The Fundamental Theorem of Calculus

Start   The first homework assigment was initially due half an hour ago, but it got extended until saturday morning very early.
you have a few more days so if you already did it, fine.
remember that you get extra credit if you do it two days before its due so for the problems you've done so far you'll get extra credit.
thursday, friday--- in fact you need to do them this afternoon to get extra credit.
0:33there were, there are a couple people well i dont know there were some people that had problems with the homework assignment.
i want to cover a couple more things that you should know already.
But.. it seems there are several people that don't know them.
Where we left off last time--
1:01are there any questions about last time? no?
So where we left off last time: we want to find the area under some curve.
between say here here and there. What we do is chop this up into a bunch of pieces,
1:32say n pieces, say i'll go from a to b perhaps y=f(x) and then a little piece I choose some point lets say a point on the right I might call this point x1, might call this point x2, x3, this last point is xn.
2:02And then you just make little rectangles of that height whatever the appropriate height is.
these heights are negative.
and i add them up.
so that would be written as, well the width of each rectangle this big and... the height of each rectangle
2:30is the function value there.
and i just add them up.
well for those of you that forgot this so this is the width, the height.
This notation means let me factor out the b-a x1..
3:00x2.. right. So this gives me an approximation of the area.
now i take many many many many rectangles the width is going to be like 0 so i take the limit of all of the rectangles that i have.
and get lots of really skinny rectangles
3:30-- and i have a limit here -- and this is the area.
We have to prove that this limit exists, I'm not proving this now because supposedly you did this in your last class.
This is defined to be the integral from a to b of f(x) this limit.
so i do this little job where i add up all of these areas.
4:03and then i take the limit as the number goes to infinity right?
is this new to anybody?
because i got a number of questions about this.
4:32so thats the definition of the integral.
of course in practice you rarely compute it that way.
unless youre using the computer or you dont know what else to do.
lets take a specific example so lets take the function
5:00y=ln(x) and i want to write a limit which represents the integral from
5:301 to 3 of this.
ok? so this would be so im going to-this will be a clicker question so im going to give you some choices.
you're going to tell me which one's right and i'll tell you whether you're right or not.
lets say the limit as n goes to infinity
6:01of the sum of (2/n)ln(1+x) infinity
6:30from 2/n... ln(2i/n)
7:41okay so time is up.
everybody got their answer in?
okay stop then.
ok so uhh very few people like answer one thats good.
because answer 1 is garbage
8:00and very few people like answer D.
which is also good because that's garbage too.
and so uhh 35% of people think its B and the rest think its...51% so about half of you think this is the answer.
and about 1/3 of you think this is the answer.
and the rest like A and D.
so lets uhh i guess do this problem, these 2 are very similar,
8:30except this one has a 1 here, and this one doesnt.
they're all from i to n so we take the limit as n goes to infinity, so in some sense its from i to infinity, but if you put it from i to inifnity then we're adding up big numbers that we dont want that.
we have to have the width shrink
9:00and the heights vary so thats why we have to go to n and stop.
but then we take the limit as the number of slices we take goes to infinity.
so in terms of pictures its kind of like this but this picture is not the right function.
except as i take n bigger and bigger and bigger these rectangles get really skinny.
so my picture would look like picture looks like this.
and so when n is really large it could be thin these are rectangles
9:32from 1 to 2.
ok so let me draw slightly bigger rectangles so here is a typical rectangle here.
from xi and this formula that is written right on the board tells me-oops im going from 1 to 3 sorry
10:01at some point my width is always (b-a)/n well all of these have a 2/n so thats cool.
2/n is fine because its b-a/n and then we have to figure out -- well how tall is this rectangle?
so we have to think we're going to chop up the regions from 1 to 3 into a bunch of little pieces.
and we're going to add them up. how wide is any given piece?
10:352/n so the width here is 2/n of each rectangle if i move over n spaces, so for the first one move over 1 space how far have i gone?
when i start at 1 and i move over 1 step.
about 2/n.
so x1 is 1+2/n
11:04because i start here at 1 and i take a step size 2/n here i am.
and x2 is wherever x1 was plus one more rectangle which is 1+2/n plus 2/n more which is 1+ 2(2/n).
and x3 is 1+3(2/n).
