Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \cos{\left(x \right)}= \frac{333 x^{3}}{500} - 7$ using $LaTeX: \displaystyle x_0=3$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{333 x_{n}^{3}}{500} + \cos{\left(x_{n} \right)} + 7}{- \frac{999 x_{n}^{2}}{500} - \sin{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 3$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{333 (3.0000000000)^{3}}{500} + \cos{\left((3.0000000000) \right)} + 7}{- \frac{999 (3.0000000000)^{2}}{500} - \sin{\left((3.0000000000) \right)}} = 2.3394077570$ $LaTeX: x_{2} = (2.3394077570) - \frac{- \frac{333 (2.3394077570)^{3}}{500} + \cos{\left((2.3394077570) \right)} + 7}{- \frac{999 (2.3394077570)^{2}}{500} - \sin{\left((2.3394077570) \right)}} = 2.1487322271$ $LaTeX: x_{3} = (2.1487322271) - \frac{- \frac{333 (2.1487322271)^{3}}{500} + \cos{\left((2.1487322271) \right)} + 7}{- \frac{999 (2.1487322271)^{2}}{500} - \sin{\left((2.1487322271) \right)}} = 2.1334723337$ $LaTeX: x_{4} = (2.1334723337) - \frac{- \frac{333 (2.1334723337)^{3}}{500} + \cos{\left((2.1334723337) \right)} + 7}{- \frac{999 (2.1334723337)^{2}}{500} - \sin{\left((2.1334723337) \right)}} = 2.1333783461$ $LaTeX: x_{5} = (2.1333783461) - \frac{- \frac{333 (2.1333783461)^{3}}{500} + \cos{\left((2.1333783461) \right)} + 7}{- \frac{999 (2.1333783461)^{2}}{500} - \sin{\left((2.1333783461) \right)}} = 2.1333783425$