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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 21 "Math 340 Midterm Exam" }}{PARA 256 "" 0 "" {TEXT -1 13 "O
ct. 22, 2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 9 "Your name" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 256 10 "Problem 1:" }{TEXT -1 13 "  (25 points)" }}{PARA 0 "" 0 
"" {TEXT -1 6 " Let  " }{XPPEDIT 18 0 "F[0] = 0;" "6#/&%\"FG6#\"\"!F'
" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "F[1] = 1;" "6#/&%\"FG6#\"\"\"F'" 
}{TEXT -1 10 "  and     " }{XPPEDIT 18 0 "F[k] = F[k-1]+F[k-2];" "6#/&
%\"FG6#%\"kG,&&F%6#,&F'\"\"\"F,!\"\"F,&F%6#,&F'F,\"\"#F-F," }{TEXT -1 
6 "  for " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 84 " = 2, 3, ...  b
e the Fibonacci numbers.  It is known that the sequence of the ratio \+
" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "s[k] = F[k-1]/F[k];" "6#/&%\"sG6#
%\"kG*&&%\"FG6#,&F'\"\"\"F-!\"\"F-&F*6#F'F." }{TEXT -1 10 ",    for  \+
" }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 12 " = 1, 2, ..." }}{PARA 0 
"" 0 "" {TEXT -1 87 "converges to the ration of golden section (0.618.
..).  Write a program that, for input " }{XPPEDIT 18 0 "n;" "6#%\"nG" 
}{TEXT -1 30 ",  output the entire sequence " }}{PARA 256 "" 0 "" 
{XPPEDIT 18 0 "s[1];" "6#&%\"sG6#\"\"\"" }{TEXT -1 3 ",  " }{XPPEDIT 
18 0 "s[2];" "6#&%\"sG6#\"\"#" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "s[3];
" "6#&%\"sG6#\"\"$" }{TEXT -1 8 ", ...,  " }{XPPEDIT 18 0 "s[n];" "6#&
%\"sG6#%\"nG" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 10 "Proble
m 2:" }{TEXT -1 14 "   (25 points)" }}{PARA 0 "" 0 "" {TEXT -1 44 "Wri
te a program that, for an input vector   " }{XPPEDIT 18 0 "a;" "6#%\"a
G" }{TEXT -1 4 " = [" }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT -1 6 "
, ... " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 23 "],  out
put the quantity" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "S = sqrt(Sum((a[i
]-mu)^2,i = 1 .. n))/(Sum(abs(a[i]),i = 1 .. n)*sqrt(Sum(a[i]^2,i = 1 \+
.. n)));" "6#/%\"SG*&-%%sqrtG6#-%$SumG6$*$),&&%\"aG6#%\"iG\"\"\"%#muG!
\"\"\"\"#F3/F2;F3%\"nGF3*&-F*6$-%$absG6#&F06#F2/F2;F3F9F3-F'6#-F*6$*$)
&F06#F2F6F3/F2;F3F9F3F5" }{TEXT -1 0 "" }{TEXT -1 0 "" }{TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 7 "where  " }{XPPEDIT 18 0 "mu = sqrt(Sum(a
[i],i = 1 .. n))/n;" "6#/%#muG*&-%%sqrtG6#-%$SumG6$&%\"aG6#%\"iG/F/;\"
\"\"%\"nGF2F3!\"\"" }{TEXT -1 74 ".    Use a randomly generated vector
 of dimension 10 to test your program." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 258 10 "Problem 3:" }{TEXT -1 12 " (25 points)" 
}}{PARA 0 "" 0 "" {TEXT -1 38 "For every pair of positive integers   \+
" }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 9 "   and   " }{XPPEDIT 18 
0 "n;" "6#%\"nG" }{TEXT -1 22 ",  there are quotient " }{XPPEDIT 18 0 
"q;" "6#%\"qG" }{TEXT -1 16 "  and remainder " }{XPPEDIT 18 0 "r;" "6#
%\"rG" }{TEXT -1 12 "  such that " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "
m = q*n+r;" "6#/%\"mG,&*&%\"qG\"\"\"%\"nGF(F(%\"rGF(" }{TEXT -1 9 ",  \+
      " }{XPPEDIT 18 0 "r <= n;" "6#1%\"rG%\"nG" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 36 "There is a simple method to compute " }
{XPPEDIT 18 0 "q;" "6#%\"qG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "r;" "
6#%\"rG" }{TEXT -1 4 ":   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 16 "   Start with   " }{XPPEDIT 18 0 "q = 0;" "6#/%\"
qG\"\"!" }{TEXT -1 9 ",  and   " }{XPPEDIT 18 0 "r = m;" "6#/%\"rG%\"m
G" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "   if  " }{XPPEDIT 18 
0 "r <= n;" "6#1%\"rG%\"nG" }{TEXT -1 9 ",  then  " }{XPPEDIT 18 0 "q \+
= 0;" "6#/%\"qG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "r = m;" "6#/
%\"rG%\"mG" }{TEXT -1 42 " is the solution, output result. Otherwise" 
}}{PARA 0 "" 0 "" {TEXT -1 16 "   (*) increase " }{XPPEDIT 18 0 "q;" "
6#%\"qG" }{TEXT -1 20 "  by 1 and compute  " }{XPPEDIT 18 0 "r = m-q*n
;" "6#/%\"rG,&%\"mG\"\"\"*&%\"qGF'%\"nGF'!\"\"" }{TEXT -1 18 ",   repe
at until  " }{XPPEDIT 18 0 "r <= n;" "6#1%\"rG%\"nG" }{TEXT -1 11 " , \+
 output " }{XPPEDIT 18 0 "q;" "6#%\"qG" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "r;" "6#%\"rG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 32 "Write a program that, for input " }
{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "n;" "
6#%\"nG" }{TEXT -1 44 ",  implement the above algorithm and output " }
{XPPEDIT 18 0 "q;" "6#%\"qG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "r;" "
6#%\"rG" }{TEXT -1 17 "  using while-do " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "For example:   Let " }{XPPEDIT 18 
0 "m = 38;" "6#/%\"mG\"#Q" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "n = 9;" 
"6#/%\"nG\"\"*" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }
{XPPEDIT 18 0 "q = 0;" "6#/%\"qG\"\"!" }{TEXT -1 5 ",    " }{XPPEDIT 
18 0 "r = 38;" "6#/%\"rG\"#Q" }{TEXT -1 3 " > " }{XPPEDIT 18 0 "n;" "6
#%\"nG" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }
{XPPEDIT 18 0 "q = 1;" "6#/%\"qG\"\"\"" }{TEXT -1 4 ",   " }{XPPEDIT 
18 0 "r = 38-1*9;" "6#/%\"rG,&\"#Q\"\"\"*&F'F'\"\"*F'!\"\"" }{TEXT -1 
8 " = 31 > " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 11 "   (repeat)" 
}}{PARA 0 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "q = 2;" "6#/%\"qG\"
\"#" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "r = 38-2*9;" "6#/%\"rG,&\"#Q\"
\"\"*&\"\"#F'\"\"*F'!\"\"" }{TEXT -1 8 " = 20 > " }{XPPEDIT 18 0 "n;" 
"6#%\"nG" }{TEXT -1 11 "   (repeat)" }}{PARA 0 "" 0 "" {TEXT -1 3 "   \+
" }{XPPEDIT 18 0 "q = 3;" "6#/%\"qG\"\"$" }{TEXT -1 4 ",   " }
{XPPEDIT 18 0 "r = 38-3*9;" "6#/%\"rG,&\"#Q\"\"\"*&\"\"$F'\"\"*F'!\"\"
" }{TEXT -1 8 " = 11 > " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 11 " \+
  (repeat)" }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "q = 4;
" "6#/%\"qG\"\"%" }{TEXT -1 4 ",   " }{XPPEDIT 18 0 "r = 38-4*9;" "6#/
%\"rG,&\"#Q\"\"\"*&\"\"%F'\"\"*F'!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 
18 0 "2 < n;" "6#2\"\"#%\"nG" }{TEXT -1 20 "     (end repeating)" }}
{PARA 0 "" 0 "" {TEXT -1 12 "Therefore,  " }{XPPEDIT 18 0 "q = 4;" "6#
/%\"qG\"\"%" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "r = 2;" "6#/%\"rG\"\"#
" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }}{PARA 0 "" 0 "" 
{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 259 10 "Problem 4:" }{TEXT -1 13 "  (25 points)" }}{PARA 0 "" 0 
"" {TEXT -1 76 "Many Fibonacci numbers are prime numbers.  Write a pro
gram that, for input  " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 32 ", \+
  find all prime numbers among" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "F[0
];" "6#&%\"FG6#\"\"!" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "F[1];" "6#&%\"
FG6#\"\"\"" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "F[2];" "6#&%\"FG6#\"\"#
" }{TEXT -1 11 ",   ...,   " }{XPPEDIT 18 0 "F[n];" "6#&%\"FG6#%\"nG" 
}{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Output those prime numbe
rs in a vector " }{XPPEDIT 18 0 "p;" "6#%\"pG" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 13 "Problem 5:   " }{TEXT -1 17 "(E
xtra 15 points)" }}{PARA 0 "" 0 "" {TEXT -1 45 "Write a program that, \+
 for an input vector   " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 4 " =
 [" }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT -1 6 ", ... " }
{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 25 "],   output two \+
vectors  " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 7 "  and  " }
{XPPEDIT 18 0 "v;" "6#%\"vG" }{TEXT -1 83 "  containing the positive e
ntries and negative entries respectively.   For example:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 58 "a := LinearAlgebra[RandomVector][row](10,
generator=-5..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%'RTABLEG6
%\"*7HBX\"-%'VECTORG6#7,\"\"\"\"\"&\"\"!!\"%!\"#!\"\"F-F/F1F-&%'Vector
G6#%$rowG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 5 "Then " }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'
RTABLEG6%\"*_5yX\"-%'VECTORG6#7&!\"%!\"#!\"\"F.&%'VectorG6#%$rowG" }}}
{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'RTABLEG6%\"*K6yX\"-%'V
ECTORG6#7&\"\"\"\"\"&F-F-&%'VectorG6#%$rowG" }}}}{MARK "1 16 8" 7 }
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