66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in ...

Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

67. Let f(x) = √(x³ + 1).

d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to find an upper bound for the absolute error in the approximation of the interval from 2 to 7 of the square root of the quantity of X2 + 4 in quantity DX using the midpoint rule with N equal to 20 subintervals. Use the fact that the absolute value of F of X is less than or equal to 1 divided by the quantity of 4 multiplied by the square root of 2 in quantity on the closed interval from 2 to 7. Round your answer to 4 decimal places, and we're given 4 possible choices as our answers. For choice A, we have 0.0023. For choice B, we have 0.0047. For choice C, we have 0.0163, and for choice D we have 0.0183. So we're asked to find the upper bound for the absolute error in the approximation of the integral that we're given the problem using the midpoint rule. So, we call that the error bound for the midpoint rule is given by the formula of E M. is less than or equal to K multiplied by the quantity of B minus a in quantity cubed divided by the quantity of 24 multiplied by n squared in quantity. So we just need to identify each of these variables in this formula. So if we look at the information that we're given, we see that k is going to be equal to 1 divided by the quantity of 4 multiplied by the square root of 2 in quantity, since we're told that the magnitude of our second derivative is less than or equal to 1 divided by the quantity of 4 multiplied by the square 2 in quantity. Next, we need to identify A and B. So A is going to be equal to our lower limit of our integration. So A is going to be equal to 2. B is going to be equal to the upper limit, that's going to be equal to 7, and N is going to be equal to 20, since that's the value that we're given in problem. So, substituting these values in our formula, we see that EM is going to be less than or equal to. One divided by. The quantity of 4 multiplied by the square root of 2 in quantity, that is multiplied by the quantity of 7 minus 2 in quantity cubed divided by the quantity of 24 multiplied by 20 squared in quantity. And so when we evaluate this expression using our calculator, we see that Em is going to be less than or equal to 0.0023. And here we've just round it to 4 decimal places. So this value is going to be the upper bound for absolute error. So that is the answer that we're looking for for this problem. And when we compare our answer here to our answer choices, we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.67. Let f(x) = √(x³ + 1).d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).

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66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
67. Let f(x) = √(x³ + 1).
d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).