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Problem 19 Medium Difficulty

(a) Find the approximations $ T_8 $ and $ M_8 $ for the integral $ \displaystyle \int_0^1 \cos (x^2)\ dx $.
(b) Estimate the errors in the approximations of part (a).
(c) How large do we have to choose $ n $ so that the approximations $ T_n $ and $ M_n $ to the integral in part (a) are accurate to within 0.0001?

Answer

a.$T_{8}=.902333$
$M_{8}=.905620$
b. $\left|E_{T}\right| \leq \frac{1}{128} \approx 0.0078125 \quad$ and $\quad\left|E_{M}\right| \leq \frac{1}{256} \approx 0.00390625$
c. $\left|E_{T}\right| : \quad n=71$
$\left|E_{M}\right| : \quad n=50$

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Video Transcript

problem. 19 party is quite easy because it's just calculation. We have the function to have the B A N is equal to eight. Right? And he's able to eight so we can calculate t eight. And I am eight through this formula here and here. Right? Just calculate and you can't get out, sir, On part B is to mate. Estimate the arrows. Okay, We have a formula here, Terrebonne. So we need to find the upper bone off the second director of back. Right. Let's start access. You go to call sign, I swear. But so I prime is able to two acts times sign negative, Cy Axe. Where and second directive is to minus two times Cy. I'm square clothes. Thanks. Pains to axe times cause I X square. So the abs absolute off burial is this is smaller, then. Science axes. What are the one two acts since axes for the one and quicker than zero. So this is 22 times, two times one times. This is one of the one. It's ah, six. Right here we get. The case is equal to six and we have this formula and you have the formula. Casey go six b A and they're all knowing. So the e t you is what are then six times one right? Guided by 12 times Age square, there's just one or 100. Okay, Yeah. You don't have to calculate am again because you know that I am is just ah, half off you cheap. So it's smaller, then 256 right? Okay, here's Parsi. How long do we have to choose in? Oh, okay. Used formula. Hey, six times one already by its wow times and square is Mari Ko 10. Since I wasn't right, um, to square it is greater than five. Sudden. Oh, there's no to here And there's quick witted and 71 because he's a ***. Right on the second while is just two in the square is acquitted and 5000 seems there's a two over here And that's where is greater than to Southern 500 they're squared And 50. Here's Alyssa, right? Yeah,