Chapter 4 - Applications of the Derivative - 4.8 Newton's Method - 4.8 Exercises - Page 316: 34

$${x_1} \approx 6.100986,{\text{ }}{x_2} \approx 6.762697$$

$$\eqalign{ & f\left( x \right) = \frac{x}{6} - \sec x{\text{ on }}\left[ {0.8} \right] \cr & f\left( x \right) = 0 \cr & \frac{x}{6} - \sec x = 0 \cr & {\text{Differentiating}} \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {\frac{x}{6} - \sec x} \right] \cr & f'\left( x \right) = \frac{1}{6} - \sec x\tan x \cr & {\text{Use the procedure Newton's Method for Approximations}} \cr & {\text{Roots of }}f\left( x \right) = 0{\text{ }}\left( {{\text{See page 311}}} \right) \cr & {x_{n + 1}} = {x_n} - \frac{{f\left( {{x_n}} \right)}}{{f'\left( {{x_n}} \right)}} \cr & {\text{Substituting }}f\left( {{x_n}} \right){\text{ and }}f'\left( {{x_n}} \right){\text{ the Newton's method takes the}} \cr & {\text{form}} \cr & {x_{n + 1}} = {x_n} - \frac{{\frac{{{x_n}}}{6} - \sec {x_n}}}{{\frac{1}{6} - \sec {x_n}\tan {x_n}}} \cr & {x_{n + 1}} = {x_n} - \frac{{{x_n} - 6\sec {x_n}}}{{1 - 6\sec {x_n}\tan {x_n}}} \cr & {\text{From the graph we can see that the first possible initial }} \cr & {\text{approximation is }}x = 6 \cr & {\text{Therefore}}{\text{,}} \cr & n = 0,{\text{ }}{x_0} = 6 \cr & {x_1} = 6 - \frac{{6 - 6\sec 6}}{{1 - 6\sec 6\tan 6}} \approx 6.092991 \cr & {\text{Computing the same calculation we obtain:}}{\text{.}} \cr & {x_2} \approx 6.101045 \cr & {x_3} \approx 6.100985 \cr & {x_4} \approx 6.100986 \cr & {x_6} \approx 6.100986 \cr & \cr & {\text{From the graph we can see that the second possible initial }} \cr & {\text{approximation is }}x = 7 \cr & {\text{Therefore}}{\text{,}} \cr & {x_1} = 7 - \frac{{7 - 6\sec 7}}{{1 - 6\sec 7\tan 7}} \approx 6.674161 \cr & {\text{Computing the same calculation we obtain:}}{\text{.}} \cr & {x_2} \approx 6.817514 \cr & {x_3} \approx 6.742460 \cr & {x_4} \approx 6.771877 \cr & {x_6} \approx 6.758919 \cr & {x_6} \approx 6.764317 \cr & {x_7} \approx 6.762016 \cr & {x_8} \approx 6.762695 \cr & {x_8} \approx 6.762697 \cr & {\text{The approximation of the roots are:}} \cr & {x_1} \approx 6.100986,{\text{ }}{x_2} \approx 6.762697 \cr} $$