Stony Brook MAT 125 Spring 2015
Lecture 02: Tangents, velocity, and limits
February 4, 2015

Start   remmeber what i told you last time the idea of calculus is, you know we were trying to figure out how to figure out the slope of curves you know how to do the slope of a line you go over a certain amount and you go up and down a certain amount and you just it doesnt change at any point on the line its always the same slope but with a curve theres different slopes, so how do we figure that out?
so one of the things that were going on back back in the day a long time ago was
0:30to figure out how we can come up with the slope of the curve so this is the first of the velocity problem you have something thats falling or moving and you want to try and figure out its velocity at a particular time so we know that so lets say you have something
1:00alright so we know this is the motion of something and then and then there, it has total different velocity's at different points so how can you figure out whats going on?
the velocity will be the slope of the curve at any spot okay if this is your location verses time so x verses t so how can i possibility figure that out?
so lets take some numbers, lets say we know
1:33that t=0 t use it as examplez slightly different the t=5 the t=6 what happened? well we know that your velocity in doing that interval
2:01was 54 meters per second but from t=5 and t=5.1 lets say it was a different number 51.1 meters per second and for t=5 and t=5.01 equals 51.001
2:32meters per second and when t=5 im making these numbers up so how would we come up with numbers like this how would we come up with velocity, what you would do is take the difference of your location dived by the difference in time you did this in physics this would be where are ypu at time 6 minus where are you at time 5 divided by 6-5
3:00seconds, so the change in distance over the change in time we are changing position over the change in time and same here you would say your position at time 5.1 minus your position at time 5 over 5.1 minus 5 so your time intervals have shrunk and you say see i wasnt going closer to 54 eters per second, i was going closer to 51 meters per second
3:31now lets shrink the time interval even narrower so 5.01 minus 5 notice whats going on in the denominator the denominator here is .01 thats a really small number the last one would be the position
4:02you get .001 in the denominator thats really close to 0 and if you think of 100 of years youll be able to work this out very hard to do because the numbers are very very bulky very unreal but each time you do a change in y over the change of x the change of position over the change of time notice the time gets closer so notice heres the question if i want to know whats going on exactly 5 seconds then the problem is gonna be 5-5
4:31thats 0 thats a problem you cant divide by 0 you get undefined so this is the question that people were trying to figure out if you think about it victorially lets say thats your thats your 5 second mark you want to know whats your slope exactly there so first you found your slope say here and then
5:00we are taking accessibily closer and closer lines im going erase this now and blow this up you have a curve and the curve is sorta doing this and thats 6 the slope kind of looks like that yea but if i say st 5.1
5:30now now the slope kind of looks like that so you get a better understanding of what the slope is and 5.01 or here on the curve the slope looks like that so were getting the tangent line closer and closer so what are the tangent lines telling you the tangent line is a tangent line that just touches the curve in one spot
6:02if you have a line that touches in 2 spots thats a secant line you might mistake this for trigonometry, 1/cosine heres the relationship theres a reason we call both of those secant if we only touch the curve in one spot, lets say right there then its called a tangent line so what we mean by the slope of a curve is what is the slope of the tangent line in exactly that spot
6:34the problem is exactly the spot remmeber you find slope, the difference in y and difference of x but x is 0 so this is the problem that people like newton were messing with and they came up with an answers, took a while but they did and heres how you do it and you do it with limits, did you watch professor sutherlands 16 minute viedoe?
who did not watch?
7:00okay make sure you watch okay don worry i wont write your names down make sure you watch, he kind of explains what it is explains what a limit is and he shows you how to do a couple limits, were gonna do a bunch of them in the next how ever many non snow days we have for this class i know youre rooting for a snow day my shoulder does not believe in snow days remember this is how you get an a folks just remmeber those shovels im not gonna tell you where i live however
7:31im not that dumb okay so if you look at the book at page 93, you dont have to have the book with you they draw a much better picture then i do the difference of the secant line and tangent line thats terrible but agian the secant line crosses in 2 spots and the tangent line
8:00is in one spot and the idea is as you shrink these two closer and close where youre trying to find your tangent line this line will approach that line and we do that through our limit process so you can do that, you can literally just take numbers and put them in your calculator and come up with an answer and you can get very close because we have a couple of problems like that, which you could do in the book but essentially you take an equation
8:32and you just start plugging in numbers until you see say we know that your location and time t is 3t squared plus 1 and you want to know
9:01what is your velocity?
at exactly t=2 you dont know, the problem is thats a parabula its a curve its not a straight line you say well what happens between time 1 the time t=1 were lcoated at you plug in 1 and you get 4
9:30time t=2 you plug in 2 and yu get 13 so you can say well my velocity is the difference in y and difference in x, the difference position over the difference in time 13-4 over 2-1 4.5 meters per second
10:01assuming x is in meters and t is in seconds you say okay what about between t=2 and t=3 well now youre at position 28 can you guys read that? you have a question?
oh yes i apologize this is why this is a raw take
10:349 meters per second thats better now what about between at time 3 youre at location 28 your velocity would be see between time 2 and time 3 you had 15 meters per second see 9 meters wasnt so good because the next second youre going 15 meters per second
11:01you can average them and come up with 12 right but you dont know if thats a good guess or not cause something could be going on in the middle you say welll now lets take smaller intervals lets figure out whats going on at times 2.1 2.1 youre location is
11:42at 14.23 so your velocity would be the location at 2.1 seconds minus your location at 2 seconds over 2.1.-2
12:02thats 14.23 minus 13 over .1 which is 12.3 meters per second
12:37what if we do this about 2.01 seconds well now your location you take 2.01 and plug it in there
13:02which is 13.1203 your velocity would be where would you be at 2.01 seconds minus where are you at minus 2 seconds ove 2.01 minus 2 thats 13.1203 minus 12 over .01
13:30which is 12.03 meters per second you get closer and closer its starting to look like 12 but the answer could be not exactly 12 you know its hard to know but as we shrink the time of what we are doing is were shrinking the secant line as we are getting closer to the tangent line that means we are getting closer and closer to the actually to the actual speed, well you can say youll never know the actual speed
14:02because youll never get exactly a 12 abd this is something that the people use to wrestle with they use to say youll never actually get that to 0 so calculus is all about figuring out how to do that this is something that people thought about for thousand of years the example for example i want to get to the wall but in order to get to the wall i have to get 1/2 way to the wall and 1/2 way to the wall i have to get 1/2 of that distance and 1/2 of that distance so i always have a little bit left
14:30so it goes on and some point you bang into the wall right doesnt make any since but if you think about it logically theres always a tiny bit left over so with the invetion of limits it helps us deal with that tiny bit of limit and trick that to 0 and that will help you find the actual speed at the time so for example here were gonna learn to shrink that to 0 and thats what he talks about, professor stewart talks about in the first part of the book
15:08so far so good? you guys must have questions yes thats a 13 but you knew that cause it still comes out 12.03 when you see that on really polished videos you know
15:31you realize that must of had more then 1 take it is really difficult to do it without a calculator were not gonna make you guys do it without a calculator this is the hard part you say well wait a second you cant do 2.0001 seconds with some craz function with es and sines all that stuff there must be another way through this thats what mat125 is all about i know thats very exciting and youreally
16:02you were wondeing when youd get to the good stuff, this is the good stuf alright so we will doing that but what is called limits and limits is figured outwhen you trig that all the way im gonna erase all this so people got the idea?
this is the movating part of the book
16:38this is how youll figure out the slope of a tangent line by finding the slope of the secant line trying to shrink that secant line down untill its the tangent line okay so what you are trying to do and dont get to hung up on this stuff okay what you are trying to do intellectually is trying to figure out what happened when the denominator is 0
17:02really what this is about is this easier to see tough, matt is this showing up betteR?
yes alright good can i erase this master piece now
17:40alright so what professor sutherland put on his 15 minute video that most of you but but not all of you watched maybe we should of made that like a super bowl commercial then all of you wouldve watched it like a puppy like a puppy of like a kardashian or maybe both maybe thats why some of you didnt watch it i guess
18:06he was talking about limits, limits is the idea behind a limit is how you get behind things that get that gett very very close to a number but not actually the number suppose yo wanted to evalute noe you write this
18:31in this notation that is the limit as x approaches a of f of x that means what happens to f of x as x gets really close to a sometimes you contribute, you have a graph
19:04you have the graph y= x squared, f of x equa;s x squared what ahppens to f of x as it gets really close to 3, to 9 nothing special going on there when you plug in 3 3 squared is nine when you plug in 3 times 3 times 3 and you square it you get really closer to 9 you plug in 3.001 and you square it youll get really closer to 9, so you get close to 9 nothing fancy is going on so you get a basic kind of limit
19:31so the first thing you want to do when you evaulate a limit is you just want to plug in a number notice does anything happen you say well here you would find the example the limit as x approaches 3 of x squared and its just 3 squared thats not very hard well of course were gonna make it harder you ccould just stop there and be really happy
20:02nut suppose instead you have a whole in the graph at the place 3,9 theres no value theres a whole there there isnt a value at 3,9 theres a value at 2.99999 theres a value at 3.0001 but theres no value at exactly 3 so how do we find the limit
20:35the answer with the limit we dont care if the value is exactly at 3 when you plug in the limit we arent actually plugging in 3 we are plugging in a number very close, infinity close to 33 as close as 3 as you want but not exactly 3 so we dont care if theres a whole at 3 because when youre at 2.99999 youre gonna be very close to 9
21:00take 2.9999 and square it in the calculator and see what happens take 3.0001 you get really close to 9 as you get closer and closer to 3 youre gonna get closer and closer to 9 you dont care that theres a whole there the limit is still gonna be 9 so if you look at this function the limit of x approaches 3 of x that equals 9 even though
21:35there is no limit at f of 3 we would write it with the famous expression of DNE in which it mesans the limit does not exist
22:03you can have a limit to the function even though you dont have a value to the funtion you can get very close you dont actually have to have a value there suppose you have a function that looks like that what happen when x gets really close to 3
22:32you say what do you mean by really close to 3 something funny is going on at 3 we write these limits and talk about one sided limits the left side and the minus side or the right side and the plus side notice how i write a little minus sign in the super script for 3 limit approaches 3 minus, what that mean is what your doing is staying as close as possible but always staying less then 3
23:01so 2.999999 etc if you imagine your on the axis your getting close to 3 cause your staying on this side so when you get very close to 3 but you get less then 3 f of x is equal to 2 so 3 plus you can guess thats the right side limit if you look at the number line your getting very close to three but your going from this side so if your getting as close as possible to 3 from this side
23:33your gonna be on the curve but youre gonna be up her at 4 cause you imagine coming down this way you dont actually get to 3 you get 3.000001 something like that just bigger then 3 so now that really equals to 4 so this is an example of the left side limit verse the right side limit, the lefdt side is 2 the right side limit is 4, they are not the same number so thats when you say the limit does not exist
24:02when the left side, one of these examples when the left side is not equal to the right side theres no limit at that spot theres a left side limit theres a right side limit but there isnt a limit at the number its self so you would say as the limit approaches 3 minus is 2 the limit as it reaches 3 plus is 4 but the limit that approaches 3 with no plus or minus
24:32does not exist you see that on webassign, here on the other han when you get really close to 3 from the minus side thats a 3 if you cant tell you get 9 and when you get really close to 3 from the plus side you also get 9, since they agree you would say the limit as x approaches 3 is 9
25:06questions? no?
you guys a lot of you have seen limits before right?
the limit as x goes to 0 of sin pi/x the book again draws a really pretty picture
25:35its doing something like that so at 0 its going up and down the sin curves iff you imagine little sine curves they are getting squished on top of each other until at 0 its going up and down in fact you can test this if you go to your calculator and plug in x just a tiny bit positive and you change the number just a little bit
26:00so if you did sin of pi over oh lets pick a god number .001 oh wow thats not and then you did it over a slightly different number youll switch from positive to negative and thats isolating so sometimes a limit wont work because you dont know if its positive or negative you have no idea where the function is
26:33you can see you can draw and say i have no idea if its positive or negatie again the limit doesnt exist cause you cant find a way to squish the two limits next to eachother lets write another type of function
27:20theres a fun one this is called a heavy side function dont really know why
27:31this says that h of t is 0 is you plug in a negative number and 1 if you plug in a non negative number 0 or a positive number so what happens to 0 well that deoends when you get really close together from the left side i get 0 cause notice 0 minus im plugging in a number less then 0
28:04now what if i take the limit and approach 0 from the other side i get 1 and theres no way to get those 2 to agree with eachother the closer i get to 0 and stay to the left of it im gonna get 0 the closer i stay to the right of it im gonna get 1 i cant get those 2 to agree so the limit at 0 does not exist
28:50so what were ginna do is put that in a graph so you can look at a graph or a function now were gonna actually calculate some limits
29:10question is if the limit does not exist does that mean that we cant calculate it or is there just nothing there when the limit does not exist it generally means is when you try to shrink your window arpund the x as narrow as possible you cannot shrink the y window
29:30to a narrow enough spot to know where it is to know where the limit is so as you are approaching x this way y is not behaving itself whether its jumping the way it did here or the way it did here because its isolating or because sometimes its going towards infinity, it hasnt gotten to infinite yet however if theres no value at the number there still can be a limit like the example before where you have a whole
30:01just because the function does not exist at a number does not mean there is not a limit at that number so you do the limit when you approach 5 you get an answer that does not mean the function has a value at 5 it just means you can find the value with just a little less then 5 the value a little more then 5 and that they agree with eachother
30:30is this because we dont love this we dont care or we dont understand a little bit of each no questions i get to move on
31:00so lets actually figure limits, so there the thing called limit laws were gonna go over this theres a bunch of these lets make sure you understand them
31:49heres something were gonna tell you thats true if you have the limit of two functions added together you can find the limit of the first function and add it to the limit of the second function so this is gonna be very handy for things like polynomials
32:05is you want to sound a little mathy the sum of the limits is the same as sum of the limts so if you wanted to find the limit of
32:30the limit as x approaches 3 of x squared plus x you already kinda know what to do but what were telling you is the limit as x approaches 3 of x squared add the limit as x approaches 3 of x now what do you think happens to x squared as the limit gets close to 3 you get 3 squared you get 9
33:00and what do you think happens to x as you get close to 3 you get 3 thats pretty straight forward so the whole thing is 12 and that the same you wouldve gotten if you plug in 3 right here you may think thats obvious but you have to understand thats a rule so if you have two functions added togethr you can find them seperatly and add them together what about subtraction
33:33same thing so if thats a minus sin you can put a minus sin
34:07the limit as x approaches a of some constant of x is the constant of x times the limit as x approaches as it approaches a of f of x so what does that mean?
34:31you want to find the limit as x approaches 3 of 5x squared you can say thats the same thing as 5 times the limit as x approaches 3 of x squared the limit as x approaches 3 is just 9 i know some of you who have not seen this before are sitting here saying and i missing something or is this kin dof obvious youre worried
35:03that theres something not obvious here that youre missing no this is pretty obvious it gets more complicated later were just laying down the rule because later you can say yes thats allowed or no thats not allowed alright what do you think happens if you multiply them together
35:34take a guess you multiply the limits
36:00that says that uf you want to find the lim of f of x times g of x you can fin the limit of f of x and multiply it by the limit of g of x so again very straight forward and what about division ?
sure same thing
36:53if you want to find the limit of the function divided by another function you can find the limit of the top function and divide it by the bottom funtion
37:03providing that that kind of function is 0 cause if its 0 then you have a problem
37:30lets do something nice and simple this is very straight forward just plug in 2 youre not gonna have a problem you can look at this and say tp yourself i dont understand this doesnt look very hard i plug in 2 and on top i got 2 and on the bottom i got negative 1 negative 2 is that right?
