Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \cos{\left(x \right)}= \frac{139 x^{3}}{200} - 9$ using $LaTeX: \displaystyle x_0=3$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{139 x_{n}^{3}}{200} + \cos{\left(x_{n} \right)} + 9}{- \frac{417 x_{n}^{2}}{200} - \sin{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 3$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{139 (3.0000000000)^{3}}{200} + \cos{\left((3.0000000000) \right)} + 9}{- \frac{417 (3.0000000000)^{2}}{200} - \sin{\left((3.0000000000) \right)}} = 2.4311369815$ $LaTeX: x_{2} = (2.4311369815) - \frac{- \frac{139 (2.4311369815)^{3}}{200} + \cos{\left((2.4311369815) \right)} + 9}{- \frac{417 (2.4311369815)^{2}}{200} - \sin{\left((2.4311369815) \right)}} = 2.2966858472$ $LaTeX: x_{3} = (2.2966858472) - \frac{- \frac{139 (2.2966858472)^{3}}{200} + \cos{\left((2.2966858472) \right)} + 9}{- \frac{417 (2.2966858472)^{2}}{200} - \sin{\left((2.2966858472) \right)}} = 2.2895884409$ $LaTeX: x_{4} = (2.2895884409) - \frac{- \frac{139 (2.2895884409)^{3}}{200} + \cos{\left((2.2895884409) \right)} + 9}{- \frac{417 (2.2895884409)^{2}}{200} - \sin{\left((2.2895884409) \right)}} = 2.2895692420$ $LaTeX: x_{5} = (2.2895692420) - \frac{- \frac{139 (2.2895692420)^{3}}{200} + \cos{\left((2.2895692420) \right)} + 9}{- \frac{417 (2.2895692420)^{2}}{200} - \sin{\left((2.2895692420) \right)}} = 2.2895692419$