{VERSION 4 0 "SUN SPARC SOLARIS" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 
257 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 14 0 
0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 
0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Out
put" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 
0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 
1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 
1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 
-1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 
0 }}
{SECT 0 {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 257 21 
"Math 2250 Spring 2001" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 258 
9 "Project I" }}{PARA 258 "" 0 "" {TEXT -1 16 "SolutionTemplate" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 47 "1)  Type in your name and student number  here:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 549 "     You are to do the section 1.5 compu
ter project, using this template.  When you are done you should print \+
out a copy to hand in to your Instructor.  A preliminary discussion, i
n which the text discussion of pages 55-57 is expanded to include Mapl
e commands, is found at the end of the tutorial which accompanies this
 project.  It is assumed that you have already been through that.  Bot
h the tutorial and this template can be downloaded by following the li
nks from our course home page, at http://www.math.utah.edu/~korevaar/2
250spring01.html.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 259 9 "SCENARIO:" }}{PARA 0 "" 0 "" {TEXT -1 421 "    It is no
t summer in Georgia.  It is winter in Salt Lake City.  We will dream i
t is early spring, (around equinox), and that the 5-day forecast in th
e newspaper says the weather will be stable, with lows of 25 degrees, \+
and highs of 55 degrees.   As in the text and for the sake of simplici
ty, we will assume a sinusoidal daily temperature oscillation, except \+
our low will be at 3 a.m, and our  high will be at 3 p.m.  " }}{PARA 
0 "" 0 "" {TEXT -1 206 "     We must leave town for 3 days, and are de
ciding whether to turn off the heat while we are gone.  The question w
e wonder about is, will the water pipes in the house freeze if we do t
urn off the heat?   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 797 "2)  Review the model which leads to equation (3) on pa
ge 56.  Use integrals #49 and 50  and the algorithm for solving first \+
order linear DE's on pages 44-45, to solve (3) by hand, keeping all pa
rameters as letters.   Staple your work onto the printout of your comp
leted project which you hand in.  Your answer should agree with equati
on (4) on page 56.  You will see that the constant c0 in that answer i
s the constant C of integration you obtain when you follow the solutio
n recipe for linear DE's.  You find its value in terms of the initial \+
condition u(0)=u0 by pluggin in u=u0, t=0, as usual.  You will discove
r that the text writes c0 incorrectly:  there is a hidden minus sign w
hich has become glued to the fraction which follows a0, in the book's \+
formula, i.e. the correct formula for c0 is" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 51 "c0:=(k*omega*b1 - k^2*a1)/(k^2 + omega^2) + u0 -a0;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c0G,(*&,&*(%\"kG\"\"\"%&omegaGF
*%#b1GF*F**&)F)\"\"#F*%#a1GF*!\"\"F*,&*$F.F*F**$)F+F/F*F*F1F*%#u0GF*%#
a0GF1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "restart:with(DEtools):with(plots):" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 318 "3)  Use dsolve to have
 Maple find the solution to (3), with u(0)=u0.  You might have already
 done this in the tutorial, in which case you can copy the appropriate
 commands  from there, paste them in here, and re-execute them.  Check
 that your solution agrees with your hand work above, as well as the (
corrected) text." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 550 "4)  Figure out th
e parameter values for the Salt Lake City temperature, as modeled abov
e.  Of course, omega will still be Pi/12, but now you have different d
aily average temperature, amplitude, and phase than was used in equati
on (2) on page 56, when summer in Georgia was being modeled.  Fill the
 new values in below.  Then work out by hand, using the cosine additio
n formula, the values you should take for a0, a1, b1, so that A(t) is \+
given by (1) on page 56, and so that the differential equation (3)  al
so has the correct parameter values.  Fill." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Average Temperature =\nTemperat
ure variation amplitude = " }}{PARA 0 "" 0 "" {TEXT -1 34 "Phase delay
 (was 4 hours before) =" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "omega:
=Pi/12;\na0:= \n   #don't forget the semicolons\na1:=\nb1:=\nA:=t->a0 \+
+ a1*cos(omega*t) + b1*sin(omega*t);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 70 "We will assume your house is moderate
ly well-insulated, so that k=0.3:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "k:=0.3;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 222 "5)  Find the solu
tion to (3) with the particular parameter choices you made above.  You
 should get a solution function like  (5) on page 56, but reflecting t
he Salt Lake City temperatures and the new insulation parameter. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 187 "
6)  Identify the part of your solution which persists as t approaches \+
infinity, i.e. the steady periodic solution.  Your formula should have
 the same character as equation (6) on page 57." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 273 "7)  Write the st
eady periodic solution in the form of equation (7) on page 57, so that
 you can see the time delay for the inside temperature.  You need to u
se the cosine addition formula.  The text has a discussion on page 315
 which may help you.  It is shown there that if" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "x(t):=t->A*c
os(omega*t) + B*sin(omega*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"
xG6#%\"tG,&*&%\"AG\"\"\"-%$cosG6#*&%&omegaGF+F'F+F+F+*&%\"BGF+-%$sinGF
.F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "then also" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "x(t):=t->C*cos(omega*t-alpha);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"xG6#%\"tGRF&6\"6$%)operatorG%&arr
owGF)*&%\"CG\"\"\"-%$cosG6#,&*&%&omegaGF/9$F/F/%&alphaG!\"\"F/F)F)F)" 
}}}{PARA 0 "" 0 "" {TEXT -1 105 "where the right triangle of Figure 5.
4.4 page 315 summarizes the relationships between A,B, C, and alpha." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 231 "8)  Assume that your heater sh
ut off at midnight, with the inside temperature equal to 70 degrees.  \+
Create a plot like Figure 1.5.10 which displays the inside temperature
 and the outside ambient temperature for the next three days." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 338 "9)  Create a picture like Figu
re 1.5.9, which also includes the slope field for this differential eq
uation, with our Salt Lake City parameters.  Choose initial temperatur
es between 40 and 70 degrees, in 5 degree increments.  Label the maxim
um and minimum temperature times by hand, on the display printout, bef
ore you hand in the project." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 187 "10)  So, based on your work in
 this project, how likely do you think it is that the pipes will freez
e if the heater is turned off for 3 days starting at midnight?  Explai
n your reasoning." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{MARK "8 0" 153 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }