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 0 to 3 of 1/√(x + 1) dx

∫ from 0 to 3 of 1/√(x + 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 4 of 1 divided by the square root of x + 3 dx with an error of magnitude less than 10-4. OK. So it appears for this particular prong we're ultimately trying to solve for the minimum number of subintervals required such that the trapezoidal rule will be able to approximate the integral that is provided to us for 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 22, B is 23, C is 72, and D is 73. OK. So, our first step that we need to take in order to solve this problem is we need to recall and use the error-bound formula for the trapezoidal rule. Which states that the absolute value of E subscript T, which I'll call ET for short, and the absolute value of ET will be less than or equal to parentheses of B minus a power of 3, all divided by 12 multiplied by n to the power of 2 multiplied by max. As, so this will be the maximum as A is 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 for this particular case, FX represents the second derivative of the function F X. And for this particular prom, we need to recall and note that FX will be equal to 1 divided by the square root of X + 3. And we also need to note that A will be equal to 1, and B will be equal to 4 for this particular problem. And we need to note that lowercase n will represent the number of subintervals for this problem. So, with that said, our second step now is we need to determine the second derivative of FX. So, let us note that we could rewrite our expression for the function FX as FX is equal to parentheses x + 3 to the power of 1/2. Which will mean that our first derivative of F X F X will be equal to 1/2 multiplied by parentheses of x + 3 to the power of 3 divided by 2. And finally, FX, the second derivative of FX, will be equal to 1/2 multiplied by 3 divided by 2 multiplied by. Parentheses of x + 3 to the power of -5 divided by 2, which will be equal to 3 divided by 4 multiplied by parentheses of x + 3 to the power of -5 divided by 2. So now, our next step is we need to find the maximum of F. So once again, the maximum of the absolute value of F. Of X on the closed interval, meaning square brackets of 1.4, is once again, square brackets indicate a closed interval. So we're asked to find the maximum of the absolute value of F X on the closed interval of 1.4. And we need to note that since parentheses X + 3 to the power of -5 divided by 2 is a decreasing function, that will mean, let's make a note, that will mean that the maximum will occur at X equals 1. Fantastic. So that will mean we can go ahead and write that the max, so once again we could write that the max. As X is an element of the closed interval 1.4 of the absolute value of F of X will be equal to 3 divided by 4, multiplied by parentheses of 1 + 3 to the power of -5 divided by 2. Which will be equal to 3 divided by 4 multiplied by 4 to the power of -5 divided by 2. Which will mean when we write our final answer in its most simplified form, it will be equal to 1 divided by 32. Fantastic. So with that said. Moving right along here. Our next step, so this will be step number 3. So step 3. We need to note that the max, so once again, the max as, so once again, this is the max as X is an element of the closed interval of 1.4 of the absolute value of F of X, which will be equal to 3 divided by. 4 multiplied by 1 divided by 32, which will be equal to 3 divided by 128. So now, For our fourth step is we now need to plug in our known variables back into our error formula. So we need to take parentheses of 4 minus 13. So in parentheses 4 minus 13 divided by 12 multiplied by n2, multiplied by 3 divided by 128. Is less than 10 to -4. So, what we need to do now is we need to begin to rearrange this expression to isolate and solve for N all by itself. And when we do that, we will find that N 2 is going to be greater than, so once again, N 2 is going to be greater than 3 of 3. So it's gonna be 3. So the power of 3 divided by 12 multiplied by 3 divided by 128 multiplied by 10 to -4. Fantastic. And now we need to take the square root of both sides to get rid of the power of 2 on N, and we will find that N is greater than the square root of. 3, so once again, N. is going to be greater than the square root of 3 to 3 divided by 12. Multiplied by. 3 divided by 128 multiplied by 10 to -4. And we need to note that the square root. Of 33 divided by 2 multiplied by 3 divided by 128 multiplied by 10 to -4. will be equal to 22.96, and we need to round up to the next whole number in order to ensure, as we should recall, that the number of sub-intervals is sufficient in order to keep the trapezoidal rule error within our desired limit, which will mean, therefore, that the minimum number of sub-intervals that are required will have to be. Equal to 23. So we need to take 22.96 and round it to the nearest whole number, which will give us 23, and that is our final answer. Hooray, we did it. So going back to look at our multiple choice answers, our correct and final answer without a doubt, will have to be multiple choice answer letter B, which once again is 23. 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 0 to 3 of 1/√(x + 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 0 to 3 of 1/√(x + 1) dx