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}{PSTYLE "" 258 269 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 "" 0 270 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 523 9 "Math 233" }{TEXT 525 22 " Fall 2002 Final Exam" }}{PARA 0 "" 0 "" {TEXT 256 2 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 258 1 "1" }{TEXT 268 2 ". " }{TEXT 536 6 " If \+ " }{XPPEDIT 558 1 "u" "6#%\"uG" }{TEXT -1 1 " " }{TEXT 526 6 " and \+ " }{TEXT -1 1 " " }{XPPEDIT 559 1 "v" "6#%\"vG" }{TEXT 527 19 " are v ectors and " }{XPPEDIT 556 1 "u=lambda*v" "6#/%\"uG*&%'lambdaG\"\"\"% \"vGF'" }{TEXT 557 21 " for some scalar " }{XPPEDIT 530 1 "lambda " "6#%'lambdaG" }{TEXT 529 31 " then which of the following " } {TEXT 531 4 "must" }{TEXT 532 11 " be true? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 5 "a) " }{XPPEDIT 545 1 " u" "6#%\"uG" }{TEXT -1 30 " has the same direction as " }{XPPEDIT 528 1 "v" "6#%\"vG" }{TEXT -1 11 " \nb) " }{XPPEDIT 546 1 "u" "6 #%\"uG" }{TEXT -1 33 " is opposite in direction to " }{XPPEDIT 561 1 "v" "6#%\"vG" }{TEXT -1 11 " \nc) " }{XPPEDIT 547 1 "u" "6 #%\"uG" }{TEXT -1 25 " is perpendicular to " }{XPPEDIT 533 1 "v" " 6#%\"vG" }{TEXT -1 16 " \nd) " }{XPPEDIT 548 1 "u" "6#%\"uG " }{TEXT -1 21 " is parallel to " }{XPPEDIT 560 1 "v" "6#%\"vG" } {TEXT -1 10 " \ne) " }{XPPEDIT 549 1 "u" "6#%\"uG" }{TEXT -1 28 " has the same length as " }{XPPEDIT 534 1 "v" "6#%\"vG" }{TEXT -1 7 " \nf) " }{XPPEDIT 550 1 "u" "6#%\"uG" }{TEXT -1 32 " has magni tude greater than " }{XPPEDIT 535 1 "v" "6#%\"vG" }{TEXT -1 2 " " }} {PARA 258 "" 0 "" {TEXT 537 4 "g) " }{XPPEDIT 551 1 "u" "6#%\"uG" } {TEXT -1 1 " " }{TEXT 538 27 " has magnitude less than " }{TEXT -1 1 " " }{XPPEDIT 540 1 "v" "6#%\"vG" }{TEXT 539 18 " \nh) \+ " }{XPPEDIT 553 1 "v=(lambda)^(-1)*u" "6#/%\"vG*&)%'lambdaG,$\"\"\"!\" \"F)%\"uGF)" }{TEXT 552 21 " \ni) " }{XPPEDIT 555 1 "l ambda = u/v" "6#/%'lambdaG*&%\"uG\"\"\"%\"vG!\"\"" }{TEXT 554 8 " \n j) " }{TEXT -1 1 " " }{XPPEDIT 541 1 "u" "6#%\"uG" }{TEXT -1 1 " " } {TEXT 542 6 " and " }{TEXT -1 1 " " }{XPPEDIT 543 1 "v" "6#%\"vG" } {TEXT 544 48 " determine a unique plane through the origin " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 562 8 "Solution " }{TEXT 563 6 ": (d)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 260 "" 0 "" {TEXT 266 4 "2. " } {TEXT 564 15 "If the vectors " }{TEXT 569 2 " " }{XPPEDIT 572 1 "`<`* (-3),b,b*`>`" "6%*&%\"GF%" } {TEXT 566 1 " " }{TEXT 570 3 "and" }{TEXT 571 2 " " }{XPPEDIT 573 1 " `<`*2,1,2*b*`>`" "6%*&%\"GF%" } {TEXT 567 1 " " }{TEXT 574 34 " are perpendicular then what might" } {TEXT 575 1 " " }{XPPEDIT 576 1 "b" "6#%\"bG" }{TEXT 568 2 " " } {TEXT 577 4 "be? " }{TEXT 578 2 " " }}{PARA 3 "" 0 "" {TEXT 565 154 " a) -1 b) -1/2 c) 0 d) 1/2 e) 1 \nf) 3/2 g) 2 h) 5/2 i) 3 \+ j) 7/2 " }{TEXT 579 7 " \n" }}{PARA 3 "" 0 "" {TEXT 581 8 "Solution" }{TEXT 582 6 ": (f)\n" }}{PARA 0 "" 0 "" {TEXT -1 40 "We \+ need the dot product - it is in the " }{TEXT 580 6 "linalg" }{TEXT -1 10 " package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn_for_b := dotprod([-3, b, b],[2, 1, 2*b]) = 0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%*eqn_for_bG/,(*$%\"bG\"\"#F)F(\"\"\" !\"'F*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve(eqn_fo r_b, b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"\"$\"\"#!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Only one of the se is in the answer list." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 257 81 "3. The l ine through the points ( 3, -1, 2) and (5, 2, -1) passes through a poi nt " }{TEXT 592 2 "(a" }{TEXT 587 2 ", " }{TEXT 588 1 "b" }{TEXT 594 5 ", -4)" }{TEXT 593 2 ". " }{TEXT 590 8 "What is " }{TEXT 591 8 " b \+ ?\n\n" }{TEXT 595 74 "a) -4 b) -3 c) -2 \+ d) -1 e) 0" }{TEXT 584 2 " " }{TEXT 596 19 " \+ " }}{PARA 0 "" 0 "" {TEXT 585 36 "f) 1 g) 2 \+ " }{TEXT -1 1 " " }{TEXT 586 39 "h) 3 i) 4 \+ j) " }{TEXT -1 1 " " }{TEXT 589 1 "5" }{TEXT -1 2 " " }} {PARA 3 "" 0 "" {TEXT 583 1 " " }}{PARA 3 "" 0 "" {TEXT 597 8 "Solutio n" }{TEXT 598 6 ": (j)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " OP := vector([3,-1,2]); \nOQ := vector([5,2,-1]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OPG-%'VECTORG6#7%\"\"$!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OQG-%'VECTORG6#7%\"\"&\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v := evalm(OQ-OP);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%\"\"#\"\"$!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "line := evalm(OP + scalarmul(v,t)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%lineG-%'VECTORG6#7%,&\"\"$\"\" \"%\"tG\"\"#,&!\"\"F+F,F*,&F-F+F,!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eqn := t = solve(line[3] = -4, t);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$eqnG/%\"tG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "b = subs(eqn, line[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"bG\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 603 82 "4. A plane passes through the three points (1 , 0, 0), (0, 2, 0), and (2, 3, 3)." }{TEXT -1 1 " " }}{PARA 3 "" 0 " " {TEXT 599 51 "Its equation is Ax + 3y + Cz = D. What is D? " } {TEXT 602 1 "\n" }}{PARA 3 "" 0 "" {TEXT 601 132 "a) 0 b) 1 c) 2 d) 3 e) 4 \nf) 5 g) 6 \+ h) 7 i) 8 j) 9" }{TEXT 600 1 "\n" }} {PARA 3 "" 0 "" {TEXT 604 8 "Solution" }{TEXT 605 6 ": (g)\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "OP := vector([1,0,0]); \nOQ \+ := vector([0,2,0]); \nOR := vector([2,3,3]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OPG-%'VECTORG6#7%\"\"\"\"\"!F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OQG-%'VECTORG6#7%\"\"!\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ORG-%'VECTORG6#7%\"\"#\"\"$F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "v := evalm(OP-OQ);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%\"\"\"!\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "w := evalm(OR-OQ);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'VECTORG6#7%\"\"#\"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "cp := crossprod(v,w);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#cpG-%'VECTORG6#7%!\"'!\"$\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "eqn1 := cp[1]*(x-1)+cp[2]*(y-0)+cp[3]*(z-0) =0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,*%\"xG!\"'\"\"'\"\"\" %\"yG!\"$%\"zG\"\"&\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn2 := map(u->-u, eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G /,*%\"xG\"\"'!\"'\"\"\"%\"yG\"\"$%\"zG!\"&\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "eqn3 := map(u->u+6, eqn2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%eqn3G/,(%\"xG\"\"'%\"yG\"\"$%\"zG!\"&F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "D = rhs(eqn3);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%\"DG\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 1 {PARA 3 "" 0 "" {TEXT 260 2 " 5" }{TEXT -1 2 ". " }{TEXT 286 15 "The function " }{XPPEDIT 285 1 "f(x,y)=2*x^2*y-3*x*y^2-6*x*y" "6 #/-%\"fG6$%\"xG%\"yG,(*(\"\"#\"\"\"*$F'F+F,F(F,F,*(\"\"$F,F'F,F(F+!\" \"*(\"\"'F,F'F,F(F,F0" }{TEXT 284 178 " has one critical point P = (a ,b) with ab < 0. Choose the ordered list [a,what] where a is the ab scissa of the critical point P = (a,b) and \"what\" describes the beha vior of " }{XPPEDIT 288 1 "f" "6#%\"fG" }{TEXT 287 8 " at P. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 2 "a)" } {TEXT 289 24 " [1,local minimum] " }{TEXT 296 2 "b)" }{TEXT 294 56 " [-1,local minimum] c) [-2,local minimum] " }{TEXT 297 7 " \nd) " }{TEXT 298 18 "[1,local maximum] " }{TEXT 299 8 " \+ e) " }{TEXT 295 22 "[-1,local maximum] " }{TEXT 292 4 " " } {TEXT 302 3 " f)" }{TEXT 303 1 " " }{TEXT 300 18 "[-2,local maximum]" }{TEXT 301 1 " " }{TEXT 290 12 " \ng) " }{TEXT 304 16 "[1,saddl e point]" }{TEXT 305 16 " h) " }{TEXT 306 17 "[-1,saddle p oint]" }{TEXT 307 19 " i) " }{TEXT 310 17 "[-2,saddle p oint]" }{TEXT 291 6 " \n " }{TEXT 311 3 "j) " }{TEXT 312 1 " " } {TEXT 308 17 "[2,local maximum]" }{TEXT 309 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 624 8 "Solution" }{TEXT 625 6 ": (d)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f := (x,y) -> 2*x^ 2*y-3*x*y^2-6*x*y:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "solve (\{diff(f(x,y),x) = 0 , diff(f(x,y),y) = 0\}, \{x,y\} );" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&<$/%\"yG\"\"!/%\"xGF&<$/F%!\"#F'<$/F(\"\"$F$<$/F %#F+F./F(\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "diff(f(x ,y),x$2)*diff(f(x,y),y$2)-(diff(f(x,y),x,y))^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\"%\"yGF&!#C*$,(F%\"\"%F'!\"'F,F&\"\"#!