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Calculate " }{XPPEDIT 608 0 "Int(x/sqrt(5- 4*x-x^2),x = -2 .. 1);" "6#-%$IntG6$*&%\"xG\"\"\"-%%sqrtG6#,(\"\"&F(*& \"\"%F(F'F(!\"\"*$F'\"\"#F0F0/F';,$F2F0F(" }{TEXT 607 1 "." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 614 4 "a) " }{XPPEDIT 266 0 "Pi-1;" "6#,&%#PiG\"\"\"F%!\"\"" }{TEXT 615 10 " \+ b) " }{XPPEDIT 267 0 "Pi-2;" "6#,&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 616 12 " c) " }{XPPEDIT 268 0 "Pi-3;" "6#,&%#PiG\"\"\"\"\"$!\" \"" }{TEXT 617 11 " d) " }{XPPEDIT 269 0 "Pi-4;" "6#,&%#PiG\"\" \"\"\"%!\"\"" }{TEXT 618 11 " e) " }{TEXT 613 1 " " }{XPPEDIT 270 0 "Pi-5;" "6#,&%#PiG\"\"\"\"\"&!\"\"" }{TEXT 612 19 " \+ " }}{PARA 0 "" 0 "" {TEXT 610 4 "f) " }{XPPEDIT 271 0 "1-Pi;" " 6#,&\"\"\"F$%#PiG!\"\"" }{TEXT 619 11 " g) " }{XPPEDIT 272 0 "2 -Pi;" "6#,&\"\"#\"\"\"%#PiG!\"\"" }{TEXT 620 6 " " }{TEXT 623 1 " " }{TEXT 611 5 "h) " }{XPPEDIT 274 0 "3-Pi;" "6#,&\"\"$\"\"\"%#PiG! \"\"" }{TEXT 621 11 " i) " }{XPPEDIT 275 0 "4-Pi;" "6#,&\"\"%\" \"\"%#PiG!\"\"" }{TEXT 622 12 " j) " }{XPPEDIT 276 0 "5-Pi;" " 6#,&\"\"&\"\"\"%#PiG!\"\"" }{TEXT 624 3 " " }}{PARA 3 "" 0 "" {TEXT 609 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 493 12 "Solution: h" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 40 "J := Int(x/sqrt(5-4*x-x^2),x = -2 .. 1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&%\"xG\"\"\",(\"\"&F** $)F)\"\"#F*!\"\"*&\"\"%F*F)F*F0#F0F//F);!\"#F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "A := numer(integrand(J)); B := denom(integran d(J));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG*$,(\"\"&\"\"\"*$)%\"xG\"\"#F(!\"\"*&\"\"%F( F+F(F-#F(F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "C := complet esquare(B,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG*$,&\"\"*\"\"\" *$),&%\"xGF(\"\"#F(F-F(!\"\"#F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn := x + 2 = 3*sin(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&%\"xG\"\"\"\"\"#F(,$*&\"\"$F(-%$sinG6#%\"tGF( F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "J1 := changevar(eqn, \+ J, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J1G-%$IntG6$,$**\"\"$\"\" \",&\"\"#!\"\"*&F*F+-%$sinG6#%\"tGF+F+F+,(\"#8F+*$)F,F-F+F.*&\"#7F+F0F +F.#F.F--%$cosGF2F+F+/F3;\"\"!,$*&F-F.%#PiGF+F+" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "J2 := map(simplify, J1 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#J2G-%$IntG6$*&,&\"\"#!\"\"*&\"\"$\"\"\"-%$sinG6#%\"t GF.F.F.-%%csgnG6#-%$cosGF1F./F2;\"\"!,$*&F*F+%#PiGF.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "plot([Re(csgn(cos(t))),Im(csgn(cos (t)))], t= 0..2*Pi, color = [RED, BLUE], discont = true);\n#In simplif ying the square root, Maple has introduced the complex sign of cos(t). 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" }{TEXT 284 1 " " }{TEXT 285 57 " The triangular region in the first qu adrant bounded by " }{XPPEDIT 19 1 "y = 2-x^2;" "6#/%\"yG,&\"\"#\"\" \"*$%\"xGF&!\"\"" }{TEXT 595 3 ", " }{XPPEDIT 19 1 "y = x;" "6#/%\"yG %\"xG" }{TEXT 596 6 ", and " }{XPPEDIT 19 1 "x = 0;" "6#/%\"xG\"\"!" } {TEXT 597 96 " is rotated about the line \ny = 3. What is the volume of the solid of revolution that results?" }}{PARA 3 "" 0 "" {TEXT 283 3 "a) " }{XPPEDIT 378 0 "10*Pi/3;" "6#*(\"#5\"\"\"%#PiGF%\"\"$!\" \"" }{TEXT 373 13 " b) " }{XPPEDIT 379 0 "14*Pi/3;" "6#*(\"#9 \"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 374 14 " c) " }{XPPEDIT 380 0 "9*Pi/2;" "6#*(\"\"*\"\"\"%#PiGF%\"\"#!\"\"" }{TEXT 375 13 " \+ d) " }{XPPEDIT 381 0 "15*Pi/2;" "6#*(\"#:\"\"\"%#PiGF%\"\"#!\"\"" } {TEXT 376 15 " e) " }{XPPEDIT 382 0 "4*Pi;" "6#*&\"\"%\"\" \"%#PiGF%" }{TEXT 377 14 " \nf) " }{XPPEDIT 387 0 "61*Pi/15; " "6#*(\"#h\"\"\"%#PiGF%\"#:!\"\"" }{TEXT 383 14 " g) " } {XPPEDIT 388 0 "67*Pi/15;" "6#*(\"#n\"\"\"%#PiGF%\"#:!\"\"" }{TEXT 384 12 " h) " }{XPPEDIT 389 0 "24*Pi/5;" "6#*(\"#C\"\"\"%#PiGF %\"\"&!\"\"" }{TEXT 385 14 " i) " }{XPPEDIT 390 0 "28*Pi/5; " "6#*(\"#G\"\"\"%#PiGF%\"\"&!\"\"" }{TEXT 386 14 " j) " } {XPPEDIT 392 0 "6*Pi;" "6#*&\"\"'\"\"\"%#PiGF%" }{TEXT 391 3 " " } {TEXT 393 6 " \n" }{TEXT 494 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 495 13 "Solution: g\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "plot([x,2-x^2,3,6-x, 4+x^2], x = 0 .. 1, color \+ = [BLUE,BLUE,BLACK,COLOR(RGB,.75,.75,.75),COLOR(RGB,.75,.75,.75)], vie w=[0..1,0..6]);" }}{PARA 13 "" 1 "" {GLPLOT2D 374 374 374 {PLOTDATA 2 "6)-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3emmm;arz@!#>F+7$$\"3[LL$e9ui2%F-F/7 $$\"3nmmm\"z_\"4iF-F27$$\"3[mmmT&phN)F-F57$$\"3CLLe*=)H\\5!#=F87$$\"3g mm\"z/3uC\"F:F<7$$\"3%)***\\7LRDX\"F:F?7$$\"3]mm\"zR'ok;F:FB7$$\"3w*** \\i5`h(=F:FE7$$\"3WLLL3En$4#F:FH7$$\"3qmm;/RE&G#F:FK7$$\"3\")*****\\K] 4]#F:FN7$$\"3$******\\PAvr#F:FQ7$$\"3)******\\nHi#HF:FT7$$\"3jmm\"z*ev :JF:FW7$$\"3?LLL347TLF:FZ7$$\"3,LLLLY.KNF:Fgn7$$\"3w***\\7o7Tv$F:Fjn7$ $\"3'GLLLQ*o]RF:F]o7$$\"3A++D\"=lj;%F:F`o7$$\"31++vV&RY2aF:Fbp7$$\"39mm;zXu9cF:Fep7$$\"3l******\\y) )GeF:Fhp7$$\"3'*)***\\i_QQgF:F[q7$$\"3@***\\7y%3TiF:F^q7$$\"35****\\P! [hY'F:Faq7$$\"3kKLL$Qx$omF:Fdq7$$\"3!)*****\\P+V)oF:Fgq7$$\"3?mm\"zpe* zqF:Fjq7$$\"3%)*****\\#\\'QH(F:F]r7$$\"3GKLe9S8&\\(F:F`r7$$\"3R***\\i? =bq(F:Fcr7$$\"3\"HLL$3s?6zF:Ffr7$$\"3a***\\7`Wl7)F:Fir7$$\"3#pmmm'*RRL )F:F\\s7$$\"3Qmm;a<.Y&)F:F_s7$$\"3=LLe9tOc()F:Fbs7$$\"3u******\\Qk\\*) F:Fes7$$\"3CLL$3dg6<*F:Fhs7$$\"3ImmmmxGp$*F:F[t7$$\"3A++D\"oK0e*F:F^t7 $$\"3A++v=5s#y*F:Fat7$$\"\"\"F)Fdt-%'COLOURG6&%$RGBGF(F($\"*++++\"!\") -F$6$7S7$F($\"\"#F)7$F+$\"3mM-2%)[_**>!#<7$F/$\"3/,)3*)RQ$)*>Ffu7$F2$ \"3'Q(4;UY9'*>Ffu7$F5$\"37#4fIWFfu7$F8$\"3YrL4L(*)*))>Ffu7$F<$\"3Q F*>;tRW)>Ffu7$F?$\"3'zr6\\H,*y>Ffu7$FB$\"3K7f'>>)Gs>Ffu7$FE$\"3u%3>_\\ +['>Ffu7$FH$\"3!p9\"4]`;c>Ffu7$FK$\"3CHJ)))ovx%>Ffu7$FN$\"3)R#)=ZZ_u$> Ffu7$FQ$\"3mVPT@2:E>Ffu7$FT$\"3*Q\\\"*))zrV\">Ffu7$FW$\"39vf%=l?H!>Ffu 7$FZ$\"3Sy*e26p$))=Ffu7$Fgn$\"3gQ$*[8tCv=Ffu7$Fjn$\"3GzZwzj1f=Ffu7$F]o $\"3)G#f\")zs\"Ffu7$Fbp$\"3\"eosfb$f2jWaaI\"Ffu7$F_s$\"35]xc7Mlp7Ffu7$Fbs$\"3jm4_9.EL7Ffu7$Fes$\" 3ur:e\\(Q!*>\"Ffu7$Fhs$\"39?*Ry8)*)e6Ffu7$F[t$\"3`.RXnW;A6Ffu7$F^t$\"3 !o+Na$R8#3\"Ffu7$Fat$\"3tpIp%p$)H/\"FfuFctFft-F$6$7S7$F($\"\"$F)7$F+Fe ^l7$F/Fe^l7$F2Fe^l7$F5Fe^l7$F8Fe^l7$Ff_(eFfu7$F?$\"3?+](o1YZ&eFfu7$FB$\"3=L$3-OJN$eFfu7$FE$\"3T+]P*o% Q7eFfu7$FH$\"3)pmm\"RFj!z&Ffu7$FK$\"3WLLe4OZrdFfu7$FN$\"3-++]n\\!*\\dF fu7$FQ$\"3n****\\ixCGdFfu7$FT$\"3<++]KqP2dFfu7$FW$\"3WL$3-TC%)o&Ffu7$F Z$\"3'pmm\"4z)em&Ffu7$Fgn$\"3#ommm`'zYcFfu7$Fjn$\"3')**\\(=t)eCcFfu7$F ]o$\"3*pmm;1J\\g&Ffu7$F`o$\"3w**\\(=[jLe&Ffu7$Fco$\"3/+]iXg#Gc&Ffu7$Ff o$\"3\\mmT&Q(RTbFfu7$Fio$\"3cm;/'=><_&Ffu7$F\\p$\"3zLLe*e$\\+bFfu7$F_p $\"3kL$3-;Y%yaFfu7$Fbp$\"3;+]73QDfaFfu7$Fep$\"3;LL3Ub_QaFfu7$Fhp$\"3E+ ++:76&Q``Ffu7$Fdq$\"3tmmmhA;L`Ffu7$Fgq$\"38++]i*p:J&Ffu7$Fjq$\"3QL$3-8/?H& Ffu7$F]r$\"3C++]2Nhq_Ffu7$F`r$\"3Wm;a)f'[]_Ffu7$Fcr$\"3%)**\\Pz\"[%H_F fu7$Ffr$\"3Emm;z#z)3_Ffu7$Fir$\"3/+](oaXt=&Ffu7$F\\s$\"3ULLL.ggm^Ffu7$ F_s$\"3/LLeCoRX^Ffu7$Fbs$\"3\"pmT&oKOC^Ffu7$Fes$\"3D+++:c.0^Ffu7$Fhs$ \"3Mmm\"H%R)G3&Ffu7$F[t$\"3/LLLB72j]Ffu7$F^t$\"3)***\\(=tY>/&Ffu7$Fat$ \"3U+]7)*ys@]Ffu7$Fdt$\"\"&F)-%&COLORG6&Fit$\"#v!\"#Fb[mFb[m-F$6$7S7$F ($\"\"%F)7$F+$\"37l(Hf6v/+%Ffu7$F/$\"3^)>\"4,;m,SFfu7$F2$\"3dE!RyNbQ+% Ffu7$F5$\"3W24%pb#)p+%Ffu7$F8$\"3)*Gm!pE55,%Ffu7$F<$\"3js+Qo-c:SFfu7$F ?$\"3F#G)30()4@SFfu7$FB$\"3o(3M!3=rFSFfu7$FE$\"3F:4y/&*>NSFfu7$FH$\"3L `)3*\\Y$Q/%Ffu7$FK$\"3wqo66VA_SFfu7$FN$\"3Cw6GDvaiSFfu7$FQ$\"3Mciey#\\ Q2%Ffu7$FT$\"3m0&36?Gc3%Ffu7$FW$\"3iCS:[$zq4%Ffu7$FZ$\"3;@5C*)3j6TFfu7 $Fgn$\"3jh1^'o_Z7%Ffu7$Fjn$\"3%4AN-iL49%Ffu7$F]o$\"3!p#e.m%zg:%Ffu7$F` o$\"3!RKN#))fetTFfu7$Fco$\"3EuQQm57\">%Ffu7$Ffo$\"3-pE5ejJ5UFfu7$Fio$ \"3zX1+PDvGUFfu7$F\\p$\"3iVt/am]\\UFfu7$F_p$\"3%=V,yS=?F%Ffu7$Fbp$\"3? 9t-WkS#H%Ffu7$Fep$\"3q9G*oc`_J%Ffu7$Fhp$\"3Dw(ycLf(RVFfu7$F[q$\"3crPyl 4ikVFfu7$F^q$\"3a.vY#R6&*Q%Ffu7$Faq$\"3&4lGWq5\"=WFfu7$Fdq$\"3N:bEpDnW WFfu7$Fgq$\"3)4DKl\"f$RZ%Ffu7$Fjq$\"3*)e?k^\"e7]%Ffu7$F]r$\"3\"GXTal/? `%Ffu7$F`r$\"3aSc'*Q.xhXFfu7$Fcr$\"3s,&o#3,v$f%Ffu7$Ffr$\"3`_=$\\*>(ei %Ffu7$Fir$\"3S#*Q=gsSgYFfu7$F\\s$\"3AS+o`ba%p%Ffu7$F_s$\"3X\\AV(eY.t%F fu7$Fbs$\"3%H.zaoRnw%Ffu7$Fes$\"3DG%=/Dh4![Ffu7$Fhs$\"3J!3g@'=5T[Ffu7$ F[t$\"3C'4YD`Ny([Ffu7$F^t$\"3U$*\\ckg'y\"\\Ffu7$Fat$\"3GIpI0j,d\\FfuF \\[mF_[m-%+AXESLABELSG6$Q\"x6\"Q!F\\em-%%VIEWG6$;F(Fdt;F(F]bl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "volume := Pi*int((3-x)^2-(3-(2-x^2))^2, x = 0 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'volumeG,$*(\"#n\"\"\"\"#:!\"\"%#PiGF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 257 36 "3. Calculate the arc length of \+ " }{XPPEDIT 520 1 "y = x^3/3+1/4/x;" "6#/%\"yG,&*&%\"xG\"\"$F(!\" \"\"\"\"*(F*F*\"\"%F)F'F)F*" }{TEXT 519 9 " for " }{XPPEDIT 521 1 "1 <= x;" "6#1\"\"\"%\"xG" }{XPPEDIT 522 1 "`` <= 2;" "6#1%!G\"\"#" } {TEXT 396 1 "." }{TEXT 395 3 " \n\n" }{TEXT 394 4 "a) " }{XPPEDIT 405 0 "58/23;" "6#*&\"#e\"\"\"\"#B!\"\"" }{TEXT 397 13 " b) \+ " }{XPPEDIT 406 0 "59/24;" "6#*&\"#f\"\"\"\"#C!\"\"" }{TEXT 398 15 " \+ c) " }{XPPEDIT 407 0 "12/5;" "6#*&\"#7\"\"\"\"\"&!\"\"" } {TEXT 399 12 " d) " }{XPPEDIT 408 0 "61/26;" "6#*&\"#h\"\"\"\" #E!\"\"" }{TEXT 400 12 " e) " }{TEXT 305 1 " " }{XPPEDIT 409 0 "62/27;" "6#*&\"#i\"\"\"\"#F!\"\"" }{TEXT 289 19 " \+ " }}{PARA 0 "" 0 "" {TEXT 287 4 "f) " }{XPPEDIT 410 0 "63/28;" "6#*& \"#j\"\"\"\"#G!\"\"" }{TEXT 401 14 " g) " }{XPPEDIT 411 0 "6 4/29;" "6#*&\"#k\"\"\"\"#H!\"\"" }{TEXT 402 9 " " }{TEXT 412 1 " " }{TEXT 288 5 "h) " }{XPPEDIT 413 0 "11/6;" "6#*&\"#6\"\"\"\"\" '!\"\"" }{TEXT 403 12 " i) " }{XPPEDIT 414 0 "66/31;" "6#*&\"# m\"\"\"\"#J!\"\"" }{TEXT 404 13 " j) " }{XPPEDIT 415 0 "67/32 ;" "6#*&\"#n\"\"\"\"#K!\"\"" }{TEXT 416 3 " " }}{PARA 3 "" 0 "" {TEXT 286 1 " " }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 496 13 "Solution: b \n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := x -> x^3/3 + 1/4/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&#\"\"\"\" \"$F/*$)9$F0F/F/F/*&#F/\"\"%F/*&F/F/F3!\"\"F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "toBeIntegrated := simplify(sqrt(1+d iff(f(x),x)^2)) assuming x > 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% /toBeIntegratedG,$*(\"\"%!\"\",&\"\"\"F**&F'F*)%\"xGF'F*F*F*F-!\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "int(toBeIntegrated, x=1.. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#f\"#C" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 364 17 "4. The graph of " }{XPPEDIT 539 1 "y = x/3;" "6 #/%\"yG*&%\"xG\"\"\"\"\"$!\"\"" }{TEXT 536 3 " " }{TEXT 537 7 " for \+ " }{XPPEDIT 257 1 "3 <= x;" "6#1\"\"$%\"xG" }{XPPEDIT 258 1 "`` <= 6 ;" "6#1%!G\"\"'" }{TEXT 538 87 " is rotated about the x-axis. What \+ is the surface area of the resulting figure? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 366 3 "a) " }{TEXT 524 2 " " } {XPPEDIT 19 1 "2*Pi*sqrt(3);" "6#*(\"\"#\"\"\"%#PiGF%-%%sqrtG6#\"\"$F% " }{TEXT 525 9 " b) " }{XPPEDIT 19 1 "4*Pi*sqrt(3);" "6#*(\"\"%\" \"\"%#PiGF%-%%sqrtG6#\"\"$F%" }{TEXT 526 9 " c) " }{XPPEDIT 19 1 "5*Pi*sqrt(3);" "6#*(\"\"&\"\"\"%#PiGF%-%%sqrtG6#\"\"$F%" }{TEXT 527 10 " d) " }{XPPEDIT 19 1 "6*Pi*sqrt(3);" "6#*(\"\"'\"\"\"%#PiGF% -%%sqrtG6#\"\"$F%" }{TEXT 528 9 " e)" }{TEXT 367 2 " " } {XPPEDIT 19 1 "2*Pi*sqrt(5);" "6#*(\"\"#\"\"\"%#PiGF%-%%sqrtG6#\"\"&F% " }{TEXT 529 1 " " }{TEXT 530 11 " " }}{PARA 0 "" 0 "" {TEXT 365 5 "f) " }{XPPEDIT 19 1 "3*Pi*sqrt(5);" "6#*(\"\"$\"\"\"%#P iGF%-%%sqrtG6#\"\"&F%" }{TEXT 523 9 " g) " }{XPPEDIT 19 1 "4*Pi*s qrt(5);" "6#*(\"\"%\"\"\"%#PiGF%-%%sqrtG6#\"\"&F%" }{TEXT 531 2 " " } {TEXT 532 7 " h) " }{XPPEDIT 19 1 "2*Pi*sqrt(10);" "6#*(\"\"#\"\"\" %#PiGF%-%%sqrtG6#\"#5F%" }{TEXT 533 10 " i) " }{XPPEDIT 19 1 "3* Pi*sqrt(10);" "6#*(\"\"$\"\"\"%#PiGF%-%%sqrtG6#\"#5F%" }{TEXT 534 10 " j) " }{XPPEDIT 19 1 "6*Pi*sqrt(2);" "6#*(\"\"'\"\"\"%#PiGF%-%%s qrtG6#\"\"#F%" }{TEXT 535 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 497 13 "Solution: i\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f := x -> x/3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&# \"\"\"\"\"$F/9$F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "answer := 2*Pi*int(f(x)*sqrt(1+diff(f(x),x)^2),x=3..6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'answerG,$*(\"\"$\"\"\"%#PiGF(\"#5#F(\"\"# F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 341 52 "5. Let R be the regio n bounded by the graph of " }{XPPEDIT 543 1 "y = 4-x^2;" "6#/%\"yG,& \"\"%\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT 541 39 ", the x-axis, and the v ertical line " }{XPPEDIT 542 1 "x = -1;" "6#/%\"xG,$\"\"\"!\"\"" } {TEXT 540 56 ". What is the x-coordinate of the center of mass of R? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 344 1 "\n" } {TEXT 345 4 "a) " }{XPPEDIT 354 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\"" } {TEXT 348 16 " b) " }{XPPEDIT 355 0 "3/4;" "6#*&\"\"$\"\" \"\"\"%!\"\"" }{TEXT 349 15 " c) " }{TEXT 550 1 " " } {XPPEDIT 256 0 "5/4;" "6#*&\"\"&\"\"\"\"\"%!\"\"" }{TEXT 551 14 " \+ d) " }{XPPEDIT 356 0 "1/5;" "6#*&\"\"\"F$\"\"&!\"\"" }{TEXT 353 13 " e) " }{XPPEDIT 357 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" } {TEXT 347 2 " " }{TEXT 346 20 " \n" }{TEXT 342 5 "f ) " }{XPPEDIT 358 0 "2/5;" "6#*&\"\"#\"\"\"\"\"&!\"\"" }{TEXT 352 15 " g) " }{XPPEDIT 359 0 "3/8;" "6#*&\"\"$\"\"\"\"\")!\"\" " }{TEXT 351 11 " " }{TEXT 360 1 " " }{TEXT 343 3 "h) " } {XPPEDIT 257 0 "3/5;" "6#*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT 549 16 " \+ i) " }{XPPEDIT 361 0 "5/8;" "6#*&\"\"&\"\"\"\"\")!\"\"" } {TEXT 350 15 " j) " }{XPPEDIT 362 0 "8/15;" "6#*&\"\")\"\" \"\"#:!\"\"" }{TEXT 363 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 498 13 "Solution: a\n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f := x -> 4-x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,& \"\"%\"\"\"*$)9$\"\"#F.!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "x_bar := int(x*f(x),x=-1..2)/int(f(x),x=-1..2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&x_barG#\"\"\"\"\"%" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 293 4 "6. " }{TEXT 544 92 "What is the y-coordi nate of the center of mass of the region R of the preceding question ? " }{TEXT 545 1 "\n" }{TEXT 292 0 "" }}{PARA 3 "" 0 "" {TEXT 291 16 " a) 1 b) " }{XPPEDIT 423 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" } {TEXT 417 14 " c) " }{XPPEDIT 424 0 "17/10;" "6#*&\"#<\"\"\" \"#5!\"\"" }{TEXT 418 12 " d) " }{XPPEDIT 425 0 "9/5;" "6#*&\" \"*\"\"\"\"\"&!\"\"" }{TEXT 419 12 " e) " }{XPPEDIT 426 0 "2; " "6#\"\"#" }{TEXT 420 8 " \nf) " }{XPPEDIT 256 0 "5/2;" "6#*&\"\"& \"\"\"\"\"#!\"\"" }{TEXT 548 12 " g) " }{XPPEDIT 257 0 "5/3;" "6#*&\"\"&\"\"\"\"\"$!\"\"" }{TEXT 547 12 " h) " }{XPPEDIT 427 0 "8/3;" "6#*&\"\")\"\"\"\"\"$!\"\"" }{TEXT 421 14 " i) \+ " }{XPPEDIT 428 0 "8/5;" "6#*&\"\")\"\"\"\"\"&!\"\"" }{TEXT 422 14 " \+ j) " }{XPPEDIT 257 0 "15/8;" "6#*&\"#:\"\"\"\"\")!\"\"" } {TEXT 546 3 " " }{TEXT 290 1 "\n" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 516 11 "Solution: c" }}{PARA 0 "" 0 "" {TEXT 517 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "int(f(x)^2,x=-1..2)/2/int( f(x),x=-1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#<\"#5" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT 307 17 "7. Suppose that " }{XPPEDIT 553 1 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT 552 71 ". On t he interval [ 1 , b ], f(x) assumes its average value when " } {XPPEDIT 627 1 "x = sqrt(19);" "6#/%\"xG-%%sqrtG6#\"#>" }{TEXT 626 15 ". \nWhat is " }{XPPEDIT 555 1 "b;" "6#%\"bG" }{TEXT 554 2 " ?" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 309 4 "a) " }{XPPEDIT 431 0 "5;" "6#\"\"&" }{TEXT 430 17 " \+ b) " }{XPPEDIT 317 0 "3*sqrt(3);" "6#*&\"\"$\"\"\"-%%sqrtG6#F $F%" }{TEXT 310 13 " c) " }{XPPEDIT 318 0 "4*sqrt(2);" "6#*& \"\"%\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 311 13 " d) " }{XPPEDIT 432 0 "6;" "6#\"\"'" }{TEXT 429 13 " e) " }{XPPEDIT 319 0 "5* sqrt(2);" "6#*&\"\"&\"\"\"-%%sqrtG6#\"\"#F%" }{TEXT 312 7 " \nf) " } {XPPEDIT 320 0 "4*sqrt(3);" "6#*&\"\"%\"\"\"-%%sqrtG6#\"\"$F%" }{TEXT 313 11 " g) " }{XPPEDIT 321 0 "2*sqrt(10);" "6#*&\"\"#\"\"\"-%% sqrtG6#\"#5F%" }{TEXT 314 12 " h) " }{XPPEDIT 322 0 "3*sqrt(5) ;" "6#*&\"\"$\"\"\"-%%sqrtG6#\"\"&F%" }{TEXT 315 13 " i) " } {XPPEDIT 323 0 "7;" "6#\"\"(" }{TEXT 316 14 " j) " } {XPPEDIT 324 0 "2*sqrt(14);" "6#*&\"\"#\"\"\"-%%sqrtG6#\"#9F%" }{TEXT 308 1 "\n" }{TEXT 325 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK } {PARA 0 "" 0 "" {TEXT 514 11 "Solution: i" }}{PARA 0 "" 0 "" {TEXT 515 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a :=1; f := x -> x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(*$)9$\"\"#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f_ave := int(f(x),x = a .. b)/(b-a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&f_aveG*&,&*&\"\"$!\"\"%\"bGF(\"\"\"#F+F(F)F+,&F*F+F+ F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "simplify(f_ave); # The average is actually quadratic in b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"\"$!\"\"%\"bG\"\"#\"\"\"*&F%F&F'F)F)#F)F%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "b = solve(f(sqrt(19)) = f_ave, b); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"bG6$!\")\"\"(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 260 1 " " }{TEXT 306 1 "8" }{TEXT -1 1 "." } {TEXT 556 30 " For what constant c is " }{XPPEDIT 558 1 "f(x) = \+ c*x^2;" "6#/-%\"fG6#%\"xG*&%\"cG\"\"\"*$F'\"\"#F*" }{TEXT 557 37 " a probability density function for" }{TEXT 561 2 " " }{XPPEDIT 259 1 " 1 <= x;" "6#1\"\"\"%\"xG" }{XPPEDIT 260 1 "`` <= 2;" "6#1%!G\"\"#" } {TEXT 560 1 "." }{TEXT 559 1 " " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 326 4 "a) " }{XPPEDIT 440 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\"" } {TEXT 433 15 " b) " }{XPPEDIT 441 0 "1/2;" "6#*&\"\"\"F$\" \"#!\"\"" }{TEXT 434 15 " c) " }{XPPEDIT 442 0 "2/3;" "6#*& \"\"#\"\"\"\"\"$!\"\"" }{TEXT 435 30 " d) 1 e) " }{XPPEDIT 257 0 "2/9;" "6#*&\"\"#\"\"\"\"\"*!\"\"" }{TEXT 628 13 " \+ \nf) " }{XPPEDIT 443 0 "4/3;" "6#*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT 436 15 " g) " }{XPPEDIT 444 0 "1/3;" "6#*&\"\"\"F$\"\"$!