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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 5 "C3M5b" }}}{EXCHG {PARA 
257 "" 0 "" {TEXT 257 53 "The Gradient and Tangent Planes for Level Su
rfaces of" }{TEXT 259 1 " " }{TEXT 258 12 "G(x,y,z) = c" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{TEXT 260 1 "D" }{TEXT 
-1 12 " in space ( " }{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" }{TEXT -1 
45 " ) is the domain of the real-valued function " }{TEXT 261 1 "G" }
{TEXT -1 16 " .  That is, G: " }{XPPEDIT 18 0 "D->R^3" "6#R6#%\"DG7\"6
$%)operatorG%&arrowG6\"*$%\"RG\"\"$F*F*F*" }{TEXT -1 8 " . If   " }
{XPPEDIT 18 0 "L[c]" "6#&%\"LG6#%\"cG" }{TEXT -1 20 "  is the subset o
f  " }{TEXT 262 1 "D" }{TEXT -1 12 "  for which " }{TEXT 263 12 "G(x,y
,z) = c" }{TEXT -1 9 " , then  " }{XPPEDIT 18 0 "L[c]" "6#&%\"LG6#%\"c
G" }{TEXT -1 31 "  is a level surface of or for " }{TEXT 264 1 "G" }
{TEXT -1 19 "  for the constant " }{TEXT 265 1 "c" }{TEXT -1 21 " .  F
or example, if  " }{XPPEDIT 18 0 "G(x,y,z)=x^2+y^2+z^2" "6#/-%\"GG6%%
\"xG%\"yG%\"zG,(*$F'\"\"#\"\"\"*$F(F,F-*$F)F,F-" }{TEXT -1 7 "  and  \+
" }{XPPEDIT 18 0 "c=9" "6#/%\"cG\"\"*" }{TEXT -1 9 " , then  " }
{XPPEDIT 18 0 "L[9]" "6#&%\"LG6#\"\"*" }{TEXT -1 74 "  is the sphere o
f radius 3 centered at the origin.  Different values of  " }{TEXT 266 
1 "c" }{TEXT -1 49 "  produce different spheres as level surfaces of \+
" }{TEXT 267 1 "G" }{TEXT -1 32 " .  Thus, for a given function  " }
{TEXT 268 1 "G" }{TEXT -1 13 "  and value  " }{XPPEDIT 18 0 "c" "6#%\"
cG" }{TEXT -1 28 "   we may define a surface  " }{XPPEDIT 18 0 "Sigma
" "6#%&SigmaG" }{TEXT -1 6 "  as  " }{XPPEDIT 18 0 "L[c]" "6#&%\"LG6#%
\"cG" }{TEXT -1 23 "  .  Now suppose that  " }{XPPEDIT 269 0 "alpha" "
6#%&alphaG" }{TEXT -1 24 "(t) is a space curve in " }{TEXT 270 1 "S" }
{TEXT -1 14 " .  That is,  " }{XPPEDIT 271 0 "alpha" "6#%&alphaG" }
{TEXT -1 2 ": " }{XPPEDIT 18 0 "[a,b]->Sigma" "6#R6#7$%\"aG%\"bG7\"6$%
)operatorG%&arrowG6\"%&SigmaGF,F,F," }{TEXT -1 14 " .  Suppose   " }
{XPPEDIT 272 0 "alpha" "6#%&alphaG" }{TEXT -1 6 "(t) = " }{TEXT 273 
17 "<f (t),g(t),h(t)>" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "By the nature of our assu
mptions, the composition " }{TEXT 274 1 "G" }{TEXT -1 1 "(" }{XPPEDIT 
275 0 "alpha" "6#%&alphaG" }{TEXT -1 1 "(" }{TEXT 276 1 "t" }{TEXT -1 
16 ")) = c for each " }{TEXT 277 1 "t" }{TEXT -1 120 " .  The overall \+
effect of our composition function is that the function is a constant.
