Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \cos{\left(x \right)}= \frac{57 x^{3}}{500} - 5$ using $LaTeX: \displaystyle x_0=3$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{57 x_{n}^{3}}{500} + \cos{\left(x_{n} \right)} + 5}{- \frac{171 x_{n}^{2}}{500} - \sin{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 3$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{57 (3.0000000000)^{3}}{500} + \cos{\left((3.0000000000) \right)} + 5}{- \frac{171 (3.0000000000)^{2}}{500} - \sin{\left((3.0000000000) \right)}} = 3.2895224475$ $LaTeX: x_{2} = (3.2895224475) - \frac{- \frac{57 (3.2895224475)^{3}}{500} + \cos{\left((3.2895224475) \right)} + 5}{- \frac{171 (3.2895224475)^{2}}{500} - \sin{\left((3.2895224475) \right)}} = 3.2762962381$ $LaTeX: x_{3} = (3.2762962381) - \frac{- \frac{57 (3.2762962381)^{3}}{500} + \cos{\left((3.2762962381) \right)} + 5}{- \frac{171 (3.2762962381)^{2}}{500} - \sin{\left((3.2762962381) \right)}} = 3.2762651442$ $LaTeX: x_{4} = (3.2762651442) - \frac{- \frac{57 (3.2762651442)^{3}}{500} + \cos{\left((3.2762651442) \right)} + 5}{- \frac{171 (3.2762651442)^{2}}{500} - \sin{\left((3.2762651442) \right)}} = 3.2762651441$ $LaTeX: x_{5} = (3.2762651441) - \frac{- \frac{57 (3.2762651441)^{3}}{500} + \cos{\left((3.2762651441) \right)} + 5}{- \frac{171 (3.2762651441)^{2}}{500} - \sin{\left((3.2762651441) \right)}} = 3.2762651441$