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0 0 0 0 0 0 }{CSTYLE "" -1 586 "" 1 14 255 0 0 1 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 587 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 588 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 589 "" 1 14 255 0 0 1 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 590 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 591 "" 1 14 255 0 0 1 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 592 "" 1 14 255 0 0 1 0 2 0 0 0 0 0 0 0 } {CSTYLE "" -1 593 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 594 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 595 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 596 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 597 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " 19 598 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 599 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 600 "" 1 12 255 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 601 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 602 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 603 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 604 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 605 "" 0 1 128 0 128 1 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 606 "" 0 1 128 0 128 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 607 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 608 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 609 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 610 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 611 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 612 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 613 "" 1 12 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 614 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 615 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 616 "" 1 18 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 617 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 618 "" 1 18 255 0 255 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 619 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 620 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 621 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 622 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 623 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 624 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 625 "" 1 18 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 626 "" 1 18 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 627 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 628 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 629 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 630 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 631 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 632 "" 1 18 128 0 128 1 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 633 "" 1 18 128 0 128 1 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 634 "" 0 1 128 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 635 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 636 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 637 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 19 638 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 639 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 640 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 641 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 642 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 643 "" 1 12 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 644 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 645 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 646 "" 1 12 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 647 "" 1 12 255 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 648 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 649 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 650 "" 1 12 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 651 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 652 "" 1 12 255 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 653 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 654 "" 1 12 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 655 "" 1 14 255 0 0 1 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 656 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 657 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "T imes" 1 18 0 0 0 1 2 1 2 2 2 2 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 }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 }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 }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 }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 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 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 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Tim es" 1 14 0 0 0 1 2 1 1 2 2 2 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 }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 }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 } 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 }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 }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 }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 } 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 }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 } 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" "/,(*$%\"xG\"\"#\"\"\"*$%\"yG\"\")!\"\"*&\"\"$F'*$%\"zG\" \"%F'F+\"\"!" }{TEXT 475 41 " at the point (2,1,1). \n What is \+ " }{XPPEDIT 482 1 "a" "I\"aG6\"" }{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#>%\"FG:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,( *$9$\"\"#\"\"\"*$9%\"\")!\"\"*$9&\"\"%!\"$F*F*" }}}{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 "NORMAL : = 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" "1%\"tG\"\"\"" }{TEXT 483 1 " " }{TEXT -1 28 "a particle follows the curve" }{TEXT 489 1 " " }{XPPEDIT 19 1 "C" "I\"C G6\"" }{TEXT 486 2 " " }{TEXT -1 24 "that is parameterized by" } {TEXT 488 1 " " }{XPPEDIT 19 1 "t -> [t,t^2,t^3]" ":6#%\"tG7\"6$%)oper atorG%&arrowG6\"7%F$*$F$\"\"#*$F$\"\"$F)F)" }{TEXT 484 3 ". " }{TEXT -1 4 "\nFor" }{TEXT 492 1 " " }{XPPEDIT 19 1 "t>1" "2\"\"\"%\"tG" } {TEXT 487 2 " " }{TEXT -1 48 "the particle follows the line that is t angent to" }{TEXT 493 1 " " }{XPPEDIT 19 1 "C" "I\"CG6\"" }{TEXT 490 2 " " }{TEXT -1 126 "at the point (1,1,1). \nThe linear part of the \+ trajectory carries the particle through the point (a, 7, c). \nWhat i s c - a ?" }}{PARA 3 "" 0 "" {TEXT 407 152 "a) 1 b) 2 \+ c) 3 d) 4 e) 5 \nf) 6 \+ g) 7 h) 8 i) 9 j) 10 " }{TEXT 408 7 " \n" }}{PARA 3 "" 0 "" {TEXT 409 8 "Solution" }{TEXT 410 6 ": (f)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "r1 := t -> [t,t^2,t^3 ];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1G:6#%\"tG6\"6$%)operatorG%& arrowGF(7%9$*$F-\"\"#*$F-\"\"$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "r2 := s -> [seq(r1(1)[j]+s*map(z->subs(t=1,diff(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#>%#r2G:6 #%\"sG6\"6$%)operatorG%&arrowGF(7#-%$seqG6$,&&-%#r1G6#\"\"\"6#%\"jGF5* &9$F5&-%$mapG6$:6#%\"zGF(F)F(-%%subsG6$/%\"tGF5-%%diffG6$F9FEF(F(-F36# FEF6F5F5/F7;F5\"\"$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " s0 := solve(r2(s)[2] = 7, s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s0 G\"\"$" }}}{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)" "-%\"rG6 #%\"tG" }{TEXT 499 12 ". At time " }{XPPEDIT 504 1 "t=13" "/%\"tG\"# 8" }{TEXT 503 57 " the particle is at point (1,2,3) \n and has vel ocity " }{XPPEDIT 502 1 "`<`*4,6,7*`>`" "6%*&%\"GF%" }{TEXT 501 15 ". Calculate " }{XPPEDIT 507 1 "g*`'` (13)" "*&%\"gG\"\"\"-%\"'G6#\"#8F$" }{TEXT 505 1 " " }{TEXT 508 6 " if " }{XPPEDIT 509 1 "f(x,y,z)=6*x^2-y^2-3*z" "/-%\"fG6%%\"xG%\"yG%\"z G,(*&\"\"'\"\"\"*$F&\"\"#F,F,*$F'F.!