 Welcome to our review for exam 3 for math 1220 calculus 2 for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. Like many of our other exam reviews, we're going to talk about the topics that will be covered on this exam and the structure of the exam itself. Specific details like time, place, and manner of the exam, because they change from semester to semester, I will not include that information in this video, consult the Canvas or the exam syllabus for more information about that. So what I do want to mention are the topics that will be covered on this exam and the structure of the questions. So this exam will go from lesson 22 all the way up to lesson 31, so there are 10 lectures covered on this exam. The topics of those lectures, as a reminder, lessons 22, 23, and 24 are applications of integration, in particular, hydrostatic force, centroids, and probability. We will see questions coming from those three sections appearing on this exam. Then the next four lessons are based upon our unit on differential equations that includes lessons 25, 26, 27, and 28. And then this exam will also cover materials about our unit on parametric equations, lessons 29, 30, and 31. Be aware that the topics of polar coordinates will not be covered on this exam, but instead will be covered on exam 4. So on this exam, there will be 11 questions total. So we have three questions, which are going to come from the multiple choice section. Each of those questions is worth five points each. And remember, with multiple choice questions, you get all five points for selecting the correct answer. And you don't get any points if you don't select the correct answer there. No partial credit is available for multiple choice questions. On the short response questions, we will have four questions on this exam from the short response. Those are going to be worth seven points each. You do not need to show work to get full credit, but you can get partial credit based upon the work you show or the answer you're included. So you can get part of the seven points if there are some mistakes there. The remaining questions here, 8, 9, 10, and 11 will be in the free response section. They do have varying amounts of points there. I do want to draw your attention to questions 9 and 11. These questions actually have multiple parts. There's a part A, part B. So together, these questions are worth 15 points each, which is kind of a lot. Oh, I guess also question 10 has two parts as well, 8 and 7. Part A is worth 8 points. Part B is worth 7 points. For which, yeah, that's also worth 15 points there. Question 8 only has the single parts worth 10 points there. One thing I want to make mention about these multiple part questions is that the reason they're broke up into parts is that if you do the first part wrong, that influences your answer to the second one. But if you get your answer wrong on part A, but you do everything on part B correctly using the wrong information you found in part A, you can still get the full points on the second part based upon whatever mistakes you made on the first part. Now clearly, you want to do this one right, so you can do that one right too. But the reason they're into parts is that as later parts are influenced by previous parts, you can still get full credit even if you made mistakes on the first part so long as you carry forward correctly all of that information. So this subdivision of points is actually to make grading easier and thus maximizing the amount of points you can get here. As usual, there'll be 100 points available on this exam. So let's talk about the specifics of this exam. There's three multiple choice questions, like I said, worth five points each. And these are the three topics you're going to see. The first question is going to be a question about differential equations for which you'll be given six differential equations. These could be first order, they could be second order. I probably won't give you more than second order because then the third derivative could be sometimes difficult to compute. The second derivative is challenging enough. What you'll be given is the general solution to one of these six differential equations. You then have to decide which differential equation does this thing solve. And so at the very least, what you could do is you can plug this solution to each of these differential equations and see if it solves or not. Only it'll be the solution to only one of them. And so then look for the correct one that is solved. We did things like this, of course, in lesson 25, which introduced us to differential equations. And particularly, we talked about solutions to differential equations. Question number two will also come from lesson 25, for which case you'll be asked to find the particular solution to a differential equation, find the particular solution given a differential equation and the initial value, so that is you're solving the initial value problem. Now, to make this into a multiple choice question, you will be given the general solution to the differential equation. So in particular, you need to plug in the initial value information into the general solution to find the particular solution. We did this in lesson 25. Remember, in lesson 25, we didn't really develop techniques of solving differential equations yet. But nonetheless, if we have a general solution, we can find a particular solution and thus solve the initial value problem. That's what we're going to do in question number two. So because you're given the general solution that dramatically simplifies question number two, define the particular solution there. Then the third and final question from the multiple choice section will be calculation involving probability. As a reminder, we discussed probability in lesson 24. In this specific example, it gives you a probability density function and it asks you to compute the probability of an event. So that does come down to a derivative, not derivative, an integral calculation. Be aware this question could also ask you to compute the expected value of a random variable, which of course you'd want to be prepared to answer that type of question as well. The calculation, the integral should not be too complicated. But like I said, it could be a probability you have to compute. It could be an expected value you have to compute. And it's also possible that these integrals could be improper. You could be going off towards infinity in one of these calculations. Be prepared to do that. The answer will be written as a decimal here. And so a calculator might be very helpful on question number three. Moving on to the short response section. Like I said, there are four questions in the short response section. They're worth seven points each. The first question number four is going to ask you to set up a hydrostatic force integral. We did examples like that in lesson 22. The thing to remember about hydrostatic force is the hydrostatic force is equal to pressure times area. Pressure is always calculated in these examples to be some density constant times the depth. Where the density constant delta here, it will probably involve something like mass per volume. You might have to also times that by gravity if it's using scientific units. If it's in pounds per cubic foot, then your row is just the number that'll be provided to you here. You do not need to memorize the density of water or the acceleration due to gravity. If you need those things, I will provide them to you. And then also area with hydrostatic force problems here. The area is always the area of a rectangle. So it's always gonna be width times a thickness, which I'll call it DX, you could call it DY if you want to. And in which case, then you put those together and make an integral. These applications are not so bad compared to some other ones we've done in the past. It's actually a breath of fresh air. But nonetheless, hydrostatic force is covered on exam three. So be prepared to set up and simplify an integral involving hydrostatic force applications. You do not need to evaluate the integral there. Some things to be aware of is when you set up your integral, you do need to include bounds. So there should be some bounds of some kind. And there should also be a differential like the DX, right? So if you're missing the bounds or the differential, you are gonna forfeit some points there. Don't do that. On the next page, we have the three remaining short response problems. Question number five will be a calculation about centroids. Remember, centroids we saw on lesson 23 for which you'll be asked to set up and simplify an integral involved in a centroid calculation. This one asks you to compute X bar, but you might also be asked to compute Y bar. I'm not gonna ask you to do both of them. You'll do one or the other. So do pay attention to the instructions there because the formula for X bar is slightly different than the formula for Y bar. If you don't know those formulas, please consult lesson 23. You probably wanna put those on your note card. You're not gonna be asked to do both. You're asked to set up and simplify the integral, but you're not asked to evaluate the integral. So I would wanna see something like the integral from A to B, X, F of X, DX, whatever it turns out to be. If you're looking for X bar, like the last one, since you're setting up an integral, you do need the differential. You do need the bounds. The omission of such would be forfeiture points. You don't wanna do that. You also need to remember that for this one, both the formula for X bar and the formula for Y bar, you have a one over A, where A is the area of the region. For the sake of simplicity, I will provide the area to you so that you don't need to compute that. But be cautious, of course, if I give you A, this needs one over A, it's reciprocal. So you might need to reciprocate in order to do that one correctly. Watch out for that. Then on to the next one, question number six. Question number six, you'll be asked to evaluate an integral involving parametric curves. So you might be given a parametric curve like this one here, X equals one plus E to the two T, Y equals E to the T, given some bounds, X equals one to X equals two. This one asks you to set up the integral, set up and simplify the integral for area of the region under the parametric curve. So you'd be getting something like A equals the integral. Again, there should be some bounds in place here, A to B, Y DX, for which you're gonna plug in your Y coordinate in here. You're gonna have to take the derivative of X to get here. And so be aware that while the bounds might be given in terms of X or in terms of Y, the integral will probably have bounds with respect to T because DX remember is the derivative of X, we'll write this as the derivative of X with respect to T times DT. So your integral does need to have the differential and have a DT in it. And then the bounds will be bounced with respect to T. So T equals something to something else. If they give you bounds like this one where it's X, you're gonna have to convert those into T. So there's just some things to be prepared for. This one asks you to do area under a curve. You might have to do area between two curves. Expect to set up an area calculation using parametric curves. Question number seven is gonna be similar to that. You're gonna be asked to set up and simplify an integral involving parametric curves much like we did before. This time it's much more generous in telling you the T bounds. But if it gives you bounds in terms of X or Y, you're gonna have to convert those into T's. You do not evaluate any of these integrals. So all four of the short response questions on this exam are setting up integrals. You do not have to evaluate any of them. You do need to simplify them as appropriate. This one, this question number seven, this one asks you about arc length, but you might also be asked to do surface area, right? So you wanna look up those formulas and both of these questions, you can find notes for them coming from lesson 31. One, and so this is the very last topic that is covered on this exam. How do you find area under the curve for parametric curves? How do you find the arc length of a parametric curve? Or if you take a parametric curve and rotate it around, how do you find the surface area of the resulting surface of revolution? Those are problems we discussed in lesson 31. You should be aware of the capable formulas, the appropriate formulas, and be able to set up and simplify these things. Again, differentials are required for full credit, so are bounds. Clearly the integral symbol is necessary, but no one forgets that one. Consult lesson 31 for all the appropriate formulas there, like area we talked about, arc length, surface area, I didn't write that on the screen. You can consult the lesson 31 or the exam syllabus for some notes on that. All right, so that then gets us to the free response section for which there were four questions there. Question 10 is worth 10 points, and this will be a question about parametric curves as well. This one, it asks you to find the equation for the tangent line of a parametric curve, which is given here, and it's also giving us the point of tangency. And you do have to write the equation of the curve in slope intercept form. So that'll look like y equals mx plus b, but that's probably not how you're gonna start. You're gonna probably start it with something like y minus y of t is equal to the derivative of y with respect to x evaluated at t times x minus x of t, for which the formulas for x and t are gonna be given to you. You'll also be given a point of tangency. So what you would have to do is you have to evaluate x at that number, in this case, t negative one. You'll have to evaluate y, again at that point of tangency, so you plug in negative one into it. You also need to calculate the derivative and evaluate at negative one in this case or whatever point of tangency they give you. Recall that for the derivative of a parametric curve. We're looking for the derivative, the slope of the tangent line is gonna be dy over dx. We don't want dy over dt or dx over dt. The slope of the tangent line is dy over dx. Now to compute dy over dx, you're gonna do dy over dt divided by dx over dt. And so doing these calculations, we can come up with the equation of the tangent line and write in slope intercept form. This question about finding a tangent line of a parametric curve, we did in lesson 30. Lesson 30 was all about derivatives involved with parametric curves and lesson 31 had to do with integrals involved with parametric curves. Moving on to the next page, question number nine, this is one that has two parts. Question number nine asks you to solve a first order linear differential equation. The technique that we solve, the technique we use to solve these linear differential equations was calculating the integrating factor. So that's actually what part A is. It's worth five points. You have to calculate the integrating factor for this function. Honestly, this is on par with a multiple choice question, but I actually asked you to compute it here explicitly. It is still worth five points, but I mean, hey, you can get some partial credit on this one if you did make a mistake there, but many of us will be able to do this correctly. This is a fairly straightforward calculation for the most part. As a reminder, linear differential equations were discussed in lesson 28 there, for which case, if your linear equation is not in standard form, you might want to standardize it and then compute the integrating factor. And so then you would say something, there'll be some calculations here, but you'll get some integrating factor, I of X equals something, we'll say F of X, put a little box around it so you know what your integrating factor is. Then on part B, you're going to use this integrating factor you just computed to then solve the differential equation, Y equals the yada yada. Now, if your integrating factor is incorrect, whatever you use as your integrating factor will be the judge on how part B works. Now, admittedly, if your integrating factor is incorrect, you might make the integral significantly harder to compute that you have to do on the next part, but nonetheless, there's 10 points available here and I will grade it based upon whatever integrating factor you use up there. So even if your integrating factor is wrong, but you do all of this part correctly, you will still get the 10 points for part B, but depending on how bad your mistake was on part A, you'll lose some of the points there, but many of us will actually get the full points this one, I'm quite confident. A lot of law students like these differential equation problems, they're really, really nice. Find the integrating factor then solve the linear differential equation using the integrating factor. Like I said, linear differential equations we did in lesson 28, please go to there if you want to see some more examples or some review of those. So then we go to question number 10. Question number 10 is also worth 15 points, although there's a lot more points put on the first part. This one, it'll give you a parametric curve. So X equals F of T here and Michael's G of T. You'll be asked to compute the derivative of the function. Like we said earlier, the derivative is gonna be dy over dt over dx over dt. And then once you've computed the derivative on part A, part two will ask you to do something related to the derivative such as compute the second derivative. That's the most likely thing you're gonna see here. For which the thing to be cautious about here is once you find the derivative, the second derivative is not the second derivative of Y with respect to T divided by the second derivative of X with respect to T. That is not what the second derivative is. No, no, no, no, no. Instead, what we have to do is we have to take the derivative of Y prime with respect to T, which we just found on the previous part. And then we have to divide that by the derivative of X with respect to T, which honestly, you should have already found that on the first part. So that second part should be pretty easy. You do need to take the derivative of Y prime with respect to T and then don't forget to divide by the derivative of X with respect to T there. So hence there's two parts there. Your second part will be dependent upon the derivative you found here. So if you've botched up the Y prime, that's clearly gonna make this thing wrong. But like I said on the previous one, I will use whatever answer you had from part A to judge whether you did part B correctly or not. So even if there's problems with part A, you can still get the full seven points on part B as long as you then do the calculation correct forward from that point, going for that point forward. So this is the derivatives and second derivatives and maybe concavity comes into play here. Monotonicity is the function of increasing, decreasing. Is it concave up, concave down? All of that stuff we discussed in less than 30, I mentioned before, it had to do with derivatives of parametric curves. So consult that one if you need some more practice or more review of any of those things. And then our last question, question number 11, you'll be asked to solve a separable differential equation. Part A is gonna ask you to separate the variables. So I'm looking for something like F of Y, D Y is equal to G of X, D X. So there might be more steps than I'm showing right here, but you're gonna separate the variables. The first step, that's worth five points. Again, it's very much on par with a multiple choice question, but as it's free response, you can get some partial credit there. Honestly, you should think of this as a short response question on par with a multiple choice question. And then the second part, you will then take the separation of variables you have from before and then you will proceed to integrate both sides and then you solve the, the separable differential equation doing that. So there's some more space here provided that I'm not showing you. That's worth 10 points. And like all these multiple part problems, depending on what answer you get here on part A will influence what part B looks like. If you do part A incorrectly, you can still get full points on part B so long as you do everything correctly with the wrong separation of variables. But honestly, most of us will be able to do part A pretty nicely if we make mistakes that generally happen on part B. And that's true for all three of these two part problems here. This exam only has 11 questions. So this is the last question on the exam. I should make some important notes about this exam. Separable differential equations were covered on section 20, lesson 26. So please consult that one if you want some more review on those topics or the exam syllabus, give some information there as well. It doesn't get everything, but it might give you some things you wanna see there. Now some two things I should notice. There were all the sections I listed because again, this goes from lesson 22 through lesson 21. Excuse me, lesson 22 through lesson 31. 21 wouldn't make any sense. There were two lessons I did not mention on this. Let's say something about that. For example, lesson 27, this was about growth models. We learned about uninhibited growth, aka natural growth. We've learned about inhibited growth such as Newton's law of cooling. Learned about logistic growth. And these were all solutions to separable differential equations that helped us model growth in various situations. As you're preparing for this exam, you really don't need to study lesson 27. That was an application of differential equations. But for the sake of this exam, what did we do in lesson 27? We solved those separable differential equations for which you're gonna show that you can solve several differential equations right here. Now by all means, the equations you have to solve either by separating variables or you could use integrating factors. We didn't use integrating factors in lesson 27, but maybe you could have. Those models, those growth models do make great examples of differential equations we can solve. So you might want to consult lesson 27 for the sake of the techniques of differential equation solutions that showed up in that one. But you do not need to memorize the logistic formula, exponential growth, inhibited growth. You don't need to solve Newton's law of cooling or whatever. There's basically two parts of each of those problems. You solve the differential equation, which the answer's only there. And then you plug in specific numbers and apply it. You probably can skip over 27 without much hurt to your study regimen whatsoever. The other one that is also worth mentioning, I never mentioned lesson 29. Lesson 29 was an introduction to parametric curves, functions given by parameters. We didn't actually do any calculus in lesson 29 whatsoever. We just were acclimating ourselves to parametric functions in that setting. The calculus of course comes into 30 and 31. Now, in order to calculate derivatives and integrals involving parametric curves, you do have to understand parametric curves. And many of us have never seen them before. And therefore we needed a lesson 29 to introduce us to them. But with that in mind, there will be no specific questions on this exam that are drafted from the types of materials we saw in lesson 29. We saw two questions in the short response section. We had to set up integrals involving parametric curves. And we saw two questions in the free response section involving taking derivatives of parametric curves. I'm not gonna ask you to graph a parametric curve or anything like that, but be aware that many of the mechanical steps you have to do with setting up the integrals or computing the derivatives of parametric curves does require techniques that were introduced into section 29, lesson 29 there. So if you're familiar with those things, you probably can focus more on lessons 30 and 31. But some of those mechanical skills from 29 do show up in other problems, which is why it's not assessed directly on this exam. So with that, that does bring us to the end of this exam. Thanks for watching. I hope that you find this review helpful as you prepare. Of course, also consult resources on Canvas like the exam syllabus, the practice exam, their solutions and other materials that might be available on Canvas. Best of luck in your studies. If you do have any questions, feel free to reach out to me. I'll be glad to answer any questions you have for you. Since if you're watching the very end of this video, good at you for watching this video. I'll give you a cute little secret that you're gonna love so much. Exam three is the easiest exam of this semester. It always is. If you're stressed out, take a little bit of burden off your shoulder. You're gonna do fantastic on this exam and you're gonna be very grateful and honestly wish all of the other exams were as easy as this one was.