QuestionLet be a continuous function on the interval , such that for all . Let be the error in approximating the integral
using the Midpoint Rule with . Which of the following estimates is guaranteed to hold?
Select one:
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Studdy Solution
STEP 1
1. The function is continuous on the interval .
2. The second derivative of , , is bounded by 240 in absolute value for all .
3. We are using the Midpoint Rule with subintervals to approximate the integral.
4. We need to determine which error bound is guaranteed to hold for the Midpoint Rule approximation.
STEP 2
1. Determine the formula for the error bound of the Midpoint Rule.
2. Calculate the error bound using the given information.
3. Compare the calculated error bound with the provided options.
STEP 3
The error bound for the Midpoint Rule is given by:
where , , and .
STEP 4
Substitute the given values into the error bound formula:
Simplify the expression:
STEP 5
Compare the calculated error bound with the provided options. The calculated bound is .
The option that matches this bound is:
The guaranteed estimate that holds is .
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