Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \cos{\left(x \right)}= \frac{39 x^{3}}{200} - 9$ using $LaTeX: \displaystyle x_0=3$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{39 x_{n}^{3}}{200} + \cos{\left(x_{n} \right)} + 9}{- \frac{117 x_{n}^{2}}{200} - \sin{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 3$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{39 (3.0000000000)^{3}}{200} + \cos{\left((3.0000000000) \right)} + 9}{- \frac{117 (3.0000000000)^{2}}{200} - \sin{\left((3.0000000000) \right)}} = 3.5077592616$ $LaTeX: x_{2} = (3.5077592616) - \frac{- \frac{39 (3.5077592616)^{3}}{200} + \cos{\left((3.5077592616) \right)} + 9}{- \frac{117 (3.5077592616)^{2}}{200} - \sin{\left((3.5077592616) \right)}} = 3.4565810710$ $LaTeX: x_{3} = (3.4565810710) - \frac{- \frac{39 (3.4565810710)^{3}}{200} + \cos{\left((3.4565810710) \right)} + 9}{- \frac{117 (3.4565810710)^{2}}{200} - \sin{\left((3.4565810710) \right)}} = 3.4559645716$ $LaTeX: x_{4} = (3.4559645716) - \frac{- \frac{39 (3.4559645716)^{3}}{200} + \cos{\left((3.4559645716) \right)} + 9}{- \frac{117 (3.4559645716)^{2}}{200} - \sin{\left((3.4559645716) \right)}} = 3.4559644836$ $LaTeX: x_{5} = (3.4559644836) - \frac{- \frac{39 (3.4559644836)^{3}}{200} + \cos{\left((3.4559644836) \right)} + 9}{- \frac{117 (3.4559644836)^{2}}{200} - \sin{\left((3.4559644836) \right)}} = 3.4559644836$