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{TEXT -1 0 "" }{TEXT 256 76 "FUNCTIONS OF TWO VARIABLES AND THEIR GRAP
HIC REPRESENTATION IN THE 3-D SPACE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 23 "Planes in the 3-D space" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f1:=(x,y)-> 3*x+2*y;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "plot3d(f1(x,y),x=-3..3,
y=-3..3, shading=ZHUE, axes=framed, orientation=[42,79]);" }}{PARA 13 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
261 "" 0 "" {TEXT 260 47 "Functions formed using the exponential funct
ion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "f2:=(x,y)->exp(-x^2)+exp(-4*y^2);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 108 "plot3d(f2(x,y),x=-3..3,y=-3..3,grid=[49,49
],axes=framed,shading=ZHUE,orientation=[-145,45],labels=[x,y,z]); " }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 96 
"We can include in the analysis of this multivariable function the stu
dy of their \"level curves\"." }}{PARA 264 "" 0 "" {TEXT -1 0 "" }}
{PARA 264 "" 0 "" {TEXT 258 12 "Level Curves" }}{PARA 266 "" 0 "" 
{TEXT -1 170 "The level curves of a function of two variables are the \+
curves where the function takes a constant proected into the xy-plane.
 The level curves have equations: f(x,y) = c" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 285 125 "The following command allows to draw the level
 curves in the xy-plane in colors. Each color represent a different va
lue of c." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 85 "contourplot3d(f2(x,y),x=-3..3,y=-3..3,shading=ZHUE,orientation
=[-145,0],axes=framed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 97 "In or
der to graph individual level curves, use the implicitplot function an
d specify the c value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "A:=implicitplot3d(f2(x,y)=1,x=-3..3
,y=-3..3,z=0..1,axes=framed,color=blue,orientation=[45,0]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "B:=implicitplot3d(f2(x,y)=.2
,x=-3..3,y=-3..3,z=0..1,axes=framed,color=red):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 77 "C:=implicitplot3d(f2(x,y)=.4,x=-3..3,y=-3..3,z
=0..1,axes=framed,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 78 "E:=implicitplot3d(f2(x,y)=.5,x=-3..3,y=-3..3,z=0..1,axes=frame
d,color=violet):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "display
3d([A,B,C,E],axes=framed,orientation=[-145,0],shading=zhue);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT 259 47 
"Function formed using trigonometric functions. " }}{PARA 264 "" 0 "" 
{TEXT -1 88 "Our particular example represents the amplitude of a vibr
ating string in space and time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f3 := (x,t)->cos(2*Pi*t)*sin
(2*Pi*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "plot3d(f3(x,t
),x=0..1,t=0..2,grid=[49,49],axes=framed,shading=ZHUE,orientation=[-14
5,45],labels=[x,t,z]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "
contourplot3d(f3(x,t),x=0..1,t=0..2,shading=zhue, orientation=[-145,0]
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 263 22 "Saddled Sh
aped surface" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "f4:=(x,y)->x*y;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 91 "plot3d(f4(x,y),x=-5..5,y=-5..5, view=-10..10,axes=box
ed,shading=zhue,orientation=[-18,65]);" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "contourplot3d(f4(x,y),x
=-5..5,y=-5..5,axes=boxed,shading=zhue,orientation=[-18,0]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 8 "Rational" }{TEXT 
266 1 " " }{TEXT 267 8 "Function" }{TEXT 268 1 " " }{TEXT 265 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 34 "g4:=unapply(-3*y/(x^2+y^2+1),x,y);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 102 "plot3d(g4(x,y),x=-5..5,y=-5..5,axes=framed,shading=Z
HUE, axes=boxed,orientation=[33,73],grid=[30,30]);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 82 "contourplot3d(g4(x,y),x=-5..5,y=-5..5,axes=
boxed,shading=zhue,orientation=[33,0]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 271 1 " " }{TEXT 273 14 "Other Examples" }{TEXT 
272 1 " " }{TEXT 270 46 "(Problem 3 and 7 Section 11.1 Garner's book  \+
)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "g5:=unapply(x^2*cos(y),x,y);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 106 "plot3d(g5(x,y),x=-10..10,y=-10..10,axes=framed,
shading=ZHUE, axes=boxed,orientation=[33,73],grid=[50,50]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "contourplot3d(g5(x,y),x=-10.
.10,y=-10..10,axes=boxed,shading=zhue,orientation=[33,0],grid=[70,70])
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "g6:=unapply(exp(-x)*co
s(y),x,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "plot3d(g6(x,
y),x=-3..3,y=-15..15,axes=framed,shading=ZHUE, axes=boxed,orientation=
[33,73],grid=[50,50]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "c
ontourplot3d(g6(x,y),x=-3..3,y=-15..15,axes=boxed,shading=zhue,orienta
tion=[33,0],grid=[70,70]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "g7:=unapply(exp(-x^2)*cos(y),x,y);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 106 "plot3d(g7(x,y),x=-10..10,y=-10..10,axes=framed,shadi
ng=ZHUE, axes=boxed,orientation=[33,73],grid=[50,50]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "contourplot3d(g7(x,y),x=-10..10,y=-
10..10,axes=boxed,shading=zhue,orientation=[33,0],grid=[70,70]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "g8:=unapply(cos(sqrt(x^2+y^2
)),x,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "plot3d(g8(x,y)
,x=-10..10,y=-10..10,axes=framed,shading=ZHUE, axes=boxed,orientation=
[33,17],grid=[70,70]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "c
ontourplot3d(g8(x,y),x=-10..10,y=-10..10,axes=boxed,shading=zhue,orien
tation=[33,0],grid=[70,70]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 32 "g9:=unapply(log(x^2+y^2+1),x,y);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 106 "plot3d(g9(x,y),x=-10..10,y=-10..10,axes=framed,shadi
ng=ZHUE, axes=boxed,orientation=[33,73],grid=[50,50]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "contourplot3d(g9(x,y),x=-10..10,y=-
10..10,axes=boxed,shading=zhue,orientation=[33,0],grid=[70,70]);" }}}
{EXCHG {PARA 263 "" 0 "" {TEXT -1 1 " " }{TEXT 262 51 "Exercises: (Gra
ph functions and their level curves)" }}{PARA 264 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 45 "1.  f(x,y) = x^2 - y^2;  c \+
= -2, -1, 0, 1, 2." }}{PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 
0 "" {TEXT -1 121 "2.  Find an equation for the level curve of f that \+
contains the given point P and graph the function and the level curve:
" }}{PARA 265 "" 0 "" {TEXT -1 32 "f(x,y) = 1 - 4x^2 - y^2;  P(1,0)" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "49" 0 }{VIEWOPTS 1 
1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }