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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 26 "Getting Started with Map
le" }}{PARA 19 "" 0 "" {TEXT -1 16 "(September 2003)" }}{PARA 0 "" 0 "
" {TEXT -1 238 "Note: This handout is actually a \"Maple Session\"; ev
erything in here was created in Maple.  The actual Maple commands that
 I entered are shown in the lines preceded by the \">\"; these lines a
re immediately followed by the resulting output." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Simple calculatio
ns:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "2+2;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Not
e the semicolon at the end of the line.  Each Maple command must end w
ith a semicolon (or a colon)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "(12+3*5)/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#F\"\"%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "Note that Maple does not convert \+
the rational number to its decimal representation.  You can find the d
ecimal representation with the \"evalf\" command." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "evalf(27/4);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"++++]n!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Maple has \+
many predefined functions.  For example, sine and cosine:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sin(1.23);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+>!))[U*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 7 "cos(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "cos(1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$cosG6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
103 "Note that Maple didn't display the numerical value of cos(1).  To
 see the numerical value, use \"evalf\":" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "evalf(cos(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+fI-.a!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "To take a square roo
t, you can use the \"sqrt\" function:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "sqrt(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "Maple has several predefined cons
tants.  The one that we will need most is pi.  The Maple name of pi th
at can be used in an input line is \"Pi\".  Maple is " }{TEXT 259 14 "
case sensitive" }{TEXT -1 86 ", so the P must be a capital letter.  In
 the output, Maple will show the Greek letter." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 3 "Pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#PiG" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(Pi);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "sin(Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "Another predefined constant that \+
you will see occasionally is \"I\".  In Maple, \"I\" is the square roo
t of -1 (a complex number).   For example: " }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 9 "sqrt(-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#\"\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sqrt(-16);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#^#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "I*I;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "The function name \"exp\" is used \+
for the exponential function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 7 "exp(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "exp(1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$expG6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
143 "Note that the number \"e\" is displayed in the output as a boldfa
ce e.  However, you should not use e^x for the exponential function; u
se exp(x)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(exp(1))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G=G=F!\"*" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 255 "Typically we want to do a lot more than calculat
e a few numbers.  We want to define functions, plot graphs, solve equa
tions, etc.  We need to be able to define and store expressions in Map
le.  This is done with the assignment statement \":=\".  For example:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "m := 355/113;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"mG#\"$b$\"$8\"" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 112 "Now the variable \"m\" has been created, and it has th
e value 355/113.  We can now use \"m\" in subsequent commands." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(m);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"+?HfTJ!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 7 "sin(m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6##\"$b$\"$8
\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 243 "The value of a variable ca
n be an expression that is more complicated than just a number. For ex
ample, suppose we want to work with the expression 4x^2-1.  We can cre
ate a variable that will hold this expression.  I will call the variab
le \"q\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "q := 4*x^2-1;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG,&*$)%\"xG\"\"#\"\"\"\"\"%F*
!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Now there are many thin
gs we can do with this expressions.  For example, we can plot it, usin
g the \"plot\" command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "
plot(q,x=-1..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 308 233 233 {PLOTDATA 
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3hLLL$=exJ$F0$!3;z)pZD#*pf&F07$$!3*RLLLtIf$HF0$!3'o^F;HC@b'F07$$!3]++]
PYx\"\\#F0$!3G'pkBmBk^(F07$$!3EMLLL7i)4#F0$!3/!*4gnbJQ#)F07$$!3c****\\
P'psm\"F0$!3%yC[By%3))))F07$$!3')****\\74_c7F0$!3_m+ey?Yo$*F07$$!3)3LL
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*F`[l-%+AXESLABELSG6$Q\"x6\"Q!Fe[l-%%VIEWG6$;F(Fhz%(DEFAULTG" 1 2 0 1 
10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 51 "We'll take a closer look at the plot comm
and later." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 145 "We can also take the derivative with respect to x, using the \+
\"diff\" command.  I will create a new variable called \"dq\" that hol
ds the derivative:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dq :=
 diff(q,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dqG,$%\"xG\"\")" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "We could have done the same thing
s without defining q.  We can give the expression itself as an argumen
t to the functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot
(4*x^2-1,x=-1..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 357 243 243 
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\"#5F)$F*F*F`[l-%+AXESLABELSG6$Q\"x6\"Q!Fe[l-%%VIEWG6$;F(Fhz%(DEFAULTG
" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "diff(4*x^2-1,x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$%\"xG\"\")" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 76 "We can also attempt to solve the equation 4x^2-1=0 with t
he \"solve\" command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "so
lve(4*x^2-1=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"\"\"\"\"##!\"
\"F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "In this case, the solve c
ommand was able to find the solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 69 "Let's take a closer look at the comma
nds \"plot\", \"diff\", and \"solve\"." }}}{EXCHG {PARA 256 "" 0 "" 
{TEXT -1 16 "The Plot Command" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 313 
"The \"plot\" command takes several arguments, and has many options.  \+
The basic command takes two arguments. The first is the expression (i.
e. function) to be plotted, and the second is the range of the indepen
dent variable of the expression.  For example, the following command p
lots x^2 on the interval -1 < x < 1:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "plot(x^2,x=-1..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 453 
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45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Note t
hat a..b is the standard method for describing a range in Maple." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can a
lso give a third argument to the plot command that specifies the verti
cal range of the plot.  For example:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "plot(x^2,x=-1..1,-1..1.5);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"\"\"\"!$\"\"
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3umm;/#)[oPF0$\"3(3L%\\M.:?9F07$$!3hLLL$=exJ$F0$\"3@IvIO>v+6F07$$!3*RL
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$!3')****\\74_c7F0$\"3KK)\\N![%)y:F]q7$$!3)3LLL3x%z#)F]q$\"3M!=Wt2u\\&
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0$\"3a([5:F?#[FF]q7$$\"3#z****\\_qn2#F0$\"3um(3N\"e(HJ%F]q7$$\"3U)***
\\i&p@[#F0$\"3!RV,qtl6;'F]q7$$\"3B)****\\2'HKHF0$\"3;&Rg9Fg$)f)F]q7$$
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3UKL$eR<*fTF0$\"3[xc,u7\\I<F07$$\"3m******\\)Hxe%F0$\"3/-\"ew^EZ5#F07$
$\"3ckm;H!o-*\\F0$\"3s%H#H+vF!\\#F07$$\"3y)***\\7k.6aF0$\"3l&3Sd]Jz#HF
07$$\"3#emmmT9C#eF0$\"3K$ySR'40!R$F07$$\"33****\\i!*3`iF0$\"3i6dN#G7,
\"RF07$$\"3%QLLL$*zym'F0$\"3CM\\`!GigW%F07$$\"3wKLL3N1#4(F0$\"3BILi![O
(H]F07$$\"3Nmm;HYt7vF0$\"35,!G3;=Tk&F07$$\"3Y*******p(G**yF0$\"3tFrt;Y
()RiF07$$\"3]mmmT6KU$)F0$\"3+j)pI?K%fpF07$$\"3fKLLLbdQ()F0$\"3$Q>x^Bqi
j(F07$$\"3[++]i`1h\"*F0$\"31F(*fd=^#R)F07$$\"3W++]P?Wl&*F0$\"3]:sFP\"o
(\\\"*F07$F+F+-%'COLOURG6&%$RGBG$\"#5F)$F*F*F^[l-%+AXESLABELSG6$Q\"x6
\"Q!Fc[l-%%VIEWG6$;F(F+;F($\"#:F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
148 "You can specify more optional arguments.  In the next example, I'
ve changed the color to black, used a dashed line, and I've given the \+
plot a title:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot(x^2,x
=-1..1,-1..1.5,title=`y=x^2`,color=black,linestyle=DASH);" }}{PARA 13 
"" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6#7S7$$!\"\"\"
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ONayFj&F07$$!3u****\\P8#\\4(F0$\"30z7`y3zL]F07$$!3+nm;/siqmF0$\"3qSopH
ns\\WF07$$!3[++](y$pZiF0$\"335mBmxO.RF07$$!33LLL$yaE\"eF0$\"3m(y?Ic&py
LF07$$!3hmmm\">s%HaF0$\"3ej\"3!Go\"z%HF07$$!3Q+++]$*4)*\\F0$\"3mUqC6(*
4)\\#F07$$!39+++]_&\\c%F0$\"3Mc-XV;)Q3#F07$$!31+++]1aZTF0$\"3?U-MW$4-s
\"F07$$!3umm;/#)[oPF0$\"3(3L%\\M.:?9F07$$!3hLLL$=exJ$F0$\"3@IvIO>v+6F0
7$$!3*RLLLtIf$HF0$\"3&y?J4F*o>')!#>7$$!3]++]PYx\"\\#F0$\"3he#)3W3%*3iF
]q7$$!3EMLLL7i)4#F0$\"3!\\_(*436US%F]q7$$!3c****\\P'psm\"F0$\"31!QHT/)
yzFF]q7$$!3')****\\74_c7F0$\"3KK)\\N![%)y:F]q7$$!3)3LLL3x%z#)F]q$\"3M!
=Wt2u\\&o!#?7$$!3KMLL3s$QM%F]q$\"3a8,Dp@*o)=Fgr7$$!3]^omm;zr)*!#@$\"3+
4t+rqAX(*!#C7$$\"3%pJL$ezw5VF]q$\"3Q>$f!R?Fe=Fgr7$$\"3s*)***\\PQ#\\\")
F]q$\"3UZsD4'35k'Fgr7$$\"3GKLLe\"*[H7F0$\"3&e?f/fV;^\"F]q7$$\"3I******
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>x^Bqij(F07$$\"3[++]i`1h\"*F0$\"31F(*fd=^#R)F07$$\"3W++]P?Wl&*F0$\"3]:
sFP\"o(\\\"*F07$F+F+-%'COLOURG6&%$RGBGF*F*F*-%&TITLEG6#%&y=x^2G-%*LINE
STYLEG6#\"\"$-%+AXESLABELSG6$Q\"x6\"Q!Fh[l-%%VIEWG6$;F(F+;F($\"#:F)" 
1 2 0 3 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "To see more examples, look up \"pl
ot\" in the Maple Help,  or enter the command \"?plot\"." }}}{EXCHG 
{PARA 18 "" 0 "" {TEXT 257 16 "The Diff Command" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 334 "You can take derivatives in Maple with the \"diff\"
 command (\"diff\" stands for \"differentiate\").  The basic command t
akes two arguments. The first is the expression to be differentiated, \+
and the second is the variable with respect to which the derivative is
 to be taken.  Maple knows the derivatives of most of the elementary f
unctions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
124 "Here are some examples.  The first finds the derivative of sin(t)
+4t, and the second finds the derivative of 1/x + exp(x^2)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(sin(t)+4*t,t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$cosG6#%\"tG\"\"\"\"\"%F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "diff(1/x + exp(x^2),x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%*$)%\"xG\"\"#F%!\"\"F**(F)
F%F(F%-%$expG6#*$F'F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "To \+
take a second or higher order derivative, just list the variable again
.  Here are some second derivatives:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "diff(sin(t)+4*t,t,t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$-%$sinG6#%\"tG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "diff(1/x + exp(x^2),x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&
\"\"\"F%*$)%\"xG\"\"$F%!\"\"\"\"#*&F+F%-%$expG6#*$)F(F+F%F%F%*(\"\"%F%
F1F%F-F%F%" }}}{EXCHG {PARA 18 "" 0 "" {TEXT 258 17 "The Solve Command
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 291 "The \"solve\" command can so
lve many algebraic equations.  We saw an example earlier where we solv
ed a quadratic.  In the simplest form of the command, the first argume
nt is the equation to be solved, and the second is the variable to be \+
solved for.   Here we find the solutions to x^2+5x+6=0:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(x^2+5*x+6=0,x);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$!\"#!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
115 "If there expression does not contain an equals sign, Maple assume
s that the expression should be set equal to zero." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 19 "solve(x^2+5*x+6,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$!\"#!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Next w
e solve a cubic polynomial:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "solve(x^3-2*x+1=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\",&#
!\"\"\"\"#F#*&#F#F'F#-%%sqrtG6#\"\"&F#F#,&F%F#*&#F#F'F#*$F*F#F#F&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Let's make one small change to the
 cubic polynomial:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve
(x^3-x+1=0,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%,&*$),&\"$3\"\"\"\"*
&\"#7F(-%%sqrtG6#\"#pF(F(#F(\"\"$F(#!\"\"\"\"'*&\"\"#F(F&#F2F0F2,(F$#F
(F**&F(F(*$)F&#F(F0F(F2F(*(^##F(F5F(-F,6#F0F(,&F$F1*&F5F(F&F6F(F(F(,(F
$F8F9F(*(^##F2F5F(F@F(FBF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 210 "
What a mess!  If you look carefully at the above output, you will see \+
that Maple has found three solutions, two of which are complex numbers
.  (Recall that the symbol I in Maple is the complex number sqrt(-1).)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "What
 happens if the equation has no solution, or if Maple can not find it?
  Here are some examples. First, a trivial example with no solution:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(1=0,x);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 103 "Maple gave no output; this means that ei
ther there is no solution, or Maple can not find any solutions." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Note that
 Maple can solve an equation like x^2=-1, because Maple allows complex
 numbers:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(x^2=-1,x
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$^#\"\"\"^#!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 41 "Sometimes Maple gives a partial solution:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "solve(x*(sin(x)+x+1/4)=
0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!-%'RootOfG6#,(%#_ZG\"\"%
*&F)\"\"\"-%$sinG6#F(F+F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 282 
"The \"RootOf\" function means the root of the expression; in the expr
ession, _Z is the variable.  In this example, Maple found one solution
 to be 0, but the other \"solution\" is just the solution to 4*_Z + 4*
sin(*_Z) +1=0.  Maple could not find an analytical solution to this eq
uation." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 94 "Let's create a function, plot it, and then use \"diff\" a
nd \"solve\" to find the critical points." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "w := (x^4+3*x^2)/(1+
x^4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG*&,&*$)%\"xG\"\"%\"\"\"
F+*&\"\"$F+)F)\"\"#F+F+F+,&F+F+F'F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "plot(w,x=-3..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 366 
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4JB;F-$\"3a8M?KnEp=F-7$$\"3u*****\\KCnu\"F-$\"3;OdGR[)3z\"F-7$$\"3s***
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0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dw := diff(w,x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#dwG,&*&,&*$)%\"xG\"\"$\"\"\"\"\"%*&\"\"'F,F*F,F,F,
,&F,F,*$)F*F-F,F,!\"\"F,**F-F,,&F1F,*&F+F,)F*\"\"#F,F,F,F0!\"#F*F+F3" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "plot(dw,x=-3..3);" }}
{PARA 13 "" 1 "" {GLPLOT2D 351 167 167 {PLOTDATA 2 "6%-%'CURVESG6$7ap7
$$!\"$\"\"!$\"3uWh27P'4)>!#=7$$!3!******\\2<#pG!#<$\"3,!\\OUTh1B#F-7$$
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