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Numerade Educator

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Problem 45 Hard Difficulty

Sketch the graph of a continuous function on $ [0, 2] $ for which the Trapezoidal Rule with $ n = 2 $ is more accurate than the Midpoint Rule.

Answer

Midpoint Rule gives $M_{2}=\frac{2-0}{2}[f(0.5)+f(1.5)]=1[0+0]=0$
Trapezoidal Rule is more accurate.

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Video Transcript

Okay, This question wants us to find a function. Such the trap is I'd rule is a better approximation in the mid point, which usually isn't the case sort of do this. Let's find a function that totally messes with the midpoint approximation. So to do that, we gotta find a function that does it not represent itself at the mid points, which means acts totally different at the mid points. And it does everywhere else, but it still has to be continuous. So let's say we have something like this. So it starts it one, then it hit zero at the midpoint goes back up. Then it goes back down a hits 1.5 again. And then you see the pattern so we can see for sure that area from midpoint, it's just gonna be Delta X plus the sum of f of mid points, which is just Delta X Times zero, which equals zero. Because the only points were using the sample from are the zeros right here. And if you're wondering what this function is, I just picked absolute value of co sign of Pi X just to get the period and positive definite nous that we wanted. So for the trap is oId. On the other hand, we get 1/2 times f of zero plus two F of one plus f of two, which is 1/2 times one plus one plus one. So area from the trap is oId is too, and actual area. Oh, and just for reference, we should write again. The area from midpoint is equal to zero, so we know the actual area is positive. So we should already have an intuition that the midpoint is a terrible approximation. But if we actually calculate this integral out, it's approximately 1.27 So the area of the trap is oId is closer.