\" \"" }{TEXT 437 15 " h) " }{XPPEDIT 445 0 "3/4;" "6#*&\"\"$ \"\"\"\"\"%!\"\"" }{TEXT 438 14 " i) " }{XPPEDIT 446 0 "3/7; " "6#*&\"\"$\"\"\"\"\"(!\"\"" }{TEXT 439 15 " j) " } {XPPEDIT 447 0 "3/8;" "6#*&\"\"$\"\"\"\"\")!\"\"" }{TEXT 327 1 "\n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 512 11 "Solution: i" }}{PARA 0 "" 0 "" {TEXT 513 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqn := int(c*x**2, x=1..2) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,$*(\"\"(\"\"\"\"\"$!\"\"%\"cGF)F)F)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "c = solve(eqn, c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG#\"\"$\"\"(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 303 54 "9. A random variable assumes values in the in terval " }{XPPEDIT 563 1 "[0, 2];" "6#7$\"\"!\"\"#" }{TEXT 562 39 " a nd has probability density function " }{XPPEDIT 567 1 "f(x) = x^3/4; " "6#/-%\"fG6#%\"xG*&F'\"\"$\"\"%!\"\"" }{TEXT 564 6 " for " }{TEXT 565 2 " " }{XPPEDIT 256 1 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 257 1 " `` <= 2;" "6#1%!G\"\"#" }{TEXT 566 27 ". What is the mean of X?" }} {PARA 0 "" 0 "" {TEXT 302 17 " " }}{PARA 0 "" 0 "" {TEXT 328 5 "a) " }{XPPEDIT 457 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT 452 12 " b) " }{XPPEDIT 458 0 "2/3;" "6#*&\"\"#\"\"\"\" \"$!\"\"" }{TEXT 453 16 " c) " }{XPPEDIT 459 0 "3/4;" "6#* &\"\"$\"\"\"\"\"%!\"\"" }{TEXT 454 13 " d) " }{XPPEDIT 460 0 "5/8;" "6#*&\"\"&\"\"\"\"\")!\"\"" }{TEXT 455 10 " e) " } {XPPEDIT 461 0 "8/5;" "6#*&\"\")\"\"\"\"\"&!\"\"" }{TEXT 456 8 " \nf) " }{XPPEDIT 462 0 "15/16;" "6#*&\"#:\"\"\"\"#;!\"\"" }{TEXT 451 13 " g) " }{XPPEDIT 463 0 "16/15;" "6#*&\"#;\"\"\"\"#:!\"\"" } {TEXT 450 11 " h) " }{XPPEDIT 464 0 "8/7;" "6#*&\"\")\"\"\"\"\" (!\"\"" }{TEXT 449 14 " i) " }{XPPEDIT 465 0 "3/2;" "6#*&\" \"$\"\"\"\"\"#!\"\"" }{TEXT 448 12 " j) " }{XPPEDIT 466 0 "4/3 ;" "6#*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT 329 1 "\n" }{TEXT 467 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 510 11 "Solution: e" }}{PARA 0 "" 0 "" {TEXT 511 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "mu = int(x*(x^3/4),x = 0 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#muG#\"\")\"\"&" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " } {TEXT 259 5 "10. " }{TEXT 568 54 "A random variable X assumes value s in the interval " }{XPPEDIT 260 1 "[0, Pi/2];" "6#7$\"\"!*&%#PiG\" \"\"\"\"#!\"\"" }{TEXT 569 39 " and has probability density function \+ " }{XPPEDIT 264 1 "f(x) = cos(x);" "6#/-%\"fG6#%\"xG-%$cosG6#F'" } {TEXT 570 6 " for " }{TEXT 571 2 " " }{XPPEDIT 256 1 "0 <= x;" "6#1 \"\"!%\"xG" }{XPPEDIT 257 1 "`` <= Pi/2;" "6#1%!G*&%#PiG\"\"\"\"\"#!\" \"" }{TEXT 572 29 ". What is the value of " }{TEXT 576 1 "P" } {TEXT 625 2 " (" }{TEXT 577 2 " " }{XPPEDIT 574 1 "Pi/6 <= X;" "6#1*& %#PiG\"\"\"\"\"'!\"\"%\"XG" }{XPPEDIT 575 1 "`` <= Pi/3;" "6#1%!G*&%#P iG\"\"\"\"\"$!\"\"" }{TEXT 573 1 " " }{TEXT 578 1 ")" }{TEXT 579 7 " \+ ? " }}{PARA 0 "" 0 "" {TEXT 265 4 "a) " }{XPPEDIT 331 1 "(sqrt(2)- 1)/2;" "6#*&,&-%%sqrtG6#\"\"#\"\"\"F)!\"\"F)F(F*" }{TEXT 294 13 " \+ b) " }{XPPEDIT 332 1 "(sqrt(2)-1)/4;" "6#*&,&-%%sqrtG6#\"\"#\"\" \"F)!\"\"F)\"\"%F*" }{TEXT 295 13 " c) " }{XPPEDIT 333 1 "(sq rt(3)-1)/2;" "6#*&,&-%%sqrtG6#\"\"$\"\"\"F)!\"\"F)\"\"#F*" }{TEXT 296 14 " d) " }{XPPEDIT 334 1 "(sqrt(3)-1)/4;" "6#*&,&-%%sqrtG6# \"\"$\"\"\"F)!\"\"F)\"\"%F*" }{TEXT 297 12 " e) " }{XPPEDIT 335 1 "(2-sqrt(2))/2;" "6#*&,&\"\"#\"\"\"-%%sqrtG6#F%!\"\"F&F%F*" } {TEXT 298 11 " \n f) " }{XPPEDIT 336 1 "(2*sqrt(2)-1)/2;" "6#*&,& *&\"\"#\"\"\"-%%sqrtG6#F&F'F'F'!\"\"F'F&F+" }{TEXT 299 10 " g) \+ " }{XPPEDIT 337 1 "(2*sqrt(2)-1)/4;" "6#*&,&*&\"\"#\"\"\"-%%sqrtG6#F&F 'F'F'!\"\"F'\"\"%F+" }{TEXT 300 10 " h) " }{XPPEDIT 338 1 "(sqrt (3)-sqrt(2))/2;" "6#*&,&-%%sqrtG6#\"\"$\"\"\"-F&6#\"\"#!\"\"F)F,F-" } {TEXT 301 11 " i) " }{XPPEDIT 339 0 "(2*sqrt(3)-1)/4;" "6#*&,&* &\"\"#\"\"\"-%%sqrtG6#\"\"$F'F'F'!\"\"F'\"\"%F," }{TEXT 330 11 " \+ j) " }{XPPEDIT 340 0 "(sqrt(3)-sqrt(2))/4;" "6#*&,&-%%sqrtG6#\"\"$\" \"\"-F&6#\"\"#!\"\"F)\"\"%F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 508 11 "Solution: c" }}{PARA 0 "" 0 "" {TEXT 509 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "int(cos(x),x = Pi/6 .. Pi/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,&*&\"\"#!\"\"\"\"$#\"\"\"F%F)#F)F%F&" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 598 9 "11. If " }{XPPEDIT 631 1 "m;" "6#%\"mG" }{TEXT 630 57 " is the median of a random variable with pdf given by " }{XPPEDIT 599 1 "f(x) = x^3/4;" "6#/-%\"fG6#%\"xG*&F'\"\"$\"\"%!\"\"" }{TEXT 600 8 " for " }{XPPEDIT 601 1 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 602 1 "`` <= 2;" "6#1%!G\"\"#" }{TEXT 603 2 ", " }}{PARA 3 "" 0 "" {TEXT 632 14 "then what is " }{XPPEDIT 634 1 "log[2](m);" "6#-&%$logG 6#\"\"#6#%\"mG" }{TEXT 633 5 "? " }{TEXT -1 3 " " }{TEXT 604 1 " \n" }}{PARA 261 "" 0 "" {TEXT -1 85 "a) 1/8 b) 1/4 \+ c) 3/8 d) 1/2 e) 5/8\nf) 3/4 " }{TEXT 605 62 " g) 7/8 h) 5/16 i) 7/16 j) 9/16" }} {PAGEBK }{PARA 3 "" 0 "" {TEXT 635 11 "Solution: f" }{TEXT 606 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn := int(x^3/4, x=0..m) \+ = 1/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,$*&\"#;!\"\"%\"mG\" \"%\"\"\"#F,\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "m = so lve(eqn, m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"mG6&*$)\"\"##\"\"$ \"\"%\"\"\"*&F'F,^#F,F,,$F&!\"\"*&^#F0F,F'F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 304 154 "12. In stretching a spring 60 cm be yond equilibrium, 1.44 J work is done. How many newtons of force are needed to maintain the spring at that position?" }}{PARA 261 "" 0 "" {TEXT -1 84 "\na) 3.8 b) 4.0 c) 4.2 d) 4.4 e) 4.6\nf) 4.8 " }{TEXT 629 61 " g) 5.0 h) 5.2 i) 5.4 j) 5.6" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 507 12 "Solution: f " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eqn1 := W = int(k*x, x = 0 .. 6/10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/%\"WG,$*(\"\"*\"\"\"\"#]!\"\"%\"kGF*F*" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqn2 := subs( W = 1.44, eq n1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/$\"$W\"!\"#,$*(\"\"* \"\"\"\"#]!\"\"%\"kGF,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn3 := k = solve(eqn2, k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn 3G/%\"kG$\"\")\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Hook esLaw := F = k*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*HookesLawG/%\" FG*&%\"kG\"\"\"%\"xGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "s ubs(\{eqn3, x=6/10\}, HookesLaw);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %\"FG$\"+++++[!\"*" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 277 234 "13. A h emispherical tank completely filled with water is pumped dry. The radi us of the disc that is the top part of the tank is 2m. At a depth y meters below the top of the tank, the horizontal cross-section of th e tank has area " }{XPPEDIT 637 1 "Pi*(4-y^2);" "6#*&%#PiG\"\"\",&\" \"%F%*$%\"yG\"\"#!\"\"F%" }{TEXT 636 26 ". What is the work done ? " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 3 "a) " } {XPPEDIT 18 0 "1200*Pi*g;" "6#*(\"%+7\"\"\"%#PiGF%%\"gGF%" }{TEXT -1 13 " b) " }{XPPEDIT 18 0 "1600*Pi*g;" "6#*(\"%+;\"\"\"%#PiGF% %\"gGF%" }{TEXT -1 14 " c) " }{XPPEDIT 18 0 "2000*Pi*g;" "6# *(\"%+?\"\"\"%#PiGF%%\"gGF%" }{TEXT -1 10 " d) " }{XPPEDIT 18 0 "2400*Pi*g;" "6#*(\"%+C\"\"\"%#PiGF%%\"gGF%" }{TEXT -1 11 " e) \+ " }{XPPEDIT 18 0 "2800*Pi*g;" "6#*(\"%+G\"\"\"%#PiGF%%\"gGF%" }{TEXT -1 20 " \nf) " }{XPPEDIT 18 0 "3200*Pi*g;" "6#*(\"%+K\" \"\"%#PiGF%%\"gGF%" }{TEXT -1 13 " g) " }{XPPEDIT 18 0 "3600* Pi*g;" "6#*(\"%+O\"\"\"%#PiGF%%\"gGF%" }{TEXT -1 12 " h) " } {XPPEDIT 18 0 "4000*Pi*g;" "6#*(\"%+S\"\"\"%#PiGF%%\"gGF%" }{TEXT -1 13 " i) " }{XPPEDIT 18 0 "4400*Pi*g;" "6#*(\"%+W\"\"\"%#PiGF% %\"gGF%" }{TEXT -1 12 " j) " }{XPPEDIT 18 0 "4800*Pi*g;" "6#*( \"%+[\"\"\"%#PiGF%%\"gGF%" }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 580 12 "Solution: h " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A := 1000*g*int(Pi*y*(4-y^2 ),y = 0 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,$*(\"%+S\"\" \"%\"gGF(%#PiGF(F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 201 " 14. A \+ thick rope hangs over the side of a 100 foot tall building, but does n ot reach the ground. The rope weighs 24 pounds per foot. The dangling \+ end of the rope is tied to a 120 pound sack of sugar. " }{TEXT 658 68 "In pulling the sack of sugar all the way to the top of the building, " }{TEXT -1 1 " " }{TEXT 659 31 "a hungry, motivated ant expends" } {TEXT -1 1 " " }{TEXT 660 25 "4992 foot-pounds of work." }{TEXT -1 1 " " }{TEXT 661 64 "How may feet of rope were hanging over the side of t he building?" }{TEXT -1 2 " \n" }}{PARA 0 "" 0 "" {TEXT 368 67 "a) 10 \+ b) 11 c) 12 d) 13 e) 14 " }{TEXT 371 1 " " }{TEXT 372 18 " " }}{PARA 0 "" 0 "" {TEXT 369 31 "f) 15 g) 16 " }{TEXT -1 1 " " }{TEXT 370 37 "h) 17 i) 18 j) 19 " }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 505 11 "Solution: g" }}{PARA 0 "" 0 "" {TEXT 506 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "eqn := int(24*y,y = 0 .. a)+120*a = 4992;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&*&\"#7\"\"\")%\"aG\"\"#F)F)* &\"$?\"F)F+F)F)\"%#*\\" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "e qn2 := map(z -> z/12, eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2 G/,&*$)%\"aG\"\"#\"\"\"F+*&\"#5F+F)F+F+\"$;%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \"#;!#E" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 15 "15. Calculate " } {XPPEDIT 19 1 "int(1/((16-x)^(1/4)),x = 0 .. 16);" "6#-%$intG6$*&\"\" \"F'),&\"#;F'%\"xG!\"\"*&F'F'\"\"%F,F,/F+;\"\"!F*" }{TEXT 587 1 "." }} {PARA 0 "" 0 "" {TEXT 278 68 "a) 21/2 b) 25/2 \+ c) 16/5 d) " }{TEXT 656 4 "32/3" }{TEXT 657 18 " \+ e) 32/5" }{TEXT 492 2 " " }{TEXT 281 18 " " }} {PARA 0 "" 0 "" {TEXT 279 43 "f) 9/2 g) 16/3 \+ " }{TEXT -1 1 " " }{TEXT 280 51 "h) 17/4 i) 21/4 \+ j) 64/5 " }{TEXT -1 1 " " }{TEXT 588 1 " " }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 503 12 "Solution: d " }}{PARA 0 "" 0 " " {TEXT 504 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "simplify(int(1/((16-x)^(1/4)),x = 0 .. 16));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#K\"\"$" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 268 5 "16. " }{TEXT 590 10 "Calculate " }{XPPEDIT 591 1 "int(sec(x)^2/(tan( x)^(1/3)),x = 0 .. Pi/4);" "6#-%$intG6$*&-%$secG6#%\"xG\"\"#)-%$tanG6# F**&\"\"\"F1\"\"$!\"\"F3/F*;\"\"!*&%#PiGF1\"\"%F3" }{TEXT 589 1 "." }} {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 643 0 "" }}{PARA 0 "" 0 "" {TEXT 644 5 "a) " }{XPPEDIT 268 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 650 12 " b) " }{XPPEDIT 269 0 "2/3;" "6#*&\"\"#\"\"\"\"\"$!\" \"" }{TEXT 651 16 " c) " }{XPPEDIT 270 0 "3/4;" "6#*&\"\"$ \"\"\"\"\"%!\"\"" }{TEXT 652 13 " d) " }{XPPEDIT 271 0 "5/8; " "6#*&\"\"&\"\"\"\"\")!\"\"" }{TEXT 653 10 " e) " }{XPPEDIT 272 0 "8/5;" "6#*&\"\")\"\"\"\"\"&!\"\"" }{TEXT 654 8 " \nf) " } {XPPEDIT 273 0 "15/16;" "6#*&\"#:\"\"\"\"#;!\"\"" }{TEXT 649 13 " \+ g) " }{XPPEDIT 274 0 "16/15;" "6#*&\"#;\"\"\"\"#:!\"\"" }{TEXT 648 11 " h) " }{XPPEDIT 275 0 "8/7;" "6#*&\"\")\"\"\"\"\"(!\"\" " }{TEXT 647 14 " i) " }{XPPEDIT 276 0 "3/2;" "6#*&\"\"$\"\" \"\"\"#!\"\"" }{TEXT 646 12 " j) " }{XPPEDIT 277 0 "4/3;" "6#* &\"\"%\"\"\"\"\"$!\"\"" }{TEXT 645 1 "\n" }{TEXT 655 0 "" }{TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 502 14 "Solution: i " }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "J := Int(sec(x)^2/(tan(x))^( 1/3), x = 0 .. Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG 6$*&-%$secG6#%\"xG\"\"#-%$tanGF+#!\"\"\"\"$/F,;\"\"!,$*&\"\"%F1%#PiG\" \"\"F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "K := changevar(u= tan(x), J, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%$IntG6$*&\" \"\"F)*$)%\"uG#F)\"\"$F)!\"\"/F,;\"\"!F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\" \"#" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 270 4 "17. " }{TEXT 276 12 " Ca lculate " }{XPPEDIT 256 0 "int(x^2/((x^3-7)^(5/3)),x = 2 .. infinity) ;" "6#-%$intG6$*&%\"xG\"\"#),&*$F'\"\"$\"\"\"\"\"(!\"\"*&\"\"&F-F,F/F/ /F';F(%)infinityG" }{TEXT 479 1 "." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 480 4 "a) " }{XPPEDIT 490 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT 488 11 " b) " }{XPPEDIT 257 0 "2/3;" "6#*&\"\"#\"\"\"\"\" $!\"\"" }{TEXT 481 18 " c) " }{XPPEDIT 258 0 "3/2;" "6#* &\"\"$\"\"\"\"\"#!\"\"" }{TEXT 482 15 " d) " }{XPPEDIT 256 0 "3/4;" "6#*&\"\"$\"\"\"\"\"%!\"\"" }{TEXT 486 21 " e ) " }{XPPEDIT 259 0 "1;" "6#\"\"\"" }{TEXT 483 13 " \nf) " } {XPPEDIT 19 1 "4/3;" "6#*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT 592 10 " \+ g) " }{XPPEDIT 261 0 "2*sqrt(2);" "6#*&\"\"#\"\"\"-%%sqrtG6#F$F%" } {TEXT 484 12 " h) " }{XPPEDIT 262 0 "sqrt(2)/2;" "6#*&-%%sqrtG 6#\"\"#\"\"\"F'!\"\"" }{TEXT 487 14 " i) " }{XPPEDIT 263 0 " (sqrt(2)-1)/2;" "6#*&,&-%%sqrtG6#\"\"#\"\"\"F)!\"\"F)F(F*" }{TEXT 485 11 " j) " }{XPPEDIT 264 0 "2;" "6#\"\"#" }{TEXT 491 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 518 14 "Solution: a \n" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "int(x^2/((x^3-7)^(5/3)),x = 2 .. infinity);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6##\"\"\"\"\"#" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 272 4 "18. " }{TEXT 593 11 " Evaluate " }{XPPEDIT 594 1 "int((x*sin(2/x)- 2*cos(2/x))/x,x = 2/Pi .. infinity);" "6#-%$intG6$*&,&*&%\"xG\"\"\"-%$ sinG6#*&\"\"#F*F)!