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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "****  MAPLE   FILE  # 1 **
**" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 87 "The purpose of this file is to  provide some general MAPLE back
ground and to summarize " }}{PARA 0 "" 0 "" {TEXT -1 64 "some MAPLE co
mmands that are useful in algebra and pre-calculus." }}{PARA 0 "" 0 "
" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 80 "In addition to an expl
anation of the usefulness and syntax of several commands, " }}{PARA 0 
"" 0 "" {TEXT -1 70 "some examples are given (which you are encouraged
 to try on your own)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 " ****   Operat
ors, Algebraic Expressions and Functions in MAPLE   ****" }}{PARA 0 "
" 0 "" {TEXT -1 337 "\n The recognized mathematical operators in MAPLE
 are similar \n to those used in many computing languages. They are + \+
for\n addition, - for subtraction, * for multiplication, / for\n divis
ion and ^ for exponentiation. To assign an algebraic \n expression to \+
a variable, we use the assignment operator : =\n as shown in the follo
wing example:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "y := (3*
x^2 + 2*x -7)/(x^3-6*x-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG*&
,(*$)%\"xG\"\"#\"\"\"\"\"$F)F*!\"(F+F+,(*$)F)F,F+F+F)!\"'!\"#F+!\"\"" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Note that I only typed in the p
art after the prompt > and ending" }}{PARA 0 "" 0 "" {TEXT -1 161 "wit
h the semi-colon....the rest is an acknowledgement from \nMAPLE. All M
APLE commands (with very few exceptions we'll\ndiscuss as necessary) e
nd in a semi-colon.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Another \+
way to define a function in MAPLE is using the \"mapping\"" }}{PARA 0 
"" 0 "" {TEXT -1 9 "operator:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f := x -> (3*x^2 + 2*x -7)/(
x^3-6*x-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)o
peratorG%&arrowGF(*&,(*$)9$\"\"#\"\"\"\"\"$F0F1!\"(F2F2,(*$)F0F3F2F2F0
!\"'!\"#F2!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "This com
mand just says that f \"maps\" a number x to the corresponding" }}
{PARA 0 "" 0 "" {TEXT -1 71 "value given by the function. We'll see a \+
major difference between these" }}{PARA 0 "" 0 "" {TEXT -1 44 "two way
s of defining a function momentarily." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 " There are several mathematic
al functions we are familiar" }}{PARA 0 "" 0 "" {TEXT -1 74 " with tha
t are known by MAPLE. These include the square\n root function...\n" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "y := sqrt(x);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"yG*$-%%sqrtG6#%\"xG\"\"\"" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 34 " ...the absolute value function..." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "y :=
 abs(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG-%$absG6#%\"xG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "...and the usual trigonometric fun
ctions sin(x), cos(x), " }}{PARA 0 "" 0 "" {TEXT -1 35 "tan(x),sec(x),
 csc(x), and cot(x).\n" }}{PARA 0 "" 0 "" {TEXT -1 64 "Note that my la
test definition of y is the only one I retain. If" }}{PARA 0 "" 0 "" 
{TEXT -1 69 "I want to use several functions at once, I must assign th
em different" }}{PARA 0 "" 0 "" {TEXT -1 6 "names." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Often we wish to \+
evaluate a function we've defined at a " }}{PARA 0 "" 0 "" {TEXT -1 
54 "particular number. If we've defined the function as an" }}{PARA 0 
"" 0 "" {TEXT -1 52 "algebraic expression, we can use the \"subs\" com
mand." }}{PARA 0 "" 0 "" {TEXT -1 62 "For instance, we can compute the
 square root of 4 as follows:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "y := sqrt(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG*$-%%sq
rtG6#%\"xG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(x=
4,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#\"\"%\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Notice that maple doesn't simplify
 that last expression. We can" }}{PARA 0 "" 0 "" {TEXT -1 10 "ask it t
o:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "The symbol % in MAPLE refers \+
to the result of the last MAPLE calculation" }}{PARA 0 "" 0 "" {TEXT 
-1 19 "that was performed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ans := subs(x=Pi,y);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$ansG*$-%%sqrtG6#%#PiG\"\"\"" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "I
n the above command, I plugged Pi (it must be capitalized)" }}{PARA 0 
"" 0 "" {TEXT -1 45 "into our sqrt funtion and assigned the result" }}
{PARA 0 "" 0 "" {TEXT -1 67 "to the variable ans. I can get a decimal \+
version of ans as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(ans);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+^QXs<!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "I
f you wish to round a decimal to a certain number of decimal places, i
t's pretty easy" }}{PARA 0 "" 0 "" {TEXT -1 78 "to do this without MAP
LE's help. However, if you know in advance that you want" }}{PARA 0 "
" 0 "" {TEXT -1 78 "ALL decimals displayed with m SIGNIFICANT DIGITS, \+
you can execute the command:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Digits := m;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "The
 default is for floating point numbers to be displayed with 10 signifi
cant digits." }}{PARA 0 "" 0 "" {TEXT -1 30 "So if we executed the com
mand:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "  " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "Digits := 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%'DigitsG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "...and then re
-evaluate the result above:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(ans);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"%t<!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 
"...we get ans to 4 significant figures. You should be aware that this
 Digits command is an" }}{PARA 0 "" 0 "" {TEXT -1 85 "\"environmental
\" command, so that all computations from now on will be affected. If \+
we" }}{PARA 0 "" 0 "" {TEXT -1 44 "wish to revert to the default, we m
ust type:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 10;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 91 "On the other hand, we could have defined the square root functi
on \nusing mapping notation:\n" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 19 " f := x -> sqrt(x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG%%sqrtG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "(
Notice the strange way MAPLE acknowledges the definition of the" }}
{PARA 0 "" 0 "" {TEXT -1 65 "function.) One benefit of this approach i
s that we can bypass the" }}{PARA 0 "" 0 "" {TEXT -1 49 "subs command \+
in evaluating f(x) at different x's:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "f(16
);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#\"\"%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 60 "Notice also that the simplify command is \+
not necessary here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 38 "****   Plotting Graphs in MAPLE   ****" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Once you'
ve defined a function using an assignment command," }}{PARA 0 "" 0 "" 
{TEXT -1 119 "you can graph it using the \"plot\" command. You need to
 specify\na range of x over which you want to graph the function:\n" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "z:= x^3 + 2*x^2 -3*x -4;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG,**$)%\"xG\"\"$\"\"\"F**&\"\"#
F*)F(F,F*F**&F)F*F(F*!\"\"\"\"%F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "plot(z,x=-6..6);" }}{PARA 13 "" 1 "" {GLPLOT2D 432 
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{PARA 0 "" 0 "" {TEXT -1 59 "MAPLE automatically adjusts the vertical \+
scale to show the " }}{PARA 0 "" 0 "" {TEXT -1 258 "graph over the spe
cified domain. In the example above, it's\nhard to see clearly the beh
avior of the function on the\ninterval [-2,2] because the function gro
ws so quickly outside\nthis interval. To fix this problem, we can plot
 it again on a smaller interval:\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "plot(z,x=-3..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 432 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Sometimes we want to plot more tha
n one graph on the same set" }}{PARA 0 "" 0 "" {TEXT -1 90 "of axes. T
his can be done by listing the functions within braces (\{\}) in the p
lot command:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "w := sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"wG-%$sinG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plo
t(\{z,w\},x=-3..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 432 432 432 
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45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
56 "If we don't like the fact that the sin(x) is compressed," }}{PARA 
0 "" 0 "" {TEXT -1 150 "but we don't want to change the interval we're
 sketching\nthe graphs over, we can restrict the y-range of the plot\n
explicitly within the plot command:\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "y := 'y';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGF$
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "(Necessary so that MAPLE will
 forget the definition we gave \"y\" above" }}{PARA 0 "" 0 "" {TEXT 
-1 39 "and consider it as a fresh variable)..." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(\{z,w
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" {TEXT -1 51 "If we've defined a function using mapping notation:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "f := x -> x^2 + 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#
%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"#\"\"\"F1F1F1F(F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "...we can still use the plot comma
nd to sketch its graph. It's essential," }}{PARA 0 "" 0 "" {TEXT -1 
81 "though, that we refer to the function as f(x) and not just f. This
 is universally" }}{PARA 0 "" 0 "" {TEXT -1 36 "true, not just for the
 plot command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
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{TEXT -1 40 "****   Solving Equations in MAPLE   ****" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "MAPLE provides a coupl
e of different ways for solving equations." }}{PARA 0 "" 0 "" {TEXT 
-1 81 "First we'll discuss the \"solve\" command.  This command is use
d by MAPLE to solve " }}{PARA 0 "" 0 "" {TEXT -1 144 "equations for wh
ich an analytic (as opposed to a numerical) solution is possible.\nA g
ood example is finding the roots of a polynomial equation:\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z := x^4 -x^3 -10*x^2 +4*x +
24;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG,,*$)%\"xG\"\"%\"\"\"F**$
)F(\"\"$F*!\"\"*&\"#5F*)F(\"\"#F*F.*&F)F*F(F*F*\"#CF*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "solve (z=0,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6&\"\"#\"\"$!\"#F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
62 "The first argument to the solve command is the ** equation ** " }}
{PARA 0 "" 0 "" {TEXT -1 236 "to  be solved, in this case z = 0. The s
econd argument is the \nvariable we are solving for, in this case x. M
APLE lists the\nroot x = -2 twice because it is a \"double zero\" of t
he \npolynomial, i.e. (x+2)^2 is a factor of the polynomial.\n" }}
{PARA 0 "" 0 "" {TEXT -1 59 "Some equations are not well-suited to an \+
analytic solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 20 "solve(cos(x)=x^3,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RootOfG6$,&-%$cosG6#%#_ZG!\"\"*$)F*\"\"$\"\"\"F//%&l
abelG%$_L1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 79 "When this command is typed in, MAPLE eith
er gets stuck and doesn't do anything," }}{PARA 0 "" 0 "" {TEXT -1 82 
"or desperately returns an answer in terms of some pretty obscure func
tions we know" }}{PARA 0 "" 0 "" {TEXT -1 15 "nothing about.\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "A command which attempts to solve \+
an equation using numerical methods " }}{PARA 0 "" 0 "" {TEXT -1 99 "(
and therefore returns a decimal answer) is \"fsolve\".\n Let's try sol
ving the above equation again:\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "fsolve(cos(x)=x^3,x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+J.ua')!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot
(\{cos(x),x^3\},x=-2..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 432 432 432 
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