 In this video, we're going to talk about exam one for math 1210 calculus one at Southern Utah University. So what you see in front of you right now is a copy of the practice exam for this course. Now, this practice exam, as the name suggests, is just a practice. This is not a graded or collected assignment. This resource is available to you to help you know what to expect as you prepare for the first exam in this course. The solutions are contained at the end of this document. You won't see them in this video, but there are other videos. If you want to see the detailed solutions of these, but also there's just the simple answers in the back of this PDF document. The first half of the document is going to look like a blank test. That is the questions are there, but none of the answers are there. That way you can look at the questions. You can even take this as a practice exam as it's suggested, right? This is a practice exam. You could take the exam, test yourself and then can check your answers later on. What I want to do in this video to help you prepare for the exam is one, talk about the structure, organization and policies with regard to the test. What are you allowed to do? What are you not allowed to do that type of stuff? And then also I want to talk about the specific contents of this exam. So some of the first things to know, there's this instructions here. These instructions you saw in the practice exam will appear exactly on the actual exam itself. So this is a write on test. That is, you're supposed to write on a piece of paper with a pen or pencil the answers to your questions, right? So there will be many questions which you are required to show your work. There will be some multiple choice questions. You do not need a scantron. You'll circle the answer A, B or C, whatever you think it is on the page. Our exams will take place in the testing center at SUU. There will be no time limit, except for those imposed by the testing center's operating hours. You have to finish the test when it closes. And you should check of course with syllabus for this course because it could be that there are some semester changes. So at the time of this video, exams are taking place in the testing center. All your answers should be simplified and you should have exact answers. That is, if the answer turns out to be the square root of two, write the square root of two. Don't give me the approximation 1.414, what have you. Just leave the exact answers exact. Some of the stuff you'll be most interested in, what study materials are you allowed for the exam. The short answer is basically nothing. No textbooks, no notes, no other study materials. You are actually allowed a single three by five no card. So what's the deal with this no card here? So with the no card, what you're going to do is you're going to write on this no card whatever notes, examples, definitions, formulas that you think would be appropriate for you to have during the test. You can put whatever you want. And by three by five, I mean this is a standard office no card. It'll be three inches by five inches rectangular. Your no card cannot be larger than that because that's not authorized. You are allowed to write on the front and back of your no card, but I do require it be handwritten. You should also put your name on it. This resource you are allowed to have. In fact, the no card is a graded portion of the exam. You'll turn it in when you turn in this exam. It's actually worth two points out of 100 pass fail right there. And again, put whatever materials you want in there. I've seen people draw funny little pictures of like Harry the platypus. I guess that makes them feel good. They put in motivational statements like you can do this. Whatever, you know, no judgment there put whatever you need to help you take the test, although it needs to have something substantive. That's for sure. A blank no card would not count. In terms of calculators, you are allowed a scientific calculator. That is to say you cannot use a graphing calculator. Graphing calculators change the nature of the questions. And so actually to allow you to have an easier exam, we're not going to allow graphing calculators. A scientific calculator, one that has the four arithmetic operations, addition, subtraction, multiplication, division, that would be useful. But also like radicals, exponents, logarithm, sine, cosine, tangent, they're inverses. You want a calculator that has those functions to help you out here. There really are no questions on the tests that require the use of a calculator, at least not that I can think of for this test. But if it would help you speed up some of your calculations, by all means you are allowed to use it, a scientific calculator. The test and center does have some that you can borrow, I believe, when you take the test, just requested when you start the test. Scratch paper will be available upon request when you get to the test and center to ask for some scratch paper. Be aware that at the end of the test, they will discard your scratch paper, they'll throw it away. Unless you specifically ask them to staple it to your test, which if it's just scratch work, don't submit that, it's not necessary. It would only be that if like, you know, you're that person who takes your math test and pen, and then after working on a long problem, you're like, dang it, I did everything wrong. Scratch, scratch, scratch, let's start over. You know, you should you should put your answers and the appropriate work that needs to be graded inside the testing packet. But in case of emergency, you can attach, you can request a scratch paper attached. But that should be that should honestly be a rarity. Normally, the scratch paper will be discarded here. And then the other thing I do want to mention about this test is that as not everyone is necessarily taking the test all at the same time. Some might be taken it earlier than others, it's a multiple date exam. It's extremely important that students don't talk about the contents of the exam during the exam window. By all means, students are welcome to study together. You know, it's like, hey, I have questions about the logarithms, let's study with a friend or a classmate or whatever. Yeah, that study groups. That's, that's perfectly fine. What what I'm saying is discourages would be like, hey, Jimbo, did you see question number one about trigometric inverse? I had a compute sign inverse of, you know, that that type of stuff like talking about specific things on the exam that is cheating and will not be tolerated. So I would encourage everyone to be above the bar there and not talk about the contents of exam until after the last day of the exam. This exam will contain two types of questions coming in two sections. The first sections will be called the multiple choice section. As the name suggests, these will be questions which are given the answer and some possible distractors. You then have to identify which of the four, five, six options, however many there are, you have to then select which single answer is correct amongst the false options. And so you'll just clearly indicate this on the page. The simplest thing I would say is just to circle your answer. I've seen other things like a checkmark or a box. That's fine. There needs to be a clear indicator. If your answer that you've indicated is all ambiguous. That is, I can't tell what, what, which, which single answer you've indicated, maybe because you've indicated nothing or you've indicated multiple answers. In that case, you wouldn't get no credit for the multiple choice section. They're typically, they'll typically be five points each. And those points will be solely upon selecting the correct answer. If you select the correct answer, then you would get the five points. If you don't select the correct answer because you selected something else or you left a blank, you wouldn't get any of the credit on a multiple choice question. No partial credit will be given to a multiple choice question, even if work is provided to it. Be aware that the multiple choice questions are going to be all or nothing. Now, for the most part, I would say that the multiple choice questions are on the easier side when it comes to questions. Usually one, two step type questions, not really combining lots of multiple topics at once. Again, fairly routine calculations. For the most part, if you studied well for the test, we should be able to, you know, work, get through the multiple choice without too much difficulty. The second section of the test is what we'll call the free response section. The free response section. This is a section for which in addition to your final answer, you do need to show all of the work necessary to get to your final answer. Show all the supporting work. Full credit for these problems will only be given if you show all of your work because the work itself will be graded. If you just wrote down the final answer, even if it's correct, you would not get full credit and it's actually possible that you might get no credit for that question. So do make sure you show all the appropriate, in the space provided, show your appropriate work. Then at the bottom of the test, you're going to see a little box right here. This is for me when it comes to grading. I'll fill out this thing. So when I hand it back to you, you'll get some good information immediately about how you scored on this test. All right, so that's what this front page is going to look like. Do write your name at the top of the test and then we should be good to go. So this is going to be the structure of how all of our midterm exams are going to work in this class and also for the final for the most part. Again, there might be some important differences between tests, which we'll talk about. But in terms of policy-wise, the organization is going to be like this. For this exam, there are 15 questions, 10 in the multiple choice section, 5 in the free response. There technically is a question 16, but that's just telling you to turn in your note card, which I did say is worth two points out of 100 there. So for this exam, I want to get into the specific contents of the exam and so you know what to expect. In terms of our lecture series, this course, or I should say this exam covers lectures one through seven. So this is our review of pre-calculus materials. So review of functions, polynomial functions, rational functions, radical functions, exponential functions, logarithmic functions. Also review of trigonometry. So include sine, cosine tangent, ck and cosecant, cotangent, they're inverses. And there are various questions we've asked about these things. How do you graph this function? What's the domain of this function? Solve this equation involving these functions. Simplify these algebraic expressions. Those are the type of things you're going to see on this exam. Now, that is a large subject matter, I admit. I mean, essentially in two weeks, we had to review the contents of two courses, which is an impossible task. We certainly were not able to cover everything. And so I tried in the homework to focus on those topics that were most relevant to calculus problems we would see this semester. And this exam is going to follow in that vein. So question number one on the exam. In this video, I'm not going to provide any of the solutions to the exam. Like I said, you should take a look at those in a separate resource on Canvas. I want to actually just expose you on this practice exam. What are the types of questions you're going to see so you know what to study? So question number one will be a question about logarithms. You'll be asked to expand or collapse the logarithmic expression. This version of the question asks you to expand it. So you have a single logarithmic expression. You want to expand it into some type of chain of logarithmic expressions. So this will involve using the three laws of logarithms. So for example, the first law tells you that when you take the log of a of a times B, if you have a log of a product, this becomes the log of a plus the log of B. So the log of a product becomes a sum of logs. The second law is similar to it. If you take the log of a quotient, whoops, then this becomes the difference of logs. And then the third law, this would tell you that if you take the log of a times A to say the Nth power, oh, that looks like an M now. Let's go with it. This becomes M times log of A. So you have your three laws of logarithms. So using those laws of logarithms, you're going to expand this logarithmic expression as much as possible. Or you could do the inverse problem where you're asked to condense them together. This question will ask you to use these three laws of logarithms. These three laws would be great things to put, I would say, on a no card. And so the laws of logarithm, logarithms we talked about in lectures seven and six. I believe lecture sevens when we talked about the laws of logarithm. So that's a good place to go if you need to study a little bit more. You also notice that this exam is formatted that there will be some space on occasion between multiple choice questions. That's a courtesy offer to you so that if you do need to do some small calculations, there's space on the paper where you can do that in close proximity to the question. But remember, for multiple choice questions, any work you show will not be great in just the final answer. So if you think the answer is A, circle something like A. All right, question number two. This will be a trigonometric question for which you will be given a right triangle diagram similar to this. So this is like a three, four, five triangle. And then given this triangle diagram, you'll be asked to compute one of the six trigonometric ratios. And this question asks you to do cotangent of theta. You could be asked to do sine cosine tangent. Like if you're doing tangent of theta, I'd be like, OK, tangent is going to be opposite over adjacent. And so you take that ratio and you get three fourths. You'll notice there's six options. These are the six possible ratios you can get there. So using knowing you're so Catoa, right? So Catoa, so Catoa, so Catoa. If you know your right triangle trigonometry, you can answer question number two. Question number three, you'll be given the graph of a very basic function and asked to recognize it. Remember, you won't have a graphing calculator. So these are things you're going to want to memorize. So some functions families that you're going to want to be familiar with. You're going to want to be familiar with the monomial functions. So things like y equals x to the n, we're in a some non-negative positive number. Like for example, you have x to the zero of the constant function. You have x to the first, the identity function, x squared, x cubed. These are examples of monomial functions. Do you know what the graph of x to the fourth, x to the fifth, f to the sixth looks like. There is a general pattern for which, in fact, parity comes into big play there. Even ones look one way, odd ones look another one. It's just the bigger the power gets slightly distorted. So you should know functions like that. You should know, of course, know the absolute value function. It is a piecewise function, but it's a very famous piecewise function. You should know it. You should know the reciprocal functions. That is functions of the form one over x to the n. One of them is showcased here, but you should also know things like one over x squared, one over x cubed, one over x to the fourth. Like I said, much like the monomials, there is a general pattern here. The evens look very different than the odds. And then they get distorted based upon how big the power is. You should be familiar with those. You should be familiar with the radical functions. So I mean things like y equals, you know, the nth root of x. So like the square root, the cube root, the fourth root, the fifth root. How do they look? Again, these come into, there's a parity here. Even ones look one way. The fourth root, the fourth root, the sixth root. The cube roots, the fifth roots, and the other odd roots look another way. Putting these function families together, we get the so-called power functions. So as you are preparing for question number three, you should know the basic graph of a power function. Things like this right here. All of the options below, with the exception of the absolute value, are power functions right here. Some other functions you should be familiar with. You should be able to graph a standard exponential function and their inverses. That is, you should be able to graph logarithms. Again, this is just the standard graph. Can you graph y equals a to the x? How does a affect the shape of the graph? Can you graph y equals log base a of x? I'm not talking about any transformations right here. It was just the fundamental graph right there. You should also know the fundamental graphs of the trig functions. So we're talking about sine, cosine, tangent, secant. You name it, it's there, right? Can you recognize the graphs of the six trigometric functions? I'm not going to require you be able to recognize by recall the graphs of the inverse trigometric functions. Inverse trigometric functions are important, but off the top of your head, I don't care if you can graph those ones. But you should be able to graph the power functions, exponentials and logs and the six trigometric functions. Can you recognize them by memory if there's no transformations there whatsoever? The last question on the first page, number four, will be about function composition. So can you compute correctly the composition of two functions? These two functions are given here f and g. So what is g of f? Be aware that the order of operations matters. Putting on your socks then your shoes is a different process than putting your shoes on then your socks. The order of operations matters when it comes to function composition. So can you compute the composite of these two functions? That's something that you'll be expected to do for this exam. So some other things to mention. I forgot to mention, if you need some more practice on trigonometry, we talked about that in lecture four, so reference there. The power functions, we reviewed those very briefly of course, but we reviewed those in lecture two. The trigonometric functions were reviewed in, of course, lecture four. I mentioned that a moment ago. Exponentials were given in lecture five and then logs were given in lecture six and seven. And then number four about composition, that was the main topic of lecture three. So reference those sections if you need some more practice on them. Moving to the next page, which of the following functions are odd? So we're talking about function symmetry right here. Remember, there's two types of symmetries we care about. There's odd functions and there's even functions. Odd functions are those which are symmetric with respect to the origin. So you can graph this if you want to. Or more importantly, you probably are going to use this algebraic test. Odd functions are those functions f of negative x that's equal to negative f of x. Even functions are those functions for which f of negative x is equal to just f of x. The negative sign disappears. So if you're testing for symmetry, whether it's odd symmetry or even symmetry, what I want you to compute is look at f of negative x. Do that for each of these functions. One, one, one, one, right? So really this question is three for the price of one, right? Or one for the price of three. However, which direction that goes, I don't know. So you have to do three tests for symmetry. And so it could be that like maybe f is symmetric and g is odd, right? So you choose something like answer D. It could be that you think all of them are symmetric, option G. It could that you think none of them are symmetric. That's the possibility as well. So do double check because although this one is odd, it could switch to even. You will be asked to test for symmetry here. We briefly reviewed this in lecture one in our course. Our question number six will be another question about function composition. But this one actually goes the other way around. This time we're going to do function decomposition. That is you'll be given the composite function. You'll be given one of the composition factors. And then you'll be asked to identify who is the missing things, right? So which function G can be plugged into little f of X to produce capital F of X. Who is that function? We talked about these type of problems likewise in lecture three. All right. And question number seven. This is a question about piecewise functions. So you'll be given the graph of a piecewise function. And then you'll be asked to identify which formula gives you this piecewise function. Let me show you all the formulas right here. And so you can treat this as a process of elimination. So you'd be like, oh, it can't be A, it can't be C, it can't be E, right? Maybe think the answer is D or something like that. Be warned though that there is an option of none of the above. I generally don't like to use none of the above questions, but it is available on this question right here. So the correct format might not be included. So do read through it cautiously. Piecewise functions, let's see, we probably introduced those. Oh boy, I don't remember. It was somewhere in lectures one, two or three where piecewise functions were introduced. Remember, these are these Frankenstein functions where you take pieces of different functions and glue them together based upon domains, different domains and different formulas like so. So number seven will ask you to recognize a piecewise function. Question number eight, it's going to be a question about inverse trigonometry, which we talked about this at the end of lecture seven, for which you'll be given the composition of a trig function with an inverse trig function. So the outside function could be like sine, cosine, tangent, secant. The inside function could be sine inverse, cosine inverse, secant inverse, what have you. And you'll be asked, can you rewrite this without the trigonometry? Can you write it purely as an algebraic function? And so I would be my recommendation to use a triangle diagram where theta equals such and such. And theta is going to be your inverse trig function right there. So that would be my recommendation. You can see some examples like this, like I said, again, at the end of lecture seven. Question number nine, can we recognize when a function is one to one? Can we recognize when a graph is a function? To be a function, your graph must pass the vertical line test. To be a one to one function, it needs to pass both the vertical line test and the horizontal line test, right? One to one is just an adjective. To be a whole one to one function needs to first be a function. So it passes vertical line test, but then it also has to pass the horizontal line test, right? That's an important property to check. It's very simple property to check, but like a previous one, there's like three parts here. So you have to be like, okay, this one's one to one, this one's one to one, but this one's not one to one. I'm not saying that's the answer, but you would then choose an answer based upon that. If you think one and two were both one to one functions, then you would choose option D. So do check for those. It's a fairly simple test, so you have to be able to do it correctly a couple of times. The vertical line test was reviewed in lecture one. The one, the horizontal line test was introduced in lecture six right before we start talking about inverse functions. And then number 10 is the last of the multiple choice section. In this question, you'll be given the graph of a function. It could be something funky like this. This is sometimes a piecewise function. You can get lots of different things here. On this graph, you'll be asked to identify either, you know, what intervals is the function increasing, whereas on what intervals is the function decreasing, on what intervals is the function constant, on what intervals is the function concave up, on what intervals is the concave down, on what intervals is the graph straight. Those are the type of questions you'd be asked. You'd be asked about the concavity of the graph or about the monotonicity of the graph. A topic we have reviewed in lecture one of this course. Again, this will be from a graphical point of view. And when I say intervals here, we're going to be describing the x-coordinate. So if you're like, hmm, it's increasing from negative six to negative four, just a negative two, just so it's clear that it's not the case. But if that's the case, you'd be like, oh, it's from negative six is what I said, negative six to negative two. Oh, then it's increasing right here. That's also not true. It would be from three to six, something like that. That's how you'd be reporting your answer. These are going to be x-coordinates, not the y-coordinates. How are the x-coordinates changing? I should say on what x-coordinates is the function increasing or decreasing or whatever the question asks. And so that's going to be the multiple choice section of this test. Moving to the free response section, let me zoom out a little bit. The first question, number 11, is going to be about graph transformations. You'll be given a basic graph like y equals the absolute value of x, y equals the x squared, y equals two to the x, y equals the natural log of x. You'll be given a basic question, excuse me, a basic graph, graph that you should know the picture of. Remember question number three, where you're supposed to recognize a function by its graph. Question 11 will also require that knowledge set, that there's some basic graphs that you need to know. And then using that basic graph, you're going to be asked to graph a different function like this one, f of x equals negative two times the absolute value of x minus two plus three. So you see the absolute value of x inside of that. There's all this other stuff here too, like there's this plus three, this minus two, this minus two. What are those numbers do to the graph? Does it do some type of reflection across the x-axis or the y-axis? Does it do some type of vertical stretch or compression or a horizontal stretch or compression? Is there a vertical shift? Is there a horizontal shift? Those are the type of things you have to indicate. So one thing you need to do is you need to list which indicate which transformations were applied to this function. So over here, you'd write things like, okay, there is a reflection, a reflection across, you know, the y-axis. If that's what you think occurs, I'm not saying that's actually what happens. There's a shift, say, down by two, you know, you would say, you would say what the transformations are. List those to the side. Then you draw your graph, you're like, oh, here's my graph, something like this. This is just the standard absolute value. The instructors also say indicate three points on it. So this will compensate in case your drawing isn't the best like minus. It's right here. It's like, okay, I'll have zero, zero. That was a point. Here's a point two, two. And then here's another point, say negative two, negative two. Now, just so we're clear, excuse me, negative two positive two. This is not the correct graph of this function, right here. I just graphed the standard absolute value function, but indicate three points on the graphs to help clarify those points are necessary for full credit. Question number 12, you'll be given a function f of x equals, in this case, negative two x cubed over x cubed minus one. You'll be asked to compute the inverse function of that, of that function, right, which inverse functions only exist for one to one functions. You may assume the functions one to one. This will be the type of thing where you're giving your function y equals negative two x cubed over x cubed minus one. You're going to switch the roles of x and y, x becomes y and y becomes x. So you get something like this. And then you have to proceed to solve, solve for y. That's what we need to do. And at the end, then you're going to tell me f inverse equals whatever turned out to be. So we did calculations like this in lecture number six about inverse functions. And then the graph transformations we did lectures three about building new functions from old functions. And question number 13. This will be a question about solving equations involving logarithms or exponentials. So use the laws of logarithms or the laws of exponentials to simplify the questions and simplify the equations just to say to solve them. So if I was looking at a question like this, I might try to condense the logarithmic expression switch maybe to like exponential form and then solve it from there. That's what those are some things I would do. I also want to be cautious because this function does involve logarithms. There are some restrictions to their domains right if I got an answer that was like negative two. Negative two is not inside the domain of this function nor this function. If it doesn't work in the original equation, it means it doesn't work. So you'll want to make sure you check your answer at the end of the problem there. Again, logarithms we talked, we talked about those in lecture six and lecture seven, exponentials were talked about in lecture four. Question number 14. This one you'll be asked to compute the domain of a function. And basically there's three problems you want to look out for. So whenever you take the square root of a negative number, that gives you a non real number. So the domain convention tells us we will accept only those real numbers in the domain that produce a well-defined real number. So the square root of negative one is a number. It's an imaginary number which belongs to the complex number family, but it's not a real number. So for calculus one, we're not going to allow for imaginary numbers. They are appropriate in some context, but calculus one, we're going to forbid them. So we'd have to check what makes any square roots become negative, right? We have to look out for division by zero. That's a concern, right? If you divide by zero, that does not give you, that doesn't give you a real number. They give you like a vertical asymptote. And then we should also be looking out for the log of zero or the log of a negative number. Those are also going to be undefined. The log of zero gives you a vertical asymptote on its graph. And then the log of negatives are also imaginary just like the square root of negative one is an imaginary number. So those are things you should look out for. There are issues of course with like tangent as well. Like tangent of x does have some places where it's undefined. But really since tangent is just sine over cosine, the issue with tangent is going to be what makes the denominator go to zero. So that's something that's already mentioned here. So be able to compute the domain of a function right in an interval notation as well. And then the last question, number 15, this will be about a difference quotient. Difference quotients were reviewed in lecture number one as we introduce functions. So you'll be given a function like x cubed right here. You need to compute or you need to plug in f of x plus h and f of x to track them factor out an h from the top and then divide that h on the bottom. So what I can tell you is the following. If you do the following, you will get no credit for this problem. If you do something like this, okay, f of x plus h is equal to f of x plus h. Just so we're clear, this is not true, right that you need to plug in instead of x and x plus h. But if you said h to f of x, that's not going to work. So f of x plus h minus f of x over h, this equals f of x plus h minus f of x. So this, just so we're clear, this right here is wrong. This is not true. I'm telling you what students often do, but it's wrong because then once you have that assumption like, oh, the f of x is cancel, you end up with h over h with them becomes one. If that's what you then put, you will get a big fat zero on this one because that is not correct. f of x plus h, you need to substitute x plus h in for x and then go from there. The algebra is going to be much more complicated than that. Take a look at the solution to this question for an example of such. And then question 16 is, again, just a place hold to remind you to turn in your no card for those two points out of 100 right there. And so that gets us to the end of this exam. Clearly, there's going to be some variability on the test specific functions and formulas and numbers will vary from question to questions. But the basic archetypes that you see in this video will be true for the actual tests you take. For example, question 15 will ask you to evaluate and simplify a difference quotient. The variability here comes down to the function and play number question 14 will ask you to compute the domain of a function. The variability is what will be that function. This function involves square roots and denominators which affect the domain. You might have one which has a square root of a logarithm or something like that. That might affect the things as well. So prepare these type of examples as you're getting ready for the test. And of course, if you do have any questions about the test whatsoever, feel free to reach out to me. Office hours or email are great opportunities to ask questions. I'd be glad to help you in whatever capacity I can. The tutoring center is also a great resource for study groups with your colleagues either just ask people in class. You can post something in the discussion board saying you're trying to make a study group and then go from there. Let me know if you have any questions and best of luck as you study for this test. Bye everyone.