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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 "" 3 284 1 {CSTYLE "" -1 -1 "" 1 14 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 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 405 4 "Math" }{TEXT 652 1 " " } {TEXT 797 3 "233" }{TEXT 651 22 " \nExam 2 Spring 2004" }}{PARA 0 " " 0 "" {TEXT 260 2 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 695 41 "1. T he position vector of a particle is " }{TEXT 699 2 " " }{XPPEDIT 702 1 "r(t)=`<`*sqrt(7)*t, t^3,2*t+ln(t)*`>`" "6%/-%\"rG6#%\"tG*(%\"GF* F*" }{TEXT 697 1 " " }{TEXT 700 22 ". What is its speed at" }{TEXT 701 2 " " }{XPPEDIT 703 1 "t=1" "/%\"tG\"\"\"" }{TEXT 698 1 " " } {TEXT 704 2 "? " }{TEXT 705 2 " " }}{PARA 277 "" 0 "" {TEXT 696 4 "a) " }{XPPEDIT 19 1 "1" "\"\"\"" }{TEXT 716 14 " b) " } {XPPEDIT 19 1 "2" "\"\"#" }{TEXT 715 14 " c) " }{XPPEDIT 19 1 "3" "\"\"$" }{TEXT 714 14 " d) " }{XPPEDIT 19 1 "4" "\"\"% " }{TEXT 713 18 " e) " }{XPPEDIT 19 1 "5" "\"\"&" } {TEXT 712 15 " \nf) " }{XPPEDIT 19 1 "6" "\"\"'" }{TEXT 711 14 " g) " }{XPPEDIT 19 1 "7" "\"\"(" }{TEXT 710 14 " \+ h) " }{XPPEDIT 19 1 "8" "\"\")" }{TEXT 709 14 " i) " } {XPPEDIT 19 1 "9" "\"\"*" }{TEXT 708 17 " j) " }{XPPEDIT 19 1 "10" "\"#5" }{TEXT 707 1 " " }{TEXT 706 7 " \n" }}{PARA 0 " " 0 "" {TEXT 717 12 "Solution: e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "magnitude := r -> sqrt(r[1]^ 2+r[2]^2+r[3]^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*magnitudeG:6#% \"rG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,(*$&9$6#\"\"\"\"\"#F4*$&F26#F 5F5F4*$&F26#\"\"$F5F4F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "x := t -> sqrt(7)*t: y := t -> t^3: z := t -> 2*t+ln(t):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "velocity := map(u->diff(u,t) , [x(t),y(t),z(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)velocityG7% *$\"\"(#\"\"\"\"\"#,$*$%\"tGF*\"\"$,&F*F)*$F-!\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(t=1, \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$\"\"(#\"\"\"\"\"#\"\"$F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "magnitude(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 263 "" 0 "" {TEXT -1 1 "2" }{TEXT 772 1 "." }{TEXT -1 2 " " }{TEXT 745 73 "This problem and \+ the next refer to a space curve that is parameterized by" }{TEXT 746 3 " " }{XPPEDIT 744 1 "r(t) = `<`*t^3+t,sqrt(2)*t^2,exp(t-1)*`>`" "6 %/-%\"rG6#%\"tG,&*&%\"GF+" }{TEXT 741 1 " " }{TEXT 743 9 " . Let " }{TEXT 759 1 " " }{XPPEDIT 758 1 "`<`*a,b,c*`>`" "6%*&%\"GF%" }{TEXT 757 50 " denote the unit \+ tangent vector to this curve at" }{TEXT 754 2 " " }{XPPEDIT 755 1 "t= 1" "/%\"tG\"\"\"" }{TEXT 742 3 ". " }{TEXT -1 8 "What is " }{TEXT 760 1 " " }{XPPEDIT 761 1 "a/c" "*&%\"aG\"\"\"%\"cG!\"\"" }{TEXT -1 3 " ? " }}{PARA 280 "" 0 "" {TEXT 723 4 "a) " }{XPPEDIT 718 0 "-1" ",$ \"\"\"!\"\"" }{TEXT 719 9 " " }{TEXT -1 1 "b" }{TEXT 781 2 ") \+ " }{TEXT 748 1 " " }{XPPEDIT 724 0 "-2" ",$\"\"#!\"\"" }{TEXT 725 11 " " }{TEXT 747 1 "c" }{TEXT 782 3 ") " }{XPPEDIT 726 0 "-3" ",$\"\"$!\"\"" }{TEXT 727 10 " " }{TEXT 749 2 " " }{TEXT -1 1 "d" }{TEXT 784 3 ") " }{XPPEDIT 728 0 "-4" ",$\"\"%!\"\"" }{TEXT 729 5 " " }{TEXT 750 5 " " }{TEXT -1 1 "e" }{TEXT 785 1 ")" } {TEXT 751 2 " " }{XPPEDIT 752 0 "-5" ",$\"\"&!\"\"" }{TEXT 753 3 " \+ " }{TEXT 732 2 " " }}{PARA 278 "" 0 "" {TEXT 722 4 "f) " }{TEXT 720 2 " " }{XPPEDIT 730 0 "1" "\"\"\"" }{TEXT 731 18 " g) \+ " }{XPPEDIT 733 0 "2" "\"\"#" }{TEXT 734 21 " h) " } {XPPEDIT 735 0 "3" "\"\"$" }{TEXT 736 23 " i) " } {XPPEDIT 737 0 "4" "\"\"%" }{TEXT 721 22 " j) " } {XPPEDIT 739 0 "5" "\"\"&" }{TEXT 740 1 " " }{TEXT 738 2 " " }}{PARA 3 "" 0 "" {TEXT 756 12 "Solution: e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "x := t -> t^3+t: y := t -> sqrt(2)*t^2: z := t -> exp(t-1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "velocity := map(u -> diff(u,t), [x(t),y(t),z(t)]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)velocityG7%,&*$%\"tG\"\"#\"\"$\"\" \"F+,$*&F)#F+F)F(F+F)-%$expG6#,&F(F+!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "unitTangent := map( u -> u/magnitude(velocity) , v elocity);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,unitTangentG7%*&,&*$% \"tG\"\"#\"\"$\"\"\"F,F,,(*$F'F*F,F(\"\")*$-%$expG6#,&F)F,!\"\"F,F*F,# F5F*,$*(F*#F,F*F)F,F-F6F**&F1F,F-F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(t=1,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$ *$,&\"#C\"\"\"*$-%$expG6#\"\"!\"\"#F(#!\"\"F.\"\"%,$*&F.#F(F.F&F/F.*&F *F(F&F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify( \" ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%#\"\"%\"\"&,$*$\"\"##\"\"\"F)#F) F&#F+F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "\"[1]/\"[3];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 263 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 43 "3. Referring to the unit tangent vector " }{XPPEDIT 777 1 "`<`* a,b,c*`>`" "6%*&%\"GF%" }{TEXT 778 39 " of the preceding problem, what is " }{XPPEDIT 771 1 "b" "I\"b G6\"" }{TEXT 780 2 " ?" }{TEXT -1 1 " " }{TEXT 324 2 " " }}{PARA 264 "" 0 "" {TEXT 274 4 "a) " }{XPPEDIT 19 1 "sqrt(2)" "-%%sqrtG6#\"\"#" }{TEXT 381 14 " b) " }{XPPEDIT 19 1 "2" "\"\"#" }{TEXT 380 23 " c) " }{XPPEDIT 19 1 "2*sqrt(2)" "*&\"\"#\"\"\" -%%sqrtG6#F#F$" }{TEXT 379 16 " d) " }{XPPEDIT 19 1 "sqrt( 2)/2" "*&-%%sqrtG6#\"\"#\"\"\"F&!