Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \cos{\left(x \right)}= \frac{167 x^{3}}{1000} - 3$ using $LaTeX: \displaystyle x_0=3$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{167 x_{n}^{3}}{1000} + \cos{\left(x_{n} \right)} + 3}{- \frac{501 x_{n}^{2}}{1000} - \sin{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 3$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{167 (3.0000000000)^{3}}{1000} + \cos{\left((3.0000000000) \right)} + 3}{- \frac{501 (3.0000000000)^{2}}{1000} - \sin{\left((3.0000000000) \right)}} = 2.4625961282$ $LaTeX: x_{2} = (2.4625961282) - \frac{- \frac{167 (2.4625961282)^{3}}{1000} + \cos{\left((2.4625961282) \right)} + 3}{- \frac{501 (2.4625961282)^{2}}{1000} - \sin{\left((2.4625961282) \right)}} = 2.3883513917$ $LaTeX: x_{3} = (2.3883513917) - \frac{- \frac{167 (2.3883513917)^{3}}{1000} + \cos{\left((2.3883513917) \right)} + 3}{- \frac{501 (2.3883513917)^{2}}{1000} - \sin{\left((2.3883513917) \right)}} = 2.3870437424$ $LaTeX: x_{4} = (2.3870437424) - \frac{- \frac{167 (2.3870437424)^{3}}{1000} + \cos{\left((2.3870437424) \right)} + 3}{- \frac{501 (2.3870437424)^{2}}{1000} - \sin{\left((2.3870437424) \right)}} = 2.3870433406$ $LaTeX: x_{5} = (2.3870433406) - \frac{- \frac{167 (2.3870433406)^{3}}{1000} + \cos{\left((2.3870433406) \right)} + 3}{- \frac{501 (2.3870433406)^{2}}{1000} - \sin{\left((2.3870433406) \right)}} = 2.3870433406$