Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \cos{\left(x \right)}= \frac{127 x^{3}}{500} - 2$ using $LaTeX: \displaystyle x_0=1$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{127 x_{n}^{3}}{500} + \cos{\left(x_{n} \right)} + 2}{- \frac{381 x_{n}^{2}}{500} - \sin{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 1$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (1.0000000000) - \frac{- \frac{127 (1.0000000000)^{3}}{500} + \cos{\left((1.0000000000) \right)} + 2}{- \frac{381 (1.0000000000)^{2}}{500} - \sin{\left((1.0000000000) \right)}} = 2.4258457606$ $LaTeX: x_{2} = (2.4258457606) - \frac{- \frac{127 (2.4258457606)^{3}}{500} + \cos{\left((2.4258457606) \right)} + 2}{- \frac{381 (2.4258457606)^{2}}{500} - \sin{\left((2.4258457606) \right)}} = 1.9627316816$ $LaTeX: x_{3} = (1.9627316816) - \frac{- \frac{127 (1.9627316816)^{3}}{500} + \cos{\left((1.9627316816) \right)} + 2}{- \frac{381 (1.9627316816)^{2}}{500} - \sin{\left((1.9627316816) \right)}} = 1.8843596621$ $LaTeX: x_{4} = (1.8843596621) - \frac{- \frac{127 (1.8843596621)^{3}}{500} + \cos{\left((1.8843596621) \right)} + 2}{- \frac{381 (1.8843596621)^{2}}{500} - \sin{\left((1.8843596621) \right)}} = 1.8821814556$ $LaTeX: x_{5} = (1.8821814556) - \frac{- \frac{127 (1.8821814556)^{3}}{500} + \cos{\left((1.8821814556) \right)} + 2}{- \frac{381 (1.8821814556)^{2}}{500} - \sin{\left((1.8821814556) \right)}} = 1.8821797905$