Expression 14: "u" left parenthesis, "x" , right parenthesis equals left parenthesis, "g" left parenthesis, "p" , right parenthesis , right parenthesis left parenthesis, "x" minus left parenthesis, "p" plus "w" , right parenthesis , right parenthesis plus "h" left brace, "m" Subscript, "i" "n" , Baseline left parenthesis, "p" , "p" plus "w" , right parenthesis less than or equal to "x" less than or equal to "m" Subscript, "a" "x" , Baseline left parenthesis, "p" , "p" plus "w" , right parenthesis , right braceux=gpx−p+w+hminp,p+w≤x≤maxp,p+w
14
Extra equations
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15
t(x) draws tangent lines of our function, which are used to draw u(x).
16
Expression 17: "t" left parenthesis, "x" , right parenthesis equals left parenthesis, "g" left parenthesis, "p" , right parenthesis , right parenthesis left parenthesis, "x" minus "p" , right parenthesis plus "f" left parenthesis, "p" , right parenthesis left brace, "p" minus StartFraction, "w" Over 4 , EndFraction less than or equal to "x" less than or equal to "p" plus StartFraction, "w" Over 4 , EndFraction , right bracetx=gpx−p+fpp−w4≤x≤p+w4
17
h calculates a list of total change in height after l approximations, where l ranges from 1 to n.
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Expression 19: "h" equals "f" left parenthesis, "a" , right parenthesis plus Start sum from "i" equals 1 to "l" , end sum, "w" left parenthesis, "g" left parenthesis, "p" , right parenthesis left bracket, "i" , right bracket , right parenthesish=fa+l∑i=1wgpi
equals=
0.1 3 1 7 9 5 0 5 2 3 6 70.131795052367
0.1 5 4 8 7 7 7 3 5 6 2 70.154877735627
0.1 8 2 2 6 4 7 2 4 2 1 60.182264724216
0.2 1 4 7 3 7 9 2 3 3 9 20.214737923392
0.2 5 3 2 1 2 7 9 7 6 4 80.253212797648
0.2 9 8 7 5 7 7 1 8 2 9 30.298757718293
0.3 5 2 6 1 4 6 6 9 3 1 90.352614669319
0.4 1 6 2 2 0 7 3 8 3 5 10.416220738351
0.4 9 1 2 2 9 4 5 0 0 10.49122945001
0.5 7 9 5 3 0 5 0 2 0 0 50.579530502005
0.6 8 3 2 6 5 8 1 9 6 1 90.683265819619
0.8 0 4 8 3 9 0 4 2 6 0 80.804839042608
0.9 4 6 9 1 4 6 1 3 5 5 40.946914613554
1.1 1 2 4 0 1 6 0 2 51.1124016025
1.3 0 4 4 1 6 3 9 7 7 61.30441639776
16 element list
19
Minimum and maximum functions that work without creating additional tables.
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Expression 21: "m" Subscript, "i" "n" , Baseline left parenthesis, "x" , "y" , right parenthesis equals StartFraction, left parenthesis, sign left parenthesis, "x" minus "y" , right parenthesis plus 1 , right parenthesis Over 2 , EndFraction times "y" plus StartFraction, left parenthesis, sign left parenthesis, "y" minus "x" , right parenthesis plus 1 , right parenthesis Over 2 , EndFraction times "x"minx,y=signx−y+12·y+signy−x+12·x
21
Expression 22: "m" Subscript, "a" "x" , Baseline left parenthesis, "x" , "y" , right parenthesis equals StartFraction, left parenthesis, sign left parenthesis, "x" minus "y" , right parenthesis plus 1 , right parenthesis Over 2 , EndFraction times "x" plus StartFraction, left parenthesis, sign left parenthesis, "y" minus "x" , right parenthesis plus 1 , right parenthesis Over 2 , EndFraction times "y"maxx,y=signx−y+12·x+signy−x+12·y