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}{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 47 "The chain rule for functions of t wo variables " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo , B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "The chain rule" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 85 " be a func tion of two variables which has continuous first order partial derivat ives " }{XPPEDIT 18 0 "g[x](x,y)=Diff(g(x,y),x)" "6#/-&%\"gG6#%\"xG6$F (%\"yG-%%DiffG6$-F&6$F(F*F(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g[y]( x,y)=Diff(g(x,y),y)" "6#/-&%\"gG6#%\"yG6$%\"xGF(-%%DiffG6$-F&6$F*F(F( " }{TEXT -1 28 " in some region of R of the " }{TEXT 281 1 "x" }{TEXT -1 1 "," }{TEXT 280 1 "y" }{TEXT -1 7 " plane." }}{PARA 0 "" 0 "" {TEXT -1 100 "Suppose that we have a curve in the region R defined par ametrically by two differentiable functions " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x=u(t),``],[y=v(t),``])" "6# -%*PIECEWISEG6$7$/%\"xG-%\"uG6#%\"tG%!G7$/%\"yG-%\"vG6#F,F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Then we can form the function " }{XPPEDIT 18 0 "h(t)=g(u(t),v(t)) " "6#/-%\"hG6#%\"tG-%\"gG6$-%\"uG6#F'-%\"vG6#F'" }{TEXT -1 62 ", which is a real valued function of the single real variable " }{TEXT 264 1 "t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 10 "chain rule" }{TEXT -1 38 " in this situation says that function " } {XPPEDIT 18 0 "h(t)" "6#-%\"hG6#%\"tG" }{TEXT -1 23 " is differentiabl e and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h*`'`(t) = \+ g[x](u(t),v(t))*u*`'`(t)+g[y](u(t),v(t))*v*`'`(t);" "6#/*&%\"hG\"\"\"- %\"'G6#%\"tGF&,&*(-&%\"gG6#%\"xG6$-%\"uG6#F*-%\"vG6#F*F&F4F&-F(6#F*F&F &*(-&F/6#%\"yG6$-F46#F*-F76#F*F&F7F&-F(6#F*F&F&" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 13 "that is, if " }{XPPEDIT 18 0 "z=g(x,y)" "6#/%\"zG-%\"gG6$%\"xG%\"yG" }{XPPEDIT 18 0 "``=g(u(t),v(t))" "6#/%!G- %\"gG6$-%\"uG6#%\"tG-%\"vG6#F+" }{TEXT -1 7 ", then " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dz/dt=``(Diff(z,x))*``( dx/dt) + \+ ``(Diff(z,y))*``( dy/dt)" "6#/*&%#dzG\"\"\"%#dtG!\"\",&*&-%!G6#-%%Diff G6$%\"zG%\"xGF&-F,6#*&%#dxGF&F'F(F&F&*&-F,6#-F/6$F1%\"yGF&-F,6#*&%#dyG F&F'F(F&F&" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 263 15 "_______________" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 284 263 263 {PLOTDATA 2 "6=-%'CURVESG6&7$7$$ \"\"\"\"\"!$\"\")F*7$F+F(-%&COLORG6&%$RGBG$\"\"%!\"\"F*F)-%*LINESTYLEG 6#\"\"#-%*THICKNESSGF7-F$6&7$7$F($!\")F*7$F+$F4F*F.F5F9-F$6&7$7$FBF+7$ F?F(F.F5F9-F$6&7$7$FBF?7$F?FBF.F5F9-%%TEXTG6$7$$F*F*$\"\"*F*Q\"t6\"-FN 6$7$$\"#$)F4FQQ\"yFU-FN6$7$$!#$)F4FQQ\"xFU-FN6$7$FQ$!\"*F*Q\"zFU-FN6$7 $$\"#bF4$\"\"'F*Q#dyFU-FN6$7$Feo$\"#[F4Q#dtFU-FN6$7$Feo$\"#fF4Q\"_FU-F N6$7$$!#bF4FgoQ#dxFU-FN6$7$FipF]pF_p-FN6$7$FipFcpFep-FN6$7$$\"#^F4$!#i F4Q\"oFU-FN6$7$Feq$!#\\F4Fiq-FN6$7$Feo$!\"'F*Q#)yFU-FN6$7$Feo$!#[F4Q#) zFU-FN6$7$$\"#aF4$!#^F4Q#__FU-FN6$7$$!#fF4FgqFiq-FN6$7$FfsF]rFiq-FN6$7 $FipFbrQ#)xFU-FN6$7$FipFhrFjr-FN6$7$$!#dF4F`sFbs-%+AXESLABELSG6$Q!FUFj t-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$%(DEFAULTGFbu" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curv e 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Cur ve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 1 7" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" " Curve 24" }}}{PARA 0 "" 0 "" {TEXT -1 64 "Roughly speaking the chain r ule follows from the approximation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta*z" "6#*&%&DeltaG\"\"\"%\"zGF%" }{TEXT -1 1 " \+ " }{TEXT 287 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "g[x]*Delta*x+g[y]*D elta*y;" "6#,&*(&%\"gG6#%\"xG\"\"\"%&DeltaGF)F(F)F)*(&F&6#%\"yGF)F*F)F .F)F)" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 57 "which we can t hink of as getting progressively better as " }{XPPEDIT 18 0 "Delta*x" "6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Delta* y" "6#*&%&DeltaG\"\"\"%\"yGF%" }{TEXT -1 13 " approach 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 291 1 "x" }{TEXT -1 5 " and " }{TEXT 289 1 "y" }{TEXT -1 33 " are differenti able functions of " }{TEXT 290 1 "t" }{TEXT -1 10 ", we have " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta*z/(Delta*t)" "6 #*(%&DeltaG\"\"\"%\"zGF%*&F$F%%\"tGF%!\"\"" }{TEXT -1 1 " " }{TEXT 288 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "g[x]*``(Delta*x/(Delta*t))+g [y]*``(Delta*y/(Delta*t))" "6#,&*&&%\"gG6#%\"xG\"\"\"-%!G6#*(%&DeltaGF )F(F)*&F.F)%\"tGF)!\"\"F)F)*&&F&6#%\"yGF)-F+6#*(F.F)F5F)*&F.F)F0F)F1F) F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 36 "where the approxim ation improves as " }{XPPEDIT 18 0 "Delta*t" "6#*&%&DeltaG\"\"\"%\"tGF %" }{TEXT -1 14 " approaches 0." }}{PARA 0 "" 0 "" {TEXT -1 24 "In the limit we obtain " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dz/dt = g[x]*``(dx/dt)+g[y]*``(dy/dt);" "6#/*&%#dzG\"\"\"%#dtG!\"\",& *&&%\"gG6#%\"xGF&-%!G6#*&%#dxGF&F'F(F&F&*&&F,6#%\"yGF&-F06#*&%#dyGF&F' F(F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "A more formal explanation is given in the following \+ subsection. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Explanation of the chain rule " }} {PARA 0 "" 0 "" {TEXT -1 53 "The continuity of the partial derivatives means that " }{XPPEDIT 18 0 "z = g(x,y);" "6#/%\"zG-%\"gG6$%\"xG%\"yG " }{TEXT -1 36 " is differentiable in the region R. " }}{PARA 0 "" 0 " " {TEXT -1 26 "Hence there are functions " }{XPPEDIT 18 0 "epsilon[1] \+ = epsilon[1](h, k);" "6#/&%(epsilonG6#\"\"\"-&F%6#F'6$%\"hG%\"kG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "epsilon[2] = epsilon[2](h,k);" "6#/&%( epsilonG6#\"\"#-&F%6#F'6$%\"hG%\"kG" }{TEXT -1 12 " such that " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "g(x+h,y+k)-g(x, y) = g[x](x,y)*h+g[y](x,y)*k+epsilon[1] *h+epsilon[2]*k;" "6#/,&-%\"gG6$,&%\"xG\"\"\"%\"hGF*,&%\"yGF*%\"kGF*F* -F&6$F)F-!\"\",**&-&F&6#F)6$F)F-F*F+F*F**&-&F&6#F-6$F)F-F*F.F*F**&&%(e psilonG6#F*F*F+F*F**&&F?6#\"\"#F*F.F*F*" }{TEXT -1 12 " ------- (i)" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "such tha t " }{XPPEDIT 18 0 "epsilon[1]" "6#&%(epsilonG6#\"\"\"" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "epsilon[2]" "6#&%(epsilonG6#\"\"#" }{TEXT -1 15 " approach 0 as " }{TEXT 282 1 "h" }{TEXT -1 5 " and " }{TEXT 283 1 "k " }{TEXT -1 13 " approach 0. " }}{PARA 0 "" 0 "" {TEXT -1 39 "We can w rite this equation in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Delta*z = Diff(z,x)*Delta*x+Diff(z,y)*Delta*y+epsilon[1 ]*Delta*x+epsilon[2]*Delta*y;" "6#/*&%&DeltaG\"\"\"%\"zGF&,**(-%%DiffG 6$F'%\"xGF&F%F&F-F&F&*(-F+6$F'%\"yGF&F%F&F1F&F&*(&%(epsilonG6#F&F&F%F& F-F&F&*(&F46#\"\"#F&F%F&F1F&F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "Delta*x=h" "6#/*&%&DeltaG\"\"\"%\" xGF&%\"hG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Delta*y=k" "6#/*&%&DeltaG \"\"\"%\"yGF&%\"kG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Delta*z=g(x+De lta*x,y+Delta*y)-g(x, y)" "6#/*&%&DeltaG\"\"\"%\"zGF&,&-%\"gG6$,&%\"xG F&*&F%F&F-F&F&,&%\"yGF&*&F%F&F0F&F&F&-F*6$F-F0!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {TEXT 284 1 "x" }{TEXT -1 5 " and " }{TEXT 285 1 "y" }{TEXT -1 33 " ar e differentiable functions of " }{TEXT 286 1 "t" }{TEXT -1 23 ", we ca n divide (i) by " }{XPPEDIT 18 0 "Delta*t" "6#*&%&DeltaG\"\"\"%\"tGF% " }{TEXT -1 11 " to obtain " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Delta*z/(Delta*t) = Diff(z,x)*``(Delta*x/(Delta*t))+Dif f(z,y)*``(Delta*y/(Delta*t))+epsilon[1]*``(Delta*x/(Delta*t))+epsilon[ 2]*``(Delta*y/(Delta*t));" "6#/*(%&DeltaG\"\"\"%\"zGF&*&F%F&%\"tGF&!\" \",**&-%%DiffG6$F'%\"xGF&-%!G6#*(F%F&F0F&*&F%F&F)F&F*F&F&*&-F.6$F'%\"y GF&-F26#*(F%F&F9F&*&F%F&F)F&F*F&F&*&&%(epsilonG6#F&F&-F26#*(F%F&F0F&*& F%F&F)F&F*F&F&*&&F@6#\"\"#F&-F26#*(F%F&F9F&*&F%F&F)F&F*F&F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "Delta *t" "6#*&%&DeltaG\"\"\"%\"tGF%" }{TEXT -1 24 " tend to zero we obtain \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dz/dt = Diff(z,x) *``(dx/dy)+Diff(z,y)*``(dy/dt);" "6#/*&%#dzG\"\"\"%#dtG!\"\",&*&-%%Dif fG6$%\"zG%\"xGF&-%!G6#*&%#dxGF&%#dyGF(F&F&*&-F,6$F.%\"yGF&-F16#*&F5F&F 'F(F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 75 "Maple's differentiation procedures \" know\" about this chain rule. Note that " }{TEXT 0 4 "D[1]" }{TEXT -1 5 " ... " }{XPPEDIT 18 0 "D[1]" "6#&%\"DG6#\"\"\"" }{TEXT -1 5 " and \+ " }{TEXT 0 4 "D[2]" }{TEXT -1 5 " ... " }{XPPEDIT 18 0 "D[2]" "6#&%\"D G6#\"\"#" }{TEXT -1 170 " are the partial differentiation operators wh ich give the partial derivatives of a function of two variables with r espect to the first and second variables respectively. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "unassig n('g','u','v'):\nDiff(g(u(t),v(t)),t);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%\"gG6$-%\"uG6#%\"tG-%\"vGF+F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*&--&%\"DG6#\"\"\"6#%\"gG6$-%\"uG6#%\"tG-%\"vG F0F*-%%diffG6$F.F1F*F**&--&F(6#\"\"#F+F-F*-F56$F2F1F*F*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Chain rule examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }{TEXT 270 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{TEXT 271 1 "z" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "g(x,y) = x^3+x*y+2*y^2+4;" "6#/- %\"gG6$%\"xG%\"yG,**$F'\"\"$\"\"\"*&F'F,F(F,F,*&\"\"#F,*$F(F/F,F,\"\"% F," }{TEXT -1 63 ", and consider the circle defined by the parametric \+ equations " }{XPPEDIT 18 0 "PIECEWISE([x = cos(t), ``],[y = sin(t), ` `])" "6#-%*PIECEWISEG6$7$/%\"xG-%$cosG6#%\"tG%!G7$/%\"yG-%$sinG6#F,F- " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "Diff(z,x)=``" "6#/-%%DiffG6$%\"zG% \"xG%!G" }{XPPEDIT 18 0 "Diff(g(x,y),x) = 3*x^2+y;" "6#/-%%DiffG6$-%\" gG6$%\"xG%\"yGF*,&*&\"\"$\"\"\"*$F*\"\"#F/F/F+F/" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 12 "At the point" }{XPPEDIT 18 0 "``(cos(t),s in(t))" "6#-%!G6$-%$cosG6#%\"tG-%$sinG6#F)" }{TEXT -1 30 ", this parti al derivative is " }{XPPEDIT 18 0 "3*cos(t)^2+sin(t)" "6#,&*&\"\"$\" \"\"*$-%$cosG6#%\"tG\"\"#F&F&-%$sinG6#F+F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Similarly " } {XPPEDIT 18 0 "Diff(z,y) = ``;" "6#/-%%DiffG6$%\"zG%\"yG%!G" } {XPPEDIT 18 0 "Diff(g(x,y),y) = x+4*y;" "6#/-%%DiffG6$-%\"gG6$%\"xG%\" yGF+,&F*\"\"\"*&\"\"%F-F+F-F-" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "At the point" }{XPPEDIT 18 0 "``(cos(t),sin(t))" "6#-%!G6 $-%$cosG6#%\"tG-%$sinG6#F)" }{TEXT -1 30 ", this partial derivative is " }{XPPEDIT 18 0 "cos(t)+4*sin(t);" "6#,&-%$cosG6#%\"tG\"\"\"*&\"\"% F(-%$sinG6#F'F(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "Also " }{XPPEDIT 18 0 "dx/dt=-sin(t)" "6#/ *&%#dxG\"\"\"%#dtG!\"\",$-%$sinG6#%\"tGF(" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "dy/dt=cos(t)" "6#/*&%#dyG\"\"\"%#dtG!\"\"-%$cosG6#%\"tG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "By the chain rule " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dz/dt = ``(Diff(z,x))*``(dx/dt)+``(Diff(z,y))*``(dy/dt) " "6#/*&%#dzG\"\"\"%#dtG!\"\",&*&-%!G6#-%%DiffG6$%\"zG%\"xGF&-F,6#*&%# dxGF&F'F(F&F&*&-F,6#-F/6$F1%\"yGF&-F,6#*&%#dyGF&F'F(F&F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(3*cos(t )^2+sin(t))*``(-sin(t))+``(cos(t)+4*sin(t))*``(cos(t));" "6#/%!G,&*&-F $6#,&*&\"\"$\"\"\"*$-%$cosG6#%\"tG\"\"#F,F,-%$sinG6#F1F,F,-F$6#,$-F46# F1!\"\"F,F,*&-F$6#,&-F/6#F1F,*&\"\"%F,-F46#F1F,F,F,-F$6#-F/6#F1F,F," } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -3*cos(t)^2*sin(t)-sin(t)^2+cos(t)^2+4*sin(t)*cos(t);" "6#/%!G,**(\" \"$\"\"\"*$-%$cosG6#%\"tG\"\"#F(-%$sinG6#F-F(!\"\"*$-F06#F-F.F2*$-F+6# F-F.F(*(\"\"%F(-F06#F-F(-F+6#F-F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "On the other hand, subs tituting " }{XPPEDIT 18 0 "x=cos(t)" "6#/%\"xG-%$cosG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=sin(t)" "6#/%\"yG-%$sinG6#%\"tG" } {TEXT -1 5 " in " }{XPPEDIT 18 0 "z = x^3+x*y+2*y^2+4;" "6#/%\"zG,**$ %\"xG\"\"$\"\"\"*&F'F)%\"yGF)F)*&\"\"#F)*$F+F-F)F)\"\"%F)" }{TEXT -1 8 " gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z = co s(t)^3+cos(t)*sin(t)+2*sin(t)^2+4;" "6#/%\"zG,**$-%$cosG6#%\"tG\"\"$\" \"\"*&-F(6#F*F,-%$sinG6#F*F,F,*&\"\"#F,*$-F16#F*F4F,F,\"\"%F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "so we can obtain " }{XPPEDIT 18 0 "dz/dt" "6#*&%#dzG\"\"\"%#dtG!\"\"" }{TEXT -1 13 " directly as " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dz/dt = -3*cos(t)^2 *sin(t)-sin(t)^2+cos(t)^2+4*sin(t)*cos(t);" "6#/*&%#dzG\"\"\"%#dtG!\" \",**(\"\"$F&*$-%$cosG6#%\"tG\"\"#F&-%$sinG6#F0F&F(*$-F36#F0F1F(*$-F.6 #F0F1F&*(\"\"%F&-F36#F0F&-F.6#F0F&F&" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "g := \+ (x,y) -> x^3+x*y+2*y^2+4;\nu := t -> cos(t);\nv := t -> sin(t);\nDiff( g('u'(t),'v'(t)),t);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"gGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),**$)9$\"\"$\"\"\"F2*&F0F2 9%F2F2*&\"\"#F2)F4F6F2F2\"\"%F2F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG%$cosG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG%$sinG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,**$)-%\"uG6#%\"tG\"\"$\"\" \"F.*&F)F.-%\"vGF+F.F.*&\"\"#F.)F0F3F.F.\"\"%F.F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&)-%$cosG6#%\"tG\"\"#\"\"\"-%$sinGF(F+!\"$*$)F,F*F+! \"\"*$F%F+F+*(\"\"%F+F&F+F,F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The following picture illustrates this example." }}{PARA 0 "" 0 " " {TEXT -1 10 "The curve " }{XPPEDIT 18 0 "PIECEWISE([x=cos(t),``],[y= sin(t),``])" "6#-%*PIECEWISEG6$7$/%\"xG-%$cosG6#%\"tG%!G7$/%\"yG-%$sin G6#F,F-" }{TEXT -1 67 " is a circle with its centre at the origin and \+ having unit radius. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "For " }{XPPEDIT 18 0 "0<=t" "6#1\"\"!%\"tG" }{XPPEDIT 18 0 "``<=2*Pi" "6#1%!G*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 16 ", the param eter " }{TEXT 272 1 "t" }{TEXT -1 51 " gives the distance along the cu rve from the point " }{XPPEDIT 18 0 "``(1,0)" "6#-%!G6$\"\"\"\"\"!" } {TEXT -1 14 " to the point " }{XPPEDIT 18 0 "``(cos(t),sin(t))" "6#-%! G6$-%$cosG6#%\"tG-%$sinG6#F)" }{TEXT -1 94 ". We can add a third coord inate of 0 to all points on this curve and place it in a horizontal " }{TEXT 278 1 "x" }{TEXT -1 1 "," }{TEXT 279 1 "y" }{TEXT -1 30 " plane of 3 dimensional space." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "proc (t) options op erator, arrow; g(cos(t),sin(t)) end proc;" "6#f*6#%\"tG7\"6$%)operator G%&arrowG6\"-%\"gG6$-%$cosG6#F%-%$sinG6#F%F*F*F*" }{TEXT -1 25 " has v alues given by the " }{TEXT 275 1 "z" }{TEXT -1 38 " coordinates of po ints on the surface " }{XPPEDIT 18 0 "z=g(x,y)" "6#/%\"zG-%\"gG6$%\"xG %\"yG" }{TEXT -1 62 ", which are in the same vertical line as points o n the curve " }{XPPEDIT 18 0 "PIECEWISE([x = cos(t), ``],[y = sin(t), ``],[z = 0, ``]);" "6#-%*PIECEWISEG6%7$/%\"xG-%$cosG6#%\"tG%!G7$/%\"y G-%$sinG6#F,F-7$/%\"zG\"\"!F-" }{TEXT -1 52 " (and directly above them ), and form a space-curve " }{XPPEDIT 18 0 "PIECEWISE([x = cos(t), `` ],[y = sin(t), ``],[z = cos(t)^3+cos(t)*sin(t)+2*sin(t)^2+4, ``])" "6# -%*PIECEWISEG6%7$/%\"xG-%$cosG6#%\"tG%!G7$/%\"yG-%$sinG6#F,F-7$/%\"zG, **$-F*6#F,\"\"$\"\"\"*&-F*6#F,F<-F26#F,Fg(cos(t),sin(t))" "6#f*6#%\"tG7\"6$%)operatorG%& arrowG6\"-%\"gG6$-%$cosG6#F%-%$sinG6#F%F*F*F*" }{TEXT -1 57 " as can b e seen by looking at the graph of this function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 477 "p1 := plot3d(x^3+x*y+2*y^2+4,x=-1.5..1.5,y=-1.5.. 1.5,\n color=COLOR(RGB,.8,.6,1),style=wireframe,gridstyle=triang ular):\np2 := plots[spacecurve]([cos(t),sin(t),0],t=0..2*Pi,\n thick ness=3,color=green):\np3 := plots[spacecurve]([cos(t),sin(t),cos(t)^3+ cos(t)*sin(t)+2*sin(t)^2+4],t=0..2*Pi,\n thickness=3,color=red):\np4 := plot3d(0,x=-1.5..1.5,y=-1.5..1.5,\n color=COLOR(RGB,.9,.9,.9 ),style=patchnogrid):\nplots[display]([p1,p2,p3,p4],axes=boxed,orienta tion=[145,60]);\n" }}{PARA 13 "" 1 "" {GLPLOT3D 449 346 346 {PLOTDATA 3 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0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000-F16&F3$\"\"*F)FbglFbgl-F<6#%,PAT CHNOGRIDG-%+AXESLABELSG6&%\"xG%\"yGQ!F/-%%FONTG6#%(DEFAULTG-%*GRIDSTYL EG6#%+TRIANGULARG-%*AXESSTYLEG6#%$BOXG-%+PROJECTIONG6%$\"$X\"F:$\"#gF: F9" 1 2 0 1 10 0 1 1 1 2 2 1.000000 60.000000 145.000000 1 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 37 "The graphs of the composite function " } {XPPEDIT 18 0 "t->g(cos(t),sin(t))" "6#f*6#%\"tG7\"6$%)operatorG%&arro wG6\"-%\"gG6$-%$cosG6#F%-%$sinG6#F%F*F*F*" }{XPPEDIT 18 0 "``=cos(t)^3 +cos(t)*sin(t)+2*sin(t)^2+4" "6#/%!G,**$-%$cosG6#%\"tG\"\"$\"\"\"*&-F( 6#F*F,-%$sinG6#F*F,F,*&\"\"#F,*$-F16#F*F4F,F,\"\"%F," }{TEXT -1 20 " a nd its derivative " }{XPPEDIT 18 0 "t->-3*cos(t)^2*sin(t)-sin(t)^2+cos (t)^2+4*cos(t)*sin(t)" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\",**(\"\" $\"\"\"*$-%$cosG6#F%\"\"#F.-%$sinG6#F%F.!\"\"*$-F56#F%F3F7*$-F16#F%F3F .*(\"\"%F.-F16#F%F.-F56#F%F.F.F*F*F*" }{TEXT -1 21 " are plotted below . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "plot([cos(t)^3+cos(t)*sin(t)+2*sin(t)^2+4,\n -3*cos( t)^2*sin(t)-sin(t)^2+cos(t)^2+4*cos(t)*sin(t)],\n t=0..2*Pi,y=-3.2.. 6.2,color=[red,brown],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7jp7$$\"\"!F)$\"\"&F)7$$\"3i]cC&eb& p8!#=$\"3H%G%\\D+qW^!#<7$$\"3YqR*pd)>hDF/$\"3ypYj\"\\9(y_F27$$\"3Zs[E^ dK,RF/$\"3HU$GkD(4KaF27$$\"3ab^NehL]_F/$\"3G9W5-)>Te&F27$$\"35*QhW&\\$ Hf'F/$\"3.N&*3hKZGdF27$$\"3zS11.fpPyF/$\"3wUGGjB-_eF27$$\"3upr'fwtl7*F /$\"3s*RPSi%\\kfF27$$\"3!QRa&4L&f/\"F2$\"39XM^bkGdgF27$$\"3_y%f^`(Q76F 2$\"3w&pJz'3%=4'F27$$\"3YjXwg<#)y6F2$\"3)\\,00X?p6'F27$$\"38ST=W'*)H@ \"F2$\"3Hz0%QL(oDhF27$$\"3!or.w_drC\"F2$\"3#3@x*=**RJhF27$$\"3W$HB5TD8 G\"F2$\"3!pHng6pQ8'F27$$\"36qGW%H$\\:8F2$\"352y#=P2H8'F27$$\"3SH22vMov 8F2$\"32()opR$4B7'F27$$\"3$*)e)pbO(eV\"F2$\"3+YJv2s`*4'F27$$\"3/]+3VNj .:F2$\"3+G!))z9b#egF27$$\"396:YIMRr:F2$\"3UIp>$=-%**fF27$$\"3!=3SCmpuq \"F2$\"3Z.)RE!fODeF27$$\"3LmSEEVgQ=F2$\"3yAHWj'Rie&F27$$\"3')4G5Xd9)*= F2$\"3o`tG\\&HbX&F27$$\"3S`:%R;(od>F2$\"33a;'[5N@J&F27$$\"3^Z,\"*pw[G? 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\"*;$!#K!\"\"$\"#iF]_n" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 2 " }{TEXT 299 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{TEXT 300 1 "z" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "g(x,y) = sin(x*y);" "6#/-%\"gG6$ %\"xG%\"yG-%$sinG6#*&F'\"\"\"F(F-" }{TEXT -1 62 ", and consider the cu rve defined by the parametric equations " }{XPPEDIT 18 0 "PIECEWISE([ x = t^2-2, ``],[y = 2*t, ``]);" "6#-%*PIECEWISEG6$7$/%\"xG,&*$%\"tG\" \"#\"\"\"F,!\"\"%!G7$/%\"yG*&F,F-F+F-F/" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "Diff(z,x)=``" "6#/-%%DiffG6$%\"zG%\"xG%!G" }{XPPEDIT 18 0 "Diff( g(x,y),x) = y*cos(x*y);" "6#/-%%DiffG6$-%\"gG6$%\"xG%\"yGF**&F+\"\"\"- %$cosG6#*&F*F-F+F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 "A t the point" }{XPPEDIT 18 0 "``(t^2-2,2*t);" "6#-%!G6$,&*$%\"tG\"\"#\" \"\"F)!\"\"*&F)F*F(F*" }{TEXT -1 30 ", this partial derivative is " } {XPPEDIT 18 0 "2*t*cos(2*t^3-4*t);" "6#*(\"\"#\"\"\"%\"tGF%-%$cosG6#,& *&F$F%*$F&\"\"$F%F%*&\"\"%F%F&F%!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Similarly " } {XPPEDIT 18 0 "Diff(z,y) = ``;" "6#/-%%DiffG6$%\"zG%\"yG%!G" } {XPPEDIT 18 0 "Diff(g(x,y),y) = x*cos(x*y);" "6#/-%%DiffG6$-%\"gG6$%\" xG%\"yGF+*&F*\"\"\"-%$cosG6#*&F*F-F+F-F-" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "At the point" }{XPPEDIT 18 0 "``(t^2-2,2*t);" "6# -%!G6$,&*$%\"tG\"\"#\"\"\"F)!\"\"*&F)F*F(F*" }{TEXT -1 30 ", this part ial derivative is " }{XPPEDIT 18 0 "(t^2-2)*cos(2*t^3-4*t);" "6#*&,&* $%\"tG\"\"#\"\"\"F'!\"\"F(-%$cosG6#,&*&F'F(*$F&\"\"$F(F(*&\"\"%F(F&F(F )F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 5 "Also " }{XPPEDIT 18 0 "dx/dt = 2*t;" "6#/*&%#dxG\"\"\"%# dtG!\"\"*&\"\"#F&%\"tGF&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dy/dt = \+ 2;" "6#/*&%#dyG\"\"\"%#dtG!\"\"\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "By the chain rule " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dz/dt = ``(Diff(z,x)) *``(dx/dt)+``(Diff(z,y))*``(dy/dt)" "6#/*&%#dzG\"\"\"%#dtG!\"\",&*&-%! G6#-%%DiffG6$%\"zG%\"xGF&-F,6#*&%#dxGF&F'F(F&F&*&-F,6#-F/6$F1%\"yGF&-F ,6#*&%#dyGF&F'F(F&F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(2*t*cos(2*t^3-4*t))*``(2*t)+``((t^2-2)*cos( 2*t^3-4*t))*2;" "6#/%!G,&*&-F$6#*(\"\"#\"\"\"%\"tGF+-%$cosG6#,&*&F*F+* $F,\"\"$F+F+*&\"\"%F+F,F+!\"\"F+F+-F$6#*&F*F+F,F+F+F+*&-F$6#*&,&*$F,F* F+F*F6F+-F.6#,&*&F*F+*$F,F3F+F+*&F5F+F,F+F6F+F+F*F+F+" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 4*t^2*cos(2* t^3-4*t)+(2*t^2-4)*cos(2*t^3-4*t);" "6#/%!G,&*(\"\"%\"\"\"*$%\"tG\"\"# F(-%$cosG6#,&*&F+F(*$F*\"\"$F(F(*&F'F(F*F(!\"\"F(F(*&,&*&F+F(*$F*F+F(F (F'F4F(-F-6#,&*&F+F(*$F*F2F(F(*&F'F(F*F(F4F(F(" }{TEXT -1 2 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(6*t^2-4)*cos(2*t^ 3-4*t)" "6#/%!G*&,&*&\"\"'\"\"\"*$%\"tG\"\"#F)F)\"\"%!\"\"F)-%$cosG6#, &*&F,F)*$F+\"\"$F)F)*&F-F)F+F)F.F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "On the other hand, subs tituting " }{XPPEDIT 18 0 "x = t^2-2;" "6#/%\"xG,&*$%\"tG\"\"#\"\"\"F( !\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 2*t;" "6#/%\"yG*&\"\"# \"\"\"%\"tGF'" }{TEXT -1 5 " in " }{XPPEDIT 18 0 "z = sin(x*y);" "6#/ %\"zG-%$sinG6#*&%\"xG\"\"\"%\"yGF*" }{TEXT -1 8 " gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z = sin(2*t^3-4*t);" "6#/% \"zG-%$sinG6#,&*&\"\"#\"\"\"*$%\"tG\"\"$F+F+*&\"\"%F+F-F+!\"\"" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "so we can obtain " } {XPPEDIT 18 0 "dz/dt" "6#*&%#dzG\"\"\"%#dtG!\"\"" }{TEXT -1 13 " direc tly as " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dz/dt = (6 *t^2-4)*cos(2*t^3-4*t);" "6#/*&%#dzG\"\"\"%#dtG!\"\"*&,&*&\"\"'F&*$%\" tG\"\"#F&F&\"\"%F(F&-%$cosG6#,&*&F/F&*$F.\"\"$F&F&*&F0F&F.F&F(F&" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 91 "g := (x,y) -> sin(x*y);\nu := t -> t^2-2;\nv : = t -> 2*t;\nDiff(g('u'(t),'v'(t)),t);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)-%$sinG6# *&9$\"\"\"9%F2F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6#%\"t G6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"#\"\"\"F1F0!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,$9 $\"\"#F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%$sinG6#*&- %\"uG6#%\"tG\"\"\"-%\"vGF,F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-% $cosG6#,$*&,&*$)%\"tG\"\"#\"\"\"F.F-!\"\"F.F,F.F-F.,&F*\"\"'\"\"%F/F. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The following picture ill ustrates this example." }}{PARA 0 "" 0 "" {TEXT -1 10 "The curve " } {XPPEDIT 18 0 "PIECEWISE([x = t^2-2, ``],[y = 2*t, ``]);" "6#-%*PIECEW ISEG6$7$/%\"xG,&*$%\"tG\"\"#\"\"\"F,!\"\"%!G7$/%\"yG*&F,F-F+F-F/" } {TEXT -1 17 " is the parabola " }{XPPEDIT 18 0 "y^2=4*x+8" "6#/*$%\"yG \"\"#,&*&\"\"%\"\"\"%\"xGF*F*\"\")F*" }{TEXT -1 15 " which has the " } {TEXT 304 1 "x" }{TEXT -1 123 " axis as its axis of symmetry. We can a dd a third coordinate of 0 to all points on this curve and place it in a horizontal " }{TEXT 305 1 "x" }{TEXT -1 1 "," }{TEXT 306 1 "y" } {TEXT -1 32 " plane of 3 dimensional space. " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "proc (t) options operator, arrow; g(t^2-2,2*t) end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6 \"-%\"gG6$,&*$F%\"\"#\"\"\"F0!\"\"*&F0F1F%F1F*F*F*" }{TEXT -1 25 " has values given by the " }{TEXT 301 1 "z" }{TEXT -1 38 " coordinates of \+ points on the surface " }{XPPEDIT 18 0 "z=g(x,y)" "6#/%\"zG-%\"gG6$%\" xG%\"yG" }{TEXT -1 62 ", which are in the same vertical line as points on the curve " }{XPPEDIT 18 0 "PIECEWISE([x = t^2-2, ``],[y = 2*t, ` `],[z=0,``])" "6#-%*PIECEWISEG6%7$/%\"xG,&*$%\"tG\"\"#\"\"\"F,!\"\"%!G 7$/%\"yG*&F,F-F+F-F/7$/%\"zG\"\"!F/" }{TEXT -1 25 ", and form a space- curve " }{XPPEDIT 18 0 "PIECEWISE([x = t^2-2, ``],[y = 2*t, ``],[z = s in(2*t^3-4*t), ``]);" "6#-%*PIECEWISEG6%7$/%\"xG,&*$%\"tG\"\"#\"\"\"F, !\"\"%!G7$/%\"yG*&F,F-F+F-F/7$/%\"zG-%$sinG6#,&*&F,F-*$F+\"\"$F-F-*&\" \"%F-F+F-F.F/" }{TEXT -1 17 " in the surface " }{XPPEDIT 18 0 "z=sin( x*y)" "6#/%\"zG-%$sinG6#*&%\"xG\"\"\"%\"yGF*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 35 "In the following picture the curve " } {XPPEDIT 18 0 "PIECEWISE([x = t^2-2, ``],[y = 2*t, ``],[z = 0, ``]);" "6#-%*PIECEWISEG6%7$/%\"xG,&*$%\"tG\"\"#\"\"\"F,!\"\"%!G7$/%\"yG*&F,F- F+F-F/7$/%\"zG\"\"!F/" }{TEXT -1 10 " ( in the " }{TEXT 302 1 "x" } {TEXT -1 1 "," }{TEXT 303 1 "y" }{TEXT -1 49 " plane) is coloured gree n, while the space-curve " }{XPPEDIT 18 0 "PIECEWISE([x = t^2-2, ``],[ y = 2*t, ``],[z = sin(2*t^3-4*t), ``]);" "6#-%*PIECEWISEG6%7$/%\"xG,&* $%\"tG\"\"#\"\"\"F,!\"\"%!G7$/%\"yG*&F,F-F+F-F/7$/%\"zG-%$sinG6#,&*&F, F-*$F+\"\"$F-F-*&\"\"%F-F+F-F.F/" }{TEXT -1 16 " in the surface " } {XPPEDIT 18 0 "z=sin(x*y)" "6#/%\"zG-%$sinG6#*&%\"xG\"\"\"%\"yGF*" } {TEXT -1 18 " is coloured red. " }}{PARA 0 "" 0 "" {TEXT -1 86 "The \" undulations\" of the red curve correspond to the values of the composi te function " }{XPPEDIT 18 0 "proc (t) options operator, arrow; g(t^2- 2,2*t) end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"-%\"gG6$,&*$F %\"\"#\"\"\"F0!\"\"*&F0F1F%F1F*F*F*" }{TEXT -1 57 " as can be seen by \+ looking at the graph of this function." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 393 "p1 := plot3d(sin(x*y),x=-3..3,y=-4..4,\n color=COLOR(RG B,.8,.6,1),grid=[40,40],gridstyle=triangular,style=wireframe):\np2 := \+ plots[spacecurve]([t^2-2,2*t,0],t=-2..2,\n thickness=3,color=green): \np3 := plots[spacecurve]([t^2-2,2*t,sin(2*t^3-4*t)],t=-2..2,\n thic kness=3,color=red):\nplots[display]([p1,p2,p3],axes=boxed,\n orient ation=[145,60]);\n#view=[-3..3,-3..3,-1..1],style=wireframe," }}{PARA 13 "" 1 "" {GLPLOT3D 663 470 470 {PLOTDATA 3 "6)-%%GRIDG6';$!\"$\"\"!$ 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Q!Fb_n-%*THICKNESSG6#F]_n-%'COLOURG6&%$RGBG$\")#)eqk!\")$\"))eqk\"F]`n F^`n-%%VIEWG6$;F(F\\_n%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 3 " }{TEXT 265 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{TEXT 266 1 "z" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "g(x,y)=1+3/(1+x^2+y^2)" "6#/-%\" gG6$%\"xG%\"yG,&\"\"\"F**&\"\"$F*,(F*F**$F'\"\"#F**$F(F/F*!\"\"F*" } {TEXT -1 62 ", and consider the curve defined by the parametric equati ons " }{XPPEDIT 18 0 "PIECEWISE([x = t, ``],[y = 1-t^2, ``]);" "6#-%* PIECEWISEG6$7$/%\"xG%\"tG%!G7$/%\"yG,&\"\"\"F/*$F)\"\"#!\"\"F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "D iff(z,x)=``" "6#/-%%DiffG6$%\"zG%\"xG%!G" }{XPPEDIT 18 0 "Diff(g(x,y), x) = -6/(1+x^2+y^2)^2*x" "6#/-%%DiffG6$-%\"gG6$%\"xG%\"yGF*,$*(\"\"'\" \"\"*$,(F/F/*$F*\"\"#F/*$F+F3F/F3!\"\"F*F/F5" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 12 "At the point" }{XPPEDIT 18 0 "``(t, 1-t^2 );" "6#-%!G6$%\"tG,&\"\"\"F(*$F&\"\"#!\"\"" }{TEXT -1 30 ", this parti al derivative is " }{XPPEDIT 18 0 "-6*t/(1+t^2+(1-t^2)^2)^2=-6*t/(2-t ^2+t^4)^2" "6#/,$*(\"\"'\"\"\"%\"tGF'*$,(F'F'*$F(\"\"#F'*$,&F'F'*$F(F, !\"\"F,F'F,F0F0,$*(F&F'F(F'*$,(F,F'*$F(F,F0*$F(\"\"%F'F,F0F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Similarly " }{XPPEDIT 18 0 "Diff(z,y) = ``;" "6#/-%%DiffG6$%\"zG% \"yG%!G" }{XPPEDIT 18 0 "Diff(g(x,y),y) = -6/(1+x^2+y^2)^2*y" "6#/-%%D iffG6$-%\"gG6$%\"xG%\"yGF+,$*(\"\"'\"\"\"*$,(F/F/*$F*\"\"#F/*$F+F3F/F3 !\"\"F+F/F5" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "At the \+ point" }{XPPEDIT 18 0 "``(t,1-t^2);" "6#-%!G6$%\"tG,&\"\"\"F(*$F&\"\"# !\"\"" }{TEXT -1 30 ", this partial derivative is " }{XPPEDIT 18 0 "- 6/(1+t^2+(1-t^2)^2)^2*(1-t^2) =6*(t^2-1)/(2-t^2+t^4)^2" "6#/,$*(\"\"' \"\"\"*$,(F'F'*$%\"tG\"\"#F'*$,&F'F'*$F+F,!\"\"F,F'F,F0,&F'F'*$F+F,F0F 'F0*(F&F',&*$F+F,F'F'F0F'*$,(F,F'*$F+F,F0*$F+\"\"%F'F,F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Als o " }{XPPEDIT 18 0 "dx/dt = 1;" "6#/*&%#dxG\"\"\"%#dtG!\"\"F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dy/dt = -2*t;" "6#/*&%#dyG\"\"\"%#dtG!