  It only assumes one value, c .  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 24 "Thus the derivative of  " }{TEXT 329 1 "G" }{TEXT -1 1 "(" }
{XPPEDIT 257 0 "alpha" "6#%&alphaG" }{TEXT -1 1 "(" }{TEXT 331 1 "t" }
{TEXT -1 20 ")) with respect to  " }{TEXT 332 1 "t" }{TEXT -1 44 "  is
  0 .  The chain rule thus tells us that" }}}{EXCHG {PARA 267 "" 0 "" 
{TEXT -1 6 "      " }{XPPEDIT 18 0 "G[x]" "6#&%\"GG6#%\"xG" }{TEXT -1 
1 "(" }{XPPEDIT 256 0 "alpha" "6#%&alphaG" }{TEXT -1 1 "(" }{TEXT 334 
1 "t" }{TEXT -1 3 ")) " }{TEXT 336 3 "f '" }{TEXT -1 1 "(" }{TEXT 335 
1 "t" }{TEXT -1 7 ")  +   " }{XPPEDIT 18 0 "G[y];" "6#&%\"GG6#%\"yG" }
{TEXT -1 1 "(" }{XPPEDIT 256 0 "alpha" "6#%&alphaG" }{TEXT -1 1 "(" }
{TEXT 338 1 "t" }{TEXT -1 3 ")) " }{TEXT 340 2 "g'" }{TEXT -1 1 "(" }
{TEXT 339 1 "t" }{TEXT -1 6 ")  +  " }{XPPEDIT 18 0 "G[z];" "6#&%\"GG6
#%\"zG" }{TEXT -1 1 "(" }{XPPEDIT 256 0 "alpha" "6#%&alphaG" }{TEXT 
-1 1 "(" }{TEXT 342 1 "t" }{TEXT -1 3 ")) " }{TEXT 344 2 "h'" }{TEXT 
-1 1 "(" }{TEXT 343 1 "t" }{TEXT -1 13 ")   =  0     " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 59 "But this can be rewritten using gradient \+
and dot product as" }}}{EXCHG {PARA 268 "" 0 "" {TEXT -1 5 "     " }
{TEXT 347 4 "grad" }{TEXT -1 3 " G(" }{XPPEDIT 256 0 "alpha" "6#%&alph
aG" }{TEXT -1 1 "(" }{TEXT 346 1 "t" }{TEXT -1 6 ")) \267  " }
{XPPEDIT 256 0 "alpha" "6#%&alphaG" }{TEXT -1 2 "'(" }{TEXT 350 1 "t" 
}{TEXT -1 14 ")  =  0       " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "R
ecall that when the dot product is zero, we know that the vectors are \+
perpendicular." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 292 12 "Conclusion: " 
}{TEXT -1 18 " The gradient at  " }{XPPEDIT 18 0 "X[0]" "6#&%\"XG6#\"
\"!" }{TEXT -1 4 " ,  " }{TEXT 293 4 "grad" }{TEXT -1 1 " " }{TEXT 
294 1 "G" }{TEXT -1 3 " ( " }{XPPEDIT 18 0 "X[0]" "6#&%\"XG6#\"\"!" }
{TEXT -1 47 " ), is orthogonal to every tangent vector (at  " }
{XPPEDIT 18 0 "X[0]" "6#&%\"XG6#\"\"!" }{TEXT -1 26 " ) of the level s
urface of" }{TEXT 295 4 "  G " }{TEXT -1 16 " that contains  " }
{XPPEDIT 18 0 "X[0]" "6#&%\"XG6#\"\"!" }{TEXT -1 70 " .  The gradient \+
vector at a point is normal to the level surface of  " }{TEXT 296 1 "G
" }{TEXT -1 27 "  that contains the point.\n" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 99 "This makes it easy to find the equation of a tangent pl
ane to a level surface for the point where  " }{XPPEDIT 18 0 "t=t[0]" 
"6#/%\"tG&F$6#\"\"!" }{TEXT -1 8 " .  If  " }{TEXT 300 4 "grad" }
{TEXT -1 1 " " }{TEXT 297 1 "G" }{TEXT -1 1 "(" }{XPPEDIT 298 0 "alpha
" "6#%&alphaG" }{TEXT -1 1 "(" }{TEXT 299 1 "t" }{TEXT -1 7 ")) = < " 
}{XPPEDIT 18 0 "m[1]" "6#&%\"mG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 
18 0 "m[2]" "6#&%\"mG6#\"\"#" }{TEXT -1 3 " , " }{XPPEDIT 18 0 " m[3]
" "6#&%\"mG6#\"\"$" }{TEXT -1 6 "  > , " }{XPPEDIT 301 0 "alpha" "6#%&
alphaG" }{TEXT -1 2 "( " }{XPPEDIT 18 0 "t[0]" "6#&%\"tG6#\"\"!" }
{TEXT -1 7 " ) = < " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT 
-1 3 " , " }{XPPEDIT 18 0 "y[0]" "6#&%\"yG6#\"\"!" }{TEXT -1 3 " , " }
{XPPEDIT 18 0 "z[0]" "6#&%\"zG6#\"\"!" }{TEXT -1 5 " > = " }{XPPEDIT 
302 0 "X[0]" "6#&%\"XG6#\"\"!" }{TEXT -1 8 " , and  " }{XPPEDIT 304 0 
"X" "6#%\"XG" }{TEXT 305 1 " " }{TEXT -1 5 " = < " }{TEXT 303 7 "x, y,
 z" }{TEXT -1 51 " > , then the equation of the tangent plane becomes
" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 2 "  " }{TEXT 311 4 "grad" }
{TEXT -1 1 " " }{TEXT 308 1 "G" }{TEXT -1 1 "(" }{XPPEDIT 309 0 "alpha
" "6#%&alphaG" }{TEXT -1 1 "(" }{TEXT 310 1 "t" }{TEXT -1 7 ")) \267 (
 " }{XPPEDIT 306 0 "X - X[1]" "6#,&%\"XG\"\"\"&F$6#F%!\"\"" }{TEXT 
307 1 " " }{TEXT -1 8 ")   =  0" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 
4 "  < " }{XPPEDIT 18 0 "m[1]" "6#&%\"mG6#\"\"\"" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "m[2]" "6#&%\"mG6#\"\"#" }{TEXT -1 3 " , " }{XPPEDIT 18 
0 " m[3]" "6#&%\"mG6#\"\"$" }{TEXT -1 9 "  > \267  < " }{XPPEDIT 18 0 
"x-x[0]" "6#,&%\"xG\"\"\"&F$6#\"\"!!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 
18 0 "y-y[0] " "6#,&%\"yG\"\"\"&F$6#\"\"!!\"\"" }{TEXT -1 3 " , " }
{XPPEDIT 18 0 "z-z[0]" "6#,&%\"zG\"\"\"&F$6#\"\"!!\"\"" }{TEXT -1 10 "
 > = 0    " }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 6 "  or  " }{XPPEDIT 
18 0 "m[1](x-x[0])+m[2](y-y[0])+m[3](z-z[0])=0" "6#/,(-&%\"mG6#\"\"\"6
#,&%\"xGF)&F,6#\"\"!!\"\"F)-&F'6#\"\"#6#,&%\"yGF)&F76#F/F0F)-&F'6#\"\"
$6#,&%\"zGF)&F@6#F/F0F)F/" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 312 8 "Example:" }{TEXT -1 82 "  It is important to see the cons
istency between this section and the case where  " }{XPPEDIT 18 0 "z=f
(x,y)" "6#/%\"zG-%\"fG6$%\"xG%\"yG" }{TEXT -1 9 " .  Let  " }{TEXT 
313 1 "G" }{TEXT -1 17 "  be defined as  " }{XPPEDIT 18 0 "G(x,y,z)=f(
x,y)-z" "6#/-%\"GG6%%\"xG%\"yG%\"zG,&-%\"fG6$F'F(\"\"\"F)!\"\"" }
{TEXT -1 15 "  and suppose  " }{XPPEDIT 18 0 "z[0]=f(x[0],y[0])" "6#/&
