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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Week 4, pp. 86-128." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Exercises
 3.2  p. 98-  10, 14, 17, 18, 25, 26." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 26 "10.  Direction field for  " }{XPPEDIT 18 0 "diff(x(t),t)=
a-b*x(t)" "/-%%diffG6$-%\"xG6#%\"tGF),&%\"aG\"\"\"*&%\"bGF,-F'6#F)F,!
\"\"" }{TEXT -1 3 ";  " }{XPPEDIT 18 0 "a>0" "2\"\"!%\"aG" }{TEXT -1 
2 ", " }{XPPEDIT 18 0 "b>0" "2\"\"!%\"bG" }{TEXT -1 20 ".  What happen
s as  " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 12 "  tends to  " }
{XPPEDIT 18 0 "infinity" "I)infinityG6\"" }{TEXT -1 2 " ?" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "a:=2:b:=1: with(DEtools): dfieldplo
t(diff(x(t),t)=a-b*x(t),x(t),t=-3..3,x=-2..4);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 33 "Looks like the solutions tend to " }{XPPEDIT 18 0 "a
/b" "*&%\"aG\"\"\"%\"bG!\"\"" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 37 "dsolve(diff(x(t),t)=aa-bb*x(t),x(t));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "14.   Let  " }{XPPEDIT 18 0 "A(t)
" "-%\"AG6#%\"tG" }{TEXT -1 89 "  denote the number of alligators.  As
sume Malthusian growth.  Assume  A(1970)=300  and  " }{XPPEDIT 18 0 "A
(1980)=1500" "/-%\"AG6#\"%!)>\"%+:" }{TEXT -1 11 ".  What is " }
{XPPEDIT 18 0 "A(2000)" "-%\"AG6#\"%+?" }{TEXT -1 1 "?" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "da:=diff(A(t),t)=alpha*A(t);\ngsol:
=dsolve(da,A(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "s0:=su
bs(t=0,rhs(gsol)); s10:=subs(t=10,rhs(gsol));" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 41 "ca:=solve(\{s0=300,s10=1500\},\{alpha,_C1\});" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "sol:=subs(ca,rhs(gsol)); s
ubs(t=30,sol); evalf(\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "18. \+
 Gompertz model." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "restart
:dg:=diff(P(t),t)=P(t)*(a-b*ln(P(t))); gsol:=dsolve(dg,P(t));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "psol:=solve(gsol,P(t));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "init:=subs(t=0,psol);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "valc:=solve(init=P0,_C1);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subs(_C1=valc,psol); simpl
ify(\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 292 "26.  Carbon dating (
do 25 first).   If you estimate a true value of 5600 and make an error
 of -50, making the estimate 5550, then that is a 1% error.  If you es
timate 2% by 3% then that is an absolute error of 1%, but no one in th
eir right mind would call that a 1% error.  What do you think?" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Exercises 3.3  p. 106-  1, 4, 5, 1
0." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "4.  Wine at 10 C reaches 15
 C in 10 min in a 23 C room; how long to reach 18 C?" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 194 "restart; dn:=diff(T(t),t)=k*(23-T(t)); s
n:=dsolve(dn,T(t));\ns0:=subs(t=0,rhs(sn)); s10:=subs(t=10,rhs(sn));\n
con:=solve(\{s0=10,s10=15\},\{k,_C1\}); sol:=subs(con,rhs(sn)); solve(
sol=18,t); evalf(\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "10.   Th
e model is very unrealistic.  Go through Example 3 to get hints." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Exercises 3
.4  p.108-  3, 4, 9, 24, 25." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "E
xample 3 -- comments." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "de
:=m*diff(v(t),t)=m*g-k*v(t); sol:=dsolve(\{de,v(0)=v0\},v(t)); # solve
 the general de" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "sol1:=su
bs(\{m=75,g=981/100,v0=0,k=15\},sol);  # sub the initial parameters" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "solx:=int(rhs(sol1),t=0..z
); # integrate to find distance at time t=z" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 39 "subs(z=60,solx); # how far in a minute?" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ht:=4000-evalf(\"); # starte
d from 4000 m" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "v1:=subs(t
=60,rhs(sol1)); # find velocity of fall when parachute opened" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "newv:=subs(\{m=75,g=981/100,
v0=v1,k=105\},sol); # find new vel with chute" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 56 "int(rhs(newv),t=0..z); # integrate to get distan
ce again" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "fsolve(\"=ht,z)
; # when is it when she fell ht more?" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "subs(t=\",rhs(newv)); # how fast?" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 85 "It is interesting to compute the accelleration at the instant t
he parachute opened.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "v
2:=subs(t=0.1,rhs(newv)); # .1 second after it opens\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "10.*(v1-v2);  # the accel (of cours
e it is really negative)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 
"evalf(\"/9.81);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "That is more \+
accelleration than most folks can stand." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 26 "Exercises 3.5  p. 125-  8." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 74 "An example of Euler's method applied
 to Stefan's law of radiative cooling." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "T:=100.;\nfor t from 1 to 20 do\n  T:=T+0.1*(20^4-T^4
)/40^4:\n  if t=10 then T1:=T: fi:od:\nT1,T; " }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 51 "It is amazing that Maple can't solve this equation." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "restart:dsolve(\{diff(T(t)
,t)=(20^4-T(t)^4)/40^4,T(0)=0\},T(t));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 50 "We can.  Let Maple do the partial fractions first:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "pf:=convert(40^4/(20^4-T^4),
parfrac,T);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sol:=int(pf,
T)=t+_C1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(sol,T);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs(t=0,sol); subs(T=1
00,\"); solve(\",_C1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "ev
alf(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subs(_C1=\",sol
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "restart;int(40^4/(20^
4-T^4),T);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "So M knows how to d
o the integral!  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 18 "Pop test solution." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "V:=10*12*8; # volume" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "init:=A(0)=V*3/100; # initial amount" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "IVP:=\{diff(A(t),t)=200*0-200*A(t)/
V,init\};  # no argon going in" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 17 "dsolve(IVP,A(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "solve(rhs(\")=0.01/100*V,t); .01 % is .01/100 of course." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Nlc:=diff(T(t),t)=k*(20-T(t)
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "temp:=dsolve(\{Nlc,T(
0)=85\},T(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ts:=subs(
t=10,rhs(temp));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(t
s=70,k);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(k=\",temp)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(rhs(\")=45,t);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 50 "The white test is the same except for the
 numbers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
14 "This is weird!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "n:=0:
for t from 0 to 2 by 0.1 do\n  if t*10=n then print(`ok`); fi;\n  t; n
:=n+1:\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "
69 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 }