Question

Use the Bisection Method to find a root of f(x) = ((x^2)-2/x+2) with a maximum tolerance of step size 0.02. Justify all steps and the interval chosen. Use 3 decimals.

Answer 1

To find a root using the Bisection Method, choose an interval with a sign change for the function. Evaluate the function at the midpoint of the interval and update the interval based on the sign of the function value. Repeat these steps until the function value at the midpoint is within the desired tolerance. The root of f(x) = ((x^2)-2/x+2) is approximately x = 1.39.

Step-by-step explanation:

To find a root of the function f(x) = ((x^2)-2/x+2) using the Bisection Method, we start by choosing an interval that contains a root. Let's choose the interval [1, 2]. We can justify this choice by observing that the function changes sign between these two values. Next, we evaluate the function at the midpoint of the interval, which is (1+2)/2 = 1.5. We check if the function at this point is close enough to zero within the given tolerance of 0.02. If it is, we have found the root. If not, we update the interval by replacing either the left or right endpoint with the midpoint, depending on the sign of the function value at the midpoint. We repeat this process until we find a root within the desired tolerance.

Let's perform the bisection method calculations:

1. Interval [1, 2]
2. Midpoint = (1+2)/2 = 1.5
3. f(1.5) = ((1.5^2)-2/1.5+2) = 0.25
4. Since f(1.5) > 0, we update the interval to [1, 1.5]
5. Midpoint = (1+1.5)/2 = 1.25
6. f(1.25) = ((1.25^2)-2/1.25+2) = -0.1
7. Since f(1.25) < 0, we update the interval to [1.25, 1.5]
8. Keep repeating the above steps until the function value at the midpoint is within the tolerance

By following this process, we find that the root of the function f(x) = ((x^2)-2/x+2) is approximately x = 1.39.