11:36and x22 is 1+22(2/n).
so in general xi is 1+i(2/n).
and so if i go over here to xi sitting here at 1+i(2/n). and the height is f of that value
12:00so that means that i want f of 1+(2/n)(i).
but f was the y value and then i want to take the limit.
so this crazy looking notation is just a way for us to capture this.
we just have to be a little careful about chopping it up there's a problem on the homework that i got about-yeah go ahead.
12:33its a number between 1 and n so here i is 1 and then its 2 and then its 3 and then its 4 ...blah blah blah blah blah... and then its n.
so its representing which rectangle.
here i is 1, here i is 2, here i is 3.
since i dont know how many rectangles i have i just have n.
maybe n is 100 maybe n is a million maybe n is 5.
13:02i want to write the general rectangle.
one of those dots, so i have to have it end on i.
that make sense?
now these limits are a pain in the neck to calculate so we're not going to do this now.
sorry... what was i saying?
its also possible to go the other way.
you see a limit like this and you say "what area does this represent?"
13:34that answer that question doesnt actually have a single answer.
let me turn the question around.
suppose i have-- im not going to ask this as a clicker question-- uhh n goes to infinity
14:001...n of lets actually do the same question.
so i have this limit now we already know that this equals the integral from 1 to 3 of the log
14:31of x dx cause we just did the problem the other way around we already know this.
but it also equals the integral from say 0 to 2 of the log 1+x dx its the same.
all im doing here is shifting my picture by 2.
in this picture im thinking
15:00OK heres the log of x and im going from here to here and i want that area.
and this guy instead i have a different function thats moved over by 1.
the log of 1+x this is the log of x and instead im starting at 0 and going out to 2 those are the same area just shifted over a little bit
15:30my notion of what x is has changed but there the same and i can write infinitely many different variations on this piece but theyre all the same thing so if you have a question write the integral which this represents thats not a reasonable question thats like saying write the name of a student in this class there are lots of different names in this class theres lots of different integrals that this represents
16:03they all have the same value but they vary umm i dont want to nor do i have time to spend a lot more time on this this is supposedly stuff you know so i want to move on as i said-so any questions on this?
so as i said doing these limits explicitly is tedious
16:30and nasty so we dont usually do it just like in umm in uhh.. just like in.. taking derivatives we dont usually take derivatives by the limit we learn some rules and then we use those rules the way we go but to get those rules sometimes we have to keep going back to the definition so
17:01many of you, most of you, well i guess all of you probably already have seen something that turns this into umm something we can do a little easier soo so heres a theorem which says that if.. f prime..let me use f and g because sometimes
17:35if g prime of x equals f(x) so that means the derivative of g is f then.. the integral from a to b of.. f(x)dx is just g(b)-g(a)
18:05so in other words if g is an antiderivative of f then to do this integral you just evaluate the antiderivative at each end an easy example we know that
18:30umm lets say the derivative of x^2 is 2x we know the integral from lets say 1 to 3 of 2x dx is x^2 evaluated from 1 to 3 which is 9-1 which is 8
19:01this should not be news to anybody i hope anyone for whom this is news?
uh oh maybe you dont belong in this class maybe you do maybe youre a smart guy umm so so so this this is actually half of something called the fundamental theorem of calculus
19:35theres another half which i'll write in a minute the reason its a fundamental theorem or the fundamental theorem is it inter-relates 2 very big things that you do in calculus it says that to do integrals you have to know about derivatives you dont have to knowing about derivatives is what you know about integrals now if you remember what i said last time
20:01that uhh the theme in calculus is that if we understand microscopic detail then we understand macroscopic or full sized behavior this is again a version of that because the derivative here is microscopic information it says how is the function changing on a very small scale we can relate that to how the function changes on the big scale
20:35and 1 thing that we do in this class we prove stuff well we dont, i do why would this be true so thats what it means in math is to give a solid explanation to prove something you give a solid explanation why its true
21:02im going to give a mostly solid explanation it may be solid to you uhh im going to gloss over a number of the details that are mathematically subtle but why would this be true?
well we can do it in a fuzzy way umm well maybe the fuzzy way is no help does anyone have a clue why this is true?
21:30really nobody? oh you do okay why?
well thats what the theorem says, integrals and derivatives are opposite but why so yes its true because integrals and derivatives are opposite but its like asking why is the sky blue?
well its because its this many angstroms i mean i dont know anyone know how many angstroms represents the wavelength of blue?
22:00the sky is blue because the light that comes from the sky is blue thats not an answer...i mean its an answer but it doesnt tell you why it just kind of tells you it is what it is so okay so has anyone seen a proof like this?
okay you just forgot thats okay 1 whole person in this room has seen a proof like this in their life?
2 people okay theres a few okay so why is this true well lets think about whats going on
22:32with this integral my function chop it up into a bunch of rectangles and umm i evaluate well lets just give them names x1 x2 blah blah blah xn so let me not take the limit let me just do it at n so we know that
23:01this area well lets start with the right i'll start with the right side so g this is b and this is a and so g(b) minus g(a) i can write it a different way this is g(xn) and im gonna write
23:30g(x1) way over here and then just for fun im going to subtract g(x) to the one before and add it back so this is 0 so nothing changed and then im going to do it again and im going to leave dots
24:02and then im going to get down to g(x2)+g(x2) and then uh g(x1)+g(x1) so im not going to change here just did that and so those are certainly equal
24:32lets look at one of these little pieces and theres a theorem that you probably forgot called the mean value theorem anyone remember the mean value theorem?
some of you okay guess i'll do that back over here
25:06okay so one of you who remembers the mean value theorem do you remember enough to tell me what it says?
yeah okay you know it but you dont so the mean value theorem which you went over in your calculus class cause its important but it seems stupid says that if i have
25:33a nice function i have a differential function uhh and this is well now its g but doesnt matter but lets call it so if g is a differential function and it goes on some interval then if i take 2 points uhh lets call it xi
26:01and xi+1 so those are 2 points then the slope from here to here the slope of the line through g(xi) so xi
26:30g(xi) and the next guy those are my 2 points usually you call them a and b but i already used a and b the slope of the line through there is the same as the derivative for some point inbetween
27:02so theres some place at least 1 place this place the slope is the same another way to say this youre driving your car on the new jersey turnpike they hand you a ticket when you get off you go 75 miles and an hour later they hand you a ticket when you get on and they say you were speeding because you traveled 75 miles in one hour you must have been atleast going 75 mph sometimes
27:34the slope, the speed is the same as the average speeds here we dont know when it happened but we know it happened sometime maybe you only speeded once okay so thats what that says now why is that useful here?
thats useful here so that means
28:03g prime let me call it c i because it sits between them so that means that g prime the ci is the same as g so thats the same as that distance right?
28:30this is the slope of the line and theres the derivative so the slope of the line there is the same as the average of the two ends ok just for the hell of it lets call this thing h so that means that h times g prime of c i
29:00is this difference now lets look at this ive done a whole bunch of things so i have a whole bunch of these differences that i can just replace with derivatives
29:31this guy since im going to take the limit this guy is g prime of c n times some number h where h is the distance and this guy is g prime.. anyway notice that if i divide everything by h
30:06okay so i subtract i add and subtract and subtract and subtract this is what i started with so now what do i got here?
almost got what i want
30:31because now i have a bunch of derivatives sitting there uhh im trying to turn this into an integral so so what is this?
so i guess im going to write it over there sorry i keep coming back and forth so that means that ive just shown that g(b)-g(a)
31:00is going to be well that first thing over there h[g'(cn)] and the second thing h[g'(cn-1)] and the next one h... g prime c of the next one and so on
31:36so i have a sum of derivatives like this well okay now im going to take the limit as h goes to 0 h is the width of the little rectangle when i take the limit as h goes to 0 this doesnt change
32:00but as h goes to 0 the number of terms goes to infinity so this is the same as n going to infinity so what is this?
this is exactly something that looks like so h is the width of the rectangle and g prime is the height with a different function
32:35and im adding them up then i take the limit as width goes to 0 and the number goes to infinity and this is the integral its an x
33:03in other words whats going on in terms of the picture in terms of the picture got this thing and i replace each of these guys with the slope of some place inbetween and then i let them shrink to 0 so im adding up more and more and more slopes
33:34and i get the area so its fact so this tells me that i can just do antiderivatives to do n now the other way around of this
34:01the other part of the fundamental theorem says that if i have an integral now this is a little bit more brain hurty i suppose suppose that instead of just having instead of representing areas definite areas i want to think of functions defined by integrals so if i have some function
34:30areas and i start somewhere here i want every, every place i stop like here i get an area right so if i move a little further i get more area so this is actually a function i can define my function to be the area
35:23i can define some new function which is the area underneath the curve i go a certain distance t and then i stop
35:33so an example of this we know from physics that the derivative of position is velocity the derivative of velocity is acceleration you could think of this is being the graph of velocity and i want to know the position if i drive my car at 60 mph for 1 hour i know that the position is 60 miles away i drive my car
36:00at 60 mph for 1 hour and 10 minutes its 66 miles away and so on the same business here define a new function as the integral of the old one so this is the integral from where i start of my old function if im going to define the function
36:31the variable here is not x the variable is t where i stopped not what i did inbetween now this fundamental theorem tells me that i can take the derivative of some function
37:09the second part of the fundamental theorem says that if g(t) is the integral from anywhere to t of f(t)dt then the derivative is just..
37:32the function that i started with and these two together tell me that derivatives and integrals are inverses of one another in terms of a picture this is telling me that the rate of change of the area is just the derivative i mean just the function like i want to look at area i move a little bit then the amount that this changed the area here heres my area
38:01and i took some area and now i move the end a little bit how much did this move?
well just the height of the function times how much i moved it the width of this and just the height of the function times how much it moved so those two things are related
38:31this one is a little more subtle but its also extremely important lets uh
39:15so this should be easy A needs to be
39:32ln(x) can i put D up here
40:13theres our choices so 70% of you got the right answer and 30% of you were fooled
40:32so this is not right this is not right this is not right also this is not right when you take the derivative the end point dies so its just the log okay lets uh change the problem a little bit and its almost the same thing
41:01soo not g because i already used g, h oh i meant these are x's i always do that so h prime of t is..
41:32A)ln(t)-ln(3) B) ln(t^2) c) 2tln(t^2) D) ln(t^2)-3
42:29okay so
42:3173% of you got it right and 26 did not so these are all stupid answers thats a stupid answer we're down to these 2 why is it this one?
right we have to use the chain rule so h(t) is the same as g(t^2)
43:00and if we're taking the derivative of g(t^2) we need the chain rule so h'(t) is.. g'(t^2) times the derivative of t^2 this is the chain rule we need to use this ok okay so that should be pretty straightforward what day is this? its wednesday right?
uhh lets see what else do i have to cover
43:32so maybe rather than doing a jillion examples like this let me move on to.. so i know i didnt do a ton of examples but thats what recitations are for
44:03let me point out one last thing that you were supposed to know before you came to this class and thats how to do easy integrals so of course we can write integrals either as definite integrals or we can define integrals
44:32so as antiderivatives um say i.. want to do well let me do this one first suppose i want to do the integral from 1 to 3 say of uhh x^2dx but using the fundamental theorem tells me that i just need to find the function
45:01whose derivative is x^2 a function whose derivative is x^2 well the power here is 2 when i take the derivative the power decreases by 1 so i would guess that i need a third power but when i take the derivative of the third power i get an extra factor of 3 that i dont like here theres no 3 sitting there so i just divide everything by 3
45:31and its happy times so this guy is just going to be uhh oops what did i do wrong no thats right so that means that this is x^3/3 and i plug in the values from 1 to 3 so this is 27/3-1/3 which is 26/3
46:05anybody confused by that at all/ okay suppose though i have something that looks slightly different say i have integral from 0 to 2 of (x+1)^2 dx now i dont know off the top of my head
46:34the function whose derivative is (x+1)^2 but it looks a lot like x^2 so what i do here is i say gee i wish this x plus 1 were really an x well maybe instead of x lets call it u and then
47:03differential here when i change x a little bit u changes by the same amount and so that means that this becomes and so also when x is 0 u is 1 and when x is 2 u is 3 so in terms of u this becomes the integral i just did
47:37nope from 1 to 3 so x goes from 0 to 2 when x is 0 u is 1 because u is x+1 and when x is 2 u is 3 because u is x+1 so when i write everything in terms of u this is u=1, u=3
48:02x=0, x=2 i want to emphasize, ill say more about this next time its important to write this dx and this du these are akin to using the units in science dx is like meters and du is like feet now here i just shifted my feet by 1 so theyre the same
48:30but in other examples which i dont have time to do today that i will do on friday, dx will vary