Science:Math Exam Resources/Courses/MATH105/April 2013/Question 01 (j)
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Question 01 (j) 

ShortAnswer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Find a bound for the error in approximating using Simpson’s rule with . Do not write down Simpson’s rule approximation . 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Given , an even integer, and an upper bound for all in the interval , the error that comes from using Simpson's Rule is as follows: where I is the integral in question. How would you find K? 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Given , an even integer, and an upper bound for all in the interval , the error that comes from using Simpson's Rule is as follows: where . We know that , and . To find an upper bound on the fourth derivative, we first compute that
As the fourth derivative is a decreasing function in x, its maximum occurs at the smallest possible value, when . Hence a value of K is given by
Hence, we have a bound on the error is given by