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(\{x=1,y=-2/3\}, \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "subs(\{x=1,y=-1\},diff(f(x,y),x$2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 636 55 "6. At an instant of time a particle's acceleration is " }{XPPEDIT 629 1 "`<`*2,1,-3*`>`" "6%*&%\"GF%!\"\"" }{TEXT 630 38 " and its principal unit normal is\n \+ " }{XPPEDIT 633 1 "N=2/3*i+2/3*j-k/3" "6#/%\"NG,(*(\"\"#\"\"\"\"\"$! \"\"%\"iGF(F(*(F'F(F)F*%\"jGF(F(*&%\"kGF(F)F*F*" }{TEXT 634 51 ". W hat is its normal component of acceleration " }{XPPEDIT 632 1 "a[N]" "6#&%\"aG6#%\"NG" }{TEXT 631 2 "?\n" }}{PARA 0 "" 0 "" {TEXT 635 169 " a) 0 b) 1/3 c) 2/3 d) 1 \+ e) 4/3\n\nf) 5/3 g) 2 h) 7/3 \+ i) 8/3 j) 3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT 637 8 "Solution" }{TEXT 638 6 ": (j)\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a := [2,1,-3]: N := [2/3,2/ 3,-1/3]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "dotprod(a,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 259 25 "7. The \+ maximum value of " }{TEXT 275 1 " " }{XPPEDIT 269 1 "f(x,y)=x+2*y" "6# /-%\"fG6$%\"xG%\"yG,&F'\"\"\"*&\"\"#F*F(F*F*" }{TEXT 270 2 " " } {TEXT 273 20 " on the unit circle " }{TEXT 274 3 " " }{XPPEDIT 271 1 "x^2+y^2=1" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT 272 3 " \+ \n" }{TEXT 610 11 "occurs at " }{TEXT 612 1 "(" }{TEXT 608 3 "a,b" } {TEXT 609 1 ")" }{TEXT 613 26 ". What is the value of " }{TEXT 606 1 "a" }{TEXT 607 1 " " }{TEXT 611 2 " ?" }}{PARA 0 "" 0 "" {TEXT 276 4 "a) " }{XPPEDIT 19 1 "1/sqrt(5)" "6#*&\"\"\"F$-%%sqrtG6#\"\"&!\"\" " }{TEXT 614 15 " b) " }{XPPEDIT 19 1 "2/sqrt(5)" "6#*&\"\" #\"\"\"-%%sqrtG6#\"\"&!\"\"" }{TEXT 615 16 " c) " } {XPPEDIT 19 1 "1/sqrt(6)" "6#*&\"\"\"F$-%%sqrtG6#\"\"'!\"\"" }{TEXT 616 21 " d) " }{XPPEDIT 19 1 "2/sqrt(6)" "6#*&\"\"#\" \"\"-%%sqrtG6#\"\"'!\"\"" }{TEXT 617 14 " e) " }{XPPEDIT 19 1 "1/sqrt(2)" "6#*&\"\"\"F$-%%sqrtG6#\"\"#!\"\"" }{TEXT 618 11 " \+ \n f) " }{XPPEDIT 19 1 "1/sqrt(3)" "6#*&\"\"\"F$-%%sqrtG6#\"\"$!\"\" " }{TEXT 619 15 " g) " }{XPPEDIT 19 1 "1/3" "6#*&\"\"\"F$\" \"$!\"\"" }{TEXT 620 20 " h) " }{XPPEDIT 19 1 "2/3" "6 #*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 621 47 " i) 1 \+ j) 0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 622 8 "Solution" }{TEXT 623 6 ": (a)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f := (x,y) -> x + 2*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 /\"\"\",$*&%'lambdaGF&%\"xGF&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn2 := diff(f(x,y),y) = lambda*diff(phi(x,y),y);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/\"\"#,$*&%'lambdaG\"\"\"%\"yG 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 11 "" 1 "" {XPPMATH 20 "6#<%/% 'lambdaG,$-%'RootOfG6#,&*$%#_ZG\"\"#\"\"&!\"\"\"\"\"#F.F-/%\"xGF'/%\"y G,$F'F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "levelCurve[1] := implicitp lot( f(x,y)=f(sqrt(5)/5,2*sqrt(5)/5), x = -3/2..3/2,y=-3/2..3/2,color= pink,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "level Curve[2] := implicitplot( x^2+y^2=1, x = -3/2..3/2,y=-3/2..3/2,color=p lum,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "level Curve[3] := contourplot(f(x,y), x = -3/2.. 3/2, y= -3/2..3/2, contours =10,color=COLOR(RGB,0.75,0.7,0.7)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display(seq(levelCurve[j],j=1..3));" }}{PARA 13 "" 1 "" {GLPLOT2D 514 443 443 {PLOTDATA 2 "6.-%'CURVESG6I7$7$$!1'********** **f'!#;$\"1&*)\\())R.[9!#:7$$!1A@+D-KRwF*$\"1,++++++:F-7$F'7$$!1uy*\\x z1c&F*$\"1*y*\\xz1'R\"F-7$7$$!1'************R&F*$\"1&*)\\())R.)Q\"F-F4 7$7$$!1'************>%F*$\"1&*)\\())R.G8F-7$$!1=@+D-KR_F*$\"1,+++++!Q \"F-7$FEF:7$F@7$$!1yy*\\xz1;$F*$\"1*y*\\xz1w7F-7$7$$!1'*************HF *$\"1&*)\\())R.o7F-FL7$7$$!1'************z\"F*$\"1&*)\\())R.37F-7$$!18 @+D-KRGF*$\"1,+++++g7F-7$FgnFR7$FX7$$!1*yy*\\xz1w!#<$\"1*y*\\xz1c6F-7$ 7$$!1h************fFao$\"1&*)\\())R.[6F-F^o7$7$$\"1R++++++gFao$\"1&*) \\())R.)3\"F-7$$!1M6-]A?$R%Fao$\"1,+++++S6F-7$F`pFeo7$F[p7$$\"1A@+D-KR ;F*$\"1*y*\\xz1O5F-7$7$$\"1/++++++=F*$\"1%*)\\())R.G5F-Fgp7$7$$\"1/+++ +++IF*$\"1X*)\\())R.o*F*7$$\"1()y*\\xz1'>F*$\"1++++++?5F-7$FhqF]q7$Fcq 7$$\"1<@+D-KRSF*$\"1!*y*\\xz1;*F*7$7$$\"1/++++++UF*$\"1Z*)\\())R.3*F*F _r7$7$$\"1/++++++aF*$\"1Z*)\\())R.[)F*7$$\"1\"*y*\\xz1O%F*$\"1/++++++! *F*7$7$Fas$\"1-++++++!*F*Fer7$F[s7$$\"1;@+D-KRkF*$\"1!*y*\\xz1'zF*7$7$ $\"1/++++++mF*$\"1Z*)\\())R.)yF*Fjs7$7$$\"1/++++++yF*$\"1[*)\\())R.G(F *7$$\"1\"*y*\\xz1w'F*Fgt7$7$F\\u$\"1-++++++yF*F`t7$7$Fgt$\"1Z*)\\())R. 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!13++++++[F*7$7$F[r$!1-++++++^F*Fajo7$7$Fcp$!1.++++++dF*7$FjhmF;7$F^[p Fgjo7$F[[p7$$\"1-++++++7F-$!15++++++gF*7$7$Fjn$!1.++++++jF*Fa[p7$7$FH$ !1/++++++pF*7$F_hmF(7$7$F_hm$!1&************f'F*Fg[p7$F[\\p7$F_jm$!18+ +++++sF*7$7$F1$!1/++++++vF*Fd\\pFejm-F$6<7$7$FejnFg]o7$FhjnFdq7$7$F^[o F^qF^]p7$7$Fe^nF]^oFa]p7$7$Fb_nF\\pFc]p7$7$Fgy$!1)[uN#e>3>FecnFe]p7$7$ Fg_lFfo7$Fa`lFh]p7$7$F(F[_oF[^p7$7$F;FY7$F^clF[_o7$7$FA$!1-++++++CF*F` ^p7$7$FSFSFc^p7$7$FY$!1-++++++OF*Fg^p7$7$FfoFAFi^p7$7$F\\p$!1-++++++[F *F]_p7$7$F^qF;F__p7$7$Fdq$!1,++++++gF*Fc_p7$7$FfrF(7$Fdq$!1-++++++gF*7 $7$F\\s$!1-++++++sF*Fi_p7$7$FatFg_l7$FfaoF_`p7$7$Fgt$!1,++++++%)F*Fb`p 7$7$FcsFgy7$Fc`m$!1-++++++%)F*7$7$F[r$!1/++++++'*F*Fi`p7$7$FcpFb_nF^ap 7$7$FjnFbaoFbap7$7$FHFe^nFdap7$7$F1$!1,++++++7F-FfapFejm-F$6P7$7$Fejn$ !1,++++++@F*7$$!1,+++++S9F-FY7$Fabp7$Fhjn$!1,++++++:F*7$F^bp7$$!1***** *******>8F-Fabn7$7$F^[o$!1,++++++FF*Fibp7$7$Fe^n$!1,++++++LF*7$FiapFS7 $FdcpF]cp7$Facp7$$!1************z5F-$!11++++++OF*7$7$Fb_nFfioFgcp7$7$F f_n$!1,++++++XF*7$$!14++++++'*F*FA7$FbdpF]dp7$7$FgyF`dp7$$!1&)******** ***R)F*$!11++++++[F*7$7$Fg_l$!1,++++++^F*Fhdp7$7$F($!1,++++++dF*7$$!11 ++++++sF*F;7$Feep7$$!1(************z(F*F_ep7$Fbep7$$!1&)************fF *$!10++++++gF*7$7$F;$!1,++++++jF*F]fp7$7$$!1&************>%F*$!1,+++++ +pF*7$F[epF(7$F\\gpFcfp7$7$FAFjfp7$$!1')***********f$F*Ffep7$7$FS$!1,+ +++++vF*F`gp7$7$FY$!1*************4)F*7$FabnFg_l7$F[hp7$FS$!1-++++++vF *7$7$FY$!1,++++++\")F*7$Ffbn$!10++++++%)F*7$7$Ffo$!1,++++++()F*Fdhp7$7 $F\\p$!1+++++++$*F*7$FccnFgy7$F_ip7$Ffo$!1-++++++()F*7$7$F\\p$!1-+++++ +$*F*7$FicnFcdp7$7$F^q$!1/++++++**F*Fhip7$7$Fdq$!1++++++]5F-7$FfdnFb_n 7$Fajp7$F^q$!10++++++**F*7$F^jp7$F^enFbao7$7$Ffr$!1++++++56F-Fgjp7$7$F \\s$!1++++++q6F-7$F[fnFe^n7$F`[q7$Ffr$!1,+++++56F-7$F][q7$FcfnFiap7$7$ Fat$!1++++++I7F-Ff[q7$7$Fgt$!1++++++!H\"F-7$F`enF^[o7$F_\\q7$Fat$!1,++ +++I7F-7$F\\\\q7$FbgnFd`o7$7$Fcs$!1,+++++]8F-Fe\\q7$7$F[r$!1++++++59F- 7$FhbnFejn7$F^]qFg\\q7$F[]q7$FchnFbbp7$7$Fcp$!1,+++++q9F-Fa]q7$7$F]\\n FhjnFc]qFejm-F$627$7$FejnF(7$FhjnFf_p7$7$F^[oF_`pF[^q7$7$Fe^nFg_lF^^q7 $7$Fb_nFf`pF`^q7$7$FgyFgy7$Fb_nF[ap7$7$Fg_lF_apFd^q7$7$F(Fb_nFg^q7$7$F ;FbaoFi^q7$7$FAFe^nF[_q7$7$FSFiapF]_q7$7$FYF^[oF__q7$7$FfoFd`oFa_q7$7$ F\\pFejnFc_q7$7$F^q$!1++++++S9F-Fe_q7$7$FdqFhjn7$F^qFbbpFejm-F$627$7$F hjnF_jp7$F[`oFbao7$7$FejnFc[qFa`q7$7$F^[oF^[q7$Fd`oFe^n7$Ff`qFc`q7$Fe` q7$Fi`oFiap7$7$Fe^nFi[qFi`q7$7$Fb_nF]\\q7$FbaoF^[o7$F^aq7$Fe^nFb\\q7$F ]aq7$$!1()***********f*F*Fd`o7$7$FgyFh\\qFbaq7$7$Fg_lF\\]q7$FeboFejn7$ FiaqFfaq7$7$Fg_l$!1,+++++59F-7$$!1#************>(F*Fh_q7$7$F($!1++++++ q9F-F_bq7$7$$!1'*************fF*FhjnFcbqFejm-%+AXESLABELSG6$%\"xG%\"yG " 1 2 0 1 0 2 9 0 4 2 1 0.000000 45.000000 45.000000 0 0 "" "" "" "" " " "" "" "" "" "" "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 639 8 "8. If " }{XPPEDIT 640 1 "x^2 +y^3- z^4=2*x*y*z+4" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yG\"\"$F(*$%\"zG\"\"%!\"\" ,&**F'F(F&F(F*F(F-F(F(F.F(" }{TEXT 641 20 " then calculate " } {XPPEDIT 642 1 "diff( z ,x)" "6#-%%diffG6$%\"zG%\"xG" }{TEXT 643 10 " \+ at " }{XPPEDIT 644 1 "``(3,2,1)" "6#-%!G6%\"\"$\"\"#\"\"\"" } {TEXT 645 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 179 "a) 1/8 b) 1/5 c) 1/4 \+ d) 1/3 e) 1/2 \nf) 2/3 g) 3/4 \+ h) 3/5 i) 4/5 j) 5/8 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 646 8 "Solution" }{TEXT 647 6 ": (a)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eqn1 := x^ 2+y^3-z(x,y)^4=2*x*y*z(x,y)+4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%e qn1G/,(*$%\"xG\"\"#\"\"\"*$%\"yG\"\"$F**$-%\"zG6$F(F,\"\"%!\"\",&*(F(F *F,F*F/F*F)F2F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn2 := \+ map(u -> diff(u,x),eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/ ,&%\"xG\"\"#*&-%\"zG6$F'%\"yG\"\"$-%%diffG6$F*F'\"\"\"!\"%,&*&F-F2F*F2 F(*(F'F2F-F2F/F2F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "eqn3 \+ := diff(z(x,y),x) = solve(eqn2, diff(z(x,y),x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%%diffG6$-%\"zG6$%\"xG%\"yGF,*&,&F,\"\"\"*&F- F0F)F0!\"\"F0,&*$F)\"\"$\"\"#*&F,F0F-F0F0F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(\{x=3,y=2\}, rhs(eqn3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"$\"\"\"-%\"zG6$F%\"\"#!\"#F&,&*$F'F%F*\"\"'F&! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(z(3,2)=1,\"); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 261 15 "9. The vector " } {XPPEDIT 277 1 "`<`*a,1,-1*`>` " "6%*&%\" GF%!\"\"" }{TEXT 278 30 " is tangent to the surface " }{XPPEDIT 279 1 "x^2+2*y^3-3*z^2=3" "6#/,(*$%\"xG\"\"#\"\"\"*&F'F(*$%\"yG\"\"$F( F(*&F,F(*$%\"zGF'F(!\"\"F," }{TEXT 280 36 " at the point (2,1,1). Wha t is a ?" }{TEXT 313 1 " " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 329 72 "a) -5 b) -3 c) -3 \+ d) -2 e) " }{TEXT 326 3 "-1 " }{TEXT 330 20 " \+ " }}{PARA 0 "" 0 "" {TEXT 327 37 "f) 0 g) 1 " }{TEXT -1 1 " " }{TEXT 328 37 "h) 2 i) 3 \+ j) " }{TEXT -1 1 " " }{TEXT 331 1 "4" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT 325 2 " \n" }{TEXT 626 8 "Solution" }{TEXT 627 5 ": (c)" }{TEXT 628 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "F := (x,y,z) -> x^2+2*y^3-3*z^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"FGf*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,(*$9$\"\"#\"\"\" *$9%\"\"$F1*$9&F1!\"$F*F*6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "grad(F(x,y,z),[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VE CTORG6#7%,$%\"xG\"\"#,$*$%\"yGF)\"\"',$%\"zG!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "v := subs(\{x=2,y=1,z=1\}, \" ); #Normal to \+ surface" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%\"\"%\" \"'!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "solve(dotprod(v, [a,1,-1])=0,a); \n#Tangent vector is perpendicular to Normal vector" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 262 16 "10. Calculate " } {XPPEDIT 319 1 "int( y*dx+2*x*dy+(z-x*y+3),z=C..``)" "6#-%$intG6$,(*&% \"yG\"\"\"%#dxGF)F)*(\"\"#F)%\"xGF)%#dyGF)F),(%\"zGF)*&F-F)F(F)!\"\"\" \"$F)F)/F0;%\"CG%!G" }{TEXT 320 11 ". where " }{TEXT 426 1 "C" } {TEXT 427 42 " is the oriented curve parameterized by " }{XPPEDIT 429 1 "r(t)=[t,t^2,t^3]" "6#/-%\"rG6#%\"tG7%F'*$F'\"\"#*$F'\"\"$" } {TEXT 428 4 ", " }{XPPEDIT 431 1 "t=0..1" "6#/%\"tG;\"\"!\"\"\"" } {TEXT 430 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 317 76 "a) 1/3 b) 2/3 c) 1 d ) 4/3 e) " }{TEXT 314 4 "5/3 " }{TEXT 318 20 " \+ " }}{PARA 0 "" 0 "" {TEXT 315 40 "f) 2 g) 8/3 \+ " }{TEXT -1 1 " " }{TEXT 316 43 "h) 3 i) 4 \+ j) 14/3" }}{PARA 3 "" 0 "" {TEXT 424 9 "\nSolution" }{TEXT 425 6 ": (j)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "r := t -> \+ [t,t^2,t^3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6#%\"tG6\"6$%) operatorG%&arrowGF(7%9$*$F-\"\"#*$F-\"\"$F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "Int( r(t)[2]*diff(r(t)[1],t)+ 2*r(t)[1]*diff (r(t)[2],t)+(r(t)[3]-r(t)[1]*r(t)[2]+3)*diff(r(t)[3],t),t=0..1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*$%\"tG\"\"#\"#9/F(;\"\"!\" \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#9\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 444 22 "11. 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Calculate " }{TEXT 447 1 " " } {TEXT 445 15 "f(1,2) - f(0,1)" }{TEXT 446 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 453 74 "a) 2 b) 4 c) 6 d) 8 e) " }{TEXT 457 2 "10" }{TEXT 460 3 " " }{TEXT 459 18 " " }}{PARA 0 "" 0 " " {TEXT 454 35 "f) 12 g) 14 " }{TEXT -1 1 " " } {TEXT 458 80 "h) 16 i) 18 j) Cannot be calculated from given information." }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 455 8 "Solution" }{TEXT 456 6 ": (i)\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "F := vector([y^2+7,2*x*y+3* y^2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'VECTORG6#7$,&*$%\"y G\"\"#\"\"\"\"\"(F-,&*&%\"xGF-F+F-F,F*\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "curl(vector([y^2+7,2*x*y+3*y^2,0]),[x,y,z]); #F is conservative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"!F' F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eqn := u(x,y) = int(y ^2+7,x) + g(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%\"uG6$%\" xG%\"yG,&*&,&*$F*\"\"#\"\"\"\"\"(F0F0F)F0F0-%\"gG6#F*F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diff(eqn,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"uG6$%\"xG%\"yGF+,&*&F*\"\"\"F+F.\"\"#-F% 6$-%\"gG6#F+F+F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "F[2] = \+ rhs(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"xG\"\"\"%\"yGF'\" \"#*$F(F)\"\"$,&F%F)-%%diffG6$-%\"gG6#F(F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "diff(g(y),y) = solve(\",diff(g(y),y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"gG6#%\"yGF*,$*$F*\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn2 := subs(g(y)=y^3+C,e qn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%\"uG6$%\"xG%\"yG,(* &,&*$F*\"\"#\"\"\"\"\"(F0F0F)F0F0*$F*\"\"$F0%\"CGF0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "subs( \{x=1,y=2\}, rhs(eqn2)) - subs(\{x= 0,y=1\}, rhs(eqn2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#=" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 263 17 "12. \+ Calculate " }{XPPEDIT 432 1 "int( (y+z)*dx+(x+1)*dy+(2*z+x),z=C..``) " "6#-%$intG6$,(*&,&%\"yG\"\"\"%\"zGF*F*%#dxGF*F**&,&%\"xGF*F*F*F*%#dy GF*F*,&*&\"\"#F*F+F*F*F/F*F*/F+;%\"CG%!G" }{TEXT 433 11 " where " }{TEXT 434 1 "C" }{TEXT 435 42 " is the oriented curve parameterized by " }{XPPEDIT 437 1 "r(t)=[sqrt(t),sin(Pi/2*t^2),t*exp(t-1)]" "6#/-% \"rG6#%\"tG7%-%%sqrtG6#F'-%$sinG6#*(%#PiG\"\"\"\"\"#!\"\"F'F2*&F'F1-%$ expG6#,&F'F1F1F3F1" }{TEXT 436 4 ", " }{XPPEDIT 438 1 "t=0..1" "6#/% \"tG;\"\"!\"\"\"" }{TEXT 281 6 ". \n\n" }{TEXT 267 71 "a) 0 \+ b) 1 c) 2 d) 3 e) 4" }{TEXT 441 21 " " }}{PARA 0 "" 0 "" {TEXT 282 34 "f) 5 \+ g) 6 " }{TEXT -1 1 " " }{TEXT 442 38 "h) 7 \+ i) 8 j) 9" }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 439 8 "Solution" }{TEXT 440 6 ": (e)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "r := (t) -> [sqrt( t), sin(Pi/2*t^2), t*exp(t-1)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "F := vector([y+z,x+1,2*z+x]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'VECTORG6#7%,&%\"yG\"\"\"%\"zG F+,&%\"xGF+F+F+,&F,\"\"#F.F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "curl(F,[x,y,z]); #F is conservative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqn := u(x,y,z) = int(y+z,x) + g(y,z);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$eqnG/-%\"uG6%%\"xG%\"yG%\"zG,&*&,&F*\"\"\"F+F /F/F)F/F/-%\"gG6$F*F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " diff(eqn,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"uG6%%\" xG%\"yG%\"zGF+,&F*\"\"\"-F%6$-%\"gG6$F+F,F+F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x + 1 = rhs(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"\"F&F&,&F%F&-%%diffG6$-%\"gG6$%\"yG%\"zGF.F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "diff(g(y,z),y) = solve(\",di ff(g(y,z),y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"gG6$% \"yG%\"zGF*\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn2 : = subs(g(y,z)=y+h(z),eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G /-%\"uG6%%\"xG%\"yG%\"zG,(*&,&F*\"\"\"F+F/F/F)F/F/F*F/-%\"hG6#F+F/" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "2*z+x = diff(rhs(eqn2),z); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"zG\"\"#%\"xG\"\"\",&F'F(-%%d iffG6$-%\"hG6#F%F%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "dif f(h(z),z) = solve( \" , diff(h(z),z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"hG6#%\"zGF*,$F*\"\"#" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "eqn3 := subs(h(z)=z^2 + C, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%\"uG6%%\"xG%\"yG%\"zG,**&,&F*\"\"\"F+F/F/ F)F/F/F*F/*$F+\"\"#F/%\"CGF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "evalm( grad(rhs(eqn3),[x,y,z]) - F ); #Check" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'VECTORG6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "subs( \{x=1,y=1,z=1\}, rhs(eqn3)) - subs(\{x=0,y=0,z= 0\}, rhs(eqn3));" }}{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 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 264 28 "13. 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::Z::xI::y]ytyx I::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::Z::xI::y^yl IC:xyyryyA:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: B" }{TEXT -1 2 "\"." }}{PARA 267 "" 0 "" {TEXT 473 13 "\nAt least one" }{TEXT -1 69 " of the implications stated in the answers may be incorrect. Ch oose " }{TEXT 472 3 "any" }{TEXT -1 22 " incorrect statement" }} {PARA 268 "" 0 "" {TEXT -1 15 "as your answer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 321 106 "a) I => II \+ b) II => III c) III => IV d) IV => I \+ e) I => III" }{TEXT 322 21 " \n" }{TEXT 324 47 "f) II => I g) I => IV " }{TEXT -1 3 " h)" } {TEXT 323 58 " II => IV i) IV => III j) IV => \+ II" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 474 8 "Sol ution" }{TEXT 475 6 ": (c)\n" }}{PARA 0 "" 0 "" {TEXT -1 107 "Properti es I, II, and IV are equivalent. They all imply property III. However, G is not simply connected so" }}{PARA 0 "" 0 "" {TEXT -1 46 "property III does not imply any of the others." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 265 1 " " }{TEXT 283 15 "14. Calcu late " }{XPPEDIT 492 1 "Int(-y^3*dx+x^3,y=C..``)" "6#-%$IntG6$,&*&%\"y G\"\"$%#dxG\"\"\"!\"\"*$%\"xGF)F+/F(;%\"CG%!G" }{TEXT 491 8 " where \+ " }{TEXT 489 1 "C" }{TEXT 490 48 " is the unit circle traversed counte r-clockwise." }{TEXT 488 2 " \n" }{TEXT 477 4 "a) " }{XPPEDIT 493 1 " 0" "6#\"\"!" }{TEXT 478 20 " b) " }{XPPEDIT 494 1 "Pi/ 4" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 479 19 " c) " } {XPPEDIT 495 1 "Pi/2" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 480 20 " \+ d) " }{XPPEDIT 496 1 "Pi" "6#%#PiG" }{TEXT 481 20 " \+ e) " }{XPPEDIT 497 1 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT 482 11 " \nf) " }{XPPEDIT 498 1 "Pi/3" "6#*&%#PiG\"\"\"\"\"$!\" \"" }{TEXT 483 18 " g) " }{XPPEDIT 499 1 "Pi/6" "6#*&%#P iG\"\"\"\"\"'!\"\"" }{TEXT 484 19 " h) " }{XPPEDIT 500 1 "2*Pi/3" "6#*(\"\"#\"\"\"%#PiGF%\"\"$!\"\"" }{TEXT 485 17 " \+ i) " }{XPPEDIT 501 1 "3*Pi/2" "6#*(\"\"$\"\"\"%#PiGF%\"\"#!\"\"" }{TEXT 486 18 " j) " }{XPPEDIT 502 1 "3*Pi/4" "6#*(\"\"$ \"\"\"%#PiGF%\"\"%!\"\"" }{TEXT 487 5 " \n" }}{PARA 3 "" 0 "" {TEXT 503 8 "Solution" }{TEXT 504 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*di ff(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 -1 31 "More easily by 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),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#\"\"$\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 261 "" 0 "" {TEXT -1 9 "15. 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:::::::::::::::::::::::5:" }}{PARA 0 "" 0 "" {TEXT 509 14 "then what i s " }{TEXT 511 1 "f" }{TEXT 513 7 "(1,2,3)" }{TEXT 514 1 " " }{TEXT 512 2 "?\n" }}{PARA 0 "" 0 "" {TEXT 516 71 "a) 0 b) 1 \+ c) 2 d) 3 e) 4" }{TEXT 518 3 " " } {TEXT 520 18 " " }}{PARA 0 "" 0 "" {TEXT 517 34 "f) \+ 5 g) 6 " }{TEXT -1 1 " " }{TEXT 519 38 "h) 7 \+ i) 8 j) 9" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 521 8 "Solution" }{TEXT 522 6 ": (g)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "diverge([x*y,2*y*z,x^2+x^2*y^3-y*z],[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"zG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(\{x=1,y=2,z=3\},\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 333 10 "16. If " }{XPPEDIT 416 1 "f(x,y,z)=x^2 - 2*y^ 2 + 3*z^2" "6#/-%\"fG6%%\"xG%\"yG%\"zG,(*$F'\"\"#\"\"\"*&F,F-*$F(F,F-! \"\"*&\"\"$F-*$F)F,F-F-" }{TEXT 415 17 " then calculate " }{TEXT 419 1 " " }{TEXT 417 11 "div(grad(f(" }{TEXT 420 5 "x,y,z" }{TEXT 421 2 ") )" }{TEXT 418 2 ". " }{TEXT 334 2 " \n" }{TEXT 332 142 "\na) 1 \+ b) 2 c) 3 d) 4 e) 5 \nf) 8 \+ g) 12 h) 15 i) 16 j) 20" }{TEXT 515 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 422 8 "Solution" }{TEXT 423 6 ": (d)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f := (x,y,z) -> x^2 - 2*y^2 + 3*z^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF *,(*$9$\"\"#\"\"\"*$9%F1!\"#*$9&F1\"\"$F*F*6\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "grad(f(x,y,z),[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%,$%\"xG\"\"#,$%\"yG!\"%,$%\"zG\"\"'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diverge(\",[x,y,z]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 335 38 "17. Calculate the surface i ntegral of " }{TEXT 338 1 " " }{XPPEDIT 339 1 "f(x,y,z)=x+z" "6#/-%\"f G6%%\"xG%\"yG%\"zG,&F'\"\"\"F)F+" }{TEXT 337 31 " over that part of \+ the plane " }{TEXT 340 1 " " }{XPPEDIT 341 1 "x+y+z=1" "6#/,(%\"xG\"\" \"%\"yGF&%\"zGF&F&" }{TEXT 336 4 " \n" }{TEXT 395 30 "that lies in t he first octant." }{TEXT 397 1 "\n" }{TEXT 396 39 "a) 0 b) \+ 1 c) " }{XPPEDIT 405 1 "sqrt(2)/2" "6#*&-%%sqrtG6#\"\"# \"\"\"F'!\"\"" }{TEXT 398 15 " d) " }{XPPEDIT 406 1 "sqrt(3 )/3" "6#*&-%%sqrtG6#\"\"$\"\"\"F'!\"\"" }{TEXT 399 13 " e) " }{XPPEDIT 407 1 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 400 7 " \nf) \+ " }{XPPEDIT 408 1 "sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT 401 14 " \+ g) " }{XPPEDIT 409 1 "2*sqrt(3)/3" "6#*(\"\"#\"\"\"-%%sqrtG6#\"\"$ F%F)!\"\"" }{TEXT 402 12 " h) " }{XPPEDIT 410 1 "2*sqrt(2)" "6 #*&\"\"#\"\"\"-%%sqrtG6#F$F%" }{TEXT 403 12 " i) " }{XPPEDIT 411 1 "sqrt(3)" "6#-%%sqrtG6#\"\"$" }{TEXT 404 14 " j) " } {XPPEDIT 412 1 "3/2" "6#*&\"\"$\"\"\"\"\"#!\"\"" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 413 8 "Solution" }{TEXT 414 6 ": (d)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "vectorLength := v -> sqrt(su m(v[i]^2,i=1..vectdim(v)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-vect orLengthGf*6#%\"vG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#-%$sumG6$*$&9$6# %\"iG\"\"#/F6;\"\"\"-%(vectdimG6#F4F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 62 "r := (x,y) -> vector([x,y,1-(x+y)]); #surface param eterization" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6$%\"xG%\"yG6\" 6$%)operatorG%&arrowGF)-%'vectorG6#7%9$9%,(\"\"\"F4F1!\"\"F2F5F)F)6\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "rx := map(z->diff(z,x), r(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rxG-%'VECTORG6#7%\"\"\" \"\"!!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ry := map(z-> diff(z,y),r(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ryG-%'VECTORG 6#7%\"\"!\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "cro ssprod(rx,ry);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"\" F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "surfaceAreaFactor : = vectorLength(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2surfaceAreaF actorG*$\"\"$#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := (x,y,z) -> x + z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf* 6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,&9$\"\"\"9&F0F*F*6\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Int(Int(f(x,y,z)*surfaceArea Factor,y=0..1-x),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$ -F$6$*&,&%\"xG\"\"\"%\"zGF+F+\"\"$#F+\"\"#/%\"yG;\"\"!,&F+F+F*!\"\"/F* ;F3F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(z=1-(x+y),\") ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$*&,&\"\"\"F*%\"yG! \"\"F*\"\"$#F*\"\"#/F+;\"\"!,&F*F*%\"xGF,/F4;F2F*" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 9 "value(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*$\"\"$#\"\"\"\"\"##F'F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 343 8 "18. 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subs(z=2-x^2-y^2, \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*IntegrandG,(*$%\"xG\"\"#\"\"$!\"#\"\"\"*$%\"yGF(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Integrand := subs(\{x=r*c os(theta),y=r*sin(theta)\},Integrand);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*IntegrandG,(*&%\"rG\"\"#-%$cosG6#%&thetaGF(\"\"$!\"#\"\"\"*&F 'F(-%$sinGF+F(F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "int(int (r*Integrand,r=0..1), theta = 0 .. 2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG!\"\"" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 376 52 "Alternative calculation using the Divergence T heorem" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "w ith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "parab := plo t3d(2-x^2-y^2, y = -sqrt(1-x^2)..sqrt(1-x^2),x=-1..1,color=aquamarine) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "disk := plot3d([r*cos( theta),r*sin(theta),1],r=0..1,theta=0..2*Pi,color=plum):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 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1qV%y.a-m)F1$\"1?++++++]F1F,7%$!1r1*GEe#f'*F1$\"1N_-^/>)e#F1F,7%$!1+++ ++++5F;$\"1uEL/^3JKF^bxF,7%$!1)o!*GEe#f'*F1$!1s^-^/>)e#F1F,7%$!1.W%y.a -m)F1$!1k************\\F1F,7%$!1,b'=\"y1rqF1$!1Ua'=\"y1rqF1F,7%$!1K+++ +++]F1$!1jV%y.a-m)F1F,7%$!1[_-^/>)e#F1$!1o1*GEe#f'*F1F,7%$!1!)Rd^IeCYF ^bxFb^alF,7%$\"1f^-^/>)e#F1$!1\"p!*GEe#f'*F1F,7%$\"1_************\\F1$ !14W%y.a-m)F1F,7%$\"1Ka'=\"y1rqF1$!16b'=\"y1rqF1F,7%$\"1aV%y.a-m)F1$!1 \\++++++]F1F,7%$\"1i1*GEe#f'*F1$!1m_-^/>)e#F1F,7%F9$!1\\`m3- " 0 "" {MPLTEXT 1 0 19 "diverge(F,[x,y,z]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 153 "Therefore the sum of the required in tegral (integrated over the aquamarine paraboloid) and the flux integr al through the bottom plum disk is equal to 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "On the plum disk the unit outward normal is " }{TEXT 378 4 " - " }{TEXT 377 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 379 3 " " }{TEXT 380 5 "z = 1" }{TEXT -1 13 ". 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