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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 0 "" }{TEXT 
258 13 "Paraboloids  " }{TEXT 265 44 "Available at www-math.bgsu.edu/~
zirbel/calc3" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "with(plots): " }{TEXT -1 25 "Needed for contour plots." }}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 19 "Elliptic Paraboloid" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 26 "f:=(x,y)->(x^2)/3 + y^2;  " }{TEXT -1 24 "Th
is defines a function " }{TEXT 259 1 "f" }{TEXT -1 19 " of two variabl
es, " }{TEXT 260 1 "x" }{TEXT -1 5 " and " }{TEXT 261 1 "y" }{TEXT -1 
45 ".  It is said to be parabolic because of the " }{XPPEDIT 18 0 "x^2
;" "6#*$)%\"xG\"\"#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y^2;" "
6#*$)%\"yG\"\"#\"\"\"" }{TEXT -1 53 " terms.  Like a parabola, the who
le surface opens up." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "plo
t3d(f,-10..10,-10..10,axes=BOXED,labels=[\"x axis\", \"y axis\", \"z a
xis\"]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "contourplot(f,-
10..10,-10..10,labels=[\"x axis\", \"y axis\"]);" }{TEXT -1 91 "  The \+
contours are ellipses in the xy plane; that's why it's called an ellip
tic paraboloid." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 21 "Hyperbolic P
araboloid" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f:=(x,y)->-(x^2
)/3 + y^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "plot3d(f,-10.
.10,-10..10,axes=BOXED,labels=[\"x axis\", \"y axis\", \"z axis\"]);" 
}{TEXT -1 78 " This surface curves both up and down, like a saddle or \+
Pringles potato chip. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "c
ontourplot(f,-10..10,-10..10,labels=[\"x axis\", \"y axis\"]);" }
{TEXT -1 28 "  This paraboloid is called " }{TEXT 262 10 "hyperbolic" 
}{TEXT -1 37 " because the contours are hyperbolae." }}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 29 "Rotated Hyperbolic Paraboloid" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f:=(x,y)->x^2 + 4*x*y + y^2; " }
{TEXT -1 23 "This paraboloid has an " }{TEXT 263 2 "xy" }{TEXT -1 101 
" term.  It isn't clear right away whether it is elliptic or hyperboli
c, but the graph makes it clear." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "plot3d(f,-10..10,-10..10,axes=BOXED,style=PATCHCONTOU
R,labels=[\"x axis\", \"y axis\", \"z axis\"]);" }{TEXT -1 115 " The P
ATCHCONTOUR style draws a surface plot with contour lines on it.  This
 can be helpful, but isn't always good." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 72 "contourplot(f,-10..10,-10..10,labels=[\"x axis\", \"y
 axis\"], contours=20);" }{TEXT -1 78 " The hyperbolae do not line up \+
with the x and y axes; they are rotated by the " }{TEXT 264 2 "xy" }
{TEXT -1 6 " term." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Parabolic
 cylinder" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f:=(x,y) -> (2*
x-3*y)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "plot3d(f(x,y)
,x=-10..10,y=-10..10,axes=boxed,style=patchcontour,labels=[\"x axis\",
 \"y axis\", \"z axis\"]);" }{TEXT -1 251 " It's hard to see unless yo
u rotate this graph, but this surface only opens up, never down, but t
here is one direction in which the height of the surface stays constan
t.  This paraboloid is balanced perfectly between the elliptic and hyp
erbolic cases." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "contourpl
ot(f,-10..10,-10..10,labels=[\"x axis\", \"y axis\"], contours=20); " 
}{TEXT -1 74 "The contours are neither ellipses nor hyperbolae, but ju
st straight lines." }}}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }