This is a homework problem using MATLAB, please help! :)
1. For 0 < x consider the equation x + ln x = 0.
a. Show analytically that the function f (x) = x + ln x has exactly one root 0 < ? < ? (i.e. show existende and uniqueness).
b. Plot a graph of the function f (x) = x + ln x on the interval [0.1, 1].
c. Explain why you can use the bisection method with an initial interval [0.5, 0.6] to approximate the root (i.e., why this choice for an initial interval is a valid one).
d. How many iterations (steps) of the bisection method are needed to guarantee that the approximate root is within 10^?8 of the exact root? (Assume the initial interval is [0.5, 0.6].)
2. Use the bisection method to write a program approximating a solution of the equation x + ln x = 0 which is within 10^?8 of the exact solution.
1. For 0 < x consider the equation x + ln x = 0.
a. Show analytically that the function f (x) = x + ln x has exactly one root 0 < ? < ? (i.e. show existende and uniqueness).
b. Plot a graph of the function f (x) = x + ln x on the interval [0.1, 1].
c. Explain why you can use the bisection method with an initial interval [0.5, 0.6] to approximate the root (i.e., why this choice for an initial interval is a valid one).
d. How many iterations (steps) of the bisection method are needed to guarantee that the approximate root is within 10^?8 of the exact root? (Assume the initial interval is [0.5, 0.6].)
2. Use the bisection method to write a program approximating a solution of the equation x + ln x = 0 which is within 10^?8 of the exact solution.