{VERSION 6 0 "IBM INTEL LINUX" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 78 "15.6 Directional Derivatives and \+ the Gradient; Tangent Planes and Normal Lines" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "restart; interface(warnlevel=0): with(plots): s etoptions3d(scaling=constrained, axes=boxed, style=patchnogrid):" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 69 "Find the equation of the tangent plane and normal line to the s phere " }{XPPEDIT 18 0 "x^2+y^2+z^2 = 6;" "6#/,(*$%\"xG\"\"#\"\"\"*$% \"yGF'F(*$%\"zGF'F(\"\"'" }{TEXT -1 61 " at the point (1,1,2). Combine all three objects in one plot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 9 "Solution." }}{PARA 0 "" 0 "" {TEXT -1 9 "Gradient:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 " grad := proc(f) end:" }}}{PARA 0 "" 0 "" {TEXT -1 15 "Tangent Plane:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "g := grad(x^2+y^2+z^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "n := eval(g,\{x=1,y=1,z=2\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "n. = n.<1,1,2>;" }}}{PARA 0 "" 0 "" {TEXT -1 12 "Normal Line:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "r := <1, 1,2> + t*n;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "graph := plo t3d(sqrt(6-x^2-y^2),x=0..1.5,y=0..1.5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "tplane := plot3d((6-x-y)/2,x=0.5..1.5,y=0.5..1.5,colo r=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "nline := spacec urve([1+2*t,1+2*t,2+4*t],t=-0.25..0.25,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "display(graph, tplane, nline, axes=normal , orientation=[-20,70]);" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Exam ple" }}{PARA 0 "" 0 "" {TEXT -1 56 "Find the equation of the plane tan gent to the graph of " }{XPPEDIT 18 0 "f(x,y) = (x^2-y^3)/6;" "6#/-% \"fG6$%\"xG%\"yG*&,&*$F'\"\"#\"\"\"*$F(\"\"$!\"\"F-\"\"'F0" }{TEXT -1 25 " at the point (3,2,1/6)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 9 "Solution." }{TEXT -1 5 " L et " }{XPPEDIT 18 0 "g(x,y,z) = z-(x^2-y^3)/6;" "6#/-%\"gG6%%\"xG%\"yG %\"zG,&F)\"\"\"*&,&*$F'\"\"#F+*$F(\"\"$!\"\"F+\"\"'F2F2" }{TEXT -1 16 " , so the graph " }{XPPEDIT 18 0 "z = f(x,y);" "6#/%\"zG-%\"fG6$%\"xG %\"yG" }{TEXT -1 22 " is the level surface " }{XPPEDIT 18 0 "g(x,y,z) \+ = 0;" "6#/-%\"gG6%%\"xG%\"yG%\"zG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "Tangent Plane:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g := grad(z-(x^2-y^3)/6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "n := eval(g,\{x=3,y=2,z=1/6\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "n. = n.<3,2,1/6>;" }}}{PARA 0 "" 0 "" {TEXT -1 12 "Normal Line:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r := <3,2,1/6> + t*n;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "g raph := plot3d((x^2-y^3)/6,x=0..6,y=0..3):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 53 "tplane := plot3d(7/6+x-2*y,x=1..5,y=1..3,color=cyan ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "nline := spacecurve([ 3-t,2+2*t,1/6+t],t=-1..1,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "display(graph, tplane, nline, orientation=[32,48]);" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 57 "Find the tangent and normal line to the curve defined by \+ " }{XPPEDIT 18 0 "sin(x*y)+x+y^2 = Pi+1;" "6#/,(-%$sinG6#*&%\"xG\"\"\" %\"yGF*F*F)F**$F+\"\"#F*,&%#PiGF*F*F*" }{TEXT -1 15 " at the point (" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 4 ",1)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 9 "Solution. " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Tangent Line:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "grad := proc(f) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "g := grad (sin(x*y)+x+y^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "n := e val(g,[x=Pi,y=1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "n. = n.;" }}}{PARA 0 "" 0 "" {TEXT -1 3 "or " }{TEXT 259 1 "y" } {TEXT -1 5 " = 1." }}{PARA 0 "" 0 "" {TEXT -1 12 "Normal Line:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "r := + t*n;" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 17 "This is the line " }{TEXT 260 1 "x" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 1 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "curve := contourplot(sin (x*y)+x+y^2,x=0..2*Pi,y=-2..2,contours=[Pi+1], numpoints=1000):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "tline := plot([x,1,x=Pi-1..P i+1],color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "nline := plot([Pi,y,y=0..2],color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display(curve, tline, nline);" }}}}}{MARK "2" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }