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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 49 "Newton's Model of Coolin
g -- A First Look at Data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 199 "The cell below enters a list of temperatures meas
ured at equally spaced times.  By cutting and pasting you can substitu
te your own data for the data below.  The same cell draws a graph of t
he data.  " }{TEXT 256 27 "Evaluate the next cell now." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 922 "with(
plots):\n\n  temp :=\n  [93.02,  90.23,  87.64,  85.57,  83.29,  81.45
,  79.70,  78.03,     \n   76.69,  75.15,  73.91,  72.48,  71.10,  69.
98,  68.90,  67.85,\n   66.62,  65.42,  64.45,  63.88,  62.95,  62.04,
  61.14,  60.27,\n   59.75,  58.90,  58.23,  57.41,  56.92,  56.28,  5
5.33,  54.71,\n   54.10,  53.49,  52.75,  52.30,  51.87,  51.29,  50.7
1,  50.14,\n   49.72,  49.30,  48.75,  48.34,  47.93,  47.52,  46.98, \+
 46.58,\n   46.05,  45.66,  45.26,  45.01,  44.62,  44.23,  43.85,  43
.59,\n   43.21,  42.83,  42.45,  42.08,  41.83,  41.46,  41.21,  40.84
,\n   40.60,  40.23,  39.86,  39.74,  39.38,  39.02,  38.90,  38.54,  \+
   \n   38.42,  38.06,  37.82,  37.59,  37.35,  37.11,  36.64,  36.52,
     \n   36.29,  36.05,  35.94,  35.70,  35.47,  35.24,  35.01,  34.7
8,     \n   34.66,  34.31,  34.20,  33.97,  33.85,  33.62,  33.39,  33
.16,     \n   32.93,  32.93,  32.82]:\n\n   plot([seq([j, temp[j]], j=
1..99)], 1..99, 0..100);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT 
-1 38 "The next cell fits a curve of the form" }}{PARA 257 "" 0 "" 
{TEXT -1 2 "\n\n" }{TEXT 257 29 "q(t) = A + k^(t-n) (q(n) - A)" }}
{PARA 0 "" 0 "" {TEXT -1 53 "\nto three data points at equally spaced \+
intervals -- " }{TEXT 258 7 "temp(n)" }{TEXT -1 2 ", " }{TEXT 259 11 "
temp(n + p)" }{TEXT -1 6 ", and " }{TEXT 260 13 " temp(n + 2p)" }
{TEXT -1 150 ".  It then draws a graph showing the original data and t
he fitted curve.  Execute the cell below to see an example with a curv
e fitted to the points  " }{TEXT 261 8 "temp(20)" }{TEXT -1 2 ", " }
{TEXT 263 7 "temp(40" }{TEXT -1 7 "), and " }{TEXT 262 8 "temp(60)" }
{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 444 "  n := \+
20:\n  p := 20:\n\n  A := (temp[n + 2 * p] * temp[n] - temp[n + p]^2)/
\n       (temp[n + 2 * p] - 2 * temp[n + p] + temp[n]):\n\n  k := (((A
 - temp[n + p])/(A - temp[n])))^(1/p):\n\n  model := t -> A + (k^(t - \+
n)) * (temp[n] - A):\n\n  blk := plot([seq([j, temp[j]], j=1..99)],\n \+
              1 .. 99, 0 .. 100, color = BLACK):\n  rd := plot([seq([j
, model(j)], j = 1 .. 99)],\n                1 .. 99, 0 .. 100, color \+
= RED):\n\n  display([blk, rd]);" }}}}{MARK "0 0 0" 49 }{VIEWOPTS 1 1 
0 1 1 1803 }