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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 11 "Maple Lab 6" }}{PARA 0 "
" 0 "" {TEXT -1 227 "Suppose we wish to solve the following problem on
 Maple: The rate of change of the population of Canada is proportional
 to its size.  In 1990 the population was 2.5 million.  It doubled in \+
1995.  What is the population in 1998?" }}{PARA 0 "" 0 "" {TEXT -1 
129 "The phrase \"rate of change... is proportional to its size\" tell
s us we will be using an exponential equation of the form y(t) = A" }
{XPPEDIT 18 0 "exp(kt);" "6#-%$expG6#%#ktG" }{TEXT -1 96 " , where y(t
) gives us the population in millions after t years.  Let's put this i
nto Maple.    " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "y:=t->A*e
xp(k*t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "A is our initial pop
ulation, i.e. our population at time t = 0.  It is given to be 2.5 (in
 millions)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "A:=2.5;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Let's see what our function looks \+
like so far:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "y(t);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 196 "We know that 5 years after our st
arting year (1990), the population doubled from what it was originally
, i.e. y(5) = 2 * A (remember, A is our original population). Let's se
e what this looks like." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "y
(5)=2*A;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "So we see we can solv
e for k in the above equation, right?" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "solve(y(5)=2*A,k);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 80 "So that's our k.  Let's assign k this value and see how our fun
ction is looking." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "k:=%;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "y(t);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 78 "Now we can figure out the population in 1998, 8 ye
ars after our starting year:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 5 "y(8);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "This is the populat
ion in millions in the year 1998." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 713 "Now let's switch gears and see how to do
 parametric equations in Maple.  First, what's the point of parametric
 equations?  We are used to seeing curves expressed as functions that \+
look like y = f(x).  But not all curves can be expressed as functions \+
of x, i.e. curves that fail the vertical line test.  So, instead of ha
ving a curve where y is expressed as a function of x, suppose we have \+
both x and y as functions of a third variable t: x = f(t), y = g(t).  \+
As t varies, we get different points (x,y) that we can plot and obtain
 a curve from.  That's it.  That's parametric equations.  All we need \+
to plot these guys in Maple is 2 extra brackets.  For example, suppose
 we wish to graph the parametric equations " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 51 "x = 3 sin(3t) + sin(4t)     y
 = 3 cos(2t) + cos(7t)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 12 "from 0 to 2 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 
-1 131 " .  Again, this just means we plot all (x,y) points of the for
m (3 sin(3t) + sin(4t), 3 cos(2t) + cos(7t)) as t varies from 0 to 2 \+
" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 28 " .  To do this, simply \+
type:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot([3*sin(3*t)+s
in(4*t),3*cos(2*t)+cos(7*t),t=0..2*Pi]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 163 "Since the above curve fails the vertical line test, it i
s not a function, and could not possibly be represented by any equatio
n y=f(x).  Parametric equations rule!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 85 "Solve the following prob
lem on Maple.  All relevant commands must be done on Maple!  " }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "IMPORTANT: " }{TEXT 295 98 " St
art a new worksheet here, and type in the command \"restart;\" at the \+
top before you do anything." }{TEXT -1 95 "  This will clear all the v
ariables and you will start fresh and clean, like a mountain spring." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The ra
te of change of a bacteria population is proportional to its size.  Af
ter 5 hours the population triples. " }}{PARA 0 "" 0 "" {TEXT -1 41 "1
) At what time will it have quadrupled? " }{TEXT 306 34 " Express your
 answer as a decimal." }{TEXT -1 40 "  (You will need 2 solve commands
 here.)" }}{PARA 0 "" 0 "" {TEXT -1 124 "2) Suppose the initial popula
tion is equal to 1000.  What is the rate of growth after 7 hours? (Rem
ember rate = derivative.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Please note THE MA
PLE LAB IS NOW OVER.  YOU ARE NOT TO HAND IN ANYTHING FURTHER.    " }}
{PARA 0 "" 0 "" {TEXT -1 102 "Try these parametric plots, they're quit
e pretty.  Also, try your own equations and see what comes up." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot([9*cos(t)-cos(9*t),9*si
n(t)-sin(9*t),t=0..2*Pi]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
58 "plot([8*cos(t)+5*cos(4*t),8*sin(t)-2*sin(6*t),t=0..2*Pi]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot([6*cos(2*t)+5*cos(6*t),
6*sin(4*t)-5*sin(6*t),t=0..2*Pi]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "plot([cos(t)-cos(80*t)*sin(t),2*sin(t)-sin(80*t),t=0.
.2*Pi]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "18 \+
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