\" \",$*&\"\"#F&%\"tGF&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "By the chain rule " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dz/dt = ``(Diff(z,x))*``(dx/dt)+`` (Diff(z,y))*``(dy/dt)" "6#/*&%#dzG\"\"\"%#dtG!\"\",&*&-%!G6#-%%DiffG6$ %\"zG%\"xGF&-F,6#*&%#dxGF&F'F(F&F&*&-F,6#-F/6$F1%\"yGF&-F,6#*&%#dyGF&F 'F(F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(-6*t/((2-t^2+t^4)^2))*`` (1)+``(6*(t^2-1)/((2-t^2+t^4)^2))*``(-2*t);" "6#/%!G,&*&-F$6#,$*(\"\"' \"\"\"%\"tGF,*$,(\"\"#F,*$F-F0!\"\"*$F-\"\"%F,F0F2F2F,-F$6#F,F,F,*&-F$ 6#*(F+F,,&*$F-F0F,F,F2F,*$,(F0F,*$F-F0F2*$F-F4F,F0F2F,-F$6#,$*&F0F,F-F ,F2F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= (-6*t-12*t^3+12*t)/((2-t^2+t ^4)^2)" "6#/%!G*&,(*&\"\"'\"\"\"%\"tGF)!\"\"*&\"#7F)*$F*\"\"$F)F+*&F-F )F*F)F)F)*$,(\"\"#F)*$F*F3F+*$F*\"\"%F)F3F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=6*t*(1-2*t^2)/(2-t^2+t^4)^2" "6#/%!G**\"\"'\"\"\"%\"tGF',&F'F '*&\"\"#F'*$F(F+F'!\"\"F'*$,(F+F'*$F(F+F-*$F(\"\"%F'F+F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "On the other hand, substituting " }{XPPEDIT 18 0 "x = t;" "6#/%\"xG%\"tG " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 1-t^2;" "6#/%\"yG,&\"\"\"F&* $%\"tG\"\"#!\"\"" }{TEXT -1 5 " in " }{XPPEDIT 18 0 "z=1+3/(1+x^2+y^2 )" "6#/%\"zG,&\"\"\"F&*&\"\"$F&,(F&F&*$%\"xG\"\"#F&*$%\"yGF,F&!\"\"F& " }{TEXT -1 8 " gives " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z = 1+3/(1+t^2+(1-t^2)^2);" "6#/ %\"zG,&\"\"\"F&*&\"\"$F&,(F&F&*$%\"tG\"\"#F&*$,&F&F&*$F+F,!\"\"F,F&F0F &" }{XPPEDIT 18 0 "`` = 1+3/(2-t^2+t^4);" "6#/%!G,&\"\"\"F&*&\"\"$F&,( \"\"#F&*$%\"tGF*!\"\"*$F,\"\"%F&F-F&" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 17 "so we can obtain " }{XPPEDIT 18 0 "dz/dt" "6#*&%#dzG\" \"\"%#dtG!\"\"" }{TEXT -1 13 " directly as " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dz/dt = -3*(-2*t+4*t^3)/((2-t^2+t^4)^2) ;" "6#/*&%#dzG\"\"\"%#dtG!\"\",$*(\"\"$F&,&*&\"\"#F&%\"tGF&F(*&\"\"%F& *$F/F+F&F&F&*$,(F.F&*$F/F.F(*$F/F1F&F.F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 6*t*(1-2*t^2)/((2-t^2+t^4)^2);" "6#/%!G**\"\"'\"\"\"%\"tGF',&F 'F'*&\"\"#F'*$F(F+F'!\"\"F'*$,(F+F'*$F(F+F-*$F(\"\"%F'F+F-" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "g := (x,y) -> 1+3/(1+x^2+y^2);\nu := t -> t;\nv := t -> 1-t^2;\nDiff(g('u'(t),'v'(t)),t);\nvalue(%);\nsimplify(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"xG%\"yG6\"6$%)operatorG%& arrowGF),&\"\"\"F.*&\"\"$F.,(F.F.*$)9$\"\"#F.F.*$)9%F5F.F.!\"\"F.F)F)F )" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6#%\"tG6\"6$%)operatorG%& arrowGF(9$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vGf*6#%\"tG6\" 6$%)operatorG%&arrowGF(,&\"\"\"F-*$)9$\"\"#F-!\"\"F(F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%%DiffG6$,&\"\"\"F'*&\"\"$F',(F'F'*$)-%\"uG6#% \"tG\"\"#F'F'*$)-%\"vGF/F1F'F'!\"\"F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(\"\"\"F&*$)%\"tG\"\"#F&F&*$),&F&F&F'!\"\"F*F&F&!\"#,&F)F** (\"\"%F&F-F&F)F&F.F&!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%\"tG \"\"\",&!\"\"F&*&\"\"#F&)F%F*F&F&F&,(F*F&*$F+F&F(*$)F%\"\"%F&F&!\"#!\" '" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "The following picture ill ustrates this example." }}{PARA 0 "" 0 "" {TEXT -1 10 "The curve " } {XPPEDIT 18 0 "PIECEWISE([x = t, ``],[y = 1-t^2, ``]);" "6#-%*PIECEWIS EG6$7$/%\"xG%\"tG%!G7$/%\"yG,&\"\"\"F/*$F)\"\"#!\"\"F*" }{TEXT -1 18 " is the parabola " }{XPPEDIT 18 0 "y = 1-x^2;" "6#/%\"yG,&\"\"\"F&*$% \"xG\"\"#!\"\"" }{TEXT -1 15 " which has the " }{TEXT 307 1 "y" } {TEXT -1 123 " axis as its axis of symmetry. We can add a third coordi nate of 0 to all points on this curve and place it in a horizontal " } {TEXT 273 1 "x" }{TEXT -1 1 "," }{TEXT 274 1 "y" }{TEXT -1 32 " plane \+ of 3 dimensional space. " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "proc (t) options operator, arrow; g(t,1-t^2) end pr oc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"-%\"gG6$F%,&\"\"\"F/*$F%\" \"#!\"\"F*F*F*" }{TEXT -1 25 " has values given by the " }{TEXT 267 1 "z" }{TEXT -1 38 " coordinates of points on the surface " }{XPPEDIT 18 0 "z=g(x,y)" "6#/%\"zG-%\"gG6$%\"xG%\"yG" }{TEXT -1 62 ", which are in the same vertical line as points on the curve " }{XPPEDIT 18 0 "P IECEWISE([x = t, ``],[y = 1-t^2, ``],[z = 0, ``]);" "6#-%*PIECEWISEG6% 7$/%\"xG%\"tG%!G7$/%\"yG,&\"\"\"F/*$F)\"\"#!\"\"F*7$/%\"zG\"\"!F*" } {TEXT -1 26 ", and form a space-curve " }{XPPEDIT 18 0 "PIECEWISE([x \+ = t, ``],[y = 1-t^2, ``],[z = 1+3/(2-t^2+t^4), ``]);" "6#-%*PIECEWISEG 6%7$/%\"xG%\"tG%!G7$/%\"yG,&\"\"\"F/*$F)\"\"#!\"\"F*7$/%\"zG,&F/F/*&\" \"$F/,(F1F/*$F)F1F2*$F)\"\"%F/F2F/F*" }{TEXT -1 17 " in the surface \+ " }{XPPEDIT 18 0 "z=1+3/(1+x^2+y^2)" "6#/%\"zG,&\"\"\"F&*&\"\"$F&,(F&F &*$%\"xG\"\"#F&*$%\"yGF,F&!\"\"F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "In the following picture the curve " }{XPPEDIT 18 0 "PIEC EWISE([x = t, ``],[y = 1-t^2, ``],[z = 0, ``]) " "6#-%*PIECEWISEG6%7$/ %\"xG%\"tG%!G7$/%\"yG,&\"\"\"F/*$F)\"\"#!\"\"F*7$/%\"zG\"\"!F*" } {TEXT -1 11 " ( in the " }{TEXT 268 1 "x" }{TEXT -1 1 "," }{TEXT 269 1 "y" }{TEXT -1 49 " plane) is coloured green, while the space-curve \+ " }{XPPEDIT 18 0 "PIECEWISE([x = t, ``],[y = 1-t^2, ``],[z = 1+3/(2-t^ 2+t^4), ``]);" "6#-%*PIECEWISEG6%7$/%\"xG%\"tG%!G7$/%\"yG,&\"\"\"F/*$F )\"\"#!\"\"F*7$/%\"zG,&F/F/*&\"\"$F/,(F1F/*$F)F1F2*$F)\"\"%F/F2F/F*" } {TEXT -1 16 " in the surface " }{XPPEDIT 18 0 "z=1+3/(1+x^2+y^2)" "6#/ %\"zG,&\"\"\"F&*&\"\"$F&,(F&F&*$%\"xG\"\"#F&*$%\"yGF,F&!\"\"F&" } {TEXT -1 19 " is coloured red. " }}{PARA 0 "" 0 "" {TEXT -1 86 "The \+ \"undulations\" of the red curve correspond to the values of the compo site function " }{XPPEDIT 18 0 "proc (t) options operator, arrow; g(t, 1-t^2) end proc;" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"-%\"gG6$F%,& \"\"\"F/*$F%\"\"#!\"\"F*F*F*" }{TEXT -1 57 " as can be seen by looking at the graph of this function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 458 "p1 := plot3d(1+3/(1+x^2+y^2),x=-2..2,y=-3..2,\n color=COLOR (RGB,.8,.6,1),grid=[40,40],gridstyle=triangular,style=wireframe):\np2 \+ := plot3d(0,x=-2..2,y=-3..2,\n color=COLOR(RGB,.95,.95,.95),gr id=[5,6],style=patchnogrid):\np3 := plots[spacecurve]([t,1-t^2,0],t=-2 ..2,\n thickness=3,color=green):\np4 := plots[spacecurve]([t,1-t^2,1 +3/(2-t^2+t^4)],t=-2..2,\n thickness=3,color=red):\nplots[display]([ p1,p2,p3,p4],axes=boxed,\n orientation=[133,60]);" }}{PARA 13 "" 1 "" {GLPLOT3D 663 470 470 {PLOTDATA 3 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"Geometrical illustration of the chain rule " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Le t " }{XPPEDIT 18 0 "g(x,y)=4-x^2-y^2" "6#/-%\"gG6$%\"xG%\"yG,(\"\"%\" \"\"*$F'\"\"#!\"\"*$F(F-F." }{TEXT -1 62 ", and consider the curve def ined by the parametric equations " }{XPPEDIT 18 0 "PIECEWISE([x=t-t^3 ,``],[y=sin(Pi*t-Pi/3)-t^2,``])" "6#-%*PIECEWISEG6$7$/%\"xG,&%\"tG\"\" \"*$F*\"\"$!\"\"%!G7$/%\"yG,&-%$sinG6#,&*&%#PiGF+F*F+F+*&F9F+F-F.F.F+* $F*\"\"#F.F/" }{TEXT -1 10 ", that is " }{XPPEDIT 18 0 "PIECEWISE([x = u(t), ``],[y = v(t), ``])" "6#-%*PIECEWISEG6$7$/%\"xG-%\"uG6#%\"tG%!G 7$/%\"yG-%\"vG6#F,F-" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u(t)=t-t^ 3" "6#/-%\"uG6#%\"tG,&F'\"\"\"*$F'\"\"$!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "v(t)=sin(Pi*t-Pi/3)-t^2" "6#/-%\"vG6#%\"tG,&-%$sinG6#,& *&%#PiG\"\"\"F'F/F/*&F.F/\"\"$!\"\"F2F/*$F'\"\"#F2" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 74 "We shall discuss the chain rule as it ap plies to these two functions when " }{XPPEDIT 18 0 "t=1/2" "6#/%\"tG*& \"\"\"F&\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "t=1/2" "6#/%\"tG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 41 " the corresponding point on the curve is " }{XPPEDIT 18 0 "``(3/8,1/4 )" "6#-%!G6$*&\"\"$\"\"\"\"\")!\"\"*&F(F(\"\"%F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 63 "First we find the equation of the tangent plane to the surface " }{XPPEDIT 18 0 "z=g(x,y)" "6#/%\"zG-%\"gG6$%\" xG%\"yG" }{XPPEDIT 18 0 "``=4-x^2-y^2" "6#/%!G,(\"\"%\"\"\"*$%\"xG\"\" #!\"\"*$%\"yGF*F+" }{TEXT -1 23 " at the point given by " }{XPPEDIT 18 0 "x=3/8" "6#/%\"xG*&\"\"$\"\"\"\"\")!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y=1/4" "6#/%\"yG*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "We have \+ " }{XPPEDIT 18 0 "Diff(g(x,y),x)=g[x](x,y)" "6#/-%%DiffG6$-%\"gG6$%\" xG%\"yGF*-&F(6#F*6$F*F+" }{XPPEDIT 18 0 "``=-2*x" "6#/%!G,$*&\"\"#\"\" \"%\"xGF(!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Diff(g(x,y),y)=g[y ](x,y)" "6#/-%%DiffG6$-%\"gG6$%\"xG%\"yGF+-&F(6#F+6$F*F+" }{XPPEDIT 18 0 "``=-2*y" "6#/%!G,$*&\"\"#\"\"\"%\"yGF(!\"\"" }{TEXT -1 6 ", so \+ " }{XPPEDIT 18 0 "g[x](3/8,1/4)=-3/4" "6#/-&%\"gG6#%\"xG6$*&\"\"$\"\" \"\"\")!\"\"*&F,F,\"\"%F.,$*&F+F,F0F.F." }{TEXT -1 5 " and " } {XPPEDIT 18 0 "g[y](3/8,1/4)=-1/2" "6#/-&%\"gG6#%\"yG6$*&\"\"$\"\"\"\" \")!\"\"*&F,F,\"\"%F.,$*&F,F,\"\"#F.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "The tangent plane t o the surface " }{XPPEDIT 18 0 "z=4-x^2-y^2" "6#/%\"zG,(\"\"%\"\"\"*$% \"xG\"\"#!\"\"*$%\"yGF*F+" }{TEXT -1 14 " at the point " }{XPPEDIT 18 0 "``(3/8,1/4,g(3/8,1/4))" "6#-%!G6%*&\"\"$\"\"\"\"\")!\"\"*&F(F(\"\"% F*-%\"gG6$*&F'F(F)F**&F(F(F,F*" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "``(3 /8,1/4,243/64)" "6#-%!G6%*&\"\"$\"\"\"\"\")!\"\"*&F(F(\"\"%F**&\"$V#F( \"#kF*" }{TEXT -1 19 " on the surface is " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z=g(3/8,1/4)+g [x](3/8,1/4)*(x-3/8)+g[y]*(y-1/4)" "6#/%\"zG,(-%\"gG6$*&\"\"$\"\"\"\" \")!\"\"*&F+F+\"\"%F-F+*&-&F'6#%\"xG6$*&F*F+F,F-*&F+F+F/F-F+,&F4F+*&F* F+F,F-F-F+F+*&&F'6#%\"yGF+,&F=F+*&F+F+F/F-F-F+F+" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "z=243/64+(-3/4)*(x-3/8)+(-1/2)*(y-1/4)" "6#/%\"zG,(*&\" $V#\"\"\"\"#k!\"\"F(*&,$*&\"\"$F(\"\"%F*F*F(,&%\"xGF(*&F.F(\"\")F*F*F( F(*&,$*&F(F(\"\"#F*F*F(,&%\"yGF(*&F(F(F/F*F*F(F(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "z = 269/64-3/4*x-y/2;" "6#/%\"zG,(*&\"$p#\"\"\"\"#k!\" \"F(*(\"\"$F(\"\"%F*%\"xGF(F**&%\"yGF(\"\"#F*F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "mtaylor(4-x^2-y^2,[x=3/8,y=1 /4],2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#\"$p#\"#k\"\"\"*&#\"\"$ \"\"%F'%\"xGF'!\"\"*&#F'\"\"#F'%\"yGF'F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Now let " }{XPPEDIT 18 0 "h(x,y)=269/64-3/4*x-y/2" "6#/-% \"hG6$%\"xG%\"yG,(*&\"$p#\"\"\"\"#k!\"\"F,*(\"\"$F,\"\"%F.F'F,F.*&F(F, \"\"#F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "Compare the two composite functions " }{XPPEDIT 18 0 "t -> g(u(t),v(t))" "6#f*6# %\"tG7\"6$%)operatorG%&arrowG6\"-%\"gG6$-%\"uG6#F%-%\"vG6#F%F*F*F*" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "t->h(u(t),v(t))" "6#f*6#%\"tG7\"6$% )operatorG%&arrowG6\"-%\"hG6$-%\"uG6#F%-%\"vG6#F%F*F*F*" }{TEXT -1 8 " , where " }{XPPEDIT 18 0 "u(t)=t-t^3" "6#/-%\"uG6#%\"tG,&F'\"\"\"*$F' \"\"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v(t)=sin(Pi*t-Pi/3)-t^ 2" "6#/-%\"vG6#%\"tG,&-%$sinG6#,&*&%#PiG\"\"\"F'F/F/*&F.F/\"\"$!\"\"F2 F/*$F'\"\"#F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 58 "The first of these composite functions ca n be pictured as " }{TEXT 259 34 "a space-curve lying in the surface" }{TEXT -1 1 " " }{XPPEDIT 18 0 "z=g(x,y)" "6#/%\"zG-%\"gG6$%\"xG%\"yG " }{TEXT -1 41 ", while the second composite function is " }{TEXT 259 40 "a space-curve lying in the tangent plane" }{TEXT -1 1 " " } {XPPEDIT 18 0 "z=h(x,y)" "6#/%\"zG-%\"hG6$%\"xG%\"yG" }{TEXT -1 49 ". \+ Both of these space curves have the projection " }{XPPEDIT 18 0 "t->(u (t),v(t),0)" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"6%-%\"uG6#F%-%\"vG 6#F%\"\"!F*F*F*" }{TEXT -1 8 " in the " }{TEXT 308 1 "x" }{TEXT -1 1 " ," }{TEXT 309 1 "y" }{TEXT -1 8 " plane. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Note that points " }{TEXT 259 25 " in the same vertical line" }{TEXT -1 47 " on all three space-curves ar e associated with " }{TEXT 259 24 "the same parameter value" }{TEXT -1 1 " " }{TEXT 312 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "If we imagine that t represents time so that the functions " } {XPPEDIT 18 0 "t -> g(u(t),v(t))" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG 6\"-%\"gG6$-%\"uG6#F%-%\"vG6#F%F*F*F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t->h(u(t),v(t))" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"-%\"hG6$ -%\"uG6#F%-%\"vG6#F%F*F*F*" }{TEXT -1 81 ", give the location of a poi nt in each of these two space-curves at time t, then " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 742 "g := (x, y) -> 4-x^2-y^2:\nh := (x,y) -> 269/64-3/4*x-y/2;\nu := t -> t-t^3:\nv := t -> sin(Pi*t-Pi/3)-t^2:\np1 := plot3d(g(x,y),x=-2..2,y=-2..2,\n \+ color=COLOR(RGB,.8,.6,1),grid=[35,35]):\np2 := plot3d(h(x,y),x=-2 ..2,y=-2..2,\n color=COLOR(RGB,.9,.8,.7),grid=[35,35]):\np3 := p lot3d(0,x=-2..2,y=-2..2,\n color=COLOR(RGB,.8,.8,.8),grid=[25, 25]):\np4 := plots[spacecurve]([u(t),v(t),0],t=-1.522..1.372,\n thic kness=3,color=green):\np5 := plots[spacecurve]([u(t),v(t),g(u(t),v(t)) ],t=-1.522..1.372,\n thickness=3,color=red):\np6 := plots[spacecurve ]([u(t),v(t),h(u(t),v(t))],t=-1.522..1.372,\n thickness=3,color=brow n):\nplots[display]([p1,p2,p3,p4,p5,p6],axes=boxed,\n view=[-2..2,- 2..2,0..6.8],gridstyle=triangular,style=wireframe);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"hGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(#\"$p #\"#k\"\"\"*&#\"\"$\"\"%F19$F1!\"\"*&#F1\"\"#F19%F1F7F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT3D 515 409 409 {PLOTDATA 3 "6--%%GRIDG6&;$!\"#\"\"!$ \"\"#F)F&X,%)anythingG6\"6\"[gl'!%\"!!#dgo\"D\"DC010000000000000C00C58 94D10D4986C008E9DACBBCAD9EC005B3D1F00E2C4AC002B67A3E01C589BFFFE3A76B2E F2B5BFFACBBCAD9E8F7FBFF625344352616FBFF1F00E2C4A6885BFEC5894D10D4982BF E5B3D1F00E2C47BFDFE3A76B2EF2AFBFD625344352616ABFCC5894D10D497ABFBFE3A7 6B2EF2A4BFAC5894D10D496DBF8C5894D10D4955B985200000000000BF8C5894D10D49 B6BFAC5894D10D499EBFBFE3A76B2EF2C9BFCC5894D10D4991BFD6253443526178BFDF E3A76B2EF2C0BFE5B3D1F00E2C50BFEC5894D10D498DBFF1F00E2C4A688BBFF6253443 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"" {TEXT -1 27 "The \+ partial derivatives of " }{XPPEDIT 18 0 "h(x,t)= 243/64+(-3/4)*(x-3/8) +(-1/2)*(y-1/4)" "6#/-%\"hG6$%\"xG%\"tG,(*&\"$V#\"\"\"\"#k!\"\"F,*&,$* &\"\"$F,\"\"%F.F.F,,&F'F,*&F2F,\"\")F.F.F,F,*&,$*&F,F,\"\"#F.F.F,,&%\" yGF,*&F,F,F3F.F.F,F," }{XPPEDIT 18 0 "``=269/64-3/4*x-y/2" "6#/%!G,(*& \"$p#\"\"\"\"#k!\"\"F(*(\"\"$F(\"\"%F*%\"xGF(F**&%\"yGF(\"\"#F*F*" } {TEXT -1 24 " coincide with those of " }{XPPEDIT 18 0 "g(x,y) = 4-x^2- y^2;" "6#/-%\"gG6$%\"xG%\"yG,(\"\"%\"\"\"*$F'\"\"#!\"\"*$F(F-F." } {TEXT -1 14 " at the point " }{XPPEDIT 18 0 "``(3/8,1/4)" "6#-%!G6$*& \"\"$\"\"\"\"\")!\"\"*&F(F(\"\"%F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "For the space-curve " }{XPPEDIT 18 0 " z= h(u(t),v(t))" "6#/%\"zG-%\"hG6$-%\"uG6#%\"tG-%\"vG 6#F+" }{XPPEDIT 18 0 "`` = 243/64+(-3/4)*(u(t)-3/8)+(-1/2)*(v(t)-1/4); " "6#/%!G,(*&\"$V#\"\"\"\"#k!\"\"F(*&,$*&\"\"$F(\"\"%F*F*F(,&-%\"uG6#% \"tGF(*&F.F(\"\")F*F*F(F(*&,$*&F(F(\"\"#F*F*F(,&-%\"vG6#F4F(*&F(F(F/F* F*F(F(" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dz/dt = (-3/4)*u*`'`(t) + (-1/2)*v*`'`(t)" "6#/*&%#dzG \"\"\"%#dtG!\"\",&*(,$*&\"\"$F&\"\"%F(F(F&%\"uGF&-%\"'G6#%\"tGF&F&*(,$ *&F&F&\"\"#F(F(F&%\"vGF&-F16#F3F&F&" }{TEXT -1 1 "," }}{PARA 0 "" 0 " " {TEXT -1 9 "which is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dz/dt=h[x](3/8, 1/4)*u*`'`(t) +h[y](3/8, 1/4)*v*`'`(t)" "6#/*&%# dzG\"\"\"%#dtG!\"\",&*(-&%\"hG6#%\"xG6$*&\"\"$F&\"\")F(*&F&F&\"\"%F(F& %\"uGF&-%\"'G6#%\"tGF&F&*(-&F-6#%\"yG6$*&F2F&F3F(*&F&F&F5F(F&%\"vGF&-F 86#F:F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "Hence the c hain rule clearly applies in this case. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The idea now is to demonstrate the \"reasonableness\" of the chain rule " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dz/dt = g[x](3/8,1/4)*u*`'`(t)+g[y](3/8,1/4)*v*` '`(t)" "6#/*&%#dzG\"\"\"%#dtG!\"\",&*(-&%\"gG6#%\"xG6$*&\"\"$F&\"\")F( *&F&F&\"\"%F(F&%\"uGF&-%\"'G6#%\"tGF&F&*(-&F-6#%\"yG6$*&F2F&F3F(*&F&F& F5F(F&%\"vGF&-F86#F:F&F&" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 311 16 "________________" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 " for the space-cur ve " }{XPPEDIT 18 0 "z=g(u(t),v(t))" "6#/%\"zG-%\"gG6$-%\"uG6#%\"tG-% \"vG6#F+" }{TEXT -1 55 " from the fact that this curve is \"close\" to the curve " }{XPPEDIT 18 0 "z=h(u(t),v(t))" "6#/%\"zG-%\"hG6$-%\"uG6# %\"tG-%\"vG6#F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 66 "Let's zoom in closer to the point of cont act of the tangent plane " }{XPPEDIT 18 0 "``(3/8,1/4)" "6#-%!G6$*&\" \"$\"\"\"\"\")!\"\"*&F(F(\"\"%F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 556 "g := (x,y) \+ -> 4-x^2-y^2: h := (x,y) -> 269/64-3/4*x-y/2:\nu := t -> t-t^3: v := t -> sin(Pi*t-Pi/3)-t^2:\np1 := plot3d(g(x,y),x=0..0.5,y=0..0.5,\n \+ color=COLOR(RGB,.8,.6,1),grid=[35,35]):\np2 := plot3d(h(x,y),x=0..0. 5,y=0..0.5,\n color=COLOR(RGB,.9,.8,.7),grid=[35,35]):\np3 := pl ots[spacecurve]([u(t),v(t),g(u(t),v(t))],t=0.38..0.96,\n thickness=3 ,color=red):\np4 := plots[spacecurve]([u(t),v(t),h(u(t),v(t))],t=0.38. .0.96,\n thickness=3,color=brown):\nplots[display]([p1,p2,p3,p4],ori entation=[122,52],\n gridstyle=triangular,style=wireframe);" }} {PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6*-%%GRIDG6&;$\"\" !F($\"\"&!\"\"F&X,%)anythingG6\"6\"[gl'!%\"!!#dgo\"D\"D401000000000000 0400FFF8E9DACBBCB400FFE3A76B2EF2B400FFC038B129A22400FF8E9DACBBCAE400FF 4ED65DE56CF400FF00E2C4A6887400FEA4C2E0FF1D4400FE3A76B2EF2B6400FDC1FE3A 76B2F400FD3B597795B3D400FCA6886A4C2E1400FC038B129A21B400FB5261707F8EA4 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contact of the tange nt plane, the two space curves (one in the surface and the other in th e tangent plane) become virtually indistinguishable. 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separation`],labeldirections=[HORIZONTAL,V ERTICAL]);" }}{PARA 13 "" 1 "" {GLPLOT2D 519 197 197 {PLOTDATA 2 "6'-% 'CURVESG6#7U7$$\"3=++++++!f%!#=$\"3iri+dB+Zf!#?7$$\"3M+]P>(3)*f%F*$\"3 UbAM.>XYcF-7$$\"3'****\\(Quh4YF*$\"3CSaot$pXN&F-7$$\"3\")\\7.`>:=YF*$ \"3TQutpRg2^F-7$$\"3A+DJnkoEYF*$\"3!\\C)*f)f5n[F-7$$\"3-+]7vB)ek%F*$\" 3/8C=K)y&\\VF-7$$\"3E+](e_0_m%F*$\"3]@DZ$4F4'QF-7$$\"3/+D1PoV%o%F*$\"3 GFr;vlE1MF-7$$\"3[+DJCnE-ZF*$\"3=ZK87I^7IF-7$$\"3S+D\")R&G2s%F*$\"3M[( z6(e^KEF-7$$\"3(**\\7ex@)RZF*$\"3OMUe \"*[1N>F-7$$\"3-++vM0VyZF*$\"3ftQ?i'y:i\"F-7$$\"3I+]P^Pn&z%F*$\"3cZ4>0 n'*p8F-7$$\"3b++DHb3:[F*$\"3BoLo0?o86F-7$$\"3[++v8qdM[F*$\"3]RqHN&Fcp7$$\"3G++v\")3q!*[F*$\"3+KYtCZ.xPFcp7$$\"3N+++-qvJ$)!#B7$$\"3P+]7`Bu-]F*$\"31ON'f7h5F#!#C7$$\"3N+DcKFX?]F*$\" 3;%G6+5QQD\"Fbr7$$\"3Q+]PpdbR]F*$\"35Z&=5d5>l%Fbr7$$\"3[+D\"eX)Rf]F*$ \"33\"=`NHz+/\"Fcp7$$\"3-+vos:nw]F*$\"3')HrCf10?<&F*$ \"3k;8I**4m$H)Fcp7$$\"3a++]kR:!>&F*$\"3)zcly3qe+\"F-7$$\"3a++vLqe4_F*$ 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" }}{PARA 0 "" 0 "" {TEXT -1 27 "Along the contour gi ven by " }{XPPEDIT 18 0 "z=3" "6#/%\"zG\"\"$" }{TEXT -1 10 ", we have \+ " }{XPPEDIT 18 0 "dz/dx=0" "6#/*&%#dzG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 8 ". Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``( 3*x^2-2*y)+``(2*y-2*x);" "6#,&-%!G6#,&*&\"\"$\"\"\"*$%\"xG\"\"#F*F**&F -F*%\"yGF*!\"\"F*-F%6#,&*&F-F*F/F*F**&F-F*F,F*F0F*" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = 0;" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx=-(3*x^2-2*y)/(2*y-2*x)" "6# /*&%#dyG\"\"\"%#dxG!\"\",$*&,&*&\"\"$F&*$%\"xG\"\"#F&F&*&F0F&%\"yGF&F( F&,&*&F0F&F2F&F&*&F0F&F/F&F(F(F(" }{XPPEDIT 18 0 "``= (3*x^2-2*y)/(2*x -2*y)" "6#/%!G*&,&*&\"\"$\"\"\"*$%\"xG\"\"#F)F)*&F,F)%\"yGF)!\"\"F),&* &F,F)F+F)F)*&F,F)F.F)F/F/" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 48 "On the other hand, differentiating the equation " }{XPPEDIT 18 0 "x^3-2*x*y+y^2 = 3" "6#/,(*$%\"xG\"\"$\"\"\"*(\"\"#F(F&F(%\"yGF(!\" \"*$F+F*F(F'" }{TEXT -1 17 " with respect to " }{TEXT 296 1 "x" } {TEXT -1 117 ", following the standard technique of implicit different iation, and using the product rule to differentiate the term " } {XPPEDIT 18 0 "-2*x*y" "6#,$*(\"\"#\"\"\"%\"xGF&%\"yGF&!\"\"" }{TEXT -1 8 ", gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*x ^2-2*y-2*x" "6#,(*&\"\"$\"\"\"*$%\"xG\"\"#F&F&*&F)F&%\"yGF&!\"\"*&F)F& F(F&F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y;" "6#,&*&%#dyG\"\"\" %#dxG!\"\"F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx \+ = 0" "6#/*&%#dyG\"\"\"%#dxG!\"\"\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 76 "which is exactly the same equation as was obtained b y using the chain rule. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 32 "For a general curve of the form " }{XPPEDIT 18 0 " g(x,y)=a" "6#/-%\"gG6$%\"xG%\"yG%\"aG" }{TEXT -1 8 ", where " }{TEXT 297 1 "a" }{TEXT -1 21 " is constant we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(g(x,y),x)+Diff(g(x,y),y)*``(dy/dx) =0" "6#/,&-%%DiffG6$-%\"gG6$%\"xG%\"yGF+\"\"\"*&-F&6$-F)6$F+F,F,F--%!G 6#*&%#dyGF-%#dxG!\"\"F-F-\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy /dx=-Diff(g(x,y),x)/Diff(g(x,y),y)" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&-%% DiffG6$-%\"gG6$%\"xG%\"yGF1F&-F,6$-F/6$F1F2F2F(F(" }{TEXT -1 3 ", " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 298 8 "________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "provided that the partial derivativ e " }{XPPEDIT 18 0 "Diff(g(x,y),y)" "6#-%%DiffG6$-%\"gG6$%\"xG%\"yGF* " }{TEXT -1 23 " is not identically 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "g := (x,y) -> x^3-2*x*y +y^2;\nDiff(g(x,y),x)/Diff(g(x,y),y);\nvalue(%);\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"gGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(*$)9$ \"\"$\"\"\"F2*(\"\"#F29%F2F0F2!\"\"*$)F5F4F2F2F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%DiffG6$,(*$)%\"xG\"\"$\"\"\"F,*(\"\"#F,F*F,%\"y GF,!\"\"*$)F/F.F,F,F*F,-F%6$F'F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *&,&*$)%\"xG\"\"#\"\"\"\"\"$*&F(F)%\"yGF)!\"\"F),&F'!\"#*&F(F)F,F)F)F- " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "implicitdiff(x^3-2*x*y+y^2=3,y,x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*&,&*$)%\"xG\"\"#\"\"\"\"\"$*&F)F*%\"yGF*!\"\"F*,&F (F*F-F.F.#F*F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "z=x^2+y^2" "6#/%\"zG ,&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "x=sinh(t)" "6#/%\"xG-%%sinhG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=t^2" "6#/%\"yG*$%\"tG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "dz/dt" "6#*&%#dzG\"\"\"%#dtG!\"\"" }{TEXT -1 79 " (a) by using the chain rule (b) by direct substitution \+ of the expressions for " }{TEXT 292 1 "x" }{TEXT -1 5 " and " }{TEXT 293 1 "y" }{TEXT -1 17 " in the equation " }{XPPEDIT 18 0 "z=x^2+y^2" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "z = exp(-x^2-y^2);" "6#/%\"zG-%$expG6#,&*$%\"xG\"\"#!\"\"*$%\"yGF+F," }{TEXT -1 7 " where \+ " }{XPPEDIT 18 0 "x = t;" "6#/%\"xG%\"tG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y = sqrt(t);" "6#/%\"yG-%%sqrtG6#%\"tG" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "dz/dt" "6#*&%#d zG\"\"\"%#dtG!\"\"" }{TEXT -1 79 " (a) by using the chain rule (b) by \+ direct substitution of the expressions for " }{TEXT 313 1 "x" }{TEXT -1 5 " and " }{TEXT 314 1 "y" }{TEXT -1 17 " in the equation " } {XPPEDIT 18 0 "z = exp(-x^2-y^2);" "6#/%\"zG-%$expG6#,&*$%\"xG\"\"#!\" \"*$%\"yGF+F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "_______ ____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "____________________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "z = x^2+x*y+y^2;" "6#/%\"zG,(*$%\"xG\"\"#\"\"\" *&F'F)%\"yGF)F)*$F+F(F)" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "x = cos (t);" "6#/%\"xG-%$cosG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = sin(t);" "6#/%\"yG-%$sinG6#%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "dz/dt" "6#*&%#dzG\"\"\"%#dtG!\"\"" }{TEXT -1 79 " (a) by using the chain rule (b) by direct substitution \+ of the expressions for " }{TEXT 315 1 "x" }{TEXT -1 5 " and " }{TEXT 316 1 "y" }{TEXT -1 17 " in the equation " }{XPPEDIT 18 0 "z = x^2+x*y +y^2;" "6#/%\"zG,(*$%\"xG\"\"#\"\"\"*&F'F)%\"yGF)F)*$F+F(F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "_____________________________ ______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "z = exp(2*x+3*y);" "6#/%\"zG-%$expG6#,&*&\"\"#\"\"\"%\"xGF+F+*&\"\"$F+% \"yGF+F+" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "x = ln(t);" "6#/%\"xG- %#lnG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = ln(t^2+1);" "6#/ %\"yG-%#lnG6#,&*$%\"tG\"\"#\"\"\"F,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "dz/dt" "6#*&%#dzG\"\"\"%#dtG!\" \"" }{TEXT -1 79 " (a) by using the chain rule (b) by direct substitut ion of the expressions for " }{TEXT 317 1 "x" }{TEXT -1 5 " and " } {TEXT 318 1 "y" }{TEXT -1 17 " in the equation " }{XPPEDIT 18 0 "z = e xp(2*x+3*y);" "6#/%\"zG-%$expG6#,&*&\"\"#\"\"\"%\"xGF+F+*&\"\"$F+%\"yG F+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "_________________ __________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 44 "Use the partial deri vatives of the function " }{XPPEDIT 18 0 "g(x,y)=x^5+y^5+x*y" "6#/-%\" gG6$%\"xG%\"yG,(*$F'\"\"&\"\"\"*$F(F+F,*&F'F,F(F,F," }{TEXT -1 23 " to find a formula for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\" \"" }{TEXT -1 17 " along the curve " }{XPPEDIT 18 0 "x^5+y^5+x*y+4=0" "6#/,**$%\"xG\"\"&\"\"\"*$%\"yGF'F(*&F&F(F*F(F(\"\"%F(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "_____________________________ ______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 44 "Use the partial deriv atives of the function " }{XPPEDIT 18 0 "g(x,y)=x^2*y+3*y^3*x^4" "6#/- %\"gG6$%\"xG%\"yG,&*&F'\"\"#F(\"\"\"F,*(\"\"$F,*$F(F.F,F'\"\"%F," } {TEXT -1 23 " to find a formula for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dy G\"\"\"%#dxG!\"\"" }{TEXT -1 17 " along the curve " }{XPPEDIT 18 0 "x^ 2*y+3*y^3*x^4=4" "6#/,&*&%\"xG\"\"#%\"yG\"\"\"F)*(\"\"$F)*$F(F+F)F&\" \"%F)F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 35 "______________ _____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_______________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 44 "Use the part ial derivatives of the function " }{XPPEDIT 18 0 "g(x,y)=y-x*exp(y)" " 6#/-%\"gG6$%\"xG%\"yG,&F(\"\"\"*&F'F*-%$expG6#F(F*!\"\"" }{TEXT -1 23 " to find a formula for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG !\"\"" }{TEXT -1 17 " along the curve " }{XPPEDIT 18 0 "y-x*exp(y)=1" "6#/,&%\"yG\"\"\"*&%\"xGF&-%$expG6#F%F&!\"\"F&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q8 " }}{PARA 0 "" 0 "" {TEXT -1 44 "Use the partial derivatives of the f unction " }{XPPEDIT 18 0 "g(x,y) = x^3+4*y*sin(x*y);" "6#/-%\"gG6$%\"x G%\"yG,&*$F'\"\"$\"\"\"*(\"\"%F,F(F,-%$sinG6#*&F'F,F(F,F,F," }{TEXT -1 23 " to find a formula for " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\" \"%#dxG!\"\"" }{TEXT -1 17 " along the curve " }{XPPEDIT 18 0 "x^3+4*y *sin(x*y)=0" "6#/,&*$%\"xG\"\"$\"\"\"*(\"\"%F(%\"yGF(-%$sinG6#*&F&F(F+ F(F(F(\"\"!" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "________ ___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_______________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 19 "Chain rule diagram " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 505 "p1 := plot([[[1,8],[8,1]],[[1,-8],[8,-1]],\n [[-1, 8],[-8,1]],[[-1,-8],[-8,-1]]],color=COLOR(RGB,.4,0,1),\n linestyle= 2,thickness=2):\nt1 := plots[textplot]([[0,9,`t`],[8.3,0,`y`],[-8.3,0, `x`],\n [0,-9,`z`],[5.5,6,`dy`],[5.5,4.8,`dt`],[5.5,5.9,`_`],\n \+ [-5.5,6,`dx`],[-5.5,4.8,`dt`],[-5.5,5.9,`_`],\n [5.1,-6.2,`o`],[ 5.1,-4.9,`o`],[5.5,-6,`)y`],[5.5,-4.8,`)z`],[5.4,-5.1,`__`],\n[-5.9,-6 .2,`o`],[-5.9,-4.9,`o`],[-5.5,-6,`)x`],[-5.5,-4.8,`)z`],[-5.7,-5.1,`__ `]]):\nplots[display]([p1,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }