| Start | The format of the exam i think its going to be 8 questions what will be on the exam will be limits definition of the derivative you dont have to write this down but you should know limits you should know definition of derivative you should be able to sketch curve very basic sketching you should be able to look at a curve and tell whats going on with the cure or the derivative |
| 0:31 | will not be concavity on the test because we have not covered that yet so if you see the word concavity you could just ignore it, but it has to do with the curvature whether its concave up or concave down what else is on the test.. it will be similar too some of the midterms but one of the midterms had a bunch of pre calc on it |
| 1:04 | i think thats the 07 one so you can ignore a lot of the 07 exam someone spent way to much time on that.. what else do you need to know thats about it.. some of you have been asking can you do the derivative short hand for those of you who had privilege of taking calculus before i think the answer is going to be yes and no if we say do it the hard way you have to do it the hard way and if we dont you can do it the easy way so there you go |
| 1:33 | gonna be a whole bunch of limits i think the test is very straight forward youve been preparing you should do niceley on the test theres a couple of questions that would sort of stretch you but you know we have to kind of do that thats what they pay us for to ruin your lives make sure you never get to be a doctor no no no im not a doctor why should you be a doctor |
| 2:07 | so im gonna go through the october 09 exam oh one last thing that i was telling some people if you go to your blackboard page and you dont have things like documents etc its gonna be a lot of emails but well figure out what to do but we dont understand why some of that is missing it shouldnt be |
| 2:30 | im gonna pull this up so if you do not have it you can see it so this is the new term professor sutherland gave in october of 09 and well do some of these feel free to ask questions raise your hands dont be shy some of you are not wearing red, remember your seawolves |
| 3:34 | so were gonna do 1a the limit as x approaches 2 x suqared minus 4 over 5x times x minus 2 so we look at that and say uh oh were gonna have 0 on the bottom 0 on the bottom is bad your gonna plug in 2 on top and your gonna get 4-4 is 0 |
| 4:03 | and youre gonna plug in 2 on the bottom and get 0 and thats a bad sign its not really a bad sign it just means youll get 0/0 which is undefined so how do you worka round that well i told my class before if you do when x goes to a number you get 0/0 that means youll be able to factor out x minus that number so if you do the limit as x goes to 10 and you get 0/0 that means the top and the bottom both contain a term x-10 |
| 4:30 | it can contain more than 1 of those but at least 1 thats what making it become 0/0 that make sense when you plug in the 10 you get 0/0 so here when you plug in 2 you get a 0 here, so you gotta go over here and take an x minus 2 out of that so its not a very hard expression to factor its the difference of 2 squares |
| 5:01 | cancel the x minus 2 opps that should be plus now you plug in 2 cause you do the limit as x goes to 2 and you dont have that problem anymore youll get 4/10 which you could leave that way you dont have to reduce to 2/5 or turn it to a decimal or anything else there you go 3 points your on your way to becoming a doctor |
| 5:31 | next one limit x goes to infinity 2cosine pi over x gotta love trigonometry you know when youre gonna need trigonometry.. well for this exam but you guys know the general answer well if youre gonna be a physics major youre gonna need trigonometry physics majors yea applied math engineering |
| 6:01 | the rest of you youre gonna run out of trigonometry at some point |
| 6:36 | now x goes to infinity what happens with pu/x well x is going to infinity so pi over infinity is going to be 0 whats the cosine of 0 you sure its 1 you should be sure. make sure you know your unit circle okay more points this is so much fun |
| 7:04 | that wasnt very hard dont be paranoid some of these limits are very straight forward some of them are not but some of the are im obviously not gonna tell you which ones that would be cheating alright limit as x goes to 4 of x squared over x minus 4 squared |
| 7:31 | now when you plug in 4 youre gonna 0 in the ottom but youre not gonna get 0 on top so when you hav a limit 0 on the bottom denominator but you dont have 0 in the numerator thats gonna be even for the limit infinity or minus infinity or something, we might get limit does not exist so you have to figure out whats going on when you approach four from the left sid from the right side and see if they agree when we do the limit as x approaches 4 |
| 8:00 | minus thats just less than 4 the top would be 16 and the bottom would be 0 would the bottom be positive or negative positive cause youre squaring it, in fact this whole this is going to be negative right square on top and square on bottom its always gonna be 0 it cant be negative Okay so when X approaches 4 from the minus side From the minus side You're going to get plus infinity |
| 8:32 | You all see that you have a number on top a number on the bottom That that's infinite And positive because it's square over square And there's other reasons why it's positive The limit as X approaches 4 from the left side of X squared over x -4 Squared. Like I always said it's positive I get plus infinity again. since they agree |
| 9:01 | The limit of this The limit is plus infinity. That's the kind of thing we are looking for We don't want to see you write positive infinity You have to do a little bit of explaining Now you can say. You could read this out and say this expression is either positive or zero And therefore the limit goes to post infinity. That would be good enough Just writing something probably wouldn't get you to full credit |
| 9:30 | It's one of professors things. He also points out my thing to If you write stuff in mathematical nonsense that's bad Even if you get the right answer you write something that sort of That's totally not true you get the right answer Who rather you write something that is true I missed the answer a little bit Okay let's move onto the next page |
| 10:02 | You see where it says name and then ID. That's ID Trying to know your ID Also I strongly suggest you where it says recitation Know what your recitation as an try know it before Thursday And don't feel like I don't know if some guy i never go If we can't find your exam at some point we will give you a zero So we suggest to be able two find your exam The best way to find it it's have the correct recitation number |
| 10:33 | You heard the speech from me before It's the same speeches last time try remember what your recitation is It's not that hard to figure out it's up on blackboard Now we have Limit as X goes to infinity X squared -9 Over five X-3 |
| 11:03 | No you look at the top and say There's an ask -3 and then the bottom has X -3 But this is not as X goes two three this is an X goes to infinity Different kind of limit So what can we do with this You can look at the power for the highest term on top highest term on the bottom You can do a little algebra you can do a bunch of things the simplest thing |
| 11:35 | Is to multiply out the bottom The highest term on top is X squared and the highest turn on the bottom is 5X squared that's a five So you can say this is just Liked the limit As X goes to infinity of X squared Over five X squared when you use that sign it means is behaving like |
| 12:04 | Kind a like And now that the X squared cancel this would be 1/5 That wasn't so bad This is the whole exam you be happy right |
| 12:30 | How about limit as x goes to two X plus H squared minus X squared Over each that sort of looks derivativeish isn't it It's not quite a derivative as X goes to two It sure is close If you're not sure what's going on dive right in and do algebra. We love algebra |
| 13:08 | So we multiply out the top X squared +2 XH Plus H squared minus X squared Over H Cancel a couple x squares Now we have the limit H approaches to 2XH Plus H squared over H |
| 13:30 | You sure it's not derivative it kind of looks like a derivative keep going Taking H out of the top And we can cancel the H's |
| 14:03 | Now when we let the limit as H goes to two we get 2X plus H it's not a derivative but close |
| 14:32 | By the way some of you are asking do I have to write the limit at every step You should like I said you shouldn't write mathematical nonsense Also you want to get as much partial credit as you can so make sure you can figure out what's going on 2X +2 yes I apologize |
| 15:05 | Alright next one |
| 15:33 | Alright how about a limit as X goes to infinity E to the x cosin x This is a fun one what are we do with this The ideas |
| 16:08 | Anyone have any idea what to do alright you are going to want to squeeze You're going to want to know the squeeze theorem how do we know to use the squeeze them you have to use your rules So when you see limits and you have signs in cosines Among other things use the squeeze theorem sin sine and cosine are between one and -1 |
| 16:35 | So what will be do here well dormant as X goes to infinity Of E to the X cosign X So it is less than or equal to The limit as X goes to minus infinity E to the X times one Because cosine is less than or equal to one this is less than or equal to |
| 17:03 | Limit E to the X, X goes to negative infinity of -1 Now would have happens to E to the x , know what it looks like That's what to the X looks like positive infinity E looks infinity when you go to the negative infinity It goes to zero So this term on the left |
| 17:31 | Is going to look like zero times -1 which is zero Here on the right well it's just positive one It's just zero again that means less limit Is somewhere between zero and zero So it's between 0 and 0 so it's zero |
| 18:04 | That is the squeeze theorem so far so good No we hating life right now Well think of what the cosine graph looks like Well since some of you took trigonometry recently well that didn't |
| 18:31 | Sad before the fortune of me Cosine Looks like that and so does sign by the way . That is one and that is negative one So no matter what cosine and sign do they'll always between one and -1 So whatever is going on here E to the X can't be bigger than one because cosine can't be bigger than one And it can't be less then e to the x times -1 because cosine cant be less than -1 |
| 19:00 | so whatever e to the x term is doing it is multiplied by cosine its never multiplied by anything greater than or multiplied by anything less than 1 its always trapped between them thats why its squeezed so if you would graph tjis you would find that the graph is somewhere between 0 and 0 |
| 19:43 | so before we do other stuff lets do some more limits so how about fall 08 exam did any of you look at the fall 08 exam yet |
| 20:01 | how bout that firsy one limit as x goes to 0 sinex/10 dont you just love trig |
| 20:53 | getting 0/ arent we, gotta be a way around it |
| 21:44 | alright thats long enough the problem is you know sine of 0 is 0 you know tangent of 0 is 0 if you dont you have four more days to remember the unit circle or what we suggest tattoo absolutely fine, nice tattoo of a unit circle make sure you can see it dont put it back here somewhere |
| 22:02 | right out front have to explain that to your friends later why you have a tatto of a unit circle, because i love trig so much that i wanted to have it on my body forever so what could we do well we all remember the tangent sine over cosine of course this is sinx over |
| 22:31 | sinx/cosx that means we can take the numerator and the denominator invert multiply that cancel the sine and now it would be cos of 0 |
| 23:00 | which is 1 be right back |
| 24:00 | im back so once again lets try some more of these these look painful dont they the second one is not to bad the limit as x goes to infinity of that we have the limit as x goes to infinity 5x squared minus 4x -1 over x squared minus 1 we look at the power of the top and the bottom and its the same |
| 24:30 | so this is going to behave as the limit as x goes to infinity
of 5x squared over x squared
which would just be 5
these arent s bad are they?
how about that third one what do you do with infinity minus infinity well you could be theres no denominator |
| 25:02 | doesnt matter so lets do this one so you guys say what am i gonna do, you multiply it by the conjugate and you multiply it by the conjugate over the conjugate |
| 25:34 | after all this is over 1 it doesnt have to have a denominator you can multiply it by 1 but you can pretend theres a denominator there the denominator is 1 so there you go so when you see that expression you should see conjugate the expression like that, radical some other stuff if you dont know wat to do cant squeeze it so maybe theres a conjugate in there |
| 26:00 | so lets see what happens square root of 4x squared to square root of four x squared plus x is 4x suqared plus x middle terms cancel when you do the filing, thats the purpose of using the conjugate you get plus 2x times the radical minus 2x times the radical and the 2x times 2x minus 4x squared |
| 26:33 | and the bottom we have this thing that we just introduced that we hate 4z squared cancel the limit as x goes to infinity of x over squared of 4x sqaured plus x |
| 27:01 | plus x now what oh 2x im sorry, good thing i brought you now what do we do i have that x on top can we take an x out of the bottom yea you can so what would that look like, well lets see |
| 27:32 | nothing happens to the top you take an x out of the radical because the square root of x squared is x in fact the absolute value of x but we dont care what about the plus x well that would become 1/x how do we know that well pretend thats a 4x squared and put that back inside you get ax square |
| 28:01 | plus x square root of x, x square root of x is x so you can take an x out of the whole thinf theres other ways around this but thats probably your best way remeber that trick cancel the x's now you have limit as x goes to infinity 1 i couldve taken out the 4 but i didnt have too |
| 28:35 | under the radical you have x squared is you take it outside the radical you get x because the square root is x so youre taking x squared out of this term that becomes the x and then you take the x squared out of this tem x is the same as x squared over x i see a lot of you guys are lost with that so we will do it again |
| 29:01 | important you understand this step cause who knows someday in life this may be the only thing that matters to you you can think of that as the square root of 4x squared plus x squared over x which you can think of as the square root of x sqaured times |
| 29:30 | 4 plu 1/x now you can take the x squared out and that becomes x square root 4 plus 1/x alright you understand that now were you just like ah ill write it down and look at it later |
| 30:01 | lot of you guys do that, watch the video over and over again now if x is infinity thats 0 the 1/x is 0, square root of 4 is 2 2 plus 2 is 4 thats 1/4, or how they say in brooklyn a quarter |
| 30:44 | we are videoing this whole thing, matt back there putting this whole thing on video remember you can get to the videos from the web page if you go to our webpage remember this thing okay all the videos are here |
| 31:00 | you have lecture 1, you have some of professor sutherlands videos all sorts of fun stuff okay thats on calculus a page lets do a few more of theses and then we do some other stuff i dont want to spend to long on limits i think we can skip d, e is just like the one we did |
| 31:30 | eh we did enough limits for the moment lets do something else because limits, you can have enough of that i know some of you, yes there will be an answer key professor sutherland is in charge of putting that up at some point hopefully before friday |
| 32:06 | where was i, i was doing the fall 09 exam back to the previous exam |
| 32:30 | fall of 09 lets look at 3 remember monday and wednesday in class ill be reviewing problems aswell |
| 33:04 | and they will be up on videos and those videos will be up so youll have lots of chances to watch me up here let f of x equal 5x cubed minus 8x plus 2 fins the slope of the secant line |
| 33:31 | you guys all know how to do that, take a second figure it out, not that hard |
| 34:57 | find the slope of the secant line, the secant line |
| 35:01 | from x equals 0 to x equals 1 would be f of 1 minus f of 0 over 1-0 thats the slope of any line, change in y over change in x you just have to find f of 1 f of 0, 1-0 f of 1 we get by plugging in 1 you get 5-8+2 is negative 1 then you plug in 0 and get 2 |
| 35:36 | so that equals -3 thats not so hard okay now find f prime of 1 so for those of you who already taken calculus you can find this the fast way if you wanrt im also going to teach that tomorrow in clas in theory professor sutherland will be teaching it so youll get to see it, yes |
| 36:03 | no you will not get points off i checked that with professor sutherland i fought for you guys, actually he said no problem wasnt much of a fight but just in case lets make sure you can do it the hard way because many of you did not take calculus in high school so we want to find f prime of 1 we going to need to do the limit |
| 36:31 | as h approaches 0 of f of x plus h minus f of x all over h and we want to do this at 1 so we really want to find the limit as h approaches 0 f of 1 plus h minus f of 1 all over h |
| 37:01 | what is f of 1 plus h would be the limit as h approaches 0 of 5(1+H) cubed minus 8(1+h) plus 2 minus f of 1 which we already found which is -1 over h |
| 37:33 | okay whats 1+h cubed well you can certainly multiply out 1+h cubed or if i showed in one of my lectures this is 1+3h +3h squared plus h cubed because x+hh cubed is x cubed plus 3 x squared h plus 3x h squared |
| 38:01 | plus h cubed you might want to memorize that not as hard as it looks as i showed before the coefficients go 1,3,3,1 x cubed, x squared x no x term no h term, h, squared cuber pattern that comes by the binomial expansion what were also going to do tomorrow in class |
| 38:30 | now that we did that minus, minus 1 all over h did i make a mistake agin i lost something in here, oh i forgot the 8 part minus 8 minus 8h, plus 2 plus 1, there we go |
| 39:06 | distribute that 5 you get 5 plus 15 h plus 15h squared plus 5h cubed minus 8 plus 2 is minus 6 minus 8h plus 1 |
| 39:31 | all over h lets see 5+1=6-6=0 15h-8h=7h this becomes the limit h goes to 0 7h plus 15h squared plus 5 over h just some algebra we love algebra |
