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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 31 "Thinking Through Some Qua
drics " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 " \+
\+
Version .8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "w
ith(plots):" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "Basic Paraboloid \+
and Saddle" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "First look at the ba
sic paraboloid vs. saddle dichotomy.:" }}{PARA 0 "" 0 "" {TEXT -1 99 "
Try rotating each (dragging with the left button on your mouse) to see
how different each can look." }}{PARA 0 "" 0 "" {TEXT -1 33 "Note how
much the second changes!" }}{PARA 0 "" 0 "" {TEXT -1 49 "Can you see \+
both algebraically and geometrically:" }}{PARA 0 "" 0 "" {TEXT -1 67 "
1. How the xz (i.e. y=0) and yz (i.e. x=0) sections compare" }
}{PARA 0 "" 0 "" {TEXT -1 46 " to each
other" }}{PARA 0 "" 0 "" {TEXT -1 53 " \+
and in the two cases?" }}{PARA 0 "" 0 "" {TEXT -1 105 " 2. How
the shape of the y=kx sections (for k a constant) vary in each exampl
e as a function of k?" }}{PARA 0 "" 0 "" {TEXT -1 89 " 3. Why t
he paraboloid graph has the 4 corners instead of being perfectly round
ed?" }}{PARA 0 "" 0 "" {TEXT -1 102 " 4. What happens to these \+
graphs as you change the coefficients of x and y but not their signs?
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "plot3d(2*x^2+y^2,x=-1..1,y=-1..
1,style=patchcontour,axes=normal,title=`paraboloid`);" }}{PARA 0 "> "
0 "" {MPLTEXT 1 0 78 "plot3d(x^2-y^2,x=-1..1,y=-1..1,style=patchcontou
r,axes=normal,title=`saddle`);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1
29 "Seeing Roundness More Clearly" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1
89 "The view=-1..1 option just shows the portion of the graph where z \+
is in this range and so" }}{PARA 0 "" 0 "" {TEXT -1 66 "restores the e
xpected circular symmetry to the paraboloid picture." }}{PARA 0 "> "
0 "" {MPLTEXT 1 0 110 "plot3d(x^2+y^2,x=-1..1,y=-1..1,view=-1..1,style
=patchcontour,axes=normal,title=`paraboloid with view option`);" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 37 "Some Trickier Paraboloids and Sad
dles" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Algebraically each of thes
e is of the form " }{XPPEDIT 19 1 "z = a*x^2+b*x*y+c*y^2;" "6#/%\"zG,(
*&%\"aG\"\"\"*$%\"xG\"\"#F(F(*(%\"bGF(F*F(%\"yGF(F(*&%\"cGF(*$F.F+F(F(
" }{TEXT -1 27 " but geometrically they are" }}{PARA 0 "" 0 "" {TEXT
-1 16 "quite different." }}{PARA 0 "" 0 "" {TEXT -1 58 " 1. Eac
h is a paraboloid or saddle. Which is which?" }}{PARA 0 "" 0 "" {TEXT
-1 78 " 2. Think about x=0 and y=0 sections. Can you tell them \+
apart this way?" }}{PARA 0 "" 0 "" {TEXT -1 68 " 3. Think about
y=x vertical sections. Does this distinguish?" }}{PARA 0 "" 0 ""
{TEXT -1 69 " 4. What do horizontal z=constant sections look li
ke for each?" }}{PARA 0 "" 0 "" {TEXT -1 117 " 5. Try thinking \+
systematically about y=kx sections as a function of k to see if you ca
n explain the pictures." }}{PARA 0 "" 0 "" {TEXT -1 105 " 6. Co
mpleting the square can be used to systematically tell paraboloids for
m saddles. For example" }}{PARA 0 "" 0 "" {TEXT -1 50 " \+
" }{XPPEDIT 19 1 "x^2+x*y+y^2 = (x+
y/2)^2+3*y^2/4;" "6#/,(*$%\"xG\"\"#\"\"\"*&F&F(%\"yGF(F(*$F*F'F(,&*$,&
F&F(*&F*F(F'!\"\"F(F'F(*(\"\"$F(*$F*F'F(\"\"%F0F(" }{TEXT -1 15 " >
= 0 always" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "plot3d(x^2- x*y+ y^2,
x=-1..1,y=-1..1,style=patchcontour,axes=normal,title=`figure 4`);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "plot3d(x^2+ 3*x*y+y^2,x=-1..1,y=-1.
.1,style=patchcontour,axes=normal,title=`figure 5`);" }}{PARA 0 "> "
0 "" {MPLTEXT 1 0 86 "plot3d(x^2+6*x*y+y^2,x=-1..1,y=-1..1,style=patch
contour,axes=normal,title=`figure 6`);" }}}}{SECT 0 {PARA 3 "" 0 ""
{TEXT -1 17 "A Borderline Case" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "
Each of these are parabolic cylinders. " }}{PARA 0 "" 0 "" {TEXT -1
64 " 1. Can you see why their shapes are so similar? "
}}{PARA 0 "" 0 "" {TEXT -1 50 " 2. How do the two examp
les differ?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plot3d(x^2,x=-1..1,y
=-1..1,style=patchcontour,axes=normal,title=`figure 7`);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plot3d(x^2- 2*x*y+ y^2,x=-1..1,y=-1
..1,style=patchcontour,axes=normal,title=`figure 8`);" }}}}{SECT 0
{PARA 3 "" 0 "" {TEXT -1 12 "Hyperboloids" }}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 65 "Can you see the relationship of the hyperboloid to the hy
perbola?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "implicitplot3d(x^2+y^2-
z^2=1,x=-2..2,y=-2..2,z=-2..2,axes=normal,title=`Hyperboloid 1`);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "implicitplot(x^2-z^2=1,x=-2..2,z=-2
..2,title=`hyperbola`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "How ab
out these two?" }}{PARA 0 "" 0 "" {TEXT -1 54 "They are also closely r
elated to the hyperbola above. " }}{PARA 0 "" 0 "" {TEXT -1 15 "Do you
see how?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "implicitplot3d(x^2+y^2
-z^2=-1,x=-2..2,y=-2..2,z=-2..2,axes=normal,title=`Hyperboloid 2`);" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "implicitplot3d(x^2+y^2-z^2=0,x=-2
..2,y=-2..2,z=-2..2,axes=normal,title=`A Cone (missing a little bit)`)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 64 "The cone can \+
be filled in more completely using the grid option" }}{PARA 0 "> " 0
"" {MPLTEXT 1 0 113 "implicitplot3d(x^2+y^2-z^2=0,x=-2..2,y=-2..2,z=-2
..2,grid=[30,30,30],axes=normal,title=`A Cone More Completely`);" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 19 "More to Think About" }}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 34 "Some other things to try include :" }}
{PARA 0 "" 0 "" {TEXT -1 74 " 1. What happens when you start ad
ding terms linear in x, y, and z?" }}{PARA 0 "" 0 "" {TEXT -1 58 " \+
2. What will a general quadric usually look like? " }}{PARA 0 ""
0 "" {TEXT -1 45 " 3. What possibilities have we missed?" }}
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