The function we are trying to approximate is P(X), plotted in black
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Avaldis 11: "P" left parenthesis, "x" , right parenthesis equals "A" positive StartFraction, "h" Over 2 , EndFraction left parenthesis, "x" positive StartRoot, 2 negative "x" squared , EndRoot , right parenthesisPx=A+h2x+2−x2
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Now we also plot the first and second derivatives (F and S) in red
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Avaldis 13: "F" left parenthesis, "x" , right parenthesis equals StartFraction, "h" Over 2 , EndFraction left parenthesis, 1 minus StartFraction, "D" Over StartRoot, left parenthesis, 2 minus "D" squared , right parenthesis , EndRoot , EndFraction , right parenthesisFx=h21−D2−D2
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Avaldis 14: "S" left parenthesis, "x" , right parenthesis equals negative StartFraction, "h" Over left parenthesis, 2 minus "D" squared , right parenthesis Superscript, 3 halves , Baseline , EndFractionSx=−h2−D232
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Now we can plot the Taylor series expansions (in blue)
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Avaldis 16: "a" equals 0a=0
00
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Avaldis 17: "y" equals "P" left parenthesis, "a" , right parenthesisy=Pa
equals=
7.4 5 3 6 6 0 5 3 0 7 27.45366053072
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Avaldis 18: "y" equals "P" left parenthesis, "a" , right parenthesis plus "F" left parenthesis, "a" , right parenthesis times left parenthesis, "x" minus "a" , right parenthesisy=Pa+Fa·x−a
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Avaldis 19: "y" equals "P" left parenthesis, "a" , right parenthesis positive "F" left parenthesis, "a" , right parenthesis times left parenthesis, "x" minus "a" , right parenthesis plus StartFraction, "S" left parenthesis, "a" , right parenthesis Over 2 , EndFraction times left parenthesis, "x" minus "a" , right parenthesis squaredy=Pa+Fa·x−a+Sa2·x−a2