\"\"" }{TEXT 378 20 " e) " }{XPPEDIT 19 1 "sqrt(2)/3" "*&-%%sqrtG6#\"\"#\"\"\"\"\"$!\"\"" } {TEXT 377 15 " \nf) " }{XPPEDIT 19 1 "2/3" "*&\"\"#\"\"\"\" \"$!\"\"" }{TEXT 376 17 " g) " }{XPPEDIT 19 1 "2*sqrt(2)/ 3" "*(\"\"#\"\"\"-%%sqrtG6#F#F$\"\"$!\"\"" }{TEXT 375 14 " h) " }{XPPEDIT 19 1 "sqrt(2)/5" "*&-%%sqrtG6#\"\"#\"\"\"\"\"&!\"\"" } {TEXT 374 18 " i) " }{XPPEDIT 19 1 "2/5" "*&\"\"#\"\"\" \"\"&!\"\"" }{TEXT 373 25 " j) " }{XPPEDIT 19 1 " 2*sqrt(2)/5" "*(\"\"#\"\"\"-%%sqrtG6#F#F$\"\"&!\"\"" }{TEXT 372 1 " " }{TEXT 325 7 " \n" }}{PARA 0 "" 0 "" {TEXT 655 12 "Solution: j" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 330 36 "4. A space curve is parameterized by" }{TEXT 793 3 " " }{XPPEDIT 791 1 "r(t) = `<`*2*t^2,t^4,t^2*`>`" "6%/-%\"rG6#%\"tG*(%\"GF*" }{TEXT 789 1 " " }{TEXT 790 66 ". W hat is the second component of its principal unit normal at " } {XPPEDIT 796 1 "t=1" "/%\"tG\"\"\"" }{TEXT 795 91 "? (You may need t o use some of your computations for the question that follows this one .)" }}{PARA 281 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 19 1 "sqrt(5)/3" " *&-%%sqrtG6#\"\"&\"\"\"\"\"$!\"\"" }{TEXT -1 16 " b) " } {XPPEDIT 19 1 "2*sqrt(5)/3" "*(\"\"#\"\"\"-%%sqrtG6#\"\"&F$\"\"$!\"\" " }{TEXT -1 14 " c) " }{XPPEDIT 19 1 "sqrt(5)" "-%%sqrtG6#\" \"&" }{TEXT -1 13 " d) " }{XPPEDIT 19 1 "sqrt(5)/5" "*&-%%sqr tG6#\"\"&\"\"\"F&!\"\"" }{TEXT -1 17 " e) " }{XPPEDIT 19 1 "2*sqrt(5)/5" "*(\"\"#\"\"\"-%%sqrtG6#\"\"&F$F(!\"\"" }}{PARA 282 " " 0 "" {TEXT -1 5 "f) " }{XPPEDIT 19 1 "3*sqrt(5)/5" "*(\"\"$\"\"\"- %%sqrtG6#\"\"&F$F(!\"\"" }{TEXT -1 12 " g) " }{XPPEDIT 19 1 "s qrt(5)/15" "*&-%%sqrtG6#\"\"&\"\"\"\"#:!\"\"" }{TEXT -1 16 " \+ h) " }{XPPEDIT 19 1 "2*sqrt(5)/15" "*(\"\"#\"\"\"-%%sqrtG6#\"\"&F$ \"#:!\"\"" }{TEXT -1 16 " i) " }{XPPEDIT 19 1 "4*sqrt(5)/1 5" "*(\"\"%\"\"\"-%%sqrtG6#\"\"&F$\"#:!\"\"" }{TEXT -1 14 " j ) " }{XPPEDIT 19 1 "sqrt(5)/25" "*&-%%sqrtG6#\"\"&\"\"\"\"#D!\"\"" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 794 12 "Solution: a" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "x := t -> 2*t^2: y := t -> t^4: z := t -> t^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "vel ocity := map(u->diff(u,t), [x(t),y(t),z(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)velocityG7%,$%\"tG\"\"%,$*$F'\"\"$F(,$F'\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "unitTangent := map( u -> u/m agnitude(velocity) , velocity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%, unitTangentG7%,$*&%\"tG\"\"\",&*$F(\"\"#\"\"&*$F(\"\"'\"\"%#!\"\"F,F,, $*&F(\"\"$F*F1F,F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "N := \+ map(u->diff(u,t), unitTangent);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% \"NG7%,&*$,&*$%\"tG\"\"#\"\"&*$F*\"\"'\"\"%#!\"\"F+F+*(F*\"\"\"F(#!\"$ F+,&F*\"#5*$F*F,\"#CF3F1,&*&F*F+F(F0F.*(F*\"\"$F(F4F6F3F1,&F'F3F2F0" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "N := simplify(subs(t=1,\") );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG7%#!#;\"#F#\"#?F(#!\")F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "principalUnitNormal := m ap( u -> u/magnitude(N), N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%4pri ncipalUnitNormalG7%,$*$\"\"&#\"\"\"\"\"##!\"%\"#:,$F'#F*\"\"$,$F'#!\"# F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 269 1 " " }{TEXT 382 2 "5." }{TEXT -1 1 " " }{TEXT 798 86 "Referring t o the curve of question 4, what is the curvature at the point for w hich " }{TEXT 800 1 " " }{XPPEDIT 801 1 "t=1" "/%\"tG\"\"\"" }{TEXT 799 1 "?" }}{PARA 265 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 813 7 "a) " }{XPPEDIT 19 1 "sqrt(5)/3" "*&-%%sqrtG6#\"\"&\"\"\"\" \"$!\"\"" }{TEXT -1 5 " " }{TEXT 814 11 " b) " }{XPPEDIT 19 1 "2*sqrt(5)/3" "*(\"\"#\"\"\"-%%sqrtG6#\"\"&F$\"\"$!\"\"" }{TEXT -1 5 " " }{TEXT 815 14 " c) " }{XPPEDIT 19 1 "sqrt(5)/27 " "*&-%%sqrtG6#\"\"&\"\"\"\"#F!\"\"" }{TEXT -1 5 " " }{TEXT 816 13 " d) " }{XPPEDIT 19 1 "2*sqrt(5)/27" "*(\"\"#\"\"\"-%%sqrt G6#\"\"&F$\"#F!\"\"" }{TEXT -1 5 " " }{TEXT 817 13 " e) \+ " }{XPPEDIT 19 1 "sqrt(5)/81" "*&-%%sqrtG6#\"\"&\"\"\"\"#\")!\"\"" } {TEXT -1 5 " " }{TEXT 818 11 " \n\n f) " }{XPPEDIT 19 1 "2*sqr t(5)/81" "*(\"\"#\"\"\"-%%sqrtG6#\"\"&F$\"#\")!\"\"" }{TEXT 819 15 " \+ g) " }{XPPEDIT 19 1 "2*sqrt(5)/5" "*(\"\"#\"\"\"-%%sqrtG6#\" \"&F$F(!\"\"" }{TEXT 820 18 " h) " }{XPPEDIT 19 1 "3*sqr t(5)/5" "*(\"\"$\"\"\"-%%sqrtG6#\"\"&F$F(!\"\"" }{TEXT 821 14 " \+ i) " }{XPPEDIT 19 1 "6*sqrt(5)/5" "*(\"\"'\"\"\"-%%sqrtG6#\"\"&F$F (!\"\"" }{TEXT 822 16 " j) " }{XPPEDIT 19 1 "12*sqrt(5)/5 " "*(\"#7\"\"\"-%%sqrtG6#\"\"&F$F(!