Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \cos{\left(x \right)}= \frac{173 x^{3}}{1000} - 7$ using $LaTeX: \displaystyle x_0=3$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{173 x_{n}^{3}}{1000} + \cos{\left(x_{n} \right)} + 7}{- \frac{519 x_{n}^{2}}{1000} - \sin{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 3$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{173 (3.0000000000)^{3}}{1000} + \cos{\left((3.0000000000) \right)} + 7}{- \frac{519 (3.0000000000)^{2}}{1000} - \sin{\left((3.0000000000) \right)}} = 3.2782572964$ $LaTeX: x_{2} = (3.2782572964) - \frac{- \frac{173 (3.2782572964)^{3}}{1000} + \cos{\left((3.2782572964) \right)} + 7}{- \frac{519 (3.2782572964)^{2}}{1000} - \sin{\left((3.2782572964) \right)}} = 3.2625083349$ $LaTeX: x_{3} = (3.2625083349) - \frac{- \frac{173 (3.2625083349)^{3}}{1000} + \cos{\left((3.2625083349) \right)} + 7}{- \frac{519 (3.2625083349)^{2}}{1000} - \sin{\left((3.2625083349) \right)}} = 3.2624531158$ $LaTeX: x_{4} = (3.2624531158) - \frac{- \frac{173 (3.2624531158)^{3}}{1000} + \cos{\left((3.2624531158) \right)} + 7}{- \frac{519 (3.2624531158)^{2}}{1000} - \sin{\left((3.2624531158) \right)}} = 3.2624531152$ $LaTeX: x_{5} = (3.2624531152) - \frac{- \frac{173 (3.2624531152)^{3}}{1000} + \cos{\left((3.2624531152) \right)} + 7}{- \frac{519 (3.2624531152)^{2}}{1000} - \sin{\left((3.2624531152) \right)}} = 3.2624531152$