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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 256 99 "Here's the equation f; let's find the de
rivative, set it to zero, and see the best k in terms of p." }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "f := ((1-p)^k +
 (1 - (1-p)^k)*(k+1))*(n/k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG
*&*&,&),&\"\"\"F*%\"pG!\"\"%\"kGF**&,&F*F*F(F,F*,&F-F*F*F*F*F*F*%\"nGF
*F*F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "df := diff(f,k);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfG,&*&*&,**&),&\"\"\"F,%\"pG!
\"\"%\"kGF,-%#lnG6#F+F,F,*(F*F,F0F,,&F/F,F,F,F,F.F,F,F*F.F,%\"nGF,F,F/
F.F,*&*&,&F*F,*&,&F,F,F*F.F,F4F,F,F,F5F,F,*$)F/\"\"#F,F.F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "k = solve(df=0,k);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/%\"kG6$,$*&-%)LambertWG6#,$*$-%%sqrtG6#,$-%#lnG6#
,&\"\"\"F5%\"pG!\"\"F7F5#F5\"\"#F5F1F7F9,$*&-F)6#,$F,#F7F9F5F1F7F9" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 259 12 "This is the " }{TEXT 261 5 "exac
t" }{TEXT 262 317 " answer to what k should be.  If anyone in the clas
s knows how to use the Lambert W equation, he/she shouldn't be in this
 class.  Instead, we need to find a simplification to this problem.  T
hrough the binomial theorem we know that (1-n)^r for small enough n is
 approximately (1-nr).  We can use this in our equation." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "g := ((1 - p*k) + (1 - (1 - p*k))*(
k+1))*n/k;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*&*&,(\"\"\"F(*&%
\"pGF(%\"kGF(!\"\"*(F*F(F+F(,&F+F(F(F(F(F(F(%\"nGF(F(F+F," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "g := simplify(g);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"gG*&*&,&\"\"\"F(*&%\"pGF()%\"kG\"\"#F(F(F(%\"nGF
(F(F,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dg := diff(g,
k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dgG,&*&%\"pG\"\"\"%\"nGF(\"
\"#*&*&,&F(F(*&F'F()%\"kGF*F(F(F(F)F(F(*$F/F(!\"\"F2" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 107 "This looks like a more reasonable answer.  We'
ll take the positive root in this case, which is 1 / sqrt(p)." }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "k = solve(dg=0,
k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"kG6$*&\"\"\"F'*$-%%sqrtG6#%
\"pGF'!\"\",$F&F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 248 "Let's check
 our answers by plugging in values for p, n, and k.  Notice how functi
ons f and g give us almost exactly the same answer (which is the expec
ted number of tests performed at this drug percentage, total number of
 people, and size of group." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "p:=0.01; n:=10000; k:=1/sqrt(p);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"pG$\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"nG\"&++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG$\"+++++5!\")" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(f);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"+]#zh&>!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 9 "evalf(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++?!\"'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "4 0 2" 146 }
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