1. a. Show that the equation x3 + 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 = x2 and y = x + 1 on the same graph.
b. Show that the equation x2 − 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 x3 − 5 = 0 to 2 decimal places given that a solution exists between 1.65 and 1.8.
4. Solve the equation x2 − 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:
x8 − 11x + 10 = 0
a. 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) = 2ex − 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.
