In Exercises 11–22, estimate the minimum number of subinterval...

Question

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (x² + 1) dx

∫ from -1 to 1 of (x² + 1) dx

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the minimum number of sub-intervals needed so that Simpson's rule approximates the integral, the interval evaluated from -1 to 2 parentheses of 3 multiplied by X2 + 2 DX with an error of magnitude less than 10 of -4. OK. So it appears for this particular prom we're asked to determine the minimum number of sub-intervals needed such that Simpson's rule approximates the interval that is provided to us by the prom itself with a specific error of magnitude that is less than 10 of -4. So now that we know what we're ultimately trying to solve for, let's take a moment to read off our multiple choice answers to see what our final answer may be. A is 1, B is 2, C is 6, and D is 58. OK. So our first step that we need to take in order to solve this particular problem. we need to recall and use the equation for the error bound for Simpson's rule, which states that the absolute value of E subscript S, which I'll just call ES for short. So the absolute value of ES is less than or equal to parenthesis of B minus A in parenthesis to the power of 5. Divided by 180. Multiplied by N to the power of 4. Multiplied by the maximum of X is an element of the closed interval of A. B. Of the absolute value of F to the power of 4 of X. Fantastic. So, we need to note that when we have square brackets, they enter, they once again square brackets represent a closed interval. If they were parentheses, then that would mean that it's an open interval. So with that said, we need to note that the square brackets of A, B, the closed interval of AB, Is the interval where A is equal to -1 and B is equal to 2. Let us also note that lowercase n will represent the number of subintervals, and we need to recall a note for this particular prong. It must be an even value, so I put a letter E for even. So with that said, we need to recall and note also based on our prior knowledge and what we're told by the problem itself. We could write that F of X is equal to 3x of the power 2 + 2, and we also need to recall a note that F to the power of 4 of X is the 4th derivative of this function of F. Fantastic. So with that said, we now need to compute the 4th derivative. So we will find that F of X is equal to once again 3 multiplied by x 2 + 2, and F of X, the first derivative of F of X is equal to 6 X. And the second derivative of F of X will be equal to simply 6, which will mean that our third derivative of F of X, so once again our third derivative of F of X will be equal to 0, which will mean that F to the power of 4 of X or 4th derivative will be simply equal to 0. Fantastic. So moving right along here, we now need to plug in our known variables into our error formula, and we need to note, let's make a note that sends F to the power of 4 U X is equal to 0. That will mean that our area once again. Based on our prior knowledge, we need to note that since the 4th derivative of FF X is equal to 0, the error formula will become, we can write it as the absolute value of ES is less than or equal to parenthesis of B minus A in parent to the power of 5 divided by 180 multiplied by N to the power of 4. Multiplied by 0, which will be equal to 0, because once again, based on our prior knowledge, any value multiplied by 0 will be 0. And also based on our prior knowledge, because the 4th derivative is 0, that means that the integrand is a quadratic value. It's quadratic in nature, so we need to recall a note that Simpson's rule gives the exact value for any even n value, even if just N is equal to 2. So therefore, without a doubt, the minimum number of subintervals that are needed will be N is equal to 2. And this is our correct and final answer. Hooray, we did it. And this corresponds to multiple choice answer letter B, which once again is 2. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)∫ from -1 to 1 of (x² + 1) dx

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In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (x² + 1) dx