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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 35 "A Difference Look at New
ton's Apple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 418 "The cell below contains data from one experiment that we did d
ropping a pillow.  You can use your own data by uploading it from your
 graphing calculator using the appropriate TI-Graph Link program and c
able and then cutting and pasting from the Graph Link window to this w
indow.  You may need to do some editing -- putting commas between entr
ies and deleting unusable data at the beginning and end of the experim
ent.  " }{TEXT 256 26 "Evaluate the next cell now" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 458 "Pillow := [5.98496, 5.97415, 5.95615, 5.92014,\n           5.88
773, 5.82651, 5.75449, 5.66086,\n           5.54203, 5.43039, 5.29715,
 5.18912,\n           5.07029, 4.94065, 4.79661, 4.63456,\n           \+
4.47611, 4.28886, 4.08000, 3.86753,\n           3.66227, 3.43541, 3.21
214, 2.98887,\n           2.76201, 2.51353, 2.27226, 2.01659,\n       \+
    1.75371, 1.48723, 2.27947, 1.76812,\n           1.45842]:\n\nn := \+
29:\n\nplot([seq([j, Pillow[j]], j = 1..n)], 1..n, 0..7);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "The next
 cell looks at the first and second order Newton divided differences. \+
 This particular data was collected at intervals of 0.02 seconds  " }
{TEXT 257 26 "Evaluate the next cell now" }{TEXT -1 2 ".." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "q :
= 0.02:\nfirst  := k -> (Pillow[k + 1] - Pillow[k])/q:\nsecond := k ->
 (first(k + 1) - first(k))/q:\n\nfor k from 1 to n-2\n    do printf(`%
4d %10.4f %10.4f %10.4f`, \n              k, Pillow[k], first(k), seco
nd(k));\n       print();\n    od;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 186 "\nNotice the velocity data isn't too bad but the acceleration \+
data (the second order divided differences) is pretty poor.  This is n
ot surprizing in view of the discussion in this module." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 194 "As we discussed in \+
the module, one way to try to avoid these kinds of errors is by using \+
measurements made at longer time intervals.  The cell below does this \+
by using every fourth data point.  " }{TEXT 258 26 "Evaluate the next \+
cell now" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "pick   := k -> Pillow[4 * k - 3]:
\nd := 0.08:\nfirst  := k -> (pick(k + 1) - pick(k))/d:\nsecond := k -
> (first(k + 1) - first(k))/d:\n\nfor k from 1 to 6\n    do printf(`%4
d %10.4f %10.4f %10.4f`, \n              k, pick(k), first(k), second(
k));\n       print();\n    od;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "This is a definite improvement but
 nothing to write home about." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}
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