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1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 66 "Tangent planes to level surfaces \+ of functions of three variables " }}{PARA 0 "" 0 "" {TEXT -1 37 "by P eter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Vers ion: 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 37 "The chain rule in te rms of gradient " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "g(x,y,z);" "6#-%\"gG 6%%\"xG%\"yG%\"zG" }{TEXT -1 87 " be a function of three variables whi ch has continuous first order partial derivatives " }{XPPEDIT 18 0 "g[ x](x,y,z) = Diff(g(x, y, z),x);" "6#/-&%\"gG6#%\"xG6%F(%\"yG%\"zG-%%Di ffG6$-F&6%F(F*F+F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "g[y](x,y,z) = Di ff(g(x,y,z),y)" "6#/-&%\"gG6#%\"yG6%%\"xGF(%\"zG-%%DiffG6$-F&6%F*F(F+F (" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g[z](x, y, z) = Diff(g(x,y,z),z );" "6#/-&%\"gG6#%\"zG6%%\"xG%\"yGF(-%%DiffG6$-F&6%F*F+F(F(" }{TEXT -1 41 " in some subset S of 3-dimensional space." }}{PARA 0 "" 0 "" {TEXT -1 98 "Suppose that we have a space-curve in S defined parametri cally by three differentiable functions " }{XPPEDIT 18 0 "PIECEWISE([ x = p(t), ``],[y = q(t), ``],[z = r(t), ``]);" "6#-%*PIECEWISEG6%7$/% \"xG-%\"pG6#%\"tG%!G7$/%\"yG-%\"qG6#F,F-7$/%\"zG-%\"rG6#F,F-" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 76 "These parametric equations can be combined to form a vector val ued function " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 322 1 "C" } {TEXT -1 2 "( " }{TEXT 266 1 "t" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "p (t);" "6#-%\"pG6#%\"tG" }{TEXT -1 1 " " }{TEXT 263 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "q(t)" "6#-%\"qG6#%\"tG" }{TEXT -1 1 " " }{TEXT 264 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "r(t)" "6#-%\"rG6#%\"tG" } {TEXT -1 1 " " }{TEXT 265 1 "k" }{XPPEDIT 18 0 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" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 47 "In terms of components i t is easily seen that " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 272 1 "C" }{TEXT -1 4 " '( " }{TEXT 357 1 "t" }{TEXT -1 10 " ) = p '( \+ " }{TEXT 301 1 "t" }{TEXT -1 3 " ) " }{TEXT 304 1 "i" }{TEXT -1 8 " + \+ q '( " }{TEXT 302 1 "t" }{TEXT -1 3 " ) " }{TEXT 305 1 "j" }{TEXT -1 8 " + r '( " }{TEXT 303 1 "t" }{TEXT -1 3 " ) " }{TEXT 306 1 "k" } {TEXT -1 10 " = [ p '( " }{TEXT 307 1 "t" }{TEXT -1 9 " ), q '( " } {TEXT 308 1 "t" }{TEXT -1 9 " ), r '( " }{TEXT 309 1 "t" }{TEXT -1 6 " ) ]. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "If we identify this position vector " }{TEXT 314 1 "C" }{TEXT -1 2 "( " }{TEXT 313 1 "t" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "p(t);" "6#-%\" pG6#%\"tG" }{TEXT -1 1 " " }{TEXT 310 1 "i" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "q(t)" "6#-%\"qG6#%\"tG" }{TEXT -1 1 " " }{TEXT 311 1 "j " }{TEXT -1 3 " + " }{XPPEDIT 18 0 "r(t)" "6#-%\"rG6#%\"tG" }{TEXT -1 1 " " }{TEXT 312 1 "k" }{TEXT -1 27 " with the coordinate triple" } {XPPEDIT 18 0 "``(p(t), q(t), r(t))" "6#-%!G6%-%\"pG6#%\"tG-%\"qG6#F)- %\"rG6#F)" }{TEXT -1 25 ", the composite function " }{XPPEDIT 18 0 "t- >g(p(t),q(t),r(t))" "6#f*6#%\"tG7\"6$%)operatorG%&arrowG6\"-%\"gG6%-% \"pG6#F%-%\"qG6#F%-%\"rG6#F%F*F*F*" }{TEXT -1 34 " can be given more s uccinctly as " }{XPPEDIT 18 0 "t -> g" "6#f*6#%\"tG7\"6$%)operatorG%& arrowG6\"%\"gGF*F*F*" }{TEXT -1 3 " ( " }{TEXT 275 1 "C" }{TEXT -1 2 " ( " }{TEXT 360 1 "t" }{TEXT -1 5 " ) )." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 10 "chain rule" } {TEXT -1 33 " in this situation has the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dt" "6#*&%\"dG\"\"\"%#dtG!\"\"" } {TEXT -1 6 " [ g( " }{TEXT 276 1 "C" }{TEXT -1 2 "( " }{TEXT 277 1 "t " }{TEXT -1 11 " ) ) ] = ( " }{XPPEDIT 18 0 "g[x]" "6#&%\"gG6#%\"xG" } {TEXT -1 1 "(" }{TEXT 273 1 "C" }{TEXT -1 2 "( " }{TEXT 274 1 "t" } {TEXT -1 10 " ) ) p '( " }{TEXT 278 1 "t" }{TEXT -1 7 " ) + ( " } {XPPEDIT 18 0 "g[y];" "6#&%\"gG6#%\"yG" }{TEXT -1 1 "(" }{TEXT 279 1 " C" }{TEXT -1 2 "( " }{TEXT 280 1 "t" }{TEXT -1 10 " ) ) q '( " }{TEXT 281 1 "t" }{TEXT -1 7 " ) + ( " }{XPPEDIT 18 0 "g[z];" "6#&%\"gG6#%\"z G" }{TEXT -1 1 "(" }{TEXT 282 1 "C" }{TEXT -1 2 "( " }{TEXT 283 1 "t" }{TEXT -1 10 " ) ) r '( " }{TEXT 284 1 "t" }{TEXT -1 4 " ). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 87 "The right han d side of this equation can be interpreted as a dot product of the vec tors" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "grad(g)" "6#- %%gradG6#%\"gG" }{TEXT -1 3 " ( " }{TEXT 299 1 "C" }{TEXT -1 2 "( " } {TEXT 300 1 "t" }{TEXT -1 8 " ) ) = " }{XPPEDIT 18 0 "g[x]" "6#&%\"gG 6#%\"xG" }{TEXT -1 1 "(" }{TEXT 285 1 "C" }{TEXT -1 2 "( " }{TEXT 286 1 "t" }{TEXT -1 3 " ) " }{TEXT 291 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "g[y];" "6#&%\"gG6#%\"yG" }{TEXT -1 1 "(" }{TEXT 287 1 "C" } {TEXT -1 2 "( " }{TEXT 288 1 "t" }{TEXT -1 3 " ) " }{TEXT 292 1 "j" } {TEXT -1 3 " + " }{XPPEDIT 18 0 "g[z];" "6#&%\"gG6#%\"zG" }{TEXT -1 1 "(" }{TEXT 289 1 "C" }{TEXT -1 2 "( " }{TEXT 290 1 "t" }{TEXT -1 3 " ) " }{TEXT 293 1 "k" }{TEXT -1 10 " and " }{TEXT 315 1 "C" }{TEXT -1 4 " '( " }{TEXT 362 1 "t" }{TEXT -1 10 " ) = p '( " }{TEXT 294 1 "t " }{TEXT -1 3 " ) " }{TEXT 297 1 "i" }{TEXT -1 8 " + q '( " }{TEXT 295 1 "t" }{TEXT -1 3 " ) " }{TEXT 298 1 "j" }{TEXT -1 8 " + r '( " } {TEXT 296 1 "t" }{TEXT -1 8 " ) ,. " }}{PARA 0 "" 0 "" {TEXT -1 24 " Hence the chain rule is " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d/dt" "6#*&%\"dG\"\"\"%#dtG!\"\"" }{TEXT -1 6 " [ g( " }{TEXT 316 1 "C" }{TEXT -1 2 "( " }{TEXT 317 1 "t" }{TEXT -1 9 " ) ) ] = " }{XPPEDIT 18 0 "grad(g)" "6#-%%gradG6#%\"gG" }{TEXT -1 3 " ( " } {TEXT 318 1 "C" }{TEXT -1 2 "( " }{TEXT 319 1 "t" }{TEXT -1 5 " ) ) " }{TEXT 340 1 "." }{TEXT -1 1 " " }{TEXT 320 1 "C" }{TEXT -1 4 " '( " } {TEXT 361 1 "t" }{TEXT -1 4 " ). " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 321 17 "_________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "The gradient vector as a normal vector \+ to level surfaces " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 9 "Suppose " }{TEXT 326 1 "C" }{TEXT -1 2 "( " }{TEXT 325 1 "t" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "p(t);" "6#-%\" pG6#%\"tG" }{TEXT -1 1 " " }{TEXT 267 1 "i" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "q(t)" "6#-%\"qG6#%\"tG" }{TEXT -1 1 " " }{TEXT 323 1 "j " }{TEXT -1 3 " + " }{XPPEDIT 18 0 "r(t)" "6#-%\"rG6#%\"tG" }{TEXT -1 1 " " }{TEXT 324 1 "k" }{XPPEDIT 18 0 "`` = [p(t), q(t), r(t)];" "6#/% !G7%-%\"pG6#%\"tG-%\"qG6#F)-%\"rG6#F)" }{TEXT -1 59 " is a space-curve which lies completely in a level surface " }{XPPEDIT 18 0 "g(x,y,z) = alpha;" "6#/-%\"gG6%%\"xG%\"yG%\"zG%&alphaG" }{TEXT -1 15 " of a func tion " }{XPPEDIT 18 0 "g(x,y,z)" "6#-%\"gG6%%\"xG%\"yG%\"zG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "Then g( " }{TEXT 327 1 "C" } {TEXT -1 2 "( " }{TEXT 328 1 "t" }{TEXT -1 8 " ) ) = " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 9 " for all " }{TEXT 339 1 "t" } {TEXT -1 12 ", so that " }{XPPEDIT 18 0 "d/dt" "6#*&%\"dG\"\"\"%#dtG !\"\"" }{TEXT -1 6 " [ g( " }{TEXT 329 1 "C" }{TEXT -1 2 "( " }{TEXT 330 1 "t" }{TEXT -1 12 " ) ) ] = 0. " }}{PARA 0 "" 0 "" {TEXT -1 7 "Si nce " }{XPPEDIT 18 0 "d/dt" "6#*&%\"dG\"\"\"%#dtG!\"\"" }{TEXT -1 6 " [ g( " }{TEXT 331 1 "C" }{TEXT -1 2 "( " }{TEXT 332 1 "t" }{TEXT -1 9 " ) ) ] = " }{XPPEDIT 18 0 "grad(g)" "6#-%%gradG6#%\"gG" }{TEXT -1 3 " ( " }{TEXT 333 1 "C" }{TEXT -1 2 "( " }{TEXT 334 1 "t" }{TEXT -1 5 " ) ) " }{TEXT 377 1 "." }{TEXT -1 1 " " }{TEXT 335 1 "C" }{TEXT -1 35 " '( t ), by the chain rule, we have" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "grad(g)" "6#-%%gradG6#%\"gG" }{TEXT -1 3 " ( " }{TEXT 336 1 "C" }{TEXT -1 2 "( " }{TEXT 337 1 "t" }{TEXT -1 5 " ) ) \+ " }{TEXT 341 1 "." }{TEXT -1 1 " " }{TEXT 338 1 "C" }{TEXT -1 4 " '( \+ " }{TEXT 356 1 "t" }{TEXT -1 7 " ) = 0 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 343 11 "___________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "for all " }{TEXT 342 1 "t" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 7 "Thus " }{XPPEDIT 18 0 "grad(g)" "6#-%%gradG6#%\"gG " }{TEXT -1 3 " ( " }{TEXT 345 1 "C" }{TEXT -1 2 "( " }{TEXT 346 1 "t " }{TEXT -1 8 " ) ) is " }{TEXT 259 32 "orthogonal to the tangent vect or" }{TEXT -1 1 " " }{TEXT 347 1 "C" }{TEXT -1 4 " '( " }{TEXT 355 1 " t" }{TEXT -1 17 " ). We say that " }{XPPEDIT 18 0 "grad(g)" "6#-%%gra dG6#%\"gG" }{TEXT -1 3 " ( " }{TEXT 348 1 "C" }{TEXT -1 2 "( " }{TEXT 349 1 "t" }{TEXT -1 8 " ) ) is " }{TEXT 259 26 "perpendicular to the c urve" }{TEXT -1 1 " " }{TEXT 350 1 "C" }{TEXT -1 2 "( " }{TEXT 351 1 " t" }{TEXT -1 4 " ). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 26 "From this we see that, if " }{XPPEDIT 18 0 "P(x[0],y[0] ,z[0])" "6#-%\"PG6%&%\"xG6#\"\"!&%\"yG6#F)&%\"zG6#F)" }{TEXT -1 33 " i s a point in the level surface " }{XPPEDIT 18 0 "g(x,y,z) = alpha;" "6 #/-%\"gG6%%\"xG%\"yG%\"zG%&alphaG" }{TEXT -1 26 ", then the gradient v ector" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "grad(g)(x[0] ,y[0],z[0]) = g[x](x[0],y[0],z[0]);" "6#/--%%gradG6#%\"gG6%&%\"xG6#\" \"!&%\"yG6#F-&%\"zG6#F--&F(6#F+6%&F+6#F-&F/6#F-&F26#F-" }{TEXT -1 1 " \+ " }{TEXT 352 1 "i" }{TEXT -1 2 " +" }{XPPEDIT 18 0 "g[y](x[0], y[0], z [0]);" "6#-&%\"gG6#%\"yG6%&%\"xG6#\"\"!&F'6#F,&%\"zG6#F," }{TEXT -1 1 " " }{TEXT 354 1 "j" }{TEXT -1 2 " +" }{XPPEDIT 18 0 "g[z](x[0], y[0], z[0]);" "6#-&%\"gG6#%\"zG6%&%\"xG6#\"\"!&%\"yG6#F,&F'6#F," }{TEXT -1 1 " " }{TEXT 353 1 "k" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " is " }{TEXT 259 60 "perpendicular to all space-curves lying in the lev el surface" }{TEXT -1 72 " and passing through this point. 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G6%&%\"xG6#\"\"!&%\"yG6#F)&%\"zG6#F)" }{TEXT -1 20 " with normal vecto r " }{TEXT 369 1 "n" }{TEXT -1 3 " = " }{TEXT 366 1 "a" }{TEXT -1 1 " \+ " }{TEXT 363 1 "i" }{TEXT -1 3 " + " }{TEXT 367 1 "b" }{TEXT -1 1 " " }{TEXT 364 1 "j" }{TEXT -1 3 " + " }{TEXT 368 1 "c" }{TEXT -1 1 " " } {TEXT 365 1 "k" }{TEXT -1 14 " has equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a*(x-x[0]) + b*(y-y[0]) + c*(z-z[0]) =0 " "6#/,(*&%\"aG\"\"\",&%\"xGF'&F)6#\"\"!!\"\"F'F'*&%\"bGF',&%\"yGF'&F1 6#F,F-F'F'*&%\"cGF',&%\"zGF'&F76#F,F-F'F'F," }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "( since, for any point" }{XPPEDIT 18 0 "``(x,y, z)" "6#-%!G6%%\"xG%\"yG%\"zG" }{TEXT -1 25 " in the plane the vector \+ " }{XPPEDIT 18 0 "``(x-x[0])" "6#-%!G6#,&%\"xG\"\"\"&F'6#\"\"!!\"\"" } {TEXT -1 1 " " }{TEXT 371 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "``(y -y[0])" "6#-%!G6#,&%\"yG\"\"\"&F'6#\"\"!!\"\"" }{TEXT -1 1 " " }{TEXT 372 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "``(z-z[0])" "6#-%!G6#,&%\" zG\"\"\"&F'6#\"\"!!\"\"" }{TEXT -1 1 " " }{TEXT 373 1 "k" }{TEXT -1 19 " is orthoganal to " }{TEXT 370 1 "n" }{TEXT -1 4 "). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Hence the tange nt plane normal to the level surface " }{XPPEDIT 18 0 "g(x,y,z)=alpha " "6#/-%\"gG6%%\"xG%\"yG%\"zG%&alphaG" }{TEXT -1 13 " at the point" } {XPPEDIT 18 0 "``(x[0],y[0],z[0])" "6#-%!G6%&%\"xG6#\"\"!&%\"yG6#F)&% \"zG6#F)" }{TEXT -1 19 " has the equation: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g[x](x[0],y [0],z[0])(x-x[0])+g[y](x[0],y[0],z[0])(y-y[0])+g[z](x[0],y[0],z[0])(z- z[0])=0" "6#/,(--&%\"gG6#%\"xG6%&F*6#\"\"!&%\"yG6#F.&%\"zG6#F.6#,&F*\" \"\"&F*6#F.!\"\"F7--&F(6#F06%&F*6#F.&F06#F.&F36#F.6#,&F0F7&F06#F.F:F7- -&F(6#F36%&F*6#F.&F06#F.&F36#F.6#,&F3F7&F36#F.F:F7F." }{TEXT -1 14 " - ------ (i). " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 374 30 "______ ________________________" }{TEXT -1 15 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "Notes" }{TEXT -1 2 " : " }}{PARA 15 "" 0 "" {TEXT -1 20 "The tangent plane at" }{XPPEDIT 18 0 "``(x[0],y[0],z[0])" "6#-%!G6%&%\"xG6#\"\"!&%\"yG6#F)&%\"zG6#F)" }{TEXT -1 24 " is defined as long as " }{XPPEDIT 18 0 "grad(g)(x[0],y [0],z[0]) <>``" "6#0--%%gradG6#%\"gG6%&%\"xG6#\"\"!&%\"yG6#F-&%\"zG6#F -%!G" }{TEXT 376 1 "0" }{TEXT -1 2 ". " }}{PARA 259 "" 0 "" {TEXT -1 16 "If the equation " }{XPPEDIT 18 0 "g(x,y,z) = alpha" "6#/-%\"gG6%% \"xG%\"yG%\"zG%&alphaG" }{TEXT -1 30 " can be solved explicitly for " }{TEXT 375 1 "z" }{TEXT -1 14 ", in the form " }{XPPEDIT 18 0 "z=h(x,y )" "6#/%\"zG-%\"hG6$%\"xG%\"yG" }{TEXT -1 38 ", the tangent plane has \+ the equation " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "z \+ = h(x[0],y[0])+h[x](x[0],y[0])*(x-x[0])+h[y](x[0],y[0])*(y-y[0])" "6#/ %\"zG,(-%\"hG6$&%\"xG6#\"\"!&%\"yG6#F,\"\"\"*&-&F'6#F*6$&F*6#F,&F.6#F, F0,&F*F0&F*6#F,!\"\"F0F0*&-&F'6#F.6$&F*6#F,&F.6#F,F0,&F.F0&F.6#F,F=F0F 0" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 " or " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h[x](x[0],y[0])*(x-x[0]) + h[y]( x[0],y[0])*(y-y[0]) - (z-z[0])=0" "6#/,(*&-&%\"hG6#%\"xG6$&F*6#\"\"!&% \"yG6#F.\"\"\",&F*F2&F*6#F.!\"\"F2F2*&-&F(6#F06$&F*6#F.&F06#F.F2,&F0F2 &F06#F.F6F2F2,&%\"zGF2&FD6#F.F6F6F." }{TEXT -1 14 " ------- (ii)," }} {PARA 0 "" 0 "" {TEXT -1 9 " where " }{XPPEDIT 18 0 "z[0]=h(x[0],y[0 ])" "6#/&%\"zG6#\"\"!-%\"hG6$&%\"xG6#F'&%\"yG6#F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 19 " Redefining g by " }{XPPEDIT 18 0 "g(x, y,z)=h(x,y)-z" "6#/-%\"gG6%%\"xG%\"yG%\"zG,&-%\"hG6$F'F(\"\"\"F)!\"\" " }{TEXT -1 22 ", so that the surface " }{XPPEDIT 18 0 "z=h(x,y)" "6#/ %\"zG-%\"hG6$%\"xG%\"yG" }{TEXT -1 27 " becomes the level surface " } {XPPEDIT 18 0 "g(x,y,z)=0" "6#/-%\"gG6%%\"xG%\"yG%\"zG\"\"!" }{TEXT -1 9 ", we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g[ x](x,y,z)=h[x](x,y)" "6#/-&%\"gG6#%\"xG6%F(%\"yG%\"zG-&%\"hG6#F(6$F(F* " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "g[y](x,y,z)=h[y](x,y)" "6#/-&%\"gG 6#%\"yG6%%\"xGF(%\"zG-&%\"hG6#F(6$F*F(" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "g[z](x,y,z)=-1" "6#/-&%\"gG6#%\"zG6%%\"xG%\"yGF(,$\"\" \"!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 51 " which show s that (ii) is a special case of (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 378 8 "Q uestion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 53 "Find the equa tion of the tangent plane to the sphere " }{XPPEDIT 18 0 "x^2+y^2+z^2 \+ = 3;" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"zGF'F(\"\"$" }{TEXT -1 14 " at the point " }{XPPEDIT 18 0 "``(1,1,1);" "6#-%!G6%\"\"\"F&F&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 379 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 46 "The sphere is a level surface of the function " }{XPPEDIT 18 0 "g(x,y ,z) = x^2+y^2+z^2;" "6#/-%\"gG6%%\"xG%\"yG%\"zG,(*$F'\"\"#\"\"\"*$F(F, F-*$F)F,F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 "The gradien t of this function is:" }}{PARA 256 "" 0 "" {TEXT -1 13 " grad(g) = \+ " }{XPPEDIT 18 0 "Diff(g(x,y,z),x);" "6#-%%DiffG6$-%\"gG6%%\"xG%\"yG% \"zGF)" }{TEXT -1 1 " " }{TEXT 386 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "diff(g(x,y,z),y);" "6#-%%diffG6$-%\"gG6%%\"xG%\"yG%\"zGF*" } {TEXT -1 1 " " }{TEXT 387 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "diff (g(x,y,z),z);" "6#-%%diffG6$-%\"gG6%%\"xG%\"yG%\"zGF+" }{TEXT -1 1 " \+ " }{TEXT 388 1 "k" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 "= \+ " }{XPPEDIT 18 0 "``(2*x);" "6#-%!G6#*&\"\"#\"\"\"%\"xGF(" }{TEXT -1 1 " " }{TEXT 380 1 "i" }{TEXT -1 2 " +" }{XPPEDIT 18 0 "``(2*y);" "6#- %!G6#*&\"\"#\"\"\"%\"yGF(" }{TEXT -1 1 " " }{TEXT 381 1 "j" }{TEXT -1 2 " +" }{XPPEDIT 18 0 "``(2*z)" "6#-%!G6#*&\"\"#\"\"\"%\"zGF(" }{TEXT -1 1 " " }{TEXT 382 1 "k" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 22 "which has the value 2 " }{TEXT 383 1 "i" }{TEXT -1 5 " + 2 " } {TEXT 384 1 "j" }{TEXT -1 5 " + 2 " }{TEXT 385 1 "k" }{TEXT -1 14 " at the point " }{XPPEDIT 18 0 "``(1, 1, 1);" "6#-%!G6%\"\"\"F&F&" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 77 "The tangent plane at t his point has this gradient vector as a normal vector. " }}{PARA 0 "" 0 "" {TEXT -1 52 "Hence the tangent plane has an equation of the form \+ " }{XPPEDIT 18 0 "2*x+2*y+2*z = d;" "6#/,(*&\"\"#\"\"\"%\"xGF'F'*&F&F' %\"yGF'F'*&F&F'%\"zGF'F'%\"dG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "Since it passes through the point" }{XPPEDIT 18 0 "``(1,1 ,1);" "6#-%!G6%\"\"\"F&F&" }{TEXT -1 11 ", we have " }{XPPEDIT 18 0 " d = 6;" "6#/%\"dG\"\"'" }{TEXT -1 6 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "The tangent plane is therefore " }{XPPEDIT 18 0 "x+y+z = 3;" "6 #/,(%\"xG\"\"\"%\"yGF&%\"zGF&\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "with(lina lg):\ng := x -> x^2+y^2+z^2;\na := 1;\nb := 1;\nc := 1;\ngrad(g(x,y,z) ,[x,y,z]);\nn := eval(%,\{x=a,y=b,z=c\});\nn[1]*(x-a)+n[2]*(y-b)+n[3]* (z-c)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)oper atorG%&arrowGF(,(*$)9$\"\"#\"\"\"F1*$)%\"yGF0F1F1*$)%\"zGF0F1F1F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,$%\"xG\"\"#,$ %\"yGF),$%\"zGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG-%'vectorG6# 7%\"\"#F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*%\"xG\"\"#\"\"'!\"\" *&F&\"\"\"%\"yGF*F**&F&F*%\"zGF*F*\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "r := sqrt(3):\np1 := plot3d([r*sin(t)*cos(s),r*sin(t)*sin(s),r*cos(t)],t=0..Pi,s=0..2*Pi): \np2 := 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{PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 400 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 54 "Find the equation of the tangent plane to the surface " }{XPPEDIT 18 0 "x^2+y^2-z^2 = 18" "6#/,(*$%\"xG\"\"#\" \"\"*$%\"yGF'F(*$%\"zGF'!\"\"\"#=" }{TEXT -1 14 " at the point " } {XPPEDIT 18 0 "``(3,5,-4);" "6#-%!G6%\"\"$\"\"&,$\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 401 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 13 "The surfa ce " }{XPPEDIT 18 0 "x^2+y^2-z^2 = 18" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"y GF'F(*$%\"zGF'!\"\"\"#=" }{TEXT -1 36 " is a level surface of the func tion " }{XPPEDIT 18 0 "g(x,y,z) = x^2+y^2-z^2;" "6#/-%\"gG6%%\"xG%\"yG %\"zG,(*$F'\"\"#\"\"\"*$F(F,F-*$F)F,!\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 33 "The gradient of this function is:" }}{PARA 256 "" 0 "" {TEXT -1 13 " grad(g) = " }{XPPEDIT 18 0 "Diff(g(x,y,z),x);" "6 #-%%DiffG6$-%\"gG6%%\"xG%\"yG%\"zGF)" }{TEXT -1 1 " " }{TEXT 407 1 "i " }{TEXT -1 3 " + " }{XPPEDIT 18 0 "diff(g(x,y,z),y);" "6#-%%diffG6$-% \"gG6%%\"xG%\"yG%\"zGF*" }{TEXT -1 1 " " }{TEXT 408 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "diff(g(x,y,z),z);" "6#-%%diffG6$-%\"gG6%%\"xG%\" yG%\"zGF+" }{TEXT -1 1 " " }{TEXT 409 1 "k" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 "= " }{XPPEDIT 18 0 "``(2*x);" "6#-%!G6#*&\"\" #\"\"\"%\"xGF(" }{TEXT -1 1 " " }{TEXT 402 1 "i" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "``(2*y);" "6#-%!G6#*&\"\"#\"\"\"%\"yGF(" }{TEXT -1 1 " \+ " }{TEXT 403 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "``" "6#%!G" } {XPPEDIT 18 0 "``(-2*z);" "6#-%!G6#,$*&\"\"#\"\"\"%\"zGF)!\"\"" } {TEXT -1 1 " " }{TEXT 410 1 "k" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 22 "which has the value 6 " }{TEXT 404 1 "i" }{TEXT -1 6 " + \+ 10 " }{TEXT 405 1 "j" }{TEXT -1 5 " + 8 " }{TEXT 406 1 "k" }{TEXT -1 14 " at the point " }{XPPEDIT 18 0 "``(3, 5, -4);" "6#-%!G6%\"\"$\"\"& ,$\"\"%!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 33 "The tangent plane to the surface " } {XPPEDIT 18 0 "x^2+y^2-z^2 = 18" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$ %\"zGF'!\"\"\"#=" }{TEXT -1 14 " at the point " }{XPPEDIT 18 0 "``(3,5 ,-4);" "6#-%!G6%\"\"$\"\"&,$\"\"%!\"\"" }{TEXT -1 14 " has equation:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "6*(x-3)+10*(y-5)+8* (z+4)=0" "6#/,(*&\"\"'\"\"\",&%\"xGF'\"\"$!\"\"F'F'*&\"#5F',&%\"yGF'\" \"&F+F'F'*&\"\")F',&%\"zGF'\"\"%F'F'F'\"\"!" }{TEXT -1 6 " or " } {XPPEDIT 18 0 "3*x+5*y+4*z=18" "6#/,(*&\"\"$\"\"\"%\"xGF'F'*&\"\"&F'% \"yGF'F'*&\"\"%F'%\"zGF'F'\"#=" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "with(linalg ):\ng := x -> x^2+y^2-z^2;\na := 3;\nb := 5;\nc := -4;\ngrad(g(x,y,z), [x,y,z]);\nn := eval(%,\{x=a,y=b,z=c\});\nn[1]*(x-a)+n[2]*(y-b)+n[3]*( z-c)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)opera torG%&arrowGF(,(*$)9$\"\"#\"\"\"F1*$)%\"yGF0F1F1*$)%\"zGF0F1!\"\"F(F(F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG! \"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,$%\"xG\"\"#,$%\" yGF),$%\"zG!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG-%'vectorG6#7 %\"\"'\"#5\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*%\"xG\"\"'\"#O! \"\"*&\"#5\"\"\"%\"yGF+F+*&\"\")F+%\"zGF+F+\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "p1 := plots [tubeplot]([0,0,z],z=-11..9,\n color=red,radius=sqrt(z^2+18),tubepoi nts=40,numpoints=30):\np2 := plot3d((18-3*x-5*y)/4,x=-5..7,y=0..9,colo r=blue,style=wireframe):\nplots[display]([p1,p2],gridstyle=triangular, color=red,\n lightmodel=light4,axes=boxed,labels=[`x`,`y`,`z`],\n \+ view=[-12..12,-12..12,-12..9]);" }}{PARA 13 "" 1 "" {GLPLOT3D 359 323 323 {PLOTDATA 3 "6*-%%MESHG6$7@7J7%$\"3_f^D7E)*y6!#;$\"\"!F-$! 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000000C013E00000000000C015C00000000000C017A00000000000C019800000000000 C01B600000000000C01D400000000000C01F200000000000C020800000000000C02170 0000000000C022600000000000C023500000000000C024400000000000C02530000000 0000C026200000000000C027100000000000C028000000000000-Fg_al6&Fi_alF,F,F j_al-%&STYLEG6#%%LINEG-%+AXESLABELSG6&%\"xG%\"yG%\"zG-%%FONTG6#%(DEFAU LTG-%'COLOURGFh_al-%*AXESSTYLEG6#%$BOXG-%*GRIDSTYLEG6#%+TRIANGULARG-%+ LIGHTMODELG6#%(LIGHT_4G-%%VIEWG6%;$!#7F-$\"#7F-F[cal;F\\calF]d`l" 1 2 0 1 10 0 1 6 6 2 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 389 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 74 "Find the equation of the tangent plane and the normal line to the sur face " }{XPPEDIT 18 0 "x^2+x*y^2+y^3+z+1 = 0;" "6#/,,*$%\"xG\"\"#\"\" \"*&F&F(*$%\"yGF'F(F(*$F+\"\"$F(%\"zGF(F(F(\"\"!" }{TEXT -1 14 " at th e point " }{XPPEDIT 18 0 "``(2,-3,4)" "6#-%!G6%\"\"#,$\"\"$!\"\"\"\"% " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 390 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 12 "The surface " }{XPPEDIT 18 0 "x^2+x*y^2+y^3+z+1 = 0" "6#/,,*$%\"xG\" \"#\"\"\"*&F&F(*$%\"yGF'F(F(*$F+\"\"$F(%\"zGF(F(F(\"\"!" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "x^2+x*y^2+y^3+z= -1" "6#/,**$%\"xG\"\"#\"\"\"* &F&F(*$%\"yGF'F(F(*$F+\"\"$F(%\"zGF(,$F(!\"\"" }{TEXT -1 36 " is a lev el surface of the function " }{XPPEDIT 18 0 "g(x,y,z) = x^2+x*y^2+y^3+ z;" "6#/-%\"gG6%%\"xG%\"yG%\"zG,**$F'\"\"#\"\"\"*&F'F-*$F(F,F-F-*$F(\" \"$F-F)F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 "The gradient of this function is:" }}{PARA 256 "" 0 "" {TEXT -1 13 " grad(g) = \+ " }{XPPEDIT 18 0 "Diff(g(x,y,z),x);" "6#-%%DiffG6$-%\"gG6%%\"xG%\"yG% \"zGF)" }{TEXT -1 1 " " }{TEXT 397 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "diff(g(x,y,z),y);" "6#-%%diffG6$-%\"gG6%%\"xG%\"yG%\"zGF*" } {TEXT -1 1 " " }{TEXT 398 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "diff (g(x,y,z),z);" "6#-%%diffG6$-%\"gG6%%\"xG%\"yG%\"zGF+" }{TEXT -1 1 " \+ " }{TEXT 399 1 "k" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 "= \+ " }{XPPEDIT 18 0 "``(2*x+y^2);" "6#-%!G6#,&*&\"\"#\"\"\"%\"xGF)F)*$%\" yGF(F)" }{TEXT -1 1 " " }{TEXT 391 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "``(2*x*y+3*y^2)" "6#-%!G6#,&*(\"\"#\"\"\"%\"xGF)%\"yGF)F)*&\"\"$ F)*$F+F(F)F)" }{TEXT -1 1 " " }{TEXT 392 1 "j" }{TEXT -1 3 " + " } {TEXT 393 1 "k" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 23 "which h as the value 13 " }{TEXT 394 1 "i" }{TEXT -1 6 " + 15 " }{TEXT 395 1 " j" }{TEXT -1 3 " + " }{TEXT 396 1 "k" }{TEXT -1 14 " at the point " } {XPPEDIT 18 0 "``(2,-3,4)" "6#-%!G6%\"\"#,$\"\"$!\"\"\"\"%" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "The tangent plane at this point has an equation " }{XPPEDIT 18 0 "13*(x-2)+15*(y+3)+``(z-4) = 0;" "6# /,(*&\"#8\"\"\",&%\"xGF'\"\"#!\"\"F'F'*&\"#:F',&%\"yGF'\"\"$F'F'F'-%!G 6#,&%\"zGF'\"\"%F+F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "The tangent plane is therefor e " }{XPPEDIT 18 0 "13*x+15*y+z+15 = 0;" "6#/,**&\"#8\"\"\"%\"xGF'F'*& \"#:F'%\"yGF'F'%\"zGF'F*F'\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The normal line has param etric equations: " }{XPPEDIT 18 0 "PIECEWISE([x=2+13*t,``],[y=-3+15 *t,``],[z=4+t,``])" "6#-%*PIECEWISEG6%7$/%\"xG,&\"\"#\"\"\"*&\"#8F+%\" tGF+F+%!G7$/%\"yG,&\"\"$!\"\"*&\"#:F+F.F+F+F/7$/%\"zG,&\"\"%F+F.F+F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 54 "Find the equation of the tangent plane to the surface " }{XPPEDIT 18 0 "3* x^2+4*y^2+5*z^2=73" "6#/,(*&\"\"$\"\"\"*$%\"xG\"\"#F'F'*&\"\"%F'*$%\"y GF*F'F'*&\"\"&F'*$%\"zGF*F'F'\"#t" }{TEXT -1 13 " at the point" } {XPPEDIT 18 0 "``(2,2,3);" "6#-%!G6%\"\"#F&\"\"$" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 32 "________________________________" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 \+ " }}{PARA 0 "" 0 "" {TEXT -1 54 "Find the equation of the tangent plan e to the surface " }{XPPEDIT 18 0 "z^2-x^2-y^2=0" "6#/,(*$%\"zG\"\"#\" \"\"*$%\"xGF'!\"\"*$%\"yGF'F+\"\"!" }{TEXT -1 13 " at the point" } {XPPEDIT 18 0 "``(3, 4, -5);" "6#-%!G6%\"\"$\"\"%,$\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "_____________________________ ___" }}{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 32 "________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 54 "Find the equation of t he tangent plane to the surface " }{XPPEDIT 18 0 "x*y+y*z+z*x=1" "6#/, (*&%\"xG\"\"\"%\"yGF'F'*&F(F'%\"zGF'F'*&F*F'F&F'F'F'" }{TEXT -1 13 " a t the point" }{XPPEDIT 18 0 "``(1,1,0);" "6#-%!G6%\"\"\"F&\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "_______________________ _________" }}{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 32 "________________________________" }}{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 t he tangent plane to the surface " }{XPPEDIT 18 0 "x^2*y^2+x*z-2*y^3=10 " "6#/,(*&%\"xG\"\"#%\"yGF'\"\"\"*&F&F)%\"zGF)F)*&F'F)*$F(\"\"$F)!\"\" \"#5" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(2,1,4);" "6#-%!G 6%\"\"#\"\"\"\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "__ ______________________________" }}{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 32 "____________________ ____________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 54 "Find the equation of the tangent plane to the surface " }{XPPEDIT 18 0 "3* x^2-2*y+z^3=9" "6#/,(*&\"\"$\"\"\"*$%\"xG\"\"#F'F'*&F*F'%\"yGF'!\"\"*$ %\"zGF&F'\"\"*" }{TEXT -1 14 " at the point " }{XPPEDIT 18 0 "``(1,1,2 )" "6#-%!G6%\"\"\"F&\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{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 32 "____________ ____________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 54 "Find the equation of the tangent plane to the surface " }{XPPEDIT 18 0 "x^(1/3)+y^(1/3)+z^(1/3)=1" "6#/,()%\"xG*&\"\"\"F(\"\"$!\"\"F()% \"yG*&F(F(F)F*F()%\"zG*&F(F(F)F*F(F(" }{TEXT -1 13 " at the point" } {XPPEDIT 18 0 "``(1, -1, 1);" "6#-%!G6%\"\"\",$F&!\"\"F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{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 32 "________________________________" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 27 "derivative of space-curve \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 614 "p1 := plot(-1/4*x^2+x+3,x=1..5,color=red,thickness=2 ):\np2 := plottools[arrow]([0,0],[2,4],.1,.1,.03,arrow,color=COLOR(RGB ,.4,0,1)):\np3 := plottools[arrow]([0,0],[4,3],.1,.1,.03,arrow,color=C OLOR(RGB,.4,0,1)):\np4 := plottools[arrow]([2,4],[4,3],.1,.1,.1,arrow, \n color=COLOR(RGB,.4,.8,0),thickness=2):\np5 := plottools[a rrow]([2,4],[3.3,4],.1,.1,.1,arrow,\n color=magenta,thicknes s=2):\nt1 := plots[textplot]([[-.1,.15,`O`],[2,4.3,`C(t)`],[4.4,3.2,`C (t+h)`]]):\nt2 := plots[textplot]([3.55,4.2,`C'(t)`],color=COLOR(RGB,. 7,0,.7)):\nplots[display]([p1,p2,p3,p4,p5,t1,t2],axes=none,view=[-.1.. 5,0..4.5]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 25 "normal vector to surface " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 904 "q1 := plot(interp([-3,-1,1],[0,3/4,1],x),x =-3..1):\nq2 := plot(interp([1,2,3],[1,1/4,-1],x),x=1..3):\np1 := plot (interp([-1,1,3],[-2,-5/4,-1],x),x=-1..3,color=black):\np2 := plot(int erp([-3,-2,-1],[0,-3/4,-2],x),x=-3..-1,color=black):\nrvrs := (_u,_v)- >evalb(_u[1]>_v[1]):\npts := [op(op(1,op(1,q2))),op(sort(op(1,op(1,p1) ),rvrs)),\n op(sort(op(1,op(1,p2)),rvrs)),op(op(1,op(1,q1)))]:\np 3 := plots[polygonplot](pts,color=COLOR(RGB,.94,.94,.94),style=patchno grid):\np4 := plot(interp([-2,0,2],[-3/4,0,1/4],x),x=-1.5..1.5,color=r ed):\np5 := plot(interp([-1,0,1],[3/4,0,-5/4],x),x=-.8..0.8,color=red) :\np6 := plottools[arrow]([0,0],[0.3,2],.1,.1,.05,arrow,\n thicknes s=1,color=COLOR(RGB,.5,0,.9)):\nt1 := plots[textplot]([[-.6,.15,`(x , y ,z )`],[-.54,.1,`o o o`],\n [1.2,2,`grad(g)(x ,y ,z )`],[1 .55,1.95,`o o o`]],color=black):\nplots[display]([p1,p2,p3,p4,p5,p 6,t1],axes=none,view=[-3..3,-2..2]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{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 "" } }}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }