Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \cos{\left(x \right)}= \frac{23 x^{3}}{200} - 5$ using $LaTeX: \displaystyle x_0=3$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{23 x_{n}^{3}}{200} + \cos{\left(x_{n} \right)} + 5}{- \frac{69 x_{n}^{2}}{200} - \sin{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 3$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{23 (3.0000000000)^{3}}{200} + \cos{\left((3.0000000000) \right)} + 5}{- \frac{69 (3.0000000000)^{2}}{200} - \sin{\left((3.0000000000) \right)}} = 3.2787966869$ $LaTeX: x_{2} = (3.2787966869) - \frac{- \frac{23 (3.2787966869)^{3}}{200} + \cos{\left((3.2787966869) \right)} + 5}{- \frac{69 (3.2787966869)^{2}}{200} - \sin{\left((3.2787966869) \right)}} = 3.2664214753$ $LaTeX: x_{3} = (3.2664214753) - \frac{- \frac{23 (3.2664214753)^{3}}{200} + \cos{\left((3.2664214753) \right)} + 5}{- \frac{69 (3.2664214753)^{2}}{200} - \sin{\left((3.2664214753) \right)}} = 3.2663941666$ $LaTeX: x_{4} = (3.2663941666) - \frac{- \frac{23 (3.2663941666)^{3}}{200} + \cos{\left((3.2663941666) \right)} + 5}{- \frac{69 (3.2663941666)^{2}}{200} - \sin{\left((3.2663941666) \right)}} = 3.2663941665$ $LaTeX: x_{5} = (3.2663941665) - \frac{- \frac{23 (3.2663941665)^{3}}{200} + \cos{\left((3.2663941665) \right)} + 5}{- \frac{69 (3.2663941665)^{2}}{200} - \sin{\left((3.2663941665) \right)}} = 3.2663941665$