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Problem 48 Hard Difficulty

Show that if $ f $ is a polynomial of degree 3 or lower, then Simpson's Rule gives the exact value of $ \int_a^b f(x)\ dx $.

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Video Transcript

Okay. This question wants us to prove that Simpson's rule gives the exact area under the curve for pollen. No meals with degree of that, most three. So initially at this problem statement, you might not know it too. D'oh! Because it seems quite complicated. But there's a little trick we know using the error expression for the Simpsons rule. So based on the book, we're given this formula that the air from the Simpson approximation is at most this constant K times B minus eight of the fifth over 1 80 end of the fourth Okay, s. So how does that help us? Well, we're also told that Kay is any constant bigger than the fourth derivative of X for all ex. So we can also say that Kay is proportional to the maximum value of F quadruple prime of X and f of X. Is any cubic excused plus B x squared plus C x plus D. So what's the fourth derivative of a cubic polynomial? Well, if you differentiate x cubed four times, he'd zero drift. Durand shed X squared four times you zero x four tons of zero. And a constant, of course, is here. So the fourth derivative of our function zero. And since r K is anything bigger than our equal to the max value of our fourth derivative, well, we could just say Okay, equal zero for Q Bix. So air of the Simpson approximation is listening equal to zero. So we can conclude that since the error zero, the rule must give the exact value of the integral. So again, a very tough looking problem. Fun. We just used the fact that this constant K has to be proportional to the max value of the fourth derivative, which we know is zero. So the error must be zero. And if there's zero air, it means that Simpson's role exactly gives us theatre girls value, which is a pretty neat fact.