The function f(x)=sin (9 x) has a Maclaurin series. Find the first 4 nonzero terms in the series, that is write down the Taylor polynomial with 4 nonzero terms.

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Step:1 ：The Maclaurin series for (sin x) is given by (sin x = x - frac{x^3}{3!} + frac{x^5}{5!} - frac{x^7}{7!} + ldots).Step:2 ：Substitute (9x) into the series for (x), we get (sin (9x) = 9x - frac{(9x)^3}{3!} + frac{(9x)^5}{5!} - frac{(9x)^7}{7!} + ldots).Step:3 ：The first four nonzero terms of the series are (9x, -frac{(9x)^3}{3!}, frac{(9x)^5}{5!}, -frac{(9x)^7}{7!}).Step:4 ：Simplify these terms, we get (9x, -frac{729x^3}{6}, frac{59049x^5}{120}, -frac{4782969x^7}{5040}).Step:5 ：So, the first four nonzero terms in the Maclaurin series for (sin (9x)) are (9x, -frac{729x^3}{6}, frac{59049x^5}{120}, -frac{4782969x^7}{5040}).Step:6 ：Therefore, the Taylor polynomial with 4 nonzero terms is (boxed{9x - frac{729x^3}{6} + frac{59049x^5}{120} - frac{4782969x^7}{5040}}).

Answer

Step: 1 ：The Maclaurin series for (sin x) is given by (sin x = x - frac{x^3}{3!} + frac{x^5}{5!} - frac{x^7}{7!} + ldots).Step: 2 ：Substitute (9x) into the series for (x), we get (sin (9x) = 9x - frac{(9x)^3}{3!} + frac{(9x)^5}{5!} - frac{(9x)^7}{7!} + ldots).Step: 3 ：The first four nonzero terms of the series are (9x, -frac{(9x)^3}{3!}, frac{(9x)^5}{5!}, -frac{(9x)^7}{7!}).Step: 4 ：Simplify these terms, we get (9x, -frac{729x^3}{6}, frac{59049x^5}{120}, -frac{4782969x^7}{5040}).Step: 5 ：So, the first four nonzero terms in the Maclaurin series for (sin (9x)) are (9x, -frac{729x^3}{6}, frac{59049x^5}{120}, -frac{4782969x^7}{5040}).Step: 6 ：Therefore, the Taylor polynomial with 4 nonzero terms is (boxed{9x - frac{729x^3}{6} + frac{59049x^5}{120} - frac{4782969x^7}{5040}}).

The function $f (x) = \sin (9 x)$ has a Maclaurin series. Find the first 4 nonzero terms in the series, that is write down the Taylor polynomial with 4 nonzero terms.

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Popular Problems

The function f(x)=sin⁡(9x) has a Maclaurin series. Find the first 4 nonzero terms in the series, that is write down the Taylor polynomial with 4 nonzero terms.

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Popular Problems

The function $f (x) = \sin (9 x)$ has a Maclaurin series. Find the first 4 nonzero terms in the series, that is write down the Taylor polynomial with 4 nonzero terms.