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"Times" 1 18 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 "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{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 "Times" 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 "T imes" 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 "" 3 260 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 261 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 263 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 3 264 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 403 30 "Math 233 Fall 2005 Fina l Exam" }}{PARA 0 "" 0 "" {TEXT 256 2 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 258 1 "1" }{TEXT 266 2 ". " }{TEXT 404 13 " The vector " } {XPPEDIT 472 1 "`<`*a,1,1*`>` " "6%*&%\"GF% " }{TEXT 473 30 " is tangent to the surface " }{XPPEDIT 474 1 "x^2- y^8-3*z^4=0" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yG\"\")!\"\"*&\"\"$F(*$%\"zG \"\"%F(F,\"\"!" }{TEXT 475 41 " at the point (2,1,1). \n What i s " }{XPPEDIT 482 1 "a" "6#%\"aG" }{TEXT 481 3 " ? " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 479 74 "a) -4 b) - 3 c) -2 d) -1 e) " }{TEXT 476 2 "0 \+ " }{TEXT 480 20 " " }}{PARA 0 "" 0 "" {TEXT 477 37 "f) 1 g) 2 " }{TEXT -1 1 " " }{TEXT 478 39 "h) 3 i) 4 j) " }{TEXT -1 1 " " }{TEXT 512 1 "5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 " " {TEXT 405 8 "Solution" }{TEXT 406 6 ": (j)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "F := (x,y,z) -> x^2-y^8-3*z^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*, (*$9$\"\"#\"\"\"*$9%\"\")!\"\"*$9&\"\"%!\"$F*F*6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "v := grad(F(x,y,z),[x,y,z]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%,$%\"xG\"\"#,$*$%\"yG\"\"(! \"),$*$%\"zG\"\"$!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "NOR MAL := subs(\{x=2,y=1,z=1\}, evalm(v) ); #Normal to surface" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'NORMALG-%'VECTORG6#7%\"\"%!\")!#7" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "solve(dotprod(NORMAL,[a,1,1] )=0,a); \n#Tangent vector is perpendicular to Normal vector" }}{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 265 4 "2. " }{TEXT -1 3 "For" }{TEXT 485 1 " " }{XPPEDIT 19 1 "t <= 1" "6#1%\"tG\"\"\"" }{TEXT 483 1 " " } {TEXT -1 28 "a particle follows the curve" }{TEXT 489 1 " " }{XPPEDIT 19 1 "C" "6#%\"CG" }{TEXT 486 2 " " }{TEXT -1 24 "that is parameteriz ed by" }{TEXT 488 1 " " }{XPPEDIT 19 1 "t -> [t,t^2,t^3]" "6#f*6#%\"tG 7\"6$%)operatorG%&arrowG6\"7%F%*$F%\"\"#*$F%\"\"$F*F*F*" }{TEXT 484 3 ". " }{TEXT -1 4 "\nFor" }{TEXT 492 1 " " }{XPPEDIT 19 1 "t>1" "6#2\" \"\"%\"tG" }{TEXT 487 2 " " }{TEXT -1 48 "the particle follows the li ne that is tangent to" }{TEXT 493 1 " " }{XPPEDIT 19 1 "C" "6#%\"CG" } {TEXT 490 2 " " }{TEXT -1 126 "at the point (1,1,1). \nThe linear pa rt of the trajectory carries the particle through the point (a, 7, c). \nWhat is c - a ?" }}{PARA 3 "" 0 "" {TEXT 407 152 "a) 1 \+ b) 2 c) 3 d) 4 e) 5 \nf) \+ 6 g) 7 h) 8 i) 9 j) 10 " }{TEXT 408 7 " \n" }}{PARA 3 "" 0 "" {TEXT 409 8 "Solution" } {TEXT 410 6 ": (f)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "r1 := t -> [t,t^2,t^3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1Gf*6#%\"tG6\"6$% )operatorG%&arrowGF(7%9$*$F-\"\"#*$F-\"\"$F(F(6\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 74 "r2 := s -> [seq(r1(1)[j]+s*map(z->subs(t=1,d iff(z,t)), r1(t))[j],j=1..3)];" }}{PARA 7 "" 1 "" {TEXT -1 42 "Warning , `j` in call to `seq` is not local" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#r2Gf*6#%\"sG6\"6$%)operatorG%&arrowGF(7#-%$seqG6$,&&-%#r1G6#\"\"\" 6#%\"jGF5*&9$F5&-%$mapG6$f*6#%\"zGF(F)F(-%%subsG6$/%\"tGF5-%%diffG6$F9 FEF(F(6\"-F36#FEF6F5F5/F7;F5\"\"$F(F(FI" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "s0 := solve(r2(s)[2] = 7, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s0G\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "r2(s0)[3] - r2(s0)[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 257 52 "3. A particle's position in space at time t is " }{XPPEDIT 500 1 "r(t)" "6#-%\"rG6#%\"tG" }{TEXT 499 12 ". At time " }{XPPEDIT 504 1 "t=13" "6#/%\"tG\"#8" }{TEXT 503 57 " the particle is at point \+ (1,2,3) \n and has velocity " }{XPPEDIT 502 1 "`<`*4,6,7*`>`" "6%*& %\"GF%" }{TEXT 501 15 ". Calculate \+ " }{XPPEDIT 507 1 "g*`'`(13)" "6#*&%\"gG\"\"\"-%\"'G6#\"#8F%" }{TEXT 505 1 " " }{TEXT 508 6 " if " }{XPPEDIT 509 1 "f(x,y,z)=6*x^2-y^2-3* z" "6#/-%\"fG6%%\"xG%\"yG%\"zG,(*&\"\"'\"\"\"*$F'\"\"#F-F-*$F(F/!\"\"* &\"\"$F-F)F-F1" }{TEXT 506 11 " \n and " }{TEXT 510 1 " " } {XPPEDIT 511 1 "g(t)=f(r(t)" "6#/-%\"gG6#%\"tG-%\"fG6#-%\"rG6#F'" } {TEXT 497 1 "." }{TEXT 498 1 "\n" }{TEXT 415 71 "\na) 1 b) 2 c) 3 d) 4 e) 5" }{TEXT 412 2 " \+ " }{TEXT 416 19 " " }}{PARA 0 "" 0 "" {TEXT 413 33 " f) 6 g) 72 " }{TEXT -1 1 " " }{TEXT 414 39 "h) \+ 8 i) 9 j) 10 " }}{PARA 3 "" 0 "" {TEXT 411 1 " \+ " }}{PARA 3 "" 0 "" {TEXT 417 8 "Solution" }{TEXT 418 6 ": (c)\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := (x,y,z) -> 6*x^2-y^2-3* z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6%%\"xG%\"yG%\"zG6\"6$%) operatorG%&arrowGF*,(*$9$\"\"#\"\"'*$9%F1!\"\"9&!\"$F*F*6\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "u := subs(\{x=1,y=2,z=3\},gr ad(f(x,y,z),[x,y,z])); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-% 'VECTORG6#7%\"#7!\"%!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "v := vector([4,6,7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VEC TORG6#7%\"\"%\"\"'\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " dotprod(u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 423 83 "4. A plan e passes through the three points (1, 0, 1), (0, 2, 1), and (2, 3, \+ 2)." }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT 419 50 "Its equation is \+ Ax + y + Cz = D. What is D? " }{TEXT 422 1 "\n" }}{PARA 3 "" 0 " " {TEXT 421 141 "a) -4 b) -3 c) -2 d) -1 e) 0 \nf) 1 g) 2 h) 3 i) 4 j) 5" }{TEXT 420 1 "\n" }}{PARA 3 "" 0 "" {TEXT 424 8 " Solution" }{TEXT 425 8 ": (b) \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "OP := vector([1,0,1]); \nOQ := vector([0,2,1]); \nOR := vector([2,3,2]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OPG-%'VECT ORG6#7%\"\"\"\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OQG-%'VECTO RG6#7%\"\"!\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ORG-%'VEC TORG6#7%\"\"#\"\"$F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v : = evalm(OP-OQ);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7 %\"\"\"!\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "w := ev alm(OR-OQ);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'VECTORG6#7%\" \"#\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "cp := crossp rod(v,w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG-%'VECTORG6#7%!\"#! \"\"\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "cp := map(z->- z,cp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG-%'VECTORG6#7%\"\"#\" \"\"!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "eqn1 := cp[1]*( x-OP[1])+cp[2]*(y-OP[2])+cp[3]*(z-OP[3])=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,*%\"xG\"\"#\"\"$\"\"\"%\"yGF*%\"zG!\"&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqn2 := map(u->u-3, eqn1) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,(%\"xG\"\"#%\"yG\"\"\"% \"zG!\"&!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "D = rhs(eqn 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"DG!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 260 1 " " }{TEXT 491 2 "5." }{TEXT -1 1 " " }{TEXT 467 3 "Let" }{TEXT -1 1 " " }{TEXT 466 1 " " }{XPPEDIT 465 1 "N" "6#%\"NG" }{TEXT 464 52 " be the number of sad dle points that the function " }{XPPEDIT 269 1 "f(x,y)=x^2*y-12*x-x*y ^2" "6#/-%\"fG6$%\"xG%\"yG,(*&F'\"\"#F(\"\"\"F,*&\"#7F,F'F,!\"\"*&F'F, *$F(F+F,F/" }{TEXT 268 12 " has. \nLet " }{TEXT 471 1 " " }{XPPEDIT 460 1 "[x[1],y[1]],[x[2],y[2]],`...`,[x[N],y[N]]" "6&7$&%\"xG6#\"\"\"& %\"yG6#F'7$&F%6#\"\"#&F)6#F.%$...G7$&F%6#%\"NG&F)6#F5" }{TEXT 459 44 " denote the saddle points. \n\nWhat is " }{XPPEDIT 461 1 "abs(y [1])+abs(y[2])+`...`+abs(y[N])" "6#,*-%$absG6#&%\"yG6#\"\"\"F*-F%6#&F( 6#\"\"#F*%$...GF*-F%6#&F(6#%\"NGF*" }{TEXT 270 34 "? \n\nNote: If yo u determine that " }{XPPEDIT 468 1 "N=0" "6#/%\"NG\"\"!" }{TEXT 469 63 ", then answer 0 because, by convention, an empty sum is 0. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 463 144 "a) 0 \+ b) 1 c) 2 d) 3 e) 4 \nf) \+ 5 g) 6 h) 7 i) 8 j) 9 " }{TEXT 462 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 428 8 "Solution" }{TEXT 429 6 ": (e)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f := (x,y) -> x^2*y-12*x-x*y^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "solve(\{diff(f(x,y),x) = 0 , diff(f(x,y), y) = 0\}, \{x,y\} ); #step-by-step below" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%<$/%\"yG,$-%'RootOfG6#,&*$%#_ZG\"\"#\"\"\"\"\"$F.F-/%\"xG\"\"!<$ /F1\"\"%/F%F-<$/F%!\"#/F1!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solns := [[4,2],[-4,-2]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%& solnsG7$7$\"\"%\"\"#7$!\"%!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "discr := (x,y) -> subs( \{u=x,v=y\} , diff(f(u,v),u$2)*diff(f( u,v),v$2)-(diff(f(u,v),u,v))^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% &discrGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)-%%subsG6$<$/%\"uG9$/% \"vG9%,&*&-%%diffG6$-%\"fG6$F2F5-%\"$G6$F2\"\"#\"\"\"-F:6$F<-F@6$F5FBF CFC*$-F:6%F " 0 "" {MPLTEXT 1 0 11 "discr(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\" %\"yGF&!\"%*$,&F%\"\"#F'!\"#F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for n from 1 to 2 do\ndiscr(op(solns[n]));\nod;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#!#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #!#[" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "abs(solns[1][2])+ab s(solns[2][2]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "---------------------------------------------------- -----------" }}{PARA 0 "" 0 "" {TEXT 470 40 "Step-by-step location of \+ critical points" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "eqn[1] := di ff(f(x,y),x) = 0 ; \neqn[2] := diff(f(x,y),y) = 0;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%$eqnG6#\"\"\"/,(*&%\"xGF'%\"yGF'\"\"#!#7F'*$F,F-! \"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$eqnG6#\"\"#/,&*$%\"xG F'\"\"\"*&F+F,%\"yGF,!\"#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 111 "From the first equation, observe that th ere is no real solution for y when x = 0. We may therefore divide by x ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eqn[3] := map(z -> simplify( z/x), eqn[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$eqnG6#\"\"$/,&% \"xG\"\"\"%\"yG!\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eqn[4] := subs( x = 2*y , eqn[1] );" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%$eqnG6#\"\"%/,&*$%\"yG\"\"#\"\"$!