\"\"F*F**&F/F*-%$cosG6#*&F/F*F)F0F*F0F*F)F0/F);*&F/F *%#PiGF0%)infinityG" }{TEXT 640 3 " . " }{TEXT 642 43 " You may find t he following formula useful:" }}{PARA 3 "" 0 "" {TEXT -1 11 " \+ " }{XPPEDIT 256 1 "Diff(x*sin(2/x),x) = (x*sin(2/x)-2*cos(2/x))/x;" "6#/-%%DiffG6$*&%\"xG\"\"\"-%$sinG6#*&\"\"#F)F(!\"\"F)F(*&,&*&F(F)-F+6 #*&F.F)F(F/F)F)*&F.F)-%$cosG6#*&F.F)F(F/F)F/F)F(F/" }{TEXT 641 3 " . \+ " }}{PARA 0 "" 0 "" {TEXT -1 18 " " }}{PARA 3 "" 0 " " {TEXT 478 5 " \na) " }{XPPEDIT 256 0 "1;" "6#\"\"\"" }{TEXT 469 16 " b) " }{XPPEDIT 262 0 "-1;" "6#,$\"\"\"!\"\"" }{TEXT 470 12 " c) " }{XPPEDIT 266 0 "2;" "6#\"\"#" }{TEXT 471 14 " \+ d) " }{XPPEDIT 267 0 "-2;" "6#,$\"\"#!\"\"" }{TEXT 472 11 " \+ e) " }{XPPEDIT 268 0 "3;" "6#\"\"$" }{TEXT 473 10 " \nf) " } {XPPEDIT 269 0 "-3;" "6#,$\"\"$!\"\"" }{TEXT 474 15 " g) " }{XPPEDIT 270 0 "4;" "6#\"\"%" }{TEXT 475 13 " h) " } {XPPEDIT 271 0 "-4;" "6#,$\"\"%!\"\"" }{TEXT 476 11 " i) " } {XPPEDIT 272 0 "5;" "6#\"\"&" }{TEXT 477 17 " j) " } {XPPEDIT 489 0 "-5;" "6#,$\"\"&!\"\"" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 468 0 "" }{TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 499 14 "Solution: c \n" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "eqn := Int((x*sin(2/x)-2*cos(2/x))/x,x = 2/Pi .. inf inity) = Limit(subs(x=N, x*sin(2/x) ) - subs(x=2/Pi, x*sin(2/x) ), N = infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%$IntG6$*&,&*& %\"xG\"\"\"-%$sinG6#,$*&\"\"#F-F,!\"\"F-F-F-*&F3F--%$cosGF0F-F4F-F,F4/ F,;,$*&F3F-%#PiGF4F-%)infinityG-%&LimitG6$,&*&%\"NGF--F/6#,$*&F3F-FCF4 F-F-F-*(F3F-F " 0 "" {MPLTEXT 1 0 40 "eqn2 := lhs(eqn) = simplify( rhs(eqn) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%$IntG6$*&,&*&%\"xG\"\"\"-%$sinG6#,$* &\"\"#F-F,!\"\"F-F-F-*&F3F--%$cosGF0F-F4F-F,F4/F,;,$*&F3F-%#PiGF4F-%)i nfinityG-%&LimitG6$*&%\"NGF--F/6#,$*&F3F-FBF4F-F-/FBF=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "eqn3 := lhs(eqn2) = Limit(sin(2*u)/ u, u = 0); #Change variable: u=1/N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%%eqn3G/-%$IntG6$*&,&*&%\"xG\"\"\"-%$sinG6#,$*&\"\"#F-F,!\"\"F-F-F-*& F3F--%$cosGF0F-F4F-F,F4/F,;,$*&F3F-%#PiGF4F-%)infinityG-%&LimitG6$*&-F /6#,$*&F3F-%\"uGF-F-F-FFF4/FF\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "eqn4 := lhs(eqn2) = Limit(Diff(sin(2*u),u)/Diff(u,u), u = 0); #L'Hopital" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/-%$Int G6$*&,&*&%\"xG\"\"\"-%$sinG6#,$*&\"\"#F-F,!\"\"F-F-F-*&F3F--%$cosGF0F- F4F-F,F4/F,;,$*&F3F-%#PiGF4F-%)infinityG-%&LimitG6$*&-%%DiffG6$-F/6#,$ *&F3F-%\"uGF-F-FIF--FC6$FIFIF4/FI\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eqn4 := lhs(eqn2) = value( rhs( eqn3) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/-%$IntG6$*&,&*&%\"xG\"\"\"-%$sinG6#,$* &\"\"#F-F,!\"\"F-F-F-*&F3F--%$cosGF0F-F4F-F,F4/F,;,$*&F3F-%#PiGF4F-%)i nfinityGF3" }}}}{SECT 0 {PARA 260 "" 0 "" {TEXT 273 42 "19. Calculate the fourth partial sum of " }{XPPEDIT 19 1 "sum(1/(2*n*(n+1)),n = 1 \+ .. infinity);" "6#-%$sumG6$*&\"\"\"F'*(\"\"#F'%\"nGF',&F*F'F'F'F'!\"\" /F*;F'%)infinityG" }{TEXT 586 1 "." }{TEXT -1 0 "" }}{PARA 260 "" 0 " " {TEXT 638 67 "a) 1/10 b) 1/5 c) 3/10 d) 2/5 \+ e) 1/2 " }}{PARA 260 "" 0 "" {TEXT 639 66 "f) 3/5 g) 7/ 10 h) 4/5 i) 9/10 j) 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT 500 14 "Solution: d \n" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a := n -> 1 /2/n/(n+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGf*6#%\"nG6\"6$%)o peratorG%&arrowGF(,$*&#\"\"\"\"\"#F/*&F/F/*&9$F/,&F3F/F/F/F/!\"\"F/F/F (F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sum(a(n), n = 1 .. 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"\"&" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 275 13 "20. Calculate" }{TEXT 582 2 " " } {XPPEDIT 583 1 "sum(3^n/(5^(n+1)),n = 1 .. infinity);" "6#-%$sumG6$*&) \"\"$%\"nG\"\"\")\"\"&,&F)F*F*F*!\"\"/F);F*%)infinityG" }{TEXT 584 4 " " }{TEXT -1 0 "" }{TEXT 581 1 "." }}{PARA 260 "" 0 "" {TEXT 274 67 "a) 1/10 b) 1/5 c) 3/10 d) 2/5 e) 1/2 " }}{PARA 260 "" 0 "" {TEXT 585 66 "f) 3/5 g) 7/10 h) 4/ 5 i) 9/10 j) 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PAGEBK }{PARA 0 "" 0 "" {TEXT 501 14 "Solution: c \n" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sum(3^n/(5^(n+1)),n = 1 .. infinity); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"#5" }}}}}{MARK "20 0 5" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }