 Welcome back, everyone, for our review for math 1210 exam number 2. As usual, I'll be a professor today, Dr. Andrew Misseldine. So with exam 2 upon us, in this video, I want to review the many important topics that you should be reviewing as you prepare for the exam. Now, the structure of the exam and the policies associated to the exam are essentially going to be the same as on exam 1. So I'll reference you to the things that we said about exam 1, either in the previous exam review video that you watched, or things that were mentioned in class or on campus, all of that stuff. So the structure again of the exam is going to be fairly much the same. We'll have 10 multiple choice questions. We'll have five free response questions. The multiple choice questions will be five points each, pass, fail. The free response will be mostly 10 points each, and you need to show full work to get full credit. You can get some partial credit. So that stuff, the structure is going to be very much the same. Things about timing, the location, the dates of the exams. Those you should refer to Canvas or the course syllabus to get that information, because those are course specific things that will change from time to time. So I don't want to talk about that in this video, because again, this is not our first rodeo now as we've already done exam 1. So what I want to do in this video is focus on the topics of exam 2, and then the corresponding sections and lectures that went along with those topics. So the topics of exam 2 are going to focus around from our lecture series chapter 2, that is limits, their interpretations, calculations of limits, and then at the end of the chapter, we've had an introduction to derivatives, which is a special type of limit, the limit of a different quotient. So we need to prepare for all of those type of calculations. You're going to see dozens and dozens, maybe not that many, but you're going to see calculations of limits over and over again on exam number 2. For example, question number 1, you will be given the graph of a function, you will not be given the formula, you'll be given the graph, and you'll be asked to compute a limit calculation from the graph of the function. This one asks for the limit as x approaches negative infinity of f of x, which is illustrated here on the graph. All right, so this will be in line with the types of calculations we did from lecture number 9, for which we learned about the intuitive notion of a limit and very much focused on numerical approximations and more importantly, on the graphical representation of a function to determine the limit. But also, as you see on this example, it might ask what's the limit as we approach a vertical asymptote, as you kind of see right here, or like I said, this question's asking about the end behavior, what happens on the far left or on the far right of the graph? So I should also mention that limit at infinity could be part of this question, so you might want to reference lectures 14 and 15 about that as well. The limit could be some finite number like 5, 1, 0, that's all acceptable. It could be the limit turns out to be infinity or negative infinity, so be aware of infinity, the limit's on infinity there. It could also be the limit doesn't exist. So for example, if I ask about the left-handed limit as you approach 0, so the limit here as x approaches 0 from the left of f of x, this would be defined to be infinity, because you have the vertical asymptote there. If I ask the limit as x approaches 0 from the right of f of x, we would see that the limit there would be 1, because you can see that value there, but if I ask about the two-handed limit, limit as x approaches 0 of f of x here, we would see in that situation infinity is not the same thing as 1, we get the limit does not exist, d and e, for which you would then choose f in that situation. So be prepared for those possibilities to be able to compute a limit graphically. Question number two will also be a question about computing limits, but this will be an algebraic calculation. So primarily look for the things that we did in section 10, our lecture 10 of our lecture series, but I also should mention things we talked about, lectures 12 and 13. 12 and 13 were about continuity, that is when a function is continuous. We learned there are situations for which since a function is continuous, we can just plug in the number, like as x is approaching two, we maybe just can just plug in x equals two into the function and compute it, that sometimes works and sometimes doesn't. So the limit laws we saw in lecture number 10 combined with continuity helps us to compute limits in very easy situations. I would say that question number two is gonna be sort of an easy limit calculation. So look at those sections for references. Question number two won't involve any limits and infinity, so no vertical or horizontal asymptotes in that consideration. For question number three, you will also be given the graph of a function, you can see the graph of the function f right here. And so from this graph, you wanna identify one of two possible questions. This one asks to determine all values of x where f is not differentiable. A variant you might see on this question is asking where the function is not continuous. These two questions are related, but they are not the same question. Things about discontinuities that we have learned about this semester, we saw those primarily in section 12, but 13 also talks about that a little bit. So things, discontinuities we could be looking for, like a remove point. So if you see something like this on the graph, this remove point, that's a discontinuity. We could have some type of jump discontinuity, maybe the graph does something like this. That would also give you a discontinuity. Maybe there's some type of vertical asymptote we go up and down, some vertical asymptote there. Those are discontinuities. If I'm asking you where's the function not continuous, those are things you wanna report. You'll notice that this function has no discontinuities. There's no places where it's discontinuous. So the answer might actually be F is always continuous. Now this question asks where the function's not differentiable, which I said those things are related. If you're looking for not differentiable, that's actually where you're gonna wanna go to lecture number 18 in our series. And we learned that a function's not differentiable if one of three things happens. One, there's a discontinuity. So if you're discontinuous, you're gonna be not differentiable. So that's how the two things are related. But also, if you have a sharp corner, the function will be discon... Or it won't be differentiable. So you look at, for example, this sharp corner at X equals three, the function is continuous there, but it's not differentiable. That's the big difference right there. And then also, we have to look out for vertical tangent lines, vertical tangents. So for example, on this graph right here, if we look at X equals negative two, there's a vertical tangent line at X equals two, that means that the function's not differentiable at X equals negative two, but it is continuous. And so there is an important distinction there. So pay attention to the question, whether it's asking for where is it not differentiable or where it's not continuous related, but not the same problem. Moving on to the next page, question number four is gonna involve a limit as X approaches positive or negative infinity or approaches the in behavior of the graph. So this is exactly the type of things that we saw in lectures 14 and 15. This one's fairly tame, I should say, because for rational functions, taking the limit as X approaches infinity or negative infinity, it's fairly straightforward. But these questions could be of more moderate difficulty. For example, when square roots start getting involved, we saw things get a lot more fishy, especially as you approach negative infinity, I have to be very careful of that. We also did limits with exponential expressions of some kind, like if I took X plus E to the X over 10 X squared minus E to the X, if I changed the question, how would that, how would affect the answer, right? We did things like that in lectures 14 and 15. So be prepared to go to the compute and limit as X approaches positive or negative infinity. Question number five is gonna be a question about piecewise functions and where, and continuity related to piecewise functions. So you'll notice in this example, in this question, there's a number C, some parameter C that's inside of the piecewise function. And we're asking what choice of C will make this thing be continuous? Well, the key there to make a piecewise function be continuous is as you go from one piece to the other, you need to be touching, right? It doesn't matter if there's a corner or whatever, they have to be touching because if the two branches are not touching, then there will be some type of jump discontinuity, which would cause it to be not continuous. So we have to see what value of C will make the switch from the lower part to the upper part touch. How do you make the left-hand limit, the right-hand limit agree at X equals four? That's what you have to investigate on question number five. This is exactly like a problem we did in lecture 13, lectures 12 and 13 about continuity. And the very first example of lecture 13 was very similar to what you're being asked to do right here. Question number six will be another limit calculation. I would say this is more of like a medium limit calculation. Some things to look out for is that this one involves some absolute values. Absolute values can be very tricky. We saw, again, examples like this in lecture 10 about limit properties, but we also saw them in like lecture 11, at least the homework in lecture 11. The lecture 11 focused on the squeeze theorem, but the homework for lecture 11, they had some more of these moderate, more medium difficulty limit calculations. But also we saw some of these in 12 and 13 because again, continuity does play a role in this, but this one's not simply just plug and show. You can't just plug in x equals three because you end up with zero over zero. You have to approach this a little bit differently. The fact that it's approaching three from the left does change the answer compared to if you're approaching from the right. Maybe, maybe it doesn't. This is sort of what I mean by it's more moderate, more medium difficult to be prepared for those type of things as well. Question number seven is also gonna involve piecewise functions. As we saw in the lecture series, limits are much more, they're very fun when you have a piecewise function. A lot of interesting things can happen in terms of limits here. So given this function f of x, can we compute the limit as x approaches zero from the left? Heck, could we approach zero from the right? Could we approach zero from either side? How does that affect the fact that the function defined to be zero at x equals zero? Does that affect it? Does it not? This function would also be interesting to ask what happens as you approach two from the left or two from the right or the two-sided limit. Maybe the limit doesn't exist because there's a jump discontinuity. So how do you decide which branch to use? And it depends on these boundaries right here. If you're asking for the limit as x approaches two from the left of f of x, well, that would put you in this domain right here. And so then you'd be looking at this branch. This is a continuous function x squared plus one on that branch. So you could then calculate this to be, oh, I'm gonna take two squared plus one, I end up with a five, which I would say is the correct answer. Of course, this one wants to do zero from the left. But this is how we can compute limits of piecewise functions. So again, the sections to be looking for here are section 10 about limit laws. That's very relevant here. Continuity, of course, comes into play. We did some examples like this. I think we did this, I think we did an example very similar to this at the end of lecture 12. So that's, I think, where the money's at. I would go there if I wanna see some more examples and more practice on this. Question number eight is gonna probably be obvious where it comes from. Question number eight comes actually from section eight. Section eight, or I should say lecture eight, was their introduction to limits. This is where we saw the precise definition of the limit. Now, my perspective when it comes to calculus one students is that the precise definition of the limit is a little too much to expect of us our first time around, and that's quite reasonable. I mean, it took me like three times when I saw these episodes on Delta Stuff before it finally clicked. So if you have a good understanding of this right now, you'll power to you. So I don't expect students to be proficient with the epsilon Delta proofs of limits, but I do expect students to have exposure to epsilon Delta and the precise definition, and thus be able to unravel statements involving specific epsilons and specific deltas. So you see right here, we have a specific function, a specific limit, a specific epsilon. It's kind of encoded in there. You have to be able to identify what is the F of X? What is the L? What is the epsilon? That notation you should be familiar with. What is the A, right? That stuff is given to you in these statements right here. What you then have to compute is what is the associated Delta, very specific Delta, for which we can unravel it from here. Now, this one is given in completely algebraic notion, algebraic notation I should say. An alternative you might see on the test is that a specific function might be given to you. Maybe this is F, in which case then we're like, oh, here's L and here's the A value, and then here's L plus epsilon, here's L minus epsilon, and so we have L plus epsilon, L minus epsilon. We have our A1 or A2. So you might be given a graphical representation of the function with this information about L, epsilon, F, and A. That's okay, if anything, we're like, I love that. That makes it a lot easier because you can basically, you can skip some of the calculations and read the information from the graph. Nonetheless, on question number eight, you do have to build a compute Delta given information about F, L, epsilon, and A. So be prepared to do that for the test. Moving on to the next page, we arrive upon question number nine, which will be our first question about derivatives. So this question, it'll give you a limit. So this case is the limit as H approaches zero of one plus H to the 10th minus one all over H. And it's gonna ask you that, hey, this is a limit of a difference quotient. How do you write that as a derivative? Recognize that this is a derivative of what? What's the function? And what's the number we're evaluating at? So remember that as we define the derivative in lecture 17, we saw two definitions of the derivative. There was the limit as H approaches zero of F of X plus H minus F of X all over H. So can we recognize this to be in that form? Or the alternative form we had, take the limit, say as T approaches X of F of T minus F of X all over T minus X. This form resembles more of the slope of a line for which we know the derivative measures the slope of a tangent line. So can we recognize this limit provided to you as one of these functions right here? You need to recognize the function F and the number A that it's being evaluated at. I got a little bit cluttered there. Let me erase these things. Question number 10, you'll be given a position function, a motion function and you're asked to compute the average velocity of that motion function on some specific interval right here. Now remember the relationship that the rate of change of position gives you velocity. So if they want the average velocity, then you'd be computing the average rate of change of your position function, you know your delta S over delta T, which is equal to S of, well, if you have some interval here A to B, this will be given to you. You have to take S of B minus S of A all over B minus A. You have to compute that average rate change. Now your function to be given as a table right here, which notice your inputs on the second line. Here's our T values. And so then S of T is on the top. So if you try to compute like, what is S of 20? Let's say you have to compute that. Well, we see from the table, that would be 80. So at after 20 minutes, the thing has traveled 80 feet from this. So we can compute the average rate change from this table or the average velocity. We should be able to compute average rates of change for a function. That's something we talked about in lecture 16 in our series. Now question number 11 will be our first free response question on the test. On this one, you'll be given a function G of X equals in this case, sine of X. You'll be given the derivative. So although we might be able to compute the derivative of this function by now using shortcuts or by the definition doesn't matter, don't care. The point is for this function right here, you'll be given the derivative. You'll be asked to compute the tangent line, the equation of the tangent line given a function, which given the derivative helps you out a lot because the tangent line will depend on the derivative, right? So the general formula is gonna be that Y minus G of A is equal to G prime of A times X minus A. You're gonna take this formula, plug in the appropriate parts. A is the X coordinate at the point of tangency. G of A will be the Y coordinate at the point of tangency. We are given A, we're gonna have to compute G of A, and then we also have to compute G prime of A, which given that they give you G of X, it shouldn't be too hard to get G prime of A. Remember to put this answer in slope intercept form. That is necessary here. Write your answer in slope intercept form. So once you plug in the appropriate parts, the three numbers there, then you have to convert this to Y equals MX plus B. So there are some formulas here maybe worth writing down. Moving on to question number 12 on the next page, you'll be given a question where you're asked to prove that the limit of some function is equal to some number, could be zero, very likely, but it could be something. And this is gonna be a much more challenging limit calculation, which is why you're actually given what the limit's gonna be. This one does require a proof of some kind. It's not just a calculation. There's a proof involved with this as well. And during that proof, there's a major theorem we're gonna have to mention. I'll give you a hint. It's on this question going to be the squeeze theorem. The appropriate point, you are gonna have to use the squeeze theorem. Now, the squeeze theorem, this will be talked about in detail in lecture number 11 in our lecture series. For the squeeze theorem to work, the idea is you have a function G of X and it's been sandwiched between two other functions. So we have some lower bound, we'll call it say like F of X and we have some upper bound, maybe call it H of X like so. And so what we've seen here is that, the limit as X approaches A of H of X is some value we'll call it L. We also know that the limit as X approaches A of F of X is likewise equal to L. So you'll notice that these values on the left and on the right, these functions approach the same value L. And therefore the function G of X which is squeezed between L and H as L and H, excuse me, as F and H come together at L, G of X would likewise have to be L. And so we then can conclude that therefore we get the limit as X approaches A of G of X is likewise equal to L. So for the squeeze theorem proof, of course you need to come up with the squeeze. You need to identify what's the function that's less than G of X, what's the function that's bigger than G of X? You have to come up with the F and the H right here. You'll have to compute the limit of F. You'll have to compute the limit of H. You'll need to observe that these two limits are the same value. And then you can invoke the squeeze theorem to get the limit of G is equal to zero, in this case or whatever the limit turns out to be. See the solutions to question number 12, to get an example of how to write one of these type of proofs or just refer to lecture 11, of course. Question number 13, the second question on this page, you'll be asked to compute the derivative of some function. And this is important note, the derivative must be computed by the definition of the derivative. So this would look something like F prime of X is equal to the limit. The limit needs to be mentioned here. The limit as H approaches zero of F of X plus H minus F of X all over H. And then plug in the specific values and go from there, simplify compute what that limit turns out to be, that will give you the derivative. So it's imperative that you compute this by the definition of the derivative like we saw in lecture 17. I should mention that the derivative, it's a limit of a difference quotient. And all throughout chapter two, we've been computing limits of difference quotients. We did it in section number 10 in the homework for number 11 and 12 and 13, just to name some of the important sections of this chapter. You do have to compute this as a limit of a difference quotient. Now it might be very tempting because if you are starting to learn chapter three stuff by now when you've watched this video, it might be tempting to use something like the power rule because you could write this as a power function and you can use that to calculate the derivative. I have to warn you that such an approach will receive no credit. If you just wrote down the derivative using the power rule, that'll be great when you get to exam number three that tests you about the power rule. But on exam number two, as that's chapter three material chapter two material is of no points, right? The instructions do say it needs to be computed by the definition. That is, you have to simplify and compute the limit of the difference quotient. Any other approach will receive no credit. Now by all means, if you compute the derivative by the definition, you write that on the page and then you just as a side note are like, I'm gonna use the power rule to double check my work. That's perfectly fine. By all means do that. Use it to check your work. But I'm saying this right here is worth no credit. It's not like you lose points for doing it, but if you omit this stuff over here, then I can't give you any credit on this one. So you have now been warned. Moving to the last page, two more questions. Question number 14, you're asked to prove that you'll be given some equation has a solution. In this case, it's cosine of X equals X. This equation is intentionally chosen to be one that you're not gonna solve by yourself. Even with your scientific calculator, you're not gonna find a solution to this thing. Cause all you're asking to do but you're not asked to solve it. You just asked to prove that it has a solution. So again, we have a proof that you have to do. And what's the major theorem you're gonna use here? The major theorem in play here is the intermediate value theorem. The intermediate value theorem we learned when we talked about continuous functions. And the intermediate value theorem was specifically mentioned in lecture number 13. So review that for an example of what to do right here. The intermediate value theorem tells us, remember that if we have some function, I mean the intermediate value theorem tells you more than what I'm gonna say here. But if we have like the X-axis, right? If we have a function that is somewhere above the X-axis and it's somewhere below the X-axis and if it's continuous, then that means the function has to somewhere across the X-axis. And so if it's function, let's say cosine of X minus X, if you move it to the right hand side, if you're like, oh, take F of X to be cosine of X minus X, I wanna find an X-intercept of F right here, that's a great thing to do. What you have to do to finish this one, what's important, you need to identify that the function in play is continuous. I don't necessarily need to prove that it's continuous, just make the observation that this is continuous. You didn't have to find one point on the graph where it's positive. You didn't have to find one point on the graph where it's negative. And then you would invoke the intermediate value theorem to say that somewhere between these points, there's an X-intercept. That X-intercept then is a solution to this equation. By all means, look at the solution to this question number 14 on the practice exam to see a template of how one should write this proof or look in lecture 13, like I mentioned earlier to see some examples of this, of course the corresponding homework as well. So the last question on the test, it's gonna be another limit calculation. This is what we'll call the hard one if we are gonna rank them by difficulty. So all the stuff we've seen in this chapter is fair game. Stuff we saw in lectures 10, in lectures 11, 12, 13, 14, I think I've listed all of them by now, haven't I right? It could be a limit as X approaches infinity. So this will be one of the more difficult ones. Most likely it's gonna be the limit of a difference quotient. So similar to the derivative calculation we did on the previous slide, right? But it doesn't exactly have the form of a derivative. Even though if you just plugged in X equals one, they're gonna end up with zero over zero. They have to remove the discontinuity. So the function's undefined at X equals one, but the limits still exist. So what type of algebraic manipulation can we do to the function to simplify it? This one's just a rational expression, but it can involve exponentials. It could evolve radicals like square roots and such. We've learned how to do these things, apply them on question number 15 to get full credit. And so that brings us to the end of the test. Like I said, this test is heavy on limit calculations. And so be prepared to do these various limit calculations. We've seen a couple of formulas that would be helpful to put on like your note card, but also there might be some good examples to include, especially if you're really worried about some of the proofs like the intermediate value theorem or the squeeze theorem. By all means, feel free to include an example like that, one of those on your note card so that if you see a question similar, which you will on the test, perhaps you can reverse engineer and particularly have that template because there are some specific points in those ones that we will need to see. Now, if there is any questions after watching this review that you have, feel free to reach out to me during office hours or email or some other method. And I'd be glad to answer your questions. The tutoring center, of course, is also a great resource, free to SU students that you can use as you're preparing for the test, form study groups and follow other good study practices. And you're gonna do just fine on this test. Let me know if you have any questions, everyone. See you next time, bye.