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  • Math 2B. Calculus. Lecture 01.

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    Watch at: 00:00 / 00:00:20Good afternoon and welcome to the fall quarter This is Math 2B: Calculusand my name is Natalia Komarova I am a professor here at the math departmentWatch at: 00:20 / 00:40and I’ll be teaching you this subject first I want to go over the basicsof this class and tell you a little bit about how it’s organizedfirst of all, I have created a website with all the information already containedWatch at: 00:40 / 01:00on the website pages but today I will still spell everything outfor you so first of all, the textbookso this is probably the most confusing part of the whole thingso it’s called calculus: Early Transcendentals so it’s 7th editionWatch at: 01:00 / 01:20of calculus by StewartWatch at: 01:20 / 01:40so the most important part is Early Transcendentalsthere is a 7th edition that does not contain these wordslook at your book, if it doesn’t say that, that’s the wrongWatch at: 01:40 / 02:00book now there’s various shapes and forms ofthis book for instance, what I have only contains singlevariable calculus so that’s good for 2A and 2Band so that will be fine for this class there’s also a bigger bookWatch at: 02:00 / 02:20that has, you know, parts that pertain to other classeslike if you’re planning to take 2D or 2E you should get the full book. The big one.There’s also another online book, So there are many questions that people askWatch at: 02:20 / 02:40me over email Is it okay to use a second hand copy?And the answer is yes. It’s okay. If you bought your book from a friend, ifit’s used, And your electronic stuff doesn’t workIf you cannot create an account, that’s okayAll you need from this book is the chapter materialWatch at: 02:40 / 03:00And the homework problems Okay? And so I do not require you to buy anew copy, of course a secondhand copy is okayany questions about the book? Okay.So now exams. We’re going to have two midtermsWatch at: 03:00 / 03:20On October 18th, and November 8th And we are going to have a common final exam.Question? Will we need to bring our book to class?Watch at: 03:20 / 03:40No, absolutely not. So common finalwhich is held on Saturday, December 7th from 1 to 3Watch at: 03:40 / 04:00so I will tell you a little bit about the common finalbut first I know that according to the policy of the mathematics departmentyou are not allowed to use books, notes, or calculators during any tests.So it is a closed book test, no cell phones, no calculators are allowed.Watch at: 04:00 / 04:20Now what’s a common final? If you’ve taken 2A, you know what that is.It means that you come here on Saturday. And it’s a common final that’s held acrossall the sections of this class So there are unique requirements for everybodyWatch at: 04:20 / 04:40You have to produce your valid UCI ID card For the midterms and the finalYou have to make sure that you have an ID And you also have to make sure that you arerecognizable in the picture so very often students produce something thatWatch at: 04:40 / 05:00looks like this with just the circle instead of a faceso make sure that you get your ID replaced such that the picture is recognizableso another thing about common final is that if you cannot make ityou should let us know early on and you don’t let me know,Watch at: 05:00 / 05:20you let the secretaries of the math department knowyou have to fill out the form that is contained onlineyou have to follow the link on the website there is a special forma standard form that you fill out and there should be no problems there.You can arrange for a make-up But of course you have to have a valid reasonWatch at: 05:20 / 05:40Not to attend the final Questions about the final?Question? Do we need a scantron or a bluebook?No, nothing like this. I will provide a paper copy of the examAnd all you need is a pencil and an eraser Yes?When you said no calculator, are we allowed to bring a simple calculatorWatch at: 05:40 / 06:00no. no but we always make sure that you can doall the math in your head there will be nothing horriblemore questions? Will a calculator be needed for the class?No, no. You can use it when you do homework, of courseWatch at: 06:00 / 06:20But in class, no Okay. So nowsome other assignments that you have apart from the midterms and the finalwe will have homework, okay? so the way it works in this class is homeworkWatch at: 06:20 / 06:40is optional which means that it’s not gradednonetheless, this is probably the most essential part of this classbecause everything that you’re tested on is basedon this optional homework. So the list of homework problems is providedWatch at: 06:40 / 07:00in the website for the class. And it goes by section numberSo for instance section 6.1 and it gives you a list of homework problemsso you sit down and do as many as you can after we have covered the materialif you’re fine with all the homework problems for each sectionWatch at: 07:00 / 07:20you’ll get an A+ okay? If you’re fine with most of themyou’ll get an A. And so on. So this is your way to study.Do the homework problems nobody will test them directly,but we will have quizzes quizzes are held once a weekWatch at: 07:20 / 07:40at the discussion sessions on Thursdaysand the quizzes are completely based on the homework assignmentsfor the previous week so if you’ve done your homeworkWatch at: 07:40 / 08:00you will know how to do the quiz problems it’s either just a homework problem takenfrom the list or something that is very very close to itso in order to prepare for the quiz, you have to do the homework.And the quizzes are graded okay, so this is gradednow another part of assignments is webwork can you raise your hand if you know what theWatch at: 08:00 / 08:20webwork is? Do you know what it is?Oh, so I see some of you are not familiar with what it isso this is an online homework assignment so on the website for the classWatch at: 08:20 / 08:40I created a link that takes you to the webwork homepagethis page is not active as of now they will activate it in about two weekswhen the first assignment is posted so the first assignment will be posted thetenth of October okay so until then you don’t have to worryWatch at: 08:40 / 09:00about it so you go there, you log in,and you do your problems. They will be 8 webwork assignments duringthe fall quarter There will be every week, posted on Thursdayand due the next Friday The first webwork assignment is based on theWatch at: 09:00 / 09:20problems from the beginning of the class And they are cumulative such thatlater on webwork problems could test your knowledge from, you know, far long ago.So everything that you have studied up to that datecan be tested with webwork each assignment has a varying number of problemsWatch at: 09:20 / 09:40and I posted the full schedule of all the webwork assignmentsthat is already known for class so you will know the due dates and the dateswhen these things are posted you have an extension for thanksgivingWatch at: 09:40 / 10:00for thanksgiving week, you have a little bit more time to finishso there are a few things that you need to knowabout webwork so among the class filesI posted a PDF file that tells youabout webwork. So one new thing that they told us about thisWatch at: 10:00 / 10:20year is that somehow it doesn’t work very wellfrom so campus. It works very well from campusif you’re away from campus, you have to use VPNto connect, and the instructions that will help you set it upare given on the website for the class. And there are various quirks associated withWatch at: 10:20 / 10:40webwork Because sometimes you may type something inthe wrong font And it thinks that you made a mistakeall these things please read what I postedit gives you a lot of information and so if you experience technical problemswith webworks, please do not email meWatch at: 10:40 / 11:00because I will not be able to help you I can only confuse you if you ask me technicalquestions About the website setup and stuffemail them there is an email address provided on thewebsite of webwork and they will be able to resolve your technicalissues if you have a problem with your mathematicsWatch at: 11:00 / 11:20then I’m your guide. Okay? Email me or come to the office hoursand I will be able to help you with any math problemsbut not with technical problems and website problems like thatWatch at: 11:20 / 11:40questions? Okay. Gradeconsists of 40% final 20% each of each of the two midtermsWatch at: 11:40 / 12:0010% webwork And 10% quizzesSo the lowest quiz and the lowest webwork are droppedWatch at: 12:00 / 12:20So one lowest dropped And here the samebecause of this policy, there is no opportunity to make-upfor quizzes or webwork there’s one seat here if you need to sitWatch at: 12:20 / 12:40down and there is also one over herethere are also two seats over there in the middleso if you miss a quiz or if you miss a webworkdon’t worry about it, because because of thisWatch at: 12:40 / 13:00you have a chance to miss one questionso is the webwork like an online quiz, or is it like an online homework assignment?it is like an online homework assignment I think you have an infinite number of attemptsMore questions? Will we be able to use a calculator on theWatch at: 13:00 / 13:20webwork? oh, yeah. You’ll be doing it at homebut not during quizzes so quizzes are like examsmore questions? Okay. Now I think the last thing is thisWatch at: 13:20 / 13:40so Early Transcendentals so what does it mean?It means that if you took calculus 2A Prior to fall 2012Watch at: 13:40 / 14:00You will have some catching up to do So the mathematics department has changedthe syllabus We used to teach things like logarithmOr arcsin or e^(x)we used to teach that in 2B now these things are taught in 2A and youWatch at: 14:00 / 14:20are supposed to know them you are supposed to know these thingsand you are supposed to know their derivatives you should know what they look like,you should be able to plot them so how many of you have not seen those beforeokay, very good. So that’s good however if you want to refresh your memoryWatch at: 14:20 / 14:40by these functionsthere is some online video material you can watch the videosand refresh your memory about these things such that you are well prepared for what wehave here instead of these things, we will be studyingWatch at: 14:40 / 15:00sequences and series which are much more fun.I’ve taught both, this is better Questions?Okay questionare the grades going to be curved? no, so there is no curveWatch at: 15:00 / 15:20there’s going to be a standard grade so the finals are graded across the wholeyou know, all the sections then if they think thatit was too hard or too easy then they are going to uniformlyadd some points or subtract some points like multiply by a factorso everybody gets the same treatment and then after that there’s no curving.Watch at: 15:20 / 15:40Also there is no extra credit There’s nothing you can doBut do the homework Okay? So everything is very straightforwardin this class Old information is in the homeworkIf you have questions, if you don’t understand somethingCome to my office hours which I will announce next weekWatch at: 15:40 / 16:00I don’t know yet when my office hours areand that will be fine. Question? You said that if the average is low, is thefinal curved or the class is curved Like overallso if they think that the common final was too hard,they are going to kind of boost that grade just for the final though?Watch at: 16:00 / 16:19just for the final. So you don’t curve this class?No. And they are not supposed to. What’s the website?I think you can find it on EEE there’s a linkalso you can go to my homepage and there’s a link therebut it’s easier to go from the EEE just to make sure, you said the webwork isWatch at: 16:19 / 16:40due Friday? That’s the following week, correct? yes, it’s posted on a Thursdayit’s due the Friday next week so it’s the following week, yesso you have a week and one day to finish it more questions?Okay. and you can always ask me questions duringWatch at: 16:40 / 17:00the class just raise your hand, okay?so now we do some review of some essential stuff that you learned in2A if you cannot see something, please let meWatch at: 17:00 / 17:20know so I avoid some parts of the whiteboardSo today we cover section 4.9 antiderivativeWatch at: 17:20 / 17:40okay ready? So let’s suppose that the derivativeof function F is given by lower case fWatch at: 17:40 / 18:00for all values of x in an integral then F capital is called the antiderivativeof f. so for exampleWatch at: 18:00 / 18:20let’s suppose that f(x) is equal to x^(2) what’s the antiderivative of this function?So there are two opposite operations you can take the derivativeWatch at: 18:20 / 18:40and you can go back and take the anti-derivative question?So this happens to be one of the antiderivatives Of this functionIn fact, how many are there? Infinitely manyPlus C So by adding a constant CWatch at: 18:40 / 19:00We create an infinite family of functions each of which is an antiderivative of thishow can we check? So this is antiderivativeTo check, we say what is F’ (x)Watch at: 19:00 / 19:20we have to differentiate so we have (3x^(2) /3)+0 is x^(2)check because it coincides with my original functionWatch at: 19:20 / 19:40question I cannot see thatyou cannot see this? That’s very bad. Okay.so are you telling me that you can only see above this line?Watch at: 19:40 / 20:00okay. so we have a theoremif F is an antiderivative of fWatch at: 20:00 / 20:20then F+C is the most general antiderivative and here C is a constantWatch at: 20:20 / 20:40okay so let’s practice and do some examplesWatch at: 20:40 / 21:00so let’s suppose that f(x) is cosx.Find F(x) so basically we are looking for a functionwhose