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If we approximate the function y=sin(x) with a0+a1 x a2 x^2 +a3 x^3, what is a0,a2,a2,a3?

Sagot :

The coefficients [tex]a_0,a_1,a_2,a_3[/tex] could be chosen to be the coefficients in the Maclaurin series of [tex]\sin(x)[/tex].

We have

[tex]y = \sin(x) \approx a_0 + a_1 x + a_2 x^2 + a_3 x^3 \\\\ \implies y(0) = 0 = a_0[/tex]

[tex]y' = \cos(x) \approx a_1 + 2a_2 x + 3a_3 x^2 \\\\ \implies y'(0) = 1 = a_1[/tex]

[tex]y'' = -\sin(x) \approx 2a_2 + 6a_3 x \\\\ \implies y''(0) = 0 = 2a_2[/tex]

[tex]y''' = -\cos(x) \approx 6a_3 \\\\ \implies y'''(0) = -1 = 6a_3[/tex]

It follows that [tex]a_0=0[/tex], [tex]a_1=1[/tex], [tex]a_2=0[/tex], and [tex]a_3 = -\frac16[/tex].

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