\"\"" }{TEXT 823 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 657 13 "Solution: d " }{TEXT 812 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "velocity;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$%\"tG\"\"%,$*$F%\"\"$F&,$F%\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "acceleration := map(u->diff( u,t), velocity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-accelerationG7% \"\"%,$*$%\"tG\"\"#\"#7F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definit ion for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition f or trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "magnitude(cross prod(velocity,acceleration))/magnitude(velocity)^3;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*(\"\"&#\"\"\"\"\"#*$%\"tG\"\"'F&,&*$F*F(F%F)\"\"%# !\"$F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(t=1,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"&#\"\"\"\"\"#\"\"*F&#F(\"#\" )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"\"&#\"\"\"\"\"##F(\"#F" }}} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " } {TEXT 265 61 "6. What is the length of the curve that is parameterize d by " }{TEXT 844 1 " " }{XPPEDIT 843 1 "r(t)=`<`*sqrt(2)*t*` `,t^2/2, ln(t)*`>` " "6%/-%\"rG6#%\"tG**%\"GF*" }{TEXT 845 1 " " }{TEXT 826 1 " " }{TEXT 827 5 "for " }{XPPEDIT 846 1 "t" "I\"tG6\"" }{TEXT 832 1 " " } {TEXT 847 18 " in the interval " }{TEXT 830 2 " " }{XPPEDIT 828 1 "[ 1,2]" "7$\"\"\"\"\"#" }{TEXT 829 3 " ?" }{TEXT 825 1 "\n" }}{PARA 0 " " 0 "" {TEXT 831 5 "a) " }{XPPEDIT 19 1 "sqrt(2)" "-%%sqrtG6#\"\"#" }{TEXT 833 21 " b) " }{XPPEDIT 19 1 "2*sqrt(2)" "*&\" \"#\"\"\"-%%sqrtG6#F#F$" }{TEXT 834 14 " c) " }{XPPEDIT 19 1 "3*sqrt(2)" "*&\"\"$\"\"\"-%%sqrtG6#\"\"#F$" }{TEXT 835 15 " \+ d) " }{XPPEDIT 19 1 "sqrt(2)/2" "*&-%%sqrtG6#\"\"#\"\"\"F&!\"\"" } {TEXT 836 15 " e) " }{XPPEDIT 19 1 "3*sqrt(2)/2" "*(\"\"$\" \"\"-%%sqrtG6#\"\"#F$F(!\"\"" }{TEXT 837 8 " \n\n f) " }{XPPEDIT 19 1 "3/2 +ln(2)" ",&*&\"\"$\"\"\"\"\"#!\"\"F%-%#lnG6#F&F%" }{TEXT 838 9 " g) " }{XPPEDIT 19 1 "3 + ln(2)" ",&\"\"$\"\"\"-%#lnG6#\"\"#F$" }{TEXT 839 9 " h) " }{XPPEDIT 19 1 "2+ln(3)" ",&\"\"#\"\"\"-%#lnG 6#\"\"$F$" }{TEXT 840 10 " i) " }{XPPEDIT 19 1 "3+ln(3)" ",&\"\" $\"\"\"-%#lnG6#F#F$" }{TEXT 841 9 " j) " }{XPPEDIT 19 1 "2*ln(3) " "*&\"\"#\"\"\"-%#lnG6#\"\"$F$" }{TEXT 842 2 " \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 848 12 "Solution: f" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "x := t -> sqrt(2)*t: y := t -> t^2/2 : z := t -> ln(t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "ve locity := map(u->diff(u,t), [x(t),y(t),z(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)velocityG7%*$\"\"##\"\"\"F'%\"tG*$F*!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "arcLength := Int(magnitude(v elocity),t=1..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*arcLengthG-%$I ntG6$*$,(*$%\"tG\"\"#\"\"\"F,F-*$F+!\"#F-#F-F,/F+;F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "value(arcLength);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"\"$\"\"#\"\"\"-%#lnG6#F&F'" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{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 -1 1 " " }{TEXT 267 7 "7. If " }{TEXT 851 1 " " }{XPPEDIT 849 1 "r(t)=`<`*t^6,6*t,t^3*`>` " "6%/-% \"rG6#%\"tG*&%\"GF*" }{TEXT 850 1 " " }{TEXT 852 79 " is the position vector of a particle, then \+ what is its tangential component, " }{TEXT 860 1 " " }{XPPEDIT 861 1 " a[T]" "&%\"aG6#%\"TG" }{TEXT 858 23 " , of acceleration at " }{TEXT 856 1 " " }{XPPEDIT 855 1 "t=1" "/%\"tG\"\"\"" }{TEXT 857 4 " ? " } {TEXT -1 1 " " }{TEXT 865 87 "(You may want to use some of your comput ations for the question that follows this one.)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 854 168 "a) 4 \+ b) 6 c) 8 d) 10 e) 12\n\nf) 14 g) 16 h) 18 i) 20 \+ j) 22 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 859 12 "Solution: j" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "x := t -> t^6: y := t -> 6*t: z := t -> t^3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "velocity := map(u -> diff(u,t), [x( t),y(t),z(t)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)velocityG7%,$*$% \"tG\"\"&\"\"'F*,$*$F(\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "speed := magnitude(velocity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&speedG,$*$,(*$%\"tG\"#5\"\"%F+\"\"\"*$F)F+F,#F,\"\"# \"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a[T] = diff(speed, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%\"TG,$*&,(*$%\"tG\"# 5\"\"%F.\"\"\"*$F,F.F/#!\"\"\"\"#,&*$F,\"\"*\"#S*$F,\"\"$F.F/#F9F3" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(t=1, \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%\"TG,$*$\"\"*#\"\"\"\"\"##\"#A\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#%\"TG\"#A" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 348 38 "8. Referring to the position vector \+ " }{TEXT 868 1 " " }{XPPEDIT 866 1 "r(t)=`<`*t^6,6*t,t^3*`>` " "6%/-% \"rG6#%\"tG*&%\"GF*" }{TEXT 867 1 " " }{TEXT 869 59 " of the preceding problem, what is the nor mal component " }{TEXT 862 1 " " }{TEXT 864 1 " " }{XPPEDIT 863 1 "a[N ]" "&%\"aG6#%\"NG" }{TEXT 404 23 " of acceleration at " }{TEXT 364 1 " " }{XPPEDIT 363 1 "t=1" "/%\"tG\"\"\"" }{TEXT 365 3 " ?" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 349 4 "a) " }{XPPEDIT 19 1 "sqrt(113)" "-%%sqrtG6#\"$8\"" }{TEXT 873 15 " b) " }{XPPEDIT 19 1 "2*sqrt(113)" "*&\"\"#\"\"\"-%%sqrtG6# \"$8\"F$" }{TEXT 872 14 " c) " }{XPPEDIT 19 1 "3*sqrt(113)" "*&\"\"$\"\"\"-%%sqrtG6#\"$8\"F$" }{TEXT 875 15 " d) " } {XPPEDIT 19 1 "4*sqrt(113)" "*&\"\"%\"\"\"-%%sqrtG6#\"$8\"F$" }{TEXT 876 14 " e) " }{XPPEDIT 19 1 "5*sqrt(113)" "*&\"\"&\"\"\"-%% sqrtG6#\"$8\"F$" }{TEXT 877 12 " \n\nf) " }{XPPEDIT 19 1 "6*sqrt (113)" "*&\"\"'\"\"\"-%%sqrtG6#\"$8\"F$" }{TEXT 879 11 " g) " } {XPPEDIT 19 1 "7*sqrt(113)" "*&\"\"(\"\"\"-%%sqrtG6#\"$8\"F$" }{TEXT 881 14 " h) " }{XPPEDIT 19 1 "8*sqrt(113)" "*&\"\")\"\"\"-%% sqrtG6#\"$8\"F$" }{TEXT 883 17 " i) " }{XPPEDIT 19 1 "9*s qrt(113)" "*&\"\"*\"\"\"-%%sqrtG6#\"$8\"F$" }{TEXT 885 15 " \+ j) " }{XPPEDIT 19 1 "10*sqrt(113)" "*&\"#5\"\"\"-%%sqrtG6#\"$8\"F$" } {TEXT 887 11 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 660 12 "Solution: b" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "velocity;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*$%\"tG\"\"&\"\"'F (,$*$F&\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "accele ration := map(u -> diff(u,t), velocity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-accelerationG7%,$*$%\"tG\"\"%\"#I\"\"!,$F(\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "acceleration := subs(t=1, accelerat ion);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-accelerationG7%\"#I\"\"!\" \"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sqrt(magnitude(accel eration)^2-22^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"$8\"#\"\"\" \"\"#F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "magnitude(cr ossprod(velocity,acceleration))/magnitude(velocity)^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,*\"$/\"\"\"\"*$%\"tG\"\"%\"#D*$F)\"\"(!#?*$F) \"#5F*#F'\"\"#,(F/F*F*F'F(F'#!\"$F2#F2\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(t=1,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*&\"$8\"#\"\"\"\"\"#\"\"*F&#F(\"$V#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "curvature := simplify(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*curvatureG,$*$\"$8\"#\"\"\"\"\"##F*\"#\")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "curvature*subs(t=1, magnitud e(velocity) )^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$\"$8\"#\"\"\" \"\"#F(" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 0 {PARA 3 "" 0 "" {TEXT 270 55 "9. At an instant of time a par ticle's acceleration is " }{XPPEDIT 406 1 "`<`*14,7,0*`>`" "6%*&%\"GF%" }{TEXT 407 38 " and its principal \+ unit normal is\n " }{TEXT 445 1 " " }{XPPEDIT 444 1 "N=`< `*2*`/`*7, 3*`/`*7,6*`/`*7*`>`" "6%/%\"NG**%#<~G\"\"\"\"\"#F'%\"/GF'\"\"(F'*(\" \"$F'F)F'F*F'**\"\"'F'F)F'F*F'%\">GF'" }{TEXT 446 50 ". What is its normal component of acceleration " }{TEXT 442 1 " " }{XPPEDIT 443 1 " a[N]" "&%\"aG6#%\"NG" }{TEXT 441 3 "?\n " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 447 162 "a) 0 b) 1 \+ c) 2 d) 3 e) 4\n\nf) 5 g) \+ 6 h) 7 i) 8 j) 9 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 661 12 "Solution: h" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a := [14,7,0]: N := [2/7,3/7,6/7]:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "dotprod(a,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"(" }}}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 276 "" 0 "" {TEXT -1 4 "10. " }{TEXT 671 75 "A part icle is moving along a space curve in such a way that its speed is \+ " }{XPPEDIT 672 1 "1+ t^6" ",&\"\"\"F#*$%\"tG\"\"'F#" }{TEXT 673 142 " at time t. The curvature of the particle's path is 2 at time \+ t = 1. What is the magnitude of its acceleration vector at that time? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 674 5 "a) " }{XPPEDIT 19 1 "3*sqrt(3)" "*&\"\" $\"\"\"-%%sqrtG6#F#F$" }{TEXT 676 18 " b) " }{XPPEDIT 19 1 "5*sqrt(3)" "*&\"\"&\"\"\"-%%sqrtG6#\"\"$F$" }{TEXT 677 16 " \+ c) " }{XPPEDIT 19 1 "10*sqrt(3)" "*&\"#5\"\"\"-%%sqrtG6#\"\"$F $" }{TEXT 678 20 " d) " }{XPPEDIT 19 1 "2*sqrt( 5)" "* &\"\"#\"\"\"-%%sqrtG6#\"\"&F$" }{TEXT 675 20 " e) " } {XPPEDIT 19 1 "3*sqrt(5)" "*&\"\"$\"\"\"-%%sqrtG6#\"\"&F$" }{TEXT 679 12 " \n\nf) " }{XPPEDIT 19 1 "3*sqrt(10)" "*&\"\"$\"\"\"-%%sqrtG 6#\"#5F$" }{TEXT 680 17 " g) " }{XPPEDIT 19 1 "4*sqrt(10) " "*&\"\"%\"\"\"-%%sqrtG6#\"#5F$" }{TEXT 681 20 " h) \+ " }{XPPEDIT 19 1 "8" "\"\")" }{TEXT 682 15 " i) " } {XPPEDIT 19 1 "10" "\"#5" }{TEXT 683 19 " j) " } {XPPEDIT 19 1 "12" "\"#7" }{TEXT 684 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 685 12 "Solution: i" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "v := 1+ t^6; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"vG,&\"\"\"F&*$%\"tG\"\"'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a_T := diff(v,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a_TG,$ *$%\"tG\"\"&\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "a_T := subs(t = 1, a_T );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a_TG\"\"'" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a_N := kappa*v^2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a_NG*&%&kappaG\"\"\",&F'F'*$%\"tG\" \"'F'\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a_N := subs( \{t=1,kappa=2\}, a_N );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a_NG\"\" )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "acc := sqrt(a_T^2 + a_ N^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$accG\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 " " 0 "" {TEXT 272 1 " " }{TEXT 353 9 "11. Let" }{TEXT 466 2 " " } {XPPEDIT 467 1 "f(x,y)=x^2*y/(x^4+2*y^2)" "/-%\"fG6$%\"xG%\"yG*(F&\"\" #F'\"\"\",&*$F&\"\"%F**&F)F**$F'F)F*F*!\"\"" }{TEXT 465 49 " for all \+ points other than the origin. Let " }{TEXT 469 1 " " }{XPPEDIT 470 1 "r(t)=`<`*t,t^2*`>`" "6$/-%\"rG6#%\"tG*&%\"GF*" }{TEXT 468 2 ". " }{TEXT 354 3 " \n\n" }{TEXT 472 13 "Calcula te " }{XPPEDIT 473 1 "limit(f(r(t)),t=0)" "-%&limitG6$-%\"fG6#-%\"r G6#%\"tG/F+\"\"!" }{TEXT 471 2 ".\n" }{TEXT 474 1 "\n" }}{PARA 0 "" 0 "" {TEXT 479 4 "a) " }{XPPEDIT 475 0 "1/2" "*&\"\"\"F#\"\"#!\"\"" } {TEXT 476 8 " " }{TEXT 488 3 "b) " }{TEXT 489 1 " " }{XPPEDIT 480 0 "1/3" "*&\"\"\"F#\"\"$!\"\"" }{TEXT 481 8 " " }{TEXT 499 1 "c" }{TEXT 500 2 ") " }{XPPEDIT 482 0 "2/3" "*&\"\"#\"\"\"\"\"$!\"\" " }{TEXT 483 7 " " }{TEXT 490 7 " d) " }{XPPEDIT 484 0 "1" " \"\"\"" }{TEXT 485 5 " " }{TEXT 491 8 " e)" }{TEXT 492 2 " \+ " }{TEXT 486 2 "0 " }}{PARA 270 "" 0 "" {TEXT 478 4 "f) " }{XPPEDIT 493 0 "4/5" "*&\"\"%\"\"\"\"\"&!\"\"" }{TEXT 494 16 " g) \+ " }{XPPEDIT 495 0 "3/5" "*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT 496 17 " \+ h) " }{XPPEDIT 497 0 "2/5" "*&\"\"#\"\"\"\"\"&!\"\"" }{TEXT 498 15 " i) " }{XPPEDIT 487 0 "1/5" "*&\"\"\"F#\"\"&!\"\"" }{TEXT 477 31 " j) Does not exist " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 275 "" 0 "" {TEXT -1 14 "Solution: b \n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f := (x,y) -> x^2*y/(x^4+2*y^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "r := t -> [t,t^2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "limit(f(r(t)[1],r(t)[2]),t=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}} {SECT 0 {PARA 266 "" 0 "" {TEXT -1 20 "12. Suppose that " } {XPPEDIT 19 1 "f(x,y)= x*y^2-8*y/x" "/-%\"fG6$%\"xG%\"yG,&*&F&\"\"\"*$ F'\"\"#F*F**(\"\")F*F'F*F&!\"\"F/" }{TEXT -1 16 " . Calculate " } {XPPEDIT 19 1 "Diff(f(x,y),x)" "-%%DiffG6$-%\"fG6$%\"xG%\"yGF(" } {TEXT -1 7 " at " }{XPPEDIT 19 1 "``(2,-1)" "-%!G6$\"\"#,$\"\"\"!\" \"" }{TEXT -1 1 "." }}{PARA 258 "" 0 "" {TEXT 501 5 "\na) " } {XPPEDIT 502 1 "-4" ",$\"\"%!\"\"" }{TEXT 503 9 " b) " }{XPPEDIT 504 1 "-3" ",$\"\"$!\"\"" }{TEXT 505 9 " c) " }{XPPEDIT 506 1 "-2 " ",$\"\"#!\"\"" }{TEXT 507 7 " d) " }{XPPEDIT 508 1 "-1" ",$\"\"\" !\"\"" }{TEXT 509 9 " e) " }{XPPEDIT 510 1 "0" "\"\"!" }{TEXT 511 9 " f) " }{XPPEDIT 512 1 "1" "\"\"\"" }{TEXT 513 10 " g) " }{XPPEDIT 514 1 "2" "\"\"#" }{TEXT 515 9 " h) " }{XPPEDIT 516 1 "3" "\"\"$" }{TEXT 517 8 " i) " }{XPPEDIT 518 1 "4" "\"\"%" }{TEXT 519 9 " j) " }{XPPEDIT 521 1 "5" "\"\"&" }{TEXT -1 1 " " } {TEXT 520 3 " " }{TEXT -1 1 " " }}{PARA 3 "" 0 "" {TEXT 355 3 " " }{TEXT 356 1 "\n" }{TEXT -1 14 "Solution: d \n" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "diff( x*y^2 - 8*y/x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&* $%\"yG\"\"#\"\"\"*&F%F'%\"xG!\"#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(\{x=2,y=-1\}, \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 359 22 "13. Suppose \+ that " }{XPPEDIT 523 1 "f(x,y) = x^2 + y^3+ x^3*y^2" "/-%\"fG6$%\"x G%\"yG,(*$F&\"\"#\"\"\"*$F'\"\"$F+*&F&F-F'F*F+" }{TEXT 524 15 " . Cal culate " }{XPPEDIT 525 1 "Diff(f(x,y),x,x)*Diff(f(x,y),y,y)-(Diff(f(x ,y),x,y))^2" ",&*&-%%DiffG6%-%\"fG6$%\"xG%\"yGF*F*\"\"\"-F%6%-F(6$F*F+ F+F+F,F,*$-F%6%-F(6$F*F+F*F+\"\"#!\"\"" }{TEXT 526 182 " at the point \+ (1,-1). \n\na) 86 b) 68 c) 44 d) 21 e) 8 \nf) -8 g) -21 h) -44 \+ i) -68 j) -86" }{TEXT 357 1 "\n" }}{PARA 0 "" 0 "" {TEXT 350 13 "Solution: i " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f := (x,y) -> x^2 + y^3+ x^3*y^2: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "A := diff(f(x,y),x,x)*d iff(f(x,y),y,y)-(diff(f(x,y),x,y))^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,&*&,&\"\"#\"\"\"*&%\"xGF)%\"yGF(\"\"'F),&F,F-*$F+\"\"$F(F)F) *&F+\"\"%F,F(!#O" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A := expand(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,*%\"yG\"#7*$%\"xG\"\"$\"\"%*&F)\"\"\"F&F*\"#O*&F)F+F&\"\" #!#C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(\{x=1,y=-1\},A );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#o" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 267 "" 0 "" {TEXT -1 11 "14. If " } {XPPEDIT 19 1 "x +y^2+z^3+4=2*x*y*z" "/,*%\"xG\"\"\"*$%\"yG\"\"#F%*$% \"zG\"\"$F%\"\"%F%**F(F%F$F%F'F%F*F%" }{TEXT -1 20 " then calculate \+ " }{XPPEDIT 19 1 "diff( z ,x)" "-%%diffG6$%\"zG%\"xG" }{TEXT -1 10 " at " }{XPPEDIT 19 1 "``(3,2,1)" "-%!G6%\"\"$\"\"#\"\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 178 "a) 1/6 b) -5/6 c) -2/3 \+ d) 1/3 e) -1 \nf) 2/3 g) -1/3 \+ h) 1 i) -1/6 j) 5/6 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 662 13 "Solution: g " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eqn1 := x +y^2+z(x,y)^3+4=2*x*y*z(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,*%\"xG\"\"\"*$%\"yG\"\"#F(*$-%\"zG6$F'F*\"\"$ F(\"\"%F(,$*(F'F(F*F(F-F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn2 := map(u -> diff(u,x),eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/,&\"\"\"F'*&-%\"zG6$%\"xG%\"yG\"\"#-%%diffG6$F)F,F'\"\"$,& *&F-F'F)F'F.*(F,F'F-F'F/F'F." }}}{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,,$*&, &!\"\"\"\"\"*&F-F2F)F2\"\"#F2,&*$F)F4!\"$*&F,F2F-F2F4F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(\{x=3,y=2\}, rhs(eqn3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&!\"\"\"\"\"-%\"zG6$\"\"$\"\"#\" \"%F',&*$F(F,!\"$\"#7F'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 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 273 15 "15. The plane \+ " }{XPPEDIT 533 1 "x=2" "/%\"xG\"\"#" }{TEXT 532 2 " " }{TEXT 534 24 " intersects the graph of" }{TEXT 535 2 " " }{XPPEDIT 536 1 "z=5-x^2+ x*y^3" "/%\"zG,(\"\"&\"\"\"*$%\"xG\"\"#!\"\"*&F(F&*$%\"yG\"\"$F&F&" } {TEXT 531 2 " " }{TEXT 537 50 " in a curve. The tangent line to th is curve at " }{XPPEDIT 539 1 "``(2,1,3)" "-%!G6%\"\"#\"\"\"\"\"$" } {TEXT 538 1 " " }{TEXT 540 27 " passes through the point " }{XPPEDIT 541 1 "``(2,0,c)" "-%!G6%\"\"#\"\"!%\"cG" }{TEXT 542 12 ". What is \+ " }{TEXT 544 1 " " }{XPPEDIT 545 1 "c" "I\"cG6\"" }{TEXT 543 5 "? \+ " }}{PARA 258 "" 0 "" {TEXT 546 5 "\na) " }{XPPEDIT 547 1 "-5" ",$\" \"&!\"\"" }{TEXT 548 10 " b) " }{XPPEDIT 549 1 "-4" ",$\"\"%!\" \"" }{TEXT 550 10 " c) " }{XPPEDIT 551 1 "-3" ",$\"\"$!\"\"" } {TEXT 552 8 " d) " }{XPPEDIT 553 1 "-2" ",$\"\"#!\"\"" }{TEXT 554 10 " e) " }{XPPEDIT 555 1 "-1" ",$\"\"\"!\"\"" }{TEXT 556 11 " \+ f) " }{XPPEDIT 557 1 "1" "\"\"\"" }{TEXT 558 11 " g) " } {XPPEDIT 559 1 "2" "\"\"#" }{TEXT 560 9 " h) " }{XPPEDIT 561 1 "3 " "\"\"$" }{TEXT 562 9 " i) " }{XPPEDIT 563 1 "4" "\"\"%" }{TEXT 564 10 " j) " }{XPPEDIT 566 1 "5" "\"\"&" }{TEXT -1 1 " " } {TEXT 565 3 " " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 665 13 "Solution: c " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "r := t -> [2,1+t,3 +t*subs(\{x=2,y=1\}, diff(5-x^2+x*y^3,y))]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG:6#%\"tG6\"6$%)operatorG%&arrowGF(7%\"\"#,&\"\"\" F/9$F/,&\"\"$F/*&F0F/-%%subsG6$<$/%\"yGF//%\"xGF--%%diffG6$,(\"\"&F/*$ F;F-!\"\"*&F;F/F9F2F/F9F/F/F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "t_0 := solve( r(t)[2]=0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$t_0G!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(t=t_0 ,r(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"#\"\"!!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 909 34 "16.Th e vector-valued function\n " }{TEXT 901 10 " " }{XPPEDIT 902 1 "r(u,v)=u^2*i+u*cos(v)*j+u*sin(v)*k" "/-%\"rG6$%\"uG%\"vG,(*&F& \"\"#%\"iG\"\"\"F,*(F&F,-%$cosG6#F'F,%\"jGF,F,*(F&F,-%$sinG6#F'F,%\"kG F,F," }{TEXT 892 32 " \n \n of the two real variables" }{TEXT 896 2 " " }{XPPEDIT 897 1 "u>0" "2\"\"!%\"uG" }{TEXT 893 2 " " }{TEXT 899 4 " and" }{TEXT 900 1 " " }{XPPEDIT 898 1 "v>=0 " "1\"\"!%\"vG" } {TEXT 894 1 " " }{TEXT 895 97 " defines a surface. Calculate the pla ne that is tangent to this surface at the point (2,1,1). \n" }}{PARA 3 "" 0 "" {TEXT 890 5 "a) " }{XPPEDIT 930 1 "2*x-2*y-z = 1" "/,(*&\" \"#\"\"\"%\"xGF&F&*&F%F&%\"yGF&!\"\"%\"zGF*F&" }{TEXT 910 28 " \+ b) " }{XPPEDIT 931 1 "x-y+z = 2" "/,(%\"xG\"\"\"%\"yG !\"\"%\"zGF%\"\"#" }{TEXT 914 14 " \nc) " }{XPPEDIT 932 1 "x- 2*y-2*z = -2" "/,(%\"xG\"\"\"*&\"\"#F%%\"yGF%!\"\"*&F'F%%\"zGF%F),$F'F )" }{TEXT 912 25 " d) " }{XPPEDIT 933 1 "x-y-z = \+ 0" "/,(%\"xG\"\"\"%\"yG!\"\"%\"zGF'\"\"!" }{TEXT 917 12 " \ne) \+ " }{XPPEDIT 934 1 "x+y-z = 2" "/,(%\"xG\"\"\"%\"yGF%%\"zG!\"\"\"\"#" } {TEXT 919 34 " f) " }{XPPEDIT 935 1 "x+2 *y-z = 3" "/,(%\"xG\"\"\"*&\"\"#F%%\"yGF%F%%\"zG!\"\"\"\"$" }{TEXT 920 8 " \ng) " }{XPPEDIT 936 1 "x+2*y+z = 5" "/,(%\"xG\"\"\"*&\"\"# F%%\"yGF%F%%\"zGF%\"\"&" }{TEXT 924 31 " h) \+ " }{XPPEDIT 937 1 "x+y-2*z = 1" "/,(%\"xG\"\"\"%\"yGF%*&\"\"#F%%\"zGF %!\"\"F%" }{TEXT 925 25 " \ni) " }{XPPEDIT 938 1 " x-y+2*z = 3" "/,(%\"xG\"\"\"%\"yG!\"\"*&\"\"#F%%\"zGF%F%\"\"$" }{TEXT 928 28 " j)" }{TEXT 906 2 " " }{XPPEDIT 939 1 "x-2*y+2*z = 2" "/,(%\"xG\"\"\"*&\"\"#F%%\"yGF%!\"\"*&F'F%%\"zGF%F%F '" }{TEXT 929 3 " " }{TEXT 941 4 " " }{TEXT 904 2 " " }{TEXT -1 3 " " }}{PARA 3 "" 0 "" {TEXT -1 15 "\nSolution: c \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "x := u^2: y := u*cos(v): z := u* sin(v):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "v1 := map(w->dif f(w,u),[x,y,z] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G7%,$%\"uG\" \"#-%$cosG6#%\"vG-%$sinGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "v2 := map(w->diff(w,v),[x,y,z] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G7%\"\"!,$*&%\"uG\"\"\"-%$sinG6#%\"vGF*!\"\"*&F)F*-%$cosGF-F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "linalg[crossprod](v1,v2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%,&*&-%$cosG6#%\"vG \"\"#%\"uG\"\"\"F/*&-%$sinGF+F-F.F/F/,$*&F.F-F)F/!\"#,$*&F.F-F1F/F5" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "N := subs(\{u=sqrt(2),v=Pi /4\},\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'VECTORG6#7%,&*&- %$cosG6#,$%#PiG#\"\"\"\"\"%\"\"#F3#F1F3F1*&-%$sinGF-F3F3F4F1,$F+!\"%,$ F6F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "N := map(simplify,N );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'VECTORG6#7%*$\"\"##\"\" \"F*,$F)!\"#F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "N := map( w->sqrt(2)*w/2,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'VECTORG 6#7%\"\"\"!\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "x := 'x ': y := 'y': z := 'z':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn := N[1]*(x-2) + N[2]*(y-1) + N[3]*(z-1) = 0 ;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$eqnG/,*%\"xG\"\"\"\"\"#F(%\"yG!\"#%\"zGF+\"\"!" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 271 "" 0 "" {TEXT -1 46 "17. The plane that is tangent to the graph of " }{TEXT 568 2 " " }{XPPEDIT 569 1 "z=5-x^2+x*y^3" "/%\"zG,(\"\"&\"\"\"*$%\"xG\"\"#!\"\"* &F(F&*$%\"yG\"\"$F&F&" }{TEXT 567 1 " " }{TEXT -1 5 " at " }{XPPEDIT 571 1 "`(`*2,1,3*`)`" "6%*&%\"(G\"\"\"\"\"#F%F%*&\"\"$F%%\")GF%" } {TEXT 570 1 " " }{TEXT -1 15 " has what " }{XPPEDIT 19 1 "x" "I\" xG6\"" }{TEXT -1 13 " intercept?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 272 "" 0 "" {TEXT 572 4 "a) " }{XPPEDIT 573 1 "0" "\"\"!" } {TEXT 574 10 " b) " }{XPPEDIT 575 1 "1" "\"\"\"" }{TEXT 576 10 " c) " }{XPPEDIT 577 1 "2" "\"\"#" }{TEXT 578 8 " d) " } {XPPEDIT 579 1 "3" "\"\"$" }{TEXT 580 10 " e) " }{XPPEDIT 581 1 "4" "\"\"%" }{TEXT 582 11 " f) " }{XPPEDIT 583 1 "5" "\"\"&" } {TEXT 584 11 " g) " }{XPPEDIT 585 1 "6" "\"\"'" }{TEXT 586 9 " \+ h) " }{XPPEDIT 587 1 "7" "\"\"(" }{TEXT 588 9 " i) " } {XPPEDIT 589 1 "8" "\"\")" }{TEXT 590 10 " j) " }{XPPEDIT 592 1 "9" "\"\"*" }{TEXT -1 1 " " }{TEXT 591 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 666 12 "Solution: b" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A := subs(\{x=2,y=1\},diff (5-x^2+x*y^3,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "B := subs(\{x=2,y=1\},diff(5 -x^2+x*y^3,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG\"\"'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eqn := A*(x-2) + B*(y-1) - ( z-3)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,*%\"xG!\"$\"\"$\" \"\"%\"yG\"\"'%\"zG!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve( subs(\{y=0,z=0\},eqn),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 593 8 "18. Let" }{TEXT 599 2 " " }{XPPEDIT 600 1 "f(x,y)= 5*sqrt(x+y^2)" "/-%\"fG6$%\"xG%\"yG*&\"\"&\"\"\"-%%sqrtG6#,&F&F**$F'\" \"#F*F*" }{TEXT 594 7 ". Let" }{TEXT 601 2 " " }{XPPEDIT 602 1 "L(x ,y)" "-%\"LG6$%\"xG%\"yG" }{TEXT 595 33 " be the linear approximatio n of" }{TEXT 603 1 " " }{XPPEDIT 604 1 "f(x,y)" "-%\"fG6$%\"xG%\"yG" } {TEXT 596 3 " at" }{TEXT 605 2 " " }{XPPEDIT 606 1 "``(5,2)" "-%!G6$ \"\"&\"\"#" }{TEXT 597 30 ". What is the approximation of" }{TEXT 667 2 " " }{XPPEDIT 668 1 "f(23/5,11/5)" "-%\"fG6$*&\"#B\"\"\"\"\"&!\"\"* &\"#6F'F(F)" }{TEXT 669 22 " that results when " }{TEXT 607 1 " " } {XPPEDIT 608 1 "L(x,y)" "-%\"LG6$%\"xG%\"yG" }{TEXT 598 13 " is use d ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 273 "" 0 "" {TEXT -1 204 "a) 41/3 b) 44/3 \+ c) 46/3 d) 72/5 e) 74/5 \+ \nf) 76/5 g) 147/10 h) 149/10 i) \+ 151/10 j) 153/10" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 670 12 "Solution: c" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f := (x,y) -> 5*sqrt(x+y^2); a := 5: b := 2:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$-%%sq rtG6#,&9$\"\"\"*$9%\"\"#F3\"\"&F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "L := (x,y) -> f(a,b) + subs(\{u=a,v=b\}, diff(f(u,v) ,u))*(x-a) + subs(\{u=a,v=b\}, diff(f(u,v),v))*(y-b); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG:6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(-% \"fG6$%\"aG%\"bG\"\"\"*&-%%subsG6$<$/%\"uGF1/%\"vGF2-%%diffG6$-F/6$F:F 6$F@F " 0 "" {MPLTEXT 1 0 36 "L(23/5,11/5); simplify(\"); \+ evalf(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"#:\"\"\"*$\"\"*#F%\" \"##F%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#Y\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LLLL:!