#7\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(eqn[4],y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 431 56 "6. A particle is accelerating \+ so that its speed at time " }{TEXT 444 1 " " }{XPPEDIT 445 1 "t" "6#% \"tG" }{TEXT 443 14 " is equal to " }{TEXT 448 1 " " }{XPPEDIT 447 1 "t^2+1" "6#,&*$%\"tG\"\"#\"\"\"F'F'" }{TEXT 446 10 ". At time" } {TEXT 450 1 " " }{XPPEDIT 451 1 "t=1" "6#/%\"tG\"\"\"" }{TEXT 449 2 " \+ " }{TEXT 452 44 "the \n acceleration of the particle is 8" } {TEXT 453 1 "i" }{TEXT 454 4 " + 6" }{TEXT 455 1 "j" }{TEXT 456 4 " + \+ 2" }{TEXT 457 1 "k" }{TEXT 458 51 ". What is the curvature of the par ticle's path at " }{TEXT 494 1 " " }{XPPEDIT 495 1 "t=1" "6#/%\"tG\"\" \"" }{TEXT 496 3 " ?\n" }}{PARA 0 "" 0 "" {TEXT 430 178 "a) 1/4 \+ b) 1/2 c) 3/4 d) 1 e) 5 /4\n\nf) 3/2 g) 7/4 h) 2 i) \+ 9/4 j) 5/2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 432 8 "Solution" }{TEXT 433 6 ": (j)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "a := [8,6,2]; #acceleration at t = 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG7%\"\")\"\"'\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "speed := t -> t^2+1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&speedGf*6#%\"tG6\"6$%)operatorG%&ar rowGF(,&*$9$\"\"#\"\"\"F0F0F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a_T := t -> D(speed)(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a_TGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,$9$\"\"#F(F( 6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a_N := t -> kappa(t) *speed(t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a_NGf*6#%\"tG6\"6$ %)operatorG%&arrowGF(*&-%&kappaG6#9$\"\"\"-%&speedGF/\"\"#F(F(6\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqn := a_T(1)^2 + a_N(1)^2 = a[1]^2+a[2]^2+a[3]^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&\" \"%\"\"\"*$-%&kappaG6#F(\"\"#\"#;\"$/\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "kappa(1) = [solve(eqn,kappa(1))]; #pick positive roo t" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&kappaG6#\"\"\"7$#\"\"&\"\"##! \"&F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 259 50 "7. By changing the order of integration , express " }}{PARA 3 "" 0 "" {TEXT 515 5 " " }{TEXT 517 6 " \+ " }{XPPEDIT 518 1 "Int(Int(psi(x,y),y=0..8*x),x=0..1)+Int(Int(psi(x,y) ,y=0..9-x^2),x=1..3)" "6#,&-%$IntG6$-F%6$-%$psiG6$%\"xG%\"yG/F-;\"\"!* &\"\")\"\"\"F,F3/F,;F0F3F3-F%6$-F%6$-F*6$F,F-/F-;F0,&\"\"*F3*$F,\"\"#! \"\"/F,;F3\"\"$F3" }{TEXT 516 19 "\n \n in the form " }{TEXT 521 5 " " }{XPPEDIT 520 1 "Int(Int(psi(x,y),x=f(y)..g(y)),y=c..d)" "6#-% $IntG6$-F$6$-%$psiG6$%\"xG%\"yG/F+;-%\"fG6#F,-%\"gG6#F,/F,;%\"cG%\"dG " }{TEXT 519 12 ". What is " }{XPPEDIT 523 1 "d+f(16)+g(5)" "6#,(%\" dG\"\"\"-%\"fG6#\"#;F%-%\"gG6#\"\"&F%" }{TEXT 522 2 " ?" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 524 161 "a) 10 b) 11 c ) 12 d) 13 e) 14 \nf) 15 g ) 16 h) 17 i) 18 j) 19 " }{TEXT 525 2 " " }{TEXT 526 5 " \n" }{TEXT -1 1 "\n" }}{PARA 3 "" 0 "" {TEXT 426 8 "Solution" }{TEXT 427 6 ": (c)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "eqn := Int(Int(13*x+17*y,y=0..8*x),x=0..1)+Int(In t(13*x+17*y,y=0..9-x^2),x=1..3) = Int(Int(13*x+17*y,x=y/8..sqrt(9-y)), y=0..8); Test proposed formula with weird function" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$eqnG/,&-%$IntG6$-F(6$,&%\"xG\"#8%\"yG\"# " 0 "" {MPLTEXT 1 0 52 "c := 0; d := 8; f := y -> y/8; g : = y -> sqrt(9-y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"yG6\"6$%)operatorG%&arrowGF(,$9$#\"\"\"\" \")F(F(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"yG6\"6$%)op eratorG%&arrowGF(-%%sqrtG6#,&\"\"*\"\"\"9$!\"\"F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "d + f(16) + g(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 434 8 "8. If " }{XPPEDIT 435 1 "x^3 + y^2 - z - 2*x*z^2 = -16" "6#/, **$%\"xG\"\"$\"\"\"*$%\"yG\"\"#F(%\"zG!\"\"*(F+F(F&F(F,F+F-,$\"#;F-" } {TEXT 436 20 " then calculate " }{XPPEDIT 437 1 "diff( z ,x)" "6#- %%diffG6$%\"zG%\"xG" }{TEXT 438 10 " at " }{XPPEDIT 439 1 "``(1, 2,3)" "6#-%!G6%\"\"\"\"\"#\"\"$" }{TEXT 440 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 212 "a) -27/13 \+ b) -23/13 c) -19/13 d) -15/13 e) -11 /13 \nf) -7/13 g) -3/13 h) \+ 1/13 i) 5/13 j) 9/13 " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 441 8 "Solution" }{TEXT 442 6 ": (d)\n" }}{PARA 0 "" 0 "" {TEXT 514 28 "By implicit differentiation: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eqn1 := x^3 + y^2 - z - 2*x* z^2 = -16; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,**$%\"xG\"\"$ \"\"\"*$%\"yG\"\"#F*%\"zG!\"\"*&F(F*F.F-!\"#!#;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "eqn2 := subs(z = z(x,y), eqn1);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%eqn2G/,**$%\"xG\"\"$\"\"\"*$%\"yG\"\"#F*-%\"z G6$F(F,!\"\"*&F(F*F.F-!\"#!#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eqn3 := map( u -> diff(u,x), eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,**$%\"xG\"\"#\"\"$-%%diffG6$-%\"zG6$F(%\"yGF( !\"\"*$F.F)!\"#*(F(\"\"\"F.F6F+F6!\"%\"\"!" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 54 "eqn4 := diff(z(x,y),x) = solve(eqn3 , diff(z(x,y),x ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/-%%diffG6$-%\"zG6$%\"x G%\"yGF,*&,&*$F,\"\"#\"\"$*$F)F1!\"#\"\"\",&F5F5*&F,F5F)F5\"\"%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn5 := lhs(eqn4) = subs( z(x,y) = z, rhs(eqn4)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/- %%diffG6$-%\"zG6$%\"xG%\"yGF,*&,&*$F,\"\"#\"\"$*$F*F1!\"#\"\"\",&F5F5* &F,F5F*F5\"\"%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "subs (\{x=1, y=2, z=3\} , rhs(eqn5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##! #:\"#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 513 14 "Alternatively:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "F := (x,y,z) -> x^3 + y^2 - z - 2*x*z^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG f*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,**$9$\"\"$\"\"\"*$9%\"\" #F29&!\"\"*&F0F2F6F5!\"#F*F*6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "expr := -diff(F(x,y,z),x)/diff(F(x,y,z),z);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%exprG,$*&,&*$%\"xG\"\"#\"\"$*$%\"zGF*!\"#\"\"\",&! \"\"F/*&F)F/F-F/!\"%F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(\{x=1, y=2, z=3\} , expr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##! #:\"#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 16 "9. Calculate " }{XPPEDIT 527 1 "int( 2*y*dx+ x*dy+2*z,z=C.. ``)" "6#-%$intG6$,(*(\"\"#\"\"\"%\"yGF)%#dxGF)F)*&%\"xGF)%#dyGF)F)*&F( F)%\"zGF)F)/F0;%\"CG%!G" }{TEXT 528 11 ". where " }{TEXT 529 1 "C" }{TEXT 530 49 " is the oriented curve parameterized by \n " } {TEXT 556 1 " " }{XPPEDIT 557 1 "r(t) = `<`*cos(t),sin(t),sqrt(t)*`>` " "6%/-%\"rG6#%\"tG*&%\"GF*" }{TEXT 531 8 " " }{TEXT 559 7 " " }{XPPEDIT 558 1 "0 <=t" "6#1\"\"!%\"tG" }{TEXT 555 1 " " }{XPPEDIT 560 1 "`` <=2 *Pi " "6#1%!G*&\"\"#\"\"\"%#PiGF'" }}{PARA 0 "" 0 "" {TEXT -1 7 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 534 4 "a) \+ " }{XPPEDIT 545 1 "0" "6#\"\"!" }{TEXT 535 20 " b) " }{XPPEDIT 546 1 "Pi/4" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 536 19 " \+ c) " }{XPPEDIT 547 1 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" } {TEXT 537 20 " d) " }{XPPEDIT 548 1 "Pi" "6#%#PiG" } {TEXT 538 20 " e) " }{XPPEDIT 549 1 "2*Pi" "6#*&\"\"# \"\"\"%#PiGF%" }{TEXT 539 11 " \nf) " }{XPPEDIT 550 1 "Pi/3" "6# *&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 540 18 " g) " }{XPPEDIT 551 1 "Pi/6" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 541 19 " \+ h) " }{XPPEDIT 552 1 "2*Pi/3" "6#*(\"\"#\"\"\"%#PiGF%\"\"$!\"\"" } {TEXT 542 17 " i) " }{XPPEDIT 553 1 "3*Pi/2" "6#*(\"\"$\" \"\"%#PiGF%\"\"#!\"\"" }{TEXT 543 18 " j) " }{XPPEDIT 554 1 "3*Pi/4" "6#*(\"\"$\"\"\"%#PiGF%\"\"%!\"\"" }{TEXT 544 5 " \n " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 532 8 "Solut ion" }{TEXT 533 6 ": (d)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "r := t -> [cos(t),sin(t),sqrt(t)]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"tG6\"6$%)operatorG%&a rrowGF(7%-%$cosG6#9$-%$sinGF/-%%sqrtGF/F(F(6\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 96 "Int( 2*r(t)[2]*diff(r(t)[1],t) + r(t)[1]*diff (r(t)[2],t)+ 2*r(t)[3]*diff(r(t)[3],t),t=0..2*Pi);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$,(*$-%$sinG6#%\"tG\"\"#!\"#*$-%$cosGF*F,\"\" \"F1F1/F+;\"\"!,$%#PiGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#PiG" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 23 "10. The vector field " }{TEXT 576 1 "F" }{TEXT 577 2 "( " }{TEXT 581 5 "x , y" } {TEXT 582 12 " ) = (y + 1)" }{TEXT 572 2 " i" }{TEXT 573 11 " + (x + 2 ) " }{TEXT 574 1 "j" }{TEXT 575 2 " " }{TEXT 561 53 "is the gradient \+ field of some \nscalar-valued function" }{TEXT 583 1 " " }{TEXT 578 1 "U" }{TEXT 580 1 "(" }{TEXT 579 3 "x,y" }{TEXT 563 1 ")" }{TEXT 562 10 " with " }{TEXT 585 1 "U" }{TEXT 587 1 "(" }{TEXT 586 8 " 1 , - 1 " }{TEXT 584 5 ") = " }{TEXT 589 1 "2" }{TEXT 588 18 ". Calculat e " }{TEXT 591 1 "U" }{TEXT 593 1 "(" }{TEXT 592 7 " 2 , 2 " } {TEXT 590 1 ")" }{TEXT 594 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 564 74 "a) 2 b) 4 c) \+ 5 d) 8 e) " }{TEXT 568 2 "10" }{TEXT 571 3 " \+ " }{TEXT 570 18 " " }}{PARA 0 "" 0 "" {TEXT 565 35 " f) 12 g) 14 " }{TEXT -1 1 " " }{TEXT 569 39 "h) 16 i) 18 j) 20" }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 566 8 "Solution" }{TEXT 567 6 ": (g)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "M := y+1; \+ N := x + 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG,&%\"yG\"\"\"F' F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG,&%\"xG\"\"\"\"\"#F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "testeq(diff(M,y) = diff(N,x) ); #F is conservative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn1 := U(x,y) = int(M,x) + \+ g(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-%\"UG6$%\"xG%\"yG,& *&,&F*\"\"\"F.F.F.F)F.F.-%\"gG6#F*F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqn2 := N = diff(rhs(eqn1),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,&%\"xG\"\"\"\"\"#F(,&F'F(-%%diffG6$-%\"gG6#% \"yGF1F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "eqn3 := diff(g( y),y) = solve(eqn2,diff(g(y),y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%eqn3G/-%%diffG6$-%\"gG6#%\"yGF,\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eqn4 := g(y) = int(rhs(eqn3),y) + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/-%\"gG6#%\"yG,&F)\"\"#%\"CG\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn5 := subs(eqn4,eqn1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/-%\"UG6$%\"xG%\"yG,(*&,&F*\" \"\"F.F.F.F)F.F.F*\"\"#%\"CGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "eqn6 := C = solve(2 = subs(\{x=1,y=-1\}, rhs(eqn5)), C);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/%\"CG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn7 := subs(eqn6, eqn5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn7G/-%\"UG6$%\"xG%\"yG,(*&,&F*\"\"\"F.F.F.F)F .F.F*\"\"#\"\"%F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs( \+ \{x=2,y=2\}, rhs(eqn7) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 340 17 "11. Calculate " }{XPPEDIT 598 1 "int( (y+1)*d x+(x+2),y=C..``)" "6#-%$intG6$,&*&,&%\"yG\"\"\"F*F*F*%#dxGF*F*,&%\"xGF *\"\"#F*F*/F);%\"CG%!G" }{TEXT 599 11 " where " }{TEXT 600 1 "C" } {TEXT 601 72 " is the oriented curve parameterized by the\nvector-va lued function " }{XPPEDIT 603 1 "r(t)= [sqrt(t/2)*` `, ` `*t*(t^3+ 6)/(t^2+24)]" "6#/-%\"rG6#%\"tG7$*&-%%sqrtG6#*&F'\"\"\"\"\"#!\"\"F.%\" ~GF.**%#~~GF.F'F.,&*$F'\"\"$F.\"\"'F.F.,&*$F'F/F.\"#CF.F0" }{TEXT 602 4 ", " }{TEXT 609 1 " " }{XPPEDIT 610 1 "0 <=t" "6#1\"\"!%\"tG" } {TEXT 608 1 " " }{XPPEDIT 604 1 "``<= 2" "6#1%!G\"\"#" }{TEXT 596 7 ". \n \n" }{TEXT 629 162 "(Although this integration can be done indep endently of problem 10, you may wish to note \nthat the integrand is c onstructed using the vector field of problem 10.)" }{TEXT 630 1 "\n" } {TEXT 635 5 "\n a)" }{TEXT 595 71 " 0 b) 1 \+ c) 2 d) 3 e) 4" }{TEXT 607 3 " \n" }{TEXT 597 78 " f ) 5 g) 6 h) 7 i) 8 \+ j) 9" }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT 605 8 "Solution" }{TEXT 606 6 ": (e)\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "r := (t) -> [sqrt(t/2),t*(t^ 3+6)/(t^2+24)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "r(0); r( 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"!F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "subs( \{x=1,y=1\} , (y+1)*x+2*y+4 ) - subs( \{x=0,y=0\} , (y+1)*x+ 2*y+4 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 631 3 "12." }}{PARA 0 "" 0 " " {TEXT -1 6 " " }{BITMAP 541 299 299 1 "?TMwITI:;JUJNjHD:QS:F:k: ;::::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 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yyY:y]:jywY:::Z:B:;Zymy;:::::::::::::::::;Z:jysy:>:<:yay=::::::::::::: ::::::::::<:;Zymy;::::::::::::;Z:jysy:::::::::::::::::Z:J:ryvY:::::::: :::::::::::::::::::::::::xCvyykyyI:ZyL:ravyy=:::::::::::::ra>:xAxI::vQ jIsyyA:::yqy;ymyjywY:::::::::::::::>:<: yay=J:B:vYxI:::::::::::::::::::::::B:>:xIyA::::::::::::>:<:yay=::::::: ::::::::::<:;Zymy;:::::::::::::::::::::::::::::::::y?xyyrivyy=y_:J:rYv yy=::::::::::::::y?>:xCy_yyYra>:<:y=yA:::<:yqy;:::::::::::::::jYsY>:<:y_yyY::::rixU:>ZY \\yyyYta;Z:jyyiy=::::::::::::::<:;:vYxIB:>:jysy:::::::ZysYryyyy::::::: :::::::;Z::xIyA:::::::::::Z:J::yay=::::::::::::::::J:B:Zymy;:::::::::: ::::::::::::::::::::::::ynyyA;jy=:y]ytyy_:jywY::ZysYryyyy:::::::::::y> >:<:rYvQxE;Zyuy::B:<:xI::y]YlyyI::::::::::::::::::::::::::vQry;:vYyA:: ::::::::::::::::::::::yoYyay?:;B:vIxeyxyyyY::::::::::::::Zytiymy<:yay= xeyxIyE:Zy=:::::::xay[yyyy=:::::::::::::vQywYxQ::yA:::::::::::Zytiymy< :ry:::::::::::::::::vQywYxQ::yA::::::::::::::::::::::::::::::::::::::: ::ZysYryxI::::::::::::::::::<:ry::vIxCxI:::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::Z::xI::y]ytyxI:::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::Z::xI::y^ylIC:xyyryyA::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::: B" }{TEXT -1 2 "\"." }}{PARA 267 "" 0 "" {TEXT 353 13 "\nAt least one" }{TEXT -1 68 " of the implications \+ stated in the answers is not correct. \nChoose " }{TEXT 352 3 "any" } {TEXT -1 38 " incorrect statement as your answer." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 271 106 "a) I => II \+ b) II => III c) III => IV d) IV => I \+ e) I => III" }{TEXT 272 21 " \n" }{TEXT 274 47 "f) II => I g) I => IV " }{TEXT -1 3 " h) " }{TEXT 273 58 " II => IV i) IV => III j) IV \+ => II" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 354 8 " Solution" }{TEXT 355 6 ": (c)\n" }}{PARA 0 "" 0 "" {TEXT -1 107 "Prope rties I, II, and IV are equivalent. They all imply property III. Howev er, G is not simply connected so" }}{PARA 0 "" 0 "" {TEXT -1 46 "prope rty III does not imply any of the others." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 264 1 " " }{TEXT 267 15 "14. Ca lculate " }{XPPEDIT 372 1 "Int(-y^3*dx+x^3,y=C..``)" "6#-%$IntG6$,&*&% \"yG\"\"$%#dxG\"\"\"!\"\"*$%\"xGF)F+/F(;%\"CG%!G" }{TEXT 371 8 " wher e " }{TEXT 369 1 "C" }{TEXT 370 48 " is the unit circle traversed coun ter-clockwise." }{TEXT 368 2 " \n" }{TEXT 357 4 "a) " }{XPPEDIT 373 1 "0" "6#\"\"!" }{TEXT 358 20 " b) " }{XPPEDIT 374 1 " Pi/4" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 359 19 " c) " }{XPPEDIT 375 1 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 360 20 " \+ d) " }{XPPEDIT 376 1 "Pi" "6#%#PiG" }{TEXT 361 20 " \+ e) " }{XPPEDIT 377 1 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" } {TEXT 362 11 " \nf) " }{XPPEDIT 378 1 "Pi/3" "6#*&%#PiG\"\"\"\" \"$!\"\"" }{TEXT 363 18 " g) " }{XPPEDIT 379 1 "Pi/6" "6 #*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 364 19 " h) " } {XPPEDIT 380 1 "2*Pi/3" "6#*(\"\"#\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 365 17 " i) " }{XPPEDIT 381 1 "3*Pi/2" "6#*(\"\"$\"\"\"%#PiGF %\"\"#!\"\"" }{TEXT 366 18 " j) " }{XPPEDIT 382 1 "3*Pi/ 4" "6#*(\"\"$\"\"\"%#PiGF%\"\"%!\"\"" }{TEXT 367 5 " \n" }}{PARA 3 "" 0 "" {TEXT 383 8 "Solution" }{TEXT 384 6 ": (i)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r := t -> [cos(t),sin(t)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Int( -r(t)[2]^3*diff(r(t)[1],t) + r (t)[1]^3*diff(r(t)[2],t),t=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%$IntG6$,&*$-%$sinG6#%\"tG\"\"%\"\"\"*$-%$cosGF*F,F-/F+;\"\"!,$%#Pi G\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"$\"\"#" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 636 35 "More easily, using Gre en's Theorem:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Int(Int(r*(d iff(x^3,x)-diff(-y^3,y)),r=0..1),theta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$*&%\"rG\"\"\",&*$%\"xG\"\"#\"\"$*$%\"yGF. F/F*/F);\"\"!F*/%&thetaG;F4,$%#PiGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(x^2=r^2-y^2,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,$*$%\"rG\"\"$F+/F*;\"\"!\"\"\"/%&thetaG;F.,$%#PiG \"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"$\"\"#" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 9 "15. Let " } {TEXT 387 1 "G" }{TEXT -1 57 " be a solid region in space. Suppose th at the boundary " }{TEXT 386 1 "S" }{TEXT -1 6 " of " }{TEXT 388 1 "G" }{TEXT -1 59 " is an orientable closed surface with outward unit \+ normal " }{TEXT 385 1 "n" }{TEXT -1 42 ". 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"6#*&\"\"#\"\"\"%#PiGF%" }{TEXT 289 17 " \+ e) " }{XPPEDIT 19 1 "5*Pi" "6#*&\"\"&\"\"\"%#PiGF%" }{TEXT 290 13 " \nf) " }{XPPEDIT 19 1 "6*Pi" "6#*&\"\"'\"\"\"%#PiGF% " }{TEXT 291 16 " g) " }{XPPEDIT 19 1 "9*Pi" "6#*&\"\"*\" \"\"%#PiGF%" }{TEXT 292 17 " h) " }{XPPEDIT 19 1 "12*Pi" "6#*&\"#7\"\"\"%#PiGF%" }{TEXT 293 14 " i) " }{XPPEDIT 19 1 "15*Pi" "6#*&\"#:\"\"\"%#PiGF%" }{TEXT 294 17 " j) " } {XPPEDIT 19 1 "18*Pi" "6#*&\"#=\"\"\"%#PiGF%" }{TEXT 295 4 " " } {TEXT 283 1 "\n" }}{PARA 3 "" 0 "" {TEXT 296 8 "Solution" }{TEXT 297 6 ": (i)\n" }}{PARA 0 "" 0 "" {TEXT 611 28 "Using the Divergence Theor em" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning , new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "F \+ := vector([0,2*y,z^3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'VE CTORG6#7%\"\"!,$%\"yG\"\"#*$%\"zG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "divergence := diverge(F,[x,y,z]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%+divergenceG,&\"\"#\"\"\"*$%\"zGF&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "int(int(int(r*divergence,z=-1..2),r =0..1),theta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG\"#: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}} {MARK "19 19 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }