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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 256 9 "intro.mws" }{TEXT -1 34 " Population Biology , \+
Fall 2005" }}{PARA 0 "" 0 "" {TEXT -1 31 "BIOL 763 / SCCC 411B / Math \+
523" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "OB
JECTIVES:\n\n - Enter mathematical expressions " }}{PARA 0 "" 0 ""
{TEXT -1 54 " - Plot functions and extract information from graphs" }
}{PARA 0 "" 0 "" {TEXT -1 63 " - Learn the solve and other commands f
or algebra and calculus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 ""
0 "" {TEXT -1 1842 "WHAT IS MAPLE AND HOW CAN IT BE USED? \n\nMuch of \+
your mathematical background has probably been focused on solving equa
tions and graphing functions. The manipulation of symbols and expressi
ons that you use to solve equations and investigate functions can also
be performed by software packages called computer algebra systems (CA
S). A CAS can be use to produce the exact symbolic solutions you obtai
ned by hand, the numerical approximations you found using a calculator
, and the graphs you have drawn. Maple is one of several CASs; other m
ajor examples are Mathematica, Macsyma, Derive, and MathCad. Probably
for students the greatest use of Maple is the graphical display of fu
nctions. In fact, this graphical information is often more useful than
a formula. One can also use Maple to find roots of equations, optimiz
e functions, estimate derivatives and integrals numerically, compute d
erivative and integral formulas, plot and analyze data, and visualize \+
geometric objects and transformations. \n\nWe will frequently use Mapl
e in class; it will often be useful in doing the homework; and it will
be essential for the projects. Some exam questions may require the u
se of Maple. For the most part, the choice whether or not to use Mapl
e will be yours to make--indeed, one of the goals of this course is fo
r you to learn how, and even more importantly, when, to use this power
ful tool.\n\nDOCUMENTATION and ON-LINE HELP\n\nThe on-line help for Ma
ple is very good. The help pages describe the syntax of each command, \+
a brief description of the algorithm that has been implemented, and a \+
few examples illustrating the use of the command. To obtain help on t
he command func, type: ?func. The Help Browser, an interactive interf
ace to the entire Maple help system, can be brought up by clicking on \+
Help, found at the top right of the Maple window.\n" }}{PARA 0 "" 0 "
" {TEXT -1 179 "RUNNING MAPLE\n\nEach installation is a little differe
nt. In CLS 303 we will be using Netscape to download files (as we did \+
this one). Here's some room for taking notes what to do." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1212 "GETTING STARTED\n\nYou have a
lready learned how to open up a worksheet: after opening Maple, positi
on the cursor arrow on the menu item File and click the left button on
ce. You can select New to obtain a fresh, empty worksheet. Or you can \+
select Open to choose an existing worksheet (the name will always end \+
with .mws), either by double clicking on the name or by selecting the \+
name and then Load. Once in a worksheet the current location is marked
by a vertical bar. You can move around by moving the mouse and then c
licking the left button once, by using the arrow keys, or by using the
scroll bar up and down arrows on the right side of this window follow
ed by finer adjustment with the mouse. A command is entered and execut
ed by positioning the cursor anywhere on the desired line and pressing
Return. Several worksheets can be open simultaneously, and if you us
e the Window option to arrange them side by side, this can be quite us
eful because you can cut and paste between them. They do, however, sha
re the computational engine, so beware: if x = 3 in one worksheet it w
ill also be 3 in all the others!\n\nIt is probably a good idea to begi
n EVERY worksheet that you create with the following two commands. " }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 14 "with(plots): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 238 "BASIC FACTS\n\nMaple is a comp
uter language; it cannot read your mind. You need to learn how to comm
unicate with Maple.\n\nThe basic symbols:\n\n - assignments are made w
ith := (plain = has a different meaning) -- think of this as giving \+
the " }{TEXT 257 5 "value" }{TEXT -1 31 " of the right hand side to th
e " }{TEXT 258 4 "name" }{TEXT -1 882 " that appears on the left hand \+
side\n - every command is terminated by a semi-colon ( ; ) or colon ( \+
: ) -- with the latter the computation is done, but the result is not \+
displayed\n - the percent symbol ( % ) refers to the result of the \+
immediately preceding computation (which might not have been the previ
ous line)\n - Maple is case sensitive -- that is, the names x and X ar
e different, pi and Pi are not the same thing\n - \{ \} -- set notat
ion (mostly used in the context of plotting a bunch of functions simul
taneously)\n - .. as in a .. b -- this is how Maple indicates the in
terval [a, b], that is, the real numbers from a to b\n - ? or help -- \+
a request to Maple for information\n\nMathematical functions have thei
r standard names, or ones that are easily guessed:\n + (plus) , - (mi
nus) , * (times) , / (divided by) , ^ (raised to the power) , sin, cos
, tan, abs, sqrt, ...\n" }}{PARA 0 "" 0 "" {TEXT -1 267 "In the follow
ing lines, use paper and pencil to first predict what you think Maple \+
will do. Then execute the command and see what actually happens! If y
ou haven't seen it before the ! notation means multiply all the number
s down to 1; for example 4!=(4)*(3)*(2)*(1)." }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "N:= 4 * 6 + 12 / 6 - 1 \+
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "power:= (-3)^3 ;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "abs( % );" }}}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 4 "Pi ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 78 "v:= sin( Pi / 4 ) : # Note that lines that end with : differ f
rom those with ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "v;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "tan( -Pi / 2 ) ;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "3 / ( 5 - sqrt( N ) ) ;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "4! ; " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 5 "50! ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 24 "Digits:= 70: evalf(Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 35 "Digits:= 10: e:= exp(1); evalf(e);" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 25 "f:= 2 * x^2 - 7 * x + 5 ;" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 20 "solve( f = 0 , x ) ;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "factor( f ) ;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 27 "solve( z = 3 * t + 4 , t );" }}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 38 "solve( a * x^2 + b * x + c = 0 , x ) ;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "solve( 25 * exp( k * 400 ) =
50 , k );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "evalf( % ); #
Hint: this is short for \"floating point evaluation.\"" }}}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 621 "\nNow enter some commands of your own! T
o produce the input prompt > you can use the Insert key on the menu \+
bar, followed by Execution Group and either After Cursor or Before Cur
sor (or, from the keyboard: Alt-i followed by g (for group) and either
a (for after) or b (for before)). Alternatively, use the [> button
in the menu bar if you want the cursor after the present location. B
e generous with the space bar to make your commands easy to read, and \+
to edit (modify after the fact). Go back to the tangent calculation ab
ove (remember how to reposition the cursor!) and change it to compute \+
the tangent of Pi / 3." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 310 "P
LOTTING\n\nGraphs of functions are produced by the plot command. In it
s simplest form, plot needs to know the function to be plotted and the
range of values for the independent variable. Note that a .. b is Map
le's way of describing the interval [a, b]. Again, try to predict the
output before tapping Return! " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "
plot( 3 * t - 2 , t = -3 .. 6 ) ; # WAIT FOR THE GRAPH TO APPEAR!!" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot( sin(3*x), x = -2*Pi \+
.. 2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "y:= 3.5 * exp \+
( 0.2 * t );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot( y, t \+
= 0 .. 10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "L:= ln(y);"
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot( L, t = 0 .. 10);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 203 "Select a couple of points from the straingt line graph
, and compute the slope. Note you can TAB to the %? places and write o
ver the highlighted characters. What does this slope have to do with \+
anything?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "t1:= %? ; L1:
= %? ; t2:= %? ; L2:= %? ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 30 "slope:= (L2 - L1) / (t2 - t1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "We illustrate how to plot data \+
points, and the usefulness of the logarithm function." }}{PARA 0 "> "
0 "" {MPLTEXT 1 0 159 "pop:= [ 508, 711, 912, 1131, 1590, 1811, 2015, \+
2249, 2509, 3008, 3610, 3967 ];\nyr:= [ 1650, 1750, 1800, 1850, 1900, \+
1920, 1930, 1940, 1950, 1960, 1970, 1975 ];" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 288 "The map command is ver
y flexible. It gives you a way to systematically do the same thing to \+
a whole list of numbers. The zip command does just what you expect: i
t zips together two lists in alternation! An alternative is to use t
he seq (sequence) command, doing the zipping \"manually\"." }}{PARA 0
"> " 0 "" {MPLTEXT 1 0 36 "log10pop:= evalf( map( log10, pop));" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "lnpop:= evalf( map( ln, pop)); \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "adjustedyr:= map( n -> n - 1650
, yr );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "data:= zip( (a,b
) -> [op( [a,b] )] , adjustedyr, pop );" }}{PARA 0 "> " 0 "" {MPLTEXT
1 0 66 "log10data:= zip( (a,b) -> [op( [a,b] )] , adjustedyr, log10po
p );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "lndata:= zip( (a,b) -> [op(
[a,b] )] , adjustedyr, lnpop );" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 55 "data2:= [ seq( [adjustedyr[k], pop[k]], k = 1 .. 12 )
];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 272 "Finally we do the actual plotting. There is a large variety of
styles that you can select, either in the command itself, or after th
e fact from the menu which you can bring up by clicking on the graph r
egion itself. Try changing style = POINT to style = LINE for example. \+
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot( data , style = POINT , ti
tle = `Population vs. time` );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 89 "plot( data2, style = LINE, title = `Population vs. time`, thic
kness = 2, color = green );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
66 "plot( log10data, style = POINT, title = `Log10 of pop vs. time`); \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot( lndata, style = P
OINT, title = `Ln of pop vs. time`);" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 341 "We have two ways to make
logarithmic plots. We can either take log of the data values, as we \+
did above, or we can use the built-in Maple command. The output is lab
eled differently, as you will see. To make the comparison easier we wi
ll give the plots names (NOTE: named plot commands must end with a co
lon!) and display them side by side. " }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 67 "PlotA:=plot(log10data, title=`Log10 of pop vs. time`,
thickness=2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "PlotB:=logplot(da
ta, title=`Logplot of pop vs. time`, thickness=2):" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 55 "Digits:= 4: # Needed to avoid bug in disp
lay( array ) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display( array( [
PlotA, PlotB] ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Digit
s:= 10:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 47 "Here are a few more functions and their graphs." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f:= sin(x); g:= 2 * x^2 ; \+
p:= f * g ;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "h := cos ( (1/4) * x
) ; s:= 0.2 * sin(2*x) + h ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 125 "p:= p; plot( p, x = -3 .. 3); # p:= p; is not necessary but i
t is a handy way for us to remember what we are plotting here." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "# For display purposes it i
s a good idea to put a blank (or comment, as here) command line after \+
a plot command line." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 340 "\nPositi
on the cursor on a point on the graph and click the left button. The n
umbers that appear on the upper-left third line of the menu bar are th
e coordinates of the current location of the cursor. Use this techniqu
e to identify the maximum and minimum values of p(x) on the interval [
-3, 3], and the x-values at which these are found. " }}}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 138 "\nTo plot more than one function in a si
ngle window, just specify the functions to be plotted as a set. Can y
ou tell which graph is which?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 59 "h:= h; s:= s; plot( \{ h , s \}, x = - 6* Pi .. 6 * Pi ) ;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "g:= g; p:= p; plot( \{ g
, - g , p \} , x= -16 .. 16 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1
0 "" }}{PARA 0 "" 0 "" {TEXT -1 195 "The following plotting commands a
re just for fun. Since we have already assigned a value to v, if we w
ant to use this letter over again, we have to reset v to be just plain
\"v\" and not a number." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "
v;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "v:= 'v';" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "fieldplot( [v, -sin(w) - 0.1*v], w \+
= -2 .. 8, v = -2 .. 2, grid=[20, 20], arrows=SLIM, color=BLUE);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 775 "HOUSECLEANING\n\nYou hav
e probably noticed that plots make scrolling up and down very slow. Go
back to a plot and click on it. A box enclosing the plot will appear
and the menu bar will change--the keys Style, Axes, and Projection ar
e very useful for making after the fact modifications to the appearanc
e of the plot--try them! Also you can expand or contract the bounding
box by dragging the little blips on the edges. To get rid of the plo
t entirely simultaneously hold down the Control (Cntrl) key and tap De
lete (Del); or use the menu bar Edit and Delete Paragraph. There's als
o a way (under Options) to get the plots to show up in new windows, ou
t of the way, but this also has disadvantages. Eventually you'll find \+
the display mode that you prefer, but this can wait. " }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 229 "DERIVATIVES AND INTEGRALS \n\n
Maple can do derivatives and integrals. For this, we first load the s
tudent package. Try to predict the effect of each command before mash
ing Return. After the given problems, try some of your own. " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 32 "f:= 3 * x^ 4 - 2 * x + 5 / x^3 ;" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff( f , x ) ;" }}}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 25 "diff( Pi * R^2 * H , R );" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 17 "F:= t * sin(t) ;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "deriv_of_F:= diff( F , t );" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 30 "easy_integral:= Int( f , x ) ;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "value( easy_integral ) ;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "g:= z^2 * exp(- z^3 ) ;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot( g , z = 0 .. 2 ) ;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "medium_integral:= Int( g ,
z = 0 .. 2 ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "value( m
edium_integral );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(
% ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "y;" }}}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 159 "Notice that y already has a value; hence
we can not use it as a variable anymore, unless we tell Maple to forg
et its value and to reestablish it as a variable." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 8 "y:= 'y';" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 48 "hard_integral:= Int( 1 / ( y * (10 - y )) , y );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "value( hard_integral );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 489 "CONGRATULATIONS! You hav
e just completed your first Maple worksheet. If you wish to save it se
lect File in the menu bar, and then either Save (which will destroy th
e original version of intro.ms) or Save As... (which will leave intro.
mws untouched, and will request a new name for this modified version).
Worksheet names should always end with .mws; also unless you really n
eed the output it is usually a good idea to remove it before saving --
to do this, select Edit, then Remove Output." }}{PARA 0 "> " 0 ""
{MPLTEXT 1 0 62 "# Have you saved your work? Check out the options un
der File." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "quit # The o
nly Maple command that doesn't need a semicolon!" }}}}{MARK "1 1 0"
31 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }