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0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 } {PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 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 }{PSTYLE "Heading 1" -1 257 1 {CSTYLE "" -1 -1 "Ti mes" 1 14 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" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 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 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 260 1 {CSTYLE "" -1 -1 "Time s" 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 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 8 "Math 233" }{TEXT 279 1 " \+ " }{TEXT -1 11 "Spring 2006" }}{PARA 18 "" 0 "" {TEXT -1 6 "Exam 3" }} {PARA 0 "" 0 "" {TEXT 259 2 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 447 4 "Load" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The following function will" }}{PARA 15 "" 0 "" {TEXT -1 77 "if x and y are variables, calculate the discriminant of f(x,y) at (x,y) ;" }}{PARA 15 "" 0 "" {TEXT -1 122 "if x and y are real constants and if (x,y) is a critical point of f, calculate the discriminant o f f(x,y) at (x,y) ;" }}{PARA 15 "" 0 "" {TEXT -1 110 "if x and y a re real constants and if (x,y) is not a critical point of f, print \+ \"Not a critical point.\" ;" }}{PARA 15 "" 0 "" {TEXT 373 72 "To use t hese functions in MapleV R5 and subsequent releases, change each" } {TEXT -1 2 " " }{TEXT 377 1 "`" }{TEXT -1 6 " to " }{TEXT 378 1 "\" " }{TEXT -1 2 " " }{TEXT 374 7 "in the " }{TEXT -1 1 " " }{TEXT 376 6 "printf" }{TEXT -1 1 " " }{TEXT 375 11 " statements" }{TEXT -1 4 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 427 "discriminant := proc()\nlocal x, y, u, v, f;\nf := a rgs[1]:\nx := args[2]:\ny := args[3]:\nif not type(x, realcons) or not type(y, realcons) then\nreturn subs(\{u=x,v=y\},diff(f(u,v),u$2)*diff (f(u,v),v$2)-(diff(f(u,v),u,v))^2);\nelif subs(\{u=x,v=y\},diff(f(u,v) ,u))<>0 or subs(\{u=x,v=y\},diff(f(u,v),v))<>0 then printf(\"Not a cri tical point.\\n\"); else return subs(\{u=x,v=y\},diff(f(u,v),u$2)*diff (f(u,v),v$2)-(diff(f(u,v),u,v))^2) fi;\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-discriminantGf*6\"6'%\"xG%\"yG%\"uG%\"vG%\"fGF&F&C&> 8(&9\"6#\"\"\">8$&F16#\"\"#>8%&F16#\"\"$@'43-%%typeG6$F5%)realconsG-FB 6$F:FDO-%%subsG6$<$/8&F5/8'F:,&*&-%%diffG6$-F/6$FMFO-%\"$G6$FMF8F3-FS6 $FU-FX6$FOF8F3F3*$)-FS6%FUFMFOF8F3!\"\"50-FI6$FK-FS6$FUFM\"\"!0-FI6$FK -FS6$FUFOFco-%'printfG6#Q7Not~a~critical~point.|+F&OFHF&F&F&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 388 "c riticalPoints := proc()\nlocal f,x,y,cp,N,j,k,returnList,nextcp;\nf := args[1]:\nreturnList := []:\ncp := [solve(\{diff(f(x,y),x) = 0 , diff (f(x,y),y) = 0\}, \{x,y\} )];\nN := nops(cp);\nfor j from 1 to N do\ni f lhs(cp[j][1]) = x then \nnextcp := [rhs(cp[j][1]),rhs(cp[j][2])];\ne lse\nnextcp := [rhs(cp[j][2]),rhs(cp[j][1])];\nfi:\nreturnList := [op( returnList),nextcp]:\nod:\nreturn returnList;\nend;\n \n\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%/criticalPointsGf*6\"6+%\"fG%\"xG%\"yG%#cp G%\"NG%\"jG%\"kG%+returnListG%'nextcpGF&F&C(>8$&9\"6#\"\"\">8+7\">8'7# -%&solveG6$<$/-%%diffG6$-F36$8%8&FI\"\"!/-FD6$FFFHFJ<$FIFH>8(-%%nopsG6 #F8,7$-%$rhsGFfn-F^o6#&Fh n6#\"\"#>F[o7$F_oF]o>F97$-%#opG6#F9F[oOF9F&F&F&" }}}{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 -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 519 "discriminantTest := pro c()\nlocal x, y, u, v, f, A, B;\nf := args[1]:\nx := args[2]:\ny := ar gs[3]:\nif subs(\{u=x,v=y\},diff(f(u,v),u))<>0 or subs(\{u=x,v=y\},dif f(f(u,v),v))<>0 then printf(`Not a critical point.\\n`); \nelse \nA := subs(\{u=x,v=y\},diff(f(u,v),u$2)*diff(f(u,v),v$2)-(diff(f(u,v),u,v)) ^2);\nB := subs(\{u=x,v=y\},diff(f(u,v),u$2)); \nfi;\nif A = 0 then pr intf(`The test is not conclusive.\\n`);\nelif A < 0 then printf(`Saddl e point\\n`);\nelif B > 0 then printf(`Local minimum\\n`);\nelse prin tf(`Local maximum\\n`);\nfi:\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%1discriminantTestGf*6\"6)%\"xG%\"yG%\"uG%\"vG%\"fG%\"AG%\"BGF&F&C'>8 (&9\"6#\"\"\">8$&F36#\"\"#>8%&F36#\"\"$@%50-%%subsG6$<$/8&F7/8'F<-%%di ffG6$-F16$FHFJFH\"\"!0-FD6$FF-FL6$FNFJFP-%'printfG6#%7Not~a~critical~p oint.|+GC$>8)-FD6$FF,&*&-FL6$FN-%\"$G6$FHF:F5-FL6$FN-F^o6$FJF:F5F5*$)- FL6%FNFHFJF:F5!\"\">8*-FD6$FFF[o@)/FfnFP-FW6#%=The~test~is~not~conclus ive.|+G2FfnFP-FW6#%.Saddle~point|+G2FPFjo-FW6#%/Local~minimum|+G-FW6#% /Local~maximum|+GF&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 514 45 "1. Calculate the directional derivative \+ of " }{XPPEDIT 515 1 "f(x,y,z)=sqrt(x+y*z)" "6#/-%\"fG6%%\"xG%\"yG% \"zG-%%sqrtG6#,&F'\"\"\"*&F(F.F)F.F." }{TEXT 516 50 " at the point ( 1, 3, 1) in the \ndirection (2/7)" }{TEXT 517 2 " i" }{TEXT 518 11 " \+ + (3/7) " }{TEXT 520 1 "j" }{TEXT 521 11 " + (6/7) " }{TEXT 519 1 "k" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 524 4 "a) " }{TEXT 527 2 "9/" }{TEXT 528 18 "14 b) \+ " }{TEXT 529 5 "19/28" }{TEXT 530 13 " c) " }{TEXT 531 2 "5/ " }{TEXT 532 16 "7 d) " }{TEXT 533 2 "3/" }{TEXT 534 18 "4 \+ e) " }{TEXT 535 3 "11/" }{TEXT 536 3 "14 " }{TEXT 525 20 " " }}{PARA 0 "" 0 "" {TEXT 523 84 "f) 23/28 \+ g) 6/7 h) 25/28 i) 13/14 j) 27/28 " }}{PARA 0 "" 0 "" {TEXT 526 1 " " }}{PARA 0 "" 0 "" {TEXT 537 12 "So lution: f" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "f := (x,y,z) -> sqrt(x+y*z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*-%%s qrtG6#,&9$\"\"\"*&9%F39&F3F3F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "A := subs( \{x=1,y=3,z=1\}, diff(f(x,y,z),x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,$*&\"\")!