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~[4 POINTSSCALCET8 6.5.011. 0/100 Submissions Used Consider the glven function and the given Interval: fx) = 12 sin(x) sin(2x) , [0 , (0) Find the average value fav...

Question

~[4 POINTSSCALCET8 6.5.011. 0/100 Submissions Used Consider the glven function and the given Interval: fx) = 12 sin(x) sin(2x) , [0 , (0) Find the average value fave of fon the glven interval. fave(b) Find such that fave ((c). (Round you} answers to threc declmal places: _ (smaller value) (larger value) (c) Sketch the graph of and rectangle whose area Is the f(x) same a5 the area under the graph of f f(x)

~[4 POINTS SCALCET8 6.5.011. 0/100 Submissions Used Consider the glven function and the given Interval: fx) = 12 sin(x) sin(2x) , [0 , (0) Find the average value fave of fon the glven interval. fave (b) Find such that fave ((c). (Round you} answers to threc declmal places: _ (smaller value) (larger value) (c) Sketch the graph of and rectangle whose area Is the f(x) same a5 the area under the graph of f f(x)



Answers

$57-58$
(a) Use a graph of $f$ to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection.
(b) Use a graph of $f^{\prime \prime}$ to give better estimates.
$$f(x)=\cos x+\frac{1}{2} \cos 2 x, \quad 0 \leqslant x \leqslant 2 \pi$$

For the fine calm. We want to use a graph of F to give a rough estimate of the intervals of con cavity and the coordinates of the inflection points. And then we're going to use the graph of F double prime to get better estimates. So we're going to consider F of X, which equals the co sign of X. Mhm. Yeah. The crossing of joy From 0 to Pi the pain of getting this graph right here. And we see that the inflection point appears to be about right here. Maybe one and then another inflection point. But it was concave down another inflection point, maybe about 2.5 perhaps and then about 3.5, maybe. A little more 3.75. Then we're gonna have another one right about here. And we can look at the second derivative and see how you're a little bit more. I'm precise by doing this method.

By a function of extra Sequels. Toe sign three Experts course called Sentry X This interval from zero to file as a computer all diverse system with differentiated used this symbol of that Come online The result we have six calls Century X minus 12 Time to x The Red Group is the function off X Why? Cause to century experts for constant tracks The blue graph is that the derivative off the APP affects that is a crime off experts equals toe six course entry X minus 12 Sign tracks preparing to their estranged from Hiroto 3.677 or toe this one the body of this before this one Yes, he had increasing function here The increasing functioning from 0 to 0.0 15 as the F of X has passed the value and then the from the from here to here Okay, from here So 3.15 45 to 0.6 from here toe zero point ir 1.267 We have a decreasing function here as as the derivative has a negative value from here to here And then it's turning up. I turned and from decreasing interest increasing up to here from 1.207 to 2.2489 It is increasing function as long as the derivative has positive value. And then from here 2.4 89 toe pie. It has the crazy function as long as the they did. But tip has a negative value here. So there, for the critical numbers are the theory goes off the derivative function. So the zeroes off derivative function is zero point. We can 45 1.2017 and 2.2489 These are the critical numbers. This is a local maximum with maximum. This is local minimum and this local maximum range from zero to fight the interval from. So we have this one. This equation from graphical toe on electrical. Who can have being a function. Six. Caused richtsmeister Sandri, ecstasy Coke zero. Set up a prime affects. It's called 20 This leads to six Cost X equals 12 Century X, and then Bird said about six so often. Mexico's two sentry X So one records 2000 and three X divide by two. Both calculations. We have done it three X equals one half so have marked on three X because our country 30.5 we have plus K pi for for period. Okay, you want to Siriwan toe? It's on Lee from zero to pi So you have except one, except to exit three SharePoint 15 1.20 and 2 25 as the grab tells that that this zeros off derivative So it's also hold a critical numbers. So here the inter ball off it ever ex prime X greater than zero less than zero. Here we have this one you can see also in the grab 15 zero a conservative and a 20 and then 2.25. When a crime of excess less than zero, you have certain 15 20 union to 24 5 Describe people. You can also see that type of X x creasing for 0.15 the opportunity 7.7 and 2.24 read decreasing from zero point between six reporting the union to 60.24 then by that's

Okay, so they give you this equation. Why? To sign. Okay. In a spiv. Co sign two x minus pi over four. All right, let me have a way to see these men's. And Max is now. We do need to include this man right here. 0.343 because it is bigger than zero, and then we can stop. I think we don't need to look at this. One could have. That's too much. So ignored. That one Here are your Max is up here. Here. Your men's down here. What they want you to see as you're getting those copy down is what happens when you type in the second equation. 10 who? Sign of correx in expire before who sign? Okay, let me, uh, very busy. It's gonna be tough for you to see. But remember, these men's on the green function match up perfectly with the zeros of the purple function so that Max matches up with this. You can't see the next one, but you can trust me. Matches up with the other. Ah, whoops. You, this one. That was wrong Point. Don't think about purple is the roots we want to match up with the green of of the max area. This is too busy for me. Uh, I did it again with the purple to match up with a green max in the green men's match up with the zeros of the purple eso before I confused myself any further. The This is what you want to copy down as I click off of the screen.

Hey, it's clear sewing name right here. So for pardon me, we're going to different, you know, equation. When we differentiate it, you got negative, too. Co selling sex time sign. That's plus one. An F prime is equal to zero when co sign of X is equal to zero. This gives us access equal to pi half and three pie half and signed X Plus one. It's equal to zero. It goes us access equal to three pi over, too, and these are critical numbers, so we're gonna split the domain into intervals at the critical numbers, then pick a number inside. So for Ciro, calm a pie or two. If we pick high fourth, it's smaller than zero, so we know that it's decreasing and for a pie has comment. Three. Pie half. It's bigger than zero if we pick pie so we know that this is increasing. And for three pi over too comma to pie. If we pick seven pi for over four, we see that this is also decreasing. For Part B. We're gonna use our first derivative equation so we see it's decreasing on decreasing when it's zero comic by half and three pie half coma to pie and increasing in pie half common. Three. Pi house So there's gonna be a maximum X equals three pi over too, and minimum upon. That's a minimum when X is equal to I go so we could see that from our first riveted sine X plus one. It's positive and negative, too, is negative. So we know that the first service negative when co sign of excess positive and it's positive or coastline of excess and negative so only depends on the sign of co sign. So the first orbit of changes when co sign is equal to zero. So it's going to be plus or minus pi. Half are the local extreme appoints so f of pie half and this gives us negative too. And after, well, three pie, huh gives us regular too. So we know that our maximum is three high half an hour minimum iss high, huh? Report? See, we're going to find our, um, second derivative to do that we get That's double prime is equal to negative, too. I think it a sign Square minus line X plus co sign square, and it's equal to zero when Xs equal to high six Clive Pi over scents and three pi over too. This smaller or equal to zero on the intervals. Zero coma. Why over six and five Piras, It's coma to pie. So efforts con cave down on these intervals and f double prime of X is bigger on the interval when X is equal to, uh excuse me Pi over six Farmer by Pirates six. So it's Kong cave up on this interval. So we see that s which is gone cavity when x is equal to here Xs equal to by six and by five or six So we know that


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