\"\"*&\"\"$F,F(F,F0" }{TEXT 506 11 " \n and " }{TEXT 510 1 " " }{XPPEDIT 511 1 "g(t)=f(r(t)" "/-% \"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 "Sol ution" }{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# >%\"fG:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,(*$9$\"\"#\"\"'*$9% F1!\"\"9&!\"$F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "u := s ubs(\{x=1,y=2,z=3\},grad(f(x,y,z),[x,y,z])); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'VECTORG6#7%\"#7!\"%!\"$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "v := vector([4,6,7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%\"\"%\"\"'\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "dotprod(u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 423 83 "4. A plane 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-%'VECTORG6#7%\"\"\"\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OQG-%'VECTORG6#7%\"\"!\"\"#\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#ORG-%'VECTORG6#7%\"\"#\"\"$F)" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 18 "v := evalm(OP-OQ);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%\"\"\"!\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "w := evalm(OR-OQ);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'VECTORG6#7%\"\"#\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "cp := crossprod(v,w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG-%'VECTORG6#7%!\"#!\"\"\"\"&" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "cp := map(z->-z,cp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cpG-%'VECTORG6#7%\"\"#\"\"\"!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "eqn1 := cp[1]*(x-OP[1])+cp[2]*(y-OP[2])+cp[ 3]*(z-OP[3])=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,*%\"xG\" \"#\"\"$\"\"\"%\"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!\"&!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "D = rhs(eqn2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%\"DG!\"$" }}}{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" "I\"NG6\"" }{TEXT 464 52 " be the number of saddle points that \+ the function " }{XPPEDIT 269 1 "f(x,y)=x^2*y-12*x-x*y^2" "/-%\"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 s addle points. \n\nWhat is " }{XPPEDIT 461 1 "abs(y[1])+abs(y[2]) +`...`+abs(y[N])" ",*-%$absG6#&%\"yG6#\"\"\"F)-F$6#&F'6#\"\"#F)%$...GF )-F$6#&F'6#%\"NGF)" }{TEXT 270 34 "? \n\nNote: If you determine that " }{XPPEDIT 468 1 "N=0" "/%\"NG\"\"!" }{TEXT 469 63 ", then answer 0 because, by convention, an empty sum is 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 463 144 "a) 0 b) 1 c) 2 d) 3 e) 4 \nf) 5 g) 6 h) 7 i) 8 j) 9" }{TEXT 462 1 "\n " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 428 8 "Solut ion" }{TEXT 429 6 ": (e)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f := (x,y) -> x^2*y-12*x-x*y^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "solve(\{diff(f(x,y),x) = 0 , diff(f(x,y),y) = 0\}, \{ x,y\} ); #step-by-step below" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%<$/%\" 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$2)-(di ff(f(u,v),u,v))^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&discrG:6$%\" xG%\"yG6\"6$%)operatorG%&arrowGF)-%%subsG6$<$/%\"uG9$/%\"vG9%,&*&-%%di ffG6$-%\"fG6$F2F5-%\"$G6$F2\"\"#\"\"\"-F:6$F<-F@6$F5FBFCFC*$-F:6%F " 0 "" {MPLTEXT 1 0 11 "discr(x,y); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"%\"yGF&!\"%*$,&F%\" \"#F'!\"#F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for n f rom 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])+abs(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] := diff(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#>&%$eqnG6#\"\"#/,&*$%\"xGF'\"\"\"*&F+F ,%\"yGF,!\"#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "From the first equation, observe that there is no real s olution 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 it s speed at time " }{TEXT 444 1 " " }{XPPEDIT 445 1 "t" "I\"tG6\"" } {TEXT 443 14 " is equal to " }{TEXT 448 1 " " }{XPPEDIT 447 1 "t^2+1 " ",&*$%\"tG\"\"#\"\"\"F&F&" }{TEXT 446 10 ". At time" }{TEXT 450 1 " " }{XPPEDIT 451 1 "t=1" "/%\"tG\"\"\"" }{TEXT 449 2 " " }{TEXT 452 44 "the \n acceleration of the particle is 8" }{TEXT 453 1 "i" } {TEXT 454 4 " + 6" }{TEXT 455 1 "j" }{TEXT 456 4 " + 2" }{TEXT 457 1 " k" }{TEXT 458 51 ". What is the curvature of the particle's path at \+ " }{TEXT 494 1 " " }{XPPEDIT 495 1 "t=1" "/%\"tG\"\"\"" }{TEXT 496 3 " ?\n" }}{PARA 0 "" 0 "" {TEXT 430 178 "a) 1/4 b) 1/2 \+ c) 3/4 d) 1 e) 5/4\n\nf) 3/2 \+ g) 7/4 h) 2 i) 9/4 j) 5/2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 432 8 "Solution" }{TEXT 433 6 ": (j)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "a := [8,6,2]; #acceleration at t = 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG7%\"\")\"\"'\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "speed := t -> t^2+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&speedG:6#%\"tG6\"6$%)operatorG%&arrowGF(,&*$9$\"\"#\"\"\"F0F0 F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a_T := t -> D(speed )(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a_TG:6#%\"tG6\"6$%)operato rG%&arrowGF(,$9$\"\"#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a_N := t -> kappa(t) *speed(t)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a_NG:6#%\"tG6\"6$%)operatorG%&arrowGF(*&-%&kappaG6#9$\"\"\"-%&sp eedGF/\"\"#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqn := a _T(1)^2 + a_N(1)^2 = a[1]^2+a[2]^2+a[3]^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,&\"\"%\"\"\"*$-%&kappaG6#F(\"\"#\"#;\"$/\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "kappa(1) = [solve(eqn,kappa (1))]; #pick positive root" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&kap paG6#\"\"\"7$#\"\"&\"\"##!\"&F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 259 50 "7. By changing \+ the order of integration, express " }}{PARA 3 "" 0 "" {TEXT 515 5 " \+ " }{TEXT 517 6 " " }{XPPEDIT 518 1 "Int(Int(psi(x,y),y=0..8*x), x=0..1)+Int(Int(psi(x,y),y=0..9-x^2),x=1..3)" ",&-%$IntG6$-F$6$-%$psiG 6$%\"xG%\"yG/F,;\"\"!*&\"\")\"\"\"F+F2/F+;F/F2F2-F$6$-F$6$-F)6$F+F,/F, ;F/,&\"\"*F2*$F+\"\"#!\"\"/F+;F2\"\"$F2" }{TEXT 516 19 "\n \n in the \+ form " }{TEXT 521 5 " " }{XPPEDIT 520 1 "Int(Int(psi(x,y),x=f(y) ..g(y)),y=c..d)" "-%$IntG6$-F#6$-%$psiG6$%\"xG%\"yG/F*;-%\"fG6#F+-%\"g G6#F+/F+;%\"cG%\"dG" }{TEXT 519 12 ". What is " }{XPPEDIT 523 1 "d+f (16)+g(5)" ",(%\"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(Int(13*x+17*y,y=0..9-x^2),x=1..3) = Int(Int(13*x+1 7*y,x=y/8..sqrt(9-y)),y=0..8); Test proposed formula with weird funct ion" }}{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#>%\"fG:6#%\"yG6\"6$%)operatorG%&arrowG F(,$9$#\"\"\"\"\")F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG:6#%\" yG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&\"\"*\"\"\"9$!\"\"F(F(" }}} {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" "/,**$%\"xG\"\"$\"\"\"*$%\"yG\"\"#F'%\"zG!\"\"*(F*F'F%F'F+F*F,,$\" #;F," }{TEXT 436 20 " then calculate " }{XPPEDIT 437 1 "diff( z ,x )" "-%%diffG6$%\"zG%\"xG" }{TEXT 438 10 " at " }{XPPEDIT 439 1 " ``(1,2,3)" "-%!G6%\"\"\"\"\"#\"\"$" }{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 differentiatio n:" }}{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 :6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,**$9$\"\"$\"\"\"*$9%\"\"# F29&!\"\"*&F0F2F6F5!\"#F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "expr := -diff(F(x,y,z),x)/diff(F(x,y,z),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,$*&,&*$%\"xG\"\"#\"\"$*$%\"zGF*!\"#\"\"\",&!\" \"F/*&F)F/F-F/!\"%F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "s ubs(\{x=1, y=2, z=3\} , expr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!#: \"#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 16 "9. Calculate " }{XPPEDIT 527 1 "int( 2*y*dx+ x*dy+2*z,z=C.. ``)" "-%$intG6$,(*(\"\"#\"\"\"%\"yGF(%#dxGF(F(*&%\"xGF(%#dyGF(F(*&F'F( %\"zGF(F(/F/;%\"CG%!G" }{TEXT 528 11 ". where " }{TEXT 529 1 "C" } {TEXT 530 49 " is the oriented curve parameterized by \n " } {TEXT 556 1 " " }{XPPEDIT 557 1 "r(t) = `<`*cos(t),sin(t),sqrt(t)*`>` " "6%/-%\"rG6#%\"tG*&%\"GF*" }{TEXT 531 8 " " }{TEXT 559 7 " " }{XPPEDIT 558 1 "0 <=t" "1\"\"!%\"tG" }{TEXT 555 1 " " }{XPPEDIT 560 1 "`` <=2*P i " "1%!G*&\"\"#\"\"\"%#PiGF&" }}{PARA 0 "" 0 "" {TEXT -1 7 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 534 4 "a) " } {XPPEDIT 545 1 "0" "\"\"!" }{TEXT 535 20 " b) " } {XPPEDIT 546 1 "Pi/4" "*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 536 19 " \+ c) " }{XPPEDIT 547 1 "Pi/2" "*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 537 20 " d) " }{XPPEDIT 548 1 "Pi" "I#PiG%*protectedG " }{TEXT 538 20 " e) " }{XPPEDIT 549 1 "2*Pi" "*&\"\"# \"\"\"%#PiGF$" }{TEXT 539 11 " \nf) " }{XPPEDIT 550 1 "Pi/3" "*& %#PiG\"\"\"\"\"$!\"\"" }{TEXT 540 18 " g) " }{XPPEDIT 551 1 "Pi/6" "*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 541 19 " h ) " }{XPPEDIT 552 1 "2*Pi/3" "*(\"\"#\"\"\"%#PiGF$\"\"$!\"\"" }{TEXT 542 17 " i) " }{XPPEDIT 553 1 "3*Pi/2" "*(\"\"$\"\"\"%#Pi GF$\"\"#!\"\"" }{TEXT 543 18 " j) " }{XPPEDIT 554 1 "3*P i/4" "*(\"\"$\"\"\"%#PiGF$\"\"%!\"\"" }{TEXT 544 5 " \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 532 8 "Solution" }{TEXT 533 6 ": (d)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "r := t -> [cos(t),sin(t),sqrt(t)];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"rG:6#%\"tG6\"6$%)operatorG%&arrowGF(7%-%$cos G6#9$-%$sinGF/-%%sqrtGF/F(F(" }}}{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+;\"\"!,$%# PiGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%#PiG" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 262 23 "10. The vector field " }{TEXT 576 1 "F" }{TEXT 577 2 "( " }{TEXT 581 5 "x , y" }{TEXT 582 12 " ) = ( y + 1)" }{TEXT 572 2 " i" }{TEXT 573 11 " + (x + 2) " }{TEXT 574 1 "j " }{TEXT 575 2 " " }{TEXT 561 53 "is the gradient field of some \nsca lar-valued function" }{TEXT 583 1 " " }{TEXT 578 1 "U" }{TEXT 580 1 "( " }{TEXT 579 3 "x,y" }{TEXT 563 1 ")" }{TEXT 562 10 " with " } {TEXT 585 1 "U" }{TEXT 587 1 "(" }{TEXT 586 8 " 1 , -1 " }{TEXT 584 5 ") = " }{TEXT 589 1 "2" }{TEXT 588 18 ". Calculate " }{TEXT 591 1 "U" }{TEXT 593 1 "(" }{TEXT 592 7 " 2 , 2 " }{TEXT 590 1 ")" } {TEXT 594 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 564 74 "a) 2 b) 4 c) 5 d ) 8 e) " }{TEXT 568 2 "10" }{TEXT 571 3 " " }{TEXT 570 18 " " }}{PARA 0 "" 0 "" {TEXT 565 35 "f) 12 \+ g) 14 " }{TEXT -1 1 " " }{TEXT 569 39 "h) 16 \+ i) 18 j) 20" }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 566 8 "Solution" }{TEXT 567 6 ": (g)\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "M := y+1; N := x + 2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG,&%\"yG\"\"\"F'F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"NG,&%\"xG\"\"\"\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "testeq(diff(M,y) = diff(N,x)); #F is conserva tive" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "eqn1 := U(x,y) = int(M,x) + g(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-%\"UG6$%\"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$-%\"gG6#%\"yGF1F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "eqn3 := diff(g(y),y) = solve(eqn2,d iff(g(y),y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%%diffG6$-% \"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..``)" "-%$intG6$,&*&,&%\"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)]" "/-%\"rG6#%\"tG7$*&-%%sqrtG6#*&F&\"\"\"\"\"#!\"\"F-%\"~G F-**%#~~GF-F&F-,&*$F&\"\"$F-\"\"'F-F-,&*$F&F.F-\"#CF-F/" }{TEXT 602 4 ", " }{TEXT 609 1 " " }{XPPEDIT 610 1 "0 <=t" "1\"\"!%\"tG" }{TEXT 608 1 " " }{XPPEDIT 604 1 "``<= 2" "1%!G\"\"#" }{TEXT 596 7 ". \n \n " }{TEXT 629 162 "(Although this integration can be done independently of problem 10, you may wish to note \nthat the integrand is construct ed 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 :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::Z:vyyuy:::: 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:::::::::::::::::;Z:jysy:>:<:yay=:::::::::::::::::::::::<:;Zymy;:::::: ::::::;Z:jysy:::::::::::::::::Z:J:ryvY:::::::::::::::::::::::::::::::: :xCvyykyyI:ZyL:ravyy=:::::::::::::ra>:xAxI::vQjIsyyA:::yqy;ymyjywY:::::::::::::::>:<:yay=J:B:vYxI:::::::::::: :::::::::::B:>:xIyA::::::::::::>:<:yay=:::::::::::::::::<:;Zymy;:::::: :::::::::::::::::::::::::::y?xyyrivyy=y_:J:rYvyy=::::::::::::::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;::::::::::::::::::::::::::::::::::y nyyA;jy=:y]ytyy_:jywY::ZysYryyyy:::::::::::y>>:<:rYvQxE;Zyuy::B:<:xI:: y]YlyyI::::::::::::::::::::::::::vQry;:vYyA::::::::::::::::::::::::yoY yay?:;B:vIxeyxyyyY::::::::::::::Zytiymy<:yay=xeyxIyE:Zy=:::::::xay[yyy y=:::::::::::::vQywYxQ::yA:::::::::::Zytiymy<:ry:::::::::::::::::vQywY xQ::yA:::::::::::::::::::::::::::::::::::::::::ZysYryxI::::::::::::::: :::<:ry::vIxCxI::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::Z::xI::y]ytyxI::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::Z::xI::y^ylIC:xyyryyA:::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::: B" }{TEXT -1 2 "\"." }}{PARA 267 "" 0 "" {TEXT 353 13 "\nAt le ast one" }{TEXT -1 68 " of the implications stated in the answers is n ot 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 "Properties I, II, and IV a re equivalent. They all imply property III. However, G is not simply c onnected so" }}{PARA 0 "" 0 "" {TEXT -1 46 "property III does not impl y any of the others." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 264 1 " " }{TEXT 267 15 "14. Calculate " } {XPPEDIT 372 1 "Int(-y^3*dx+x^3,y=C..``)" "-%$IntG6$,&*&%\"yG\"\"$%#dx G\"\"\"!\"\"*$%\"xGF(F*/F';%\"CG%!G" }{TEXT 371 8 " where " }{TEXT 369 1 "C" }{TEXT 370 48 " is the unit circle traversed counter-clockwi se." }{TEXT 368 2 " \n" }{TEXT 357 4 "a) " }{XPPEDIT 373 1 "0" "\"\"! " }{TEXT 358 20 " b) " }{XPPEDIT 374 1 "Pi/4" "*&%#PiG \"\"\"\"\"%!\"\"" }{TEXT 359 19 " c) " }{XPPEDIT 375 1 "Pi/2" "*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 360 20 " d) " }{XPPEDIT 376 1 "Pi" "I#PiG%*protectedG" }{TEXT 361 20 " \+ e) " }{XPPEDIT 377 1 "2*Pi" "*&\"\"#\"\"\"%#PiGF$" }{TEXT 362 11 " \+ \nf) " }{XPPEDIT 378 1 "Pi/3" "*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 363 18 " g) " }{XPPEDIT 379 1 "Pi/6" "*&%#PiG\"\"\"\"\"' !\"\"" }{TEXT 364 19 " h) " }{XPPEDIT 380 1 "2*Pi/3" "* (\"\"#\"\"\"%#PiGF$\"\"$!\"\"" }{TEXT 365 17 " i) " } {XPPEDIT 381 1 "3*Pi/2" "*(\"\"$\"\"\"%#PiGF$\"\"#!\"\"" }{TEXT 366 18 " j) " }{XPPEDIT 382 1 "3*Pi/4" "*(\"\"$\"\"\"%#PiGF$ \"\"%!\"\"" }{TEXT 367 5 " \n" }}{PARA 3 "" 0 "" {TEXT 383 8 "Solut ion" }{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+;\"\"!,$%#PiG\"\"#" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 9 "value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$%#PiG#\"\"$\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 636 35 "More easily, using Green's Theorem:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Int(Int(r*(diff(x^3,x)-diff(-y^3,y)),r=0..1),the ta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$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 reg ion in space. Suppose that the boundary " }{TEXT 386 1 "S" }{TEXT -1 6 " of " }{TEXT 388 1 "G" }{TEXT -1 59 " is an orientable closed su rface with outward unit normal " }{TEXT 385 1 "n" }{TEXT -1 42 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 44 "On the plum disk the unit outward normal \+ is " }{TEXT 314 4 " - " }{TEXT 313 1 "k" }{TEXT -1 6 " and" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dotprod(F,vector([0,0,-1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% \"zG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 " But on the plum disk" }{TEXT 315 3 " " }{TEXT 316 5 "z = 1" }{TEXT -1 13 ". Therefore:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "required_integral = -int(int(1*r,r=0..1), theta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2required_integra lG,$%#PiG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 285 4 "20. 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::B:yay=::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::<:xI:::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::B:1:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT 284 4 "a) " }{XPPEDIT 19 1 "-4*Pi" ",$*&\"\"%\"\"\"%#PiGF% !\"\"" }{TEXT 286 15 " b) " }{XPPEDIT 19 1 "-Pi" ",$%#PiG! \"\"" }{TEXT 287 18 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