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{SECT 0 {PARA 257 "" 0 "" {TEXT -1 11 "Math 2250-1" }}{PARA 258 "" 0 "
" {TEXT 256 53 "Newton's law of cooling application - Maple Project 1
" }}{PARA 265 "" 0 "" {TEXT 258 20 "Notes and assignment" }}{PARA 260 
"" 0 "" {TEXT -1 18 "Friday September 4" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 127 "Our first Maple project comes from \+
section 1.5, First Order Linear Differential Equations.  We shall buil
d this project out of " }{TEXT 257 48 "1.5 Application: Indoor Tempera
ture Oscillations" }{TEXT -1 487 ", pages 56-58, and from Professor Gu
stafson's version of this project for his Math 2250 section.  I recomm
end you read our notes in conjunction with the text.  As you do the Ma
ple project you may also want to consult the hints on Professor Gustaf
son's page (even though the questions may not look exactly the same), \+
along with the Maple Introduction notes I recommended to you last week
.  Links to all this material are on our Maple home page, which is lin
ked from our course home page, " }{URLLINK 17 "http://math.utah.edu/~k
orevaar/2250fall08" 4 "http://math.utah.edu/~korevaar/2250fall08" "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 318 "Here's \+
the set-up in the text, page 56. You have no air conditioner or evapor
ative cooler for your home, it's summer, you live in the authors' home
 town of Athens Georgia, and the outside temperatures are oscillating \+
periodically, with a period of 24 hours.  In other words, the ambient \+
temperature outside is give by " }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "A(
t) = a[0]+a[1]*cos(omega*t)+b[1]*sin(omega*t);" "6#/-%\"AG6#%\"tG,(&%
\"aG6#\"\"!\"\"\"*&&F*6#F-F--%$cosG6#*&%&omegaGF-F'F-F-F-*&&%\"bGF0F--
%$sinGF3F-F-" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "where the \+
angular frequency " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 4 "
 is " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 28 " r
adians per 24 hours, i.e. " }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "omega =
 1/12*Pi;" "6#/%&omegaG,$*&\"#7!\"\"%#PiG\"\"\"F*" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 347 "radians per hour.  In the book example t
he 24-hour average temperature is 80 degrees F, with a minimum of 70 d
egrees at 4 a.m., and a maximum of 90 degrees at 4 p.m.  This is consi
stent with the fact that the hottest part of the day occurs in the aft
ernoon, and the coolest part is in the early morning hours.  Using thi
s information, we may write" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "A(t) =
 80-10*cos(omega*(t-4));" "6#/-%\"AG6#%\"tG,&\"#!)\"\"\"*&\"#5F*-%$cos
G6#*&%&omegaGF*,&F'F*\"\"%!\"\"F*F*F4" }{TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 119 "and expand using the cosine addition angle formula, wh
ich you should have memorized and which is in your Calculus book:" }}
{PARA 264 "" 0 "" {XPPEDIT 18 0 "cos(alpha+beta) = cos(alpha)*cos(beta
)-sin(alpha)*sin(beta);" "6#/-%$cosG6#,&%&alphaG\"\"\"%%betaGF),&*&-F%
6#F(F)-F%6#F*F)F)*&-%$sinGF.F)-F3F0F)!\"\"" }{TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 11 "This yields" }}{PARA 266 "" 0 "" {XPPEDIT 18 0 "A(
t) = 80-5*cos(omega*t)-5*3^(1/2)*sin(omega*t);" "6#/-%\"AG6#%\"tG,(\"#
!)\"\"\"*&\"\"&F*-%$cosG6#*&%&omegaGF*F'F*F*!\"\"*(F,F*\"\"$#F*\"\"#-%
$sinGF/F*F2" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 9 "(Check!!)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "A(t)=80
-5*cos(omega*t)-5*sqrt(3)*sin(omega*t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"AG6#%\"tG,(\"#!)\"\"\"*&\"\"&F*-%$cosG6#*&%&omegaGF*F'F*F*!
\"\"*(F,F*\"\"$#F*\"\"#-%$sinGF/F*F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "A(t)=80-10*cos(omega*(t-
4));;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"AG6#%\"tG,&\"#!)\"\"\"*&
\"#5F*-%$cosG6#*&%&omegaGF*,&F'F*\"\"%!\"\"F*F*F4" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "omega=Pi/12;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/%&omegaG,$*&\"#7!\"\"%#PiG\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warnin
g, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 62 "a0:=80.:\na1:=-5.:\na2:=-5*sqrt(3.):\nomega:=eva
lf(Pi/12):\nk:=.2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "c1:=(
a1*k^2-a2*k*omega)/(k^2+omega^2);\nc2:=(a1*k*omega+k^2*a2)/(k^2+omega^
2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1G$\"+Dc5NB!\"*" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#c2G$!+Hzg.c!\"*" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 31 "evalf(12/Pi*arctan(-2.36/5.6));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$!+*=jM_\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 110 "u:=(t,u0)->a0+(u0-a0-c1)*exp(-k*t)+c1*cos(omega*t)\n
+c2*sin(omega*t);\nA:=t->a0+a1*cos(omega*t)+a2*sin(omega*t);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6$%\"tG%#u0G6\"6$%)operatorG%&arrowG
F),*%#a0G\"\"\"*&,(9%F/F.!\"\"%#c1GF3F/-%$expG6#,$*&%\"kGF/9$F/F3F/F/*
&F4F/-%$cosG6#*&%&omegaGF/F;F/F/F/*&%#c2GF/-%$sinGF?F/F/F)F)F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AGf*6#%\"tG6\"6$%)operatorG%&arrow
GF(,(%#a0G\"\"\"*&%#a1GF.-%$cosG6#*&%&omegaGF.9$F.F.F.*&%#a2GF.-%$sinG
F3F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "usp:=t->80+
2.36*cos(omega*t)-5.6*sin(omega*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%$uspGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,(\"#!)\"\"\"*&$\"$O#!\"#F.
-%$cosG6#*&%&omegaGF.9$F.F.F.*&$\"#c!\"\"F.-%$sinGF5F.F<F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "plot(\{u(t,70),u(t,60),u(t,
65),usp(t),\nA(t)\},t=0..72,color=black,\ntitle=`broken air conditione
r in Athens`);" }}{PARA 13 "" 1 "" {GLPLOT2D 434 227 227 {PLOTDATA 2 "
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