Stony Brook MAT 125 Spring 2015
Lecture 01: Course info and precalculus review
January 28, 2015

Start   This is MAT125 thats me professor Kahn note the spelling khan is not me kahn
0:30are usually from afghanistan pakistan or mongolia try and spell it correctly, maaybe thats why my emails are not reaching you if you sent it to the other professors khan they would enjoy your emails but i wont get them my office is in s 240c thats the basement of the math towere
1:00next to the math learning center i have office hours before this class on mondays and wednesdays something like 3-4 and i also have it on wednesday mornings and by appointment whoch means you send me an email and i say meet me in my office or i send you an email that says meet me in my office youre in trouble you dont want that
1:30i am almost never here on tuesdays and thursdays because tuesday and thursday im in nyc that way, right the big city here we go.. were gonna do stuff up on blackboard in a few minutes this is the room, im gonna be in here monday and wednesdays for the duration of the semester we are video taping this class just in case if you guys were wondering and eventually we will set it up for you guys to watch the videos and maybe there will be stuff on video that you are suppose to watch
2:02i dont know thats why its called experienments well find out, stony brook wants to get more involved with online teaching for soem crazy reason those videos are useful i cant imagine why a lot of you already took 123
2:30some of you did not take 123 with me, nd that im sorry for you so in theory i know what you know coming into this class today were not gonna do anything about calculus normally on the first day i dont do anything besides hi for ten minutes and then you leave but yea that was monday and monday youll have that 3 a day hope you all enjoy your day, hope monday there will be another storm, they already are talking about sunday
3:08this is calculus a lot of you already did calculus a lot of you took this in high school and showed up to the plaement test an didnt understand why you were suppose to take the placement test seriously how many have already taken calculus already alrught well some of you might find this as a review and some might say they dont need to come to this
3:36thats part why we have it on video most people who had me both know the most import thing to do is come to my class because in general theres nothing on test and quiz that i wont cover in class so if youre not here thats not my problem in general i try and make sure you see everything you need to see i am however not in charge of this course
4:00the person in charge of course the coordinator is professor scott sutherland he teaches from tuesday-thursday 4-5;20 and scott and I are organizing eveythingr were doing it as a team but hes in charge hes also the associate chairmen so he over rules me
4:31if you said well professor kahn said hell say thats to bad im in charge and hes not he writes the exams he picks the homework problems hes great, were trying to work this out as a team thing sometimes youll email me and i cant do that and ill say email him also we have webassign
5:00which ill do in a couple minutes, ill work through with everybody i know how much you love webassign so much fun im not in charge none the less we pick this so you guys have to work with this semester, webassign is a specific grade so make sure you get the problems done you should get good grades on the webassign because after all you have a week or two at a time to work on the problems have someone help you if you want you can learn from webassign
5:32we have exams, the exams are the first midterm is febuary 26 at 8.5 pm notice i did not say am
6:02pm thats at night 8.5 at night i dont know where it will be yet maybe some place beautiful like javists or frey thats exam number 1 febuary 26 make sure you know that dont email me on febuary 27th and say oh i didnt realize there was a midterm yesterday clearly i havent been coming to class okay we will post announcements about all this stuff in advance dont worry second midterm is on march 31st
6:33thats about a month later with a week of spring break in the middle im not in charge of that that means the first midterm is gonna be in a month were not gonna get that much material in you in a month, and the second midterm we can only get 3 weeks of material in a that means theres more on the final were not excited about that schedule either you should be prepared now final exam is cumalative and a big chuck of the exam will be what we learn after exam 2
7:01alright sorry about that folks but i got to keep pointing out im not in charge you have paper homework also and i think its one or two questions that should all be posted on blackboard so theres something you should know you guys,any of you new to stony brook this semester
7:31okay so we have something called blackboard and sign in with all of that we have announcements and the syllabus up there and all that fun stuff if you go into courses all of MAT125 it is one giant blackboard
8:05if you click where it says theses are not the droids you are looking for and you click over here, thats the course discribtion and by the way notice the final the final is at 8am not pm yes im not in charge of that either so that will be all the good stuff so make sure you all go there
8:35this is the book now we, professor sutherland and i had a nice chat about the book, this is about 250 bucks to buy this book with the webassign pack and all that thats a lot of money ont buy the book you should buy a used copy rent a copy, this can be used it doesnt have to be this version it could be a slightly earlier version 10 bucks oonline go buy yourself a used copy, you have to buy a webassign
9:01you can buy webassign through webassign you just click on them, it will say hey youre not suppose to use this without paying for it you give them you credit card, anyone do it yet? its less then 50 bucks 47? something like that okay 47 bucks, like i said you can get used copies of this book all over the place i wouldnt know but apparently there are places where you can download them that of course would be illegal
9:36the older additions the problem numbers may not match up so you should make sure that that you have the correct problem, i would general post them on blackboard once i know what the problems are so for example\ i know we have a webassign up there already
10:03remember homework number 1 is not from the book i went and did that very quickly so lets go back so this is our course page see where it says schedules and homework assignments do that click thing and then take the homework and the webassign so the webassign is due and the paper homework, the paper homework will be the third week
10:31your tas are in charege of grading so again i cant do much about that you should know who your tas are these are your various tas you should know now when all your stuff is one thing i strongly suggest you do
11:00know when your recitation is because when we give you an exam you write your recitation number on it and you write the wrong recitation number on it its kind of hard to figure out where your exam goes sometimes they get misplaced, theres son many peope of course last semester it took fie weeks to find your final for 5 weeks they had an incomplete course thats because the put the wrong tas name on it its sitting in some oter tas pile and that ta went on a five week vacation just like you went on a five week vaction
11:31so if you want us to loose your final the best thing to do is put the wrong recitaion on it dont write down i dont know some guy at 10 oclock in the morning were not gonna be able to find your test its not that hard to find out who your recitation person is doesnt me you have to go well we suggest you go to recitation thats where we go over things like homework practoce problems etc i will have a review session before each exam, dont know where it will be, good chance it will be in this room
12:03but i dont know ill find out any questions so fa back to webassign so if you have not done webassign before the important thing to do, do not use safari all you guys that use apple products they email me and say i cant get webassign to work often its because you use safari so dont, use chrome
12:32firefox, theres something called optimum very popular maybe not so much america, use those they do much better safari doesnt work, also you might have to do something complete 3rd party cookies you may have a cookie thats getting in the way webassigns then you go to blackboard
13:01go to our course go to where it says tools access webassign now i of course cant access webassign you would get a slightly different page at this point, amoung other things find your name or put in your code but the at that point, dont pay attention to this stuff
13:33thats webassign, from there you have all the problems for webassign so you go to course page tools access webassign dont use safari make sure you pay, you can use webassign for free for about two weeks and then they are gonna ask you for money and that point youll probably have to pay any questions so far
14:11what else so what is calculus thats the other thing what is calulus and why do you care?
ahh we got a couple people who care by the way any of you plan to be math majors no physic majors?
14:31applied math majors?
1, engineers?
generally this is not the right sequence for you but you can take it this is the 3 semester calc course if you want to do those majors you want to do this in two semesters feel free to stay its always fun to have you, we look forward to your visits but if youre serious about math physics and engineering you may want to do
15:01mat131 before applied mat161 you get through calc in 2 semesters instead of 3 and its a little harder that may want you to stay, you may not use calculators on the exam for this course i am sorry thats not my call, whos not in charge im not in charge, just like you cant use calculators in 123 you cant use them in 125 i dont know why, you can use them on your homework in fact some of the webassigns you will need it
15:35the good thing about not having a calculator is you wont need to find the sin of 70 degrees is were only gonna ask you about sin cosin and tangent of basic angles bad news of course youre gonna have to memorize again remember the last time you memorized it you forgot the time before that you memorized it and forgot
16:00thats not memorizing if you forget it you havent actually memorized it like at that point when i forget my name and to put pants on so far so good you want to try and remmeber these things so what is calculus calculus is all about figuring out how things change over time so if you leanr up until now what you have done in math
16:30a line and a line is a slope go over a bit and up a but go over a bit and go up a but when you have a curve the slope is always changing calculus you learn how that slope changes you also from calculus you learn how to add things up but i can already add things this now you learn how to add with more complex stuff many of you have taken physics remember how you learned that middle g is 9.8 gravity is 9.8 sstuff like that
17:00not actually 9,8 9.8 at sea level if youre way up high say your say your 30 thousand feet up then its not 9.8 anymore it changes on the way down see you use calculus to find out how gravity is effecting something all the way down which of course is harder see thats the main thing to use calculus ofr is to how curves are changing and how things that are being totaled up believe it or not they are reflections of the same thign
17:31you have gone yet, yea awesome why do we care about something like that? say you have a curve that looks like that, you dont have to write this down you have some curve, right here you can say the curve is flat the chnage before hand is going up its sloping its getting less steep and less steep imagine its a misle or baseball whatever football you catch it, right there
18:01you are at the top of the arc, so the exact same spot its not moving in that direction does go up and down and in calculus you can find exactly where that spot is so you can find a maximum or a minimum something thats going down just by using calculus thats a very useful rule, suppose thats the problem curve for your company for those of you who are business majors this is the place where you want to stop making what you are making because after that you have less profiy
18:31this is your maximum profit thats a very nice thing to figure out you use calculus for that in fact if you take business calc or economics you learn about marginal change thats where the margial change equals 0 your best time calculus is looking how something change also suppose you wanted to figure out what this area is
19:00we can find areas of rectangles but this you can use calculus and find the area of that thtas second semester calc i dont thing we will do any of that this semester if we do it will be at the fvery end thats the neat thing we use calculus for, we use it again to see how things are changing
19:35and as they change, and what we can leanr from them you see this a lot in physics and a lot in engineering economics medicine believe it or not but whats going on in medice is something else figuring it out for you but if your in medical research you really need to know this stuff so thats what is calculus, why do we care
20:00well mainly of you care cause you have to take this class right if this was not required i bet this room would be a lot less crowded alright so all im gonna do today is do some review and then we will learn but nit yet
20:45remember this from last semester? radians and degrees in calculus we do nothing in degrees when you do your homework with a calculator and you keep getting it wrong its probably in degrees
21:00so in all the trigonometry in caculus its all in radians maybe there will be some random problem thats the exception but pretty much everything is in radians so you need to be able to convert from radians to degrees you remember how to do that?
just in case that is something like s circle close enough okay so once around the circle
21:31is 360 degrees alright well this circle has a radius of one 1 whatever and the circumference of the circle will be 2pi which is 2pi times 1 so radians what we do is we have some angle and we know that angle in degrees we can look at the arc length
22:02and we call the radians the length of the arc and goes aroudn the radians as you make the angle bigger or smaller you make the arc bigger or smaller they correspond exactly all the way around is 2pi and all the way around is 360 degrees and we call that and this is how you convert it you divide by 2
22:31so 180 degrees is pi radians if you have something in degrees to put it in radians you just do this so the simplest way to convert is if youre going from degreees to radians
23:07you multiply your angle in degrees by pi/180 so once again youre gonna have to get comfortable with this so for example if you have 60 degrees and you want to convert that radians you times that by pi/180
23:3860/180 is 1/3 so this because pi over 3 60 degres is pi/3 you guys sort of remember this right?
i know 123 was a long time ago so it was a blure it should still be in your brain somewhere in general you know gonna have a calculator so your not gonna have to hard of angles
24:00so this is the stuff you should make sure you know okay 0 degrees is 0 radians you all can take 0 and multiply it by something and get 0 very easy we all know how to multiply by 0 right?
24:3630 degrees is pi/6 radians 45 degrees is pi/4 60 degrees you just learned is pi/3 you get each of these by taking these and multiplying it by pi and dividing it by 180 90 degrees is pi/2
25:02most problems you would deal with will be these values in radians but just in case 180 degrees iss pi 270 degrees is 3pi/2
25:30now there will be others generally were not gonna do anything to annoying its just annoying to not have to do it in degrees were gonna use these why are we doing this? for the people who havent done calculus
26:00you have in your back years theres 3 areas where people general do bad in calculus it doesnt have anything to do with calculus they do badly because they are reaching their trigonometry so they get 1the problem on the exam and they can trick them so you just took the course mat123, you have a lot of trigonometry there after trigonmetry, your gonna have a lot of problems with sin cosine and tangent all that the way you use trigonometry in real life
26:30now after this well the engineers but you need to know first area where people have a lot of problem the second area where people have problems is sgraphing so ill put up some graphs and make sure you know your graphs third is in your algebraic manipulate factoring combinding the admit terms all that stuff thise are the three places people tend to have trouble actual calculus is not very hard
27:01its what you do with calculus that tends to mess people up okay so we will give you some problems there will be 5 steps to solve a problem, 1 step would be calculus and the other five will be algebra so lets review some trigonometry oh before i forget one another thing here if you want to know radians in degrees
27:38its unlike to go from radians to degrees thats why i didnt write it but well multiply by 180/pi so when you want to go this way you multiply by
28:01180/pi over here very rarely will we make you do that but you can choose to do that get a problem and you say pi/6 and you want to turn that back to degrees memorixe these so trigonometry theres a lot of right triangle trig, remember soh cah toa you have a triangle
28:32you get some angle you call t theta and you need to know the sin cosin and tangent the sin of theta
29:33alright so sin of theta is gonna be opposite over hypotenuse is a/c you remember the soh cah toa stuff right?
i not gonna teach you im just gonna remind you cos theta b/c tan of theta
30:00is a/b you guys all see that?
so for example is thats a 3 4 5 triangle sin of theta 3/5
30:33cosin theta adjacent over hypotenuse which is 4/5 and tan of theta is 3/4, someone give me a problem with a triangle in it and youll have to find sin and cosin and set things up sin in it and at the end you plug in
31:03pi/6 or pi/4 something like that that leads up to a circle, you might know the circle you memorized it you forgot it you memorized it you forgot it theres a fun thing you can do with your finger that some of your teachers taught you last semester, i dont really know it
31:33some of you may doy, may make a chart everybody see that?
32:08you have sin cosin and tangent and you 5/6 which is 30 degrees, pi/4 which is 45 degrees pi/3 which is 60 degrees remember sin and cosin all of your fractions have 2 in the denominator
32:33and the numerator starts at 1.. 1, 2, 3 as with precalc i recommend you memorize this again and at each exam you write this in the corner somewhere so you dont have to think about throughout the exam scribble it get it right and then put it in there sin goes 1,2,3 cosine goes 3,2,1
33:01sin is going up cosin is going down sin pi/4 is the same thing as cosine pi/4, its radical 2 over 2 the tangent you find by dividing the numerator of the sin adn dividing it by the numerator of the cosine because tangent is sin/cosin so its 1/radical 3 you get the same thing so 1, and radical 3/1 is radical 3 so i recommend
33:33you memorize that chart if you can memorize that chart or tattoo it even better and the you dont have to do it again anyone got a trig tattoo?
so thats one set of things you can memoriz4
34:02now theres another set of things you can memorize im gonna erase this youu just sin cosin and tangent of 0, pi, pi/2 and 3pi/2 sin cosin tangent, 0
34:30pi/2 also known as 90 degrees?
and 3pi/2 by the way a lot of this stuff is in the book theres very few of you who can find a copy of the book somehow very front theres a couple pages that tell you all this stuff theres a really fun part in the beginning to explain why you need to learn calculus
35:10so remember sin goes up while cosine goes down sin strts at 0 and ends up at 1 cosin starts at 1 and goes down to 0 sin goes up cosin goes down tangent is sin divided by cosine so 0/1 is 0 1 divided by 0 is
35:31undefined because you cannot divide by 0 tangent actually gets infinity big as you approach pi/2 equals 90 degrees you get a really big number the sin of pi is 0 and and the sin of 3pi/2 is -1 so cosine of negative 1 is 0
36:040 divided by -1 is 0 -1 divided by 0 is undifined now we are gonna ask you stuff about secant cosecant and cotangent but so we will do that in a minute with those i flip sin cosin and tangent upside down so most of these are easy to turn upside down
36:33but this last row is a little tricky this is the same what what does it mean by undefined i put one over that, you turn that upside down and you get 0 you want to add those 4 okay when we say undefined what really happens is tangent approaches infinity
37:00is -2, 1 over those is 0 very soon in a few minutes youll learn about that one of the other things that will happen later in first semester calc or second semester calc you have problems where you have you have sin cosin and tangent in the denominator of the problem you want to have the in the denominator of the problem because the numerator
37:34because you take things out of the bottom and out it in the top csc cocsc and cotangent so cocsc of an angle you take the sin of an angle and do one over so 1/1/2 is 2 take the radical 2 a turn it around
38:00that should equal square root of 2 but it can be in either form\ secant thet is 1/cosine theta and cot of theta is 1/ the tangent of theta
38:30again you have avaalue and turn it upside down these are universes of eachother the problem is the cosin of that the cosine in the denominator the answer to that is sec instead thats because you forgot the denominator and and moved it up to the numerator youll get 1/ cosine square and and the answer squared thats the same thing
39:00you should also know on this list tangent of theta is sin over cosine cot theta is cosin/sin this is almost all the trig you will need for the course
39:31were gonna need some relationship for these so some other relationships you want to know okay take the sin of the angle and square it take the cosin of the same angle
40:00squared you get 1 thats the pythagorean theorem this is not a trigonometry class so im not gonna show you where it came from the sin squared of an angle plus cosine squared of an agle, which means 1-sin squared is cosine and 1-cosine squared is sin remember when you did all those trig identities you all arranged the trig stuff this is where youll actually use it because youll have a problem it will be messy and you can use an identity and turn it to something easier
40:31if ill be able to do that youll be able to do that i hope youll be able to do thag couple other things you should know 1+tan squared of an angle is sec squared of an angle
41:01you get that by taking the top of the equation by cosin squared and you do 1+cot squared is csc squared sitting here saying hes not so great i dont understand why im here but this is as fun as its gonna get
41:56this is something unfornately we did not cover in mat123
42:00would happen if we had the time the sin cosin of angle addition and subtraction formula is a fancy way of saying the sin two angles added together it is not sin a plus sin b you dont just distribute thats to easy it is sin a cosin b
42:30plus sina cosin b plus cosAsinB and if this is a mins sign you just minus sign there we said find the sign of theta
43:00plus pi/2 sin of theta cosin pi/2 plus cosin theta plus pi/2 alright find sin of theta plus pi/2 that would be sin theta cosin pi/2 plus cosin theta sin pi/2 then you look at your chart and go hey cosine of pi/2 is 0
43:32sin of pi/2 is 1 its just gonna come out cosin of theta i repeat cosin pi/2 is 0 thats why the left hand sign goes away sin pi/2 1, so this is just cosin stheta is 1 this is very useful imformation to know if theres a minus sign in the middle you just use the minus sign
44:01okay thats one of those and theres cosin a plus b cosin a cosin b minus sina sinb so notice cosin has what in the middle or a minus in the middle okay we add these and we subtract these two
44:32and if this is a minus this is a plus so we want to find cosin pi/2 plus theta in reverse it doesnt mean anything cosin pi/2 cosin theta minus sin pi/2 sin theta
45:01cosin pi/2 is 0 pi/2 is 0 sin over pi/2 is 1 cosin pi/2 is 0 this is 0 because youre multiplying 0 1 minus sin theta so you look at this problem your gonna have this in it you convert it to this which is easier
45:31how much fun we have?
good amount really?
i dont know why you use calculus in chemistry used for rates and stuff llike that by the way for those of you who are interested
46:05for those of youwho are interested were gonna have a bio we gonna have special tas we will have a review session do what we can with calculus help specifically make you better at bio i dont have full details but you can always email professor sutherland and he will help the idea is find a way to intergrade all this stuf
46:32its gonna be a spacial session, for an hour once a wekek maybe longer for the undergrade tas want to help you guys out, kinda help you last thing to review logarithms and exponentials
47:04actually theres two things to review i lies by the way this video will be up sometime so watch this
47:30were gonna do this as a reality series logarithms and exponentials are inverses of eachother
48:03so if you have something 2 to the power of x you use the logarithm and figure out out how to solve it or the other way around so if you have e to the x equals y you can say also x equals the natural log of 5
48:30which i would say log 5 pretty much in calculus except when you use radians pretty much what you use is e to the x is the natural log of x sorry about that folks here we go if you dont have a calculator you have to get comfortable to find the natural log of 5 we dont care what the natural log of 5 is its just log 5
49:00so some rules you should be comfortable with the natural log of 1 is always 0 doesnt matter what phase is natural log of e 1 natural log e to the x is x so if i said whats the natural log of e to the 5 its 5
49:34similairly e to the 0 is 1 because anything to 0 is 1 e to the that log is notice these are the inverse functions of each other which gives you solve problem, you have a problem
50:01you have e to the x equals something and then you just write e is the natural log of x or natural log will equal something and then the answer will be e to the something this is something you need to know and then you have the log laws you have the log of two things multiplied together
50:33its the log of a plus the log of b if you have the log of two things divided log of a minus log of b and you have the log
51:00of something to the power you leave the power in the front thats what log is very useful for to get things out of the power back down to everything else in the equation you have the problem you cant solve it because you have a unknown power by using logs you can
51:34so if you have 7 to the x equals 20 you want to solve that you take the log of both sides and you say how does that help me now i can take the x and put it infront and i can isolate the x
52:02thats the real value of logarithms because you get to some problem and this is where you are and this is where we would want you to get i have no idea if were gonna have stuff like this but this s what you should be comfortable doing okay last top to make sure you guys are good
52:44graphing you know how much you love graphing theres a lot of it in calculus will just kinda write stuff down the whole on curves actually
53:06mid april good so it will be on the final and there are a lot of graphs you learned in high school and you may wonder where they came from with calculus we will be much better at it belive it or not you want to mqake sure you get comfortable with graphing but theres some basic graphs you want to make sure that you have memorized
53:32graph y equals x is a line through the origin divide everything nicely in half this is 45 degrees this is 45 degrees thats the line y equals x guess whats next y equals x squared
54:02y equals x squared is a pralabula goes up the origin goes equally up both sides what about y equals x cubed
54:30kinda look like that] so far so good? were gonna have a problemw ith x cubed and im just gonna im gonna draw a picture on the board so you can get an idea of what yx cubed looks like heres some others y equals 1/x
55:09what does 1/x do, 1/2 does the vertical asymptote goes all the way up to infinity down here it gets close to 0 1/x will be very useful youll see it shows up a lot then we get lets see y equals e to the x
55:34e to the x goes through 0,1 i have a horizontal asymptote so itsnever negative its never even 0 youll hae a lot of problems with e to the x in the problem and youre trying to find 0's you can jut take the e to the x term out e can never be 0 and never be negative e to the x is always positive
56:07one more y equals natural log of x e to the x goes through 0.1 natural log of x goes through 1,0 e tot he x has a horizontal asymptote log over x has a vertical asymptote
56:30notice you get a negative answer with log of x because you have because log of 1 is 0 however you can not take log of a negative number you cant find the log of negative something can fin the log of negative numbers and it goes up to infinity x very slowly one of the things you do with limits how slow something goes up e to the x goes up very quickly
57:00goes up to infinity natural log of x goes up a very long time it gets there but it takes a very long time what you can do it compare the rate you have two equation one of them is an equation on in the numerator one in te denominator you try and figure out which unit is going faster it means you plug in a number between the two values so when you say e to the x grows very quickly and natural log of x goes very slowly\
57:32e to the 10 is a really big number natural long of 10 is not a very big number in fact it takes a long time e to the 10 is very long so far so good? 3 graphs left
58:17the equation y equals a squared minus x squared that believe it or not is a semi circle this is a and negative a and thats a up there thats because you take the equation
58:31x squaed plus squared is b sqaured and you solve it for y you get y equals plus or minus the square root of a squared minus x squared so what you get is you get a semi circle center the origin we just take the upper half because of the positive square root so minus a is a so for example\
59:05y is the square root of 25 minus x squared all that means iis a semi circle negative 5 and 5 okay were does 5 come from, square root of 25 not so bad when you see this you have to remember semi circle
59:30not 25 minus x squared is a parabula the square root is a semi circle alright i promise last three everybodys favorite graph, sin\
60:09thats what sin looks like sin starts at the origin it doesnt actually start anywhere but the graph of sin of x goes through the origin goes up to one down to 0 down to -1 back to 0] keeps going forever and those are important values we had a question like this on the final
60:30we had some trig graph i hope some of you are happy with your 123 grades not everybody okay y equals cosine of x y of cosin x is very similar to cosin of x
61:02except you start it here in another words you start it an the y axis instead of being at the origin its on the y axis its at 1 okay it goes down to 0 and down to negative 1 back up to 0 back up to one and keep going
61:31thats what cosin looks like last at least favorite y equals tangent of x y equals tangent of x looks a lot like cubed graph except it has a vertical asymptote and the graph does not
62:02graph goes out forever but close to the origin y equals tangent of x were able to do the graph you learned this in 3rd semester calc or if you took the AMS the second semester it has vertical asymptotes at pi/2 and negtaive pi/2 so it goes up down so you want to be comfortable being able to sketch these graphs we will show you these problems and expect you to draw a picture
62:35so on that note that enough for today see you all on monday