38:03okay no mistakes, i dont want to do something wrong so thats not a problem right? where potentially would i have a problem?
when x is 3 thats when things would get complicated, alright so what would i do when x approaches 3 you go well now i have a problem cause you plug 3 on top
38:33i need to get 9 minus 21 is minus 12 plus 12 is 0 i plug in 3 on the bottom i get 0, 0/0 is bad remeber that back when you first learned fractions and your teacher said you cant divide by 0 now you can go back to 6th grade and tell your teacher youre wrong you can divide by 0 cause heres what you do
39:09what we could do we could say wel whats the problem her factor the top the top factors to x-3 times x-4
39:32i plug in 3 on top and thats why im getting 0 but if i plug in 3 on the bottom this is the problem now i really want to cancel the x-3 on the top and the x-3 on the bottom but im not allowed to do that 0/0 youre not allowed to just cancel 0 but remember when you do the limit as x goes to 3 x isnt actually 3 its infinite to 3 but its not actually 3 so these arent actually 0
40:02theyre very tiny numbers they might as well be 0 but they are not 0 which means we can cancel them because they would both come out .0000000 something 1 not actually 0 so we cancel now when we plug in 3 i get negative 1 this is the currecial thing that we do with limits
40:47so thats how you would evaluate that so if you went to your calculator and you plug this equation in and you plugged in 2.99999 you would get really close to negative 1
41:02if youre close to 3 your calculator would go but if you plugged in 2.99999 or 3.00001 youd get an answer because its evaulating it but not exactly 3 so with a limit what we do is say these are the problems we can get rid of those how did i know we can factor that by the way when you have one of these polynomials over polynomails and if you have x approaches a number whatever that number is
41:31and you get 0/0 that means you have a factor of that number so another words lets say its limit as x approaches 10 you have a polynomial on the top and a polynomial on the bottom plug in 10 you get 0/0 that mens you have a factor 10 in the top and a factor of x-10 on the bottom youre gonna be able to take out a x minus 10 from both parts okay thats your clue very important remember that
42:01okay lets do another one what did i say was the clue notice i said when x approaches 5 plug 5 in the top wjat happens you get 0 you plug 5 in the bottom you get 0 that means im gonna be ale to factor out x-5 in the top and the bottom sure enough
42:36the top factors into x-5 x-1 the bottom factors in to x-5 and x+5 no you say aha now i see why i choose getting 0/0 cause when i put 5 on the top i get 0 when i plug 5 in the bottom i get 0 i can cancel theses terms remember i can cancel these terms because theyre not
43:01actually 0 so now when i plug in 5 i get 4 on top and 10 on the bottom so i get 4/1-0 2/5 or 24 if this was an exam question you could leave it as 4/10 we dont care about those things alright what about something like that notice when i plug in 0 in the top im gonna get 0 when i plug in 0 in the bottom im gonna get 0
43:30so theres gotta be a way to figure this one out so lets do this one so when you plug in thats a 4+h for those of you who cant read the perfect hand writing when you plug in 0 you get 4 squared thats 16 minus 16 over h thats annoying because you are getting 0/0 well lets multiply out the top then 4 plus h squared is 4 squared
44:00two times 4 plus h h squared you get that so far so good?l lets get rid of those 16 niw now the problem is when you plug in h as 0 you still get 0/0, now what?
44:30i dont know copy off the person next to you, factor we can factor as they say so you factor that now you look and say aha this is the problem remember when i told you if youre getting 0/0 that means theres a common factor that you can get rid of
45:01now we cancel the hs now when we plug in 0 you get 8 and if you put this in the calculator and you shrunk h down to a very tiny number and you calculated it you would get very close to 8 okay these are what you are going to see a lot of lets try another one
45:51that says the limit as x approaches 0 of 2plus x cubed minus 8 over x
46:05who remembers how to do 2+X cubed well if your not sure you can do 2+x +2+x and then when you are done you can multiply that by 2+x and you get 2 times 2 times 2 cubed plus 3 times 2 squared times x
46:30plus 3 times 2 times x squared plus x cubed you guys get that?
so that means im right?
now if oyu cant do that you have to work it out the hard way so now you cancel the 8s and you get the limit
47:03x approaches 0 of 12x plus 6x squared plus x cubed over x and you say okay i still have a problem when i plug in 0 i get 0 factor the x out of the top
47:3312 plus 6x plus x squared all over x and cancel the x's is that what we did?
yes or we were just waiting for me?
plug in 0 you get 12
48:04alright how bout one more thats really like annoying?
48:50you should of had a webassign problem that looks something liek this so lets find those if f of x is 3/x+4
49:01you want to find f of a and you want to find f of a plus h then you want to find f of a+h minus f of a all over h so much fun f of x is 3/x+4 well lets find f of x+$ so if you want to find f of a you literally plug in a thats not very hard webassign you put that one and you say yes
49:33now you do f of a plus h okay so if f of a is 3/a+4, then f of a is 3/a plus 4 the f of a plus h is 3/a+h plus 4 it doesnt matter what this is okay its f of whatever is 3/whatever plus 4
50:01doesnt matter what goes there so now i want to find f of a plus h minus f of a over h this is something refered to as the difference quation and if you think about it were gonna do more about this on monday but if we have clzass on moday this is the difference in the y's over the difference in the xs remember before we were doing the infinity velocity thing
50:31this is gonna be the same formula you do 3/a+h+4 minus 3/a+4 all over h the problem is you put that in webassign and you get a big red x you say i dont understand and you email me and you cry and you say what will work you can do more with that
51:01okay that can be simplified why would you want to simplify because were gonna make you thats why okay so how would we simplify that well we have two fractions on top we have 3/a+h+4 and 3/a+4 so if you want to combine fractions you want to have a common denominator so here the common denominator would be a+h+4 times a plus 4 so
51:31this is the left hand fraction multiply the top and bottom by a plus 4 take the right hand fraction multiply top and bottom by a+h+4 whole thing over h notice they now have a common denominator
52:06divide those into one fraction and it looks like that you all following me all i did was get a common denominator
52:30and combine two fractions together if i plug in 0 im still gonna get 0/0 youll see but im not plugging in 0 im simplifying this as much as i can so lets multiply the top and do some canceling so you get 3a plus 12 minus 3a minus 3h minus 12
53:01over a plus 4 times a plus h plus 4 all over h you say why cant i cancel this? i just put that in so if i cancel that i get back to where i started also if i cancel it you mean
53:30cancel these and cancel these thats subtraction i can take this a cancel this one fraction and keep the a plus 4 right this is a different fraction adn this is a plus 4 but then i end up back here and the whole this is to put it back together thats what people typically do in exam they go back to where they started and go i dont know what to do and when you grade it you have this huge mess so becareful with that now i cancel the 3as and the 12s
54:02and the only thing left on top is minus 3h over a plus 4 a plus h plus 4 the whole thing is over h so far so good?
notice this is actually h over 1 right so h is h over 1 so if invert the 5 and multiply i know get minus 3h
54:32over a+4 a plus h plus 4 times 1/h and those hs now cancel and i get negative 3 over a+4 a plus h plus 4 thats as good as its gonna get but the trick is were gonna ask you to do it its a typical exam question
55:02to get from here to here alright thats enough limits for one day so well see you on monday maybe