| 40:02 | pull out an h and then cancel now when you take the limit as h goes to 0 you get 7 |
| 40:39 | okay part c, write the equation of the tangent line the equation of any line y-y1 equals y-y1=m(x-x1), we know have all the pieces we need |
| 41:05 | what is y1, f of 1, which is negative 1 the slope is 7 and that is x minus also 1 and you can leave it like that you dont have to simplify the equation of a line we always say dont do extra algebra and get yourself in trouble |
| 41:30 | okay part d talks about concavity which is not on the test so you ignore that oh i think thats enough for the moment continuity |
| 42:01 | how are we doing on this we understand this one were finding f prime of 1, remember every time you do the derivative you are shrinking the denominator as close to 0 as possible this is always going to be the limit as h goes to 0, the one part is up here |
| 42:33 | how did we get -1, what is f of negative 1 plus in 1, i lost the equation now its 5x cubed minus 8x+2 you plug in 1, you get negative 1, we found that here thats y1 which is f of 1 okay other questions |
| 43:04 | were good okay we can come back to stuff we can always rewind the ideo at this point so for what values of x is this continuous |
| 43:33 | any ideas, we have f of x e to the x e to the x looks like this, thats continuos everywhere so we want to know what values of x is the function continuous you can also ask where is the function not continuos and then its all reals except those values |
| 44:05 | were definetely going to have a problem at 0, arent we?
cause then we have e to the 1 over x, so we cant have a 0 here that would be 0 and that would be bad so its not continuous at x equals 0 any other problem spots |
| 44:30 | any other places you can think of when a teacher ask that, that means yes theres another spot got any ideas or do you want to jus wait for me when 2 equals e1/x its equal to 0 when the denominators whole thing is 0 |
| 45:01 | because again youll be dividing by 0 so where is that, well when 2 equals e1/x take the natural log of both sides the natural log is 2 which equals 1/x or x equals the natural log of 1/2 so thats the other place its not continuous |
| 45:36 | other than that its continuous at all reals so if youre gonna write its not continuous at x=2, natural log of 1/2 you should also write all reals except x equals 0, or 1/log2 its not continuous because you can not divide by 0 |
| 46:02 | other wise its continuous everywhere we good should i do that again people understand, we have two problem spots the first is here when x is 0 we do either 1/0 you cant do e to the 1/0, cause it to the 1/0 is undefined thats problem spot number 1 problem spot number 2 is when the whole denominator is s0 so you set 2=e to the x equals 0 put the 2 on the other side |
| 46:30 | take the log of bot sides, natural og and solve for x its always asier when i do it yes e to the 1/x, i didnt say e to the 1/x is 0 you want to know where e to the x equals 2 because then you have 2-2 in the denominator so e to the 1//x equals 2 when x eqauls 1/x log 2 |
| 47:01 | natural log of 2 |
| 47:41 | okay lets scorll around this thing is awesome got to tell ya, 8 points write a limit that represents the slope of the graph y= absolute value of x to the x when x is not equal to 0, and 1 when x is equal to 0 |
| 48:01 | what do you do with that, any ideas its not as bad as it looks now remember slope is derivative so we would have to do the definition of the derivative here right so whats the definition of the derivative, its |
| 48:30 | the limit as h goes to 0 f of x plus h minus f of x all over h write that down in a lot of places you never know what youll pick up partial credit for remember life is about partial credit |
| 49:03 | well what is f of x plus h, were gonna do limit h goes to 0 absolute value of x plus h to the x plus h, that kind of looks like an x plus h minus absolute value of x to the x all over h and your done, thats the whole problem thats it nothing else |
| 49:35 | thats 8 points, the whole test is nine points thats pretty good |
| 50:02 | what happens with the part when y=1 and x=0 you dont care, youre only doing the limit you dont actually care what happens at 0 you just want the limit i mean you can replace this with 1 if you want but you dont actually have to evaluate it i dont know maybe thats what professor sutherland wants but we dont have to evaluate the limit |
| 50:32 | and by the way yea the limit as h approaches, yea id leave it alone ill check with it and see if he changes that now we move onto graphing which we love at the right is a graph of the derivative of a function you can ignore 6a theres no concavity on the test |
| 51:01 | you know what a concavity is right , i have a hole in my tooth kahn-cavity, |
| 51:32 | lets just draw that graph again so thats approximately what the graph looks like, thats the derivative of the fucntion which of the following best represents the function thats f prime of x, what do we know, well |
| 52:04 | you look at the derivative of the function we say the derivative is negative up until this spot and then its 0, and if its negative that means the function is going down and then it gets to this spot and stops going down because its 0 and now its going up, its positive all in this zone over here remember negative means going down and positive means going up |
| 52:35 | so you need the graph to be going down and get that spot and go up for a while and then it goes up until you get to here another 0 and then you go down for a while until you get to here where it makes a bottom again and then its positive and goes up again so its doing something like that |
| 53:00 | so the closest one is the second graph remember theres more than one picture that can be that derivative but this is the one that you want graph number b, graph whatever that is so lets do that again, look at the derivative function the only difference is i didnt go down farther, you can go down as far as you want doesnt matter |
| 53:30 | these are 0's the derivatives are 0 means it has a maximum and minimum or a flat spot okay so before hand we have a negative derivative so the function is going down then they have a positive derivative and the function is going up, so going down get to there, going up at the origin where 0 so once again we have to have a max or a min at 0, at the y axis they were negative so we go down for a while and get to that spot |
| 54:01 | so we go down for a while till we get to there go up another clue this is a polynomial and it has 2 maxima and minima so the original function will have 3 maxima and minima the derivative every time you take the derivative you lose one if you started with 10 maxima and minimas then the first derivative will have 9, the second will have 8 third derivative will have 7 and so on |
| 54:32 | minor exceptions should i repeat that you have a function of f of x, and you have max and mins the first derivative will have 1 fewer and the 2nd derivative will have 2 fewer and so on so since i told you that if this is f prime of x f double prime of x will only have one max or min so what would that look like |
| 55:00 | for the second derivative we would have to have a 0 here and a 0 there youd have to have a 0 somewhere around there somewhere around there the function is going up until we get to that spot, so its positive and its going down till it gets to this spot so its negative its going up again so its positive negative positive so that would look like the third graph |
| 55:35 | this one right here these are fun, should i repeat that you look at the derivative graph now you preten this is a regular graph because we want the second derivative so the derivative of this, where this graph have maximums and minimums the derivative of it will have 0's it has a max there and a in there |
| 56:00 | and 0 there the graph is going up so it would be positivw its going down so it can be negative its going up so it would be positive so you have positive negative positive oh your right its the first one, its the left one sorry i was just looking for a parabula, its the left one not the right one |
| 56:33 | because the location of the 0s alright lets do a little more stuff |
| 57:04 | actually i dont think we need to do this so lets pick a different test lets do another continuity one |
| 57:40 | take a second and see if you can do 2 this si the oct 08 exam, fall 08 i did that kind of fast, ill back up for a second so you can see what i did this is here under fall of 08 i remember when i was doing the fall of 08 |
| 58:01 | that was a long time ago for some of you, you were stil in elementary school oh my god so well do two, try and figure out if its continuous |
| 59:43 | alright thats long enough, we suffered enough is this function continuous and if not where is it not continuous |
| 60:14 | alright we go 3 things going on here x is less than negative 1, we have 3x squared if x is between 1 and negative 1 we have 3tan(pi/4)x and x is gretaer than 1 is 3x cubed |
| 60:31 | so first of all its going to be continuous everywhere 1 and negative 1, thats the place we have to check but polynomials are continuous everywhere and the 3tan pi/4 x isnt going to be a problem between 1 and negative 1 because youre between tan pi/-4 and tan positive pi/4 its continuos there tangent doesnt give you a problem till you get to pi/2 so you could write that |
| 61:01 | so part of your answer is going to be all reals except you have to figure out where its not continuous so how would we test at negative 1, well first of all you just test in your head you plug in negative 1 in the top and you get 3 you take -3 in the middle part you get 3 tan negative pi/4 now youre really glad you tattooed the unit circle on your wrist and you look and you remember tan of pi/4 is 1 so the tan of -pi/4 is negative 1 |
| 61:30 | so thats negative 3 3 is not the same thing as negative 3 its not going to be continuous at negative 1 so how do you show that, you do the limit x approaches negative 1 from the left side of f of x 3 times -3 squared is 3 and the limit as x approaches 1 from the left side of x |
| 62:03 | is 3tanpi/4, which is negative 3 so not continuous at x equals negative 1 alright lets test it at 1 well 1, this would be 3 3tan pi/4 would be 3 its going to be continuous at 1 but lets check |
| 62:32 | we do the limit x approaches 1 rom the minus side of f of x 3 tan pi/4, 3tan pi/4 is 1 see you get 3 limit as x approaches 1 from the plus side of x is just 3 your not done though, you also have to show that |
| 63:02 | f of 1 is 3 so its continuous at x equals 1, so you would say this is continuos at all reals except x=-1, or something like that |
| 63:30 | whatd you say f of 1 is 3 continuity you have to show the limit from the left plus the limit from the right equals the value of the function do bot parts, show that the limit exist and the two sides are the same and that equals the function its self alright we can skip it number 3, dont do that lets do number 5 |
| 64:02 | 5a heres a fun function list all values where it has a maximum lets see if i can do a reasonable approximation of this curve |
| 64:30 | something like that thats the graph of the derivative of x the derivative of f sorry |
| 65:01 | thats the derivative of f prime what is the maximum that maximum means the graph is going up and the graph is going down so if the graph is going up the derivative would be positive, the graph is going down the derivative would be negative and at the to[ you get 0 so we have a 0 right here positive above that negative below that, so we have a maximum at 0 here we have a 0 |
| 65:30 | but were negative to the left positive to the right so this will be a minimum not a maximum so the only place is at x equals 0 thats it just that one can i explain that one more time sure okay a maximum if oyu think about the maximum on a graph |
| 66:06 | whats going on right there, well to the left of it the slope is positive, the curve is going up, so you have a positive slope right here the slope is 0 and now we get negative slope so if we look at our graph of our derivative |
| 66:31 | we know that here its positive, were above the x axis we get a zero then we are below the x axis so that would be a maximum as oppose to here where at first were below the axis a negative value and then 0 and above the x axis so this would be a minimum because a minimum you are going down and you have 0 and your going up to the negative and you have a 0 and a poitive this one, good |
| 67:02 | one more time, what are you missing the originally graph has to go up and then down to be a maximum the maximum value you reached the highest value for some zone its called a local max were going up till we get 0 were going down that means we have a slope on the left thats positive the slope on the right gets negative |
| 67:30 | and right in the middle has to be 0 so if we look at the derivative graph positive meets above the x axis so positive then were 0 and then were negative |
| 68:01 | if you want a maximum or a minimum you have to cross the x axis okay it has to have a 0, the derivative graph has to have a 0 the original graph well lets see here only positive so this will be getting steeper so your graph would be sort of going up and then it will be getting shallower like that and then here it would be the maximum and then it would turn arund and come down, thats what this piece would look like |
| 68:35 | but im not really going to do that, thats painful if we go back and ask you to graph the orginal derivative, i dont think we would do that, we might i dont know, i didnt write the exam exam is really nice, well written, printed |
| 69:01 | we are giving you guys a review session we have to save some stuff heres something we can do number 8 no wrong test, hang on lets go to spring of 09 although February is not exactly the spring heres a fun one |
| 69:35 | sketch the graph of a function that does all that kind of stuff |
| 72:30 | we had enough should i start drawing this?
alright we got f of 0 equals 0 pick up a point f of 7 equals 11 |
| 73:01 | so 7, 11 okay so far so good and you should sort of cross these off as you are doing the, look at the bottom we got f of 1 is 3 and f of 2 is 3 alright those are the easy ones now lets do some limits the limit as x apporaches 7 from the |
| 73:30 | minus side is 3 but when we get it from the plus side its negative 3 so put a circle there and lets see thats something, when we get it from the negative side but from the plus side youre doing something like that now the limit as x goes to infinity is 0 so maybe that |
| 74:00 | theres lots of ways you can get to 0 you can go straight up like this you can dance for a bit doesnt really matter now we get from 2 to the plus side we go up to infinity but notice at 2 were at infinity and 2 from the minus side were at negative infinity |
| 74:38 | and as f apporahces 1 from the plus side is that right, that piece doesnt look right make sure i get all that correct 2 from the plus side up to positive infinity 2 from the minus side |
| 75:00 | negative infinity something like that so on the left side of 2 we are going down to negative infinity and the right side of 2 we are going up to positive infinity im sorry the right side of 1 on the left side of 1 were at 5 and we have to go through the origin and to negative infinity |
| 75:31 | lets check all our conditions again f of 0 is 0 f of 7 is 11, when were at 7 were at 11 when we approach 7 from the minus side we get to 3 when we approach 7 from the plus side thats this way we get to -3 as we got to infinfity we get to 0, so far so good now when we approach 2 form the plus side, over here |
| 76:00 | were going up to positive infinity when we approach two from the left side were going to negative infinity so thats a vertical asymptote except we actuallly have a value at 2 so its kind of weird, you dont actually have to draw the dotted line if you dont want to because its not actually correct alright when we go up from 1, from the right side we go from positive infinifty 1 from the left side we have to go ot 5 then we have to go down, as we go down we go to negative infinity |
| 76:31 | that hurt no you dont connect the points howd we like that one oh i went through the origin because f of 0 is 0 you do need to go through that at the origin correct lets see if therees another one like thi |
| 77:04 | february 07, we might have to do one of those october 08, do we have to do one of those take a picture it last longer february 07 |
| 77:31 | no okay so that was one like tht see if theres other good stuff to go through how are we on the e;s heres a good one |
| 78:03 | limit as x approaches 0 plus this is which test this is the february 09 the spring |
| 78:39 | what si the limit as x approaches 0 plus e to the negative 1/x if youre not sure what e to the -1 looks like for now put it in your calculator and take a picture of it sor of plant taht in your brain e to the x e to the negative x |
| 79:01 | e to the 1/x, you should know what those graphs look like so what happens to the graph og e to the -1/x when x approaches 0 from the positive side well 1/0 would be infinite 1 over positive number youd get positive but this is 1/x so this would look like e to the negative infinity e to the negative something is something over e positive infinity |
| 79:31 | that would be 0 because e to the infinity is a really big number so its 0 1/infinity is 0 so far so good as 0 plus means are denominator is a positive number approaching 0 so were getting infinity but this is negative 1/x so the whole thing becoes negative so you get negative infinity |
| 80:02 | okay because you get negative 1 over a positive number e to the negative infinity is 1 over e to the positive infinity, e to the positive infinity is positive infinity right thats our e graph so this is 0 how bout the limit when x approaches 0 minus e to the -1/x |
| 80:30 | well now this becomes positive -1/x so this would look like e to the positive infinity which is just infinity |
| 82:03 | who has questions for me well do one more well do one more question and then well put an end to this pain lets go to the spring of 09 one more time |
| 82:32 | explain why the function is continuous at 3 or not ill give you a clue its either continuous or its not |
| 84:02 | explain whether the function is continuous or not well lets do it, f of x equals x squared minus 3x over x squared minus 9 when x is not 3 and 21 when x is 3 |
| 84:36 | well lets see were going to have to do the limit arent we so the limit when x goes to 3 x squared minus 3x over x squared minus 9 0/0 that means there must be a x-3 factor in these |
| 85:00 | we can pull an x minus 3 out of the top and the bottom look at that, now the x minus 3's cancel and so the limit is x approaches 3 is now of x plus 3 which is 3/6 or a half |
| 85:36 | but f of 3 is 21 so f of 3 does not equal the limit of x approaches 3 of f of x therefore therefore not ontinuous at x equals 3, its continuous everywhere else |
| 86:00 | do that again so the limit you get a half but f of 3 is 21 which means you have a little whole a a little dot at 21 there not equal itsnot continuous at x equals 3 good alright i think i had enough for one day |