\"\"\"\"%#\"\"\"\"\"#F +" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "B := subs( \{x=1,y=3,z =1\}, diff(f(x,y,z),y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,$*& \"\")!\"\"\"\"%#\"\"\"\"\"#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "C := subs( \{x=1,y=3,z=1\}, diff(f(x,y,z),z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG,$*(\"\"$\"\"\"\"\")!\"\"\"\"%#F(\"\"#F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "simplify(A*(2/7) + B*(3/7) + C*(6/7));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#B\"#G" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 38 "2. The \+ tangent plane to the surface " }{XPPEDIT 19 1 "2*x*y + y^3 + z + x^2 *z^3=9" "6#/,**(\"\"#\"\"\"%\"xGF'%\"yGF'F'*$F)\"\"$F'%\"zGF'*&F(F&F,F +F'\"\"*" }{TEXT -1 32 " at the point (-1, 1, 2) has " }}{PARA 260 "" 0 "" {TEXT -1 10 "equation " }{XPPEDIT 19 1 "A*x+y+C*z = D;" "6#/, (*&%\"AG\"\"\"%\"xGF'F'%\"yGF'*&%\"CGF'%\"zGF'F'%\"DG" }{TEXT -1 14 ". What is D?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 540 77 "a) 35 b) 36 c) 37 d) 3 8 e) 39 " }{TEXT 541 20 " " }}{PARA 0 "" 0 "" {TEXT 538 35 "f) 40 g) 41 " }{TEXT 543 1 " " }{TEXT 539 39 "h) 42 i) 43 j) " }{TEXT 544 1 " " }{TEXT 542 2 "44" }{TEXT 545 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 546 12 "Solution: g" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "F := (x,y, z) -> 2*x*y + y^3 + z + x^2*z^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"FGf*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,**(\"\"#\"\"\"9$F19% F1F1*$)F3\"\"$F1F19&F1*&)F2F0F1)F7F6F1F1F*F*F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "P := -1,1,2;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"PG6%!\"\"\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "testeq( F(P) = 9 ); #Check that the given point is on the level s urface" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 133 "N := subs(\{x=P[1],y=P[2],z=P[3]\},[diff(F( x,y,z),x),diff(F(x,y,z),y),diff(F(x,y,z),z)]); # Calculates the gradi ent at the given point" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG7%!#9 \"\"\"\"#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "tangentPlane \+ := N[1]*(x-P[1])+N[2]*(y-P[2])+N[3]*(z-P[3])=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-tangentPlaneG/,**&\"#9\"\"\"%\"xGF)!\"\"\"#TF+%\"yGF )*&\"#8F)%\"zGF)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " map(w -> w + 41, tangentPlane);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,( *&\"#9\"\"\"%\"xGF'!\"\"%\"yGF'*&\"#8F'%\"zGF'F'\"#T" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 " " 0 "" {TEXT 301 3 "3. " }{TEXT -1 1 " " }{TEXT 260 15 "The function \+ " }{XPPEDIT 257 1 "f(x,y) = 3*x^3-x*y+y;" "6#/-%\"fG6$%\"xG%\"yG,(*& \"\"$\"\"\"*$F'F+F,F,*&F'F,F(F,!\"\"F(F," }{TEXT 256 58 " has one cr itical point P = (a , b) . What is a + b ?" }{TEXT 310 2 " " }} {PARA 0 "" 0 "" {TEXT 259 79 "a) 1 b) 2 c) \+ 3 d) 4 e) 6 " }{TEXT 260 20 " \+ " }}{PARA 0 "" 0 "" {TEXT 257 36 "f) 8 g) 9 \+ " }{TEXT 302 1 " " }{TEXT 258 40 "h) 10 i) 12 \+ j) " }{TEXT 303 1 " " }{TEXT 261 2 "15" }{TEXT 304 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 405 12 "Solution: h" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f := (x,y) ->3*x^3-x*y+y; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operato rG%&arrowGF),(*&\"\"$\"\"\")9$F/F0F0*&F2F09%F0!\"\"F4F0F)F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqn1 := diff( f(x,y), x ) = \+ 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&*&\"\"*\"\"\")%\"xG\" \"#F)F)%\"yG!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "e qn2 := diff( f(x,y), y ) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eq n2G/,&%\"xG!\"\"\"\"\"F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "cp := solve( \{eqn1,eqn2\} , \{x,y\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG<$/%\"xG\"\"\"/%\"yG\"\"*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "rhs(cp[1]) + rhs(cp[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "crit icalPoints(f); # Verification of critical point" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$\"\"\"\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 305 19 "4. The function " }{XPPEDIT 307 1 "f(x,y) = x*y+1/x+64/y" "6#/-%\" fG6$%\"xG%\"yG,(*&F'\"\"\"F(F+F+*&F+F+F'!\"\"F+*&\"#kF+F(F-F+" }{TEXT 257 2 " " }{TEXT 306 41 " has a critical point at ( 1/4 , 16 ). " } }{PARA 3 "" 0 "" {TEXT 418 102 "At this critical point, calculate the \+ expression made up of second derivatives of f that is used to " }} {PARA 3 "" 0 "" {TEXT 419 88 "determine the nature of the critical poi nt. (This expression, which is often called the" }}{PARA 3 "" 0 "" {TEXT 420 57 "discriminant of f, is denoted by D in the textbook.) \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 258 77 "a) 2 b) 3 c) \+ 4 d) 5 e) 6 " }{TEXT 259 20 " \+ " }}{PARA 0 "" 0 "" {TEXT 256 77 "f) 7 g) 8 \+ h) 9 i) 10 j) " }{TEXT 308 1 " " } {TEXT 260 2 "11" }{TEXT 309 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 406 12 "Solution : b" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f := (x,y) -> x*y+1/x + 64/y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(*&9$\" \"\"9%F0F0*&F0F0F/!\"\"F0*&\"#kF0F1F3F0F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "cp := map(allvalues, criticalPoints(f));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG7%7$#\"\"\"\"\"%\"#;7$,&#F(\"\") !\"\"*&^##F(F.F(\"\"$#F(\"\"#F(,&F.F/*&^#F.F(F3F4F(7$,&#F(F.F/*&^##F/F .F(F3F4F(,&F.F/*&^#!