"\"#F(,&*$F*\"\"$F(F (F(F./F*;\"\"!F(" }{TEXT 461 2 " ." }}{PARA 0 "" 0 "" {TEXT 371 17 " \+ " }}{PARA 0 "" 0 "" {TEXT 463 157 "a) 3 b ) 4 c) 5 d) 6 e) 7 \nf) 8 \+ g) 9 h) 10 i) 11 j) 12" }{TEXT 464 1 "\n" }{TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 373 8 "Soluti on" }{TEXT 374 8 ": ( c )\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "int(4*x^2*(x^3+1)^3,x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "student[changev ar](u=(x^3+1), Int(4*x^2*(x^3+1)^3,x = 0 .. 1), u);\n# This is the req uired change of variable" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$, $*(\"\"%\"\"\"\"\"$!\"\"%\"uGF*F)/F,;F)\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "value( % );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 259 19 "11. Calculate " }{XPPEDIT 258 0 "in t(x*sqrt(x-1),x = 1 .. 2);" "6#-%$intG6$*&%\"xG\"\"\"-%%sqrtG6#,&F'F(F (!\"\"F(/F';F(\"\"#" }{TEXT 465 2 " ." }}{PARA 0 "" 0 "" {TEXT 269 4 " a) " }{XPPEDIT 467 1 "8/15;" "6#*&\"\")\"\"\"\"#:!\"\"" }{TEXT 359 15 " b) " }{XPPEDIT 468 1 "2/3;" "6#*&\"\"#\"\"\"\"\"$!\"\" " }{TEXT 360 16 " c) " }{XPPEDIT 469 1 "4/5;" "6#*&\"\"%\" \"\"\"\"&!\"\"" }{TEXT 361 17 " d) " }{XPPEDIT 470 1 "1; " "6#\"\"\"" }{TEXT 362 14 " e) " }{XPPEDIT 471 1 "4/3;" "6# *&\"\"%\"\"\"\"\"$!\"\"" }{TEXT 363 11 " \n f) " }{XPPEDIT 472 1 "16/15;" "6#*&\"#;\"\"\"\"#:!\"\"" }{TEXT 364 15 " g) " } {XPPEDIT 473 1 "6/5;" "6#*&\"\"'\"\"\"\"\"&!\"\"" }{TEXT 365 16 " \+ h) " }{XPPEDIT 474 1 "5/3;" "6#*&\"\"&\"\"\"\"\"$!\"\"" } {TEXT 366 17 " i) " }{XPPEDIT 475 0 "7/5;" "6#*&\"\"(\"\" \"\"\"&!\"\"" }{TEXT 466 15 " j) " }{XPPEDIT 476 0 "8/5;" " 6#*&\"\")\"\"\"\"\"&!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 " " 0 "" {TEXT 367 8 "Solution" }{TEXT 368 8 ": ( f )\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "int(x*sqr t(x-1),x = 1 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#;\"#:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "J := student[changevar](u=( x-1), Int(x*sqrt(x-1),x = 1 .. 2), u);\n# This is the required change \+ of variable" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&,&\"\" \"F*%\"uGF*F*F+#F*\"\"#/F+;\"\"!F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "J1 := Int(expand(student[integrand](J)), u = 0 .. 1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J1G-%$IntG6$,&*$%\"uG#\"\"\"\" \"#F,*$)F*#\"\"$F-F,F,/F*;\"\"!F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "value(J1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#;\" #:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 375 17 "12. Calculate " }{XPPEDIT 256 0 "int(exp(x)/(1+exp(x)),x = 0 .. 1) ;" "6#-%$intG6$*&-%$expG6#%\"xG\"\"\",&F+F+-F(6#F*F+!\"\"/F*;\"\"!F+" }{TEXT 477 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 18 0 "ln(exp(1)/2);" "6#-%#lnG6#*&-%$expG6 #\"\"\"F*\"\"#!\"\"" }{TEXT -1 18 " b) " }{XPPEDIT 18 0 "ln((1+exp(1))/2);" "6#-%#lnG6#*&,&\"\"\"F(-%$expG6#F(F(F(\"\"#!\"\"" }{TEXT -1 15 " c) " }{XPPEDIT 18 0 "ln(1+exp(1)/2);" "6#-%# lnG6#,&\"\"\"F'*&-%$expG6#F'F'\"\"#!\"\"F'" }{TEXT -1 12 " d) \+ " }{XPPEDIT 18 0 "ln(1/2+exp(1));" "6#-%#lnG6#,&*&\"\"\"F(\"\"#!\"\"F( -%$expG6#F(F(" }{TEXT -1 12 " e) " }{XPPEDIT 18 0 "ln(2);" "6# -%#lnG6#\"\"#" }{TEXT -1 20 " \nf) " }{XPPEDIT 18 0 "ex p(1)/((1+exp(1))^2);" "6#*&-%$expG6#\"\"\"F'*$,&F'F'-F%6#F'F'\"\"#!\" \"" }{TEXT -1 14 " g) " }{XPPEDIT 18 0 "exp(1)/(1+exp(1));" "6#*&-%$expG6#\"\"\"F',&F'F'-F%6#F'F'!\"\"" }{TEXT -1 22 " \+ h) " }{XPPEDIT 18 0 "1/(1+exp(1));" "6#*&\"\"\"F$,&F$F$-%$expG6 #F$F$!\"\"" }{TEXT -1 22 " i) " }{XPPEDIT 18 0 "1-ln (2);" "6#,&\"\"\"F$-%#lnG6#\"\"#!\"\"" }{TEXT -1 15 " j) " }{XPPEDIT 18 0 "exp(1)-1;" "6#,&-%$expG6#\"\"\"F'F'!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 376 8 "Sol ution" }{TEXT 377 8 ": ( b )\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "int(exp(x)/(1+exp(x)),x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#,&\"\"\"F(-%$expG6#F(F(F(-F%6#\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "J := student[changevar](u=1+exp(x) , Int(exp(x)/(1+exp(x)),x = 0 .. 1), u);\n# This is the required chang e of variable" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&\"\" \"F)%\"uG!\"\"/F*;\"\"#,&F)F)-%$expG6#F)F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "value(J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#ln G6#,&\"\"\"F(-%$expG6#F(F(F(-F%6#\"\"#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 315 33 "13. Calculate the area between " }{XPPEDIT 535 0 "y = \+ x^2-1;" "6#/%\"yG,&*$%\"xG\"\"#\"\"\"F)!\"\"" }{TEXT 533 12 " and \+ " }{XPPEDIT 536 0 "y = x+1;" "6#/%\"yG,&%\"xG\"\"\"F'F'" }{TEXT 534 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 316 82 "a) 8/3 b) 3 c) 10/3 d ) 7/2 e) " }{TEXT 322 3 " " }{TEXT 321 1 "4" }{TEXT 537 18 " " }}{PARA 0 "" 0 "" {TEXT 317 39 "f) 21/4 \+ g) 14/3 " }{TEXT -1 1 " " }{TEXT 320 48 "h) 9/2 \+ i) 5 j) 16/3" }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 318 8 "Solution" }{TEXT 319 8 ": ( h )\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve( x^2-1 = x+1, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot([x+1,x^2-1], x = -1 .. 2);" }} {PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6&-%'CURVESG6$7S7$ $!\"\"\"\"!$F*F*7$$!3[*****\\P&3Y$*!#=$\"3F0++]i9Rl!#>7$$!3C++Dcx6x()F /$\"3x***\\PC#)GA\"F/7$$!3b++]iTDP\")F/$\"3W****\\Peui=F/7$$!3A****\\P \"\\J\\(F/$\"3y++]i3&o]#F/7$$!3g***\\7V0@&oF/$\"3S++voX*y9$F/7$$!3w++D cexdiF/$\"3B***\\P9CAu$F/7$$!3j***\\i+#QUcF/$\"3O++v$*zhdVF/7$$!3$**** \\i!3%f+&F/$\"31++v$>fS*\\F/7$$!3;++D\"oS:P%F/$\"3$)***\\(=$f%GcF/7$$! 3h*****\\<#)*=PF/$\"3Q+++Dy,\"G'F/7$$!3#*****\\(G3U9$F/$\"33++]7&)\\F/$\"3)****\\7B>&)\\\"F ip7$$\"3w***\\P>:mk&F/$\"3)***\\P>:mk:Fip7$$\"3d***\\iv&QAiF/$\"3'*** \\iv&QAi\"Fip7$$\"3j++]PPBWoF/$\"31++vtLU%o\"Fip7$$\"3%*)*****\\Nm'[(F /$\"3!******\\Nm'[Fip7$$\"3z*****\\@80+\"Fip$\"3z*****\\@80+#Fip7$$\"31++]7,Hl5Fip$\"3 1++]7,Hl?Fip7$$\"3()**\\P4w)R7\"Fip$\"3()**\\P4w)R7#Fip7$$\"3;++]x%f\" )=\"Fip$\"3;++]x%f\")=#Fip7$$\"3!)**\\P/-a[7Fip$\"3!)**\\P/-a[AFip7$$ \"3/+](=Yb;J\"Fip$\"3/+](=Yb;J#Fip7$$\"3')****\\i@Ot8Fip$\"3')****\\i@ OtBFip7$$\"3')**\\PfL'zV\"Fip$\"3')**\\PfL'zV#Fip7$$\"3>+++!*>=+:Fip$ \"3>+++!*>=+DFip7$$\"3-++DE&4Qc\"Fip$\"3-++DE&4Qc#Fip7$$\"3=+]P%>5pi\" Fip$\"3=+]P%>5pi#Fip7$$\"39+++bJ*[o\"Fip$\"39+++bJ*[o#Fip7$$\"33++Dr\" [8v\"Fip$\"33++Dr\"[8v#Fip7$$\"3++++Ijy5=Fip$\"3++++Ijy5GFip7$$\"31+]P /)fT(=Fip$\"31+]P/)fT(GFip7$$\"31+]i0j\"[$>Fip$\"31+]i0j\"[$HFip7$$\" \"#F*$\"\"$F*-%'COLOURG6&%$RGBG$\"#5F)F+F+-F$6$7SF'7$F-$!3776K;)o]E\"F /7$F4$!3E'4#H*Q?iH#F/7$F9$!3kTw[p%4&yLF/7$F>$!3K-$=.gr_Q%F/7$FC$!3ZDM! f6l[I&F/7$FH$!3[dMHLT-%3'F/7$FM$!3XBYbHDN;oF/7$FR$!3!*4@Vkc0%\\(F/7$FW $!3uGw@2K'*)3)F/7$Ffn$!3zFK?er\"ph)F/7$F[o$!3*=j\"[CaR6!*F/7$F`o$!3'[: %puYUw$*F/7$Feo$!3]BpBx\"*pe'*F/7$Fjo$!3eaWB]*R3&)*F/7$F_p$!3)evPO0%Rd **F/7$Fdp$!3;d!3$=a%*****F/7$F[q$!3]z//9gYk**F/7$F`q$!31U,Lm-lS)*F/7$F eq$!3cdbxcV)pl*F/7$Fjq$!3U))3KY@Xv$*F/7$F_r$!33(G^zGT&H!*F/7$Fdr$!3ab( fdk%o(e)F/7$Fir$!3_r=W2074\")F/7$F^s$!35\"4R2w&y9vF/7$Fcs$!3.l6P&ot:\" oF/7$Fhs$!3z7GhF/7$F]t$!3^xmkakk:`F/7$Fbt$!3wT,\"*))o)\\R%F/7$Fgt $!3BF0YaYU9MF/7$F\\u$!3)fnCaO$[!R#F/7$Fau$!3Q49UE[#p;\"F/7$Ffu$\"32-fj *Q$pE5Ffp7$F[v$\"3%pi-zBI%[8F/7$F`v$\"3'=F&Gg9[LEF/7$Fev$\"32M2tR%Hs6% F/7$Fjv$\"3EiwS>k_)e&F/7$F_w$\"3;'>lm]+W?(F/7$Fdw$\"3/hn'QHO7'))F/7$Fi w$\"3'\\.0HiQx1\"Fip7$F^x$\"3!3O?J+Y0D\"Fip7$Fcx$\"3W\\-RM-]X9Fip7$Fhx $\"3!4Ih0yOok\"Fip7$F]y$\"3TaewV\\')Q=Fip7$Fby$\"3c>2%pT?s1#Fip7$Fgy$ \"3%*o[\"H8Z*yAFip7$F\\z$\"3WQ\\Ls\\Z7DFip7$Faz$\"3ktB^OT^VFFipFez-F[[ l6&F][lF+F^[lF+-%+AXESLABELSG6$Q\"x6\"Q!Ffdl-%%VIEWG6$;F(Ffz%(DEFAULTG " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "int((x+1)-(x^2-1 ), x = -1 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"*\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 1 " " }{TEXT 270 26 "14. Tle Lorenz function " }{XPPEDIT 541 0 "L;" "6#%\"LG" }{TEXT 538 49 " of a certain country has the fol lowing values: " }}{PARA 3 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 539 0 "L (0) = 0,` `*L(20) = 5,` `*L(40) = 15,` `*L(60) = 30,` `*L(80) = 50 ,` `*L(90) = 70,` `*L(100) = 100;" "6)/-%\"LG6#\"\"!F'/*&%#~~G\"\"\" -F%6#\"#?F+\"\"&/*&F*F+-F%6#\"#SF+\"#:/*&F*F+-F%6#\"#gF+\"#I/*&F*F+-F% 6#\"#!)F+\"#]/*&F*F+-F%6#\"#!*F+\"#q/*&F*F+-F%6#\"$+\"F+FL" }{TEXT 540 1 "." }}{PARA 3 "" 0 "" {TEXT 543 22 "Using trapezoids and " } {TEXT 545 3 "all" }{TEXT 546 57 " the given data, obtain an estimate \+ for the area under " }{XPPEDIT 544 0 "y = L(x);" "6#/%\"yG-%\"LG6#%\" xG" }{TEXT 542 1 "." }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 547 82 "a) 2910 b) 2920 c) 2930 \+ d) 2940 e) 2950 " }{TEXT 550 1 " " }{TEXT 551 18 " \+ " }}{PARA 0 "" 0 "" {TEXT 548 38 "f) 2960 g) 297 0 " }{TEXT -1 1 " " }{TEXT 549 45 "h) 2980 i) 299 0 j) 3000 " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 323 8 "Solut ion" }{TEXT 324 8 ": ( e )\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "(0+5)/2*20 + (5+15)/2*20 + ( 15+30)/2*20 + (30+50)/2*20 + (50+70)/2*10+(70+100)/2*10;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"%]H" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 105 "15. By applying Simpson's Rule with four subintervals, what approximation of the area under the graph of " }{XPPEDIT 18 0 "y = 16-x^4;" "6#/%\" yG,&\"#;\"\"\"*$%\"xG\"\"%!\"\"" }{TEXT -1 36 " and over the x-axis \+ is obtained? " }}{PARA 0 "" 0 "" {TEXT 326 86 "a) 301/6 b) \+ 307/6 c) 158/3 d) 209/4 e) " } {TEXT 328 5 "107/2" }{TEXT 557 3 " " }{TEXT 330 18 " \+ " }}{PARA 0 "" 0 "" {TEXT 327 42 "f) 252/5 g) 152/3 \+ " }{TEXT -1 1 " " }{TEXT 329 43 "h) 201/4 i) 103/2 \+ j)" }{TEXT -1 1 " " }{TEXT 556 5 "256/5" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 331 8 "Solution " }{TEXT 332 8 ": ( g )\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f := x -> 16 - x^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"#;\" \"\"*$)9$\"\"%F.!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "(f(-2)+4*f(-1) + 2*f(0) + 4*f(1) + f(2))/3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$_\"\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 272 11 "16. If " }{XPPEDIT 565 0 "y( x);" "6#-%\"yG6#%\"xG" }{TEXT 564 58 " is the unique solution of th e initial value problem " }{XPPEDIT 560 0 "diff(y(x),x) = 2*x*y(x); " "6#/-%%diffG6$-%\"yG6#%\"xGF**(\"\"#\"\"\"F*F--F(6#F*F-" }{TEXT 558 6 " , " }{XPPEDIT 561 0 "y(0) = e;" "6#/-%\"yG6#\"\"!%\"eG" }{TEXT 559 21 " \n then what is " }{XPPEDIT 563 0 "y(2);" "6#-%\"yG6#\" \"#" }{TEXT 562 3 " ? " }{TEXT 273 2 " \n" }{TEXT 271 4 "\na) " } {XPPEDIT 257 0 "2*exp(1);" "6#*&\"\"#\"\"\"-%$expG6#F%F%" }{TEXT 566 12 " b) " }{XPPEDIT 568 0 "exp(2);" "6#-%$expG6#\"\"#" }{TEXT 567 11 " c) " }{XPPEDIT 576 0 "3*exp(1);" "6#*&\"\"$\"\"\"-%$ex pG6#F%F%" }{TEXT 569 12 " d) " }{XPPEDIT 577 0 "exp(3);" "6#-% $expG6#\"\"$" }{TEXT 570 14 " e) " }{XPPEDIT 578 0 "4*exp(1) ;" "6#*&\"\"%\"\"\"-%$expG6#F%F%" }{TEXT 571 9 " \nf) " }{XPPEDIT 579 0 "exp(4);" "6#-%$expG6#\"\"%" }{TEXT 572 14 " g) " } {XPPEDIT 580 0 "5*exp(1);" "6#*&\"\"&\"\"\"-%$expG6#F%F%" }{TEXT 573 10 " h) " }{XPPEDIT 581 0 "exp(5);" "6#-%$expG6#\"\"&" }{TEXT 574 14 " i) " }{XPPEDIT 582 0 "6*exp(1);" "6#*&\"\"'\"\"\"-% $expG6#F%F%" }{TEXT 575 12 " j) " }{XPPEDIT 583 0 "exp(6);" "6 #-%$expG6#\"\"'" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 325 0 "" }{TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 313 8 "Solution " }{TEXT 314 8 ": ( h )\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "eqn := int(1/y,y) = int(2*x , x) + \+ C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%#lnG6#%\"yG,&*$)%\"xG \"\"#\"\"\"F/%\"CGF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn 2 := y = solve(eqn,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%\" yG-%$expG6#,&*$)%\"xG\"\"#\"\"\"F/%\"CGF/" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 39 "eqn3 := subs(\{y= exp(1), x = 0\}, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%$expG6#\"\"\"-F'6#%\"CG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn4 := C = solve(eqn3, C); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/%\"CG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn5 := subs(eqn4, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/%\"yG-%$expG6#,&*$)%\"xG\"\"#\"\"\" F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eqn6 := simplify( subs(x=2,eqn5) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/%\"yG-% $expG6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 274 4 "17. " }{TEXT 295 12 " The height " }{XPPEDIT 590 0 "y(t);" "6#-%\"yG6#%\"tG " }{TEXT 584 67 " of water in a leaking tank is given by the different ial equation " }{XPPEDIT 591 0 "diff(y(t),t) = -r^2*sqrt(y(t));" "6#/ -%%diffG6$-%\"yG6#%\"tGF*,$*&%\"rG\"\"#-%%sqrtG6#-F(6#F*\"\"\"!\"\"" } {TEXT 585 8 " where " }{XPPEDIT 592 0 "r;" "6#%\"rG" }{TEXT 586 105 " is the (constant) radius of the hole through which the water leaks. If the initial height (i.e., at " }{XPPEDIT 593 0 "t = 0;" "6#/%\"t G\"\"!" }{TEXT 587 58 ") of the water was 144 and if the tank becomes empty at " }{XPPEDIT 594 0 "t = 8;" "6#/%\"tG\"\")" }{TEXT 588 32 ", then what is the value of " }{XPPEDIT 595 0 "r;" "6#%\"rG" } {TEXT 589 2 " ?" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 294 44 "a) 0 b) 1 c) " }{XPPEDIT 303 1 "sqrt(2)/2" "6#*&-%%sqrtG6#\"\"#\"\"\"F'!\"\"" }{TEXT 296 19 " \+ d) " }{XPPEDIT 304 1 "sqrt(3)/3" "6#*&-%%sqrtG6#\"\"$\" \"\"F'!\"\"" }{TEXT 297 16 " e) " }{XPPEDIT 305 1 "1/2" "6 #*&\"\"\"F$\"\"#!\"\"" }{TEXT 298 7 " \nf) " }{XPPEDIT 306 1 "sqrt(2 )" "6#-%%sqrtG6#\"\"#" }{TEXT 299 16 " g) " }{XPPEDIT 307 1 "2*sqrt(3)/3" "6#*(\"\"#\"\"\"-%%sqrtG6#\"\"$F%F)!\"\"" }{TEXT 300 14 " h) " }{XPPEDIT 308 1 "2*sqrt(2)" "6#*&\"\"#\"\"\"-%%sqr tG6#F$F%" }{TEXT 301 16 " i) " }{XPPEDIT 309 1 "sqrt(3)" " 6#-%%sqrtG6#\"\"$" }{TEXT 302 18 " j) " }{XPPEDIT 310 1 "3/2" "6#*&\"\"$\"\"\"\"\"#!\"\"" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT 311 8 "Solution" }{TEXT 312 8 ": ( i )\n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "eqn := dsolve( diff(y(t),t) = -r^2*sqrt(y(t)) , y(t) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,(*$-%\"yG6#%\"tG#\"\"\"\"\"#F-*(F.! \"\"%\"rGF.F+F-F-%$_C1GF0\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eqn1 := subs( \{t=0,y(t)=144\}, eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&*$\"$W\"#\"\"\"\"\"#F*%$_C1G!\"\"\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqn2 := _C1 = solve(eqn1, _C 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%$_C1G\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn3 := subs(eqn2, eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,(*$-%\"yG6#%\"tG#\"\"\"\"\"#F-*(F. !\"\"%\"rGF.F+F-F-\"#7F0\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn4 := y(t) = solve(eqn3, y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/-%\"yG6#%\"tG,(*(\"\"%!\"\"%\"rGF,F)\"\"#\"\"\"*(\"#7F0 )F.F/F0F)F0F-\"$W\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "su bs(\{y(t) = 0, t = 8\}, eqn4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\" \"!,(*&\"#;\"\"\")%\"rG\"\"%F(F(*&\"#'*F()F*\"\"#F(!\"\"\"$W\"F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(%,r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&*$\"\"$#\"\"\"\"\"#,$F#!\"\"F#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 276 17 "18. Calculate " }{XPPEDIT 597 0 "int(1/x,x = -3 \+ .. -1);" "6#-%$intG6$*&\"\"\"F'%\"xG!\"\"/F(;,$\"\"$F),$F'F)" }{TEXT 596 1 "." }}{PARA 3 "" 0 "" {TEXT 277 1 "a" }{TEXT 275 3 ") " } {XPPEDIT 603 0 "-ln(2);" "6#,$-%#lnG6#\"\"#!\"\"" }{TEXT 598 16 " \+ b) " }{XPPEDIT 604 0 "-ln(3);" "6#,$-%#lnG6#\"\"$!\"\"" } {TEXT 599 16 " c) " }{XPPEDIT 605 0 "-ln(6);" "6#,$-%#lnG6 #\"\"'!\"\"" }{TEXT 600 13 " d) " }{XPPEDIT 256 0 "-ln(9);" " 6#,$-%#lnG6#\"\"*!\"\"" }{TEXT 611 18 " e) " }{XPPEDIT 606 0 "-ln(1/2);" "6#,$-%#lnG6#*&\"\"\"F(\"\"#!\"\"F*" }{TEXT 601 12 " \nf) " }{XPPEDIT 607 0 "-ln(1/3);" "6#,$-%#lnG6#*&\"\"\"F(\"\" $!\"\"F*" }{TEXT 602 15 " g) " }{XPPEDIT 613 0 "-ln(1/6);" "6#,$-%#lnG6#*&\"\"\"F(\"\"'!\"\"F*" }{TEXT 608 11 " h) " } {XPPEDIT 614 0 "-ln(1/9);" "6#,$-%#lnG6#*&\"\"\"F(\"\"*!\"\"F*" } {TEXT 612 14 " i) " }{XPPEDIT 615 0 "1/exp(1)-1/exp(3);" "6# ,&*&\"\"\"F%-%$expG6#F%!\"\"F%*&F%F%-F'6#\"\"$F)F)" }{TEXT 609 15 " \+ j) " }{XPPEDIT 616 0 "1/exp(3)-1/exp(1);" "6#,&*&\"\"\"F%-%$e xpG6#\"\"$!\"\"F%*&F%F%-F'6#F%F*F*" }{TEXT 610 8 " " }{TEXT 291 3 "\n " }}{PARA 3 "" 0 "" {TEXT 292 8 "Solution" }{TEXT 293 8 ": \+ ( b )\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "int(1/x,x = -3 .. -1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6#\"\"$!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT 278 16 "19. Calculate " }{XPPEDIT 18 0 "int(tan(x),x = 0 .. Pi/4);" "6#-%$intG6 $-%$tanG6#%\"xG/F);\"\"!