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0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 269 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 "" 0 270 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 "" 258 273 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 "" 0 277 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 810 6 "Math " }{TEXT 811 4 "233 1" }{TEXT 812 22 " Fall 2002 Final Exam" }}{PARA 0 "" 0 "" {TEXT 259 2 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 263 1 "1" }{TEXT 303 2 ". " } {TEXT 837 6 " If " }{XPPEDIT 859 1 "u" "I\"uG6\"" }{TEXT -1 1 " " } {TEXT 825 6 " and " }{TEXT -1 1 " " }{XPPEDIT 860 1 "v" "I\"vG6\"" } {TEXT 826 19 " are vectors and " }{XPPEDIT 857 1 "u=lambda*v" "/%\"u G*&%'lambdaG\"\"\"%\"vGF&" }{TEXT 858 21 " for some scalar " } {XPPEDIT 829 1 "lambda" "I'lambdaG6\"" }{TEXT 828 31 " then which of \+ the following " }{TEXT 830 4 "must" }{TEXT 831 11 " be true? " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 273 "" 0 "" {TEXT -1 5 "a) " } {XPPEDIT 846 1 "u" "I\"uG6\"" }{TEXT -1 30 " has the same direction as " }{XPPEDIT 827 1 "v" "I\"vG6\"" }{TEXT -1 11 " \nb) " } {XPPEDIT 847 1 "u" "I\"uG6\"" }{TEXT -1 33 " is opposite in directi on to " }{XPPEDIT 865 1 "v" "I\"vG6\"" }{TEXT -1 11 " \nc) " } {XPPEDIT 848 1 "u" "I\"uG6\"" }{TEXT -1 25 " is perpendicular to \+ " }{XPPEDIT 833 1 "v" "I\"vG6\"" }{TEXT -1 16 " \nd) " } {XPPEDIT 849 1 "u" "I\"uG6\"" }{TEXT -1 21 " is parallel to " } {XPPEDIT 863 1 "v" "I\"vG6\"" }{TEXT -1 10 " \ne) " }{XPPEDIT 850 1 "u" "I\"uG6\"" }{TEXT -1 28 " has the same length as " } {XPPEDIT 834 1 "v" "I\"vG6\"" }{TEXT -1 7 " \nf) " }{XPPEDIT 851 1 " u" "I\"uG6\"" }{TEXT -1 32 " has magnitude greater than " } {XPPEDIT 836 1 "v" "I\"vG6\"" }{TEXT -1 2 " " }}{PARA 258 "" 0 "" {TEXT 838 4 "g) " }{XPPEDIT 852 1 "u" "I\"uG6\"" }{TEXT -1 1 " " } {TEXT 839 27 " has magnitude less than " }{TEXT -1 1 " " }{XPPEDIT 841 1 "v" "I\"vG6\"" }{TEXT 840 18 " \nh) " }{XPPEDIT 854 1 "v=(lambda)^(-1)*u" "/%\"vG*&)%'lambdaG,$\"\"\"!\"\"F(%\"uGF(" } {TEXT 853 21 " \ni) " }{XPPEDIT 856 1 "lambda = u/v" " /%'lambdaG*&%\"uG\"\"\"%\"vG!\"\"" }{TEXT 855 8 " \nj) " }{TEXT -1 1 " " }{XPPEDIT 842 1 "u" "I\"uG6\"" }{TEXT -1 1 " " }{TEXT 843 6 " an d " }{TEXT -1 1 " " }{XPPEDIT 844 1 "v" "I\"vG6\"" }{TEXT 845 48 " d etermine a unique plane through the origin " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 866 8 "Solution" }{TEXT 867 6 ": (d)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 260 "" 0 "" {TEXT 273 4 "2. " }{TEXT 868 15 "If th e vectors " }{TEXT 873 2 " " }{XPPEDIT 876 1 "`<`*(-3),b,b*`>`" "6%*& %\"GF%" }{TEXT 870 1 " " }{TEXT 874 3 "and" }{TEXT 875 2 " " }{XPPEDIT 877 1 "`<`*2,1,2*b*`>`" "6%*&% \"GF%" }{TEXT 871 1 " " }{TEXT 878 34 " are perpendicular then what might" }{TEXT 879 1 " " }{XPPEDIT 880 1 "b" "I\"bG6\"" }{TEXT 872 2 " " }{TEXT 881 4 "be? " }{TEXT 882 2 " " }}{PARA 3 "" 0 "" {TEXT 869 154 "a) -1 b) -1/2 \+ c) 0 d) 1/2 e) 1 \nf) 3/2 g) 2 h) 5/2 i) 3 j) 7/2 " }{TEXT 883 7 " \n" }}{PARA 3 "" 0 "" {TEXT 887 8 "Solution" }{TEXT 888 6 ": (f )\n" }}{PARA 0 "" 0 "" {TEXT -1 40 "We need the dot product - it is in the " }{TEXT 886 6 "linalg" }{TEXT -1 10 " package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(li nalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn_for_b := dotp rod([-3, b, b],[2, 1, 2*b]) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% *eqn_for_bG/,(*$%\"bG\"\"#F)F(\"\"\"!\"'F*\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "solve(eqn_for_b, b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"\"$\"\"#!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 40 "Only one of these is in the answer list. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 261 81 "3. The line through the points ( 3, -1, 2) and (5, 2, -1) passes through a point " }{TEXT 905 2 "(a" } {TEXT 897 2 ", " }{TEXT 898 1 "b" }{TEXT 907 5 ", -4)" }{TEXT 906 2 ". " }{TEXT 903 8 "What is " }{TEXT 904 8 " b ?\n\n" }{TEXT 910 74 "a) -4 b) -3 c) -2 d) -1 e) \+ 0" }{TEXT 891 2 " " }{TEXT 911 19 " " }}{PARA 0 " " 0 "" {TEXT 892 36 "f) 1 g) 2 " }{TEXT -1 1 " " }{TEXT 893 39 "h) 3 i) 4 j) " }{TEXT -1 1 " " }{TEXT 900 1 "5" }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT 890 1 " " }}{PARA 3 "" 0 "" {TEXT 912 8 "Solution" }{TEXT 913 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#>%#O PG-%'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,!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eqn : = t = solve(line[3] = -4, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq nG/%\"tG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "b = subs(e qn, line[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"bG\"\"&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 920 82 "4. A plane passes through the three points (1, 0, 0), (0, 2, 0), a nd (2, 3, 3)." }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT 914 51 "Its equ ation is Ax + 3y + Cz = D. What is D? " }{TEXT 917 1 "\n" }} {PARA 3 "" 0 "" {TEXT 916 132 "a) 0 b) 1 c) 2 \+ d) 3 e) 4 \nf) 5 g) 6 h) 7 \+ i) 8 j) 9" }{TEXT 915 1 "\n" }}{PARA 3 "" 0 "" {TEXT 921 8 "Solution" }{TEXT 922 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-%'VECT ORG6#7%\"\"\"\"\"!F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#OQG-%'VECTO RG6#7%\"\"!\"\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ORG-%'VECTORG 6#7%\"\"#\"\"$F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v := ev alm(OP-OQ);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7%\" \"\"!\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "w := evalm (OR-OQ);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'VECTORG6#7%\"\"# \"\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "cp := crossp rod(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 267 2 " 5" }{TEXT -1 2 ". " }{TEXT 384 15 "The function " }{XPPEDIT 383 1 "f(x,y)=2*x^ 2*y-3*x*y^2-6*x*y" "/-%\"fG6$%\"xG%\"yG,(*(\"\"#\"\"\"*$F&F*F+F'F+F+*( \"\"$F+F&F+F'F*!\"\"*(\"\"'F+F&F+F'F+F/" }{TEXT 382 178 " has one cri tical point P = (a,b) with ab < 0. Choose the ordered list [a,what] where a is the abscissa of the critical point P = (a,b) and \"what\" \+ describes the behavior of " }{XPPEDIT 386 1 "f" "I\"fG6\"" }{TEXT 385 8 " at P. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 391 2 "a)" }{TEXT 387 24 " [1,local minimum] " }{TEXT 394 2 "b)" }{TEXT 392 56 " [-1,local minimum] c) [-2,local minimum] " }{TEXT 395 7 " \nd) " }{TEXT 396 18 "[1,local maximum] " } {TEXT 397 8 " e) " }{TEXT 393 22 "[-1,local maximum] " }{TEXT 390 4 " " }{TEXT 400 3 " f)" }{TEXT 401 1 " " }{TEXT 398 18 "[-2,lo cal maximum]" }{TEXT 399 1 " " }{TEXT 388 12 " \ng) " }{TEXT 402 16 "[1,saddle point]" }{TEXT 403 16 " h) " }{TEXT 404 17 "[-1,saddle point]" }{TEXT 405 19 " i) " }{TEXT 408 17 "[-2,saddle point]" }{TEXT 389 6 " \n " }{TEXT 409 3 "j) " } {TEXT 410 1 " " }{TEXT 406 17 "[2,local maximum]" }{TEXT 407 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 950 8 "Solution " }{TEXT 951 6 ": (d)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f := (x,y) -> 2*x^2*y-3*x*y^2-6*x*y:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "solve(\{diff(f(x,y),x) = 0 , diff(f(x,y),y) = 0\}, \{ x,y\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&<$/%\"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$2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"%" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 968 55 "6. At an instant of time a particle 's acceleration is " }{XPPEDIT 957 1 "`<`*2,1,-3*`>`" "6%*&%\"GF%!\"\"" }{TEXT 958 38 " and its principal unit normal is\n " }{XPPEDIT 962 1 "N=2/3*i+2/3*j-k/3" "/%\"NG,(*( \"\"#\"\"\"\"\"$!\"\"%\"iGF'F'*(F&F'F(F)%\"jGF'F'*&%\"kGF'F(F)F)" } {TEXT 964 51 ". What is its normal component of acceleration " } {XPPEDIT 961 1 "a[N]" "&%\"aG6#%\"NG" }{TEXT 959 2 "?\n" }}{PARA 0 "" 0 "" {TEXT 965 169 "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 969 8 "Solution" }{TEXT 970 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 "dot prod(a,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 265 25 "7. The maximum value of " }{TEXT 319 1 " " }{XPPEDIT 313 1 "f(x,y )=x+2*y" "/-%\"fG6$%\"xG%\"yG,&F&\"\"\"*&\"\"#F)F'F)F)" }{TEXT 314 2 " " }{TEXT 317 20 " on the unit circle " }{TEXT 318 3 " " }{XPPEDIT 315 1 "x^2+y^2=1" "/,&*$%\"xG\"\"#\"\"\"*$%\"yGF&F'F'" }{TEXT 316 3 " \+ \n" }{TEXT 927 11 "occurs at " }{TEXT 929 1 "(" }{TEXT 925 3 "a,b" } {TEXT 926 1 ")" }{TEXT 930 26 ". What is the value of " }{TEXT 923 1 "a" }{TEXT 924 1 " " }{TEXT 928 2 " ?" }}{PARA 0 "" 0 "" {TEXT 320 4 "a) " }{XPPEDIT 19 1 "1/sqrt(5)" "*&\"\"\"F#-%%sqrtG6#\"\"&!\"\"" } {TEXT 931 15 " b) " }{XPPEDIT 19 1 "2/sqrt(5)" "*&\"\"#\"\" \"-%%sqrtG6#\"\"&!\"\"" }{TEXT 932 16 " c) " }{XPPEDIT 19 1 "1/sqrt(6)" "*&\"\"\"F#-%%sqrtG6#\"\"'!\"\"" }{TEXT 934 21 " \+ d) " }{XPPEDIT 19 1 "2/sqrt(6)" "*&\"\"#\"\"\"-%%sqrtG6#\"\" '!\"\"" }{TEXT 936 14 " e) " }{XPPEDIT 19 1 "1/sqrt(2)" "*& \"\"\"F#-%%sqrtG6#\"\"#!\"\"" }{TEXT 938 11 " \n f) " }{XPPEDIT 19 1 "1/sqrt(3)" "*&\"\"\"F#-%%sqrtG6#\"\"$!\"\"" }{TEXT 940 15 " \+ g) " }{XPPEDIT 19 1 "1/3" "*&\"\"\"F#\"\"$!\"\"" }{TEXT 942 20 " h) " }{XPPEDIT 19 1 "2/3" "*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT 944 47 " i) 1 j) 0" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 946 8 "Solution " }{TEXT 947 6 ": (a)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f := (x,y) -> x + 2*y: phi := (x,y) -> x^2 + y^2:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 49 "eqn1 := diff(f(x,y),x) = lambda*diff(phi(x,y ),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/\"\"\",$*&%'lambdaGF &%\"xGF&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn2 := di ff(f(x,y),y) = lambda*diff(phi(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/\"\"#,$*&%'lambdaG\"\"\"%\"yGF*F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eqn3 := phi(x,y) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,&*$%\"xG\"\"#\"\"\"*$%\"yGF)F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "solve( \{eqn1,eqn2,eqn3\}, \{x,y,la mbda\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%'lambdaG,$-%'RootOfG6# ,&*$%#_ZG\"\"#\"\"&!\"\"\"\"\"#F.F-/%\"xGF'/%\"yG,$F'F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 114 "levelCurve[1] := implicitplot( 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 "levelCurve[2] := implic itplot( x^2+y^2=1, x = 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1)************\\\"F*7$$!13++++++OF*F^q7$FfdoF_do7$Fcdo7$$!1%)********* **R#F*$\"1#************>\"F*7$7$FY$\"1!)*************)FaoF[eo7$7$Ffo$ \"1\")************HFao7$$!13++++++7F*F\\p7$7$Fieo$\"1S++++++gFaoFaeo7$ Feeo7$$\"1E\"z$G6Dh:!#I$!1'4wB++++++IFaoF`fo7$7$F ^q$!1>++++++!*Fao7$F^eoFfo7$F^goFgfo7$F[go7$$\"1;++++++CF*Fieo7$7$Fdq$ !1-++++++:F*Fago7$7$Ffr$!1-++++++@F*7$F\\coFY7$F\\hoFego7$Figo7$$\"1:+ +++++[F*$!13++++++CF*7$7$F\\s$!1-++++++FF*F_ho7$7$Fat$!1-++++++LF*7$$ \"1#*************fF*FS7$F\\ioFeho7$Fiho7$$\"1:++++++sF*Fgdo7$7$Fgt$!1, ++++++RF*Faio7$7$Fcs$!1-++++++XF*7$$\"1#************R)F*FA7$F\\joFeio7 $Fiio7$$\"1:++++++'*F*$!13++++++[F*7$7$F[r$!1-++++++^F*Fajo7$7$Fcp$!1. ++++++dF*7$FjhmF;7$F^[pFgjo7$F[[p7$$\"1-++++++7F-$!15++++++gF*7$7$Fjn$ !1.++++++jF*Fa[p7$7$FH$!1/++++++pF*7$F_hmF(7$7$F_hm$!1&************f'F *Fg[p7$F[\\p7$F_jm$!18++++++sF*7$7$F1$!1/++++++vF*Fd\\pFejm-F$6<7$7$Fe jnFg]o7$FhjnFdq7$7$F^[oF^qF^]p7$7$Fe^nF]^oFa]p7$7$Fb_nF\\pFc]p7$7$Fgy$ 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****f$F*Ffep7$7$FS$!1,++++++vF*F`gp7$7$FY$!1*************4)F*7$FabnFg_ l7$F[hp7$FS$!1-++++++vF*7$7$FY$!1,++++++\")F*7$Ffbn$!10++++++%)F*7$7$F fo$!1,++++++()F*Fdhp7$7$F\\p$!1+++++++$*F*7$FccnFgy7$F_ip7$Ffo$!1-++++ ++()F*7$7$F\\p$!1-++++++$*F*7$FicnFcdp7$7$F^q$!1/++++++**F*Fhip7$7$Fdq $!1++++++]5F-7$FfdnFb_n7$Fajp7$F^q$!10++++++**F*7$F^jp7$F^enFbao7$7$Ff r$!1++++++56F-Fgjp7$7$F\\s$!1++++++q6F-7$F[fnFe^n7$F`[q7$Ffr$!1,+++++5 6F-7$F][q7$FcfnFiap7$7$Fat$!1++++++I7F-Ff[q7$7$Fgt$!1++++++!H\"F-7$F`e nF^[o7$F_\\q7$Fat$!1,+++++I7F-7$F\\\\q7$FbgnFd`o7$7$Fcs$!1,+++++]8F-Fe \\q7$7$F[r$!1++++++59F-7$FhbnFejn7$F^]qFg\\q7$F[]q7$FchnFbbp7$7$Fcp$!1 ,+++++q9F-Fa]q7$7$F]\\nFhjnFc]qFejm-F$627$7$FejnF(7$FhjnFf_p7$7$F^[oF_ `pF[^q7$7$Fe^nFg_lF^^q7$7$Fb_nFf`pF`^q7$7$FgyFgy7$Fb_nF[ap7$7$Fg_lF_ap Fd^q7$7$F(Fb_nFg^q7$7$F;FbaoFi^q7$7$FAFe^nF[_q7$7$FSFiapF]_q7$7$FYF^[o F__q7$7$FfoFd`oFa_q7$7$F\\pFejnFc_q7$7$F^q$!1++++++S9F-Fe_q7$7$FdqFhjn 7$F^qFbbpFejm-F$627$7$FhjnF_jp7$F[`oFbao7$7$FejnFc[qFa`q7$7$F^[oF^[q7$ Fd`oFe^n7$Ff`qFc`q7$Fe`q7$Fi`oFiap7$7$Fe^nFi[qFi`q7$7$Fb_nF]\\q7$FbaoF ^[o7$F^aq7$Fe^nFb\\q7$F]aq7$$!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$%\"xG%\"yG" 2 514 443 443 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 3037 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 972 8 "8. If " }{XPPEDIT 973 1 "x^2 +y^3-z^ 4=2*x*y*z+4" "/,(*$%\"xG\"\"#\"\"\"*$%\"yG\"\"$F'*$%\"zG\"\"%!\"\",&** F&F'F%F'F)F'F,F'F'F-F'" }{TEXT 974 20 " then calculate " } {XPPEDIT 975 1 "diff( z ,x)" "-%%diffG6$%\"zG%\"xG" }{TEXT 976 10 " \+ at " }{XPPEDIT 977 1 "``(3,2,1)" "-%!G6%\"\"$\"\"#\"\"\"" }{TEXT 978 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 277 "" 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 979 8 "Solution" }{TEXT 980 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#>%%eqn1G/,(* $%\"xG\"\"#\"\"\"*$%\"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*F2F(*(F'F 2F-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&,&*$F'F%F*\"\"'F&! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(z(3,2)=1,\"); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 268 15 "9. The vector " } {XPPEDIT 332 1 "`<`*a,1,-1*`>` " "6%*&%\" GF%!\"\"" }{TEXT 333 30 " is tangent to the surface " }{XPPEDIT 334 1 "x^2+2*y^3-3*z^2=3" "/,(*$%\"xG\"\"#\"\"\"*&F&F'*$%\"yG\"\"$F'F' *&F+F'*$%\"zGF&F'!\"\"F+" }{TEXT 335 36 " at the point (2,1,1). What \+ is a ?" }{TEXT 439 1 " " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 504 72 "a) -5 b) -3 c) -3 \+ d) -2 e) " }{TEXT 501 3 "-1 " }{TEXT 505 20 " \+ " }}{PARA 0 "" 0 "" {TEXT 502 37 "f) 0 g) 1 \+ " }{TEXT -1 1 " " }{TEXT 503 37 "h) 2 i) 3 \+ j) " }{TEXT -1 1 " " }{TEXT 506 1 "4" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT 500 2 " \n" }{TEXT 953 8 "Solution" }{TEXT 954 5 ": (c)" }{TEXT 955 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "F := (x,y,z) -> x^2+2*y^3-3*z^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"FG:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,(*$9$\"\"#\"\"\"* $9%\"\"$F1*$9&F1!\"$F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "grad(F(x,y,z),[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG 6#7%,$%\"xG\"\"#,$*$%\"yGF)\"\"',$%\"zG!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "v := subs(\{x=2,y=1,z=1\}, \" ); #Normal to surfa ce" }}{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#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 269 16 "10. Calculate " } {XPPEDIT 446 1 "int( y*dx+2*x*dy+(z-x*y+3),z=C..``)" "-%$intG6$,(*&%\" yG\"\"\"%#dxGF(F(*(\"\"#F(%\"xGF(%#dyGF(F(,(%\"zGF(*&F,F(F'F(!\"\"\"\" $F(F(/F/;%\"CG%!G" }{TEXT 447 11 ". where " }{TEXT 679 1 "C" } {TEXT 680 42 " is the oriented curve parameterized by " }{XPPEDIT 682 1 "r(t)=[t,t^2,t^3]" "/-%\"rG6#%\"tG7%F&*$F&\"\"#*$F&\"\"$" } {TEXT 681 4 ", " }{XPPEDIT 685 1 "t=0..1" "/%\"tG;\"\"!\"\"\"" } {TEXT 683 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 443 76 "a) 1/3 b) 2/3 c) 1 d ) 4/3 e) " }{TEXT 440 4 "5/3 " }{TEXT 445 20 " \+ " }}{PARA 0 "" 0 "" {TEXT 441 40 "f) 2 g) 8/3 \+ " }{TEXT -1 1 " " }{TEXT 442 43 "h) 3 i) 4 \+ j) 14/3" }}{PARA 3 "" 0 "" {TEXT 677 9 "\nSolution" }{TEXT 678 6 ": (j)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "r := t -> \+ [t,t^2,t^3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG:6#%\"tG6\"6$%)o peratorG%&arrowGF(7%9$*$F-\"\"#*$F-\"\"$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "Int( r(t)[2]*diff(r(t)[1],t)+ 2*r(t)[1]*diff(r( t)[2],t)+(r(t)[3]-r(t)[1]*r(t)[2]+3)*diff(r(t)[3],t),t=0..1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$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 704 22 "11. 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Calculate " }{TEXT 708 1 " " } {TEXT 705 15 "f(1,2) - f(0,1)" }{TEXT 707 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 714 74 "a) 2 b) 4 c) 6 d) 8 e) " }{TEXT 718 2 "10" }{TEXT 721 3 " " }{TEXT 720 18 " " }}{PARA 0 "" 0 " " {TEXT 715 35 "f) 12 g) 14 " }{TEXT -1 1 " " } {TEXT 719 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 716 8 "Solution" }{TEXT 717 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 270 17 "12. \+ Calculate " }{XPPEDIT 688 1 "int( (y+z)*dx+(x+1)*dy+(2*z+x),z=C..``) " "-%$intG6$,(*&,&%\"yG\"\"\"%\"zGF)F)%#dxGF)F)*&,&%\"xGF)F)F)F)%#dyGF )F),&*&\"\"#F)F*F)F)F.F)F)/F*;%\"CG%!G" }{TEXT 689 11 " where " } {TEXT 690 1 "C" }{TEXT 691 42 " is the oriented curve parameterized \+ by " }{XPPEDIT 693 1 "r(t)=[sqrt(t),sin(Pi/2*t^2),t*exp(t-1)]" "/-%\"r G6#%\"tG7%-%%sqrtG6#F&-%$sinG6#*(%#PiG\"\"\"\"\"#!\"\"F&F1*&F&F0-%$exp G6#,&F&F0F0F2F0" }{TEXT 692 4 ", " }{XPPEDIT 695 1 "t=0..1" "/%\"tG; \"\"!\"\"\"" }{TEXT 337 6 ". \n\n" }{TEXT 274 71 "a) 0 b) 1 c) 2 d) 3 e) 4" }{TEXT 699 21 " " }}{PARA 0 "" 0 "" {TEXT 340 34 "f) 5 \+ g) 6 " }{TEXT -1 1 " " }{TEXT 700 38 "h) 7 i) 8 j) 9" }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 " " }}{PARA 3 "" 0 "" {TEXT 697 8 "Solution" }{TEXT 698 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\"\"\"%\"zGF+,&%\"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$F+F,F+F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x + 1 = rhs (\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"\"F&F&,&F%F&-%%di ffG6$-%\"gG6$%\"yG%\"zGF.F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "diff(g(y,z),y) = solve(\",diff(g(y,z),y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"gG6$%\"yG%\"zGF*\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn2 := subs(g(y,z)=y+h(z),eqn);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%\"uG6%%\"xG%\"yG%\"zG,(*&,& F*\"\"\"F+F/F/F)F/F/F*F/-%\"hG6#F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "2*z+x = diff(rhs(eqn2),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"zG\"\"#%\"xG\"\"\",&F'F(-%%diffG6$-%\"hG6#F%F%F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "diff(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 "eqn 3 := subs(h(z)=z^2 + C, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%e qn3G/-%\"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(e qn3),[x,y,z]) - F ); #Check" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VEC TORG6#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 271 28 "13. Consider a v ector field " }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{BITMAP 320 27 27 1 "?TMgUSB:n@>?>b;jGj:>=>::::vyyyyy=:::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::jyxqysyywiyyyyA:::::::::::: :::::::::::::::::::::::::::::::::::jy:jy;::::::::::::::::::::::::::::: ::::::::::::::::vI:vY::::::::::::::yvayA;:vmYtyyA::::::::::::::::::::: :::::::::::::::::::xCywivY:y]ykYsavyy=:::y^ylyyI:::::::::::::::::::::: :::::::::::::::::ZY:<:y_:J:ravyy=::::::::: ::::::::::::::::::::::::rivMvY:::::y^YlyyI::vMvY:::ravMxyy;::::vMxEyyy :::::::::::::::::::::::::::xEy^yyY:::::y^YlyyI::vMvY:::ravMxyy;::::vMx Eyyy:::::::::::::::::::::::::xEy^yyY:::::y^YlyyI::vMvY:::ravMxyy;::::v MxEyyy:jyB:vYyI:>:jywY:::::::::::::::::::::::::::::::::vMxAyyy::::::<: vY::Zy[y=:::;:xI:::::xCy^yyY::::::::::::::::::::::::::y^y[y=:::::B:jy; 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::ZysYryxI::::::::::::::::::<:ry::vIxCxI:::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::Z::xI::y]ytyxI:::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::Z::xI::y^ylIC:xyyryyA::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::: B" }{TEXT -1 2 "\"." }}{PARA 269 "" 0 "" {TEXT 737 13 "\nAt least one" }{TEXT -1 69 " of the implications \+ stated in the answers may be incorrect. Choose " }{TEXT 736 3 "any" } {TEXT -1 22 " incorrect statement" }}{PARA 270 "" 0 "" {TEXT -1 15 " as your answer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 456 106 "a) I => II b) II => III c) III => IV d) IV => I e) I => III" }{TEXT 457 21 " \+ \n" }{TEXT 459 47 "f) II => I g) I => I V " }{TEXT -1 3 " h)" }{TEXT 458 58 " II => IV \+ i) IV => III j) IV => II" }}{PARA 3 "" 0 "" {TEXT -1 0 " " }}{PARA 3 "" 0 "" {TEXT 742 8 "Solution" }{TEXT 743 6 ": (c)\n" }} {PARA 0 "" 0 "" {TEXT -1 107 "Properties 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 o thers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 272 1 " " }{TEXT 341 15 "14. Calculate " }{XPPEDIT 776 1 "Int(- y^3*dx+x^3,y=C..``)" "-%$IntG6$,&*&%\"yG\"\"$%#dxG\"\"\"!\"\"*$%\"xGF( F*/F';%\"CG%!G" }{TEXT 774 8 " where " }{TEXT 772 1 "C" }{TEXT 773 48 " is the unit circle traversed counter-clockwise." }{TEXT 771 2 " \+ \n" }{TEXT 758 4 "a) " }{XPPEDIT 777 1 "0" "\"\"!" }{TEXT 759 20 " \+ b) " }{XPPEDIT 778 1 "Pi/4" "*&%#PiG\"\"\"\"\"%!\"\"" } {TEXT 760 19 " c) " }{XPPEDIT 779 1 "Pi/2" "*&%#PiG\"\" \"\"\"#!\"\"" }{TEXT 761 20 " d) " }{XPPEDIT 780 1 "Pi " "I#PiG%*protectedG" }{TEXT 762 20 " e) " }{XPPEDIT 781 1 "2*Pi" "*&\"\"#\"\"\"%#PiGF$" }{TEXT 763 11 " \nf) " } {XPPEDIT 782 1 "Pi/3" "*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 764 18 " \+ g) " }{XPPEDIT 783 1 "Pi/6" "*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 765 19 " h) " }{XPPEDIT 784 1 "2*Pi/3" "*(\"\"#\"\"\"%# PiGF$\"\"$!\"\"" }{TEXT 766 17 " i) " }{XPPEDIT 785 1 "3* Pi/2" "*(\"\"$\"\"\"%#PiGF$\"\"#!\"\"" }{TEXT 767 18 " j) " }{XPPEDIT 786 1 "3*Pi/4" "*(\"\"$\"\"\"%#PiGF$\"\"%!\"\"" }{TEXT 768 5 " \n" }}{PARA 3 "" 0 "" {TEXT 788 8 "Solution" }{TEXT 789 6 " : (i)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r := t -> [cos(t) ,sin(t)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Int( -r(t)[2]^ 3*diff(r(t)[1],t) + r(t)[1]^3*diff(r(t)[2],t),t=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*$-%$sinG6#%\"tG\"\"%\"\"\"*$-%$cosG F*F,F-/F+;\"\"!,$%#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 ea sily 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(I nt(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. Let " } {TEXT 792 1 "G" }{TEXT -1 57 " be a solid region in space. Suppose th at the boundary " }{TEXT 791 1 "S" }{TEXT -1 6 " of " }{TEXT 793 1 "G" }{TEXT -1 59 " is an orientable closed surface with outward unit \+ normal " }{TEXT 790 1 "n" }{TEXT -1 42 ". 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If " }{XPPEDIT 659 1 "f( x,y,z)=x^2 - 2*y^2 + 3*z^2" "/-%\"fG6%%\"xG%\"yG%\"zG,(*$F&\"\"#\"\"\" *&F+F,*$F'F+F,!\"\"*&\"\"$F,*$F(F+F,F," }{TEXT 658 17 " then calculat e " }{TEXT 664 1 " " }{TEXT 662 11 "div(grad(f(" }{TEXT 665 5 "x,y,z" }{TEXT 666 2 "))" }{TEXT 663 2 ". " }{TEXT 520 2 " \n" }{TEXT 518 142 "\na) 1 b) 2 c) 3 d) 4 e) 5 \nf) 8 g) 12 h) 15 i) 16 j ) 20" }{TEXT 800 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 667 8 "Solution" }{TEXT 668 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#>%\"fG:6%%\"xG%\"yG%\"zG6\"6$%)operato rG%&arrowGF*,(*$9$\"\"#\"\"\"*$9%F1!\"#*$9&F1\"\"$F*F*" }}}{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!\"%,$%\"zG \"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diverge(\",[x,y,z] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 529 38 "17. Calculate the sur face integral of " }{TEXT 532 1 " " }{XPPEDIT 533 1 "f(x,y,z)=x+z" "/- %\"fG6%%\"xG%\"yG%\"zG,&F&\"\"\"F(F*" }{TEXT 531 31 " over that part \+ of the plane " }{TEXT 535 1 " " }{XPPEDIT 536 1 "x+y+z=1" "/,(%\"xG\" \"\"%\"yGF%%\"zGF%F%" }{TEXT 530 4 " \n" }{TEXT 631 30 "that lies in the first octant." }{TEXT 633 1 "\n" }{TEXT 632 39 "a) 0 b ) 1 c) " }{XPPEDIT 648 1 "sqrt(2)/2" "*&-%%sqrtG6#\"\"# \"\"\"F&!\"\"" }{TEXT 634 15 " d) " }{XPPEDIT 649 1 "sqrt(3 )/3" "*&-%%sqrtG6#\"\"$\"\"\"F&!\"\"" }{TEXT 635 13 " e) " } {XPPEDIT 650 1 "1/2" "*&\"\"\"F#\"\"#!\"\"" }{TEXT 637 7 " \nf) " } {XPPEDIT 651 1 "sqrt(2)" "-%%sqrtG6#\"\"#" }{TEXT 639 14 " g) \+ " }{XPPEDIT 652 1 "2*sqrt(3)/3" "*(\"\"#\"\"\"-%%sqrtG6#\"\"$F$F(!\" \"" }{TEXT 641 12 " h) " }{XPPEDIT 653 1 "2*sqrt(2)" "*&\"\"# \"\"\"-%%sqrtG6#F#F$" }{TEXT 643 12 " i) " }{XPPEDIT 654 1 "sq rt(3)" "-%%sqrtG6#\"\"$" }{TEXT 646 14 " j) " }{XPPEDIT 655 1 "3/2" "*&\"\"$\"\"\"\"\"#!\"\"" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT 656 8 "Solution" }{TEXT 657 6 ": (d)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "wi th(linalg):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "vectorLength := v -> sqrt(sum(v[i]^2,i=1..vectdim( v)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-vectorLengthG:6#%\"vG6\"6$ %)operatorG%&arrowGF(-%%sqrtG6#-%$sumG6$*$&9$6#%\"iG\"\"#/F6;\"\"\"-%( vectdimG6#F4F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "r := (x ,y) -> vector([x,y,1-(x+y)]); #surface parameterization" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"rG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)-%' vectorG6#7%9$9%,(\"\"\"F4F1!\"\"F2F5F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "rx := map(z->diff(z,x),r(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rxG-%'VECTORG6#7%\"\"\"\"\"!!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ry := map(z->diff(z,y),r(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ryG-%'VECTORG6#7%\"\"!\"\"\"!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "crossprod(rx,ry);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"\"F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "surfaceAreaFactor := vectorLength(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%2surfaceAreaFactorG*$\"\"$#\"\"\"\" \"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := (x,y,z) -> x + \+ z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6%%\"xG%\"yG%\"zG6\"6$%)o peratorG%&arrowGF*,&9$\"\"\"9&F0F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Int(Int(f(x,y,z)*surfaceAreaFactor,y=0..1-x),x=0..1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$*&,&%\"xG\"\"\"%\"zG F+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 539 8 "18. 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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 integral (integrate d over the aquamarine paraboloid) and the flux integral through the bo ttom 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 604 4 " - " }{TEXT 603 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 608 3 " " }{TEXT 610 5 "z = 1" }{TEXT -1 13 ". 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