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "f(19/4,21/10); simplify(\"); evalf(\"); # actual" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"$H##\"\"\"\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*$\"$H##\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+&fuK^\"!\")" }}}}{SECT 0 {PARA 284 "" 0 "" {TEXT 630 16 "19. Sup pose that" }{TEXT 960 2 " " }{XPPEDIT 19 1 "x(t)" "-%\"xG6#%\"tG" } {TEXT 942 1 " " }{TEXT 958 3 "and" }{TEXT 959 3 " " }{XPPEDIT 19 1 " y(t)" "-%\"yG6#%\"tG" }{TEXT 944 2 " " }{TEXT 956 7 "satisfy" }{TEXT 957 13 "\n\n " }{XPPEDIT 19 1 "x(1)=6" "/-%\"xG6#\"\"\"\"\"' " }{TEXT 946 4 ", " }{XPPEDIT 19 1 "x*`'`(1)=2" "/*&%\"xG\"\"\"-%\"' G6#F%F%\"\"#" }{TEXT 948 4 ", " }{XPPEDIT 19 1 "y(1)=5" "/-%\"yG6#\" \"\"\"\"&" }{TEXT 950 2 ", " }{TEXT 961 3 "and" }{TEXT 962 4 " " } {XPPEDIT 19 1 "y*`'`(1)=4" "/*&%\"yG\"\"\"-%\"'G6#F%F%\"\"%" }{TEXT 952 5 ". \n\n" }{TEXT 963 2 "If" }{TEXT -1 12 " \n " } {XPPEDIT 19 1 "z[x](6,5)=-2" "/-&%\"zG6#%\"xG6$\"\"'\"\"&,$\"\"#!\"\" " }{TEXT -1 5 " , " }{XPPEDIT 19 1 "z[y](6,5)=3" "/-&%\"zG6#%\"yG6$ \"\"'\"\"&\"\"$" }{TEXT -1 2 " ," }{TEXT 686 1 " " }{TEXT 687 4 " and " }{TEXT 964 2 " " }{XPPEDIT 19 1 "phi(t)=z(x(t),y(t))" "/-%$phiG6#% \"tG-%\"zG6$-%\"xG6#F&-%\"yG6#F&" }{TEXT 953 4 " \n\n" }{TEXT 965 5 " find " }{TEXT 966 1 " " }{XPPEDIT 19 1 "phi*`'`(1)" "*&%$phiG\"\"\"-% \"'G6#F$F$" }{TEXT 955 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 609 4 "a) " } {XPPEDIT 610 1 "2" "\"\"#" }{TEXT 611 9 " b) " }{XPPEDIT 612 1 "4 " "\"\"%" }{TEXT 613 9 " c) " }{XPPEDIT 614 1 "6" "\"\"'" }{TEXT 615 7 " d) " }{XPPEDIT 616 1 "8" "\"\")" }{TEXT 617 9 " e) " } {XPPEDIT 618 1 "10" "\"#5" }{TEXT 619 8 " f) " }{XPPEDIT 620 1 "12 " "\"#7" }{TEXT 621 9 " g) " }{XPPEDIT 622 1 "14" "\"#9" }{TEXT 623 9 " h) " }{XPPEDIT 624 1 "16" "\"#;" }{TEXT 625 9 " i) \+ " }{XPPEDIT 628 1 "18" "\"#=" }{TEXT 629 9 " j) " }{XPPEDIT 627 1 "20" "\"#?" }{TEXT -1 1 " " }{TEXT 626 3 " " }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 663 12 "Solution : d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "(-2)*2 + (3)*4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 636 10 "20. Let " }{XPPEDIT 637 1 "z=x ^3*y^4 " "/%\"zG*&%\"xG\"\"$%\"yG\"\"%" }{TEXT 638 5 ", " } {XPPEDIT 639 1 "x=2*s^2+t-1" "/%\"xG,(*&\"\"#\"\"\"*$%\"sGF&F'F'%\"tGF 'F'!\"\"" }{TEXT 640 9 ", and " }{XPPEDIT 641 1 "y=exp(s+t-1)" "/% \"yG-%$expG6#,(%\"sG\"\"\"%\"tGF)F)!\"\"" }{TEXT 642 13 ". Calculate " }{TEXT 646 1 " " }{XPPEDIT 647 1 "diff(z,s)" "-%%diffG6$%\"zG%\"sG" }{TEXT 643 1 " " }{TEXT 648 8 " when " }{XPPEDIT 649 1 "s=1" "/%\"sG \"\"\"" }{TEXT 644 8 " and " }{XPPEDIT 650 1 "t=0" "/%\"tG\"\"!" } {TEXT 645 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 274 "" 0 "" {TEXT -1 181 "a) -2 b) 2 c) -4 d) 4 e) - 8 \+ \nf) 8 g) -16 h) 16 i) -3 2 j) 32" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 664 12 "Solution: h" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Direct" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "z \+ := x^3*y^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG*&%\"xG\"\"$%\"yG \"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs(\{x=2*s^2+t-1 ,y=exp(s+t-1)\}, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*$%\"sG\" \"#F'%\"tG\"\"\"!\"\"F)\"\"$-%$expG6#,(F&F)F(F)F*F)\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(\",s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(,(*$%\"sG\"\"#F(%\"tG\"\"\"!\"\"F*F(-%$expG6#,(F'F* F)F*F+F*\"\"%F'F*\"#7*&F%\"\"$F,F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "simplify(subs(\{s=1,t=0\}, \"));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Chain Rule" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "z := \+ x^3*y^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG*&%\"xG\"\"$%\"yG\" \"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "A := diff(z,x)*Diff (x,s)+diff(z,y)*Diff(y,s); \n#Comment: The capital D in \"Diff\" is us ed to delay \n the differentiation until after the substituti ons " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,&*(%\"xG\"\"#%\"yG\" \"%-%%DiffG6$F'%\"sG\"\"\"\"\"$*(F'F0F)F0-F,6$F)F.F/F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A := subs(\{x=2*s^2+t-1,y=exp(s+t-1 )\}, A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,&*(,(*$%\"sG\"\"#F* %\"tG\"\"\"!\"\"F,F*-%$expG6#,(F)F,F+F,F-F,\"\"%-%%DiffG6$F'F)F,\"\"$* (F'F6F.F6-F46$F.F)F,F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 " A := value(A);\n#Comment: The command \"value\" forces Maple to perfor m\n the differentiations that we have delayed\n by u sing \"Diff\" instead of \"diff\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"AG,&*(,(*$%\"sG\"\"#F*%\"tG\"\"\"!\"\"F,F*-%$expG6#,(F)F,F+F,F-F, \"\"%F)F,\"#7*&F'\"\"$F.F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A := simplify(subs(\{s=1,t=0\}, A));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "14 2 0" 139 }{VIEWOPTS 1 1 0 1 1 1803 }