Answer:
D. 0.391
Explanation:
You want the approximate solution to f(x) = x² -3x +1 = 0 on the interval [0, 1] using 5 iterations of the Bisection Method.
Iteration
The Bisection Method makes one iteration by finding the function value at the midpoint of the interval. The midpoint replaces the end of the interval whose function value has the same sign. At the end of one iteration, the midpoint of the halved interval is calculated.
Start:
f(0) = 1, f(1) = -1. Interval: [0, 1]; midpoint: 1/2
First iteration:
f(1/2) < 0. Interval: [0, 1/2]; midpoint: 1/4
Second iteration:
f(1/4) > 0. Interval: [1/4, 1/2]; midpoint: 3/8
Third iteration:
f(3/8) > 0. Interval: [3/8, 1/2]; midpoint: 7/16
Fourth iteration:
f(7/16) < 0. Interval: [3/8, 7/16]; midpoint: 13/32
Fifth iteration:
f(13/32) < 0. Interval: [3/8, 13/32]; midpoint: 25/64
Solution
The approximate solution after 5 iterations is x ≈ 25/64 ≈ 0.391.
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Additional comments
The approximate solution in the interval to full calculator precision is 0.38196601125. The exact solution is 1.5-√1.25.
You will notice that the function values for the ends of the interval [0, 1] are [positive, negative]. So, when the function value at the midpoint is negative, that point replaces the right end of the interval.
Of course, the midpoint is the average of the interval end values.
On average, it takes about 3.3 iterations to improve the accuracy of the solution by 1 decimal place.