Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \cos{\left(x \right)}= \frac{171 x^{3}}{500} - 9$ using $LaTeX: \displaystyle x_0=3$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{171 x_{n}^{3}}{500} + \cos{\left(x_{n} \right)} + 9}{- \frac{513 x_{n}^{2}}{500} - \sin{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 3$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{171 (3.0000000000)^{3}}{500} + \cos{\left((3.0000000000) \right)} + 9}{- \frac{513 (3.0000000000)^{2}}{500} - \sin{\left((3.0000000000) \right)}} = 2.8694424716$ $LaTeX: x_{2} = (2.8694424716) - \frac{- \frac{171 (2.8694424716)^{3}}{500} + \cos{\left((2.8694424716) \right)} + 9}{- \frac{513 (2.8694424716)^{2}}{500} - \sin{\left((2.8694424716) \right)}} = 2.8644713492$ $LaTeX: x_{3} = (2.8644713492) - \frac{- \frac{171 (2.8644713492)^{3}}{500} + \cos{\left((2.8644713492) \right)} + 9}{- \frac{513 (2.8644713492)^{2}}{500} - \sin{\left((2.8644713492) \right)}} = 2.8644643525$ $LaTeX: x_{4} = (2.8644643525) - \frac{- \frac{171 (2.8644643525)^{3}}{500} + \cos{\left((2.8644643525) \right)} + 9}{- \frac{513 (2.8644643525)^{2}}{500} - \sin{\left((2.8644643525) \right)}} = 2.8644643525$ $LaTeX: x_{5} = (2.8644643525) - \frac{- \frac{171 (2.8644643525)^{3}}{500} + \cos{\left((2.8644643525) \right)} + 9}{- \frac{513 (2.8644643525)^{2}}{500} - \sin{\left((2.8644643525) \right)}} = 2.8644643525$