 Welcome to the final exam review for Math 1220 Calculus II for students at Southern Utah University. If you're watching this video, then this is probably the last video for you of the whole semester. You probably don't need me to remind you, but in case you forgot, this is Professor Andrew Missildine like usual. This is probably not your first rodeo either. We've had several other exams that we've talked about this semester, and so as you probably can expect in this video, we're going to talk about the structure and the topics that are covered on the final exam for Math 1220. There's many things in this video that we're not going to talk about. We're not going to talk about the place, the time, other semester specific details of this exam. Please consult the final exam syllabus or other information you can find on Canvas to learn more about those, because even section to section, there might be different times that you're taking the exam. I can't put that in the video. Look to the syllabus. Look to Canvas for information about that. What we can talk about, of course, in this video is the specific types of things that we're going to see on this exam. In particular, I want to give you heads up on things you should be studying and things you might not have to. What should we know about this exam? This is, in fact, a final exam for Math 1220. As such, this is a comprehensive final exam. This exam covers essentially all of the topics that we have covered in this semester on Calculus II. There are some notable exceptions that I will make mention. Unless I specifically say that a topic's omitted, you should be prepared to study for it. This exam is going to be 15 questions long, just so you're aware. There are five questions in the multiple choice section, usual format as usual, as in the past midterms that we've taken. There should be, I think, five questions in the short response section. Then you count them five questions in the free response section. Again, these are the same structure as in previous exams, so you probably don't need some explanations going about that. Let's focus on the topics that are going to be on this exam. As a comprehensive final exam, it covers everything. This exam is going to cover lessons one through 48, everything we've done this semester. You might have noticed, wait, isn't there a lesson 49 and lesson 50? These will be the first omissions that I should make mention that 49 and 40, 49 and 50, while they are excellent topics, they are optional topics that are not required from for the course description. As such, they won't be manifest on the final exam. We do want to cover topics through from lesson one all the way to lesson 48. As we've summarized this semester, there are a few general topics that we can mention that are going to be represented on this final exam. So first and foremost, calculus two is a class about integrals. We calculate so many integrals throughout this semester. And so when it comes to integration, you must be prepared to set up and evaluate numerous integrals. There's going to be a lot of integrals on this test. So things to be aware of when it comes to integration, we should be able to use various techniques of integration. So things like U substitution trigonometric substitution, rationalizing substitutions, we should be able to use integration by parts integration by cycles integration by hope, we should be able to use algebraic and trigonometric identities, such as partial fraction decomposition. So algebraic trigonometric identities and techniques to help us transform integrals, it transform functions into integrants that are more favorable for integration. These are things we've done throughout the whole semester. A lot on this exam will be focused on techniques of integration, you have to be able to find anti derivatives, no doubt about it. Now, on this exam, sometimes you'll be asked to evaluate an indefinite integral, that means you're going to have to find an anti derivative, don't forget your plus C. There will be some situations where you might have to evaluate an indefinite integral, excuse me. So indefinite integrals were the anti derivatives plus the C. The definite integrals is when you have to then plug in the numbers. Once you found an anti derivative, typically, if you're using the fundamental theorem calculus, I should say, we should be able to set up and evaluate both indefinite and definite integrals. This also includes improper integrals. So we might allow our bounds to approach infinity or negative infinity. We might be approaching vertical asymptotes, we should be prepared to compute in proper integrals, as well as proper integrals. So techniques of integration, calculation of integrals, this will be paramount for this exam. I should also mention that applications of integration will also be covered on this final exam. Now, we have done a lot, a lot of applications of integration, ranging from a variety of topics here. While I'm interested in calculus students understanding the method of integration, this process of accumulation, that is, can we subdivide the problem to small linear problems, and then take limits forming integrals of these Riemann sums. This is things we've done with volume problems, area problems, arc length problems, work problems, hydrostatic force, just to name some of them. Now, as we're preparing for this exam, I do need you to still be able to set up and evaluate integrals for many of these applications. But first and foremost, this is a mathematics class, we've done a lot of scientific and engineering applications. But for the purpose of the final exam, I am not going to ask you any of these scientific or engineering applications of integrals. So nothing to do with work, nothing to do with hydrostatic force, those are the two main ones we're going to throw out. But applications of integrals that have direct consequence to geometric topics, that is fair game. So area, arc length, centroids, surface area, volume, so like the washer method, the shell method, all of those geometric applications, yep, I could possibly ask you one of those types of questions on this exam. So be prepared to find area under a curve or area between two curves, find the arc length or surface area, find volumes of solid of revolution, or finding centroids like X bar or Y bar. Those are all applications that are fair game for this for this for this exam. I should also throw out two other ones to mention the average value and probability examples that we did before. Probability could fall under the umbrella of statistics, but we definitely can put it under the umbrella of mathematics. So we should be looking at applications which are geometric, and you should look at applications which are statistical. So things like the average value of a function, the centroid of a region, the probability that such an event would happen the expected value of a random variable. These are all topics we've done. They are applications of integrals that all have a mathematical flavor to them. And I think this is a compromise that we all can live with. Because honestly, if all the applications we did work in hydrostatic force were probably some of the hardest ones, mostly because they transcend the mathematics, there's you need a good understanding of science there as well. So we are going to have to do some applications of integrals that we will see, like so, I should also say that with respect to integrals, we have some new worlds that we had we explored this semester as well. These are going to be manifests on the exam. For example, we played around with parametric functions, parametric equations. How do you find derivatives and integrals involving parametric equations, a function given by a parameter? This is a slightly different way of approaching functions. And there's the calculus is modified. So in particular, how do you do integrals and derivatives for these parametric functions? Send can also be said about polar functions. Polar functions are equals f of theta. How do you find the tangent line of a polar function? How do you find the arc length of a polar function like the circumference of a limouson? Is an example we could consider? How do you find the area under a polar curve or the area between two polar curves? So this is a slightly new world that we explored. And particularly we focused on the calculus of this new world. And in particular, the focus was on integrals in that situation. And then the last thing I'm going to put under this umbrella of new worlds, I'd also throw in differential equations. Because differential equations, as we said, many times was very much like finding implicit anti derivatives. So like in calculus one, you have differentiation and the explicit differentiation and implicit differentiation, solving a differential equations very much like solve solving an implicit anti derivative. And so it's a new world to calculus. And so we'll put that under the new world umbrella there. So that's the main topics, integrals, techniques of integration, applications of integration, and our exploration into new realms of integration. That's them. Those are the main things that are covered on this exam. But if you were hoping something was omitted, there's other big umbrella to put sequences and series is another extremely important topic that we've explored in this semester. Honestly, series we can put still under this the integral label if you want to. And so while I put in a bullet point, yes, I'm now continuing the list that we started here before, series are just discrete integrals. It's a different type of integral than the continuous integrals we spent most of the semester exploring. But a series is just a discrete improper integral. And we do have to be able to know how to evaluate them or approximate them. Or oftentimes you're just interested in convergence or divergence. And so we are very much interested in convergence test. Can we determine whether a series is convergent or divergent? And why can we justify it? Also, we are interested on power series. This is a very important example to mention, because power series have not appeared on any of the previous midterm exams that we've taken in this course, but they will be manifest on the final exam. So the topics of power series showed up in lessons 43 through 48. 49 and 50 does also involve power series. But like I said, that doesn't actually manifest itself on this final exam. So as you're studying for this final exam, there is this practice exam you see on the screen right now. If you haven't already done so, go find the exit final exam syllabus, go find the practice final exam on canvas pauses video right now to do so if you don't already have them because you want those to help you study. But in addition to the final exam materials, the materials you've used for all of the previous exams are also helpful practice exams, pre exams, the actual exam that you took, you probably have access to all of the solutions to these practice and actual graded exams. And hopefully you've been handed back your graded exams for the previous ones in this course. A lot of the questions you're going to see on this final exam are drafted from the question pools for the other exams you've already taken. So anything you can study from the previous exams will be beneficial for this final exam. But like I said, there are a few topics that will be dropped that won't be seen on the final exam. I am trying to make this into a two hour exam. So we're capping it at 15 questions. But even though there's 15 questions, there is a large variety of topics that we can see on this exam. So we have to be prepared for everything, even though you aren't going to see everything on the test. I also want to point out that when you take the final exam, there will be different versions of the final exam in play. So if you're taking the test and someone next to you is also taking the final exam, you might not have the same exam. Don't plan on it. And so, you know, if luck is a thing, you know, rub your rabbit foot before you get handed out your exam to make sure you get a good version, I guess, because all of the topics we're going to talk about right now could be on the exam, you're not going to see all of them, but you could see all of them. And even if you don't see one of these topics, your neighbor probably will. And saying could go for topics your neighbor doesn't see you probably are seeing. So be aware, this is a truly comprehensive exam. We need to be prepared for all of these different things. I should also mention numerical some numerical integration techniques. This will be very limited. But how do we approximate integrals or series? There will be a little bit on the exam very, very little, but there will be some things. And so I will point your attention to where you should focus on that when we get to it. So with that said, let's now jump into the question by questions, what you might see on the final exam and get really specific on what you should be preparing for. So we're going to start with the multiple choice section. Like I said, there's five questions in the multiple choice. Each of these questions, we five points each. Alright, so question one, you can consider this an integral question. As in question number one is going to ask you to evaluate a definite integral. It could be a proper integral like this one is, it could be an improper integral like we have seen also. So like maybe like this thing goes off towards infinity or something. It could be an application of some kind like it might ask you to find the area under a curve. It might ask you to find the area between two curves. That's what question one is going to ask you to do. It's going to ask you to evaluate a integral proper and proper. So probably the fundamental theorem calculus will be a useful tool here. Other techniques of integration might be helpful. Now question one, the integration shouldn't get too complicated. Now maybe like a basic use substitution, maybe like a very basic integration by part. Those are possible. Like with this one right here, it looks like an algebraic manipulation is probably your best bet to find the antiderivative. But you know, you could do a use substitution. There are options. It's a multiple choice question, you're graded only upon your final answer. So it doesn't matter how you get it. So long as you do it correctly. Alright, so question number one will be an integral question that asks you to evaluate an integral of the different types of continuous integrals we have seen this semester. So question number two, this is one that's very important to focus on because unlike question one, question one, this version appeared on exam one, I believe. And the other possibilities will be grabbed from definite integrals we did like on exam one and exam two, right? It'll be one of something like that. So you can go to exam one and two to study different versions of this problem you've already seen. Question number two, we're going to put a nice star next to it. Question number two is a novel question with regard to the final exam because it's about Maclaurin series. None of the previous exams have covered Maclaurin series, infinite series where they are but power series were not. So can we find a Maclaurin series for the function in this case? It's x times e to the x. What I'm looking for in question number two is that we use the table of important Maclaurin series. Things that like the Maclaurin series for e to the x, sine of x, cosine of x, one over one minus x, that table. You should know that table or be able to produce it on the fly. You should know these standard Maclaurin series. Like for this one, you need to know the Maclaurin series for e to the x. That's most of the heavy lifting. There is this x that's also here which will modify the Maclaurin series for which then you take the Maclaurin series for e to the x and modify it by some techniques and then you're going to find the Maclaurin series for this function here. You of course can use Taylor's formula and produce it from scratch. That's not the intent of this question. There will be a question, the free response that expects you to do something like that. This one wants you to use the standard Maclaurin series like e to the x, cosine, sine and all of its friends that were on that list. Like a binomial series might be an example that would show up here. You should use a standard formula and then slightly modify it to find the correct answer to number two. But number two will ask you about a Maclaurin series of some kind. So let me label this is going to be a power series type question. Alright question number three. Question number three is going to focus on applications. Remember these were the umbrellas that I had mentioned earlier applications of integration. So the version you see on the screen right now this is very similar to something we saw on exam two. This is going to be a numerical integration problem for which you're given the function in a tabular format and using either the trapezoid rule, the midpoint rule, or Simpsons rule you have to approximate the area under the curve. So this is a numerical problem that you could see on the exam where you're approximating the area of the curve using the trapezoid midpoint or Simpsons rule. What I can tell you is what I what you can rest assure is that these are the only numerical topics that you are going to see on this exam. Okay so we've done approximations of series you're not going to see anything like that. We've done some other some other type of numerical stuff. No no no no error bounds. Well am I going to ask you on the final exam if you get a numerical approximation which this is just a possibility for number three it would be trapezoid rule, midpoint rule, Simpsons rule. I mean I guess it could be left or right that's not really focused on in Calc 2. It's probably trapezoidal rule, midpoint rule, or Simpsons rule so you should know how to do those calculations. That's if this is a numerical application. Some other things you should be prepared for so in addition to this numerical approximation you should also be prepared to do an average value. Average value of a function we talked about those on exam one. You should also be able to do some applications with respect to probability. So what's the expected value of this continuous random variable? What's the probability of an event? We talked about probability on exam three. So feel free to go back to those exams. The numerical and the numerical approximations is exam two. This is exam one and this is exam three. So these these will be a multiple choice integral application similar to one of those three multiple choice questions you saw on exams one, two, and three. Alright question number four which you can see on the screen here this will be a series question. Not a serious question, a series question for which much like we saw in exam four this will be a multiple choice question that asks us to determine the convergence or divergence of various series by various tests and in some situations if it's convergent it'll ask you to find the sum of the series. This version asks you which of the following six series is absolutely convergent but some other things we could see here is which of these series is divergent by the divergence test. Which of these series is convergent or divergent by the p test? Which of these series is convergent by the alternating series test? Which of these series is a geometric series and convergent and if it's convergent find the sum of the geometric series. Which of these is a telescoping series? Is it convergent or divergent and if it is convergent what does the sum add up to be? So we saw many multiple choice questions like that on exam four. Question number four on the final exam will ask you one of those series multiple choice questions that we saw on exam four. And now to the last question in the multiple choice section this is the first page of the exam. Question number five will ask you a question about polar coordinates polar functions. On exam four we had three multiple choice questions that are about polar functions. One of them which is about recognizing polar graphs, polar coordinates. One of them was about computing derivatives involving polar coordinates. Like maybe finding the slope of a tangent line to a polar curve. Another question was about finding integrals of polar functions like finding the area of a polar region or finding the area between two polar curves. Finding the arc length of a polar curve. Something like that the circumference of a limouson was an example of those things. So polar functions were covered on exam four. Question number five will ask you a question about polar functions and as every question we saw on exam four that was about polar functions was formatted as a multiple choice question. One of those questions will be grabbed for the final exam. And that then gets us through the multiple choice section. Let's move on to the free response section. Remember free response there are worth more points than the multiple choice but you can also get some partial credit. All that you need for full credit is to have the correct answer on the line but there is space provided to show your work if work is appropriate and you can get some partial credit with that one. Question number six this is also a star problem it's a novel problem because this is a problem that's going to be involving power series. Something that did not appear on previous exams so put special attention to this problem here. Question number six is going to ask you to find a power series representation of a rational function or a function that is related to a rational function perhaps because it's derivative is a rational function like take for example arc tangent of x. It's derivative is a rational function and so utilizing utilizing geometric series we can come up with power series representations for various functions. So this one is going to ask you to come up with a power series representation of the following function in this case you get 3 over 2 plus x minus x squared but remember those sections we talked about 43 43 through 48 this is where we go into depth about power function power series excuse me particularly less than 44 was the main was the main culprit for these type of problems power series representations. So focus on less than 44 as you're trying to prepare for this one. The previous question number two was about the chlorine series and we spent actually three lessons talking about the chlorine series didn't we so you had 47 46 and 45 44 a 40 40 which was a 46 no 47 was the one that has the the chlorine series table you should know that table so you'll definitely want to go to lessons 44 and 47 those are big ones in this final fifth unit about power series all right so then getting to question number seven question number seven is going to ask you to calculate an anti-derivative now question one similar to that so like uh so this is going to be an integral problem again so there was an integral problem in the multiple choice there's going to be an integral problem in the short response there's going to be an integral problem in the free response they're all over the place now while the the problem we saw in question number one the multiple choice who wants you to evaluate a definite integral question number seven will instead focus on improper integrals so we are only looking for the we're only looking for the anti-derivative so don't forget the plus c you do need that for full credit on these indefinite integrals and then these ones the techniques might be a little bit more involved like with this example right here you see there's some trigonometric function in play here cosine of the fourth cosine of the fourth of 2x so trigonometric identities would be very relevant how you should know those trigonometric identities and be prepared to be able to evaluate these integrals using those integration by parts might be relevant so yeah and focus of course on trig identities for this one but any of any of the types of problems we've seen could manifest here because there are going to be some questions in the in the free response to have more involved integrals so while the technique of anti anti differentiation here will be a little bit more advanced than what you saw on question number one it won't it's not as hard as what you're going to see in the free response so this is like the moderately difficult integral question number one was the easy integral question number seven is going to be the moderate integral and then we'll see the free response there's going to be one that's a little bit more challenging so in particular focus on trig stuff here that's probably what you're going to see some type of trigonometric integral you probably don't need more advanced techniques like trig substitution or partial fractions you'll use those later okay focus probably for trig integrals for number number seven be aware of those identities you need to know them all right so then coming to question number eight question number eight is going to be an application type question we saw one of these in the multiple choice we're going to see it in the short response as well what are the type to integrate integration applications we've seen I will list them on the screen explicitly right now so arc length is a problem you should be prepared for and if I read through this correctly yeah this problem in its current form is set up as an arc length problem we should be able to do area problems that includes area between two curves the boundaries might not be given they might be implicit you might have to find them out centroid problems can you set up a centroid problem to find x bar or y bar can we do surface area can we do volume and when I talk about volume I talk about volume the solid of revolution we're focused on the disk method or the shell method I'm not going to do any of those volume problems where you have like a base and you start stacking polygons on top of them or something like that we've done some of those in the past this question won't ask you to do anything like that if it asks you to do volume it'll be a standard application of the washer method or the shell method you should know both of those methods in preparation for this problem so like I said earlier this will be a geometric integration application this problem like we've seen on previous exams so we saw examples like this on exams one and two this will be one where you set up the integral you do not need to evaluate it it will be a definite integral so as you set up your integral there will be bounds you do need to include the bounds if you don't include them you'll be missing something there should be a function you should appropriately simplify the function and there also needs to be a differential of some kind if you're missing the differential you're losing points if you don't have your bounds you don't you've lost some points if you don't have the integral symbol that's also losing points if you don't appropriately simplify the function you're losing some points there so do make sure you have all those things and this applies to all of these set up and do not evaluate integrals like question number nine question number nine is going to be a question about parametric equations we saw in the multiple choice a question about pair of polar functions that is the only question on the exam that'll be about polar functions likewise question number nine will be the only question on the exam about parametric functions now this one is a set up as an integral problem so this one wants you to find the area of some parametric region you might also have to do arc length of a parametric region you might have to do surface area a surface that's rotated around an axis using parametric curves that's possibilities but you also need to be prepared for derivatives right so integral problems and be aware parametric equations we saw this on exclusively on exam three you had a couple of questions in the shore response section that were integral problems involving parametric functions but there were some questions in the free response where we had to set up derivatives like find the tangent line of a polar curve find the second derivative of a polar curve you had some questions like that they showed up in the free response section be aware that for the final exam those questions could be reformatted into short response questions as opposed to the free response questions they were before so be prepared for that question number nine the one guarantees it'll ask you about parametric functions and it'll ask you to either set up an evaluated integral involving parametric functions or a derivative involving parametric functions if it tells you not to evaluate you don't have to evaluate but if it doesn't then you better so be prepared for that remember parametric equations were exclusively on exam three now we get to the last question on the shore response this is another integral technique question I told you there's a lot of integrals on this exam count two was about integrals question number 10 is going to be out integrals you probably should be using integration by parts on question number 10 like if you look this one I hear x times e to the x the best technique is integration by parts just be prepared to use integration by parts it will be manifest on the exam probably question number 10 all right so now let's move to the free response section of the exam free response section of course these questions have different amounts of points based upon their difficulty but we want to be prepared for each and every one of these question number 11 okay question number 11 it's worth eight points it gives you a power series and it asks you to find the interval of convergence given the power series so this is like what we did in section 43 a lesson 43 you're given a power series you have to find the interval of convergence probably the way you want to solve this problem is start using the ratio test to find the radius of convergence once you have the radius of convergence you get most of the interval of convergence but you have to test the end points individually like let's say hypothetically the radius of convergence turned out to be four on this one it's not just so you're aware it's centered at zero so it's like okay from negative four to four you have those and then we have to determine if I plug in x equals four is it convergent or divergent that's like a street fighter series type problem there same thing so maybe like negative four is not included but positive four is again I'm not claiming that's the right answer but you need to find the interval of convergence you do have to check the end points if you do not check all of the end points you aren't going to get full credit on this one okay you do and you have to also check and give some justification on what happens at those end points there speaking of street fighter questions this gets us to question number 12 which is worth nine points in this case you're given a infinite series no x's in play there you have to determine whether it's absolutely convergent conditionally convergent or divergent and you can use any integral test that we are any series test that we know about so we got the alternating series test we have the comparison test the limit comparison test the integral test the geometric series test the ratio test the root test I see that has the same acronym there and maybe I forgot someone I don't know it's always the case you always forget someone it feels like alternate series test comparison test integral test geometric series I feel like we're pretty good the only other one you can talk about like a telescoping series so we have to be able to determine the convergence of these series this is a street fighter question this is street fighter 2 determine the convergence of any series whatsoever now this question is formatted in the same way that the street fighter questions were formatted for exam for exam exam 4 that's the word I'm looking for but there were some short response questions on exam 4 that could be modified into this question as well those are excellent ones to look at as well as you prepare for question number 12 so this will be a question about series infinite series there all right so then we go to the next page of the free response section we get to question number 13 which is worth 10 points this will be an integral question in particular we want to think of this as our as our street fighter integral problem you could be given any integral under the sun and you can use any of the techniques we've learned about so I've said them before but I'll say them again integration by parts use substitution trigonometric substitution partial fraction decomposition rationalize the substitution just to name a few we've had a lot of techniques we've learned question number 13 will be your chance to shine in which case you can evaluate or you'll be asked to evaluate some indefinite integral don't forget the plus C on the previous question about the street fighter series you need to also make sure you include the convergence test that you're using if you believe it's convergent by the alternate series test you need to be explicit and say those things there with number 13 show all the work find the antiderivative here question number 14 is a question about differential equations this will be the only question on the exam involving differential equations in any form whatsoever there's basically two things you should be prepared for with regard to question number 14 can you solve a linear differential equation well like this one right here it asks you to find the integrating factor can you solve a separable differential equation by separation of variables those are two problems we saw on exam 3 question number 14 will ask you to do one thing involving these differential equations and so then we get to the last question which this one gets another star this is a question about power series which have not appeared on previous exams and in particular this will be a question which asks you to to find a Taylor series for a function and so in this case you want to use Taylor's formula so that if you take your series cn x to the x minus a to the n right we want to use Taylor's formula where cn is equal to the nth derivative of f evaluated at the center a divided by n factorial so you're going to have to take the higher derivatives of our function I will give you a function whose higher derivatives become somewhat predictable like the natural log of x there's a pattern that'll be established you could do something like centricosh the hydrophonic functions there's a pattern that'll be established eventually so you can predict what these higher derivatives are going to be without too much difficulty I'll give you a center that's not too pathological and you'll be then asked to come up with the power series the Taylor series for that function you do not have to find the radius of convergence on question number 15 we already did that on question 11 you do not even find the interval of convergence you do not need to prove that the Taylor series is convergent for anything I mean clearly it's conversion on the integral of convergence but you don't need to do that you don't need to use Taylor's inequality to prove that a function is equal to its- It's Taylor series on the interval of convergence you don't need Taylor series ah sorry you don't need Taylor's inequality for this exam because again, you don't need to use any error bounds for this exam whatsoever. So this gets us to question number 15, and there we have it. The basic test feels like the following way. There are three sections, multiple choice, short response, free response. Each of them has five questions each. Each of those questions for the most part have the following feel. You have a question to ask you to do an integration technique. You have a question that asks you to do an application of integrals. You have a question that asks you to explore a new world. On the multiple choice section, we have polar functions. On the short response section, we have parametric functions. And on the free response, we had differential equations. You have a question that asks you to do something with series. So convergence of series, evaluation of series. And then you have a question about power series. Each of the three sections on the exam with their five questions follows this format, clearly not necessarily in this order. The only exception of course in the free response and the free response, there was no application problem. Instead, we got two power series questions. So power series is the tail end of our course, but you can see that there is a high distribution of power series questions on this final exam. You need to prepare for it. And if you feel like, well, that seems unfair, that's just a small part of the semester. Well, it's actually a big chunk of the semester because it hasn't been tested on any previous exams, I do have to cover them so they're gonna show up in the final exam. And the most important thing is that power series as a topic is comprehensive. It is. Everything we learn kind of points towards power series. Believe it or not, power series is essentially the main objective of Calculus II. It's the cumulative topic that covers everything. If we have a good understanding of power series, then we have a good understanding of Calculus II, which is why it's not unfair to ask many questions about power series on this exam. All right, so with that said, we're now at the end of our exam review and essentially at the end of the semester, right? Whether you're gonna take the final exam in a couple of days or in a couple of seconds, hopefully you're prepared, hopefully you do really well and hopefully I don't see you next semester in Calculus II. I definitely want you to pass this exam. It's an important exam and it does have a big impact on your final grade so we all want you to do well on this. Of course, if you have any questions whatsoever, reach out to the resources that you need. Reach out to me, I'll be glad to help you out either during office hours or even outside of office hours. I wanna help you as much as I can. Utilize the tutoring center, work with classmates. You can do this. You're gonna have to put in the effort. This final exam is a big exam. I mean, if I'm being honest, it's probably gonna take 10 hours of study to get a good grade on this exam. I'm trying to be honest and frank. It takes a long time to get a good score on an exam and this is 10 hours assuming you've been, of course, active in lessons and other assignments as well. If you are grossly behind, you're missing some work, it might take a lot more effort to make up what you didn't get from those assignments. So I'm assuming if you're watching this video, you've made it to the end because we're about 40 minutes right now, you probably are a dutiful student who is eager to work hard and take this exam and do fantastic. Keep on doing it. If you've done well this semester, I expect you'll do well on this final exam as well. Best of luck. Let me know if you have any questions and let the, and may the force be with you.