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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 105 "The tangent plane to a surface, \+
linear approximation and differentiabilty for functions of two variabl
es " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Can
ada" }}{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 70 "Linear approximation of a function of two variable
s and tangent planes" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}
}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the function " }{XPPEDIT 18 0 "
g(x,y) = 3/(1+x^2+x*y+2*y^2);" "6#/-%\"gG6$%\"xG%\"yG*&\"\"$\"\"\",*F+
F+*$F'\"\"#F+*&F'F+F(F+F+*&F.F+*$F(F.F+F+!\"\"" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
129 "g := (x,y) -> 3/(1+x^2+x*y+2*y^2);\nplot3d(g(x,y),x=-3..3,y=-3..3
,\n        color=COLOR(RGB,.8,.6,1),axes=boxed,lightmodel=light4);" }}
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03FDC71C71C71C71C3FD8F9C18F9C18FA3FD5A240E6C2B4483FD29E4129E4129E3FD00
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5555555553FD80000000000003FDA7B9611A7B9613FDC71C71C71C71C3FDD89D89D89D
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55555555555553FD2BB512BB512BC3FD0572620AE4C413FCC71C71C71C71C3FC8C6318
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B143FC167B493F17EEA3FBEDFD6D58C389A3FBB7D6C3DDA338B3FB89374BC6A7EFA3FB
60FA066F397363FC435E50D79435E3FC6C16C16C16C173FC999999999999A3FCCB5D4E
F40991E3FD00000000000003FD1A7B9611A7B963FD33333333333343FD47AE147AE147
B3FD55555555555553FD5A240E6C2B4483FD55555555555553FD47AE147AE147B3FD33
333333333343FD1A7B9611A7B963FD00000000000003FCCB5D4EF40991E3FC99999999
9999A3FC6C16C16C16C173FC435E50D79435E3FC1F7047DC11F703FC00000000000003
FBC93A581C93A583FB999999999999A3FB702E05C0B81703FB4C1BACF914C1C-%&COLO
RG6&%$RGBG$\"\")!\"\"$\"\"'F6$\"\"\"F)-%+LIGHTMODELG6#%(LIGHT_4G-%*AXE
SSTYLEG6#%$BOXG-%+AXESLABELSG6%%\"xG%\"yGQ!F/" 1 2 0 1 10 0 2 1 6 2 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "We shall investigate how the va
lues of the function change in the neighbourhood of the point" }
{XPPEDIT 18 0 "``(1,2)" "6#-%!G6$\"\"\"\"\"#" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 46 "First we calculate the change in the valu
e of " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 6 " wh
en " }{TEXT 270 1 "x" }{TEXT -1 21 " increases from 2 to " }{XPPEDIT 
18 0 "2+h" "6#,&\"\"#\"\"\"%\"hGF%" }{TEXT -1 8 ", where " }{XPPEDIT 
18 0 "h = 1/1000;" "6#/%\"hG*&\"\"\"F&\"%+5!\"\"" }{TEXT -1 3 ".  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
128 "g := (x,y) -> 3/(1+x^2+x*y+2*y^2):\nh := 0.001;\n'g(2,1)' = evalf
(g(2,1));\n'g(2+h,1)'=g(2+h,1);\n'g(2+h,1)-g(2,1)'=g(2+h,1)-g(2,1);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"$" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%\"gG6$\"\"#\"\"\"$\"+LLLLL!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"gG6$,&\"\"#\"\"\"%\"hGF)F)$\"+T@[JL!#5" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/,&-%\"gG6$,&\"\"#\"\"\"%\"hGF*F*F*-F&6$F)F*
!\"\"$!(#>^=!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 27 "The partial derivatives of " }{XPPEDIT 18 0 "g(x,y);" "6#
-%\"gG6$%\"xG%\"yG" }{TEXT -1 17 " with respect to " }{TEXT 268 1 "x" 
}{TEXT -1 5 " and " }{TEXT 269 1 "y" }{TEXT -1 31 " can be obtained as
 functions  " }{TEXT 262 3 "g_x" }{TEXT -1 5 " and " }{TEXT 262 3 "g_y
" }{TEXT -1 31 " respectively by using Maple's " }{TEXT 0 1 "D" }
{TEXT -1 11 " operator. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "g := (x,y) -> 3/(1+x^2+x*y+2*y^2);
\ng_x := D[1](g);\ng_y := D[2](g);  " }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"gGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&\"\"\"F/,*F/F/*$)9
$\"\"#F/F/*&F3F/9%F/F/*&F4F/)F6F4F/F/!\"\"\"\"$F)F)F)" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$g_xGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*
&,*\"\"\"F0*$)9$\"\"#F0F0*&F3F09%F0F0*&F4F0)F6F4F0F0!\"#,&F3F4F6F0F0!
\"$F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$g_yGf*6$%\"xG%\"yG6\"6
$%)operatorG%&arrowGF),$*&,*\"\"\"F0*$)9$\"\"#F0F0*&F3F09%F0F0*&F4F0)F
6F4F0F0!\"#,&F3F0*&\"\"%F0F6F0F0F0!\"$F)F)F)" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g_x(2,1);\ng
_y(2,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"&\"#F" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6##!\"#\"\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "g[x](2,1)=Limit((g
(2+h,1)-g(2,1))/h,h=0)" "6#/-&%\"gG6#%\"xG6$\"\"#\"\"\"-%&LimitG6$*&,&
-F&6$,&F*F+%\"hGF+F+F+-F&6$F*F+!\"\"F+F4F7/F4\"\"!" }{TEXT -1 23 ", we
 would expect that " }{XPPEDIT 18 0 "g(2+h,1)-g(2,1))/h" "6#*&,&-%\"gG
6$,&\"\"#\"\"\"%\"hGF*F*F*-F&6$F)F*!\"\"F*F+F." }{TEXT -1 1 " " }
{TEXT 318 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "g[x](2,1)" "6#-&%\"gG6
#%\"xG6$\"\"#\"\"\"" }{TEXT -1 6 " when " }{TEXT 319 1 "h" }{TEXT -1 
20 " is small, that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "g(2+h,1)-g(2,1))" "6#,&-%\"gG6$,&\"\"#\"\"\"%\"hGF)F)F)
-F%6$F(F)!\"\"" }{TEXT -1 1 " " }{TEXT 320 1 "~" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "g[x](2,1)*h" "6#*&-&%\"gG6#%\"xG6$\"\"#\"\"\"F+%\"hGF+
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 36 "h := 0.001;\n'g_x(2,1)*h'=g_x(2,1)*h;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&-%$g_xG6$\"\"#\"\"\"F)%\"hGF)$!+_=&=&=!#8" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "This estimated change agrees wi
th the actual change to 3 figures. " }}{PARA 0 "" 0 "" {TEXT -1 44 "No
w we calculate the change in the value of " }{XPPEDIT 18 0 "g(x,y)" "6
#-%\"gG6$%\"xG%\"yG" }{TEXT -1 6 " when " }{TEXT 321 1 "y" }{TEXT -1 
21 " increases from 1 to " }{XPPEDIT 18 0 "1+k;" "6#,&\"\"\"F$%\"kGF$
" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "k = 1/2000;" "6#/%\"kG*&\"\"
\"F&\"%+?!\"\"" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since \+
" }{XPPEDIT 18 0 "g[y](2,1)=Limit((g(2,1+k)-g(2,1))/k,k=0)" "6#/-&%\"g
G6#%\"yG6$\"\"#\"\"\"-%&LimitG6$*&,&-F&6$F*,&F+F+%\"kGF+F+-F&6$F*F+!\"
\"F+F4F7/F4\"\"!" }{TEXT -1 23 ", we would expect that " }{XPPEDIT 18 
0 "g(2,1+k)-g(2,1))/k" "6#*&,&-%\"gG6$\"\"#,&\"\"\"F*%\"kGF*F*-F&6$F(F
*!\"\"F*F+F." }{TEXT -1 1 " " }{TEXT 322 1 "~" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "g[y](2,1)" "6#-&%\"gG6#%\"yG6$\"\"#\"\"\"" }{TEXT -1 6 
" when " }{TEXT 323 1 "k" }{TEXT -1 20 " is small, that is, " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(2,1+k)-g(2,1)" "6#,&-%\"g
G6$\"\"#,&\"\"\"F)%\"kGF)F)-F%6$F'F)!\"\"" }{TEXT -1 1 " " }{TEXT 324 
1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "g[y](2,1)*k" "6#*&-&%\"gG6#%\"yG
6$\"\"#\"\"\"F+%\"kGF+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "k := 0.0005;\n'g(2,1)'=e
valf(g(2,1));\n'g(2,1+k)'=g(2,1+k);\n'g(2,1+k)-g(2,1)'=g(2,1+k)-g(2,1)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG$\"\"&!\"%" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"gG6$\"\"#\"\"\"$\"+LLLLL!#5" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/-%\"gG6$\"\"#,&\"\"\"F)%\"kGF)$\"+1CAKL!#5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%\"gG6$\"\"#,&\"\"\"F*%\"kGF*F*-F&
6$F(F*!\"\"$!(F46\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "'g_y(2,1)*k'=g_y(2,1)*k;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*&-%$g_yG6$\"\"#\"\"\"F)%\"kGF)$!+66666!#8
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "This
 estimated change again agrees with the actual change to 3 figures. " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Now con
sider " }{TEXT 259 14 "changing both " }{TEXT 276 1 "x" }{TEXT 259 5 "
 and " }{TEXT 275 1 "y" }{TEXT 259 15 " simultaneously" }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 119 "h := 0.001; \nk := 0.0005;\n'g(2,1)' = evalf(g(2,1));\n'g(2+h
,1+k)' = g(2+h,1+k);\n'g(2+h,1+k)-g(2,1)' = g(2+h,1+k)-g(2,1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"kG$\"\"&!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
\"gG6$\"\"#\"\"\"$\"+LLLLL!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"
gG6$,&\"\"#\"\"\"%\"hGF),&F)F)%\"kGF)$\"+hAPIL!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&-%\"gG6$,&\"\"#\"\"\"%\"hGF*,&F*F*%\"kGF*F*-F&6$F)F*
!\"\"$!(s5'H!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 24 "If we consider changing " }{TEXT 271 1 "x" }{TEXT -1 19 "
 first followed by " }{TEXT 272 1 "y" }{TEXT -1 45 ", the total change
 is approximately equal to " }{XPPEDIT 18 0 "g[x](2,1)*h+g[y](2+h,1)*k
" "6#,&*&-&%\"gG6#%\"xG6$\"\"#\"\"\"F,%\"hGF,F,*&-&F'6#%\"yG6$,&F+F,F-
F,F,F,%\"kGF,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "h := 0.001; \nk := 0.0005;\n
'g_x(2,1)*h+g_y(2+h,1)*k'=g_x(2,1)*h+g_y(2+h,1)*k;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"kG$\"\"&!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%$g_xG6$\"\"
#\"\"\"F*%\"hGF*F**&-%$g_yG6$,&F)F*F+F*F*F*%\"kGF*F*$!+dT\">'H!#8" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "If we con
sider changing " }{TEXT 273 1 "y" }{TEXT -1 19 " first followed by " }
{TEXT 274 1 "x" }{TEXT -1 45 ", the total change is approximately equa
l to " }{XPPEDIT 18 0 "g[y](2,1)*k+g[x](2,1+k)*h;" "6#,&*&-&%\"gG6#%\"
yG6$\"\"#\"\"\"F,%\"kGF,F,*&-&F'6#%\"xG6$F+,&F,F,F-F,F,%\"hGF,F," }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 76 "h := 0.001; \nk := 0.0005;\n'g_y(2,1)*k+g_x(2,1+
k)*h'=g_y(2,1)*k+g_x(2,1+k)*h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
hG$\"\"\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG$\"\"&!\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%$g_yG6$\"\"#\"\"\"F*%\"kGF*F**&
-%$g_xG6$F),&F*F*F+F*F*%\"hGF*F*$!+oQ\">'H!#8" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "There is very little diff
erence between these two estimates." }}{PARA 0 "" 0 "" {TEXT -1 110 "I
n fact, a simpler estimate can be obtained by evaluating each of the p
artial derivatives at the initial point" }{XPPEDIT 18 0 "``(2,1)" "6#-
%!G6$\"\"#\"\"\"" }{TEXT -1 14 " and computing" }}{PARA 256 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "g[x](2,1)*h+g[y](2,1)*k" "6#,&*&-&%\"g
G6#%\"xG6$\"\"#\"\"\"F,%\"hGF,F,*&-&F'6#%\"yG6$F+F,F,%\"kGF,F," }
{TEXT -1 13 " ------- (i)." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 
277 10 "__________" }{TEXT -1 13 "             " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "h := 0.001; \+
\nk := 0.0005;\n'g_x(2,1)*h+g_y(2,1)*k'=g_x(2,1)*h+g_y(2,1)*k;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"kG$\"\"&!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&
*&-%$g_yG6$\"\"#\"\"\"F*%\"kGF*F**&-%$g_xGF(F*%\"hGF*F*$!+jH'H'H!#8" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Looking
 at a small section of the surface in the neighbourhood of the point" 
}{XPPEDIT 18 0 "``(1,2)" "6#-%!G6$\"\"\"\"\"#" }{TEXT -1 37 ", we see \+
that it looks like a plane. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "g := (x,y) -> 3/(1+x^2+x*y+
2*y^2);\nplot3d(g(x,y),x=1.9..2.1,y=0.9..1.1,\n     color=COLOR(RGB,.8
,.6,1),axes=boxed,lightmodel=light4);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"gGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&\"\"\"F/,*F/F/*$)
9$\"\"#F/F/*&F3F/9%F/F/*&F4F/)F6F4F/F/!\"\"\"\"$F)F)F)" }}{PARA 13 "" 
1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6&-%%GRIDG6&;$\"#>!\"\"$\"#@F)
;$\"\"*F)$\"#6F)X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\":\":3FD82E6D9F4649
833FD80ACAD15F1F533FD7E75A447D14163FD7C41C0B3FCFD63FD7A1103535982D3FD7
7E36CEF050F63FD75B8FE21A291D3FD7391B7589F30B3FD716D98D57285F3FD6F4CA2A
ED98C83FD6D2ED4D20C3DB3FD6B142F03EDDCF3FD68FCB0E237F413FD66E859E49FFF3
3FD64D7295DF7CE73FD62C91E7D489EE3FD60BE384EE8F113FD5EB675BD8D2253FD5CB
1D59352D0C3FD5AB0567AC710D3FD58B1F6FFE77D23FD56B6B5911E2AB3FD54BE90803
88A03FD52C98603594203FD50D79435E50DA3FD80FFCF84C3BC13FD7ECA5D726BB203F
D7C98046C323563FD7A68C5A6158123FD783CA223905FD3FD76139AB8E40C63FD73EDB
00C5CF943FD71CAE297927763FD6FAB32A8A13673FD6D8EA063609C53FD6B752BC292E
F83FD695ED499105543FD674B9A92ECA1D3FD653B7D3697FD53FD632E7BE5FA5F63FD6
12495DF89E4C3FD5F1DCA3F5C0423FD5D1A180031A683FD5B197DFC7E2B53FD591BFAE
F695DD3FD57218D75CC6453FD552A340F29B2D3FD5335ED1EA009C3FD5144B6EBD88B0
3FD4F568FA3EFF0A3FD7F1BD663DFD383FD7CEB137154FE03FD7ABD5EAF57ABE3FD789
2B95B9024E3FD766B2483B486F3FD7446A106CC98E3FD72252F9670A7A3FD7006D0B80
36863FD6DEB84C5E6D7E3FD6BD34BF0AC1393FD69BE26403E2AB3FD67AC139507E4B3F
D659D13A9147D23FD639126112B55C3FD61884A3DE6A1A3FD5F827F7CC50A93FD5D7FC
4F93657C3FD5B8019BDA317A3FD59837CB46F5533FD5789ECA8F85DC3FD559368488DA
0C3FD539FEE2364AF73FD51AF7CAD886763FD4FC2123FC350C3FD4DD7AD18853A83FD7
D3AE976B7AA63FD7B0ECA15CDFA13FD78E5AE31EAC4D3FD76BF971208E693FD749C85C
DC0E6E3FD727C7B4E86DBF3FD705F7850E37873FD6E457D65A843C3FD6C2E8AF31EF0E
3FD6A1AA13633D3A3FD6809C0439B6E73FD65FBE808F31923FD63F1184DDCBD83FD61E
950B515AC63FD5FE490BD888C63FD5DE2D7C35A6343FD5BE42500F2C063FD59E8778FF
F0A23FD57EFCE6A70F663FD55FA286B783103FD54078450773B23FD5217E0B9F38893F
D502B3C2C80E393FD4E419511A82273FD4C5AE9B8C936C3FD7B5D03A7336CE3FD79357
C679C36F3FD7710EE18F83D03FD74EF5A0B3F9C63FD72D0C15F968623FD70B52509854
B63FD6E9C85D02BB8F3FD6C86E44F6FBE03FD6A7440F9275553FD68649C163DAD23FD6
657F5C7D38A13FD644E4E085AE433FD6247A4ACADBAA3FD6043F965201FD3FD5E434BB
E8D8003FD5C459B23612243FD5A4AE6DC99E8A3FD58532E12C953B3FD565E6FCF0DCE7
3FD546CAAFC084653FD527DDE66CD1A03FD509208BFD061E3FD4EA9289BCD9DA3FD4CC
33C74AACDB3FD4AE042AA570243FD79821FE43F03B3FD775F25734537E3FD753F198E0
B0E33FD7321FD8D32AEE3FD7107D29B175093FD6EF099B4FF8893FD6CDC53AC4B0F03F
D6ACB01279BEE13FD68BCA2A3FB1503FD66B13875F84B93FD64A8C2CAC58143FD62A34
1A94D7563FD60A0B4F345B853FD5EA11C663C0363FD5CA4779C9EEA53FD5AAAC60EC1E
6D3FD58B40713DCC193FD56C039E3065D23FD54CF5D942AE4D3FD52E17120FD6873FD5
0F67365E4E843FD4F0E6322E4D8E3FD4D293EFC8127A3FD4B47057C9DC583FD4967B51
359C3F3FD77AA3921E35B23FD758BC04A27E143FD73702BBF460193FD71577CE238C96
3FD6F41B4E63FE123FD6D2ED4D20C3DB3FD6B1EDD80E865C3FD6911CFA3DC42E3FD670
7ABC2CC7703FD6500723D953293FD62FC234D208653FD60FABF04782D63FD5EFC4551D
2CFC3FD5D00B5FF9CBA93FD5B0810B57C1083FD591254F9507203FD571F82302E21C3F
D552F979F54A643FD5342946D20EFE3FD515877A1FB04D3FD4F7140293F3AE3FD4D8CE
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)$\"\"'F)$\"\"\"\"\"!-%+LIGHTMODELG6#%(LIGHT_4G-%*AXESSTYLEG6#%$BOXG-%
+AXESLABELSG6%%\"xG%\"yGQ!F4" 1 2 0 1 10 0 2 1 6 2 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 61 "The estimate (i) for the change in the value of the funct
ion " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 15 " wh
en changing " }{TEXT 326 1 "x" }{TEXT -1 11 " from 2 to " }{XPPEDIT 
18 0 "2+h" "6#,&\"\"#\"\"\"%\"hGF%" }{TEXT -1 5 " and " }{TEXT 327 1 "
y" }{TEXT -1 11 " from 1 to " }{XPPEDIT 18 0 "1+k;" "6#,&\"\"\"F$%\"kG
F$" }{TEXT -1 55 " amounts to replacing the surface near the point whe
re " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y=1" "6#/%\"yG\"\"\"" }{TEXT -1 21 " by a plane which i
s " }{TEXT 259 25 "tangential to the surface" }{TEXT -1 13 " at the po
int" }{XPPEDIT 18 0 "``(2, 1, 1/3);" "6#-%!G6%\"\"#\"\"\"*&F'F'\"\"$!
\"\"" }{TEXT -1 25 " in 3 dimensional space. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "In detail, let " }
{XPPEDIT 18 0 "h=x-2" "6#/%\"hG,&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 5 " \+
and " }{XPPEDIT 18 0 "k=y-1" "6#/%\"kG,&%\"yG\"\"\"F'!\"\"" }{TEXT -1 
44 ", then the estimate for the value at a point" }{XPPEDIT 18 0 "``(x
,y)" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "z=g(2,1)+g[x](2,1)*(x-2)+g[y](2,1)*(y-1
)" "6#/%\"zG,(-%\"gG6$\"\"#\"\"\"F**&-&F'6#%\"xG6$F)F*F*,&F/F*F)!\"\"F
*F**&-&F'6#%\"yG6$F)F*F*,&F7F*F*F2F*F*" }{TEXT -1 2 ", " }}{PARA 0 "" 
0 "" {TEXT -1 8 "that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "z=1/3+``(-5/27)*(x-2)+``(-2/9)*(y-1)" "6#/%\"zG,(*&\"\"
\"F'\"\"$!\"\"F'*&-%!G6#,$*&\"\"&F'\"#FF)F)F',&%\"xGF'\"\"#F)F'F'*&-F,
6#,$*&F4F'\"\"*F)F)F',&%\"yGF'F'F)F'F'" }{TEXT -1 2 ", " }}{PARA 0 "" 
0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "z = -5/27*x-2/9*y+25/27" "6#/%\"zG,(*(\"\"&\"\"\"\"#F!\"\"%\"xGF(F*
*(\"\"#F(\"\"*F*%\"yGF(F**&\"#DF(F)F*F(" }{TEXT -1 14 " ------- (ii).
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "A gen
eral " }{TEXT 259 32 "linear function of two variables" }{TEXT -1 13 "
 of the form " }{XPPEDIT 18 0 "s(x,y) = a*x+b*y+c;" "6#/-%\"sG6$%\"xG%
\"yG,(*&%\"aG\"\"\"F'F,F,*&%\"bGF,F(F,F,%\"cGF," }{TEXT -1 39 " has a \+
graph which is a (non-vertical) " }{TEXT 259 5 "plane" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "Diff(s(x,y),
x) = s[x](x,y);" "6#/-%%DiffG6$-%\"sG6$%\"xG%\"yGF*-&F(6#F*6$F*F+" }
{XPPEDIT 18 0 "``=a" "6#/%!G%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 
"Diff(s(x,y),y)=s[y](x,y)" "6#/-%%DiffG6$-%\"sG6$%\"xG%\"yGF+-&F(6#F+6
$F*F+" }{XPPEDIT 18 0 "``=b" "6#/%!G%\"bG" }{TEXT -1 48 ", that is, th
e partial derivatives are constant." }}{PARA 0 "" 0 "" {TEXT -1 64 "Th
e plane which is tangential to our surface at the point where " }
{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 
0 "y=1" "6#/%\"yG\"\"\"" }{TEXT -1 65 " must have partial derivatives \+
which match those of the function " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6
$%\"xG%\"yG" }{TEXT -1 16 " at this point. " }}{PARA 0 "" 0 "" {TEXT 
-1 32 "Hence its equation has the form " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "z=g[x](2,1)*x+g[y](2,1)*y+c" "6#/%\"zG,(*&-&%\"g
G6#%\"xG6$\"\"#\"\"\"F.F+F.F.*&-&F)6#%\"yG6$F-F.F.F3F.F.%\"cGF." }
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "z = -5/27*x-2/9*y+c;" "6#/%\"zG,
(*(\"\"&\"\"\"\"#F!\"\"%\"xGF(F**(\"\"#F(\"\"*F*%\"yGF(F*%\"cGF(" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "The equation (ii) certa
inly has this form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 19 "g_x(2,1);\ng_y(2,1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##!\"&\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"#\"\"*
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "If w
e had not aleady found the constant " }{TEXT 325 1 "c" }{TEXT -1 82 ",
 it could be determined by using the fact that the plane passes throug
h the point" }{XPPEDIT 18 0 " ``(2,1,1/3)" "6#-%!G6%\"\"#\"\"\"*&F'F'
\"\"$!\"\"" }{TEXT -1 25 " in 3 dimensional space. " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "g(2,1)=g_x
(2,1)*2+g_y(2,1)*1+c;\nsolve(%,c);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/#\"\"\"\"\"$,&#!#;\"#FF%%\"cGF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
#\"#D\"#F" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 37 "The tangent plane at the point where " }{XPPEDIT 18 0 "x=2" "6#
/%\"xG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=1" "6#/%\"yG\"\"\"
" }{TEXT -1 14 " is therefore " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "z = -5/27*x-2/9*y+25/27;" "6#/%\"zG,(*(\"\"&\"\"\"\"#F!
\"\"%\"xGF(F**(\"\"#F(\"\"*F*%\"yGF(F**&\"#DF(F)F*F(" }{TEXT -1 1 " " 
}}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " 
}{XPPEDIT 18 0 "5*x+6*y+27*z = 25;" "6#/,(*&\"\"&\"\"\"%\"xGF'F'*&\"\"
'F'%\"yGF'F'*&\"#FF'%\"zGF'F'\"#D" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 245 "g := (x,y) \+
-> 3/(1+x^2+x*y+2*y^2);\ns := (x,y) -> -5/27*x-2/9*y+25/27;\np1 := plo
t3d(g(x,y),x=1..3,y=0..2,color=COLOR(RGB,.8,.6,1)):\np2 := plot3d(s(x,
y),x=1..3,y=0..2,color=COLOR(RGB,.8,1,.6)):\nplots[display]([p1,p2],ax
es=boxed,lightmodel=light4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG
f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&\"\"\"F/,*F/F/*$)9$\"\"#F/
F/*&F3F/9%F/F/*&F4F/)F6F4F/F/!\"\"\"\"$F)F)F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"sGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(9$#!\"&
\"#F*&#\"\"#\"\"*\"\"\"9%F6!\"\"#\"#DF1F6F)F)F)" }}{PARA 13 "" 1 "" 
{GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%%GRIDG6&;$\"\"\"\"\"!$\"\"$F);$
F)F)$\"\"#F)X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\":\":3FF80000000000003F
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The value
s of " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 5 " an
d " }{XPPEDIT 18 0 "s(x,y)" "6#-%\"sG6$%\"xG%\"yG" }{TEXT -1 18 " coin
cide at when " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 5 " and \+
" }{XPPEDIT 18 0 "y=1" "6#/%\"yG\"\"\"" }{TEXT -1 52 ", and are approx
imately equal at neighbouring values" }{XPPEDIT 18 0 " ``(x,y)" "6#-%!
G6$%\"xG%\"yG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "g := (x,y) -> 3/(1+x^2+x*y+
2*y^2);\ns := (x,y) -> -5/27*x-2/9*y+25/27;\n'g(2,1)'=g(2,1);\n's(2,1)
'=s(2,1);\nh := 0.001; \nk := 0.0005;\n'g(2+h,1+k)'=g(2+h,1+k);\n's(2+
h,1+k)'=s(2+h,1+k);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%
\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&\"\"\"F/,*F/F/*$)9$\"\"#F/F/*&F
3F/9%F/F/*&F4F/)F6F4F/F/!\"\"\"\"$F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"sGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(9$#!\"&\"#F*&#\"
\"#\"\"*\"\"\"9%F6!\"\"#\"#DF1F6F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"gG6$\"\"#\"\"\"#F(\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%\"sG6$\"\"#\"\"\"#F(\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG
$\"\"\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG$\"\"&!\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6$,&\"\"#\"\"\"%\"hGF),&F)F)%\"
kGF)$\"+hAPIL!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"sG6$,&\"\"#\"
\"\"%\"hGF),&F)F)%\"kGF)$\"+q.PIL!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 74 "Constructing a tangent plane via tangent lines to \+
two cross-section curves" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";
" }}}{PARA 0 "" 0 "" {TEXT -1 52 "We shall determine the tangent plane
 to the surface " }{XPPEDIT 18 0 "z=g(x,y)" "6#/%\"zG-%\"gG6$%\"xG%\"y
G" }{XPPEDIT 18 0 "`` = x^2-3*x*y-y^2+30;" "6#/%!G,**$%\"xG\"\"#\"\"\"
*(\"\"$F)F'F)%\"yGF)!\"\"*$F,F(F-\"#IF)" }{TEXT -1 20 " at the point w
here " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y = 2;" "6#/%\"yG\"\"#" }{TEXT -1 93 " by first determi
ning the tangent lines to the curves of intersection of the vertical p
lanes " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y = 2;" "6#/%\"yG\"\"#" }{TEXT -1 18 " with the surface
 " }{XPPEDIT 18 0 "z=g(x,y)" "6#/%\"zG-%\"gG6$%\"xG%\"yG" }{TEXT -1 2 
". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT3D 400 300 300 
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{TEXT -1 5 " The " }{TEXT 294 1 "x" }{TEXT -1 5 " and " }{TEXT 295 1 "
z" }{TEXT -1 51 " coordinates of curve of intersection of the plane " 
}{XPPEDIT 18 0 "y = 2;" "6#/%\"yG\"\"#" }{TEXT -1 18 " with the surfac
e " }{XPPEDIT 18 0 "z = x^2-3*x*y-y^2+30;" "6#/%\"zG,**$%\"xG\"\"#\"\"
\"*(\"\"$F)F'F)%\"yGF)!\"\"*$F,F(F-\"#IF)" }{TEXT -1 23 " satisfy the \+
equation  " }{XPPEDIT 18 0 "z = x^2-6*x+26;" "6#/%\"zG,(*$%\"xG\"\"#\"
\"\"*&\"\"'F)F'F)!\"\"\"#EF)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 5 "Then " }{XPPEDIT 18 0 "dz/dx = 2*x-6;" "6#/*&%#dzG\"\"\"%#dxG!\"
\",&*&\"\"#F&%\"xGF&F&\"\"'F(" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" 
{TEXT -1 31 "The tangent line to the curve  " }{XPPEDIT 18 0 "z = x^2-
6*x+26;" "6#/%\"zG,(*$%\"xG\"\"#\"\"\"*&\"\"'F)F'F)!\"\"\"#EF)" }
{TEXT -1 20 " at the point where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"
\"" }{TEXT -1 14 " has gradient " }{XPPEDIT 18 0 "-4" "6#,$\"\"%!\"\"
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "The equation of this
 tangent line is " }{XPPEDIT 18 0 "z = 21+(-4)*(x-1);" "6#/%\"zG,&\"#@
\"\"\"*&,$\"\"%!\"\"F',&%\"xGF'F'F+F'F'" }{TEXT -1 5 " or  " }
{XPPEDIT 18 0 "z = 25-4*x;" "6#/%\"zG,&\"#D\"\"\"*&\"\"%F'%\"xGF'!\"\"
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 48 "plot([x^2-6*x+26,25-4*x],x=-1..3,z,thickness=
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"" {TEXT -1 4 "The " }{TEXT 296 1 "x" }{TEXT -1 5 " and " }{TEXT 297 
1 "z" }{TEXT -1 51 " coordinates of curve of intersection of the plane
 " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 18 " with the su
rface " }{XPPEDIT 18 0 "z = x^2-3*x*y+y^2+30;" "6#/%\"zG,**$%\"xG\"\"#
\"\"\"*(\"\"$F)F'F)%\"yGF)!\"\"*$F,F(F)\"#IF)" }{TEXT -1 23 " satisfy \+
the equation  " }{XPPEDIT 18 0 "z = 31-3*y-y^2;" "6#/%\"zG,(\"#J\"\"\"
*&\"\"$F'%\"yGF'!\"\"*$F*\"\"#F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 5 "Then " }{XPPEDIT 18 0 "dz/dy = -3-2*y;" "6#/*&%#dzG\"\"\"%
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0 "" {TEXT -1 31 "The tangent line to the curve  " }{XPPEDIT 18 0 "z =
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{TEXT -1 20 " at the point where " }{XPPEDIT 18 0 "y = 2;" "6#/%\"yG\"
\"#" }{TEXT -1 14 " has gradient " }{XPPEDIT 18 0 "-7;" "6#,$\"\"(!\"
\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "The equation of th
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#@\"\"\"*&,$\"\"(!\"\"F',&%\"yGF'\"\"#F+F'F'" }{TEXT -1 5 " or  " }
{XPPEDIT 18 0 "z = 35-7*y;" "6#/%\"zG,&\"#N\"\"\"*&\"\"(F'%\"yGF'!\"\"
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
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e 6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "mtaylor(g(x,y),[x=1,y=2],2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(\"#R\"\"\"*&\"\"(F%%\"yGF%!\"\"*&\"\"%F%%\"xGF%F)" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 65 "Using a tangent plane
 to help in illustrating partial derivatives" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We il
lustrate the partial derivatives of  " }{XPPEDIT 18 0 "g(x,y)=x^2+2*y^
2-x*y" "6#/-%\"gG6$%\"xG%\"yG,(*$F'\"\"#\"\"\"*&F+F,*$F(F+F,F,*&F'F,F(
F,!\"\"" }{TEXT -1 14 " at the point " }{XPPEDIT 18 0 "``(1,1)" "6#-%!
G6$\"\"\"F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 906 "#
 define a function of two variables\ng := (x,y) ->  x^2+2*y^2-x*y;\n\n
# define a point (a,b) at which to investigate the partial derivatives
\na := 1; b := 1;\n\n# give the width of the plotting intervals \nxwid
th := 1: ywidth := 1:\n\nx0 := a-xwidth/2: x1 := a+xwidth/2:\ny0 := b-
ywidth/2: y1 := b+ywidth/2:\ndx := xwidth/2: dy := ywidth/2:\nh := (x,
y) -> g(a,b)+ D[1](g)(a,b)*(x-a) + D[2](g)(a,b)*(y-b):\np1 := plot3d(g
(x,y),x=x0..x1,y=y0..y1,color=COLOR(RGB,.6,.3,.8),\n                  \+
           style=wireframe,gridstyle=triangular):\np2 := plot3d(h(x,y)
,x=a..a+dx,y=b..b+dy,\n                      color=COLOR(RGB,.8,.6,.6)
,style=patchnogrid):\np3 := plots[polygonplot3d]([[a,b,h(a,b)],[a+dx,b
,h(a,b)], \n                      [a+dx,b,h(a+dy,b)]],color=green):\np
4 := plots[polygonplot3d]([[a,b,h(a,b)],[a,b+dy,h(a,b)], \n           \+
           [a,b+dy,h(a,b+dy)]],color=red):\nplots[display]([p1,p2,p3,p
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OLOURG6&F4$FEFE$\"*++++\"!\")Fgn-FM6$7%FP7%FCFTFQ7%FCFT$\"+++++NFV-Fen
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ONTG6#%(DEFAULTG-%*GRIDSTYLEG6#%+TRIANGULARG" 1 2 0 1 10 0 1 1 1 3 2 
1.000000 64.000000 -20.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve
 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Th
e " }{TEXT 260 3 "red" }{TEXT -1 73 " right-angled triangle represents
 the partial derivative with respect to " }{TEXT 301 1 "x" }{TEXT -1 
192 ", where the hypotenuse is a tangent to the surface at the given p
oint and the partial derivative is the gradient of this tangent line g
iven by the ratio of the lengths of the other two sides. " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Similarly, the " }
{TEXT 298 5 "green" }{TEXT -1 73 " right-angled triangle represents th
e partial derivative with respect to " }{TEXT 302 1 "y" }{TEXT -1 3 ".
  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We illu
strate the partial derivatives of  " }{XPPEDIT 18 0 "g(x,y) = 5/(3+x^2
+y^2);" "6#/-%\"gG6$%\"xG%\"yG*&\"\"&\"\"\",(\"\"$F+*$F'\"\"#F+*$F(F/F
+!\"\"" }{TEXT -1 14 " at the point " }{XPPEDIT 18 0 "``(1/3,1/4);" "6
#-%!G6$*&\"\"\"F'\"\"$!\"\"*&F'F'\"\"%F)" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 913 "# de
fine a function of two variables\ng := (x,y) ->  5/(3+x^2+y^2);\n\n# d
efine a point (a,b) at which to investigate the partial derivatives\na
 := 1/3; b := 1/4;\n\n# give the width of the plotting intervals\nxwid
th := 1.5: ywidth := 1.5:\n\nx0 := a-xwidth/2: x1 := a+xwidth/2:\ny0 :
= b-ywidth/2: y1 := b+ywidth/2:\ndx := xwidth/2:\ndy := ywidth/2:\nh :
= (x,y) -> g(a,b)+ D[1](g)(a,b)*(x-a) + D[2](g)(a,b)*(y-b):\np1 := plo
t3d(g(x,y),x=x0..x1,y=y0..y1,color=COLOR(RGB,.6,.3,.8),\n             \+
                style=wireframe,gridstyle=triangular):\np2 := plot3d(h
(x,y),x=a..a+dx,y=b..b+dy,\n                      color=COLOR(RGB,.8,.
6,.6),style=patchnogrid):\np3 := plots[polygonplot3d]([[a,b,h(a,b)],[a
+dx,b,h(a,b)], \n                      [a+dx,b,h(a+dy,b)]],color=green
):\np4 := plots[polygonplot3d]([[a,b,h(a,b)],[a,b+dy,h(a,b)], \n      \+
                [a,b+dy,h(a,b+dy)]],color=red):\nplots[display]([p1,p2
,p3,p4],axes=framed);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"
xG%\"yG6\"6$%)operatorG%&arrowGF),$*&\"\"\"F/,(\"\"$F/*$)9$\"\"#F/F/*$
)9%F5F/F/!\"\"\"\"&F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG#\"
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6#%,PATCHNOGRIDG-%)POLYGONSG6$7%7%$\"+LLLLLF)$\"+++++DF)$\"+TB\\v:F,7%
F*FenFgn7%F*Fen$\"+HZFF8F,-%'COLOURG6&F9$\"\"!Fao$\"*++++\"!\")F`o-FU6
$7%FX7%FYF0Fgn7%FYF0$\"+K\"H$*Q\"F,-F^o6&F9FboF`oF`o-%*AXESSTYLEG6#%&F
RAMEG-%+AXESLABELSG6&%\"xG%\"yGQ!F5-%%FONTG6#%(DEFAULTG-%*GRIDSTYLEG6#
%+TRIANGULARG" 1 2 0 1 10 0 1 1 1 3 2 1.000000 45.000000 45.000000 0 
0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 4 "The " }{TEXT 260 3 "red" }{TEXT -1 73 " right-angled tri
angle represents the partial derivative with respect to " }{TEXT 299 
1 "x" }{TEXT -1 192 ", where the hypotenuse is a tangent to the surfac
e at the given point and the partial derivative is the gradient of thi
s tangent line given by the ratio of the lengths of the other two side
s. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Si
milarly, the " }{TEXT 298 5 "green" }{TEXT -1 73 " right-angled triang
le represents the partial derivative with respect to " }{TEXT 300 1 "y
" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 62 "The concept of differen
tiabilty for functions of two variables" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }
{XPPEDIT 18 0 "z= g(x,y)" "6#/%\"zG-%\"gG6$%\"xG%\"yG" }{TEXT -1 73 " \+
is a function of two variables which has continuous partial derivative
s " }{XPPEDIT 18 0 "Diff(g(x,y),x) = g[x];" "6#/-%%DiffG6$-%\"gG6$%\"x
G%\"yGF*&F(6#F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Diff(g(x,y),y) = \+
g[y];" "6#/-%%DiffG6$-%\"gG6$%\"xG%\"yGF+&F(6#F+" }{TEXT -1 12 " at a \+
point " }{XPPEDIT 18 0 "``(x[0], y[0]);" "6#-%!G6$&%\"xG6#\"\"!&%\"yG6
#F)" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 
18 0 "Delta*x;" "6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "Delta*y;" "6#*&%&DeltaG\"\"\"%\"yGF%" }{TEXT -1 35 " be
 small changes in the variables " }{TEXT 278 1 "x" }{TEXT -1 5 " and \+
" }{TEXT 279 1 "y" }{TEXT -1 15 " respectively. " }}{PARA 0 "" 0 "" 
{TEXT -1 31 "If the corresponding change in " }{XPPEDIT 18 0 "z=g(x,y)
" "6#/%\"zG-%\"gG6$%\"xG%\"yG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "Delt
a*z" "6#*&%&DeltaG\"\"\"%\"zGF%" }{TEXT -1 8 ", then  " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta*z" "6#*&%&DeltaG\"\"\"%\"z
GF%" }{TEXT -1 1 " " }{TEXT 280 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
g[x]*Delta*x+g[y]*Delta*y;" "6#,&*(&%\"gG6#%\"xG\"\"\"%&DeltaGF)F(F)F)
*(&F&6#%\"yGF)F*F)F.F)F)" }{TEXT -1 16 "  ------- (i),  " }}{PARA 0 "
" 0 "" {TEXT -1 28 "where it is understood that " }{XPPEDIT 18 0 "g[x]
" "6#&%\"gG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g[y]" "6#&%\"g
G6#%\"yG" }{TEXT -1 33 " are to be evaluated at the point" }{XPPEDIT 
18 0 "``(x[0],y[0]);" "6#-%!G6$&%\"xG6#\"\"!&%\"yG6#F)" }{TEXT -1 2 ".
 " }}{PARA 0 "" 0 "" {TEXT -1 117 "We can build on the approximation (
i) to obtain a definition of the differentiability of a function of tw
o variables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 53 "Consider first the corresponding 1 dimensional case. " }}{PARA 
0 "" 0 "" {TEXT -1 29 "The derivative of a function " }{XPPEDIT 18 0 "
g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 12 " at a point " }{XPPEDIT 18 0 "x \+
= x[0];" "6#/%\"xG&F$6#\"\"!" }{TEXT -1 18 " is defined to be " }
{XPPEDIT 18 0 "g*`'`(x[0]) = Limit((g(x[0]+h)-g(x[0]))/h,h = 0);" "6#/
*&%\"gG\"\"\"-%\"'G6#&%\"xG6#\"\"!F&-%&LimitG6$*&,&-F%6#,&&F+6#F-F&%\"
hGF&F&-F%6#&F+6#F-!\"\"F&F8F=/F8F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 17 "We say that g is " }{TEXT 259 14 "differentiable" }
{TEXT -1 4 " at " }{XPPEDIT 18 0 "x=x[0]" "6#/%\"xG&F$6#\"\"!" }{TEXT 
-1 24 ", if this limit exists. " }}{PARA 0 "" 0 "" {TEXT -1 20 "For sm
all values of " }{XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT -1 9 " we have " }
}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "g(x[0]+h)-g(x[0])" 
"6#,&-%\"gG6#,&&%\"xG6#\"\"!\"\"\"%\"hGF,F,-F%6#&F)6#F+!\"\"" }{TEXT 
-1 1 " " }{TEXT 281 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "g*`'`(x[0])*
h" "6#*(%\"gG\"\"\"-%\"'G6#&%\"xG6#\"\"!F%%\"hGF%" }{TEXT -1 14 " ----
--- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 
-1 3 "If " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 
62 ", this is equivalent to either of the standard approximations " }
{XPPEDIT 18 0 "Delta*y" "6#*&%&DeltaG\"\"\"%\"yGF%" }{TEXT -1 1 " " }
{TEXT 289 1 "~" }{XPPEDIT 18 0 "``(dy/dx)*Delta*x" "6#*(-%!G6#*&%#dyG
\"\"\"%#dxG!\"\"F)%&DeltaGF)%\"xGF)" }{TEXT -1 4 " or " }{XPPEDIT 18 
0 "Delta*y/(Delta*x)" "6#*(%&DeltaG\"\"\"%\"yGF%*&F$F%%\"xGF%!\"\"" }
{TEXT -1 1 " " }{TEXT 290 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx
" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "epsi
lon(h)=``" "6#/-%(epsilonG6#%\"hG%!G" }{XPPEDIT 18 0 "epsilon(x[0],h) \+
= (g(x[0]+h)-g(x[0]))/h-g*`'`(x[0]);" "6#/-%(epsilonG6$&%\"xG6#\"\"!%
\"hG,&*&,&-%\"gG6#,&&F(6#F*\"\"\"F+F5F5-F06#&F(6#F*!\"\"F5F+F:F5*&F0F5
-%\"'G6#&F(6#F*F5F:" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "g(x[0]+
h)-g(x[0]) = g*`'`(x[0])*h+epsilon(h)*h;" "6#/,&-%\"gG6#,&&%\"xG6#\"\"
!\"\"\"%\"hGF-F--F&6#&F*6#F,!\"\",&*(F&F--%\"'G6#&F*6#F,F-F.F-F-*&-%(e
psilonG6#F.F-F.F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Th
us " }{XPPEDIT 18 0 "epsilon(h);" "6#-%(epsilonG6#%\"hG" }{TEXT -1 8 "
 is the " }{TEXT 259 5 "error" }{TEXT -1 28 " in the approximation (ii
). " }}{PARA 0 "" 0 "" {TEXT -1 91 "An equivalent definition of differ
entiability of a function of one variable is as follows. " }}{PARA 0 "
" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%
\"xG" }{TEXT -1 4 " is " }{TEXT 259 14 "differentiable" }{TEXT -1 4 " \+
at " }{XPPEDIT 18 0 "x=x[0]" "6#/%\"xG&F$6#\"\"!" }{TEXT -1 17 " with \+
derivative " }{XPPEDIT 18 0 "g*`'`(x[0])" "6#*&%\"gG\"\"\"-%\"'G6#&%\"
xG6#\"\"!F%" }{TEXT -1 35 " provided that there is a function " }
{XPPEDIT 18 0 "epsilon(h) = epsilon(x[0],h);" "6#/-%(epsilonG6#%\"hG-F
%6$&%\"xG6#\"\"!F'" }{TEXT -1 11 " such that " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "g(x[0]+h)-g(x[0]) = g*`'`(x[0])*h+epsil
on(h)*h;" "6#/,&-%\"gG6#,&&%\"xG6#\"\"!\"\"\"%\"hGF-F--F&6#&F*6#F,!\"
\",&*(F&F--%\"'G6#&F*6#F,F-F.F-F-*&-%(epsilonG6#F.F-F.F-F-" }{TEXT -1 
1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }{XPPEDIT 18 0 "Limit(epsilo
n(h),h = 0) = 0;" "6#/-%&LimitG6$-%(epsilonG6#%\"hG/F*\"\"!F," }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
36 "By analogy we can define a function " }{XPPEDIT 18 0 "g(x,y)" "6#-
%\"gG6$%\"xG%\"yG" }{TEXT -1 24 " of two variables to be " }{TEXT 259 
14 "differentiable" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(x[
0],y[0]);" "6#-%!G6$&%\"xG6#\"\"!&%\"yG6#F)" }{TEXT -1 33 " provided t
hat there are numbers " }{XPPEDIT 18 0 "g[x] = g[x](x[0],y[0]);" "6#/&
%\"gG6#%\"xG-&F%6#F'6$&F'6#\"\"!&%\"yG6#F." }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "g[y] = g[y](x[0],y[0]);" "6#/&%\"gG6#%\"yG-&F%6#F'6$&%
\"xG6#\"\"!&F'6#F/" }{TEXT -1 16 " and  functions " }{XPPEDIT 18 0 "ep
silon[1] = epsilon[1](h, k);" "6#/&%(epsilonG6#\"\"\"-&F%6#F'6$%\"hG%
\"kG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "epsilon[2] = epsilon[2](h,k);" 
"6#/&%(epsilonG6#\"\"#-&F%6#F'6$%\"hG%\"kG" }{TEXT -1 12 " such that  \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "g(x[0]+h,y[0]+k)-g(x[0],y[0]) = g[x]*h+g[y]*k+epsilon[1
]*h+epsilon[2]*k;" "6#/,&-%\"gG6$,&&%\"xG6#\"\"!\"\"\"%\"hGF-,&&%\"yG6
#F,F-%\"kGF-F--F&6$&F*6#F,&F16#F,!\"\",**&&F&6#F*F-F.F-F-*&&F&6#F1F-F3
F-F-*&&%(epsilonG6#F-F-F.F-F-*&&FD6#\"\"#F-F3F-F-" }{TEXT -1 1 " " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 293 23 "______________________
_" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 10 "such that " }{XPPEDIT 18 0 "epsilon[1]" "6#&%(epsilonG6#
\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "epsilon[2]" "6#&%(epsilonG
6#\"\"#" }{TEXT -1 15 " approach 0 as " }{TEXT 287 1 "h" }{TEXT -1 5 "
 and " }{TEXT 288 1 "k" }{TEXT -1 13 " approach 0. " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The main result in this
 connection is that a function " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%
\"xG%\"yG" }{TEXT -1 4 " is " }{TEXT 259 14 "differentiable" }{TEXT 
-1 11 " at a point" }{XPPEDIT 18 0 "``(x[0],y[0])" "6#-%!G6$&%\"xG6#\"
\"!&%\"yG6#F)" }{TEXT -1 15 " provided that " }{XPPEDIT 18 0 "g(x,y)" 
"6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 4 " is " }{TEXT 259 10 "continuous" }
{TEXT -1 8 " and has" }{TEXT 259 31 " continuous partial derivatives" 
}{TEXT -1 24 "  in a neighbourhood of " }{XPPEDIT 18 0 "``(x[0],y[0])
" "6#-%!G6$&%\"xG6#\"\"!&%\"yG6#F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Explanation  " }}{PARA 0 "" 0 "" 
{TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 " g(x,y)" "6#-%\"gG6$%\"xG%
\"yG" }{TEXT -1 76 " is continuous and has continuous partial derivati
ves in a neighbourhood of " }{XPPEDIT 18 0 "``(x[0],y[0])" "6#-%!G6$&%
\"xG6#\"\"!&%\"yG6#F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 68 
"Then, by the definition of partial derivatives, there is a function \+
" }{XPPEDIT 18 0 "epsilon[1](h)" "6#-&%(epsilonG6#\"\"\"6#%\"hG" }
{TEXT -1 11 " such that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "g(x[0]+h,y[0])-g(x[0],y[0]) = g[x](x[0],y[0])*h+epsilon
[1](h)*h;" "6#/,&-%\"gG6$,&&%\"xG6#\"\"!\"\"\"%\"hGF-&%\"yG6#F,F--F&6$
&F*6#F,&F06#F,!\"\",&*&-&F&6#F*6$&F*6#F,&F06#F,F-F.F-F-*&-&%(epsilonG6
#F-6#F.F-F.F-F-" }{TEXT -1 15 "  ------- (iii)" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "and " }{XPPEDIT 18 0 "Limit(epsilon[1](h),h=0)=0" "6#/-%&L
imitG6$-&%(epsilonG6#\"\"\"6#%\"hG/F-\"\"!F/" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 31 "Similarly, there is a function " }
{XPPEDIT 18 0 "epsilon[2](k);" "6#-&%(epsilonG6#\"\"#6#%\"kG" }{TEXT 
-1 11 " such that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"g(x[0]+h,y[0]+k)-g(x[0]+h,y[0]) = g[y](x[0]+h,y[0])*k+epsilon[2](k)*k
;" "6#/,&-%\"gG6$,&&%\"xG6#\"\"!\"\"\"%\"hGF-,&&%\"yG6#F,F-%\"kGF-F--F
&6$,&&F*6#F,F-F.F-&F16#F,!\"\",&*&-&F&6#F16$,&&F*6#F,F-F.F-&F16#F,F-F3
F-F-*&-&%(epsilonG6#\"\"#6#F3F-F3F-F-" }{TEXT -1 15 "  ------- (iv) " 
}}{PARA 0 "" 0 "" {TEXT -1 4 "and " }{XPPEDIT 18 0 "Limit(epsilon[2](k
),k = 0) = 0;" "6#/-%&LimitG6$-&%(epsilonG6#\"\"#6#%\"kG/F-\"\"!F/" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "Then, by adding (iii) a
nd (iv), we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
g(x[0]+h,y[0]+k)-g(x[0],y[0]) = g[x](x[0],y[0])*h+g[y](x[0]+h,y[0])*k+
epsilon[1](h)*h+epsilon[2](k)*k;" "6#/,&-%\"gG6$,&&%\"xG6#\"\"!\"\"\"%
\"hGF-,&&%\"yG6#F,F-%\"kGF-F--F&6$&F*6#F,&F16#F,!\"\",**&-&F&6#F*6$&F*
6#F,&F16#F,F-F.F-F-*&-&F&6#F16$,&&F*6#F,F-F.F-&F16#F,F-F3F-F-*&-&%(eps
ilonG6#F-6#F.F-F.F-F-*&-&FR6#\"\"#6#F3F-F3F-F-" }{TEXT -1 14 "  ------
- (v)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 
"By the continuity of " }{XPPEDIT 18 0 "g[x]" "6#&%\"gG6#%\"xG" }
{TEXT -1 14 " implies that " }{XPPEDIT 18 0 "g[y](x[0]+h,y[0])" "6#-&%
\"gG6#%\"yG6$,&&%\"xG6#\"\"!\"\"\"%\"hGF.&F'6#F-" }{TEXT -1 10 " tends
 to " }{XPPEDIT 18 0 "g[y](x[0],y[0])" "6#-&%\"gG6#%\"yG6$&%\"xG6#\"\"
!&F'6#F," }{TEXT -1 4 " as " }{XPPEDIT 18 0 "h->0" "6#f*6#%\"hG7\"6$%)
operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Hence we may write " }
{XPPEDIT 18 0 "g[y](x[0]+h,y[0])=g[y](x[0],y[0])+epsilon[3](h)" "6#/-&
%\"gG6#%\"yG6$,&&%\"xG6#\"\"!\"\"\"%\"hGF/&F(6#F.,&-&F&6#F(6$&F,6#F.&F
(6#F.F/-&%(epsilonG6#\"\"$6#F0F/" }{TEXT -1 8 ", where " }{XPPEDIT 18 
0 "Limit(epsilon[3](h),h=0)=0" "6#/-%&LimitG6$-&%(epsilonG6#\"\"$6#%\"
hG/F-\"\"!F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "Then (v)
 can be written as " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "g(x[0]+h,y[0]+k)-g(x[0],y[0]) = g[x](x[0],y[0])*h+g[y](x[0],y[0])*k
+epsilon[1](h)*h+(epsilon[3](h)+epsilon[2](k))*k" "6#/,&-%\"gG6$,&&%\"
xG6#\"\"!\"\"\"%\"hGF-,&&%\"yG6#F,F-%\"kGF-F--F&6$&F*6#F,&F16#F,!\"\",
**&-&F&6#F*6$&F*6#F,&F16#F,F-F.F-F-*&-&F&6#F16$&F*6#F,&F16#F,F-F3F-F-*
&-&%(epsilonG6#F-6#F.F-F.F-F-*&,&-&FQ6#\"\"$6#F.F--&FQ6#\"\"#6#F3F-F-F
3F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 51 "This equation can be given in the symmetrical form \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x[0]+h,y[0]+k)-
g(x[0],y[0]) = g[x](x[0],y[0])*h+g[y](x[0],y[0])*k+epsilon[4](h,k)*h+e
psilon[5](h,k)*k" "6#/,&-%\"gG6$,&&%\"xG6#\"\"!\"\"\"%\"hGF-,&&%\"yG6#
F,F-%\"kGF-F--F&6$&F*6#F,&F16#F,!\"\",**&-&F&6#F*6$&F*6#F,&F16#F,F-F.F
-F-*&-&F&6#F16$&F*6#F,&F16#F,F-F3F-F-*&-&%(epsilonG6#\"\"%6$F.F3F-F.F-
F-*&-&FQ6#\"\"&6$F.F3F-F3F-F-" }{TEXT -1 15 " ------- (vi), " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }
{XPPEDIT 18 0 "Limit(epsilon[4](h,k),``(h,k)=(0,0))=0" "6#/-%&LimitG6$
-&%(epsilonG6#\"\"%6$%\"hG%\"kG/-%!G6$F-F.6$\"\"!F4F4" }{TEXT -1 5 " a
nd " }{XPPEDIT 18 0 "Limit(epsilon[5](h,k),``(h,k)=(0,0))=0" "6#/-%&Li
mitG6$-&%(epsilonG6#\"\"&6$%\"hG%\"kG/-%!G6$F-F.6$\"\"!F4F4" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
116 "The converse is clear, since the existence (and continuity) of th
e partial derivatives is implied by taking each of " }{TEXT 291 1 "h" 
}{TEXT -1 5 " and " }{TEXT 292 1 "k" }{TEXT -1 28 " in turn equal to 0
 in (vi)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Roughly speaking d
ifferentiability means that the approximation (i): " }}{PARA 256 "" 0 
"" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta*z" "6#*&%&DeltaG\"\"\"%\"zGF%
" }{TEXT -1 1 " " }{TEXT 282 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "g[x
]*Delta*x+g[y]*Delta*y;" "6#,&*(&%\"gG6#%\"xG\"\"\"%&DeltaGF)F(F)F)*(&
F&6#%\"yGF)F*F)F.F)F)" }{TEXT -1 3 ",  " }}{PARA 0 "" 0 "" {TEXT -1 
29 "gets progressively better as " }{XPPEDIT 18 0 "Delta*x" "6#*&%&Del
taG\"\"\"%\"xGF%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Delta*y" "6#*&%&
DeltaG\"\"\"%\"yGF%" }{TEXT -1 13 " approach 0. " }}{PARA 0 "" 0 "" 
{TEXT -1 38 "This result leads to a version of the " }{TEXT 259 10 "ch
ain rule" }{TEXT -1 43 " for a function of two variables since, if " }
{XPPEDIT 18 0 "z = g(x,y);" "6#/%\"zG-%\"gG6$%\"xG%\"yG" }{TEXT -1 41 
" has continuous partial derivatives, and " }{TEXT 284 1 "x" }{TEXT 
-1 5 " and " }{TEXT 285 1 "y" }{TEXT -1 33 " are differentiable functi
ons of " }{TEXT 286 1 "t" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta*z/(Delta*t)" "6#*(%&DeltaG\"\"
\"%\"zGF%*&F$F%%\"tGF%!\"\"" }{TEXT -1 1 " " }{TEXT 283 1 "~" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "g[x]*``(Delta*x/(Delta*t))+g[y]*``(Delta*y/(D
elta*t))" "6#,&*&&%\"gG6#%\"xG\"\"\"-%!G6#*(%&DeltaGF)F(F)*&F.F)%\"tGF
)!\"\"F)F)*&&F&6#%\"yGF)-F+6#*(F.F)F5F)*&F.F)F0F)F1F)F)" }{TEXT -1 2 "
, " }}{PARA 0 "" 0 "" {TEXT -1 36 "where the approximation improves as
 " }{XPPEDIT 18 0 "Delta*t" "6#*&%&DeltaG\"\"\"%\"tGF%" }{TEXT -1 15 "
 approaches 0. " }}{PARA 0 "" 0 "" {TEXT -1 24 "In the limit we obtain
  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dz/dt = g[x]*``
(dx/dt)+g[y]*``(dy/dt);" "6#/*&%#dzG\"\"\"%#dtG!\"\",&*&&%\"gG6#%\"xGF
&-%!G6#*&%#dxGF&F'F(F&F&*&&F,6#%\"yGF&-F06#*&%#dyGF&F'F(F&F&" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 84 "A general \+
formula for the tangent plane to the graph of a function of two variab
les " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "
" {TEXT -1 93 "The argument in the first section generalises to show t
hat the tangent plane to the graph of " }{XPPEDIT 18 0 "g(x,y)" "6#-%
\"gG6$%\"xG%\"yG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "x = a;" "6#/%
\"xG%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = b;" "6#/%\"yG%\"bG
" }{TEXT -1 5 " is  " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "s(x,y) = g(a,
 b)+g[x](a, b)*(x-a)+g[y](a, b)*(y-b);" "6#/-%\"sG6$%\"xG%\"yG,(-%\"gG
6$%\"aG%\"bG\"\"\"*&-&F+6#F'6$F-F.F/,&F'F/F-!\"\"F/F/*&-&F+6#F(6$F-F.F
/,&F(F/F.F6F/F/" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{TEXT 263 25 "_________________________" }{TEXT -1 1 " " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 11 "Example 1  " }}{PARA 0 "" 0 "" {TEXT 266 8 "Question" }{TEXT 
-1 3 ":  " }}{PARA 0 "" 0 "" {TEXT -1 58 "(a) Find the equation of the
 tangent plane to the surface " }{XPPEDIT 18 0 "z = 8-2*x^2-y^2;" "6#/
%\"zG,(\"\")\"\"\"*&\"\"#F'*$%\"xGF)F'!\"\"*$%\"yGF)F," }{TEXT -1 7 " \+
where " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 7 "  and  \+
" }{XPPEDIT 18 0 "y = -1;" "6#/%\"yG,$\"\"\"!\"\"" }{TEXT -1 3 ".  " }
}{PARA 0 "" 0 "" {TEXT -1 81 "(b) Construct a 3d plot to show the surf
ace and the tangent plane from part (a). " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 267 8 "Solution" }{TEXT -1 2 ": " }}
{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 43 "First
 we calculate the partial derivatives." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "g := (x,y) -> 8-2*x^2-y^
2;\ng_x := D[1](g);\ng_y := D[2](g); " }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"gGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(\"\")\"\"\"*&\"\"#F
/)9$F1F/!\"\"*$)9%F1F/F4F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$g
_xGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$9$!\"%F)F)F)" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$g_yGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)
,$9%!\"#F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 74 "plot3d(g(x,y),x=-2..2,y=-2..2,\n      color=CO
LOR(RGB,.8,.6,1),axes=boxed);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 
300 {PLOTDATA 3 "6%-%%GRIDG6&;$!\"#\"\"!$\"\"#F)F&X,%)anythingG6\"6\"[
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45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 36 "Now we construct the tangent plane. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
75 "a := 1;\nb := -1;\ng(a,b)+g_x(a,b)*(x-a)+g_y(a,b)*(y-b);\ns := una
pply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"bG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,(\"#6\"\"\"*&\"\"%F%%\"xGF%!\"\"*&\"\"#F%%\"yGF%F%" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"sGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(\"
#6\"\"\"*&\"\"%F/9$F/!\"\"*&\"\"#F/9%F/F/F)F)F)" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The tangent plane has equ
ation   " }{XPPEDIT 18 0 "z = 11-4*x+2*y;" "6#/%\"zG,(\"#6\"\"\"*&\"\"
%F'%\"xGF'!\"\"*&\"\"#F'%\"yGF'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 68 ": Thi
s result can be obtained more easily using the Maple procedure " }
{TEXT 0 7 "mtaylor" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "g := (x,y) -> 8-2*x^2-y^2:
\nmtaylor(g(x,y),[x=1,y=-1],2);\ns := unapply(%,x,y);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,(\"#6\"\"\"*&\"\"%F%%\"xGF%!\"\"*&\"\"#F%%\"yGF%F
%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGf*6$%\"xG%\"yG6\"6$%)operat
orG%&arrowGF),(\"#6\"\"\"*&\"\"%F/9$F/!\"\"*&\"\"#F/9%F/F/F)F)F)" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(b)" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "p1 := plot3d(g(x,y),x=-2..2
,y=-2..2,color=COLOR(RGB,.8,.6,1)):\np2 := plot3d(s(x,y),x=-2..2,y=-2.
.2,color=green,\n    grid=[50,20],style=wireframe,gridstyle=triangular
):\nplots[display]([p1,p2],axes=boxed,lightmodel=light4);" }}{PARA 13 
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505401BFFFFFFFFFFF8-F16&F3$F)F)$\"*++++\"!\")F@-%&STYLEG6#%%LINEG-%*AX
ESSTYLEG6#%$BOXG-%+AXESLABELSG6&%\"xG%\"yGQ!F/-%%FONTG6#%(DEFAULTG-%*G
RIDSTYLEG6#%+TRIANGULARG-%+LIGHTMODELG6#%(LIGHT_4G" 1 2 0 1 10 0 1 1 
6 2 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 " }}{PARA 0 "" 0 "
" {TEXT 264 8 "Question" }{TEXT -1 3 ":  " }}{PARA 0 "" 0 "" {TEXT -1 
58 "(a) Find the equation of the tangent plane to the surface " }
{XPPEDIT 18 0 "z=(2-y^2)/(1+x^2)" "6#/%\"zG*&,&\"\"#\"\"\"*$%\"yGF'!\"
\"F(,&F(F(*$%\"xGF'F(F+" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "x=1/4" 
"6#/%\"xG*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "y
=1/3" "6#/%\"yG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 
"" {TEXT -1 81 "(b) Construct a 3d plot to show the surface and the ta
ngent plane from part (a). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 265 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" 
{TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 43 "First we calculate th
e partial derivatives." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "g := (x,y) -> (2-y^2)/(1+x^2);\ng_x
 := D[1](g);\ng_y := D[2](g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"g
Gf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&,&\"\"#\"\"\"*$)9%F/F0!\"\"
F0,&*$)9$F/F0F0F0F0F4F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$g_xG
f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*(,&\"\"#\"\"\"*$)9%F0F1!\"
\"F1,&*$)9$F0F1F1F1F1!\"#F9F1F:F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$g_yGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&9%\"\"\",&*$)9$
\"\"#F0F0F0F0!\"\"!\"#F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 36 "Now we construct the tangent plane. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
78 "a := 1/4;\nb := 1/3;\ng(a,b)+g_x(a,b)*(x-a)+g_y(a,b)*(y-b);\ns := \+
unapply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG#\"\"\"\"\"%
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG#\"\"\"\"\"$" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,(#\"$7\"\"#^\"\"\"*&#\"#KF&F'%\"yGF'!\"\"*&#\"$G
\"\"$`\"F'%\"xGF'F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGf*6$%\"xG
%\"yG6\"6$%)operatorG%&arrowGF),(#\"$7\"\"#^\"\"\"*&#\"#KF0F19%F1!\"\"
*&#\"$G\"\"$`\"F19$F1F6F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 33 "The tangent plane has equation   " }
{XPPEDIT 18 0 "z=112/51-128/153*x-32/51*y" "6#/%\"zG,(*&\"$7\"\"\"\"\"
#^!\"\"F(*(\"$G\"F(\"$`\"F*%\"xGF(F**(\"#KF(F)F*%\"yGF(F*" }{TEXT -1 
2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "
Note" }{TEXT -1 68 ": This result can be obtained more easily using th
e Maple procedure " }{TEXT 0 7 "mtaylor" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "g := \+
(x,y) -> (2-y^2)/(1+x^2):\nmtaylor(g(x,y),[x=1/4,y=1/3],2);\ns := unap
ply(%,x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#\"$7\"\"#^\"\"\"*&#
\"#KF&F'%\"yGF'!\"\"*&#\"$G\"\"$`\"F'%\"xGF'F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"sGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(#\"$7\"
\"#^\"\"\"*&#\"#KF0F19%F1!\"\"*&#\"$G\"\"$`\"F19$F1F6F)F)F)" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(b)" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "p1 := plot3d(g(x,y),x=-1..1,y=-1..
1,color=COLOR(RGB,.8,.6,1)):\np2 := plot3d(s(x,y),x=-1..1,y=-1..1,colo
r=green,\n    grid=[50,20],style=wireframe,gridstyle=triangular):\nplo
ts[display]([p1,p2],axes=boxed,lightmodel=light4);" }}{PARA 13 "" 1 "
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C213-F16&F3$F)F)$\"*++++\"!\")F=-%&STYLEG6#%%LINEG-%*AXESSTYLEG6#%$BOX
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IANGULARG-%+LIGHTMODELG6#%(LIGHT_4G" 1 2 0 1 10 0 1 1 6 2 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Linear approximation example " }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
303 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 198 "A cy
linder has height 5 cm. and radius 2 cm. Use partial derivatives to es
timate the change in the volume of the cylinder when the height increa
ses by  0.02 cm. and the radius increases by 0.01 cm. " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 304 8 "Solution" }{TEXT -1 
2 ": " }}{PARA 0 "" 0 "" {TEXT -1 11 "The volume " }{TEXT 305 1 "V" }
{TEXT -1 27 " of a cylinder with height " }{TEXT 307 1 "h" }{TEXT -1 
12 " and radius " }{TEXT 306 1 "r" }{TEXT -1 14 " is given by  " }
{XPPEDIT 18 0 "V=Pi*r^2*h" "6#/%\"VG*(%#PiG\"\"\"*$%\"rG\"\"#F'%\"hGF'
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{XPPEDIT 18 0 "Delta*r = 1/100;" "6#/*&%&DeltaG\"\"\"%\"rGF&*&F&F&\"$+
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{XPPEDIT 18 0 "Delta*V;" "6#*&%&DeltaG\"\"\"%\"VGF%" }{TEXT -1 1 " " }
{TEXT 311 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Pi*``(2)^2/50+2*Pi*``(
2)*``(5)/100 = 7/25*Pi;" "6#/,&*(%#PiG\"\"\"*$-%!G6#\"\"#F,F'\"#]!\"\"
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{TEXT -1 1 " " }{TEXT 310 1 "~" }{TEXT -1 11 "  0.87965. " }}{PARA 0 "
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2/50+2*Pi*2*5/100;\nevalf(evalf(%),5);" }}{PARA 11 "" 1 "" {XPPMATH 
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" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "A mo
re accurate value is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "cyl_vol := (r,h) -> Pi*r^2*h;\ncyl_
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" {XPPMATH 20 "6#>%(cyl_volGf*6$%\"rG%\"hG6\"6$%)operatorG%&arrowGF)*(
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}{PARA 0 "" 0 "" {TEXT -1 60 "The change in the volume can be thought \+
of as comprising of " }{TEXT 259 9 "two parts" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "1. A disk
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$F%F%" }{TEXT -1 27 " and height (or thickness) " }{XPPEDIT 18 0 "Delt
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ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1
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256 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 27 "The volume
 of this disk is " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 
"Pi*(r+Delta*r)^2*`.`*Delta*h = Pi*r^2*`.`*Delta*h+2*Pi*r*`.`*Delta*r*
`.`*Delta*h+Pi*`.`*Delta*r^2*`.`*Delta*h;" "6#/*,%#PiG\"\"\"*$,&%\"rGF
&*&%&DeltaGF&F)F&F&\"\"#F&%\".GF&F+F&%\"hGF&,(*,F%F&*$F)F,F&F-F&F+F&F.
F&F&*4F,F&F%F&F)F&F-F&F+F&F)F&F-F&F+F&F.F&F&*0F%F&F-F&F+F&F)F,F-F&F+F&
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0 "" {TEXT -1 22 "This is approximately " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "Pi*r^2*`.`*Delta*h" "6#*,%#PiG\"\"\"*$%\"rG\"
\"#F%%\".GF%%&DeltaGF%%\"hGF%" }{TEXT -1 17 "   ------- (ii), " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "if we neg
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\"\"#\"\"\"%#PiGF%%\"rGF%%\".GF%%&DeltaGF%F'F%F(F%F)F%%\"hGF%" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "Pi*`.`*Delta*r^2*`.`*Delta*h" "6#*0%#PiG
\"\"\"%\".GF%%&DeltaGF%%\"rG\"\"#F&F%F'F%%\"hGF%" }{TEXT -1 34 ", whic
h will be small compared to " }{XPPEDIT 18 0 "Pi*r^2*`.`*Delta*h" "6#*
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}{PARA 0 "" 0 "" {TEXT -1 57 "This amounts to taking the radius of the
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-1 37 "2. A cylindrical \"shell\" with height " }{TEXT 313 1 "h" }
{TEXT -1 15 ", inner radius " }{TEXT 312 1 "r" }{TEXT -1 19 ", and out
er radius " }{XPPEDIT 18 0 "r + Delta*r" "6#,&%\"rG\"\"\"*&%&DeltaGF%F
$F%F%" }{TEXT -1 3 ".  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
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}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 31 "_______________________________" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 54 "Find the equation of
 the tangent plane to the surface " }{XPPEDIT 18 0 "z=g(x,y)" "6#/%\"z
G-%\"gG6$%\"xG%\"yG" }{XPPEDIT 18 0 "`` = arctan(y/x);" "6#/%!G-%'arct
anG6#*&%\"yG\"\"\"%\"xG!\"\"" }{TEXT -1 13 " at the point" }{XPPEDIT 
18 0 "``(1, 1, Pi/4);" "6#-%!G6%\"\"\"F&*&%#PiGF&\"\"%!\"\"" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "_____________________________
__" }}{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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 31 "_______________________________" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 68 "Find the equation of
 the tangent plane to the graph of the function " }{XPPEDIT 18 0 "g(x,
y) = x^2+2*x*y+6*y^2;" "6#/-%\"gG6$%\"xG%\"yG,(*$F'\"\"#\"\"\"*(F+F,F'
F,F(F,F,*&\"\"'F,*$F(F+F,F," }{TEXT -1 33 " at two points of your choo
sing. " }}{PARA 0 "" 0 "" {TEXT -1 31 "_______________________________
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 31 "_______________________________" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 68 "Find the equation of t
he tangent plane to the graph of the function " }{XPPEDIT 18 0 "g(x,y)
 = 3*x-x^3+y^3-3*y;" "6#/-%\"gG6$%\"xG%\"yG,**&\"\"$\"\"\"F'F,F,*$F'F+
!\"\"*$F(F+F,*&F+F,F(F,F." }{TEXT -1 15 " at the points " }{XPPEDIT 
18 0 "``(-1, -1), ``(-1, 1), ``(1, -1), ``(1,1)" "6&-%!G6$,$\"\"\"!\"
\",$F'F(-F$6$,$F'F(F'-F$6$F',$F'F(-F$6$F'F'" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 22 "Explain what you find." }}{PARA 0 "" 0 "" 
{TEXT -1 31 "_______________________________" }}{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 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 31 "__
_____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "
" {TEXT -1 76 "Use partial derivatives to estimate the change in the v
alue of the function " }{XPPEDIT 18 0 "g(x,y) = x^2*y+4*x*y^2" "6#/-%
\"gG6$%\"xG%\"yG,&*&F'\"\"#F(\"\"\"F,*(\"\"%F,F'F,F(F+F," }{TEXT -1 6 
" when " }{TEXT 328 1 "x" }{TEXT -1 30 " changes from 1 to 1.001 and  \+
" }{TEXT 329 1 "y" }{TEXT -1 43 " (simultaneously) changes from 2 to 2
.002. " }}{PARA 0 "" 0 "" {TEXT -1 31 "_______________________________
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 31 "_______________________________" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 3 "Q8 " }}{PARA 0 "" 0 "" {TEXT -1 76 "Use partial derivative
s to estimate the change in the value of the function " }{XPPEDIT 18 
0 "g(x,y)=arctan(y/x)" "6#/-%\"gG6$%\"xG%\"yG-%'arctanG6#*&F(\"\"\"F'!
\"\"" }{TEXT -1 6 " when " }{TEXT 330 1 "x" }{TEXT -1 28 " changes fro
m 1 to 1.01 and " }{TEXT 331 1 "y" }{TEXT -1 43 " (simultaneously) cha
nges from 1 to 0.99.  " }}{PARA 0 "" 0 "" {TEXT -1 31 "_______________
________________" }}{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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 31 "_______________________________
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 3 "Q9 " }}{PARA 0 "" 0 "" {TEXT -1 276 "The base radiu
s and height of a right circular cone are 5 cm. and 10 cm. respectivel
y. There is a possible error of as much as 0.03 cm. in each measuremen
t. Use partial derivatives to esimate the maximum resulting error that
 might occur when computing the volume of the cone. " }}{PARA 0 "" 0 "
" {TEXT -1 31 "_______________________________" }}{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 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 31 "_______________________________" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Q10  " }}
{PARA 0 "" 0 "" {TEXT -1 146 "(a) A horizontal beam with a rectangular
 cross-section is supported at both ends, and when loaded exhibits a d
eflection at its mid point given by " }{XPPEDIT 18 0 "S=k/(w*h^3)" "6#
/%\"SG*&%\"kG\"\"\"*&%\"wGF'*$%\"hG\"\"$F'!\"\"" }{TEXT -1 8 ", where \+
" }{TEXT 332 1 "w" }{TEXT -1 5 " and " }{TEXT 333 1 "h" }{TEXT -1 52 "
 are the width and height of the cross-section, and " }{TEXT 338 1 "k
" }{TEXT -1 82 " is a constant that depends on the length of the beam \+
and the amount of the load. " }}{PARA 0 "" 0 "" {TEXT -1 28 "Show that
 for small changes " }{XPPEDIT 18 0 "Delta*w" "6#*&%&DeltaG\"\"\"%\"wG
F%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Delta*h" "6#*&%&DeltaG\"\"\"%\"hG
F%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Delta*S" "6#*&%&DeltaG\"\"\"%
\"SGF%" }{TEXT -1 18 " in the variables " }{TEXT 334 1 "w" }{TEXT -1 
1 "," }{TEXT 335 2 " h" }{TEXT -1 5 " and " }{TEXT 336 1 "S" }{TEXT 
-1 2 ", " }{XPPEDIT 18 0 "Delta*S" "6#*&%&DeltaG\"\"\"%\"SGF%" }{TEXT 
-1 1 " " }{TEXT 337 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-S*(Delta*w/
w+3*Delta*h/h)" "6#,$*&%\"SG\"\"\",&*(%&DeltaGF&%\"wGF&F*!\"\"F&**\"\"
$F&F)F&%\"hGF&F.F+F&F&F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
183 "(b) If a sag of 1 cm. occurs when the width is 2 cm. and the heig
ht is 4 cm., use the formula obtained in (a) to estimate the sag when \+
the width is 2.05 cm. and the height is 4.05 cm." }}{PARA 0 "" 0 "" 
{TEXT -1 31 "_______________________________" }}{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 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 31 "__
_____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 18 
"Code for pictures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{SECT 1 {PARA 0 "" 0 "" {TEXT -1 27 "code for surface and point " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
706 "g := (x,y) -> x^2-3*x*y-y^2+30:\na := 1: b := 2:\np1:=plot3d(g(x,
y),x=-2..3.5,y=-2..3.5,color=red):\np2:=plots[spacecurve]([a+.1*cos(t)
,b+.1*sin(t),\n g(a+.1*cos(t),b+.1*sin(t))],t=0..2*Pi,color=green,thic
kness=2):\np3:=plots[spacecurve]([a+.2*cos(t),b+.2*sin(t),\n  g(a+.2*c
os(t),b+.2*sin(t))],t=0..2*Pi,color=yellow,thickness=1):\np4:=plot3d(0
,x=-2..4,y=-2..4,grid=[7,7],\n    color=COLOR(RGB,.9,.9,.9)):\np5:=plo
ts[pointplot3d]([[a,b,0],[a,b,g(a,b)]],style=line,\n     color=green,t
hickness=2):\nt1:=plots[textplot3d]([1.5,2.3,25,`x=1,y=2,z=21`],color=
black,\n     font=[HELVETICA,BOLD,10]):\nplots[display]([p1,p2,p3,p4,p
5,t1],axes=boxed,lightmodel=light4,\n    orientation=[-26,55],labels=[
`x`,`y`,`z`],view=0..44);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 692 "g := (x,y) -> x^2-3*x*y-y^2+30:\np
1:=plot3d(g(x,y),x=-2..3.5,y=-2..3.5,color=red):\np2:=plot3d(39-7*y-4*
x,x=-2..3.5,y=-2..3.5,color=blue,\n              style=wireframe):\np3
:=plots[spacecurve]([x,2,g(x,2)],x=-2..3.5,\n                 color=gr
een,thickness=3 ):\np4:=plots[spacecurve]([x,2,25-4*x],x=-2..3.5,\n   \+
              color=magenta,thickness=3):\np5:=plots[spacecurve]([1,y,
g(1,y)], y=-2..3.5, \n                 color=yellow, thickness=3):\np6
:=plots[spacecurve]([1,y,35-7*y],y=-2..3.5, \n                 color=c
yan, thickness=3):    \nplots[display]([p1,p2,p3,p4,p5,p6],gridstyle=t
riangular,\n   axes=boxed,lightmodel=light4,orientation=[-44,49],\n   \+
         labels=[`x`,`y`,`z`],view=0..60);" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 0 "" 0 "" {TEXT -1 30 "code for pictures of cylinder " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2108 "
r := 1: h := 2:\np1 := plot([r*cos(t),r*0.3*sin(t),t=-Pi..0],color=nav
y):\np2 := plot([r*cos(t),r*0.3*sin(t)+h,t=0..2*Pi],color=navy):\nA1:=
[r,0]: B1:=[r,h]: C1:=[-r,h]: D1:=[-r,0]:\np3 := plot([[A1,B1],[D1,C1]
],color=navy):\na := evalf(Pi/60):\nc1 := seq([r*cos(j*a),r*0.3*sin(j*
a)],j=-12..0):\nc2 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-27..-12):\nc3
 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-33..-27):\nc4 := seq([r*cos(j*a
),r*0.3*sin(j*a)],j=-48..-33):\nc5 := seq([r*cos(j*a),r*0.3*sin(j*a)],
j=-60..-48):\nc6 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=0..12):\nc7 :=
 seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=12..27):\nc8 := seq([r*cos(j*a),h
-r*0.3*sin(j*a)],j=27..33):\nc9 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j
=33..48):\nc10 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=48..60):\np4 := \+
plots[polygonplot]([c1,c6],\n    style=patchnogrid,color=COLOR(RGB,.95
,.89,1)):\np5 := plots[polygonplot]([c2,c7],\n    style=patchnogrid,co
lor=COLOR(RGB,.95,.91,1)):\np6 := plots[polygonplot]([c3,c8],\n    sty
le=patchnogrid,color=COLOR(RGB,.953,.923,1)):\np7 := plots[polygonplot
]([c4,c9],\n    style=patchnogrid,color=COLOR(RGB,.95,.91,1)):\np8 := \+
plots[polygonplot]([c5,c10],\n    style=patchnogrid,color=COLOR(RGB,.9
5,.89,1)):\nc11 := seq([r*cos(2*j*a),r*0.3*sin(2*j*a)+h],j=0..60):\np9
 := plots[polygonplot]([c11],\n    style=patchnogrid,color=COLOR(RGB,.
953,.933,1)):\np10 := plot([[A1,[r+.2,0]],[B1,[r+.2,h]]],\n           \+
            color=COLOR(RGB,.5,.35,.75),linestyle=2):\np11 := plottool
s[arrow]([r+.13,h/2-.13],[r+.13,0],\n          0,.06,.08,arrow,color=C
OLOR(RGB,.5,.35,.75)):\np12 := plottools[arrow]([r+.13,h/2+.13],[r+.13
,h],\n          0,.06,.08,arrow,color=COLOR(RGB,.5,.35,.75)):\np13 := \+
plottools[arrow]([r/2+.1,h],[r,h],\n          0,.05,.15,arrow,color=CO
LOR(RGB,.5,.35,.75)):\np14 := plot([[0,h],[r/2-0.1,h]],color=COLOR(RGB
,.5,.35,.75)):\np15 := plot([[[0,h]]$3],style=point,\n     symbol=[cro
ss,diamond,circle],color=COLOR(RGB,.5,.35,.75)):\nt1 := plots[textplot
]([[r+.13,h/2,'h'],[r/2,h,'r']],\n                             font=[H
ELVETICA,9],color=navy):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,
\n           p10,p11,p12,p13,p14,p15,t1],axes=none);" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3087 "r := 1:
 h := 2:\nrp := 1.15: hp := 2.25:\np1 := plot([r*cos(t),r*0.3*sin(t),t
=-Pi..0],color=navy):\np2 := plot([r*cos(t),r*0.3*sin(t)+h,t=0..2*Pi],
color=navy):\nA1:=[r,0]: B1:=[r,h]: C1:=[-r,h]: D1:=[-r,0]:\nAp:=[rp,0
]: Bp:=[rp,hp]: Cp:=[-rp,hp]: Dp:=[-rp,0]:\np3 := plot([[A1,B1],[D1,C1
]],color=navy):\na := evalf(Pi/60):\nc1 := seq([r*cos(j*a),r*0.3*sin(j
*a)],j=-12..0):\nc2 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-27..-12):\nc
3 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-33..-27):\nc4 := seq([r*cos(j*
a),r*0.3*sin(j*a)],j=-48..-33):\nc5 := seq([r*cos(j*a),r*0.3*sin(j*a)]
,j=-60..-48):\nc6 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=0..12):\nc7 :
= seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=12..27):\nc8 := seq([r*cos(j*a),
h-r*0.3*sin(j*a)],j=27..33):\nc9 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],
j=33..48):\nc10 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=48..60):\np4 :=
 plots[polygonplot]([c1,c6],\n    style=patchnogrid,color=COLOR(RGB,.9
5,.89,1)):\np5 := plots[polygonplot]([c2,c7],\n    style=patchnogrid,c
olor=COLOR(RGB,.95,.91,1)):\np6 := plots[polygonplot]([c3,c8],\n    st
yle=patchnogrid,color=COLOR(RGB,.953,.923,1)):\np7 := plots[polygonplo
t]([c4,c9],\n    style=patchnogrid,color=COLOR(RGB,.95,.91,1)):\np8 :=
 plots[polygonplot]([c5,c10],\n    style=patchnogrid,color=COLOR(RGB,.
95,.89,1)):\nc11 := seq([r*cos(2*j*a),r*0.3*sin(2*j*a)+h],j=0..60):\np
9 := plots[polygonplot]([c11],\n    style=patchnogrid,color=COLOR(RGB,
.953,.933,1)):\np10 := plot([[A1,[r+.8,0]],[B1,[r+.4,h]],[Bp,[r+.8,hp]
]],\n                  color=COLOR(RGB,.5,.35,.75),linestyle=2):\np11 \+
:= plottools[arrow]([r+.33,h/2-.13],[r+.33,0],\n          0,.06,.08,ar
row,color=COLOR(RGB,.5,.35,.75)):\np12 := plottools[arrow]([r+.33,h/2+
.13],[r+.33,h],\n          0,.06,.08,arrow,color=COLOR(RGB,.5,.35,.75)
):\np13 := plottools[arrow]([r/2+.1,h],[r,h],\n          0,.05,.15,arr
ow,color=COLOR(RGB,.5,.35,.75)):\np14 := plot([[0,h],[r/2-0.1,h]],colo
r=COLOR(RGB,.5,.35,.75)):\np15 := plot([[[0,h]]$3,[[0,hp]]$3],style=po
int,\n     symbol=[cross,diamond,circle],color=COLOR(RGB,.5,.35,.75)):
\np16 := plot([rp*cos(t),rp*0.3*sin(t),t=-Pi-0.5..0.5],\n             \+
                           linestyle=2,color=navy):\np17 := plot([rp*c
os(t),rp*0.3*sin(t)+hp,t=0..2*Pi],\n                                  \+
      linestyle=2,color=navy):\np18 := plot([[Ap,Bp],[Dp,Cp]],linestyl
e=2,color=navy):\np19 := plottools[arrow]([rp+.53,hp/2-.13],[rp+.53,0]
,\n          0,.06,.08,arrow,color=COLOR(RGB,.5,.35,.75)):\np20 := plo
ttools[arrow]([rp+.53,hp/2+.13],[rp+.53,hp],\n          0,.06,.08,arro
w,color=COLOR(RGB,.5,.35,.75)):\np21 := plottools[arrow]([rp/2+.25,hp]
,[rp,hp],\n          0,.05,.15,arrow,color=COLOR(RGB,.5,.35,.75)):\np2
2 := plot([[0,hp],[rp/2-0.25,hp]],color=COLOR(RGB,.5,.35,.75)):\nt1 :=
 plots[textplot]([[r+.33,h/2,'h'],[r/2,h,'r'],\n           [rp+.53,hp/
2,`h +   h`],[rp/2,hp,`r +   r`]],\n                font=[HELVETICA,9]
,color=navy):\nt2 := plots[textplot]([[rp+.58,hp/2,`D`],[rp/2+0.05,hp,
`D`]],\n                      font=[SYMBOL,9],color=navy):\nplots[disp
lay]([p1,p2,p3,p4,p5,p6,p7,p8,p9,\n           p10,p11,p12,p13,p14,p15,
p16,p17,p18,\n             p19,p20,p21,p22,t1,t2],axes=none);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
2240 "r := 1.15: h := 0.2:\np1 := plot([r*cos(t),r*0.3*sin(t),t=-Pi..0
],color=navy):\np2 := plot([r*cos(t),r*0.3*sin(t)+h,t=0..2*Pi],color=n
avy):\nA1:=[r,0]: B1:=[r,h]: C1:=[-r,h]: D1:=[-r,0]:\np3 := plot([[A1,
B1],[D1,C1]],color=navy):\na := evalf(Pi/60):\nc1 := seq([r*cos(j*a),r
*0.3*sin(j*a)],j=-12..0):\nc2 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-27
..-12):\nc3 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-33..-27):\nc4 := seq
([r*cos(j*a),r*0.3*sin(j*a)],j=-48..-33):\nc5 := seq([r*cos(j*a),r*0.3
*sin(j*a)],j=-60..-48):\nc6 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=0..
12):\nc7 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=12..27):\nc8 := seq([r
*cos(j*a),h-r*0.3*sin(j*a)],j=27..33):\nc9 := seq([r*cos(j*a),h-r*0.3*
sin(j*a)],j=33..48):\nc10 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=48..6
0):\np4 := plots[polygonplot]([c1,c6],\n    style=patchnogrid,color=CO
LOR(RGB,.95,.89,1)):\np5 := plots[polygonplot]([c2,c7],\n    style=pat
chnogrid,color=COLOR(RGB,.95,.91,1)):\np6 := plots[polygonplot]([c3,c8
],\n    style=patchnogrid,color=COLOR(RGB,.953,.923,1)):\np7 := plots[
polygonplot]([c4,c9],\n    style=patchnogrid,color=COLOR(RGB,.95,.91,1
)):\np8 := plots[polygonplot]([c5,c10],\n    style=patchnogrid,color=C
OLOR(RGB,.95,.89,1)):\nc11 := seq([r*cos(2*j*a),r*0.3*sin(2*j*a)+h],j=
0..60):\np9 := plots[polygonplot]([c11],\n    style=patchnogrid,color=
COLOR(RGB,.953,.933,1)):\np10 := plot([[A1,[r+.2,0]],[B1,[r+.2,h]]],\n
                       color=COLOR(RGB,.5,.35,.75),linestyle=2):\np11 \+
:= plottools[arrow]([r+.13,h+.15],[r+.13,h],\n          0,.05,.5,arrow
,color=COLOR(RGB,.5,.35,.75)):\np12 := plottools[arrow]([r+.13,-.15],[
r+.13,0],\n          0,.05,.5,arrow,color=COLOR(RGB,.5,.35,.75)):\np13
 := plottools[arrow]([r/2+.2,h],[r,h],\n          0,.05,.15,arrow,colo
r=COLOR(RGB,.5,.35,.75)):\np14 := plot([[0,h],[r/2-0.2,h]],color=COLOR
(RGB,.5,.35,.75)):\np15 := plot([[[0,h]]$3],style=point,\n     symbol=
[cross,diamond,circle],color=COLOR(RGB,.5,.35,.75)):\nt1 := plots[text
plot]([[r+.17,h/2+.02,'h'],[r/2,h,'`r +    r`']],\n                   \+
          font=[HELVETICA,9],color=navy):\nt2 := plots[textplot]([[r+.
1,h/2+.02,`D`],[r/2+.05,h,`D`]],\n                             font=[S
YMBOL,9],color=navy):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,\n  \+
         p10,p11,p12,p13,p14,p15,t1,t2],axes=none);" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2991 "r := 1.1
5: h := 2: rp := 1:\np1 := plot([r*cos(t),r*0.3*sin(t),t=-Pi..0],color
=navy):\np2 := plot([r*cos(t),r*0.3*sin(t)+h,t=0..2*Pi],color=navy):\n
A1:=[r,0]: B1:=[r,h]: C1:=[-r,h]: D1:=[-r,0]:\nAp:=[rp,0]: Bp:=[rp,h]:
 Cp:=[-rp,h]: Dp:=[-rp,0]:\np3 := plot([[A1,B1],[D1,C1]],color=navy):
\na := evalf(Pi/60):\nc1 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-12..0):
\nc2 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-27..-12):\nc3 := seq([r*cos
(j*a),r*0.3*sin(j*a)],j=-33..-27):\nc4 := seq([r*cos(j*a),r*0.3*sin(j*
a)],j=-48..-33):\nc5 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-60..-48):\n
c6 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=0..12):\nc7 := seq([r*cos(j*
a),h-r*0.3*sin(j*a)],j=12..27):\nc8 := seq([r*cos(j*a),h-r*0.3*sin(j*a
)],j=27..33):\nc9 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=33..48):\nc10
 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=48..60):\np4 := plots[polygonp
lot]([c1,c6],\n    style=patchnogrid,color=COLOR(RGB,.95,.89,1)):\np5 \+
:= plots[polygonplot]([c2,c7],\n    style=patchnogrid,color=COLOR(RGB,
.95,.91,1)):\np6 := plots[polygonplot]([c3,c8],\n    style=patchnogrid
,color=COLOR(RGB,.953,.923,1)):\np7 := plots[polygonplot]([c4,c9],\n  \+
  style=patchnogrid,color=COLOR(RGB,.95,.91,1)):\np8 := plots[polygonp
lot]([c5,c10],\n    style=patchnogrid,color=COLOR(RGB,.95,.89,1)):\npp
 := seq([[r*cos(2*j*a),r*0.3*sin(2*j*a)+h],\n           [r*cos(2*(j+1)
*a),r*0.3*sin(2*(j+1)*a)+h],\n           [rp*cos(2*(j+1)*a),rp*0.3*sin
(2*(j+1)*a)+h],\n           [rp*cos(2*j*a),rp*0.3*sin(2*j*a)+h]],j=0..
60):\np9 := plots[polygonplot]([pp],\n    style=patchnogrid,color=COLO
R(RGB,.953,.933,1)):\np10 := plot([[A1,[r+.2,0]],[B1,[r+.2,h]]],\n    \+
               color=COLOR(RGB,.5,.35,.75),linestyle=2):\np11 := plott
ools[arrow]([r+.13,h/2-.13],[r+.13,0],\n          0,.06,.08,arrow,colo
r=COLOR(RGB,.5,.35,.75)):\np12 := plottools[arrow]([r+.13,h/2+.13],[r+
.13,h],\n          0,.06,.08,arrow,color=COLOR(RGB,.5,.35,.75)):\np13 \+
:= plottools[arrow]([rp/2+.1,h],[rp,h],\n          0,.05,.15,arrow,col
or=COLOR(RGB,.5,.35,.75)):\np14 := plot([[0,h],[rp/2-0.1,h]],color=COL
OR(RGB,.5,.35,.75)):\np15 := plot([[0,h]],style=point,symbol=cross,\n \+
                               color=COLOR(RGB,.5,.35,.75)):\np16 := p
lot([rp*cos(t),rp*0.3*sin(t),t=0..2*Pi],\n                            \+
         linestyle=2,color=navy):\np17 := plot([rp*cos(t),rp*0.3*sin(t
)+h,t=0..2*Pi],\n                                     linestyle=1,colo
r=navy):\np18 := plot([[Ap,[rp,h+0.4]],[Dp,Cp],[B1,[r,h+0.4]]],\n     \+
                                linestyle=2,color=navy):\np19 := plott
ools[arrow]([rp-.2,h+.3],[rp,h+.3],\n          0,.06,.3,arrow,color=CO
LOR(RGB,.5,.35,.75)):\np20 := plottools[arrow]([r+.2,h+.3],[r,h+.3],\n
          0,.06,.3,arrow,color=COLOR(RGB,.5,.35,.75)):\nt1 := plots[te
xtplot]([[r+.14,h/2,'h'],[rp+.11,h+.3,'`r`'],\n            [rp/2,h,'`r
`']],font=[HELVETICA,9],color=navy):\nt2 := plots[textplot]([[rp+.06,h
+.3,`D`]],\n                      font=[SYMBOL,9],color=navy):\nplots[
display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,\n  p10,p11,p12,p13,p14,p15,p16,p
17,p18,p19,p20,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1574 "c := 7.2: h := 2:\nA2 := [0,0]: B2 := [0,h]: C2 := [c,h]: D2 := \+
[c,0]:\nB3 := [.15,h+.15]: C3 := [c+.15,h+.15]: D3 := [c+.15,.15]: \np
1 := plot([[A2,B2,C2,D2,A2],[B2,B3,C3,C2],[C3,D3,D2]],color=navy):\np2
 := plots[polygonplot]([A2,B2,C2,D2],\n         style=patchnogrid,colo
r=COLOR(RGB,.95,.91,1)):\np3 := plots[polygonplot]([B2,B3,C3,C2],\n   \+
      style=patchnogrid,color=COLOR(RGB,.953,.923,1)):\np4 := plots[po
lygonplot]([C2,C3,D3,D2],\n         style=patchnogrid,color=COLOR(RGB,
.95,.89,1)):\np5 := plot([[A2,[0,-.3]],[D2,[c,-.3]],[A2,[-.5,0]],\n   \+
     [B2,[-.5,h]],[C2,[c+.7,h]],[C3,[c+.9,h+.15]]],\n              col
or=COLOR(RGB,.5,.35,.75),linestyle=2):\np6 := plottools[arrow]([c/2-.3
,-.2],[0,-.2],\n          0,.08,.03,arrow,color=COLOR(RGB,.5,.35,.75))
:\np7 := plottools[arrow]([c/2+.3,-.2],[c,-.2],\n          0,.08,.03,a
rrow,color=COLOR(RGB,.5,.35,.75)):\np8 := plottools[arrow]([-.3,h/2+.1
3],[-.3,h],\n          0,.13,.07,arrow,color=COLOR(RGB,.5,.35,.75)):\n
p9 := plottools[arrow]([-.3,h/2-.13],[-.3,0],\n          0,.13,.07,arr
ow,color=COLOR(RGB,.5,.35,.75)):\np10 := plottools[arrow]([c+.3,h-.2],
[c+.5,h],\n          0,.1,.3,arrow,color=COLOR(RGB,.5,.35,.75)):\np11 \+
:= plottools[arrow]([c+.85,h+.35],[c+.65,h+.15],\n          0,.1,.3,ar
row,color=COLOR(RGB,.5,.35,.75)):\nt1 := plots[textplot]([[c/2,-.2,'`2
    r`'],[c+.65,h+.085,'`r`'],\n        [-.3,h/2,'`h`']],font=[HELVETI
CA,9],color=navy):\nt2 := plots[textplot]([[c/2+.01,-.2,`p`],[c+.5,h+.
085,`D`]],\n                            font=[SYMBOL,9],color=navy):\n
plots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,t1,t2],axes=none);
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 
2 33 1 1 }