Question 05 (c)
Full-Solution Problem. Justify your answer and show all your work. Simplification of the answer is not required.
An unknown function has the following values: , , , ,
It is known that the fourth derivative lies between and on the interval . What is the largest possible value that could have? You may use the fact that if on the interval , then the error in using to approximate has absolute value less than or equal to .
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!
The previous part of this question gives us the value of the Simpson's rule approximation. Find the worst value the error can be and then sum this worst value with the approximation from the previous part.
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.
Using the hint, we plug in into the error bound and notice that the error lies between
Evaluating these bounds, we get that
and one last simplification yields
Now, let's use the the approximation and notice that by definition, we have that
Notice that in magnitude, the value
and so we have that the largest the integral can be is the approximation added onto this largest error bound. This gives
completing the question.