 Welcome to our review for Exam 3 for Math 1210, Calculus 1 at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. As we're studying for Exam 3, it's important to realize that this test is going to focus on derivatives. We saw derivatives a little bit on Exam 2, but that was mostly just the definition of the derivative. In the Exam 3, we're going to be focusing on some applications of the derivative. We'll see some more on the next exam, but this test will look at applications to science and related rates problems. But this test will predominantly be focusing on techniques of computing derivatives, like the power rule, the chain rule, the product rule, and things like that. The structure and policies and procedures for this exam will essentially be the same as with the previous two exams. Now, some of those details do change from semester to semester. If you have questions about the dates of the exam, time limits, calculator use, and any other policies like that, I would refer to you to the course syllabus, which can be found on Canvas. This test will have 15 questions. 10 of them will be in the multiple choice section. Multiple choice section, each of those questions are worth five points each. You'll select the single correct answer, and full credit will only be given upon selecting the correct answer. If you select anything else or you don't clearly select anything, you would get no credit on those ones. The last five questions, 11 through 15, are in the free response section. These are questions which you must show all of your work. A final answer is not good enough. That will get you essentially no credit. You have to show all of the appropriate work on those questions. Most of them are worth 10 points, but there's also one question 12 is worth eight points. There is a 16th question, but this is just a reminder here to turn in your note card with this exam. So let's get into the details of this test. Like I said, specifically this exam is going to cover the topics about derivatives and such. The topics of this exam will coincide with chapter three of Jane Stewart's calculus textbook, which with our lecture series that coincides with lectures 19 through 30. So like I mentioned earlier, this will cover all the different derivative rules we've learned about. So power rule, the product rule, the chain rule, the quotient rule, implicit differentiation, logarithmic differentiation. These are all techniques that will be on there. We have to also be able to calculate the derivative of various function families like power functions, exponential functions, logarithmic functions, trigonometric functions, hyperbolic functions. This test will also cover the derivatives of the inverse trigonometric functions. I won't ask questions that require you know the derivative of the inverse hyperbolic functions. You do need to know the derivatives of the six hyperbolic functions, and you do need to know the derivatives of the six inverse trigonometric functions, but I will not require anyone memorize the derivatives of the six inverse hyperbolic functions. That's an important omission for this test. And like I also mentioned, this test will cover the topics of related rates and scientific applications of rates of change. So let's look at the questions one by one to give you an expectation of what we need to be preparing for. So question number one, you see here that you're going to be asked to compute the derivative in this case of a polynomial function. When you get your version of the test, it might not actually be a polynomial, but this question will require the use of the power rule. The power rule which we saw back in lecture 19 will be necessary for question number one. Question number two, you'll be asked to compute the limit of a trigonometric function. Before we learned about derivatives of trigonometric functions, we had to first learn about some trigonometric limits. This was necessary so that we could compute limits of difference quotients involving trigonometric functions. That's a necessary step to calculate the derivative of sine and cosine. So two limits you definitely need to know. These are ones that you should memorize. You need to know that the limit as theta approaches zero of sine theta over theta, you need to know that this is equal to one. That's going to be very useful information. You should also know that the limit as theta approaches zero of cosine theta minus one over h, excuse me, over theta. This is likewise equal to zero. This is important information. These both of these limits sine theta over theta and cosine theta minus one over theta both have the form zero over zero. But using some trigonometric forms, we can actually compute those. I don't expect you to know the derivation of these limits. You should memorize them and use them to compute more complicated trigonometric limits like we see here in question number two. If you need some more practice, please consult lecture 21 to see more about some trigonometric limits. Question number three, you can see here in the practice test, question number three is asking you to compute a second derivative. So here we have the function e to the x times cosine of x has to find the second derivative. That's what the notation here is all about. This is a second derivative. Now what techniques will be necessary to compute the second derivative? It depends, right? It could be the power rule. It could be the product rule. We might need to know derivatives of exponentials, logs. It could be any of those things. But do anticipate that question number three is going to ask you to compute the second derivative of a function. Calculate the second derivative just means you take the derivative of the derivative. So make sure you take the derivative twice and apply whatever rules are necessary. This one's a little bit of wildcard. I can't predict for you what that technique is going to be. Just be ready to calculate a second derivative on the test. Question number four is going to involve inverse trigonometric functions to some degree. You see here you have to find the derivative of y equals sine inverse of the square root of sine. So can we compute the derivative of an inverse trigonometric function? We did learn about those. And like I mentioned on the previous slide, that you do need to know the six derivatives of the inverse trigonometric functions. We learned about them when we did implicit differentiation, in particular the lecture 25, which was the second lecture implicit differentiation, demonstrates how you can compute and use the derivative of general inverse functions, but trigonometric inverses were included in lecture 25. So consult that one for some more examples. When you look at example number five, question number five, I should say, this one's actually take the derivative of e to the x times the natural log of x. What you will need on question number five is knowledge of the product rule. So be able to compute a derivative of a product. Now in this example, you need to also know how to take the derivative of exponentials and logarithms, but there are lots of different product functions that you could use. While the product rule will be guaranteed, other techniques like logarithms, maybe the power function, trig functions could be showing up. So you really need to know all of these techniques, but I've made sure that all of the bases are covered. So there will be a question about the product rule, and that will be question number five. The product rule was introduced in lecture 20 in our series. Question number six, you look at this one, and if you're thinking along how we did with question number five, this is a question that's going to involve the quotient rule. This question right here is, we have to take the derivative of a fraction, e to the x divided by x squared plus 2x. So in addition to the quotient rule, and so this is our typical low d high minus high d low square of the bottom, here we go, you should know the quotient rule. And by all means, use the poem to help you remember it. The quotient rule is also introduced in lecture 20, right after the product rule. You will need to know the quotient rule for this test. Question number seven, which we can still see on the screen here, this is a question that will involve the chain rule to some degree. There are other questions that will definitely use the chain rule, but question number seven will require the chain rule. We see in this one that we're taking the derivative of e to the x squared. There are two functions in play, one inside the other, and because of that composition of functions, the chain rule is necessary. This example also points out that we have to know how to take the derivative of any exponential. We definitely need to know how to take the derivative of the natural exponential, the derivative of e to the x, of course, it's itself. But in this example, we need to also know that if you take the derivative of any base, a to the x here, its derivative will look like the natural log of a times a to the x. Both this exponential rule and the chain rule were introduced in lectures 22 and 23. 22 and 23 were about the chain rule. In lecture 23, we also used the chain rule to compute the derivative of an arbitrary exponential function, which is necessary in this question. Now, that might not be the case for your version of this exam. Question seven might not require exponentials at all. But one, question number seven will use the chain rule. I can promise you that. And then two, there probably will be a question that uses the derivative of an exponential, whether it's the natural exponential or two to the x, three to the x, doesn't matter, you should know that one. Question number eight will be a question involves trigonometric derivatives. We talked about trigonometric limits earlier, but we have to also compute the derivatives of trigonometric functions. And as a reminder, the trigonometric derivatives were talked about earlier in section 21. Question number eight. I want you to know here that this gives you a slightly different phrasing of essentially the same question. A lot of the questions were like, find the derivative, okay? But some variations of that would be things like the following. Find the slope of the tangent line of a function, which this function is given as a trigonometric function at a specific point. So the fact that the function is trigonometric is why the trigonometric derivatives is appropriate here. But the other thing to point out here is that the slope of the tangent line is the derivative. So we're not looking for the derivative in general. We don't want the general formula. We're looking for a specific evaluation of the derivative. It's going to be a specific number, a, b, c, d, e here, or it could be it's a vertical tangent line, then the slope might be undefined. So f might be a possibility there. But it's important to know that the slope of the tangent line is the derivative. So even though it's phrased as tangent line and slope, we're looking for derivatives. Another question could be phrased in such a way that f of x is some motion function. It's a motion function. And then you're asked to find the velocity of that motion function. Well, velocity is the derivative of the position function. Therefore, knowing that vocabulary tells you you need to take a derivative. So look for some of these variations. Pretty much all these questions are asking you to calculate a derivative for the most part. So look for the variation in vocabulary there. Question number nine will ask you something about hyperbolic functions. Okay. So this one is asking for the limit of hyperbolic secant as x approaches infinity. So this is just a limit calculation. But as we covered everything about hyperbolic functions in one lecture, we introduced the hyperbolic functions. We did limits. We did derivatives of these things. Question number nine will ask you something about hyperbolic functions. It could just be like, what is cinch of one? That could be just evaluated. This one could be a limit calculation or most likely you'll be asked to compute a derivative of a hyperbolic function. It's important to remember that the derivative of cinch is going to equal cosh. The derivative of cosh, I need to put an x in there, is equal to cinch. And then the other four, you have hyperbolic tangent, cotangent, secant and cosh secant as well. You can calculate their derivatives using the course, the quotient rule, using these two observations. It's probably best just memorizing them or putting them on some approved note card if you're allowed that for your test. So be prepared to do that. Lecture 30 is the lecture that we talked about hyperbolic functions. Like I mentioned earlier, I will not require you have any memorized information about the inverse hyperbolic functions. My philosophy is if you can do well with the inverse trigimetric functions and you can do well with the hyperbolic functions and put those together, you can do something with inverse hyperbolic if necessary, but I won't require that. Now there is one potential exception to that idea about inverse hyperbolic that I'll talk about later when it comes up. You don't have to worry about it in the multiple choice at least. And then question 10 is the last question, the multiple choice section. This will be a question that requires you to take the derivative of a logarithm. So it's important to remember the derivative of the natural log is going to equal 1 over x. And in fact, if we take the derivative of the natural log of the absolute value of x that likewise is 1 over x, there is of course a domain difference there. And in general, if we take the derivative of the log base a of the absolute value of x, that's going to equal 1 over the natural log of a times x. Of course, this could also involve the chain rule like this example happens here, it demonstrates here. Also look at some variations of the question. This one doesn't ask for the formula. I want you to evaluate the derivative at zero. So what's the y-intercept of the derivative? So there could be different ways of phrasing these questions, but for the most part, these multiple choice questions will ask you to compute the derivative of various function families using various derivative rules that we developed in unit three of our course. Moving on to the free response section. Question number 11 is going to ask you to provide the proof of some derivative rule of some kind. So for example, this one gives us the question, can we compute or it tells us the derivative of cosecant is negative cotangent cosecant starting again in three. This is my second edit starting again and starting again. In three, two, one. I should mention that the topic of derivatives of logarithms was introduced in lecture 26. Please consult that one if you need some more practice. Moving on to the free response section now, which there's five questions there. Question number 11 will be asking you to provide a proof of some derivative rule of some kind. The example I provided in this practice test is verify that the derivative of cosecant is equal to negative cotangent times cosecant. And it gives you the hint that you should use the quotient rule because after all, cosecant is just one over sine of x. So if we know the derivative of sine, if we use the quotient rule, we can derive with a little bit of trig identity in there as well. We can derive the formula for the derivative of cosecant. So with that in mind, what are some other derivative proofs that you might be expected to compute for this test as you're preparing for it? Well, any of the six trigometric functions would be possibilities here, the six trig functions. Cosecant is the one that's on this practice test. We prove sine and tangents derivative in the lecture series. I'm pretty certain in the homework you were asked to prove the derivative of secant, but I could be wrong about that one. So you, as you're preparing for the test, should be able to prove that the derivatives of sine, cosine, tangent, cotangent, cosecant, and secant are what they claim to be. Tangent, secant, cotangent, and cosecant, you can use the quotient rule combined with the derivative of sine and cosine. To prove the derivative of sine and cosine, you do need to use the definition of the derivative as a difference quotient. We did sine in our lectures. Copsine is very similar and you're encouraged to do that. All right. So another one, could you do the, could you provide a proof for the derivative of the six hyperbolic functions? Turns out these proofs are very, very similar. The proofs of hyperbolic tangent, cotangent, secant, and hyperbolic cosecant will be very similar to what you're expected to do right here. You could use the derivative of sine and cosine combined with the quotient rule and you get all of those. How do you prove the derivative of sine and cosine? Well, we prove the derivative of cinch in our lecture series. We just use the definition of cinch because cinch, remember, is just the function e to the x minus e to the negative x over 2. So just using standard derivative properties, it's very easy to prove that the derivative of this is Cauch. And you could also prove the derivative of Cauch's cinch. So could you prove the six hyperbolic functions are their derivatives? That is, could you prove what the derivatives are? The derivative will be provided to you, so you don't have to have to memorize, but you have to verify it. You should also be able to prove the derivative for logarithms. So we saw on question number 10, the formula for the derivative of logarithms. Can you use implicit differentiation to prove the derivative formula for the natural log or for general logarithms? That's something we proved in class. That could be a question, that could be a variant of question number 11. Similar, can you do the inverse trigometric functions? Inverse trig. Also, could you do the inverse hyperbolic functions? Okay, so this was the one, this is the one exception I made to, do you need to know the inverse hyperbolic functions? The reason is you don't have to know what the derivative is. It'll be provided to you. You just have to verify the formula using implicit differentiation. We did several examples of this in our lecture series. We proved the derivative of arc sine was such and such, using implicit differentiation. We did that for logarithms. We did that for the inverse hyperbolic functions. I think we did it for arc cinch. So you should be able to replicate one of those, because if I give you the formula for the inverse of hyperbolic secant, even though the formula is like whatever it is, don't worry about it, you don't have to have it memorized, the derivation of which is the same strategy using implicit differentiation. The idea is if your function f of x equals y, you can switch it to implicit differentiation. So x equals f inverse of y. Then you take the derivative, or if you start with the inverse, go the other way. It doesn't really matter who's f and who's f inverse, but to take the derivative implicitly, then solve for the derivative of the function you need. That's the general strategy. Some other things that you should be able to do, could you prove the power rule using logarithmic differentiation? That's something we did in our lecture series. Could you prove the quotient rule by combining the product rule and the chain rule together? I will not expect you to prove the chain rule. I will not expect you to prove the product rule, although we did see the proofs of both of those in our lecture series. Those ones are a little bit more intense. The power rule, if you use logarithmic differentiation, is fairly straightforward. We did that one in class. You should be able to reproduce one of those type of proofs for question number 11. So there are a lot of different lectures you could consult for this one. So I guess I could try to write them all down, but we saw some of these in lecture 21. Excuse me, lecture. Yeah, we did see that in 21 about trigonometric functions. That's a good one to look at. You should also take a look at lecture 25 about implicit differentiation. I would also include 26 about logarithmic differentiation and definitely 30 about hyperbolic. That's where most of those are going to come from, but the proof of the quotient rule is actually in a different lecture that's not listed right here. Let's move on to question number 12. We talked a lot about 11 right there. 11 is sort of a sticky one for many students, understandably, which is important, that the test isn't going to only be easy questions. You have to have easy questions, moderate questions, and difficult questions so that I as the grader can really gauge you as a student how much knowledge you have. If I only give you easy questions, then I won't be able to tell the difference between a C, B, and A student. I'd only be able to tell the difference between passing students and failing students. But as I have to differentiate between those who pass as A, B, and C, I need some various ranging questions. So do prepare for those. Do your best you can. Don't sell yourself short, right? You can do hard things. I know you can, so still study them. Question number 12. This is going to be a question related to the topics we saw in lecture 27 about rates of change in scientific applications. Two examples that are extremely important to know is the position motion problem that we've talked about before. So remember that if you have your position function, S of t, its derivative is going to be velocity. So velocity is the first derivative of position. You should also know that acceleration is going to be the derivative of velocity and hence is the second derivative of position. That's what this version of question 12 is going through. It's asking things about position, velocity, and acceleration. So if you have to compute the acceleration of the position function, what does that mean? What intervals is the thing moving forward? Where is it backwards? So pay attention to things like that. Is it speeding up? Is it slowing down? There's various questions in the homework that are related to that. So prepare for that accordingly. So this motion problem is a very important problem to know. Be aware of that. Some other vocabulary should also be aware of is with economics. You have things like revenue. So remember that revenue is going to equal the quantity times the price, right? Cost, profit, let's say that one, profit is equal, of course, to revenue take away cost. Those are things you should know. And then also in economic problems, if I use the word marginal, marginal revenue means the derivative of revenue. Marginal cost means the derivative of cost. Marginal profit means the derivative of profit. You get the idea. In an economic sense, marginal is a synonym for derivatives. That's something you should be expected to know. This question number eight is not going to be too difficult because I only gave it eight points for a reason. The actual calculations are not going to be too difficult. It just comes down to interpretation. Knowing that acceleration is the second derivative of position is an important interpretation you need to know as you're preparing for question 12. These were questions we saw in lecture 27. I already mentioned that one. All right. Next page. So we get question number 13. Question number 13 is going to ask you to find the equation of a tangent line. This is very similar to a question we saw on exam two. But on exam two, we weren't very good at computing derivatives yet. So the derivative is actually given to you. So the general formula that you need to know is that the formula of the tangent line is going to look like y minus f of a is equal to f prime at a times x minus a, where here a is the specific x-coordinate, f of a here is the specific y-coordinate at the point of tangency. You'll notice that for this equation right here, the point of tangency is given to you. And then the derivative gives you the slope of the tangent line. So you need to know that tangent formula. But what's different on this question compared to what we saw on the first test is in this question, you will have to compute the tangent line. Excuse me. You have to compute the derivative. You will have to compute the tangent line, but you have to compute the derivative. But the equation or the tangent line you have to find, this will not be some explicit function relationship. This will be some implicit equation. You'll notice that it doesn't look like y equals whatever. The y's and the x's are actually cohabitating inside of this equation. So you need to compute the derivative here implicitly. So implicit differentiation might be a useful tool to use on lots of questions on this test, but to make sure that you do, that I can't accurately assess your knowledge of implicit differentiation, question number 13 will require implicit differentiation. Again, you can use this on other questions on the test. Implicit differentiation might be a great tool to use, but question number 13 will require it. Implicit differentiation was a topic we talked about in lectures 24 and 25. Please consult those seconds if you need some more practice with implicit differentiation. Question number 14 is similar vein as we saw with the previous one. This question will mandate that you use logarithmic differentiation, which honestly is one of the coolest derivative rules there is, but it's not always required. And sometimes we will use a different technique. Logarithmic differentiation was covered in section 26 and you will be provided a function which requires logarithmic differentiation. This one right here, if you take the function y equals x to the x and you have to take the derivative of said thing, so y prime, many of us are very tempted to be like, oh, I can use the power rule. So it's x times x to the x minus one. Nope, that's not true. You can't use the power rule. Some of you might be like, oh, it's an exponential. So I'm gonna get x to the x and then I get the inner derivative, which is like a one or something. Nope, doesn't work either. This is not an exponential function. This is not a power function. So those rules do not apply. This function right here is some type of power exponential function. It's a hybrid between the two. It's kind of like in Starcraft 2 when you take the hybrid of the Protoss and the Zerg. It's neither of them. It's the combination of the two. So those rules do not apply. Logarithmic differentiation is the only technique that we have learned that would apply to this power exponential function. So you do need to use it and it is mandated that you use it on this one. Okay, and honestly, it's the best technique to use for this one anyways. And I would, I would recommend if appropriate use logarithmic differentiation on any of the previous questions. Use it on the multiple choice section. It's a good technique, but question number 14 will require its usage. Now we arrive to the last question on the test. If you've noticed that one topic has been missed so far, this is that topic. Question number 15 will be about related rates, which is one of the last topics we learned about in unit three. Related rates were talked about in lectures 28 and 29. In which case, you'll be given some type of story problem for which you'll be given some information about quantities. You'll be given some information about their derivatives, the rates of change, and then you'll have to then infer information about an unknown rate of change. So you have to solve for some derivative here. Now what type of story problems should you be prepared for as we're getting ready for this test? Well, look at examples similar to the lecture notes and to the homework. I'm not saying this question will be identical to those, but it'll be a similar style. A lot of these questions have a very geometric flair to it. So you look at something like this, I see this triangle here. So this makes me think of like the Pythagorean equation. That is sometimes a useful identity. I'm not saying that's necessary for this question. You might need to make an argument using similar triangles. That is a very important technique for various calculus students, us basically. You might have some other type of trigonometric ratio going on. Maybe a tangent argument, a sine argument, what have you. We probably only want to use cosine tangent if we know something about the angle or we want to know something about the angle. But the Pythagorean identity, similar triangles is useful. You should know like the area of a triangle, one-half base times height. You should, of course, you should basically know all of the basic area formulas. How do you find the area of a square or a rectangle? How do you find the area of a circle? Those are things you should know. In fact, with some of those water tank filling problems we did, that's a possibility for this test. We might have to do a little bit more exotic things. Like we had to find the area of an esosceles trapezoid. I mean, that wasn't too horrible. We basically dissect it into triangles and quadrilatals would have you. But we should be able to do some of these basic two-dimensional area problems. Also perimeter problems. The Pythagorean equation is an example of a perimeter-type problem. So can we do related rates with regard to perimeter and things like that? Those two-dimensional formulas of area and perimeter are things you should know. Can you find the circumference of a circle? You should know that one by memory. On the other hand, we did see questions in the homework and in the lectures about three-dimensional problems. Maybe the three-dimensional volume of a cone or a pyramid or a prism or maybe surface area. We could have done some things like that. I do not expect you to have memorized formulas for three-dimensional figures. So you don't need to memorize the area of like a cone. I mean, maybe a cube. A rectangular prism is pretty obvious. But I won't expect you to have memorized the volume of a pyramid or things like that. If it is involving a three-dimensional figure, I will provide to you on the test the formulas you need. And also, there might be scientific applications here for which the formula will be provided to you. You'll probably jump up and down, which is not approved during the test. So you can't do that. But you can shout inside your heart if the formula is provided to you. But more likely than not, you'll be given probably a two-dimensional problem for which you need to create the relationship, however that is. Again, it'll be very similar to the homework in the lecture. So if you've been following along with our class and doing your homework, you'll probably be well-suited for this question. It's not as difficult as it might seem. And so that's question number 15. Like I said, question number 16 is just a reminder to turn in your authorized no-card. And that is then the test. If you have any questions, please reach out to me. I'd be glad to answer your questions. Of course, use all the other appropriate study tools that exist out there. You know, your notes, the textbook, the lecture notes and lecture videos, the tutoring center, form study groups. There are lots of things you can do. And if you're not sure how to study for this test, then please reach out for me with those questions, right? This is our third exam. So we've taken other exams in this class. So we kind of know what to expect. But some of us have had the realization by now that what we have been doing for the test isn't good. I mean, if you're getting good grades on the test, then keep it up. Do what you're doing. But if you have struggled to get the grade you want on this test, it's not too late to course correct. We do have one more midterm after this and then the final exam. So there's a lot of points available still in the course. Please reach out to me. And I would be glad to help you offer some study tips and suggestions, strategies you could use to better prepare for this exam. I can only help you as much as you're willing to ask. So please ask. And I'm glad to help you. All right. So that'll then conclude our review for exam number three. Please consult some of the other study resources on Canvas, the exam syllabus, the practice test itself, the solutions to the practice test to help you with your studies. Best of luck, everyone. Bye.