ROOT FINDING A GIVEN FUNCTION

Given the function $f(x)=3e^x-10(2{)}^x,$ find the root using the following methods:

a$)$ Use bisection with $10$ iterations. Look in the region $x_{r}in[0,5].$ After updating the bracket $10$ times, save the final bracket midpoint $x_m$ on $HW\{1\}.$

b$)$ Use Newton's method with $5$ iterations. Use an initial guess of $x_0=0.$ Save the newest guess $($after $5$ iterations$)$ on $HW\{2\}$

c$)$ Use Secant method with $5$ iterations. Use a set of initial guesses $x_{-1}=0,x_0=1.$ Save the newest guess $($after $5$ iterations$)$ on $HW\{3\}$

ROOT FINDING WITH TOLERANCE

Given the equation $3xcos(x)=x^{2}sin(x),$ find the first positive solution using

a$)$ Bisection method. Search on $x_{r}in[1,2].$ Plot both sides of the equation to demonstrate that a solution exists there. Then, iterate the bisection method until $x_u-x_l{10}^{-4}.$ Save the root, $x_m$ on HW$\{4\}.$ Add a marker to your plot for ordered pair of the root location that you found.

b$)$ Newton's method. Use an initial guess of $x_0=2.$ Iterate until Save the final guess on $HW\{5\}$

c$)$ Secant Method. Use initial guesses of $x_{-1}=1,x_0=2.$ Iterate using the same tolerances as part b$.$ Save the final guess on $HW\{6\}$