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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "A particle is moving towa
rd the origin. Its speed is always twice the remaining distance. Does \+
it ever reach the origin?" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "War
ning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 18 0 "dx/dt = -2*x;" "6#/*&%#dxG\"\"\"%#dtG!\"\",$*&\"
\"#F&%\"xGF&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "field:=fi
eldplot([1,-2*x],x=0..2,t=0..1):display(field);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "h:=t->exp(-2*t);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"hGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#,$9$!\"#F(F(F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "D(h)(t)=-2*h(t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$-%$expG6#,$%\"xG!\"#F*F$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "solution:=plot(h,0..2,0..1):
display(solution);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "displ
ay(\{field,solution\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "
attemptedsolution:=plot(h+.5,0..2,0..1):display(\{attemptedsolution,fi
eld\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "D(h+.5)(x)=-2*(h
(x)+.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$-%$expG6#,$%\"xG!\"#F*,
&F%F*$\"#5!\"\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "soluti
ons:=plot(\{seq(i*h(t)/5,i=0..20)\},t=0..2,x=0..1,color=red):display(
\{solutions,field\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 166 "Lets say instead that this particle has a speed that i
s proportional both to the inverse of its distance from the origin, an
d to the amount of time that has elapsed. " }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "dx/dt = t/x;" "6#/*&%#dxG\"\"\"%
#dtG!\"\"*&%\"tGF&%\"xGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
62 "field:=fieldplot([1,t/x],t=0..10,x=0.0001..10):display(field);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "field:=fieldplot([x,t],t=0.
.10,x=00001..10):display(field);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "h:=t->sqrt(t^2+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"hGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&*$)9$\"\"#\"\"\"F
4F4F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "D(h)(t)=t/h(
t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"tG\"\"\"*$-%%sqrtG6#,&*$)
F%\"\"#F&F&F&F&F&!\"\"F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 
"solution:=plot(h(t),t=0..10,x=0..10):display(solution);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "display(\{field,solution\});" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Now consider a particle whose spe
ed is proportional to both the amount of time that has elapsed, and to
 the function g(x) where " }{XPPEDIT 18 0 "g(x) = sqrt(1-x^2);" "6#/-%
\"gG6#%\"xG-%%sqrtG6#,&\"\"\"F,*$)F'\"\"#F,!\"\"" }{MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "dx/dt = t*sqrt(1-x^2);" "6#/*&
%#dxG\"\"\"%#dtG!\"\"*&%\"tGF&-%%sqrtG6#,&F&F&*$)%\"xG\"\"#F&F(F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "field:=fieldplot([1,t*(sqrt(
1-x^2))],t=9..10,x=-1..1):display(field);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "diff(arcsin(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#*&\"\"\"F$*$-%%sqrtG6#,&F$F$*$)%\"xG\"\"#F$!\"\"F$F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "h:=t->sin(t^2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"hGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%$sinG6#*$)9
$\"\"#\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "D(h)(
t)=t*sqrt(1-x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&-%$cosG6#*$)
%\"tG\"\"#\"\"\"F-F+F-F,*&F+F--%%sqrtG6#,&F-F-*$)%\"xGF,F-!\"\"F-" }}}
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