%\"zG6#\"\"!-%\"fG6$&%\"xG6#F'&%\"yG6#F'" }{TEXT -1 4 " ,  " }
{XPPEDIT 18 0 "f[x](x[0],y[0])=m[1]" "6#/-&%\"fG6#%\"xG6$&F(6#\"\"!&%
\"yG6#F,&%\"mG6#\"\"\"" }{TEXT -1 4 " ,  " }{XPPEDIT 18 0 "f[y](x[0],y
[0])=m[2]" "6#/-&%\"fG6#%\"yG6$&%\"xG6#\"\"!&F(6#F-&%\"mG6#\"\"#" }
{TEXT -1 8 " , and  " }{XPPEDIT 18 0 "X[0]=(x[0],y[0],z[0])" "6#/&%\"X
G6#\"\"!6%&%\"xG6#F'&%\"yG6#F'&%\"zG6#F'" }{TEXT -1 8 " .  Then" }}}
{EXCHG {PARA 262 "" 0 "" {TEXT -1 3 "   " }{TEXT 314 6 "grad G" }
{TEXT -1 1 "(" }{XPPEDIT 18 0 "X[0])" "6#&%\"XG6#\"\"!" }{TEXT -1 6 ")
 = < " }{XPPEDIT 18 0 "G[x](X[0]" "6#-&%\"GG6#%\"xG6#&%\"XG6#\"\"!" }
{TEXT -1 3 " , " }{XPPEDIT 18 0 "G[y](X[0])" "6#-&%\"GG6#%\"yG6#&%\"XG
6#\"\"!" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "G[z](X[0]" "6#-&%\"GG6#%\"z
G6#&%\"XG6#\"\"!" }{TEXT -1 7 " > = < " }{XPPEDIT 18 0 "f[x](x[0],y[0]
)" "6#-&%\"fG6#%\"xG6$&F'6#\"\"!&%\"yG6#F+" }{TEXT -1 3 " , " }
{XPPEDIT 18 0 "f[y](x[0],y[0])" "6#-&%\"fG6#%\"yG6$&%\"xG6#\"\"!&F'6#F
," }{TEXT -1 3 " , " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 
8 " > =  < " }{XPPEDIT 18 0 "m[1]" "6#&%\"mG6#\"\"\"" }{TEXT -1 2 ", \+
" }{XPPEDIT 18 0 "m[2]" "6#&%\"mG6#\"\"#" }{TEXT -1 3 " , " }{XPPEDIT 
18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 1 ">" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 13 "The vector < " }{XPPEDIT 18 0 "m[1]" "6#&%\"mG6#\"\"\"" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "m[2]" "6#&%\"mG6#\"\"#" }{TEXT -1 3 " \+
, " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 5 "> =  " }
{XPPEDIT 315 0 "N[1]" "6#&%\"NG6#\"\"\"" }{TEXT 316 1 " " }{TEXT -1 
187 "  is exactly the normal vector that we used to establish the equa
tion for the tangent plane in the previous section and we repeat that \+
process here.  The equation for the tangent plane is" }}}{EXCHG {PARA 
263 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 319 0 "N[1] " "6#&%\"NG6#\"\"
\"" }{TEXT -1 5 " \267 ( " }{XPPEDIT 317 0 "X - X[1]" "6#,&%\"XG\"\"\"
&F$6#F%!\"\"" }{TEXT 318 1 " " }{TEXT -1 24 ")   =  0   which implies
" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "m[1](x-x
[0] )+ m[2](y-y[0])-(z-z[0]) = 0" "6#/,(-&%\"mG6#\"\"\"6#,&%\"xGF)&F,6
#\"\"!!\"\"F)-&F'6#\"\"#6#,&%\"yGF)&F76#F/F0F),&%\"zGF)&F;6#F/F0F0F/" 
}{TEXT -1 24 "   which in turn implies" }}}{EXCHG {PARA 265 "" 0 "" 
{TEXT -1 3 "   " }{XPPEDIT 18 0 "z-z[0]=m[1](x-x[0]) + m[2](y-y[0])" "
6#/,&%\"zG\"\"\"&F%6#\"\"!!\"\",&-&%\"mG6#F&6#,&%\"xGF&&F26#F)F*F&-&F.
6#\"\"#6#,&%\"yGF&&F;6#F)F*F&" }{TEXT -1 2 "  " }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 320 15 "Maple Example: " }{TEXT -1 78 " Use Maple to find a
n equation of the plane tangent to the level surface of   " }{XPPEDIT 
18 0 "G(x,y.z)=(2*x^2+4*y^2+6*z^2+x*y+y*z)/36" "6#/-%\"GG6$%\"xG-%\".G
6$%\"yG%\"zG*&,,*&\"\"#\"\"\"*$F'F0F1F1*&\"\"%F1*$F+F0F1F1*&\"\"'F1*$F
,F0F1F1*&F'F1F+F1F1*&F+F1F,F1F1F1\"#O!\"\"" }{TEXT -1 18 "   that cont
ains  " }{XPPEDIT 18 0 "X[0]=(1,2,3)" "6#/&%\"XG6#\"\"!6%\"\"\"\"\"#\"
\"$" }{TEXT -1 27 "  .  Plot the surface near " }{XPPEDIT 18 0 "X[0]" 
"6#&%\"XG6#\"\"!" }{TEXT -1 40 " ,  the tangent plane, and the gradien
t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "with(linalg): with(pl
ots): with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "x0:
=1: y0:=2: z0:=3: X0:=vector([x0,y0,z0]);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 43 "G:=(x,y,z)->(2*x^2+4*y^2+6*z^2+x*y+y*z)/36;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "K:=G(x0,y0,z0);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "gradG:=grad(G(x,y,z),[x,y,z]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "N:=subs(x=x0,y=y0,z=z0,op(gr
adG));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "X:=vector([x,y,z]
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "tplane1:=evalm(innerp
rod(N,X)=innerprod(N,X0));\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "O
bviously, tplane1 is an equation for the tangent plane at " }{XPPEDIT 
18 0 "X[0]" "6#&%\"XG6#\"\"!" }{TEXT -1 3 "  ." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "zee:=solve(tplane1,z);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 58 "Now let\222s set up the plots, including labels for \+
the axes." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "tplane:=plot3d
(zee,x=0..2,y=1..3,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 67 "graphG:=implicitplot3d(G(x,y,z)=K,x=0..2,y=0..3,z=2..4,color=c
yan):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "xaxis:=spacecurve(
[t,0,0],t=0..3,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 46 "yaxis:=spacecurve([0,t,0],t=0..3,color=black):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 46 "zaxis:=spacecurve([0,0,t],t=0..5,color=bl
ack):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Nvec:=spacecurve(e
valm(X0+t*N),t=0..1,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "xlabel:=textplot3d([3,-.3,.2,'x'],color=black):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "ylabel:=textplot3d([-.3,3,.3
,'y'],color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "disp
lay(tplane,graphG,xaxis,yaxis,zaxis,Nvec,xlabel,ylabel);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 321 14 "C3M5b Pr
oblem:" }{TEXT -1 9 "  Given: " }{XPPEDIT 18 0 "G(x,y,z)=x^2+y+z^2-3" 
"6#/-%\"GG6%%\"xG%\"yG%\"zG,**$F'\"\"#\"\"\"F(F-*$F)F,F-\"\"$!\"\"" }
{TEXT -1 9 "    and  " }{XPPEDIT 18 0 "X[0]=(1,1,1)" "6#/&%\"XG6#\"\"!
6%\"\"\"F)F)" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "
(a) Use Maple to find the gradient of " }{TEXT 322 3 " G " }{TEXT -1 
5 " at  " }{XPPEDIT 18 0 "X[0]" "6#&%\"XG6#\"\"!" }{TEXT -1 65 "  and \+
an equation for the tangent plane to the level surface of  " }{TEXT 
323 1 "G" }{TEXT -1 17 "  that contains  " }{XPPEDIT 18 0 "X[0]" "6#&%
\"XG6#\"\"!" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "(
b) Plot the level surface of " }{TEXT 324 1 "G" }{TEXT -1 24 " , the t
angent plane at " }{XPPEDIT 18 0 "X[0]" "6#&%\"XG6#\"\"!" }{TEXT -1 
47 " , and a line that represents the gradient of  " }{TEXT 325 1 "G" 
}{TEXT -1 6 "  at  " }{XPPEDIT 18 0 "X[0]" "6#&%\"XG6#\"\"!" }{TEXT 
-1 29 "  .  Include coordinate axes." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 34 "Suggestions:  For the surface, use" }{TEXT 326 15 " implicitplo
t3d" }{TEXT -1 7 " with  " }{XPPEDIT 18 0 "x=0..2.5" "6#/%\"xG;\"\"!$
\"#D!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "y=0..2.5" "6#/%\"yG;\"\"!
$\"#D!\"\"" }{TEXT -1 7 " , and " }{XPPEDIT 18 0 "z=0..2.5" "6#/%\"zG;
\"\"!$\"#D!\"\"" }{TEXT -1 31 " .  For the tangent plane, use " }
{TEXT 327 6 "plot3d" }{TEXT -1 7 " with  " }{XPPEDIT 18 0 "x=.5..1.5 \+
" "6#/%\"xG;$\"\"&!\"\"$\"#:F(" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 "y
=.5..1.5 " "6#/%\"yG;$\"\"&!\"\"$\"#:F(" }{TEXT -1 3 "  ." }}}}{MARK "
8 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 
}