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
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Extra equations
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t(x) draws tangent lines of our function, which are used to draw u(x).
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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
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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=
1.7 9 2 8 7 9 8 0 0 0 81.79287980008
1.1 1 5 2 6 3 8 9 8 2 81.11526389828
0.9 7 2 1 2 7 2 0 4 9 1 70.972127204917
1.4 9 6 7 2 7 8 4 3 9 11.49672784391
2.3 1 9 6 2 6 5 6 8 1 62.31962656816
3.2 4 7 4 9 3 4 5 1 6 13.24749345161
4.2 1 8 9 3 2 0 0 1 1 34.21893200113
5.2 1 1 7 9 9 3 5 6 6 75.21179935667
6.2 1 6 6 0 7 3 2 4 5 76.21660732457
7.2 2 8 6 9 9 5 3 1 47.2286995314
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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
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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