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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 15 "Maple Project 8" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 379 "
Since you are learning about conics, we might as well introduce the \"
implicitplot\" command.  When we plotted things on Maple, we usually h
ad them in the form y = f(x).  But what if you want to plot a curve, b
ut you can't write it in the form y = f(x) (or don't want to)?  In cas
es like these we use the \"implicitplot\" command.  For instance, supp
ose we'd like to plot the curve  " }{XPPEDIT 18 0 "2*y^2-3*x^2-4*y+12*
x+8 = 0;" "6#/,,*&\"\"#\"\"\"*$%\"yG\"\"#F'F'*&\"\"$F'*$%\"xG\"\"#F'!
\"\"*&\"\"%F'F)F'F0*&\"#7F'F.F'F'\"\")F'\"\"!" }{TEXT -1 195 ".  We ca
n't solve for y nicely here, so we'll use the \"implicitplot\" command
, which we must first load up from the plots package.  In this command
 we MUST specify a range for x AND a range for y." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "implicitplot(2*y^2-3*x^2-4*y+12*x+8=0,x=-5..10,y=-5..
6);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "It's a hyperbola!  This is
 a nice and easy way to plot conics.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 153 "Now let's switch gears and consider \+
the \"solve\" command again.  First off, \"solve\" can handle systems \+
of equations.  Suppose we wish to solve the system " }}{PARA 257 "" 0 
"" {TEXT -1 9 "a + b = 3" }}{PARA 258 "" 0 "" {TEXT -1 10 "a - b = -4
" }}{PARA 0 "" 0 "" {TEXT -1 87 "for a and b.  We will assign the equa
tions to variables and then use a \"solve\" command." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "eqn1:= a+b = 3;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "eqn2:= a-b = -4;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "soln:=solve(\{eqn1,eqn2\},\{a,b\});" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 112 "In the above command, we labelled the solution
s with the variable \"soln\", so we can refer to them if we want to:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "soln[2];" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 219 "There is also a command called \"fsolve\".  Th
is command works like \"solve\", but it gives us solutions in decimal \+
format.  It is also useful when \"solve\" will not give us an answer. \+
 For example, let's find a solution to " }{XPPEDIT 18 0 "x^7+3*x^4+2*x
-1 = 0;" "6#/,**$%\"xG\"\"(\"\"\"*&\"\"$F(*$F&\"\"%F(F(*&\"\"#F(F&F(F(
\"\"\"!\"\"\"\"!" }{TEXT -1 20 " .  Trying solve...." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 23 "eqn:=x^7+3*x^4+2*x-1=0;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "solve(eqn,x);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 175 "Solve fails here.  The reason why is that there is no fo
rmula for finding roots of polynomials of degree 5 or more.  But \"fso
lve\" will find a pretty good APPROXIMATE solution:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 14 "fsolve(eqn,x);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 121 "We can also tell \"fsolve\" where to look for solution
s to an equation.  For example, if we want to look for solutions to  \+
" }{XPPEDIT 18 0 "x^2-1 = 0;" "6#/,&*$%\"xG\"\"#\"\"\"\"\"\"!\"\"\"\"!
" }{TEXT -1 30 "  in the interval [0,2], type:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 26 "fsolve(x^2-1 = 0 ,x=0..2);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 9 "Problems:" }}{PARA 0 "" 0 "" {TEXT -1 40 "
1) Suppose we wish to graph the ellipse " }{XPPEDIT 18 0 "25*x^2+9*y^2
 = 225;" "6#/,&*&\"#D\"\"\"*$%\"xG\"\"#F'F'*&\"\"*F'*$%\"yG\"\"#F'F'\"
$D#" }{TEXT -1 5 " .   " }{TEXT 264 32 "First do it with \"implicitplo
t\"." }{TEXT -1 112 "  What if we didn't have this command, but we sti
ll wanted to do it quick and easy on Maple?  Here's what we do:" }
{TEXT 262 152 " first solve the equation for y and label the solutions
.  Now plot each of the 2 solutions simultaneously on the same plot (o
ver the interval [-5,5]).  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 139 "2) Since we are fooling around with plots, her
e is a quick and easy problem that once again shows us not to complete
ly trust Maple plots.  " }{TEXT 258 7 "Define " }{TEXT -1 4 "f(x)" }
{TEXT 267 2 " =" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^3-x^2-x+1.001;" "6#
,**$%\"xG\"\"$\"\"\"*$F%\"\"#!\"\"F%F*$\"%,5!\"$F'" }{TEXT -1 3 " , " 
}{TEXT 266 9 "i.e. as a" }{TEXT -1 1 " " }{TEXT 268 9 "function." }
{TEXT -1 2 "  " }{TEXT 259 31 "Plot it on the interval [-2,2]." }
{TEXT -1 93 "  Looking at this plot, guess how many real solutions the
re are to the equation  f(x) = 0 .  " }{TEXT 260 57 "Now solve the equ
ation f(x) = 0 with the \"solve\" command." }{TEXT -1 154 "  How many \+
REAL solutions are there (\"I\" signifies imaginary numbers)?  Ah.  So
mething must be not quite right with our last plot.  To see what happe
ned, " }{TEXT 261 36 "plot f(x) on the interval [0.9,1.1]." }{TEXT -1 
161 "  The graph does not cross the x-axis at x=1!  So we see our firs
t Maple plot did not give us enough detail to actually let us see that
 f(1) is not equal to 0.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 169 "3) The command \"fsolve\" is an extrememly usefu
l one, but it does have its limitations.  Sometimes it will not give u
s all solutions.  For example, consider the equation " }{XPPEDIT 18 0 
"x^2+1/x-1/(x^2) = 0;" "6#/,(*$%\"xG\"\"#\"\"\"*&\"\"\"F(F&!\"\"F(*&\"
\"\"F(*$F&\"\"#F+F+\"\"!" }{TEXT -1 5 " .   " }{TEXT 269 23 "Solve it \+
with \"fsolve\"." }{TEXT -1 26 "  It gives one solution.  " }{TEXT 
256 187 "But now plot the left-hand-side of the equation from -2 to 2,
 and put the restriction \"y=-10..10\" in your plot command to restric
t the y-values so we can see it nicely (it blows up at 0)." }{TEXT -1 
48 "  The plot tells us there is another solution.  " }{TEXT 257 78 "N
ow use a modification of \"fsolve\" to find the other solution to our \+
equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
40 "4) The general equation of a circle is  " }{XPPEDIT 18 0 "x^2+y^2+
a*x+b*y = c;" "6#/,**$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F(*&%\"aGF(F&F(F(*&%
\"bGF(F*F(F(%\"cG" }{TEXT -1 4 " .  " }{TEXT 263 100 "Find the equatio
n of the circle that passes through the points (1,1), (3,-1), and (6,4
) and plot it." }{TEXT -1 144 "  (Hint: Assign the above equation to t
he variable \"eq\".  Since the circle contains the point (1,1), the fo
llowing equation must be satisfied:  " }{XPPEDIT 18 0 "1^2+1^2+a*1+b*1
 = c;" "6#/,**$\"\"\"\"\"#\"\"\"*$\"\"\"\"\"#F(*&%\"aGF(\"\"\"F(F(*&%
\"bGF(\"\"\"F(F(%\"cG" }{TEXT -1 344 "  .  To get this equation on Map
le, use the command  \"eq1:=subs(\{x=1,y=1\},eq);\"  .  Now you need 2
 more equations before you can solve for a,b, and c, right?  When you \+
solve for a,b, and c, label your solutions with the variable \"soln\" \+
as we did before.  To substitute them into eq, you can use the command
 \"eq:=subs(soln,eq);\".  Then plot eq.)" }}}}{MARK "18 4 2" 0 }
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