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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 258 24 "An Introduction to Mapl
e" }}{PARA 0 "" 0 "" {TEXT -1 40 "We will always work in what is calle
d a " }{TEXT 259 9 "worksheet" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 20 "In this environment " }{TEXT 263 3 "red" }{TEXT -1 44 " t
ext after the Maple prompt (>) represents " }{TEXT 260 11 "Maple input
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 126 "It can be executed \+
by moving the cursor anywhere into that line and hitting the enter key
 (try this with any red line below). " }}{PARA 0 "" 0 "" {TEXT -1 50 "
Maple input usually is terminated by a semicolon (" }{TEXT 264 1 ";" }
{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 74 "The execution of a Map
le command will usually be followed by some kind of " }{TEXT 261 6 "ou
tput" }{TEXT -1 14 " displayed in " }{TEXT 262 4 "blue" }{TEXT -1 2 ".
 " }}{PARA 0 "" 0 "" {TEXT -1 111 "The display of the output (but not \+
the actual calculation) can be suppressed by ending a command with a c
olon (" }{TEXT 265 1 ":" }{TEXT -1 26 ") instead of a semicolon (" }
{TEXT 266 1 ";" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 132 "Use the following to get acquainted with
 Maple's syntax and some of the commands needed more frequently in the
 assignments to come. " }}{PARA 0 "" 0 "" {TEXT -1 55 "Other commands \+
will be introduced when they are needed." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "restart; " }}{PARA 0 "" 0 "" {TEXT -1 85 "This command
 is useful at the beginning of every worksheet. It clears Maple's memo
ry." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Maple can do simple arithm
etic. Here are a few examples:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "2
-3+4/5*6^7;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "If you need to use
 floating point arithmetic then do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "evalf(1/3 + 1/2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "By defa
ult 10 digits are displayed, although this can be changed easily by ad
ding the desired number of digits to the evalf command." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "evalf(1/3 + 1/2, 50);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 49 "Maple also knows certain mathematical constants. " }}
{PARA 0 "" 0 "" {TEXT -1 25 "(Note the capitalization)" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 3 "Pi;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "As \+
a floating point number" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(Pi
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Another constant that you m
ight use is the number " }{XPPEDIT 18 0 "e =2.7182" "6#/%\"eG$\"&#=F!
\"%" }{TEXT -1 4 " ..." }}{PARA 0 "" 0 "" {TEXT -1 31 "This is how to \+
get it in Maple:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "exp(1);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "evalf(exp(1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "A useful \+
" }{TEXT 273 8 "shortcut" }{TEXT -1 17 " is the operator " }{TEXT 268 
1 "%" }{TEXT -1 39 ". It refers to the most recent output. " }}{PARA 
0 "" 0 "" {TEXT -1 62 "For example (note again that Maple treats the f
ollowing as an " }{TEXT 272 5 "exact" }{TEXT -1 7 " value)" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 8 "sqrt(2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "%^3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "does the \+
same as" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "(sqrt(2))^3;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 26 "Alternatively, we can use " }{TEXT 269 
11 "assignments" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "In th
e previous example this would mean" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "root2 := sqrt(2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Here we \+
assigned the " }{TEXT 270 5 "value" }{TEXT -1 9 " sqrt(2)=" }{XPPEDIT 
18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 8 " to the " }{TEXT 271 
8 "variable" }{TEXT -1 8 " root2. " }}{PARA 0 "" 0 "" {TEXT -1 65 "Thu
s, the following command will also produce the desired answer:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "root2^3;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 59 "The next Maple concept we need to understand is that of a
n " }{TEXT 274 10 "expression" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 48 "All of the examples used above were expressions." }}
{PARA 0 "" 0 "" {TEXT -1 93 "To illustrate the use of expressions bett
er we make use of the real strength of Maple - it's " }{TEXT 298 21 "s
ymbolic capabilities" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 61 "W
e know that the area of a circle is given by the expression " }
{XPPEDIT 18 0 "Pi*r^2" "6#*&%#PiG\"\"\"*$%\"rG\"\"#F%" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 43 "Let's assign this expression to a var
iable " }{TEXT 275 7 "area_e " }{TEXT -1 5 "(the " }{TEXT 282 2 "_e" }
{TEXT -1 58 " being used here to emphasize that we are dealing with an
 " }{TEXT 281 1 "e" }{TEXT -1 10 "xpression)" }{TEXT 280 1 " " }{TEXT 
-1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "area_e := Pi*r^2;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "So the expression Pi*r^2 is now s
tored under the name area_e. \nIf we want to know the area of a specif
ic circle, say the one with r=3, we can do this as follows:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "subs(r=3,area_e);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 74 "Note that we had to enter Pi*r^2 even though Maple d
isplays the result as " }{XPPEDIT 18 0 "Pi*r^2" "6#*&%#PiG\"\"\"*$%\"r
G\"\"#F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "If we omit t
he multiplication operator " }{TEXT 303 1 "*" }{TEXT -1 28 " - this is
 what will happen:" }}{PARA 0 "" 0 "" {TEXT -1 86 "(You always have to
 include arithmetic operators when you are formulating Maple input." }
}{PARA 0 "" 0 "" {TEXT -1 10 "This is a " }{TEXT 304 25 "common source
 of mistakes" }{TEXT -1 2 "!)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ju
nk := Pir^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "junk := Pi r^2;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 121 "As an alternative to using expressions (whose evaluation
 is sometimes a bit cumbersome) we can (and most often will) use " }
{TEXT 276 9 "functions" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
66 "Let's repeat the area example using function notation (now we use \+
" }{TEXT 283 6 "area_f" }{TEXT -1 5 " for " }{TEXT 279 1 "f" }{TEXT 
-1 9 "unction)." }}{PARA 0 "" 0 "" {TEXT -1 23 "Read this as \"to ever
y " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 39 " the function area_f as
signs the value " }{XPPEDIT 18 0 "Pi*r^2" "6#*&%#PiG\"\"\"*$%\"rG\"\"#
F%" }{TEXT -1 2 "\":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "area_f := r
 -> Pi*r^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Then the area of t
he circle with radius 3 is obtained by asking for" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "area_f(3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "
The difference between " }{TEXT 278 11 "expressions" }{TEXT -1 5 " and
 " }{TEXT 277 9 "functions" }{TEXT -1 40 " might seem confusing at the
 beginning. " }}{PARA 0 "" 0 "" {TEXT -1 100 "Let's illustrate the dif
ference in their use in connection with some other frequently used com
mands." }}{PARA 0 "" 0 "" {TEXT -1 57 "First we could plot the area as
 a function of the radius." }}{PARA 0 "" 0 "" {TEXT -1 6 "Using " }
{TEXT 299 11 "expressions" }{TEXT -1 6 " we do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "plot(area_e, r=0..4);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 3 "In " }{TEXT 300 17 "function notation" }{TEXT -1 30 " the \+
same is accomplished via " }}{PARA 0 "" 0 "" {TEXT -1 103 "(note the m
issing r; Maple knows from the function definition above that r is the
 independent variable)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(area
_f, 0..4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "However, the follow
ing also works" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(area_f(r), r
=0..4); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "This shows us that we
 can easily get an " }{TEXT 285 10 "expression" }{TEXT -1 8 " from a \+
" }{TEXT 284 8 "function" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "area_f(r);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "is
 the same as" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "area_e;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 80 "How do we go the other way, i.e., how can
 we turn an expression into a function?" }}{PARA 0 "" 0 "" {TEXT -1 
42 "This is done with the help of the command " }{TEXT 286 7 "unapply
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "Here is how it works
 (for a similar example):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "circum
ference := 2*Pi*r;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "So circumfe
rence holds the " }{TEXT 288 11 "expression " }{TEXT -1 7 "2*Pi*r." }}
{PARA 0 "" 0 "" {TEXT -1 9 "To get a " }{TEXT 287 8 "function" }{TEXT 
-1 8 "  we do " }}{PARA 0 "" 0 "" {TEXT -1 108 "(i.e., we tell Maple w
hich expression to convert to a function, and what the independent var
iable will be): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "circumf := unap
ply(circumference, r);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "We end \+
this introduction with a simple calculus problem." }}{PARA 0 "" 0 "" 
{TEXT -1 40 "First we will define a simple function  " }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 25 "f := x -> sin(x/2)*(x-1);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 24 "Let's plot the graph of " }{XPPEDIT 18 0 "f" "6#%\"f
G" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-2*Pi,2*Pi]" "6#7
$,$*&\"\"#\"\"\"%#PiGF'!\"\"*&\"\"#F'F(F'" }{TEXT -1 1 "." }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "plot(f, -2*Pi..2*Pi);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 48 "To find the intersections of the graph with the " }
{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 35 "-axis we can attempt to use \+
either " }{TEXT 289 5 "solve" }{TEXT -1 4 " or " }{TEXT 290 6 "fsolve
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 103 "(note that both func
tions cannot be blindly trusted; they often find only some - but not a
ll solutions)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(f(x)=0, x);
" }}{PARA 0 "" 0 "" {TEXT -1 61 "We missed the intersections at the en
dpoints of the interval." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Using
 fsolve on the entire interval even gives only one solution" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 31 "fsolve(f(x)=0, x, -2*Pi..2*Pi);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "However, some fine tuning helps" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fsolve(f(x)=0, x, 1/2..2);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "and (e.g.)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "fsolve(f(x)=0, x, -6.5..-6);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 46 "To locate critical points we need to know the " }
{TEXT 291 10 "derivative" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f" "6#%\"
fG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "Since we are using
 " }{TEXT 294 17 "function notation" }{TEXT -1 17 " this is done via" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(f);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 293 11 "Expressions" }{TEXT -1 37 " are differentiated with the \+
help of " }{TEXT 292 4 "diff" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dif
f(f(x), x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "We can plot the gr
aph of the derivative on the same interval as above" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 24 "plot(D(f), -2*Pi..2*Pi);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 28 "Or even both graphs together" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "plot(\{f, D(f)\}, -2*Pi..2*Pi);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 34 "Finally, we can compute integrals." }}{PARA 0 "" 
0 "" {TEXT -1 40 "The antiderivative of the derivative of " }{XPPEDIT 
18 0 "f" "6#%\"fG" }{TEXT -1 11 " should be " }{XPPEDIT 18 0 "f" "6#%
\"fG" }{TEXT -1 7 " again " }}{PARA 0 "" 0 "" {TEXT -1 63 "(note that \+
Maple doesn't bother with the additive constant):   " }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "int(D(f)(x), x);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 27 "Note that we integrated an " }{TEXT 296 11 "expression " 
}{TEXT -1 45 "above, and Maple also returned an expression." }}{PARA 
0 "" 0 "" {TEXT -1 55 "It does not seem to be possible to antidifferen
tiate a " }{TEXT 295 8 "function" }{TEXT -1 10 " in Maple " }}{PARA 0 
"" 0 "" {TEXT -1 86 "(or obtain the result of integration as a functio
n - if you want this you need to use " }{TEXT 305 7 "unapply" }{TEXT 
-1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 68 "Also note that the answer doe
s not look quite like what we expected." }}{PARA 0 "" 0 "" {TEXT -1 
26 "Another useful command is " }{TEXT 301 8 "simplify" }{TEXT -1 40 "
. It might help in situations like this." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "
It doesn't here. " }}{PARA 0 "" 0 "" {TEXT -1 73 "Since we really star
ted with a factored version of this answer let's try " }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 10 "factor(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
13 "To compute a " }{TEXT 302 17 "definite integral" }{TEXT -1 46 " we
 simply add the limits of integration as in" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "int(f(x), x=-2*Pi..2*Pi);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 78 "As a simple exercise you might want to compute the zeros \+
of the derivative of " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 7 " abov
e:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "42" 0 }{VIEWOPTS 
1 1 0 1 1 1803 }