\")F(F3F4F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "discriminant(f,1/4,16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 300 2 "5." }{TEXT 441 1 " " }{TEXT 298 12 "The function" }{TEXT 299 3 " \+ " }{XPPEDIT 263 1 "f(x,y) = 12*x*y + x^2*y + 2*x*y^2" "6#/-%\"fG6$%\"x G%\"yG,(*(\"#7\"\"\"F'F,F(F,F,*&F'\"\"#F(F,F,*(F.F,F'F,F(F.F," }{TEXT 262 150 " has one local maximum. \nWhat is the value of f at this point? \n\n(The next question also pertains to this function so save \+ your calculations.) " }}{PARA 3 "" 0 "" {TEXT 297 2 "a)" }{TEXT 415 3 " 12" }{TEXT 421 5 " " }{TEXT 416 110 "b) 14 c) 15 \+ d) 16 e) 16 \nf) 21 g) 27 h) 30 i) 32 j) 34 " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 417 13 "Solution: i" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f \+ := (x,y) -> 12*x*y + x^2*y + 2*x*y^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"xG\" \"\"%\"yGF&\"#7*&F%\"\"#F'F&F&*&F%F&F'F*F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "cp := criticalPoints(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG7&7$\"\"!F'7$!#7F'7$F'!\"'7$!\"%!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "discriminantTest(f,0,0);" }}{PARA 6 "" 1 "" {TEXT -1 12 "Saddle point" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "discriminantTest(f,-12, 0);" }}{PARA 6 "" 1 "" {TEXT -1 12 "Saddle point" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "disc riminantTest(f,0, -6);" }}{PARA 6 "" 1 "" {TEXT -1 12 "Saddle point" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "discriminantTest(f,-4, -2) ;" }}{PARA 6 "" 1 "" {TEXT -1 13 "Local maximum" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "f(-4,-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" #K" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 3 " 6. " }{TEXT 424 12 "The function" }{TEXT 426 2 " " }{XPPEDIT 423 1 "f (x,y) = 12*x*y + x^2*y + 2*x*y^2" "6#/-%\"fG6$%\"xG%\"yG,(*(\"#7\"\"\" F'F,F(F,F,*&F'\"\"#F(F,F,*(F.F,F'F,F(F.F," }{TEXT 422 140 " of the p receding exercise has other critical points. \nWhich statement below b est describes the quantity and nature of the critical points" }{TEXT 425 19 " of f other than " }}{PARA 3 "" 0 "" {TEXT 442 18 "the local maximum?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 320 2 "a)" }{TEXT 318 1 " " }{TEXT -1 1 " " }{TEXT 319 388 "two local \+ minima \nb) one local minimum, one saddle point \nc) two sadd le points \nd) three local minima \ne) two local minima, one saddle point \nf) one local minimum, two saddle points \+ \ng) three saddle points \nh) three local minima, one sad dle point \n i) two local minima, two saddle points \n j) o ne local minimum, three saddle points" }{TEXT 321 3 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 9 "Solution:" }{TEXT 282 4 " g" }{TEXT -1 2 " \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "cpList := criticalPoints(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'cp ListG7&7$\"\"!F'7$F'!\"'7$!#7F'7$!\"%!\"#" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 88 "for n from 1 to nops(cpList) do\nprint(cpList[n]); \ndiscriminantTest(f,op(cpList[n]));\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"!F$" }}{PARA 6 "" 1 "" {TEXT -1 12 "Saddle point " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"!!\"'" }}{PARA 6 "" 1 "" {TEXT -1 12 "Saddle point" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$!#7\"\" !" }}{PARA 6 "" 1 "" {TEXT -1 12 "Saddle point" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$!\"%!\"#" }}{PARA 6 "" 1 "" {TEXT -1 13 "Local maximu m" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 269 33 "7. What is the maximum value of " }{TEXT 326 1 " " }{XPPEDIT 322 1 "f(x,y)=3*x+4*y" "6#/-%\"fG6$%\"xG%\"yG,&*&\"\"$ \"\"\"F'F,F,*&\"\"%F,F(F,F," }{TEXT 323 9 " if " }{XPPEDIT 324 1 "x^2+y^2=9" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(\"\"*" }{TEXT 325 3 " \+ ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 329 3 " a) " }{TEXT 327 1 " " }{TEXT 427 1 "5" }{TEXT -1 13 " " } {TEXT 328 133 "b) 8 c) 10 d) 12 e) 15 \n f) 36/5 g) 44/5 h) 48/5 i) 56/5 \+ j) 64/5" }{TEXT 330 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 407 9 "Solution:" } {TEXT 280 5 " e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 50 "f := (x,y) -> 3*x+4*y: phi := (x,y) -> x^2 \+ + y^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn1 := diff(f(x, y),x) = lambda*diff(phi(x,y),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% %eqn1G/\"\"$,$*&%'lambdaG\"\"\"%\"xGF*\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn2 := diff(f(x,y),y) = lambda*diff(phi(x,y),y); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/\"\"%,$*&%'lambdaG\"\"\"% \"yGF*\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eqn3 := phi( x,y) = 9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,&*$%\"xG\"\"#\" \"\"*$%\"yGF)F*\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sol ve( \{eqn1,eqn2,eqn3\}, \{x,y,lambda\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<%/%\"xG#\"\"*\"\"&/%'lambdaG#F(\"\"'/%\"yG#\"#7F(<%/F.#!#7F(/F% #!\"*F(/F*#!\"&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(9/5, 12/5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 124 "with(plots):#The following lines are to verif y that the level curve of f is tangent to the constraint curve at the \+ solution " }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoor ds has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 " levelCurve[0] := implicitplot( f(x,y)=15, x = -3..3, y=-3..3, color=pi nk, thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "constr aintCurve := implicitplot( phi(x,y) = 9, x = -3..3, y=-3..3, color=AQU AMARINE, thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "p ointPlot := plot([ [9/5,12/5], [9/5+.01,12/5+.01] ],style=POINT,symbol =DIAMOND, color=BLUE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "d isplay(levelCurve[0],pointPlot,constraintCurve,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 272 247 247 {PLOTDATA 2 "6(-%'CURVESG617$ 7$$\"3u++++++!3\"!#<$\"3]************RHF*7$$\"2c(**************F*$\"3y ,++++++IF*7$7$$\"3t++++++?8F*$\"3c,+++++gFF*F'7$7$$\"3]++++++g:F*$\"3= ************zDF*F37$7$$\"3q+++++++=F*$\"3Y*************R#F*7$$\"3!z*** ********R;F*$\"3N,+++++?DF*7$FDF97$7$$\"3#4++++++/#F*$\"3v************ >AF*7$$\"3=)***********f>F*$\"39,+++++!G#F*7$7$$\"3S)***********f>F*FS F?7$7$FSFL7$FL$\"3J************>AF*7$7$FG$\"3)*)***********f=F*FZ7$7$F 6$\"3#))***********z;F*7$$\"3?*************f#F*F@7$F`oFin7$7$F0$\"3A)* ***********\\\"F*7$$\"3/************>HF*$\"3s++++++g:F*7$7$FioF:F]o-%' COLOURG6&%$RGBG\"\"\"$\"*w6%Hv!\"*$\"+9Vygz!#5-%*THICKNESSG6#\"\"#-F$6 &7$7$$\"3/+++++++=F*$\"3!**************R#F*7$$\"30++++++5=F*$\"39+++++ +5CF*-F`p6&Fbp$\"\"!F^rF]r$\"*++++\"!\")-%&STYLEG6#%&POINTG-%'SYMBOLG6 #%(DIAMONDG-F$6du7$7$$!3!)************fFF*$!3e++++++q6F*7$$!3K+++++](z #F*$!3=************z5F*7$Fbs7$$!3.,++++D;GF*$!3h)********\\P-\"F*7$7$$ !3Q+++++]xGF*$!3#>************R)!#=Fhs7$F^t7$$!3'pmmmmmm\"HF*$!3QALLLL LLoFct7$7$$!3++++++]PHF*$!3+#*************fFctFet7$F[u7$$!3K++++++tHF* $!3A\"***********pQFct7$7$$!3/+++++]xHF*$!33#************f$FctFau7$7$$ !3[+++++]xHF*Fju7$$!3aFFFFFF(*HF*$!3j>FFFFFF7Fct7$7$$!31+++++](*HF*$!3 =#************>\"FctF`v7$7$$!3]+++++](*HF*Fiv7$Fgv$\"3c2+++++v6Fct7$7$ Fgv$\"3s2++++++7FctF_w7$7$F]w$\"3'y++++++?\"Fct7$$!3ep2Bp2BzHF*$\"3-+x I#p2BR$Fct7$7$Fhu$\"3i2++++++OFctFjw7$F`x7$$!3ir&G9dGk%HF*$\"3S?dG9dGk aFct7$7$F\\u$\"3a2++++++gFctFdx7$Fjx7$$!38++++++-HF*$\"3f1+++++?uFct7$ 7$$!3]**********\\xGF*$\"3Y2++++++%)FctF^y7$7$$!3%***********\\xGF*Fgy 7$$!3e********\\7[GF*$\"3W1++++D\"G*Fct7$7$$!3U**********\\(z#F*F(F]z7 $Fcz7$$!3#QHN#)eqky#F*$\"3w%HN#)eqk5\"F*7$7$F^s$\"3O++++++q6F*Fgz7$7$$ !3d************>DF*$!3+:dG9dGC;F*7$$!3Y\"444444c#F*$!3;************f:F *7$Ff[l7$$!3@,+++++5EF*$!35(***********p9F*7$7$$!3K#======p#F*$!3<**** ********>8F*F\\\\l7$Fb\\lF]s7$F][l7$$!3z********\\78FF*$\"3]++++]7t7F* 7$7$$!3W\"======p#F*F4Fi\\l7$F_]l7$$!3,N#)eqkF*7$$!3 C++++++(R#F*$!3;*************z\"F*7$F\\_l7$Fb[l$!3y9dG9dGC;F*7$Fc^l7$$ !31HN#)eqkZCF*$\"3)*HN#)eqkFF*F``l7$7$$!3 :************R?F*$!3knmmmmm'>#F*7$F]alF[al7$F_al7$Fh^l$!39,++++DY>F*7$ 7$Fh^l$\"3#4++++]i%>F*7$$!3*fF*7$7$$!3Kmmmmm m'>#F*FLFhal7$7$$!3'emmmmmm>#F*FL7$$!3HKLLLLL=@F*$\"32MLLLLL=@F*7$7$F[ al$\"3knmmmmm'>#F*Febl7$7$F__lF]_l7$Fj^lFh^l7$7$FbalFh^lFj`l7$F[cl7$$! 3Kv6%HN#)e*>F*$\"3xx6%HN#)eB#F*7$7$$!3q)********\\i%>F*FSFdcl7$7$$!3#* )********\\i%>F*FS7$$!3)))**********\\'=F*$\"3k++++++XBF*7$7$F__l$\"3o ++++++(R#F*Fadl7$7$Fi[lFg[l7$Fd[lFb[l7$7$Fc_lFb[lF_cl7$7$$!3#*)******* *****z\"F*$\"3C++++++(R#F*7$$!3?GN#)eqkF444444c#F*7$$!3%RB)eqk uFct$\"3e++++++-HF*7$7$F^u$\"3++++++]PHF*Fhjl7$Ffjl7$$!3y\"*********** pQFctFbu7$7$FjuFhuFb[m7$F^[m7$$!3m#p&G9dGkaFct$\"31s&G9dGk%HF*7$7$Fju$ \"3/+++++]xHF*Fh[m7$7$FjuF^v7$$!3x>FFFFFF7FctFav7$7$FivFgvFc\\m7$F^\\m 7$$!3#ym2Bp2BR$Fct$\"3ep2Bp2BzHF*7$7$Fiv$\"31+++++](*HF*Fi\\m7$7$FivF] w7$F`wF]w7$7$FdwFgvFd]m7$F_]m7$$!3Ut*********\\<\"FctF`]m7$7$FdwF`]mFh ]m7$7$FhwF]w7$F]xF[x7$7$FaxFhuF_^m7$F\\^m7$$\"3WbFFFFFF7Fct$\"3kEFFFFF (*HF*7$7$FaxF_\\mFc^m7$Fa^m7$Fgx$!3=r&G9dGk%HF*7$7$F[yF\\uF[_m7$Fi^m7$ $\"3_H+++++qQFct$\"3))***********H(HF*7$7$F[yF_[mFa_m7$F__m7$FayF_y7$7 $FgyFeyFi_m7$Fg_m7$$\"3olLLLLLLoFct$\"32mmmmmm;HF*7$7$Fgy$\"3%******** ***\\xGF*F]`m7$7$FgyF[z7$F`zF^z7$7$F(FdzFh`m7$Fc`m7$$\"3F.++++vB5F*$\" 3\"))********\\i\"GF*7$7$F($\"3))**********\\(z#F*F\\am7$Fj`m7$FjzFhz7 $7$F^[lF^sFfam7$Fham7$F\\]lFj\\l7$7$F4F`]lFjam7$7$F4$\"3W\"======p#F*7 $$\"3g'***********p6F*F67$FabmFbam7$F\\bm7$Ff]lFd]l7$7$F[pFj]lFfbm7$F^ bm7$$\"3'[++++++Z\"F*$\"3l(***********4EF*7$7$F:$\"39!444444c#F*Fjbm7$ Fhbm7$F`^lF^^l7$7$Fd^lFb[lFdcm7$Ffcm7$Fi_l$!3iGN#)eqkZCF*7$7$F@F]`lFhc m7$7$F@$\"3M***********pR#F*7$$\"3M7dG9dGC;F*FG7$7$$\"367dG9dGC;F*FG7$ F[pFacm7$F\\dm7$Fc`lFa`l7$7$Fg`lFh^lFjdm7$7$FfalFh^l7$F[blFial7$7$FLF_ blF_em7$7$FL$\"3'emmmmmm>#F*7$$\"3q)********\\i%>F*FS7$7$$\"3#*)****** **\\i%>F*FSF^dm7$7$FLFcbl7$FhblFfbl7$7$F\\clF[alF_fm7$Fafm7$Fgcl$!3yv6 %HN#)e*>F*7$7$FSF[dlFcfm7$7$FSFgem7$FdemFL7$FjfmFcem7$7$FSF_dl7$FddlFb dl7$7$FhdlF__lF^gm7$7$FcelFael7$FhelFfel7$7$FGF\\flFcgm7$7$FGFbdm7$F_d mF@7$Fhgm7$FSF[fm7$Fegm7$Fbfl$!3k)**********\\e\"F*7$7$FfflFi[lF\\hm7$ 7$F`glFi[l7$$\"3!fB)eqk " 0 "" {MPLTEXT 1 0 26 "f := (x,y) -> x^2+x*y+y^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(*$9$\" \"#\"\"\"*&F/F19%F1F1*$F3F0F1F)F)6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqn1 := diff( f(x,y), x ) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&%\"xG\"\"#%\"yG\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqn2 := diff( f(x,y), y ) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,&%\"xG\"\"\"%\"yG\"\"#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "cp := solve( \{eqn1,eqn2\} , \{x,y\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG<$/%\"xG\"\"!/%\" yGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "criticalPoints(f); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$\"\"!F%" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 75 "discriminantTest(f,0,0); # This interior loca l minimum is a global minimum" }}{PARA 6 "" 1 "" {TEXT -1 13 "Local mi nimum" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " phi := (x,y) -> x ^2 + y^2: #Now we look on the boundary" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn1 := diff(f(x,y),x) = lambda*diff(phi(x,y),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&%\"xG\"\"#%\"yG\"\"\",$*&%' lambdaGF*F'F*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn2 := \+ diff(f(x,y),y) = lambda*diff(phi(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,&%\"xG\"\"\"%\"yG\"\"#,$*&%'lambdaGF(F)F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eqn3 := phi(x,y) = 1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,&*$%\"xG\"\"#\"\"\"*$%\"yGF) F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solve( \{eqn1,eqn2, eqn3\}, \{x,y,lambda\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$<%/%\"xG,$ -%'RootOfG6#,&*$%#_ZG\"\"#F-!\"\"\"\"\"F./%'lambdaG#F/F-/%\"yGF'<%/F%F '/F1#\"\"$F-F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f(1/sqrt( 2),-1/sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f(1/sqrt(2),1/sqrt(2));" }} {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 260 "" 0 "" {TEXT -1 8 "9. Let " }{XPPEDIT 19 1 "f (x,y)=x+2*y" "6#/-%\"fG6$%\"xG%\"yG,&F'\"\"\"*&\"\"#F*F(F*F*" }{TEXT -1 7 " and " }{XPPEDIT 19 1 "phi(x)=int(f(x,y),y=x..x^2)" "6#/-%$phi G6#%\"xG-%$intG6$-%\"fG6$F'%\"yG/F.;F'*$F'\"\"#" }{TEXT -1 11 ". What is " }{XPPEDIT 19 1 "phi(3)" "6#-%$phiG6#\"\"$" }{TEXT -1 1 "?" }} {PARA 3 "" 0 "" {TEXT 511 128 "a) 10 b) 20 c) 30 \+ d) 40 e) 50 \nf) 60 g) 70 h) 80 i ) 90 j) 100" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 512 9 "Solution:" }{TEXT 513 6 " ( i )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f := (x,y) - > x+2*y; phi := (x)-> int(f(x,y),y=x..x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&9$\"\" \"*&\"\"#F/9%F/F/F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiGf*6# %\"xG6\"6$%)operatorG%&arrowGF(-%$intG6$-%\"fG6$9$%\"yG/F3;F2*$)F2\"\" #\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "phi(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"xG\"\"\",&*$)F%\"\"#F&F&F%!\"\" F&F&*$)F%\"\"%F&F&F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ph i(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#!*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 334 3 "10." }{TEXT -1 1 " " }{TEXT 331 12 "Calculate " }{XPPEDIT 332 1 "int(int(`(`*3*x^2- y*`)`,x=-1..1),y=0..3)" "6#-%$intG6$-F$6$,&*(%\"(G\"\"\"\"\"$F+%\"xG\" \"#F+*&%\"yGF+%\")GF+!\"\"/F-;,$F+F2F+/F0;\"\"!F," }{TEXT 333 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 337 75 "a) -3 \+ b) -2 c) -1 d) 0 e) 1 " } {TEXT 338 19 " " }}{PARA 0 "" 0 "" {TEXT 335 36 "f) \+ 2 g) 3 " }{TEXT -1 1 " " }{TEXT 336 43 "h) \+ 4 i) 5 j) 6 \n\n" }{TEXT 522 11 "Answer (a) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "int(int(3*x^2-y, x = -1 .. 1),y = 0 .. 3);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 339 113 "11. Let \+ a and b be the constants and f and g the functions of \+ one variable such that\n\n " }{XPPEDIT 445 1 "int(int(psi(x,y ),y=(x+1)^2..13-(x-4)^2),x=1..2) = int(int(psi(x,y),x=f(y)..g(y)),y=a. .b)" "6#/-%$intG6$-F%6$-%$psiG6$%\"xG%\"yG/F-;*$,&F,\"\"\"F2F2\"\"#,& \"#8F2*$,&F,F2\"\"%!\"\"F3F9/F,;F2F3-F%6$-F%6$-F*6$F,F-/F,;-%\"fG6#F-- %\"gG6#F-/F-;%\"aG%\"bG" }{TEXT 346 32 "\nfor every continuous functio n " }{XPPEDIT 19 1 "psi" "6#%$psiG" }{TEXT 347 137 ". (In other wor ds, reverse the order of integration of the \niterated integral on the left side of the identity.) \n\nWhat is the sum " }{TEXT 353 1 " \+ " }{XPPEDIT 354 1 "a+b+f(12)+g(16)" "6#,*%\"aG\"\"\"%\"bGF%-%\"fG6#\"# 7F%-%\"gG6#\"#;F%" }{TEXT 352 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 348 72 "a) 16 b) 17 c) \+ 18 d) 19 e) " }{TEXT 349 2 "20" }{TEXT 444 1 " " }{TEXT 351 20 " " }}{PARA 0 "" 0 "" {TEXT 350 76 "f ) 21 g) 22 h) 23 i) 24 j ) 25" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 443 10 "Answer (d)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "plot([(x+1)^2,13-(x-4)^2],x=1..2,color=[PLUM,BLUE],thickness=2,num points=500);" }}{PARA 13 "" 1 "" {GLPLOT2D 293 211 211 {PLOTDATA 2 "6' -%'CURVESG6$7`jl7$$\"\"\"\"\"!$\"\"%F*7$$\"3\"4[S-s'4-5!#<$\"3x7R 3SF07$$\"3?@1Jl5#R+\"F0$\"30ac*fj*p:SF07$$\"3i)GW@tsf+\"F0$\"3&>U1@gER -%F07$$\"3*\\^d()*z.35F0$\"3.&3aWg;A.%F07$$\"37Q!>&\\M455F0$\"3(pM*zvc ZSSF07$$\"3)>)4\\:\"**>,\"F0$\"3g$\\O2WS\"[SF07$$\"3+\\WAAB(R,\"F0$\"3 GRho9X3cSF07$$\"3s,4X:I,;5F0$\"3gQuYy%3V1%F07$$\"3m'H[T;Z!=5F0$\"3/$)p scV^sSF07$$\"3_CA6O&R,-\"F0$\"3k7POXP'43%F07$$\"3)Q&pZ)\\#)>-\"F0$\"3] /r9CKT)3%F07$$\"3T!>&fPs0C5F0$\"3%QZh5q2o4%F07$$\"3<$e\"z&\\Sh-\"F0$\" 3dilo3`C0TF07$$\"3Ek@3,\"[\"G5F0$\"3b)=B*>ZQ8TF07$$\"3-O!=!*>r*H5F0$\" 3%HsU)oIy?TF07$$\"3G&f(zQ!R@.\"F0$\"3]9_+t!*eHTF07$$\"3s$*oW$[vR.\"F0$ \"3AEIEni0PTF07$$\"3g+.:0<6O5F0$\"3'e1]e(3vXTF07$$\"3F`mKBE+Q5F0$\"3%) >jo#paM:%F07$$\"30@0Egs2S5F0$\"3!G2KyA:>;%F07$$\"3&4YI_!G0U5F0$\"3-&f \\$f'z*pTF07$$\"3b;$e\"zS6W5F0$\"3)HD8'oBSyTF07$$\"3EoT3sp+Y5F0$\"3WDq \")HX9'=%F07$$\"3E^c#Gr[![5F0$\"3U2o5IN]%>%F07$$\"3iq_j2&p,0\"F0$\"3A[ u.5]>.UF07$$\"3'H]^dm:?0\"F0$\"3i#e*[#Ho2@%F07$$\"3[)=%4n&4S0\"F0$\"3a aWr,`&*=UF07$$\"3ApXGi%pg0\"F0$\"3pd\"RPj@uA%F07$$\"3U6d&ym%3e5F0$\"3N `F1+DrNUF07$$\"3aJd'G[M+1\"F0$\"3F\")efq?uVUF07$$\"3++++?%*>i5F0$\"3#e L[yWmED%F07$$\"3G4YI7Z9k5F0$\"39y]L$R$pgUF07$$\"373T0FI% F07$$\"3=bvx)y*4w5F0$\"3@5A'HL!>5VF07$$\"3Uz'R)p6*f:m,3\"F0$\"3#pYMlB\"4FVF07$$\"3))\\\\Z#)3\"F0$\"3#[ltp,h1O%F07$$\"3[tnQ8a7!4\"F0$\"3% )H_oVUioVF07$$\"3W%>(fGu:#4\"F0$\"3mCW.1F7xVF07$$\"3wlGVEB5%4\"F0$\"3w T:d`XE&Q%F07$$\"3=i5`&)=E'4\"F0$\"3'HCIF*QJ%R%F07$$\"3=g+.0>8)4\"F0$\" 3Op]!4\\d@S%F07$$\"3dvx)QRa-5\"F0$\"3#o!\\/>&o5T%F07$$\"3LGT17))=-6F0$ \"3X\"fn:!y>>WF07$$\"35'3V:w/V5\"F0$\"3$e!HXH&)4GWF07$$\"3J#>'4o/816F0 $\"3s&3V'[byNWF07$$\"3-*\\\\Zt4#36F0$\"3L`Dp&G[XW%F07$$\"3nuto.#=-6\"F 0$\"3qqN&pC??6F0$\"3)fia<^e_\\%F07$$\"3y\"*e%Hl &GA6F0$\"3)*p**p#R'4/XF07$$\"3F#=\"fN9o#eXHXF07$$\"3`B<'3@V- 8\"F0$\"3\\mCUPh$z`%F07$$\"3?)3W?ABB8\"F0$\"3m#[oKK-oa%F07$$\"37`lF))* QU8\"F0$\"39C(z!fg(\\b%F07$$\"3!***)\\\\*4?O6F0$\"3]U\\#3paLc%F07$$\"3 'Q*oW$Gn$Q6F0$\"3e\\g\"*QYhsXF07$$\"31\"[S-oG.9\"F0$\"3cf*f%fo+\"e%F07 $$\"3R.<&edMB9\"F0$\"31(p*)[V(f*e%F07$$\"39S+-?XPW6F0$\"3))y66#3U$)f%F 07$$\"3eU8nD'\\i9\"F0$\"38xVpbuQ1YF07$$\"3M]^dF%[#[6F0$\"3$\\5zlIr\\h% F07$$\"3)z$*oW,K-:\"F0$\"3KZr/;x\\BYF07$$\"3cV<(e$QN_6F0$\"3#)3=zMqiKY F07$$\"3I&oU8DEU:\"F0$\"3.eR(>u!pSYF07$$\"3Q+-5ImQc6F0$\"3-)*[Z)H.+l%F 07$$\"3cT2P0iLe6F0$\"3#=<9a<:%eYF07$$\"3!HR'>3_Eg6F0$\"3Q/QupdumYF07$$ \"3uj=$f]RB;\"F0$\"3WcO\"*Q@rvYF07$$\"3ce#HY>AW;\"F0$\"3COPgOMs%o%F07$ $\"3'=$f'H8ZF07$ $\"31$\\YK=kU<\"F0$\"3tR&[&QZUFZF07$$\"3QJd'G#)zj<\"F0$\"3qsqPL\"Hmt%F 07$$\"3Eh1LN'o$y6F0$\"3WnBR6**GXZF07$$\"31sg.Q#Q.=\"F0$\"3sVACK[(Qv%F0 7$$\"3M]^d2%4B=\"F0$\"3K/5R]VZiZF07$$\"3!GYJd()>V=\"F0$\"3M8b^>LDrZF07 $$\"33$f'HQ4X'=\"F0$\"3W_!ybqn0y%F07$$\"3)f(z)R\"\\S)=\"F0$\"3#>smv1; \"*y%F07$$\"3Y&z(*)QPL!>\"F0$\"3s%*Rv()=c(z%F07$$\"3%R'>)4i&R#>\"F0$\" 30\"*3!*e&)f1[F07$$\"3;e!HX,`W>\"F0$\"3xqc(H.Cc\"[F07$$\"3uH[TZ'*H'>\" F0$\"3W1XBT@tB[F07$$\"3')oWB\"F0$\"3)[#fk/cML[F07$$\"30#3T0\\<.? \"F0$\"3r/5=fqRT[F07$$\"3)3V:xJeC?\"F0$\"3_&)QFTE#3&[F07$$\"3[>(f)H:]/ 7F0$\"3'G%Q:&*p#)f[F07$$\"3xf)H\\(eK17F0$\"3=6]QmQ(y'[F07$$\"3'p_jX? 7F0$\"3!393y'pSI\\F07$$\"3(GYJdMWDA\"F0$\"3G(*GuoLqR\\F07$$\"3X#>'43tQ C7F0$\"3QC%>k*))*y%\\F07$$\"3vGW@Z?YE7F0$\"3=kioZK8d\\F07$$\"3_@3T0`aG 7F0$\"3w6'o#yTTm\\F07$$\"3#GS,2\"HbI7F0$\"3'*)\\QuGm`(\\F07$$\"3fusj3g PK7F0$\"3A)GRRk-N)\\F07$$\"3jLoT[QaM7F0$\"3K8v\\5i=$*\\F07$$\"3HKh1$H! 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{XPPMATH 20 "6$,&\"\"%\"\"\"*$,&\"#8F%%\"yG!\"\" #F%\"\"#F%,&F$F%F&F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "f : = y -> 4-(13-y)^(1/2); g := y -> y^(1/2)-1; a := 4; b := 9;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"yG6\"6$%)operatorG%&arrow GF(,&\"\"%\"\"\"*$,&\"#8F.9$!\"\"#F.\"\"#F3F(F(6\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"gGf*6#%\"yG6\"6$%)operatorG%&arrowGF(,&*$9$#\"\" \"\"\"#F0!\"\"F0F(F(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 57 "int(int(x^3*y^5,y=(x+1)^2..13-(x-4)^2),x=1..2) ; #example " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"(t@N*\"$&Q" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "int(int(x^3*y^5,x=f(y)..g(y) ),y=a..b); \n#evidence supporting correctness of answer " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"(t@N*\"$&Q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "a := 4; b:= 9; f := y -> 4-sqrt(13-y); g := y -> sq rt(y)-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"bG\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"fGf*6#%\"yG6\"6$%)operatorG%&arrowGF(,&\"\"%\"\"\"-%%sqrtG6#,&\"#8F .9$!\"\"F5F(F(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"yG6 \"6$%)operatorG%&arrowGF(,&-%%sqrtG6#9$\"\"\"!\"\"F1F(F(6\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a + b + f(12) + g(16);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 270 18 "12. Calcul ate " }{TEXT 341 1 " " }{XPPEDIT 342 1 "Int( Int( sqrt(1+8*y^3), y= sqrt(x)..1),x=0..1)" "6#-%$IntG6$-F$6$-%%sqrtG6#,&\"\"\"F,*&\"\")F,*$% \"yG\"\"$F,F,/F0;-F)6#%\"xGF,/F6;\"\"!F," }{TEXT 340 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 344 73 "a) 5/18 \+ b) 1/3 c) 7/18 d) 4/9 e) " }{TEXT 345 3 "1/2" }{TEXT 446 20 " " }}{PARA 0 "" 0 "" {TEXT 343 78 "f) 5/9 g) 11/18 h) 2/3 i) 13/18 j) 7/9\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 9 "Solution:" }{TEXT 284 5 " (i)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Answer := int( int ( sqrt(1+8*y^3), x=0..y^2),y=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'AnswerG#\"#8\"#=" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 61 "This answer is based on the following reversal of integration" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Int( Int( sqrt(1+8*y^3), y=sqrt(x). .1),x=0..1) = Int( Int( sqrt(1+8*y^3), x=0..y^2),y=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-F%6$*$,&\"\"\"F+*&\"\")F+)%\"yG\" \"$F+F+#F+\"\"#/F/;*$%\"xGF1F+/F6;\"\"!F+-F%6$-F%6$F)/F6;F9*$)F/F2F+/F /F8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 264 103 "13. Let a and b be the constants and f and g \+ the functions of one variable such that\n\n " }{TEXT 364 5 " \+ " }{XPPEDIT 365 1 "int(int(psi(x,y),x=-sqrt(2-y)..y),y=-2..1) + int( int(psi(x,y),x=-sqrt(2-y)..sqrt(2-y)),y=1..2) = int(int(psi(x,y),y=f( x)..g(x)),x=a..b)" "6#/,&-%$intG6$-F&6$-%$psiG6$%\"xG%\"yG/F-;,$-%%sqr tG6#,&\"\"#\"\"\"F.!\"\"F8F./F.;,$F6F8F7F7-F&6$-F&6$-F+6$F-F./F-;,$-F3 6#,&F6F7F.F8F8-F36#,&F6F7F.F8/F.;F7F6F7-F&6$-F&6$-F+6$F-F./F.;-%\"fG6# F--%\"gG6#F-/F-;%\"aG%\"bG" }{TEXT 355 33 "\n\nfor every continuous fu nction " }{XPPEDIT 19 1 "psi" "6#%$psiG" }{TEXT 356 13 ". What is \+ " }{TEXT 362 1 " " }{XPPEDIT 363 1 "a+b+f(3)^2+g(3)" "6#,*%\"aG\"\"\"% \"bGF%*$-%\"fG6#\"\"$\"\"#F%-%\"gG6#F+F%" }{TEXT 361 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 357 69 "a) 1 \+ b) 2 c) 3 d) 4 e) 5" }{TEXT 358 1 " \+ " }{TEXT 360 20 " " }}{PARA 0 "" 0 "" {TEXT 359 72 "f) 6 g) 7 h) 8 i) 9 j) \+ 12" }{TEXT -1 2 " " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 285 1 " " }}{PARA 0 "" 0 "" {TEXT 286 9 "Solution:" }{TEXT 287 5 " (a)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "pl1 := plot( [-sqrt(2-y),y,y=-2..2],color=BLUE):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "pl2 := plot([y,y,y=-2..1],color=NAVY):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "pl3 := plot([sqrt(2-y),y,y=1..2],co lor=MAROON):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plots[displ ay](pl1,pl2,pl3);" }}{PARA 13 "" 1 "" {GLPLOT2D 374 374 374 {PLOTDATA 2 "6'-%'CURVESG6$7W7$$!\"#\"\"!F(7$$!3Yy)>Wv#3y>!#<$!3MLLL$Q6G\">F.7$$ !3a0hzpJ\")e>F.$!3bmm;M!\\p$=F.7$$!3C!*\\J&\\8p$>F.$!3MLLL))Qj^F.$!3ALLL=Kvl;F.7$$!3Y'Qa](H;#*=F.$!3wmm;C2G!e\"F.7$$!3'Q! 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F0F-F7/F.F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "psi := (x,y) -> 3*x+2*y; #Get evidence of correctness with specific psi" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrow GF),&*&\"\"$\"\"\"9$F0F0*&\"\"#F09%F0F0F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "int(int(psi(x,y),x=-sqrt(2-y)..y),y=-2..1) + in t(int(psi(x,y),x=-sqrt(2-y)..sqrt(2-y)),y=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!#j\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " int(int(psi(x,y),y=x..2-x^2),x=-2..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!#j\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a := -2; \+ b := 1; f := x -> x; g := x -> 2-x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(9$F(F(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%) operatorG%&arrowGF(,&\"\"#\"\"\"*$9$F-!\"\"F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "a+b+f(3)^2+g(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 265 52 "14. Calculate the area \+ enclosed by the polar curve " }{TEXT 367 1 " " }{XPPEDIT 368 1 "r=(2+s in(theta))/sqrt(Pi)" "6#/%\"rG*&,&\"\"#\"\"\"-%$sinG6#%&thetaGF(F(-%%s qrtG6#%#PiG!\"\"" }{TEXT 366 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 276 75 "a) 4 b) 17/4 c ) 9/2 d) 19/4 e) " }{TEXT 277 21 "5 \+ " }}{PARA 0 "" 0 "" {TEXT 274 36 "f) 21/4 g) 11/2 \+ " }{TEXT -1 1 " " }{TEXT 275 44 "h) 23/4 i) 6 \+ j) 27/4" }}{PARA 0 "" 0 "" {TEXT 268 1 " " }}{PARA 0 "" 0 "" {TEXT 288 9 "Solution:" }{TEXT 289 5 " (c)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "simplify(int(((2+sin(theta))/sqrt(Pi))^2 ,theta = 0 .. 2*Pi)/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"*\"\"#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 266 21 "15. Calculate " }{XPPEDIT 449 1 "1/Pi" "6#*&\"\"\"F$%#PiG!\"\" " }{TEXT 448 2 " " }{XPPEDIT 370 1 "int(int( `(`*x^2+y^2*`)`,y=x..sqr t(1-x^2)),x=0..1/sqrt(2))" "6#-%$intG6$-F$6$,&*&%\"(G\"\"\"*$%\"xG\"\" #F+F+*&%\"yGF.%\")GF+F+/F0;F--%%sqrtG6#,&F+F+*$F-F.!\"\"/F-;\"\"!*&F+F +-F56#F.F9" }{TEXT 271 3 ". " }}{PARA 0 "" 0 "" {TEXT 452 72 "\na) 1 /64 b) 1/32 c) 1/16 d) 1/8 e) " } {TEXT 453 1 " " }{TEXT 454 3 "1/4" }{TEXT 455 1 " " }{TEXT 456 13 " \+ " }}{PARA 0 "" 0 "" {TEXT 450 36 "f) 3/64 g) 3/3 2 " }{TEXT -1 1 " " }{TEXT 451 41 "h) 3/16 i) 3/8 \+ j) 3/4" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 371 9 "Solution:" }{TEXT 372 5 " (c)\n" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "(1/Pi)* int(int((x^2+y^2),y=x..sqrt(1-x^2)),x=0..1/sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "(1/Pi)*int(int(r^2*r,r=0..1),theta=Pi/4..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"#;" }}}{PARA 3 "" 0 "" {TEXT 369 1 "\n" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 457 43 "16. Calculate the area inside both circles " }{XPPEDIT 460 1 "r=co s(theta)" "6#/%\"rG-%$cosG6#%&thetaG" }{TEXT 458 7 " and " } {XPPEDIT 461 1 "r=sin(theta)" "6#/%\"rG-%$sinG6#%&thetaG" }{TEXT 459 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT 462 4 "a) " }{XPPEDIT 466 1 "(Pi-1)/32" "6#*&,&%#PiG\"\"\"F&!\"\"F&\"#KF'" }{TEXT 463 5 " " }{TEXT 484 2 "b)" }{TEXT 485 2 " " }{XPPEDIT 467 1 "(Pi-2)/32" "6#*&,&%#PiG\"\"\"\"\"#!\"\"F&\"#KF(" }{TEXT 464 6 " " }{TEXT 486 2 "c)" }{TEXT 487 3 " " }{XPPEDIT 468 1 "(Pi-1)/1 6" "6#*&,&%#PiG\"\"\"F&!\"\"F&\"#;F'" }{TEXT 465 4 " " }{TEXT 488 2 "d)" }{TEXT 489 3 " " }{XPPEDIT 469 1 "(Pi-2)/16" "6#*&,&%#PiG\"\" \"\"\"#!\"\"F&\"#;F(" }{TEXT 470 8 " e)" }{TEXT 477 1 " " } {XPPEDIT 478 1 "(Pi-1)/8" "6#*&,&%#PiG\"\"\"F&!\"\"F&\"\")F'" }{TEXT 471 3 " \n" }{TEXT 498 2 "f)" }{TEXT 499 3 " " }{XPPEDIT 479 1 "(Pi -2)/8" "6#*&,&%#PiG\"\"\"\"\"#!\"\"F&\"\")F(" }{TEXT 472 5 " " } {TEXT 496 2 "g)" }{TEXT 497 3 " " }{XPPEDIT 480 1 "(Pi-1)/4" "6#*&,& %#PiG\"\"\"F&!\"\"F&\"\"%F'" }{TEXT 473 4 " " }{TEXT 494 2 "h)" } {TEXT 495 2 " " }{XPPEDIT 481 1 "(Pi-2)/4" "6#*&,&%#PiG\"\"\"\"\"#!\" \"F&\"\"%F(" }{TEXT 474 6 " " }{TEXT 492 2 "i)" }{TEXT 493 1 " " }{XPPEDIT 482 1 "(Pi-1)/2" "6#*&,&%#PiG\"\"\"F&!\"\"F&\"\"#F'" }{TEXT 475 7 " " }{TEXT 490 2 "j)" }{TEXT 491 1 " " }{XPPEDIT 483 1 "(P i-2)/2" "6#*&,&%#PiG\"\"\"\"\"#!\"\"F&F'F(" }{TEXT 476 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 500 9 "Solution:" } {TEXT 501 4 " (f)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "Area = (1/2)*int(sin(theta)^2,theta=0..Pi/4)+ (1/2)*int(cos(theta)^2,theta=Pi/4..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%%AreaG,&#!\"\"\"\"%\"\"\"%#PiG#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 267 21 " 17. The gr aph of " }{XPPEDIT 383 1 "y=sin(x)" "6#/%\"yG-%$sinG6#%\"xG" }{TEXT 380 2 ", " }{XPPEDIT 384 1 "x" "6#%\"xG" }{TEXT 382 2 " " }{TEXT 387 2 "in" }{TEXT 388 1 " " }{XPPEDIT 385 1 "[0,Pi/6]" "6#7$\"\"!*&%#PiG\" \"\"\"\"'!\"\"" }{TEXT 381 1 " " }{TEXT 386 129 " is rotated about the x-axis. Which of the following integrals represents the surface area \+ of the resulting surface of revolution" }{TEXT 379 3 "?\n\n" }{TEXT 272 4 "a) " }{XPPEDIT 504 1 "2*Pi*Int(cos(x)*sqrt(1+sin(x)^2),x=0..Pi /6)" "6#*(\"\"#\"\"\"%#PiGF%-%$IntG6$*&-%$cosG6#%\"xGF%-%%sqrtG6#,&F%F %*$-%$sinG6#F.F$F%F%/F.;\"\"!*&F&F%\"\"'!\"\"F%" }{TEXT 505 15 " \+ b) " }{XPPEDIT 390 1 "2*Pi*Int(sin(x)*sqrt(1+sin(x)^2),x=0..Pi/6 )" "6#*(\"\"#\"\"\"%#PiGF%-%$IntG6$*&-%$sinG6#%\"xGF%-%%sqrtG6#,&F%F%* $-F,6#F.F$F%F%/F.;\"\"!*&F&F%\"\"'!\"\"F%" }{TEXT 389 12 " \nc) \+ " }{XPPEDIT 503 1 "2*Pi*Int(sin(x)*sqrt(1+cos(x)^2),x=0..Pi/6)" "6#*( \"\"#\"\"\"%#PiGF%-%$IntG6$*&-%$sinG6#%\"xGF%-%%sqrtG6#,&F%F%*$-%$cosG 6#F.F$F%F%/F.;\"\"!*&F&F%\"\"'!\"\"F%" }{TEXT 502 13 " d) " } {XPPEDIT 392 1 "2*Pi*Int(cos(x)*sqrt(1+cos(x)^2),x=0..Pi/6)" "6#*(\"\" #\"\"\"%#PiGF%-%$IntG6$*&-%$cosG6#%\"xGF%-%%sqrtG6#,&F%F%*$-F,6#F.F$F% F%/F.;\"\"!*&F&F%\"\"'!\"\"F%" }{TEXT 391 16 " \ne) " } {XPPEDIT 394 1 "2*Pi*Int(sin(x)*sqrt(1+cos(x)),x=0..Pi/6)" "6#*(\"\"# \"\"\"%#PiGF%-%$IntG6$*&-%$sinG6#%\"xGF%-%%sqrtG6#,&F%F%-%$cosG6#F.F%F %/F.;\"\"!*&F&F%\"\"'!\"\"F%" }{TEXT 393 21 " f) " } {XPPEDIT 395 1 "2*Pi*Int(sin(x)*sqrt(1+sin(x)),x=0..Pi/6)" "6#*(\"\"# \"\"\"%#PiGF%-%$IntG6$*&-%$sinG6#%\"xGF%-%%sqrtG6#,&F%F%-F,6#F.F%F%/F. ;\"\"!*&F&F%\"\"'!\"\"F%" }{TEXT 396 13 " \ng) " }{XPPEDIT 398 1 "2*Pi*Int(cos(x)*sqrt(1+sin(x)),x=0..Pi/6)" "6#*(\"\"#\"\"\"%#Pi GF%-%$IntG6$*&-%$cosG6#%\"xGF%-%%sqrtG6#,&F%F%-%$sinG6#F.F%F%/F.;\"\"! *&F&F%\"\"'!\"\"F%" }{TEXT 397 21 " h) " }{XPPEDIT 399 1 "2*Pi*Int(cos(x)*sqrt(1+cos(x)),x=0..Pi/6)" "6#*(\"\"#\"\"\"%#Pi GF%-%$IntG6$*&-%$cosG6#%\"xGF%-%%sqrtG6#,&F%F%-F,6#F.F%F%/F.;\"\"!*&F& F%\"\"'!\"\"F%" }{TEXT 400 17 " \ni) " }{XPPEDIT 402 1 "2* Pi*Int(sin(x)^2*sqrt(1+cos(x)),x=0..Pi/6)" "6#*(\"\"#\"\"\"%#PiGF%-%$I ntG6$*&-%$sinG6#%\"xGF$-%%sqrtG6#,&F%F%-%$cosG6#F.F%F%/F.;\"\"!*&F&F% \"\"'!\"\"F%" }{TEXT 401 18 " j) " }{XPPEDIT 404 1 "2*Pi *Int(cos(x)^2*sqrt(1+sin(x)),x=0..Pi/6)" "6#*(\"\"#\"\"\"%#PiGF%-%$Int G6$*&-%$cosG6#%\"xGF$-%%sqrtG6#,&F%F%-%$sinG6#F.F%F%/F.;\"\"!*&F&F%\" \"'!\"\"F%" }{TEXT 403 6 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 290 9 "Solution:" }{TEXT 291 5 " (c)\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := x -> sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%$sinG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "2*Pi*Int(f(x)*sqrt(1+diff(f(x),x)^2),x=0..Pi/6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&-%$si nG6#%\"xGF&,&F&F&*$)-%$cosGF.F%F&F&#F&F%/F/;\"\"!,$*&\"\"'!\"\"F'F&F&F &F&" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 35 "18. Calculate the surface area " }{XPPEDIT 19 1 "S" "6#%\"S G" }{TEXT -1 34 " of that part of the paraboloid " }{XPPEDIT 19 1 "z =2*(x^2+y^2)" "6#/%\"zG*&\"\"#\"\"\",&*$%\"xGF&F'*$%\"yGF&F'F'" } {TEXT -1 42 " that lies under the \nhorizontal plane " }{XPPEDIT 19 1 "z=6" "6#/%\"zG\"\"'" }{TEXT -1 12 ". What is " }{XPPEDIT 19 1 "S/Pi" "6#*&%\"SG\"\"\"%#PiG!\"\"" }{TEXT -1 2 " ?" }}{PARA 3 "" 0 "" {TEXT 278 166 "a) 23/2 b) 25/2 c) 27/2 \+ d) 29/2 e) 31/2 \nf) 55/4 g) 57/4 h) \+ 59/4 i) 61/4 j) 63/4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 293 9 "Solution:" }{TEXT 294 4 " g " }{TEXT 408 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "int(in t(sqrt(1+16*r^2)*r,r=0..sqrt(3)),theta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"#d\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalf( \" ); # for verification" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K&pnZ%!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "int(int(sqrt(1+diff(2*(x^2+ y^2),x)^2 + diff(2*(x^2+y^2),y)^2),y=-sqrt(3-x^2)..sqrt(3-x^2)),x=-sqr t(3)..sqrt(3)); # for verification" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# -%$intG6$,,*$,&\"\"$\"\"\"*$%\"xG\"\"#!\"\"#F*F-\"\"(-%#lnG6#,&F'\"\"% F0F*#F*\"\")*&F1F*F,F-F--F26#,&F'!\"%F0F*#F.F7*&F9F*F,F-!\"#/F,;,$*$F) F/F.FC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalf( \" ); # fo r verification" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+J&pnZ%!\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 273 4 "19. " }{TEXT 295 65 "What is the \+ surface area of the surface that is parameterized by\n" }}{PARA 3 "" 0 "" {TEXT 506 1 " " }{XPPEDIT 410 1 "x=u+v,y=u-v,z=u-sqrt(5)*v" "6%/% \"xG,&%\"uG\"\"\"%\"vGF'/%\"yG,&F&F'F(!\"\"/%\"zG,&F&F'*&-%%sqrtG6#\" \"&F'F(F'F," }{TEXT 409 8 " \n\nfor " }{XPPEDIT 414 1 "u" "6#%\"uG" } {TEXT 413 19 " in [0,1] and " }{XPPEDIT 412 1 "v" "6#%\"vG" } {TEXT 411 12 " in [2,5]. " }{TEXT 296 1 "\n" }}{PARA 3 "" 0 "" {TEXT 507 128 "a) 8 b) 9 c) 10 d) 12 e) 1 5 \nf) 16 g) 18 h) 20 i) 24 j) 30" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 508 9 "Solutio n:" }{TEXT 509 4 " d" }{TEXT 547 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "r := (u,v) -> [u+v,u-v,u-sqrt(5)*v];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF)7%,& 9$\"\"\"9%F0,&F/F0F1!\"\",&F/F0*&-%%sqrtG6#\"\"&F0F1F0F3F)F)6\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "v1 := map(z->diff(z,u), r(u, v));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G7%\"\"\"F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "v2 := map(z->diff(z,v), r(u,v));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G7%\"\"\"!\"\",$*$\"\"&#F&\"\"#F '" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "cp := linalg[crossprod ](v1,v2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG-%'VECTORG6#7%,&*$ \"\"&#\"\"\"\"\"#!\"\"F-F-,&F*F-F-F-!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "integrand := simplify(sqrt(simplify(cp[1]^2+cp[2]^2+c p[3]^2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*integrandG\"\"%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "int(int(integrand,u=0..1),v= 2..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 292 55 "20. Let S be the solid region that that lies und er " }{XPPEDIT 313 1 "z=8-(x^2+y^2)" "6#/%\"zG,&\"\")\"\"\",&*$%\"xG \"\"#F'*$%\"yGF+F'!\"\"" }{TEXT 311 13 " and over " }{XPPEDIT 314 1 "z=x^2+y^2" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)" }{TEXT 312 3 " .\n " }{TEXT 315 38 "The volume of S is what multiple of " } {XPPEDIT 317 1 "Pi" "6#%#PiG" }{TEXT 316 144 "?\n\na) 1 b) \+ 2 c) 3 d) 4 e) 5 \nf) 8 g ) 12 h) 15 i) 16 j) 20" }{TEXT 510 1 " \n" }}{PARA 3 "" 0 "" {TEXT -1 10 "Answer i" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "J := Int(Int (r*(8-r^2 - r^2),r=0..2),theta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$-F&6$*&%\"rG\"\"\",&\"\")F,*&\"\"#F,)F+F0F,!\"\"F ,/F+;\"\"!F0/%&thetaG;F5,$*&F0F,%#PiGF,F," }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "value( J );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*& \"#;\"\"\"%#PiGF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "int( int(8-2*(x^2+y^2),y=-sqrt(4-x^2)..sqrt(4-x^2)),x=-2..2); #Verify" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "19 0 1" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }