{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 } {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 "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }} {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?" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "8" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }