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:
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 in .
3. We are using the Midpoint Rule with subintervals to approximate the integral.
4. We need to use the error bound formula for the Midpoint Rule to determine which estimate is guaranteed to hold.
STEP 2
1. Recall the error bound formula for the Midpoint Rule.
2. Calculate the width of each subinterval.
3. Apply the error bound formula using the given information.
4. Compare the calculated error bound with the given options.
STEP 3
Recall the error bound formula for the Midpoint Rule:
The error in approximating the integral using the Midpoint Rule is given by:
where is the interval of integration, and is the number of subintervals.
STEP 4
Calculate the width of each subinterval:
The interval is divided into subintervals. The width of each subinterval is:
STEP 5
Apply the error bound formula using the given information:
Given that , the maximum value of is 240. Substitute , , , and into the error bound formula:
STEP 6
Compare the calculated error bound with the given options:
The calculated error bound is . Therefore, the estimate that is guaranteed to hold is:
The correct choice is .
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