Find best element of continuous approximation for the f ( x ) = sin ⁡ ( x ) for x ∈ [ 0 , π / 4 ] .

Question

Find best element of continuous approximation for the  f  (  x  )  =  sin  ⁡  (  x  )   for   x  ∈  [  0  ,  π      /    4  ]  .

Alexzander Bowman · Accepted Answer

If you want the &quot;best&quot; approximation of   sin  ⁡  (  x  ) for       x    1    ≤  x  ≤      x    2   using (say) a cubic polynomial, you could minimize with respect to   (  a  ,  b  )  I  =      ∫                  x        1                            x        2                        (      sin      ⁡      (      x      )      −      (      a      x      +      b              x        3            )      )        2  This would correspond to a linear regression with an infinite number of data points.Using       x    1    =  0 and       x    2    =      π    4  , this would lead to  a  =            80640              2            −      20160              2            π      −      1920              2                    π        2            +      60              2                    π        3                    π      5        ≈  0.999259  b  =  −            2150400              2            −      537600              2            π      −      53760              2                    π        2            +      2240              2                    π        3                    π      7        ≈  −  0.161035For these values   I  ≈  6.81  ×      10          −      9       while using Taylor coefficients   (  a  =  1  ,  b  =  −      1    6    ) we should get   I  ≈  4  32  ×      10          −      7       that is to say almost   64 times larger.

Find best element of continuous approximation for the f ( x ) = sin ⁡<!-- ⁡ -

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