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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 35 "  A Surface and its Tangent Plane\n\n" }}{PARA 0 "
" 0 "" {TEXT -1 13 "We need plots" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Look at the f
unction f(x,y)=x^2 + y^2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:=x^2 \+
+ y^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*$)%\"xG\"\"#\"\"\"
\"\"\"*$)%\"yGF)F*F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Set S to \+
be the surface" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "S := plot3d(f,x=0
..2,y=0..2,color='red'):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "To ge
t the equation of the tangent plane, we need to find\nthe partial deri
vatives of f" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "dfdx:=diff(f,x);  d
fdy:=diff(f,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dfdxG,$%\"xG\"\"
#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dfdyG,$%\"yG\"\"#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 138 "Let's look at the tangent plane at (x,y)
=(1,1):\nThe equation of the tangent plane is\n\nz = f(1,1)  + dfdx(1,
1) * (x-1) + dfdx(1,1) * (y-1)\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 
"z := subs(x=1,y=1,f)+subs(x=1,y=1,dfdx)*(x-1)+subs(x=1,y=1,dfdy)*(y-1
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG,(!\"#\"\"\"%\"xG\"\"#%\"y
GF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Call the tangent plane P" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "P := plot3d(z,x=0..2,y=0..2,color
='yellow'):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Show the surface a
nd its tangent plane on the same graph" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 17 "display3d(\{S,P\});" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}
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