Answer:
C. 2.19
Step-by-step explanation:
You want to find the root of f(x) = sin(1+x) in the interval [1, 3] using the Bisection Method for up to 5 iterations.
Bisection method
The Bisection Method approximates the root of a function by repeatedly halving the interval in which the root is found. One iteration of the method evaluates the function at the midpoint of the interval, then replaces the end of the interval that gives a function value with the same sign.
Here, the starting interval is [1, 3], and we have f(1) > 0 and f(3) < 0. The midpoint of the interval is (1+3)/2 = 2. Evaluating f(2), we find f(2) > 0, so the midpoint (2) replaces the interval endpoint (0) that had f(x) > 0. The new interval is [2, 3], and the new midpoint is x = 5/2. Thus ends the first iteration.
Iterations
2nd iteration: f(5/2) < 0 ⇒ interval is [2, 5/2]; midpoint is 9/4
3rd iteration: f(9/4) < 0 ⇒ interval is [2, 9/4]; midpoint is 17/8
4th iteration: f(17/8) > 0 ⇒ interval is [17/8, 9/4]; midpoint is 35/16 ≈ 2.19
5th iteration: f(35/16) < 0 ⇒ interval is [17/8, 35/16]; midpoint is 69/32 ≈ 2.16
The fifth iteration gives an approximate root of 2.16, but that is not an answer choice. The fourth iteration gives an approximate root of 2.19, so we believe this is the answer you're looking for.
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Additional comment
The actual root is x = π-1 ≈ 2.14159265359.
Another iteration beyond those shown gives an approximate root of 2.14. We're calling that the 6th iteration, so its result is out of scope for this problem.
The attachment has an extra column to show the decimal values of the fractions used for the interval ends.