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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT -1 34 "
Variables, Functions and Equations" }}{PARA 19 "" 0 "" {TEXT -1 28 "Dr
. Saccone, Revised 1/18/01" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 
"restart;" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Variables" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "In a worksheet
 you can define your own variables. One must use caution here. Haphaza
rdly defined variables can lead to unexpected results in a worksheet.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 204 "Mapl
e has some built-in variables. Note that variable names are case-sensi
tive. To display a value that has been assigned to a variable name ent
er the name of the variable on a command line and hit enter." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 9 "
Examples." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "Pi;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf(Pi,10);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 9 "infinity;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "gamma; " }{TEXT -1 16 "Euler's constant" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(gamma,10);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 132 "Some variable names have no value assign
ed to them. In this case Maple manipulates expressions involving that \+
variable symbolically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 257 9 "Example. " }{TEXT -1 87 "In order to \+
be safe, execute the following command by clicking on it and hitting E
nter." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 112 "The Maple environment should now be restored to t
he way it was at startup time. At the moment the variable name " }
{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 139 " should have no assigned v
alue. If this is the case Maple will simply echo the name of the varia
ble when it is entered into a command line." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "x;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(x,10);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 11 "x^2+3*x+10;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 22 "Expressions involving " }{XPPEDIT 18 0 "x;" "6#%\"xG" }
{TEXT -1 34 " will now be treated symbolically." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "(x+3)^3;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expand((x+3)^3);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 4 "The " }
{TEXT 302 4 "eval" }{TEXT 303 1 " " }{TEXT -1 8 "command." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We can evaluate ex
pression for given values of the variable using the command " }{TEXT 
301 6 "eval. " }{TEXT -1 47 "This will not assign any value to the var
iable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }{TEXT 304 8 "Example." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "
eval(x^2+3*x+10,x=1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "x;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 39 "We haved evaluated the expression with " }{XPPEDIT 18 0 "
x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 22 " without changing the " }{TEXT 
316 5 "value" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 143 "We can now assign a value to this v
ariable. When we do this the value we assign will be substituted for t
hat variable whenever it is used. The " }{TEXT 258 20 "assignment oper
ator " }{TEXT -1 86 "is the symbol \":=\". Be careful not to use \"=\"
, this symbol is reserved other purposes." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "x:=5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "x;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x^2+3*x+10;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "(x+3)^3;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 16 "expand((x+3)^3);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 214 "Thus, this object is no longer treated symbolically. One
 must be careful in assigning values to variables. If you lose track o
f what variables have been assigned you may incur unexpected results i
n your worksheets." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 259 8 "Example." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "plot(x^2,x=-1..1);" }}{PARA 8 "" 1 "" {TEXT -1 35 "Er
ror, (in plot) invalid arguments\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 267 "The plot command does not respond well to our request. T
he problem is that Maple now reads the second argument as \"5=-1..1\".
 The first argument is read as \"25\", but that's ok as can be seen fr
om the next command (assuming you haven't assigned a value to the vari
able " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 10 " somehow)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "plot(25,t=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "So, as
signments can wreak havoc if they are not done carefully. The value of
 5 is now assigned to the variable " }{XPPEDIT 18 0 "x;" "6#%\"xG" }
{TEXT -1 171 ". This will be the case in any new worksheet you open. I
t will also be the case if you try to go back to the beginning of this
 worksheet and re-execute the commands above." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "To " }{TEXT 260 9 "unassig
n " }{TEXT -1 54 "a variable you can restart Maple, you can execute th
e " }{TEXT 261 8 "restart " }{TEXT -1 37 "command, or you can do the f
ollowing." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "x := 'x';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 
"x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The first statement makes
 of use of something called " }{TEXT 262 16 "forward quotes, " }{TEXT 
-1 43 "a topic we need not go into at this moment." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 18 "Naming Co
nventions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
105 "Variable names cannot begin with a number. They may only contain \+
alphanumeric characters and underscores." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 29 "Example of legal d
efinitions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "myName := 5;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "thisIs4You := 2/3 + Pi;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "the_Last_Name := sqrt(1
21);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 257 "" 0 
"" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 31 "Example of illegal d
efinitions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "2theLastDrop \+
:= 23;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "this&that := 0;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Pi := -1;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 13 "Functions or " }{TEXT 271 10 "Procedures" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Suppose we have some kin
d of computation that we wish to repeatedly. For example, suppose we w
ant to evaluate the expression " }{XPPEDIT 18 0 "x^2+2*x+2;" "6#,(*$%
\"xG\"\"#\"\"\"*&F&F'F%F'F'F&F'" }{TEXT -1 23 " for various values of \+
" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 111 " throughout the workshe
et. One method of doing so would be to assign a function, or, as Maple
 would call it, a " }{TEXT 264 10 "procedure " }{TEXT -1 61 "to a vari
able name. Maple's syntax for defining a function is" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 265 24 "var -
> expression in var" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 6 "where " }{TEXT 266 4 "var " }{TEXT -1 17 "is some variab
le." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 267 8 "Example." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f :
= x -> x^2+2*x+2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 7 "Caution" 
}{TEXT 305 106 ": the left hand side is simply \"f\", it is not \"f(x)
\" the way we normally define functions in math classes." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 304 "If you click on t
he blue output above you will see Maple's internal representation of t
his function assignment, namely  \"f := proc (x) options operator, arr
ow; x^2+2*x+2 end proc\" where \"proc\" stands for \"procedure\". Fort
unately we do not have to worry about this internal language, at least
 not for now." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 53 "We can now evaluate this function at various numbers." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(1);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 6 "f(Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 306 90 "Note tha
t in the above calculation we used the \"%\" symbol. This represents t
he last output" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 20 "We can also graph f." }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 20 "plot(f(x), x=-1..1);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 102 "Note that we do not have to use the same independen
t variable in the plot as we did when we defined f." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(f(t),
 t=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 14 "Mo
re examples." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := x -> a*
x^2+b*x+c; " }{TEXT -1 54 "A function definition with undefined parame
ters a,b,c." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 11 "f(sin(x)); " }{TEXT -1 24 "Composition of func
tions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f(f(x)); " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g := (s,t) -> s^2 + t^2; " }
{TEXT -1 27 "A function of two variables" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g(x,y);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g(1,3);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "h(x) := sin(x)*cos(x);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 5 "h(0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 33 "Wha
t happened in the last example" }{TEXT 307 53 "? This is related to th
e caution given above. You do " }{TEXT -1 0 "" }{TEXT 308 0 "" }{TEXT 
309 4 "not " }{TEXT 310 0 "" }{TEXT -1 0 "" }{TEXT 311 92 "define func
tions using the notation \"(x)\" on the left hand side of the  assignm
ent operator." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 312 8 "Example." }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart;" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f(x):=x^2+3
*x+10;" }}}{PARA 0 "" 0 "" {TEXT -1 105 "All this does is define an ex
pression whose name is \"f(x)\". The variable does not behave like a f
unction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "f;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(0);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 102 "The exp
ression f(0) is unassigned. If we want to evaluate the expression f(x)
 when x=0 we can use the " }{TEXT 313 5 "eval " }{TEXT -1 8 "command.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 15 "eval(f(x),x=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "It i
s probably best to " }{TEXT 314 6 "never " }{TEXT -1 30 "make assignme
nts for the form " }{XPPEDIT 18 0 "f(x) := expr;" "6#>-%\"fG6#%\"xG%%e
xprG" }{TEXT -1 35 ". The results can be unpredictable." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := x \+
-> x^3+20;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f(x) := x^2+3
*x+10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(0);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "f(y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 172 "When we entered f(x) Maple did not eva
luate the function at the unassigned variable x because there is a var
iable name \"f(x)\", and that takes precedence. However, there is " }
{TEXT 315 4 "not " }{TEXT -1 84 "a variable named f(y) so Maple evalua
te the function f at the unassigned variable y." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 134 "If you option-click on the output of a function assignment you
 do not get a menu with many options. Compare the following two output
s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "x^2+3*x+10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "f := x -> x^2+3*x+10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 190 "
If you option-click on the output from the first command you get lots \+
of options, such as graphing. However, the same procedure applied to s
econd output yields very little, possibly nothing." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Equations" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The equal
s sign \"=\" is reserved for " }{TEXT 273 11 "equations. " }{TEXT -1 
76 "Equations differ from assignments in that no information is actual
ly stored." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 274 8 "Example." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "x:=1; y:=2; " }{TEXT -1 53 "Note that we can do more than one op
eration per line." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "x=y;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x; y;" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 177 "The input \"x=y\" did not modify x or y. So what \+
did it do? Well, nothing. So what can we do with equations? We can ask
 if they are true and, in some instances, we can solve them." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ev
alb(x=y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The \"b\" in the M
aple function " }{TEXT 275 6 "evalb " }{TEXT -1 104 "stands for Boolea
n which refers to an object of a true/false natures. The function retu
rns the value of " }{TEXT 276 5 "true " }{TEXT -1 28 "when the equatio
n holds and " }{TEXT 277 6 "false " }{TEXT -1 17 "when it does not." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "S
uppose we unassign the variables " }{XPPEDIT 18 0 "x;" "6#%\"xG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 17 " and st
art over: " }{MPLTEXT 1 0 15 "x:='x'; y:='y';" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalb(x=y);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Still not true since thes
e variables are undefined. Now try" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "x:=2; y:=2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "evalb(x=y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 278 18 "Solving Equations." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The Mapl
e command " }{TEXT 279 6 "solve " }{TEXT -1 48 "will solve an equation
 for a specified variable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }{TEXT 280 8 "Example." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "First, let's reinitializ
e the environment: " }{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "solve(x^2-4=0,x);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 49 "Maple returns the solutions as a list of numbers." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "solve(x^3-6*x^2+11*x-6=0,x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "solve(x^3+1=2-x,x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "solve(a*x+4=5,x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 44 "Maple can work with complex numbers as well." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve
(x^2+1=0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "so
lve(a*x^2+b*x+c=0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Look fa
miliar?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }{TEXT 300 115 "Try solving some of your own equations and see what
 happens. You way want to open a blank worksheet and work there." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "If you om
it \"=0\" Maple assumes this is what you want." }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "solve(t^2+3*t+1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 33 "When the root of a polynomial is " }{TEXT 281 9 "repeated
 " }{TEXT -1 54 "Maple lists this root as many times as it is repeated
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 282 8 "Example." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "sol
ve(x^2+2*x+1,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve((
x-2)^3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "It is possible to wo
rk with more than one equation. In this case we pass a " }{TEXT 283 4 
"set " }{TEXT -1 112 "of equations to the solve command. A set is a li
st of expressions with braces \"\{\}\" that are separated by commas." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "solve(\{x+y=0,y=1\},\{x,y\});" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 22 "There answer gives us " }{TEXT 284 4 "one " }{TEXT -1 38 
"solution. What we have passed on is a " }{TEXT 285 7 "system " }
{TEXT -1 16 "of equations. A " }{TEXT 286 9 "solution " }{TEXT -1 92 "
to a system of equations is a list of values for the variables that so
lve all the equations " }{TEXT 287 15 "simultaneously." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(
\{x^2+3*y=1,y=x+1\},\{x,y\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 
"In this case we have found two solutions; the output is two pairs, or
 " }{TEXT 288 5 "sets," }{TEXT -1 12 " of numbers." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "solve(\{a*x+
b*y=A,c*x+d*y=B\},\{x,y\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "T
his solution of a linear system is known as " }{TEXT 289 14 "Cramer's \+
Rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }{TEXT 290 20 "Numerical solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{TEXT 291 6 "solve " 
}{TEXT -1 56 "does not find a satisfactory symbolic answer every time.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "solve(cos(x)=x,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "T
his answer is mathematically sophisticated way of saying " }{TEXT 292 
14 "I don't know. " }{TEXT -1 38 "However, you can ask Maple to find a
n " }{TEXT 293 12 "approximate " }{TEXT -1 65 "solution, rather than a
n exact. This is usually referred to as a " }{TEXT 294 10 "numerical \+
" }{TEXT -1 161 "solution. Maple has built-in procedures for find appr
oximate solutions. You may have seen cruder, but similar, methods in y
our calculus course. For example, the " }{TEXT 295 13 "tangent line " 
}{TEXT -1 17 "approximation or " }{TEXT 296 15 "Newton-Raphson " }
{TEXT -1 7 "method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 54 "The Maple function for finding numerical solutions is \+
" }{TEXT 297 8 "fsolve. " }{TEXT -1 61 "This function returns a floati
ng point answer, hence the \"f\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fsolve(cos(x)=x,x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "cos(%);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 72 "Note that the last command verified that this is a
 pretty good solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 95 "The nice thing about fsolve is that you can specify \+
a place to look. For example, the equation " }{XPPEDIT 18 0 "sin(x) = \+
0;" "6#/-%$sinG6#%\"xG\"\"!" }{TEXT -1 86 " has infintely many solutio
ns. We can ask fsolve to find solutions in a certain range." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fs
olve(sin(x)=0,x,10..13);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "The \+
function fsolve may find no solution (this will happen, for example, w
hen no solution exists)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fsolve(sin(x)=0,x,10..12);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 185 "Suppose we wish to increase th
e accuracy. We can attempt to do this by increasing the number of sign
ificant digits Maple uses in its computations. This number is stored i
n the variable " }{TEXT 298 7 "Digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Digits;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 133 "This is the number of digits Maple is cu
rrently using. You should be able to verify this by looking at output \+
above. Digits is not a " }{TEXT 299 10 "protected " }{TEXT -1 30 "vari
able, it can be redefined." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 20;" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 19 "fsolve(cos(x)=x,x);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 7 "cos(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
167 "Oops, it's just a little off. When you increase the number of sig
nificant digits there will still be error, such as rounding error. How
ever, you do get more accuracy. " }}}}{MARK "1 0" 0 }{VIEWOPTS 1 1 0 
1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }