Problem $1.[60$ points$]$ Use Jupyter notebook for this problem. Please solve all the problems i$)$ a $-$ii$)$ d and provide complete Python Code solutions.

Consider the equation

$tanx=x$

a$.[15$ points$]$ Find the first $5$ digits of the smallest positive root $($i$.$e$.,$ the smallest root greater

than $0)$ using Bisection method. What was the initial interval used for finding the root and

why did you choose it$?$

b$.[15$ points$]$ Find the first $5$ digits of the same root in part $($a$)$ using Newton's method. What

was the initial value used for finding the root?

c$.[15$ points$]$ Modify the initial value found in part $($b$)$ until Newton's method no longer con$-$

verges. Discuss why Newton's method fails to converge to the smallest positive root when

starting at the modified initial value.

d$.[15$ points$]$ Create a plot comparing the convergence rate of Bisection method and Newton's

method. Discuss your result.