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Question

Solve the given problems by finding the appropriate derivative.
By Newton's method, find the value of $x$ for which $y=e^{\cos x}$ is minimum for $0<x<2 \pi$. Check the result by displaying the graph on a calculator and then finding the minimum point.

Stanley Enemuo · Accepted Answer

So we need to find the value of X. I need his method with wide. Which one is equal to meet the co sign and uh arrange domain. Mhm. Zero is less than an access list part. So now if this is a function why? So we can you can take them of that promise that is equal to minus climatic times E. To the coast side. So minimizing or maximizing the friendship that time we&#x27;re back to zero. Therefore some sign of access equal to zero. It&#x27;s of course I&#x27;m and it&#x27;s equal to seal. So actually put zero and uh hi And one X. is also equal to infinity. So now we take the second derivative that&#x27;s double time it back and that&#x27;s equal to be sort of course on that. Well, yes Science gradeck mind is close on it. So now if we substitute the value of exits are given equation we have a lot that&#x27;s equal to E. To co sign next The Coast and X is -1. So we have eight and a negative one. And that is zero 368

Sam Sohn · Answer

All right. So this question is asking to find the first local minimum in the positive side of the X axis of Ethel Back says he could the coastline of X, divided by X. So we&#x27;re going to first start off by posting a copy of the graph here like this. And Resino&#x27;s actually has a convenient feature where you can highlight over 80 points and it will show you the coordinates so we can see that it&#x27;s at about 2.798 Of course, we want to be a little more precise than that. So this is where the finding local extreme a calculator is going to come in now. This calculator is very similar to toe finding zeros one but she can see here. The major difference is in the slight modification of the formula, which is over here this time, instead of the voting. F of X, divided by Prime Amax, were dividing f prime of X by F double prime of X, as in the first derivative divided by the second derivative. So that means we find a coast the derivative of coast out of X, divided by X and the second derivative of coastline of X, divided by X. I&#x27;ve already done the work out here, so I&#x27;m just got posted right here, though the first derivative is going to be negative of X times. The sign of X plus coast out of X old about it by X squared and F double Prime of X is equal to two ex con side of acts minus and it&#x27;s four minus two times codes on the X all the matter by excuse. So let&#x27;s plug those in. So first we need to find our initial point as well and says 2.7 98 8 is fairly close to three. Let&#x27;s start. Let&#x27;s go with X of zero is equal to three. Did you get three here? And then we also get free a single negative of the two times sign of C two plus co sign of C two old abided by He two squared and then for a double prime of X. We get is equal to prophecies to time. See two times science about C two minus currencies. See two squared minus two. It&#x27;s coz I see too all this I see too cute. But then again we do C to minus t two divided by need to. And then so this is gonna be our new except one. So we do is equals f two and then we dragged them down. Oh, and this one actually goes down to 20 just for extra precision, because these tend to be a little war finicky with how they behave. But usually tenants good enough. So that&#x27;s up to you then was being zoom this set get in approximately 2.798386 as answer two points of 93 86 We could see that it actually corresponds with the 2.798 that we had shown on the graph earlier.

Sam Sohn · Answer

All right. So those questions, once again asking for the local extreme, Um, and this time to graphics F of X is equal to eat through the X, divided by X. So we are going to start by computing the derivative and the second derivative of Iraq&#x27;s. I&#x27;ve already done this part for you, so you can weekend to go for it is quickly, but you should check this on your own. So f primal backs is going to be either. The X Times X minus one divided by X squared and F double prime of axes. He could do either X Times X squared minus two X plus two, all divided by X cubed. So now we&#x27;ve got now have to choose a starting point. And before we could do that, I&#x27;m going to post a copy of the graph. No, here, and as we can see, the graft is fairly interesting. It has only one no go minimum on the positive side of the X axis, and there is not. And there is no local explain logo extreme, no minimum or maximum in the extra less than zero area. And that&#x27;s because the denominator the X becomes negative and either eggs always has to be positive, so we can basically disregard the left side and only focus on the right side. Now it does say one, but we need to restore me to stop check it. So let&#x27;s run on different number like, for example, 0.1. And then we run the F Kramek cracks, which is equal to uh, X Hun&#x27;s ex miners. One running dry, Greg&#x27;s squared. And then this will be rex of X Times X squared minus two X plus two bettered by excuse. And then we do next minus of karma backs provided by have double comebacks. That is the way copyright X Valley over. Then we do the same thing as we always do, which is just drag these columns down and have a computer automatically for us. And as we can see, the valley converges to indeed exactly one. Now, just make sure I&#x27;m gonna try on a few different values. Okay, this one convergence a little slowly, but if we were to go on, we can see that it&#x27;s going to still converge. Downs one just like that

Carson Merrill · Answer

So you&#x27;re looking at the derivative graph. We see that at Pi over four and three. Power for that, there is going to be a relative maxims or minimums. Um, we look at this graph. We see that when x is right here, that f double prime of X is, um, negative. That tells us that this is going to be con cave down. So at pi over four, that would be a relative maximum and then three power for, um, we see that after Bill Prime of X is positive, which tells us that, um, it is Kong cave up and, uh, ex equaling three pira for be a coordinate for a relative minimum. So as we see here going back, um, from zero to pi, we do, in fact, see that it is a relative maximum at pyro for an hour out of minimum at three power for in each of those has a coordinate y coordinate of one and negative one, respectively.

Sherrie Fenner · Answer

question 37. We are looking at a function of why in court ex clear sign. So I took my group. Why? Ride and I had to use the, um Brad, It rule crying G plus g prime f So I have X sine X plus x where co sign IX and then I&#x27;m gonna take my Why Devil Crying and again I have to apply the chain of the product rule for both of these. No, I have to sign X plus two x coz I necks And then I&#x27;m looking at the second part plus two x coz my next And then when I take the g prime get a negative sign So it x weird sign Thanks. Combining my light terms I am going to get combine these two I have to sign of X plus we&#x27;re ex sign Thanks My ex weird sign of necks And now I want my people why people cried So I take the derivative of to sign next which is just to sign Thanks. Then I have to do my product rule or four ex co scientific. So I get four co sign X minus were X sign of minus No on the last piece that minus two x sign of X minus X weird. Who? Sign X Combining my like terms. I am going to get six co sign of X minus six x sign of X minus X square Code signing Vicks. So I&#x27;m using, um, 1.5 is my approximate starting point. So I am putting in, um X minus a devil prime over F triple prime. And in your graphing calculator, if you put these in like l won our story, Why, 1123 white, four. And then you can just go to bars. And in white five, you can just have the wife. Why Double Brian? 1/4 bye bye. Why Triple crime? It&#x27;s a lot easier on that. Right Did. So I got, um when I plugged in 1.5 into this right here, one x one, of course, people. Approximately 1.5. My first inflection point. And then I got accepted to 1.520092 ex of three. One point 519 855 And I also got that for, except for. So that becomes my ex value. And I put my ex value into my original wife function. And when I do that, when I do, uh, 1.519855 I get my value. 2.3 06 90 Yes, it&#x27;s 90 six, Yeah.

Rebecca Polasek · Answer

I want to find the absolute extreme A on the interval 0 to 2 pipe for sine X Plus Co Cenex. So, first of all, I actually want to find the critical numbers. So I need to take the derivative. So I get co sign of X minus sign of X and then I want to let this equal to zero, so co sign of X minus sign of eggs equals zero. So to solve this, I would add over sine X to the question is where it is. Co sign Equal sign. Well, co sign and sign are equal at hi over for and at five pi over four. That&#x27;s where they are squared up to over two and negative squared up to over two. So with those two options in mind, I&#x27;m only doing those cause I&#x27;m between urine to pied. I&#x27;m going to now list out the X values and the Y values of the original function to see which is the absolute maximum and which is the absolute minimum. So I have to check the endpoints and then my critical numbers and then the other end point. So zero power before five power foreign too high are the values that I want to check in the original ffx. So figuring out in the original when X is you are the interest one When X is pi over four, it&#x27;s a decimal 1.414 When it&#x27;s five pi over for its negative 1.414 and then when it&#x27;s Dubai, it&#x27;s one again. So on this interval, the lowest value compared to all four of them is at five choir before and then the highest value happens, uh, pi over four. So that&#x27;s our absolute mint and absolute max on the given interval. I could change that interval like maybe I want to look from high over to two pi. If I change the interval, then I have to change the endpoints. So this becomes pi over two and my critical numbers need to be within the interval. So actually noticed. Hi, over four is not in the interval. That&#x27;s less than pi Over two and five fiber for is not in the interval. That&#x27;s greater than pie. So actually, as I&#x27;m checking for this next interval, I don&#x27;t use either of my critical numbers. I&#x27;m just checking the endpoints, so checking these end points back in the original pi over two yields an answer of one and then pie yields an answer of negative one. So obviously the value that is more is one that&#x27;s our absolute maximum on the interval in the value that is less. Is this negative one? So that&#x27;s gonna be the absolute minimum on

Problem

Solve the given problems by finding the appropria…

Problem 22 Easy Difficulty

Solve the given problems by finding the appropriate derivative.
By Newton's method, find the value of $x$ for which $y=e^{\cos x}$ is minimum for $0<x<2 \pi$. Check the result by displaying the graph on a calculator and then finding the minimum point.

Answer

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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Anna Marie V.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus