Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \cos{\left(x \right)}= \frac{67 x^{3}}{1000} - 2$ using $LaTeX: \displaystyle x_0=3$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{67 x_{n}^{3}}{1000} + \cos{\left(x_{n} \right)} + 2}{- \frac{201 x_{n}^{2}}{1000} - \sin{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 3$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{67 (3.0000000000)^{3}}{1000} + \cos{\left((3.0000000000) \right)} + 2}{- \frac{201 (3.0000000000)^{2}}{1000} - \sin{\left((3.0000000000) \right)}} = 2.5902854730$ $LaTeX: x_{2} = (2.5902854730) - \frac{- \frac{67 (2.5902854730)^{3}}{1000} + \cos{\left((2.5902854730) \right)} + 2}{- \frac{201 (2.5902854730)^{2}}{1000} - \sin{\left((2.5902854730) \right)}} = 2.5815897545$ $LaTeX: x_{3} = (2.5815897545) - \frac{- \frac{67 (2.5815897545)^{3}}{1000} + \cos{\left((2.5815897545) \right)} + 2}{- \frac{201 (2.5815897545)^{2}}{1000} - \sin{\left((2.5815897545) \right)}} = 2.5815859185$ $LaTeX: x_{4} = (2.5815859185) - \frac{- \frac{67 (2.5815859185)^{3}}{1000} + \cos{\left((2.5815859185) \right)} + 2}{- \frac{201 (2.5815859185)^{2}}{1000} - \sin{\left((2.5815859185) \right)}} = 2.5815859185$ $LaTeX: x_{5} = (2.5815859185) - \frac{- \frac{67 (2.5815859185)^{3}}{1000} + \cos{\left((2.5815859185) \right)} + 2}{- \frac{201 (2.5815859185)^{2}}{1000} - \sin{\left((2.5815859185) \right)}} = 2.5815859185$