Note your answer should be justified and will depend on the

Question

Problem 1: Show graphically the e^-x = sin(x/2) has at least one solution. Find an interval [a, b] containing the smallest solution, ?, for which the bisection method will converge to ?. Use the bisection method with a hand calculator or a computer to find the smallest root accurate to within ∈ = 0.01. Note: your answer should be justified and will depend on the choice of [a, b]. Produce a table of your iterates with the following columns: n, a_n, b_n, c_n, b_n - c_n. With a₁ = a and b₁ = b (Hand-written is fine.) for example:

n	an	bn	cn	bn - cn
. . .	. . .	. . .	. . .	. . .

Problem 2: Let ? be the unique root of f(x). Find an interval [a, b] containing ? for which the bisection method will converge to ?. Then, estimate the number of iterates needed to find ? within an accuracy of ∈ = 10^-9. Note: your answer should be justified and will depend on the choice of [a, b].

F(x) = x-3/1+x⁴

Problem 3: Show graphically that x = e^-x has one solution. Use the bisection method with a hand calculator or computer to find the root accurate to within ∈ = 0.01. Produce a table of your iterates with the following columns: n, a_n, b_n, c_n, b_n - c_n. (Hand written is fine.) For example:

n	an	bn	cn	bn - cn
. . .	. . .	. . .	. . .	. . .

Problem 4: Let ? be the unique root of f(x). Find an interval [a, b] containing ? for which the bisection method will converge to ?. Then, estimate the number of iterates needed to find ? within an accuracy or ∈ = 10^-9. Note: Your answer should be justified and will depend on the choice of [a, b].

(a). f(x) = 31x³ - x² + 27x - 2125

(b). f(x) = e² - x - 2