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Question 610 ptsConsider a periodic function of period 4:f(c) = € + 1, for 0 < c < 4 Calculate the Fourier coefficient bz for this function. Give your a...

Question

Question 610 ptsConsider a periodic function of period 4:f(c) = € + 1, for 0 < c < 4 Calculate the Fourier coefficient bz for this function. Give your answer to 2 decimal places.

Question 6 10 pts Consider a periodic function of period 4: f(c) = € + 1, for 0 < c < 4 Calculate the Fourier coefficient bz for this function. Give your answer to 2 decimal places.



Answers

For the function
f(x)=?sin(x?3?/4) , determine its amplitude and period, and then graph it for two periods.
Enter the exact answers.
Amplitude: A=
Period:
P=
Using your answers for the amplitude and period, select the correct graph of the function
f(x)=?sin(x?3?/4)

Problem, you're given the graph of a sine function and clearly it has some transformations happen. So the first thing we want to do is find the amplitude, the midline in the period. And I actually think finding the midline is going to be the easiest to find first. So remember when sine is equal to your typical sign curve would start the origin an increase from here and then it was, it would go through its cycle until it gets back to the origin. So we just need to determine what this uh cycle would look like. Well, if you know this, it looks like our midline is going to be at this line, Y equals negative two. Because if we were going to actually to go through and trace this starting at um are y intercept? We would go up, we go back through our midline and then we would come up and hit our midline again. So now we have determined are middling, it's wide equals negative two. Next, we're going to find our ample too well remember the amplitude represents the height of the curve, meaning it's the distance from our midline to the max or to the men. Well, if I find our max that happens when y is equal to two and our midline is down here at negative two. So this distance we represent our amplitude which is four units, so our amplitude is four. Alright, the next thing we need to do is we need to find the period. Remember the period represents how long it takes to go through one complete cycle. So if we were again to start the Y intercept, how long does it take to go through one cycle? Well, it goes along, goes down and comes back up and it goes from zero till five, which means our period would be equal to five because it takes five units to go through one full cycle. Okay well now do we know the midline to amplitude in the period? We can start to write an equation for this function. So remember our general form is why equals A. Times the sine of B times X minus C. And then re add t. Well remember A represents the amplitude. Which we already found two before now to get B. We actually used the periods of the graph. So remember the period is equal to two pi divided by B. Because your typical sine function goes through a period of two pi. Well remember we just found a period to be five. So we'll substitute that in for the period. And now we just need to solve for B. So I multiply both sides by B. So these peace cancel. So we get five B. Is equal to two pi. So the sulfur be. We just divide both sides of our equation by five. So now we've found that B. Is equal to two pi divided by five. All right, C. Represents the horizontal shift. We'll remember our typical graph starts at the origin. But all we did in this case was go down. We didn't move left or right. So R. C. Value is zero. Indeed represents the vertical shift. And as I just mentioned, we went down to units. So our devalue will be negative too. So now we just have to substitute these into our formula. So when we do that we'll have Y equals four times the sine of B. Which is two pi over five times X minus C. Which is zero. And then we have to add our devalue which is negative tip. So now we can clean this up a little bit. So we'll have Y. Equals four times to sign. Oh well if we distribute to pi over five, that just leaves us with two pi over five times X. And then we have plus negative two, which is really just minus two. So now we have found an equation to represent the ground.

We have a question in which we have been given affects equal to for because x minus by by three. Bless one. So we have to find certain things literature this. Okay, so if we just open it up it will become for cause X miles by by three plus four. This is the value of fx. So if you compare this with basic extended cause I'm the standard. A question of course I in it will become all the astrological sign function. It will become for because or are a course A because B x minus C plus D. So now let us get the value required values. First aid amplitude amplitude which is a if we compare these two we will be getting amplitude as four. Okay, period. So period is always to buy buy baby and if we compare here here it is one. So value of B is one. So to buy buy one that is to buy superior is to buy amplitude before and part three which is midline. So we have to find the value of the middle light. Remember this is the value for P that is period. Okay, Okay, so midline is always the main portion of maximum and minimum values. So if you look at it, maximum value will be will always be four plus one. That is five. So why max is five? Why minimum will be minus four plus four? That is zero. Okay, so midline will be plus five, zero by two, that is five by two. So middle line is why call to 2.5. Thank you so much.

In this problem let our and denote the right endpoint some using and sub intervals right, compute the indicated right? Some for the given function on the indicated intervals right? Surround your answer to four decimal places here are 64 ethics Equals to one by X in two X -1 on three and 6 Internet is given to us. Right? So we are asked to compute the indicated right some for the given function. So let us see how do we solve this problem? So here fx is given to us. This is interval. Okay, so this is a a closed interval. Right? So fx this is function of X is given by one by X into X -1 here. All right. So now we can see that this is on this interval. Okay, now also right. Endpoint will be bored. So therefore right endpoints approximation is done by award. Let us see. So this is done by when we put integration of fx dx and the limits are a to be here. All right. So now here it comes. It will give us almost. This is e questo Delle legs into F X one. Right? This is for f this is a big backer tree. So it will be FX two plus FX three plus dash dash dash, It will be F x n minus one. Right plus F ex in All right. So this is how you have to report the values. Right? So now what is the Aleks now? So delicate is nothing but this is given by b minus c upper limit minus lower limit upon number of some interference. This is the basic think about right endpoint approximation. Okay, so now what we are given with we are given with three and six intervals. Right, so here the Lexus 10 X All right, so here is equals to three and b is equal to six. Mhm and n is equal to six is used. Right? So that means this implies that the legs will be what the legs will be equals 26 minus three by six. Okay, so this will be three by six, that is one by two. Okay, so now okay, so now what will be the length of the sub intervals? So the legs is nothing but this is giving us the length of the sub intervals So how we will make 17 minutes. So therefore this will be sub intervals will be seven intervals will be work. This was three, three plus one by two. Is seven by two. Right, this is worst in german, Now we have to make six interviews laundry. So this will be again this will be seven by two plus half you have to add. Right, so that means eight by two is for only. Right, So now 3rd interval will be what? four plus one by two? That means 4 to 8 plus nine by two. That means nine by two. Political. Okay, no, moving forward We can see nine x 2 will be dead nine by two plus one by two, that will be equals two. 10 x two. That is five. Right, so 1234 for is there now to more southern terminus has to be made so Yeah. Okay, so let us see what it is giving us. So now five and 11 by two is one interval some interval. Right. And the last in last interval will be 11 by two and this is six. So is that the same thing we are talking about? This was three and this was this was the major in southern travels. Right. This was the intervals and we have used seven troubles. Okay, so now moving forward, what we can do here is we will just put all the values for we will evaluate for the function at the right end points. So evaluating functions function at right end point. Right. So what does it mean? It means that we will start from seven by two. Right, we will take seven by 24 the nine by two and then 5 10, 11 by two then six. Finally, so this is what we are going to do. So All right, so now I'll let us see if of Excellent, let us say f of X men is request to F seven x 2. Let us write these uh interface first So that we can easily solve it in 49-5 49 x two then 5, 11 by two. And then finally it will be six years. So seven x 2, four, nine x 2, five. And then 11 by two. And again, this is six. Okay, so this will be F X two. This will be F four. F X three will be F nine by two and F X four will be eight first to F five. Right? & F X five will be equals two. Okay, FX five will be close to F 1192. Then F X six is equals to F six again, is this the right thing? Yes. So now what we can do here is we will just calculate for it. We will just put the values instead of eggs. We have function F X equals two, one by X into X minus one. So we will just put seven by two instead of X is equals to seven by two. So, we will just replace these values seven by two into 7 to 7 by two minus one. Let us do for all the values we will get here Finally. All right. So, we will get FX one as 4 x 35. Right? And fx two, that is F four S one by 12. F nine by two is four by 63 F five is won by 20. Right? And F 11 by two is four by 99. So F six will be lastly 1956. So our six will be what? Our six week was 21 by two into four by 35 plus one by 12 plus four by 63 plus one by 20 plus four by nine plus one by 56. That will give us 0.38 67063492. Right, so this will be the answer for this problem. I hope you understood. Thank you for watching.

Okay. What we're gonna do right now is kind of step through the process of being able to identify some key features of a triggered a metric graph. And so we have the graph F. Of X. Or the function F. Of X is equal to negative sign of x minus three pi over four. Okay. Um And so we know that generically ethel vax is equal to a sign of B. X minus pleasure minus some see value plus or minus some devalue. So let's talk about what these are. A or the absolute value of A or the absolute value of A is called your amplitude. Um B is going to help define your period. So typically for a sine function, your period is equal to two pi divided by B. Yeah this right here is called a phase shift. Mhm. Or a excuse me Or a vertical shift. And this plus or minus D. Is a note. This is a horizontal shift. Uh Don't know my directions, you can tell it's a friday. So this is going to be a horizontal shift and this will be a vertical shift. Okay so let's talk about what we see here. So um there is not a number in front of sign which is implied. This negative implies to be that A. Is negative one. So my amplitude is equal to one and my period my period B. Is one, so my period is equal to two pi. Okay so now let's kind of developed kind of a graph. And so what I like to do especially for grabbing this by hand is I know five key points for the sine function. So zero pi over two pi three pi over two and two pi. And I know just a sign of X. Just a sign of X. Um This is equal to 010 negative one and zero. So a negative sine of X would be flipped zero negative 101 and zero. Okay. So now what is this phase shift do for me? And so it is actually moving, it's this space shift is moving too right? Three pi or four units. Okay so this row now becomes three pi were four, then I would do pie or two plus three pi over four which actually gives me um 55 or four. Right 55 or four because that's 22 or two plus three pies. So that's five pi over four. Um This value would be pi plus three five or four which gives me +75 or four. And then I have this would give me 95 or four and then this would give me 11 pie before. Okay so now let's kind of look at our, just try to draw a graph and I'm looking at I'm gonna be doing these two things. Okay so what we have here we have an amplitude of one and negative one here. I have let's just do pies over fourth. So I have pi over four. I have two pi before which is pi or two. I have three pi over four. I have a four pi before which is pi five pi over four. 65 or four which is three pi over two 75 or four eight pi before which is two pi 95 or four, 10 5 or four which is 55 or two. And now 11 5 or four. So I'm just counting by pies of the force. And so I do know here at 35 or four were at zero at 55 or four were down here at negative one. So let me do this in red then 75 before we're back up here at zero, and then 95 or four were up here at one, and then 11 5 before I'm back down here at zero. So if I just kind of graft this out, Yes. Mhm. Um like this, and so I also noticed that um at a distance I've got to go back up here, so at here where I would do um from this is a distance of power. Before then that would be a distance of 54 and then at 55 So I'm back up here at one, if I have to distances. And so this is going to go back and then back down. So I believe this looks like graph B or the second graf. I hope this helped um kind of understanding amplitude periods, face shifts and so forth for a sine function. Yeah.


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