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Problem 25 Medium Difficulty

Find the approximations $ L_n $, $ R_n $, $ T_n $, and $ M_n $ for $ n = 5 $, 10, and 20. Then compute the corresponding errors $ E_L $, $ E_R $, $ E_T $, and $ E_M $. (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) What observations can you make? In particular, what happens to the errors when $ n $ is doubled?

$ \displaystyle \int_0^1 xe^x\ dx $

Answer

$$\begin{array}{|r|c|c|c|c|} \hline n & L_{n} & R_{n} & T_{n} & M_{n} \\ \hline 5 & 0.742943 & 1.286599 & 1.014771 & 0.992621 \\ 10 & 0.867782 & 1.139610 & 1.003696 & 0.998152 \\ 20 & 0.932967 & 1.068881 & 1.000924 & 0.999538 \\ \hline \end{array}$$
$$\begin{array}{|r|c|c|c|c|}\hline n & E_{L} & E_{R} & E_{T} & E_{M} \\ \hline 5 & 0.257057 & -0.286599 & -0.014771 & 0.007379 \\ 10 & 0.132218 & -0.139610 & -0.003696 & 0.001848 \\ 20 & 0.067033 & -0.068881 & -0.000924 & 0.000462 \\ \hline \end{array}$$

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Video Transcript

problem. 25 for this problem, you should not. Is that he wants you to computer corresponding arrows. Now, that estimate estimate harassed you can use this. Asked with arrows. Ira Bond. Right. But the corresponding arrest it needs to It needs the I might just the same because, like Al or are Or, um or teeth just minus minus. Then you can find Yell yard. You can yet, right? I don't know. And you need to know how to compute our Artie and PNM. You can use the here, the formula here and how its left hand side by our is the right hand side. We have ah formula here. Ah, here's here's Al How is equal to this one? Ours even knew this one. Just confused. There's no tricks here. Just compute for ending for any single 5 10 or and 20 computer attacks future tax here. And just all the same thing is no one. So compute those things. And of course, Bonnie era. You need to compute I how to compute. I hear, uh, you can find a You find a bunking from 0 to 1. Yeah, it is equal toe X minus one that guys That's because the directive off dysfunction is equal to that one. That's a new tin likeness formula, right? If we go to Daryl minus minus, one is just one. So the eyes one and you can compute this one. But just about I cast the Hera is smaller because you created the interval into more small intervals. So there are approximately is more close to the real value I