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 (a) the Trapezoidal 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 subintervals needed so that the trapezoidal rule approximates the integral, the integral evaluated from 1 to 23 multiplied by x 2 + 2 dx with an error of magnitude less than 10 4. OK, so it appears for this particular prompt, we're ultimately asked to determine what the minimum number of sub-intervals are that are required, such that the trapezoidal rule approximates the integral that is provided to us by the prom itself, that has an error of magnitude that is less than 10-4. 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 116, B is 117, C is 367, and D is 368. OK. So, our first step that we need to take is we need to recall how to determine the error for a trapezoidal rule, and we need to note that the error for the trapezoidal rule, which is bounded by, means we have to consider the boundaries that are given to us by the problem itself. So we go ahead and write that the absolute value of capital E subscript T, which I'll call ET for short, so the absolute value of ET. Is less than or equal to parentheses of b minus a to the power of 3 divided by 12 multiplied by n to the power of 2 multiplied by the max of a less than or equal to x and x is less than or equal to b of the absolute value of f of x. And we need to note that F of X, we need to note where F of X is equal to 3 multiplied by x to the 2 + 2, and that A is equal to -1, and B is equal to 2. And we need to recall a note based on the prom itself that lowercase n will represent the number of subintervals. So, with that said, our second step now is we need to compute FX, where FX is the second derivative of FX. So we need to recall once again that F of X is equal to 3 multiplied by X to the power of 2 + 2. Which will mean that FX, the first derivative of FX, will be equal to 6X, which will mean that the second derivative F of X will be simply equal to 6. So that will mean that the max of the absolute value of FX will be equal to 6, and once again, this will be 4. X is an element of square brackets of -1.2. So, this will be true for X is an element of the closed interval of -1.2, because square brackets represent a closed interval. And our third step now is we need to plug the values into our error bound, so we need to take A equal to -1 and B is equal to 2. So that would mean that B minus A. Will be equal to 3. And once again, we need to note that max of the absolute value of FX will be equal to 6. So that will mean we can go ahead when we plug in our values into our error bound equation. We can go ahead and write that the absolute value of e oft is less than or equal to parentheses of 33 divided by 12 multiplied by n 2 multiplied by 6. And we will find that this is equal to, when we write it in its most simplified form, it will be equal to 27 divided by 2 multiplied by n to the power of 2. However, we want this to be less than 10-4, so we need to go ahead and write that 27 divided by. 2 multiplied by n to the 2 is less than 10-4. So, our fourth step now, so this is step number 4, is we now need to isolate and solve for N. So when we begin to do that, we will find that 27 divided by 2 multiplied by n to the power of 2 will be less than 10 to -4. And when we rearrange to get N all by itself on one side, we will find that N to the power of 2. Will be greater than 27 divided by 2 multiplied by 10 to -4. Which on the square root both sides, we will find that N will be greater than the square root of 135,000. Awesome, so. So now what we need to do is we need to take the square root of 135,000. So we need to note, so let's make a note here in blue, that the square root of 135,000 is equal to 367.42. And we need to note that when we round up to the next whole number, and we need to do this in order to ensure that the error bound is satisfied, so once again we need to round up to the nearest or the next whole number in order to ensure that the error bound is satisfied. So that will mean that the approximation error will be less than the desired tolerance of 10 to -4, because when we once again round to the nearest whole number, we will find that it's approximately equal to 368. So therefore, once again, to recap, because this is important, the approximation error will be less than the desired tolerance of 10 to -4. So therefore, as a result, the minimum number of subintervals needed will be 368, and that is our final answer. Hooray, we did it. And this corresponds to multiple choice answer letter D, which once again is 368. 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 (a) the Trapezoidal 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 (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (x² + 1) dx