Find best element of continuous approximation for the f ( x ) = sin &#x2061;<!-- ⁡ -

logiski9s 2022-07-09 Answered
Find best element of continuous approximation for the
f ( x ) = sin ( x )  for  x [ 0 , π / 4 ] .
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Answers (1)

Alexzander Bowman
Answered 2022-07-10 Author has 19 answers
If you want the "best" 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 0.161035
For 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.

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