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{SECT 0 {PARA 18 "" 0 "" {TEXT 327 34 "DISCOVERING MATHEMATICS WITH MA
PLE" }{TEXT 326 0 "" }}{PARA 258 "" 0 "" {TEXT 328 77 "An Interactive \+
Exploration for Mathematicians, Engineers, and Econometricians" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 259 28 "Chapte
r 1. A Tour of Maple V" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0
"" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "Just a Maple Session" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 247 "In this \+
session we shall run through a number of examples in order to show the
use of Maple as a calculator. But we shall not attempt to give a comp
lete picture of the Maple commands we use. The details will be left to
the introductory worksheets " }{TEXT 312 11 "TourW1a.mws" }{TEXT -1
5 " and " }{TEXT 311 11 "TourW1b.mws" }{TEXT -1 327 ". This session is
merely meant to provide an overall view of the possibilities, without
giving much attention to individual commands and all their special fe
atures. So, immediate and complete understanding is unlikely and not s
trictly required. In the worksheets that follow this session, more att
ention is paid to the details." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Numerical Calculations" }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 228 "We shall adopt
the usual convention that Maple commands, constants and variables are
recognizable by the font and the style. In the text, a Maple command,
a variable, expression or parameter is usually printed in boldface, l
ike " }{TEXT 260 10 "expression" }{TEXT -1 207 ". Maple input lines ar
e preceded by a Maple prompt (>) and the Maple instructions are always
given in the typewriter font. Finally, Maple variables in an output r
egion are printed in italics, like variable. " }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 188 "After starting a new wor
ksheet, the cursor (|) blinks directly to the right of the Maple promp
t. This means that Maple waits for an instruction to be given at this \+
position. We might try " }{TEXT 262 5 "?trig" }{TEXT -1 187 " in ord
er to find out what Maple knows about the trigonometric functions. To \+
indicate that we have finished typing an instruction and wait for Mapl
e to act on it, we hit the key." }}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "?trig" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "A help page opens with
information on the trigonometric functions. From this information we \+
learn that the Maple command " }{TEXT 272 3 "sin" }{TEXT -1 88 " perta
ins to the ordinary sine function. Verification of a few values may co
nvince you. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 58 "sin(0),sin(Pi/6), sin(Pi/4), sin(Pi/3),sin(Pi/2), \+
sin(Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 331 "Yes, we recognize t
he correct values corresponding to the familiar arguments of the sine \+
function. The commas are used to separate the Maple answers and place \+
them in a single row instead of each on an output row of its own. This
Maple output row of sine values is made into a so-called 'list' by pu
tting square brackets around it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "whattype([%]);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 71 "Every complete Maple instruction must en
d in a semicolon (;) or colon (" }{TEXT 265 1 ":" }{TEXT -1 164 "). Th
e semicolon causes the result of the instruction to appear as output o
n the screen. This output is suppressed when the colon is used. When t
he closing symbol (" }{TEXT 266 1 ";" }{TEXT -1 4 " or " }{TEXT 267 1
":" }{TEXT -1 443 ") is omitted, nothing happens after hitting the key, simply because Maple is waiting for the instruction to be co
mpleted. Newcomers to the Maple system often make the mistake of repea
ting the instruction in such a situation, with the most likely result \+
of an error message (syntax error) being issued, because Maple tries t
o read the double instruction as a single one. The best advise is to t
ype a ; when Maple appears to be waiting. " }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The ditto symbol or percentage \+
sign (" }{TEXT 306 1 "%" }{TEXT -1 90 ") refers to the last instructio
n processed by Maple. This symbol should be repeated once (" }{TEXT
307 3 "%%)" }{TEXT -1 11 " or twice (" }{TEXT 273 3 "%%%" }{TEXT -1
140 ") if you wish to refer to the one but last Maple instruction or t
he one before that, respectively. To go back any further requires Mapl
e's " }{TEXT 274 7 "history" }{TEXT -1 41 " command; see Exercise 1 \+
of section 1.5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 58 "Further, in the previous lines, the use of constants like
" }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 48 " (=Pi) catches the eye,
and also the fact that " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#
" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "sqrt(3)" "6#-%%sqrtG6#\"\"$" }
{TEXT -1 204 " remain unevaluated. Apparently, Maple does not automat
ically replace such constants by approximating decimal fractions, but \+
instead keeps working with the symbolic (and thus exact) representatio
ns like " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 3 ", " }{XPPEDIT
18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 6 " and " }{XPPEDIT 18
0 "sqrt(3)" "6#-%%sqrtG6#\"\"$" }{TEXT -1 45 ". Consider the following
symbolic expression:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 51 "symbolic_expression := 19*Pi^2 - ln(3)/(1
+sqrt(2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 185 "If we happen to n
eed a numerical value for this expression, Maple obliges on request by
evaluating it to any precision required, that is to say, rounded to a
s many decimals as we wish. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%,50);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 17 "The instruction " }{TEXT 275 11 "evalf(
%,50)" }{TEXT -1 15 " - this means: " }{TEXT 268 1 "f" }{TEXT -1 14 "l
oating point " }{TEXT 269 4 "eval" }{TEXT -1 195 "uation; a floating p
oint number, 'float' briefly, is a real number represented by a decima
l fraction with a fixed but freely chosen number of digits - produces \+
a numerical value for the quantity " }{TEXT 270 19 "symbolic_expressio
n" }{TEXT -1 121 " rounded to 50 decimal digits. The same result can b
e achieved by clicking the right mouse button on the Maple output of \+
" }{TEXT 313 20 "symbolic_expression " }{TEXT -1 14 "and selecting " }
{TEXT 314 11 "Approximate" }{TEXT -1 26 " in the menu that pops up." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The inst
ruction " }{TEXT 276 12 "evalf(%, 50)" }{TEXT -1 53 " has no effect \+
on the exact value of the variable " }{TEXT 271 19 "symbolic_express
ion" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 20 "symbolic_expression;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 47 "Maple knows a few mathematical constants, like " }
{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "gamm
a" "6#%&gammaG" }{TEXT -1 132 ", the Euler constant. The names of thes
e constants are protected, so that it is impossible to accidently assi
gn new values to them. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Pi := 3.14159;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0
"" 0 "" {TEXT -1 103 "You may assign new values or expressions to such
protected names but only after using Maple's command " }{TEXT 277 9
"unprotect" }{TEXT -1 34 "; this is not really recommended. " }}{PARA
0 "" 0 "" {TEXT -1 31 "Another well-known constant is " }{XPPEDIT 18
0 "e" "6#%\"eG" }{TEXT -1 36 ", the base of the natural logarithm " }
{TEXT 278 2 "ln" }{TEXT -1 88 ". This constant is not a Maple constant
. In fact, Maple knows the mathematical constant " }{XPPEDIT 18 0 "e"
"6#%\"eG" }{TEXT -1 4 " as " }{TEXT 279 6 "exp(1)" }{TEXT -1 47 ". To \+
avoid having to type the full expression " }{TEXT 280 6 "exp(1)" }
{TEXT -1 27 " every time we wish to use " }{XPPEDIT 18 0 "e" "6#%\"eG
" }{TEXT -1 19 " , we could define " }{XPPEDIT 18 0 "e" "6#%\"eG" }
{TEXT -1 62 " to have this value throughout the Maple session. But bew
are, " }{XPPEDIT 18 0 "e" "6#%\"eG" }{TEXT -1 61 " is not a protected \+
constant: we can redefine it if we like. " }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "e := exp(1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(e), ln(e);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "e := 2.718; ln(e);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 74 "It is possible to protect a newly define
d expression in the following way:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "e := exp(1): protect('e');"
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "e := 2.718;" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "ln(e);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 58 "Without doubt you remember that the mathematical constant
" }{XPPEDIT 18 0 "e" "6#%\"eG" }{TEXT -1 50 " is also equal to the li
mit value of the sequence " }{XPPEDIT 18 0 "(1+1/n)^n" "6#),&\"\"\"F%*
&F%F%%\"nG!\"\"F%F'" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "n" "6#%\"nG" }
{TEXT -1 78 " tends to infinity. Let us check to see if this is known \+
to Maple. Note that " }{TEXT 281 8 "infinity" }{TEXT -1 28 " is also
a Maple constant. " }}{PARA 0 "" 0 "" {TEXT -1 40 "As you will see be
low, we first use the " }{TEXT 282 5 "Limit" }{TEXT -1 331 " command w
ith capital L. Use of a capital letter is not accidental but an indic
ation of the so-called `inert' form of a Maple command, which is retur
ned unevaluated, or rather, evaluated to its name. The use of the lowe
r case command causes the value (in this case of the limit) to appear \+
as output. We shall return to this later." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Limit((1+1/n)^n,n=inf
inity) = limit((1+1/n)^n,n=infinity);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 39 "rhs(%) - e; # rhs means right-hand side" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "evalb(rhs(%%) - exp(1) = 0);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "This confirms that Maple knows \+
about this limit. " }}{PARA 0 "" 0 "" {TEXT -1 130 "We repeat that Map
le distinguishes between lower case and upper case, but not only in Ma
ple commands, also in general. Therefore, " }{XPPEDIT 18 0 "e" "6#%\"e
G" }{TEXT -1 153 " and E are totally different names with possibly tot
ally different meanings. In fact, E is a new name, so that any value c
an be assigned to it, whereas " }{TEXT 325 1 "e" }{TEXT -1 3 " = " }
{TEXT 283 6 "exp(1)" }{TEXT -1 20 " is still protected." }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "E := 2.
178;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "In one of the preceding
input lines we used the " }{TEXT 315 5 "evalb" }{TEXT -1 21 " command
. The letter " }{TEXT 316 1 "b" }{TEXT -1 42 " comes from the name Boo
le, so that with " }{TEXT 258 5 "evalb" }{TEXT -1 64 " Boolean expre
ssions are evaluated with two possible outcomes: " }{TEXT 259 4 "true
" }{TEXT -1 5 " and " }{TEXT 260 5 "false" }{TEXT -1 38 ". Both are Ma
ple constants.The sharp (" }{TEXT 317 1 "#" }{TEXT -1 68 ") is used by
Maple to add comments to an input line. Text following " }{TEXT 318
1 "#" }{TEXT -1 214 " is ignored by Maple. In the current worksheet en
vironment this option is not as useful as it used to be in previous Ma
ple releases, because since Release 4 plain text can be put in text r
egions anywhere we wish. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 651 "You may have noticed that there is a difference b
etween the equality sign (=) and the assignment symbol (:=). This is n
o surprise for those who are familiar with Pascal. The symbol := is us
ed to assign a value (or a name) to an arbitrary string of symbols, bu
t remember, no interspacing. The assigned value faces the side of the \+
equality sign. It is this feature that allows Maple (and any other CA \+
system for that matter) to do calculations with symbols, where purely \+
numerical packages can only handle numbers. This distinguishing featur
e is an essential characteristic of a CAS and that is why it will run
through all the forthcoming discussions. " }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 ""
{TEXT -1 21 "Symbolic Calculations" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}
{PARA 4 "" 0 "" {TEXT 305 194 "We continue our tour with examples in w
hich symbols are manipulated like numbers. Apart from numbers, other o
bjects like functions, polynomials and matrices can take part in these
computations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "expand((x+y)^4);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 348 "This is a familiar result; you recognize Newton's Binomi
um, don't you? If the exponent 4 is increased to 100, precisely 101 te
rms will be printed to the screen instead of a mere five. This is rath
er a lot, acknowledging that we most likely are not really interested \+
in every individual coefficient. Therefore, it seems best to suppress \+
the output. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 18 "expand((x+y)^100):" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 14 "coeff(%,x,31);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
216 "It is good practice not to accept any result face value, we check
ed a more or less random coefficient, say number 31. Realizing that we
are dealing with binomial coefficients, the outcome should be - apart
from the " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 44 "-power of cour
se - the binomial coefficient " }{TEXT 257 16 "binomial(100,31)" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "Let us verify this. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
33 "evalb(binomial(100,31) = %/y^69);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 31 "This is rather convincing. Also" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "100!/31!/69!;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "should give the same result, ag
ain neglecting the " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 64 "-powe
r. Factorizing this large integer into prime numbers gives:" }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "if
actor(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 197 "Clearly, there ar
e exactly five factors 2 occurring in this decomposition. Did Maple do
a good job? We may gain confidence by counting the number of factors \+
2 occurring in the binomial coefficient " }{TEXT 308 16 "binomial(100,
31)" }{TEXT -1 52 " in a different way, and comparing the two results.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 56 "Sum(floor(100/2^i),i=1..6) = sum(floor(100/2^i),i=1..
6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "What does the answer mea
n? According to Maple, 97 is the precise number of factors 2 occurring
in 100! (100 factorial). Recall that floor(" }{XPPEDIT 18 0 "x" "6#%
\"xG" }{TEXT -1 49 ") is the largest integer less than or equal to \+
" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 20 " . Hence, floor(100/" }
{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 20 ") is the number of " }
{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 35 "-multiples not larger than 1
00. As " }{XPPEDIT 18 0 "2^6" "6#*$\"\"#\"\"'" }{TEXT -1 9 " < 100 < \+
" }{XPPEDIT 18 0 "2^7" "6#*$\"\"#\"\"(" }{TEXT -1 32 ", which implies \+
that floor(100/" }{XPPEDIT 18 0 "2^i" "6#)\"\"#%\"iG" }{TEXT -1 15 " \+
) = 0 for all " }{XPPEDIT 18 0 "i = 7;" "6#/%\"iG\"\"(" }{TEXT -1 33 "
or larger, the summation index " }{XPPEDIT 18 0 "i" "6#%\"iG" }
{TEXT -1 82 " does not need to run beyond 6. As a result, the number o
f factors 2 occurring in " }{TEXT 309 16 "binomial(100,31)" }{TEXT -1
4 " is " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 61 "Sum(floor(100/2^i) - floor(31/2^i) - floor(69/2^i),i=
1..6) = " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "sum(floor(100/2^i) - fl
oor(31/2^i) - floor(69/2^i),i=1..6);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 8 "Like in " }{TEXT 284 5 "Limit" }{TEXT -1 253 " before, we \+
used the inert form of the Maple command (capital first letter) in the
left-hand side of the equality to enhance the readability. As a conse
quence, evaluation takes place to the name, so that the command's full
name is printed to the screen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 230 "You probably won't be surprised to learn
that Maple is able to differentiate and integrate. Let us choose a di
fferentiable function (any will do), and let us instruct Maple to calc
ulate its derivative and second order derivative. " }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f := exp(x*l
n(1+x^2) - x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "`f'` := d
iff(f,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "`f''` := diff(
f,x$2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 187 "Clearly, the form o
f the second order derivative is rather involved. Maple does not have \+
any trouble plotting such complicated functions. In the figure below \+
the graphs of the function " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1
22 " and its derivative " }{XPPEDIT 18 0 "`f'`" "6#%#f'G" }{TEXT -1
31 " are plotted on the interval [" }{XPPEDIT 18 0 "-3" "6#,$\"\"$!\"
\"" }{TEXT -1 48 ",1] and displayed together in a single picture. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "Note that
those names which have to be taken literally should be surrounded by \+
left quotes as in `" }{XPPEDIT 18 0 "`f'`" "6#%#f'G" }{TEXT -1 8 " ` a
nd `" }{XPPEDIT 18 0 "`f\"`" "6#%#f\"G" }{TEXT -1 284 "`. This is nece
ssary here, because, as we shall see, the single quote ' has a special
meaning in the Maple language - this is also the case for the double \+
quote \" in Release 4. So, if we wish to use quotes as part of a name \+
(or string), we have to put left quotes around this string. " }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "pl
ot(\{f,`f'`\},x=-3..1,color=black);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 64 "It is not so difficult to distinguish the two graphs, be
cause " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 47 " has positive val
ues only and its derivative " }{XPPEDIT 18 0 "`f'`" "6#%#f'G" }{TEXT
-1 73 " has positive and negative values. Judging by the graphs, the \+
function " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 60 " has a local \+
maximum and a local minimum on the interval [" }{XPPEDIT 18 0 "-3" "6#
,$\"\"$!\"\"" }{TEXT -1 193 ",1]. Indeed, the derivative vanishes at t
wo different points which are positioned symmetrically with respect to
the origin. This symmetry is obvious from the analytical form of the \+
derivative: " }{XPPEDIT 18 0 "`f'`" "6#%#f'G" }{TEXT -1 1 "(" }
{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 2 ")/" }{XPPEDIT 18 0 "f" "6#%\"
fG" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 20 ") is a \+
function of " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }{TEXT -1 2 ". "
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 25 "fsolve(`f'` = 0,x=-3..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
12 "We see that " }{TEXT 285 6 "fsolve" }{TEXT -1 78 " (the letter f \+
again comes from 'float') finds only one of the two zeros of " }
{XPPEDIT 18 0 "`f'`" "6#%#f'G" }{TEXT -1 208 ". This is not a mistake \+
in the Maple program as you might think, but an unavoidable consequenc
e of the numerical process of calculating roots of equations. At a lat
er stage we shall try to clarify this point." }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "minimum, maximum =
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evalf(subs(x=-%,f)),evalf(subs
(x=%,f));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 159 "To verify Maple's
computation of the derivative, we could try to reverse the process an
d calculate the indefinite integral by giving the following instructio
n:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 26 "Int(`f'`,x) = int(`f'`,x);" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 153 "Apparently, this is quite easy, as Maple recovers the \+
original function very quickly. Indefinite integration - or finding a \+
primitive - of the function " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1
51 " itself may prove much harder, if not impossible. " }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(f,x)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "As was to be expected, Ma
ple is unable to find a primitive function for " }{XPPEDIT 18 0 "f" "
6#%\"fG" }{TEXT -1 84 " and shows this by echoing the integration for
mula. Note that the first letter of " }{TEXT 286 3 "int" }{TEXT -1
232 " is lower case. Therefore, the failure to produce a result can n
ot be attributed to an inert command. In fact, Maple's lack of success
does not come as a surprise, because it is impossible to express a pr
imitive for this function " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1
135 " in terms of elementary functions or other special functions. On
the other hand, Maple is able to calculate the definite integral of \+
" }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 21 " over the interval [" }
{XPPEDIT 18 0 "-3" "6#,$\"\"$!\"\"" }{TEXT -1 61 ",1] without difficul
ty, if instructed to proceed numerically." }}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int(f,x=-3..1) = eval
f(Int(f,x=-3..1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The capit
al letter " }{TEXT 319 1 "I" }{TEXT -1 4 " of " }{TEXT 288 3 "Int" }
{TEXT -1 279 " in the right-hand side of the last input line prevents \+
Maple from first trying to integrate symbolically before finally turni
ng to numeric integration methods. This makes sense, because we alread
y know that Maple is unable to integrate this function symbolically. T
he procedure " }{TEXT 289 9 "evalf/int" }{TEXT -1 119 " from the Maple
library has the same objective. It is not available at startup, but h
as to be read from the library by " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "### WARNING: persistent stor
e makes one-argument readlib obsolete\nreadlib(`evalf/int`);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "Again, note the back quotes; h
ere the forward slash (/) is not the symbol for 'division', but merely
a simple connector for joining the two commands " }{TEXT 290 5 "evalf
" }{TEXT -1 5 " and " }{TEXT 291 3 "int" }{TEXT -1 143 ". In other wor
ds, it is part of the name of the procedure. The actual advantage of t
his procedure over the sequence of the individual commands " }{TEXT
292 3 "int" }{TEXT -1 5 " and " }{TEXT 293 5 "evalf" }{TEXT -1 146 " i
s that it gives more control over the precision and the numerical inte
gration procedure we wish Maple to use. In fact Maple automatically ca
lls " }{TEXT 294 9 "evalf/int" }{TEXT -1 15 " when we apply " }{TEXT
295 5 "evalf" }{TEXT -1 7 " after " }{TEXT 296 3 "Int" }{TEXT -1 17 " \+
(with a capital " }{TEXT 320 1 "I" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "To complete this Maple
session we give a few examples of the use of matrices under Maple. Wi
th this in mind, we first load the " }{TEXT 298 6 "linalg" }{TEXT -1
498 " package of linear algebra procedures. A Maple package is just a \+
collection of Maple procedures written for a common type of applicatio
n. In order to avoid making the Maple system top-heavy by including to
o many instantly available procedures in the default Maple library, a \+
selection was made of generally applicable routines, that is to say, o
f procedures that are useful in more than one specific situation. Proc
edures of a less general nature are grouped in special packages. The M
aple command " }{TEXT 299 6 "with()" }{TEXT -1 260 ", in which the nam
e of the particular package is put between the parentheses, is used to
load all procedures of this package simultaneously. During the active
session all procedures of the package thus loaded into the Maple libr
ary remain instantly available. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 29 "These warnings mean that the " }{TEXT
300 4 "norm" }{TEXT -1 5 " and " }{TEXT 301 5 "trace" }{TEXT -1 85 " p
rocedures which are part of the standard Maple library are replaced by
the special " }{TEXT 302 6 "linalg" }{TEXT -1 27 " routines of the sa
me name." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 39 "A := matrix([[1,2,3],[2,3,5],[3,4,6]]);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 46 "Let us compute the determinant of the ma
trix " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 48 ", its trace, and it
s characteristic polynomial. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "`determinant of A` := det(A)
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "`trace of A` := trace(A);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "`characteristic polynomial of A` :=
charpoly(A,lambda);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "The val
ues of the determinant and the trace agree with the coefficients of th
e characteristic polynomial in as much as " }{XPPEDIT 18 0 "-1 = -det(
A);" "6#/,$\"\"\"!\"\",$-%$detG6#%\"AGF&" }{TEXT -1 25 " and the coeff
icient of " }{XPPEDIT 18 0 "lambda^2" "6#*$%'lambdaG\"\"#" }{TEXT -1
5 " is " }{XPPEDIT 18 0 "-10 = -trace(A" "6#/,$\"#5!\"\",$-%&traceG6#
%\"AGF&" }{TEXT -1 25 ". With the Maple command " }{TEXT 303 6 "fsolve
" }{TEXT -1 62 " we can compute all zeros of the characteristic polyno
mial of " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 30 ", including the c
omplex ones. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 57 "fsolve(`characteristic polynomial of A` ,lambda,
complex);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "It is clear that \+
" }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 96 " has one real and one pai
r of complex conjugate eigenvalues. Be warned that the Maple constant \+
" }{TEXT 304 1 "I" }{TEXT -1 112 " is not used as the standard notat
ion for the general identity matrix, but instead denotes the complex n
umber " }{XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT -1 3 " = " }{XPPEDIT 18
0 "sqrt(-1)" "6#-%%sqrtG6#,$\"\"\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "The matrix " }
{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 68 " is invertible, because its \+
determinant does not vanish, and hence " }{XPPEDIT 18 0 "A" "6#%\"AG
" }{TEXT -1 74 " is non-singular. Let us verify that Maple computes th
e inverse correctly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 45 "evalm(A)*inverse(A) = multiply(A,inverse(
A));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA
0 "" 0 "" {TEXT -1 72 "There is no doubt about the correctness of this
result. By using evalm (" }{TEXT 256 1 "m" }{TEXT -1 6 "atrix-" }
{TEXT 310 4 "eval" }{TEXT -1 77 "uation) the matrix and all its elemen
ts are printed on the screen; the input " }{TEXT 258 2 "A;" }{TEXT -1
23 " echoes only the name " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1
50 " and thus hides the matrix elements. The asterisk " }{TEXT 297 1 "
*" }{TEXT -1 122 " is the symbol for multiplication. However, it can n
ot be used for matrix multiplication, for which the composite symbol \+
" }{TEXT 287 2 "&*" }{TEXT -1 60 " is reserved. Here we have applied t
he equivalent procedure " }{TEXT 259 8 "multiply" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We close \+
by mentioning that a Maple session can be ended with either one of the
instructions " }{TEXT 260 4 "quit" }{TEXT -1 2 ", " }{TEXT 261 4 "don
e" }{TEXT -1 5 ", or " }{TEXT 262 4 "stop" }{TEXT -1 104 ". A closing \+
semicolon or colon is not needed. Of course, a Maple session can also \+
be ended through the E" }{TEXT 264 1 "x" }{TEXT -1 17 "it option in th
e " }{TEXT 263 1 "F" }{TEXT -1 9 "ile menu." }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
159 "We have come to the end of this Maple session. We hope that this \+
introductory Maple session has whetted your appetite. You can now work
through the worksheets " }{TEXT 321 11 "TourW1a.mws" }{TEXT -1 5 " an
d " }{TEXT 322 11 "TourW1b.mws" }{TEXT -1 67 ", and try your hand at t
he corresponding assignments in worksheets " }{TEXT 323 11 "TourA1a.mw
s" }{TEXT -1 5 " and " }{TEXT 324 11 "TourA1b.mws" }{TEXT -1 1 "." }}}
}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2
33 1 1 }