You are planning to use Simpson’s Rule to estimate the value o...

Question

You are planning to use Simpson’s Rule to estimate the value of the integral Estimate ∫ from 1 to 2 of f(x) dx with an error magnitude less than 10⁻⁵ using Simpson’s Rule.

You have determined that |f⁽⁴⁾(x)| ≤ 3 throughout the interval of integration. How many subintervals should you use to ensure the required accuracy?

(Remember that for Simpson’s Rule the number of subintervals must be even.)

Accepted Answer

Hello. In this video we are told to suppose that we know or we want to approximate the value of the integral from 0 to 3 of a function g of x by using Simpson's rule so that the absolute error is at most 104. You know that throughout 0 to 3, the fourth derivative satisfies that it is less than or equal to 5. How many subintervalsn are you required to guarantee the errors criterion? So let's go ahead and just recall the error bound for the Simpson's rule. The error bound for Simpson's rule eS has to be less than or equal to b minus a raise to the power of 5 divided by 180 multiplied by n to the 4th, and this will be multiplied by the maximum value. Of x between our defined boundaries. Of the fourth derivative of g. So, let's go ahead and take a look at the given information. First, we are given an interval and we are working on the interval from 0 to 3, so we know that A is equal to 0 and B is equal to 3. We are also told that the value of the 4th derivative of g has to be less than or equal to 5. So what this means is that the maximum value. Of x between the values of 0 and 3 of the fourth derivative of g of x has to equal to 5. So by plugging this in the error boundary, we know that we want. Well, by plugging in the information, we can write the inequality to be 3 minus 0, raise to the power of 5, divided by 180, multiplied by n 4. And this is going to be multiplied by 5. Now, if we look back at the at the problem, we don't want this value to exceed 104. So what this means is that this quantity has to be less than or equal to 104. So all we need to do in this case is we need to simplify and solve for n. Starting with the numerator 3 minus 0 to the fifth power is going to simplify to be 243, and 243 multiplied by 5 is going to leave us with 1,215 divided by 180 and to the 4th power, and this is less than or equal to 10 rates to the 4th. Now the next thing we're going to do is we're going to cross multiply. Cross-multiplying allows us to rewrite the inequality to be 1,215 divided by 180. Multiply by 10 to the 4th and this will be less than or equal to n to the 4th. Further simplifying is going to give us 67,500 less than are equal to n to the 4th power. And if we were to take the 4th root of both sides of this inequality and reorder, we will have n must be greater than equal to the 4th root of 67,500. Now here with the help of a calculator this is going to approximate to be 16.12 and since Simpson's rule requires n to be an even value, we are going to round up to the next even integer, and that means the value of n that we want is going to be 18, which will be the solution to the problem. And with that being said, the overall solution is going to be option B. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

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