*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 617 9 ".\n\n\n a) " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 627 1 " \+ " }{XPPEDIT 18 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT 626 14 " b ) " }{XPPEDIT 265 0 "ln(2);" "6#-%#lnG6#\"\"#" }{TEXT 618 16 " \+ c) " }{XPPEDIT 266 0 "2*ln(2);" "6#*&\"\"#\"\"\"-%#lnG6#F$F%" } {TEXT 619 12 " d) " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#! \"\"" }{TEXT 628 1 " " }{XPPEDIT 256 0 "ln(3);" "6#-%#lnG6#\"\"$" } {TEXT 624 14 " e) " }{XPPEDIT 267 0 "ln(3);" "6#-%#lnG6#\"\" $" }{TEXT 620 12 " \nf) " }{XPPEDIT 268 0 "2*ln(3);" "6#*&\"\"# \"\"\"-%#lnG6#\"\"$F%" }{TEXT 621 17 " g) " }{XPPEDIT 273 0 "3*ln(3);" "6#*&\"\"$\"\"\"-%#lnG6#F$F%" }{TEXT 622 12 " \+ h) " }{XPPEDIT 274 0 "1;" "6#\"\"\"" }{TEXT 625 23 " \+ i) " }{XPPEDIT 275 0 "2;" "6#\"\"#" }{TEXT 623 25 " \+ j) " }{XPPEDIT 276 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }}{PARA 3 " " 0 "" {TEXT 289 9 "\nSolution" }{TEXT 290 8 ": ( a )\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "int(tan (x),x = 0 .. Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\" #F&-%#lnG6#F'F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 281 16 "20. Calculate \+ " }{XPPEDIT 630 0 "int(sec(x),x = 0 .. Pi/3);" "6#-%$intG6$-%$secG6#% \"xG/F);\"\"!*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 629 2 ".\n" }}{PARA 260 " " 0 "" {TEXT 280 5 "a) " }{XPPEDIT 636 1 "ln(sqrt(2));" "6#-%#lnG6#- %%sqrtG6#\"\"#" }{TEXT 631 16 " b) " }{XPPEDIT 637 1 "ln(s qrt(3));" "6#-%#lnG6#-%%sqrtG6#\"\"$" }{TEXT 282 18 " c) \+ " }{XPPEDIT 638 1 "ln(1+sqrt(3));" "6#-%#lnG6#,&\"\"\"F'-%%sqrtG6#\" \"$F'" }{TEXT 283 12 " d) " }{XPPEDIT 639 1 "ln(2+sqrt(3));" " 6#-%#lnG6#,&\"\"#\"\"\"-%%sqrtG6#\"\"$F(" }{TEXT 284 12 " e) \+ " }{XPPEDIT 640 0 "ln(1+sqrt(2));" "6#-%#lnG6#,&\"\"\"F'-%%sqrtG6#\"\" #F'" }{TEXT 632 10 " \nf) " }{XPPEDIT 641 1 "ln(2+sqrt(2));" "6#- %#lnG6#,&\"\"#\"\"\"-%%sqrtG6#F'F(" }{TEXT 633 9 " g) " } {XPPEDIT 642 1 "ln(3+sqrt(2));" "6#-%#lnG6#,&\"\"$\"\"\"-%%sqrtG6#\"\" #F(" }{TEXT 285 10 " h) " }{XPPEDIT 643 0 "ln(2*sqrt(2));" "6#-% #lnG6#*&\"\"#\"\"\"-%%sqrtG6#F'F(" }{TEXT 286 16 " i) " } {XPPEDIT 644 0 "ln(2*sqrt(3));" "6#-%#lnG6#*&\"\"#\"\"\"-%%sqrtG6#\"\" $F(" }{TEXT 634 18 " j) " }{XPPEDIT 645 1 "ln(sqrt(2)+sq rt(3));" "6#-%#lnG6#,&-%%sqrtG6#\"\"#\"\"\"-F(6#\"\"$F+" }{TEXT 635 2 " " }{TEXT 279 1 "\n" }}{PARA 3 "" 0 "" {TEXT 287 8 "Solution" } {TEXT 288 8 ": ( d )\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "int(sec(x),x = 0 .. Pi/3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&\"\"#\"\"\"*$\"\"$#F(F'F(" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "20 5 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }