1. a. Show that the equation x^{3} + 3x − 2 = 0 has a root between x = 0 and x = 1.

b. Use the bisection method to find this root to 2 decimal places.

2. a. Sketch the functions y = x^{2} and y = x + 1 on the same graph.

b. Show that the equation x^{2} − x − 1 = 0 has a root between 0 and -1.

c. This equation also has a positive root. Between which two positive integers does this root lie?

d. Use the bisection method to find the positive root to 1 decimal places.

3. Use the bisection method to solve the equation x^{3} − 5 = 0 to 2 decimal places given that a solution exists between 1.65 and 1.8.

4. Solve the equation x^{2} − 6x + 3 = 0 to 2 decimal places using the bisection method and given starting interval x = [5, 6].

5. Louise invests some of the profits from a business. In order to understand one particular investement she needs to find a solution greater than 1 to the equation:

x

^{8}− 11x + 10 = 0a. Show that this equation has a solution between x = 1.05 and x = 1.15

b. Calculate two iterations of the bisection method to solve the equation starting with the interval [1.05, 1.15]

6. Consider the function f(x) = 2e^{x} − 2x − 3

a. Show that there is a root to the equation f(x) = 0 in the interval [0, 2]

b. Using [0, 2] as the starting interval, calculate two iterations of the bisection method to solve the equation f(x) = 0. State the interval obtained at the end of the second iteration.