derivative is equal to cosine so what is F(x)?Watch at: 21:00 / 21:20very good sinx+Canother example f(x) is given by x^(n)where n is not equal to minus 1 so may I ask you please do not talk duringWatch at: 21:20 / 21:40this class if you have questions ask me, but do not talk.So we have a rule to evaluate the antiderivative of thisWatch at: 21:40 / 22:00It’s called the power rule power ruleokay so it tells us it tells us that the antiderivativeis given by this functionplus C and in fact this is the rule that we usedWatch at: 22:00 / 22:20here to calculate the derivative in our very firstexample so I have a questionwhy is it that we have to require n is not equal to -1what happens to this formula when n equals -1?Yes. You divide by zero So this formula is obviously not applicableWatch at: 22:20 / 22:40Something else goes on there so that’s my next examplea very special case where n is equal to -1Watch at: 22:40 / 23:00f is 1/x what’s the antiderivative?Natural log of x In fact, it’s like thisNatural log of the absolute value of x plus CWatch at: 23:00 / 23:20and we actually have to say that this formula holds for any interval that does not contain0 so if I give youWatch at: 23:20 / 23:40if I give you an interval from one to five firstWatch at: 23:40 / 24:00on this interval any such function with a constant Cwill serve as an antiderivative of the log if I want to write down the most general antiderivativeon the whole real line,it’s something slightly more complicated so f(x) is ln(x) +CWatch at: 24:00 / 24:20for positive values of x and this logarithm of –xoh—I’m sorry this is noted C¬1Watch at: 24:20 / 24:40+ C2 When x is negativeand the antiderivative is not defined for x=0because the original function is not defined for x=0question? can C1 and C2 be equal?they can be equal, but in general they don’t have to beWatch at: 24:40 / 25:00so this is the most general form of the antiderivative of the function 1/xso this function can have a discontinuity it goes like lnx, ln(-x),but here, when you equal x=0, you can have a different constant.Watch at: 25:00 / 25:20So it’s in general a discontinuous function. We also have something like thisFor negative values of n So let’s supposeWatch at: 25:20 / 25:40That f is x^(-4) So now we have a very similar situationSo by using the power rule My power is -4Watch at: 25:40 / 26:00So I have to So x^(-4+1)-4+1 It’s x^(-3)Watch at: 26:00 / 26:20Divided by –x So the most generalantiderivative is given by the following discontinuous functionWatch at: 26:20 / 26:40so the antiderivative again is not defined for x=0Watch at: 26:40 / 27:00because the function itself is not defined thereand the experience it’s discontinuating, as it goes to 0because we’re allowed to take different constants for negative and positive valuesof x so what we need to know for this classWatch at: 27:00 / 27:20is a list of antiderivatives it’s best to know those by heartso we have a table in the textbookthat goes like this. We have a function and we have a particular antiderivativeWatch at: 27:20 / 27:40so what are the most common functions that you should know by heart?So first I list a ruleif you have a function f, any function f multiplied by a constant CWatch at: 27:40 / 28:00so the antiderivative also gets multiplied by Cyou know that, right? And similarlywith the summation the antiderivative of a sumis the sum of two antiderivatives nowWatch at: 28:00 / 28:20particular examples of functions we’ll list some of themhere I do this for completenessWatch at: 28:20 / 28:40so I’ll continue this table here let’s do cosine xgives me sinx so this is f, this is Fso this is sinx -cosxsecant squared gives me tangentWatch at: 28:40 / 29:00secant tangent gives me secantthese two follow from the definition of the derivatives of secant and tangentwe know if the derivative of this is equal to thisWatch at: 29:00 / 29:20then the antiderivative of this is thisthis table works both ways to go from here to here you have to take aderivative to go from here to here, you have to takethe antiderivative what else?Watch at: 29:20 / 29:40here we have Watch at: 29:40 / 30:00something that is associated with arcsin and arctangentso this is the best one why is it the best one?Watch at: 30:00 / 30:20It’s equal to itself, right? it's the easiest one to remember, the exponentis equal to its own derivative also its equal to its own antiderivativequestions? So let’s practiceSo a simple common problem That we encounter isWatch at: 30:20 / 30:40find the antiderivatives of functions so find all functions gsuch that g prime isWatch at: 30:40 / 31:005+cosx + 3x^(2)So look at this function So we go to the table and of course we don’tWatch at: 31:00 / 31:20find that function in the table However, if we simplify this function we’llfind its components in the table, right?so the first thing we do here is simplify and then we’ll be able to use the tableso I’m going to say that this is 5 plus cosxand here I divide through by x, so I have 3x + xWatch at: 31:20 / 31:40-1/2 Now each of the components hereAnd we found them in the table and I’m going to use the tableso G so I’m sorry Gthe derivative of G is this so therefore I have to find the derivativeWatch at: 31:40 / 32:00the antiderivative so it’s 5x+ sinx how did I get the first term?we will learn how to integrate in this class but we have to find an antiderivativeWatch at: 32:00 / 32:20of a constant. So where can you find that?for example, here if n equals 0that’s a constant 1, right? So n equals 0 gives me x to the power of 1Watch at: 32:20 / 32:39Divided by 1, so that’s x so the antiderivative of 1 is xand here this tells me that multiplication by a constant jus carries throughso this number 5 appears in front of the antiderivativesine is the antiderivative of cosine just pulls right from the tableWatch at: 32:39 / 33:00this one is easy again, it’s a power function, soit’s 3x^(2) over 2 this one is also easybecause you use the same rule. Power rule. Now we don’t have an integer power,the power is equal to -1/2 so we have x^(1/2)divided by ½ and thenWatch at: 33:00 / 33:20don’t forget +C this usually costs one pointon any test questions?do we have to simplify the ½ to a radical or can we leave itwhen it’s easy, like this probablyI wouldn’t take a point off for this but different graders are differentWatch at: 33:20 / 33:39okay the next question is a little bit more sophisticatedwe can talk about differential equations in itself it’s a huge topicWatch at: 33:39 / 34:00and there are whole courses taught on this but I will just show you what it isso the problem is like this: find f if f’ is equal to x^6 and f(1) is equalWatch at: 34:00 / 34:20to 3 so I need to find the function f given thisinformation two pieces of informationWatch at: 34:20 / 34:40the first piece of information pertains to its derivativeand the second one tells me what the value of the function f isat one point x is equal to 1so that’s what I need to find. So from this equationI can find f By looking at the most general antiderivativeWatch at: 34:40 / 35:00So general antiderivative I take the antiderivative of x^6Which is x^7 over 7 +C So I found a whole bunch of functions fWatch at: 35:00 / 35:20they all differ by this constant and that is why I’m given this second conditionthis condition will help eliminate most of theseand 0 on the relative one so use f(1) equals 3Watch at: 35:20 / 35:40how do I use them? I plug it in. Exactly.So I go f(1) is 1^7 over 7 +C and that’s supposed to be equal to 3Watch at: 35:40 / 36:00so I can say that 1/7 +C equals 3 where C equals 3-(1/7)which is 20/7 therefore my function fWatch at: 36:00 / 36:20not the most general one, but the actual one that solves both of these,okay, that’s given by x^7 over 7plus 20/7 or ((x^7)+20)/7 so out of all of these functions, I identifiedWatch at: 36:20 / 36:40the one that satisfies not only the first equation, but the second one, tooquestions? okayWatch at: 36:40 / 37:00so now we will refresh our memory with regards to graphic antiderivativesand we will talk about the notion of velocity solet’s suppose that the function f is given graphicallyWatch at: 37:00 / 37:20something like this. One second. So it starts off here,Watch at: 37:20 / 37:40goes negative, like this,and like this okay, something like thisWatch at: 37:40 / 38:00let us sketch the graph of the antiderivative so no formula there givenand I want to draw F capital so how do I do this in principle?This is the derivative of this function Now remember, what is an antider—Watch at: 38:00 / 38:20what is a derivative? The derivative is the rate of changeIt’s the rate of change It's the rate at which the function changes.if we think of the independent variable as time, the derivative is how quickly thatfunction changes. it tells you the slope, or the rate of changeWatch at: 38:20 / 38:40and the rate of change can be interpreted as a velocityso let’s suppose that this is velocity of motionand as you can see, as time goes by, it changes sometimes it will go faster, then it willslow down, okay, at this point, the velocity is equal to zeroWatch at: 38:40 / 39:00and here it becomes negative. Which means thatwe go backward then again at this point, we turn and startgoing forward so by using this information, I am going todraw the position, given the velocity okay? I have to recreate the positiongiven the velocity I start with some arbitrary pointWatch at: 39:00 / 39:20okay, let’s suppose that we know we start at 1and now, so look the velocity here is positivewhich means that I am going forward I go forward means that my position, the coordinateof my position, increases So for a whileWatch at: 39:20 / 39:40between time equals 0 and time equals 1 I go forwardslower and slower and slower, but I move forward this is my positive direction, according toincreases at this point I stopWatch at: 39:40 / 40:00at this point my derivative is equal to zero which means that I’m going to have a maximumhere, right? and now my velocity becomes negativeat this point I start going backward and that’s exactly what I’m drawing hereI start going backward Faster and faster and fasterWatch at: 40:00 / 40:20at point 2, my velocity is the fastest negativeand then it becomes slower and slower and slower.so at point 2 I get something like this and I stop at point 3Watch at: 40:20 / 40:40because my velocity again is 0 after 3, I continue to go forwardso my coordinate increases and then the velocity decreasesso somehow I level off I start going slower and slower and slowerAnd eventually almost stop but I don’t quite stopWatch at: 40:40 / 41:00Questions? so you should be able to take the graph ofa function and draw its antiderivativebut I want you to think about velocity I want you to think if this is positive,This increases If this is negative,this decreases if this is 0, it means I experience eitherWatch at: 41:00 / 41:20a maximum or a minimum I don’t change at that pointQuestions? very good so now in the last problemI think it’s the hardest of all we will talk not only about velocity but alsoWatch at: 41:20 / 41:40acceleration because they’re both connected to derivativesand antiderivatives so the problem is like thissuppose that the acceleration of a particleis given by this function so here is my vocabularyWatch at: 41:40 / 42:00a is acceleration v is velocityand s is position these are the common notationsWatch at: 42:00 / 42:20and you know that the velocity is the derivative of theposition and the acceleration is the derivative ofthe velocity do you know this?Okay. So what is given is the acceleration and also some information about the positionWatch at: 42:20 / 42:40at the beginning, the position is 2 and the velocity at the beginning is -1find the position as a function of time given this informationWatch at: 42:40 / 43:00I start by saying that acceleration is a derivative of the velocityso v’ okayis t+1 if I know the derivative of the velocityI can find the velocity by taking the antiderivativeWatch at: 43:00 / 43:20so v(t) is found by calculating the antiderivative of this functionwhich is really easy t^2+t +C so given the accelerationWatch at: 43:20 / 43:40so let me just I don’t want to misstepso this is a a is the same as v’and it’s given by t+1 so if I know v’ I know vand it’s given by this unfortunately I have this unknown constantWatch at: 43:40 / 44:00here but I can fix that by saying thatoh, look v(0) is equal to -1that tells me find the appropriate C so we use v(0)is -1 so what is v(0)?it is 0/2 + 0 + C and it’s supposed to be equal to -1Watch at: 44:00 / 44:20therefore C is -1 therefore v is (t^2)/2+ t -1 so I completed the first stepWatch at: 44:20 / 44:40using the information about the acceleration I found the velocity.and I also used this condition about the initial velocitythat helped me identify this particular constant, Cokay, so that’s step number one step number 2, I know the velocity nowWatch at: 44:40 / 45:00but I need to know the position so step number 2I go here The velocity is the derivative of the positionSo v is the same as s’ and that’s given by this formula that IWatch at: 45:00 / 45:20just derived t^2 over 2+ t - 1 From this, I can find the positionby taking the antiderivativeWatch at: 45:20 / 45:40so if I know the derivative of s, I can find the antiderivative Again I have an unknown constant called AWatch at: 45:40 / 46:00So what’s this constant? That’s the last stepI’m going to use this piece of information, the initial positionS(0) is given by 0/6 + 0/2 – 0 + A is supposed to be equal to 2Watch at: 46:00 / 46:20Which means that A equals 2 So I can write down the answers(t) is given by t^3 over 6 plus t^2 over 2minus t plus 2 so this is the formula that defines the positionWatch at: 46:20 / 46:40of that particle as a function of time questions?Okay, thank you very much