
English: 
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Okay so I'd like to begin the
second lecture by reminding you
what we did last time.
So last time, we
defined the derivative

French: 
Materyèl ou pral gade a se yon
materyèl nou ofri gras a
Lisans Kreyativite non-komèsyal
pou pataje
Sipò ou ap ede Klas Ouvè M.I.T.
kontinye ofri
bon resous edikasyon gratis pou
tout moun.
Pou bay yon kontribisyon pa w, oubyen
pou w gade lòt materyèl
nan plis pase yon santenn kou,
al vizite Klas Ouvè M.I.T.
nan sit entènèt: ocw.mit.edu
Tou dabò, pou dezyèm leson sa a
mwen ta renmen fè nou sonje
sa nou te fè nan dènye leson an.
Nan dènye leson an nou te defini
derive kòm

French: 
pant liy tanjant.
Sa se te aspè jeyometrik la.
Epi tou,
nou te fè kèk kalkil.
Nou te rive montre ki jan derive 1/x
se te -1/ x^2.
Epi tou nou te kalkile
derive x nan n-yèm puisans
lè n=1, 2, elatriye. Se te bay x,
eskize m, nx^(n-1).
Alò, se sa nou te fè nan dènye leson an.
Jodi a, mwen vle
fin gade kèk lòt aspè
sou sa yon derive ye.
Sa enpòtan anpil.
Se ka bagay pi enpòtan
mwen di nan leson sa a.

English: 
as the slope of a tangent line.
So that was our
geometric point of view
and we also did a
couple of computations.
We worked out that the
derivative of 1 / x was -1 /
x^2.
And we also computed the
derivative of x to the nth
power for n = 1, 2, etc.,
and that turned out to be x,
I'm sorry, nx^(n-1).
So that's what we did
last time, and today I
want to finish up with
other points of view
on what a derivative is.
So this is extremely
important, it's
almost the most important thing
I'll be saying in the class.

English: 
But you'll have to think about
it again when you start over
and start using calculus
in the real world.
So again we're talking
about what is a derivative
and this is just a
continuation of last time.
So, as I said last time,
we talked about geometric
interpretations, and today
what we're gonna talk about
is rate of change
as an interpretation
of the derivative.
So remember we drew graphs
of functions, y = f(x)
and we kept track of the change
in x and here the change in y,
let's say.

French: 
Men, n ap oblije refleshi sou sa
ankò lè nou rekòmanse
epi lè nou kòmanse sèvi ak kalkil diferansyèl
pou nou rezoud pwoblèm pratik nan lavi.
Alò, n ap kontinye pale sou
ki sa yon derive ye,
e sa se suit esplikasyon
nou te bay nan dènye leson an.
Nan dènye leson an, nou te pale
de entèpretasyonjeyometrik.
Jodi a nou pral pale de
to varyasyon kòm
entèpretasyon derive.
Sonje nou te trase graf
fonksyon y=f(x).
Epi nou te swiv varyasyon nan x,
epi bò isit la
nou te suiv varyasyon nan y.

French: 
Nou pral sèvi ak nouvo pèspektiv sa a
pou nou suiv,
to varyasyon nan x
ak to varyasyon nan y.
Se to varyasyon relatif sa a
ki enterese nou.
Sa se delta y / delta x
e sa gen yon lòt
entèpretasyon.
Sa se to varyasyon mwayenn lan.
Dabitid, nou ta panse kon sa:
Si x t ap mezire yon peryòd tan,
alò mwayenn lan lè sa a,
li tounen yon to,
e mwayenn lan kouvri
yon entèval tan delta x.
Lè sa a, pou limit la,
ou ekri dy/dx, e sa se
mwayenn to varyasyon an,
e sa se to ki
kalkile vitès enstantane a.

English: 
And then from this new point
of view a rate of change,
keeping track of the rate
of change of x and the rate
of change of y, it's the
relative rate of change
we're interested in, and that's
delta y / delta x and that
has another interpretation.
This is the average change.
Usually we would think of that,
if x were measuring time and so
the average and that's
when this becomes a rate,
and the average is over
the time interval delta x.
And then the limiting
value is denoted dy/dx
and so this one is the
average rate of change
and this one is the
instantaneous rate.

French: 
Bon, sa se lide mwen vle
diskite kounyea.
Mwen pral ban nou kèk egzanp.
Ann gade pou nou wè.
Tou dabò, kèk egzanp
Nan syans fizik.
Dabitid, Q se non pou yon chaj,
epi dq/dt
se sa ki kouran.
Alò, sa se yon egzanp
nan syans fizik.
Yon lòt egzanp, youn ki pi
fasil pou n konprann,
se pou n sèvi ak lèt “s” pou n reprezante distans/
Nan ka sa a, to varyasyon an,
se sa nou rele vitès.

English: 
Okay, so that's
the point of view
that I'd like to
discuss now and give you
just a couple of examples.
So, let's see.
Well, first of all, maybe some
examples from physics here.
So q is usually the
name for a charge,
and then dq/dt is
what's known as current.
So that's one physical example.
A second example, which is
probably the most tangible one,
is we could denote the
letter s by distance
and then the rate of change
is what we call speed.

English: 
So those are the
two typical examples
and I just want to
illustrate the second example
in a little bit more
detail because I think
it's important to have some
visceral sense of this notion
of instantaneous speed.
And I get to use the example of
this very building to do that.
Probably you know,
or maybe you don't,
that on Halloween
there's an event that
takes place in this
building or really
from the top of
this building which
is called the pumpkin drop.
So let's illustrates this
idea of rate of change
with the pumpkin drop.
So what happens is,
this building-- well

French: 
Sa se 2 egzanp ki sèvi anpil,
e mwen ta renmen
dekri dezyèm egzanp lan
ak plis detay
paske mwen panse li enpòtan
pou nou konprann tout bon vre
zafè vitès enstantane a.
E mwen ka sèvi ak bilding kote
nou ye la a pou mwen fè sa.
Nou ka deja konnen, petèt tou nou ka pa
konnen, chak ane pou fèt Latousen (Alowin),
gen yon bagay ki fèt nan
tèt bilding sa a.
Se sa yo rele “lage joumou.”
Yo lage yon joumou
soti nan tèt bilding lan rive atè anba a.
Nou pral esplike lide to varyasyon sa a
ak egzanp joumou k ap tonbe a.
Men sa ki rive nan bilding sa a.
Ann gade bilding lan.

English: 
let's see here's the building,
and here's the dot, that's
the beautiful grass out on
this side of the building,
and then there's
some people up here
and very small
objects, well they're
not that small when
you're close to them, that
get dumped over the side there.
And they fall down.
You know everything at MIT
or a lot of things at MIT
are physics experiments.
That's the pumpkin drop.
So roughly speaking,
the building
is about 300 feet
high, we're down here
on the first usable floor.
And so we're going to
use instead of 300 feet,
just for convenience
purposes we'll
use 80 meters because that makes
the numbers come out simply.

French: 
Men yon pwen. Sa se bèl
gazon devan bilding lan.
Epi men yon moun ak yon objè
nan men li. Objè a genlè piti.
Men, obè pa vrèman piti
lè ou pre li.
Yo jete objè a
soti depi sou arebò a la a.
Epi li tonbe.
Nou konnen tout bagay nan M.I.T,
tout bagay oswa prèske tout bagay,
se pwojè nan syans fizik.
Se sa ki menmen zafè
lage joumou an.
Alò bilding lan apeprè 300 pye
nan wotè. Epi nou-menm nou bò isit la
nan premye etaj ki ka sèvi a.
Olye de 300 pye, nap sèvi
ak 80 mèt
pou kalkil yo
ka pi fasil.

French: 
Alò nou gen wotè a ki kòmanse
a 80 mèt a lè t = 0.
Epi, pou nou kalkile
akselerasyon akoz pezantè nou gen
fòmil sa a pou h. h se wotè a.
Alò, lè t=0, nou anwo nèt,
h se 80 mèt,
nou mezire sa an mèt.
Lè t=4, nou wè (5 * 4^2)
se 80.
Mwen te chwazi chif sa yo
espre pou
nou ka rive anba nèt.
Alò, lide varyasyon mwayenn sa a,
ki fè varyasyon mwayenn lan, oubyen
vitès mwayenn la, petèt nou ka rele l

English: 
So we have the height
which starts out
at 80 meters at time 0 and then
the acceleration due to gravity
gives you this formula
for h, this is the height.
So at time t = 0, we're up
at the top, h is 80 meters,
the units here are meters.
And at time t = 4 you
notice, 5 * 4^2 is 80.
I picked these numbers
conveniently so
that we're down at the bottom.
Okay, so this notion
of average change here,
so the average change, or
the average speed here,
maybe we'll call it
the average speed,

French: 
vitès mwayenn, se tan sa a
joumou an ap pran
pou l tonbe. Se pral
varyasyon nan h / varyasyon nan t
E sa kòmanse kòm…
ak ki sa li kòmanse?
Li kòmanse a 80, pa vre?
E li fini a 0.
Men, an reyalite, fòk nou fè sa al lanvè.
Fòk nou konsidere 0-80 paske
premye chif la se dènye pozisyon an,
epi dezyèm chif la,
se premye pozisyon an.
E sa dwe divize pa 4 – 0;
4 segonn mwens 0 segonn.
Alò, natirèlman sa se -20 mèt
pa segonn.
Alò vitès mwayenn nèg sa a
se 20 mèt pa segonn.

English: 
since that's-- over this time
that it takes for the pumpkin
to drop is going to be
the change in h divided
by the change in t.
Which starts out at, what
does it start out as?
It starts out as 80, right?
And it ends at 0.
So actually we have
to do it backwards.
We have to take 0 - 80
because the first value is
the final position
and the second value
is the initial position.
And that's divided by
4 - 0; times 4 seconds
minus times 0 seconds.
And so that of course is
-20 meters per second.
So the average speed of this
guy is 20 meters a second.

English: 
Now, so why did I
pick this example?
Because, of course, the
average, although interesting,
is not really what
anybody cares about who
actually goes to the event.
All we really care about
is the instantaneous speed
when it hits the pavement
and so that's can
be calculated at the bottom.
So what's the
instantaneous speed?
That's the derivative,
or maybe to be
consistent with the notation
I've been using so far,
that's d/dt of h.
All right?
So that's d/dt of h.
Now remember we have
formulas for these things.
We can differentiate
this function now.
We did that yesterday.
So we're gonna take the rate of
change and if you take a look
at it, it's just the rate
of change of 80 is 0,

French: 
Pou ki sa mwen chwazi egzanp sa a?
Paske, se nòmal, mwayenn lan,
menm si sa enteresan,
sa pa vreman enteresan pou moun
k ap gade joumou sa a.
Sèl sa ki enterese nou vreman,
se vitès enstantane
lè joumou a frape atè.
Alò, nou ka kalkile vitès sa a
lè li rive anba nèt.
Alò, ki sa vitès enstantane a ye?
Se derive a, oubyen pou mwen kenbe menm
notasyon mwen te konmanse sèvi avè l deja a,
se d/dt de h.
Dakò?
Alò se d/dt de h.
Sonje nou gen fòmil pou
bagay sa yo.
Nou ka kalkile derive.
fonksyon sa a kounyea.
Nou te deja fè sa yè.
Alò nou pral pran to varyasyon an
Epi si ou gade sa byen,
to varyasyon 80 se 0,

French: 
mwens to varyasyon pou sa a
-5t^2, sa se -10t.
Alò, nou ka sèvi ak fòmil sa yo:
d/dt de 80 egal 0,
epi d/dt de t^2 egal 2t.
Ka sa a espesyal.
Men, m ap fè koken la a,
Gen yon ka espesyal
tout moun ka wè.
Mwen pa t mete li bò isit la.
Ka n=2 se dezyèm ka ki la a.
Men, lè n=0 nou ka kalkile derive a tou.
Paske sa se yon konstant.
Derive yon konstan se 0.
Apre sa, faktè n lan la a
se 0, e nou toujou wè sa.
Kounyea, si ou gade fòmil ki sou tèt li
w ap wè
se ka kote n=-1.
Alò, nap gen yon pi gwo echantiyon tale,
lè nou rive nan kalkil ak puisans.
Oke.
Annou retounen bò isit la kote
nou gen to varyasyon nou an.
Men sa li ye.

English: 
minus the rate change for
this -5t^2, that's minus 10t.
So that's using the fact
that d/dt of 80 is equal to 0
and d/dt of t^2 is equal to 2t.
The special case...
Well I'm cheating
here, but there's
a special case that's obvious.
I didn't throw it in over here.
The case n = 2 is that
second case there.
But the case n = 0 also works.
Because that's constants.
The derivative of
a constant is 0.
And then the factor n there's
0 and that's consistent.
And actually if you look
at the formula above it
you'll see that it's
the case of n = -1.
So we'll get a larger pattern
soon enough with the powers.
Okay anyway.
Back over here we have
our rate of change
and this is what it is.
And at the bottom, at
that point of impact,

French: 
Anba a, nan pwen kontak la,
nou gen
t=4 epi h’, ki se derive a,
egal -40 mèt pa segonn.
Derive a 2 fwa pi vit ke vitès
mwayenn ki la a Si ou bezwen
konvèti l, se apeprè
90 mil a lè.
Se pou sa polis yo la a minui
lavèy fèt Latousen (“Alowin”)
pou yo asire tout moun.
Epi tou, se pou sa
fòk ou pare pou benyen
apre sa.
An tou ka, se sa k pase.
Joumou a frape 90 mil a lè.
An reyalite, bilding la
yon ti jan pi wo,
gen reziztans van epi mwen sèten
nou ka fè yon rechèch
ki pi konplè sou egzanp sa a.
Bon. Kounyea mwen vle ban nou
de twa lòt egzanp
paske tan ak kalite paramèt ak
varyab sa yo

English: 
we have t = 4 and so h',
which is the derivative,
is equal to -40
meters per second.
So twice as fast as
the average speed here,
and if you need to convert that,
that's about 90 miles an hour.
Which is why the police are
there at midnight on Halloween
to make sure you're all safe
and also why when you come
you have to be prepared
to clean up afterwards.
So anyway that's what happens,
it's 90 miles an hour.
It's actually the
buildings a little taller,
there's air resistance
and I'm sure you
can do a much more thorough
study of this example.
All right so now I want to give
you a couple of more examples
because time and these kinds
of parameters and variables

French: 
se pa sèl sa ki enpòtan nan
kalkil diferansyèl ak entegral.
Si se te sèl kalite syans
fizik sa yo ki te ladan l,
sijè sa a t ap pi
espesyalize anpil.
Alò mwen vle ban nou kèk egzanp
ki pa gen tan kòm varyab.
Konsa twazyèm egzanp mwen
pral bay la se...
Lèt t repranzate tanperati,
dt/dx reprezante
sa ki rele gradyan tanperati.
E sa enpòtan anpil lè
n ap prevwa meteyo
paske se diferans tanperati sa yo
ki lakoz van ak lòt chanjman nan atmosfè a.
E gen yon lòt tèm enpòtan
ki nan tout syans ak jeni

English: 
are not the only ones that
are important for calculus.
If it were only this kind of
physics that was involved,
then this would be a much more
specialized subject than it is.
And so I want to give you a
couple of examples that don't
involve time as a variable.
So the third example
I'll give here
is-- The letter T often
denotes temperature,
and then dT/dx would be what
is known as the temperature
gradient.
Which we really care
about a lot when
we're predicting the weather
because it's that temperature
difference that causes air flows
and causes things to change.
And then there's
another theme which

French: 
Mwen pral pale nan diskisyon
sou sansibilite mezi.
Ban m esplike sa.
Mwen pa vle rete twò lontan sou sa
paske m ap montre nou sa
jis pou mwen ka entwodui
kèk lide ki nan devwa nou yo.
Nan premye pwoblèm nan devwa a,
gen yon egzanp ki baze sou
yon sistèm pozisyònman global (“GPS”)
ki senplifye nèt. Se yon GPS senp
kote latè tou plat.
Epi nan sitiyasyon sa a,
si latè tou plat,
se yon liy orizontal konsa.
Apre sa, ou gen yon satelit,
ki bò isit la.

English: 
is throughout the sciences
and engineering which
I'm going to talk about under
the heading of sensitivity
of measurements.
So let me explain this.
I don't want to belabor
it because I just
am doing this in
order to introduce you
to the ideas on your
problem set which
are the first case of this.
So on problem set one
you have an example
which is based on a
simplified model of GPS,
sort of the Flat Earth Model.
And in that situation,
well, if the Earth is flat
it's just a horizontal
line like this.
And then you have a satellite,
which is over here, preferably

English: 
above the earth, and the
satellite or the system
knows exactly where the point
directly below the satellite
is.
So this point is
treated as known.
And I'm sitting here
with my little GPS device
and I want to know where I am.
And the way I
locate where I am is
I communicate with this
satellite by radio signals
and I can measure this distance
here which is called h.
And then system will compute
this horizontal distance which
is L. So in other
words what is measured,

French: 
Li ta pi bon si satelit la sou tèt late,
e si satelit la ka lokalize
egazkteman pwen ki dirèkteman anba
satelit la.
Alò, pwen sa a se yon pwen
nou ka lokalize.
E mwen chita la, avèk ti sistèm
pozisyonman global (“GPS”) mwen,
e mwen vle konnen ki kote mwen ye.
E jan pou mwen konnen ki kote mwen ye,
se pou m kominike
ak satelit sa a gras ak siyal radyo.
Se kon sa mwen ka mezire distans sa a
ki rele h.
Epi sistem la ap kalkile
distans orizontal la
ki se L.
Alò, sa nou ka mezire se:

French: 
h gras a siyal radio ak yon mont
oubyen plizyè mont.
Apre sa, L depann de h.
Epi, men sa ki enpòtan nan tout
sistèm sa yo:
nou pa fouti konnen
ki sa h ye egzakteman.
Gen yon erè nan h
nou pral reprezante ak delta h.
Nou pa fouti konnen
ki sa h ye egzakteman.
Sa ki bay pi gwo dout nan GPS
se nan iyonosfè a sa soti.
Men, gen anpil koreksyon
tout kalite ki fèt.
Epi tou si ou nan
bilding lan,
li difisil pou mezire l.
Men se yon sijè ki
enpòtan anpil.
Mwen pral eksplike sa talè.

English: 
so h measured by radios,
radio waves and a clock,
or various clocks.
And then L is deduced from h.
And what's critical in
all of these systems
is that you don't
know h exactly.
There's an error in h
which will denote delta h.
There's some degree
of uncertainty.
The main uncertainty in
GPS is from the ionosphere.
But there are lots
of corrections
that are made of all kinds.
And also if you're
inside a building
it's a problem to measure it.
But it's an extremely
important issue,
as I'll explain in a second.
So the idea is we
then get at delta

English: 
L is estimated by considering
this ratio delta L/delta
h which is going
to be approximately
the same as the derivative
of L with respect to h.
So this is the thing that's
easy because of course it's
calculus.
Calculus is the
easy part and that
allows us to deduce something
about the real world that's
close by over here.
So the reason why you should
care about this quite a bit
is that it's used all the
time to land airplanes.
So you really do care
that they actually
know to within a few feet or
even closer where your plane is
and how high up it
is and so forth.
All right.
So that's it for the
general introduction
of what a derivative is.
I'm sure you'll be
getting used to this
in a lot of different contexts
throughout the course.

French: 
Lide a se pou nou konsidere
delta L.
Nou estime l gras a konsiderasyon rapò
delta L / delta h ki
apeprè menm ak derive L
pa rapò a h.
Alò sa byen fasil paske
se kalkil diferansyèl
ak entegral.
Kalkil diferansyèl ak entegral
se pati ki pi fasil la e li pèmèt nou
konprann plizyè pwoblèm pratik
ki tou pre nou la a.
Konsa, rezon ki fè sa dwe enterese nou anpil,
se paske
sa sèvi tout tan pou
fè avyon ateri.
Se pou sa fòk yo konnen kote avyon w lan ye,
ak ki wotè li ye, elatriye.
E fòk erè nan kalkil distans sa yo
pa depase kèk mèt.
Dakò. Kounyea, nou fini
ak entwodiksyon jeneral
sou ki sa derive ye.
Mwen sèten w ap fin abitye
ak sa
nan anpil lòt diskisyon nan kou sa a.

French: 
E kounyea fòk nou retounen
nan bon jan detay.
Èske tout moun satisfè ak sa nou
rive fè jouk kounyea?
Wi?
Etidyan: Kòman ou te rive kalkile ekwazyon wotè a?
Pwofesè: Aaa... Bèl kesyon.
Kesyon an se koman mwen rive
kalkile ekwasyon wotè sa a?
Mwen envante l paske se fòmil
nan syans fizik
ou pral aprann lè ou pran klas 8.01.
Sa gen rapò ak vitès la.
Si ou kalkile derive vitès,
sa ba w akselerasyon.
Epi akselerasyon akoz pezantè,
se 10 mèt pa segonn.
E sa se dezyèm derive sa a.
An tou ka,
mwen annik rale li
nan kou syans fizik nou.
Alò ou ka annik di:
al gade 8.01.

English: 
And now we have to get back
down to some rigorous details.
Okay, everybody happy with
what we've got so far?
Yeah?
Student: How did you get
the equation for height?
Professor: Ah good question.
The question was how did I
get this equation for height?
I just made it up because
it's the formula from physics
that you will learn when
you take 8.01 and, in fact,
it has to do with the fact
that this is the speed if you
differentiate
another time you get
acceleration and
acceleration due to gravity
is 10 meters per second.
Which happens to be the
second derivative of this.
But anyway I just pulled it
out of a hat from your physics
class.
So you can just say see 8.01 .

English: 
All right, other questions?
All right, so let's go on now.
Now I have to be a little bit
more systematic about limits.
So let's do that now.
So now what I'd like to talk
about is limits and continuity.
And this is a warm
up for deriving
all the rest of the formulas,
all the rest of the formulas
that I'm going to
need to differentiate
every function you know.
Remember, that's our goal
and we only have about a week
left so we'd better get started.
So first of all there is
what I will call easy limits.

French: 
Oke. Lòt kesyon?
Dakò, ann kontinye.
Kounyea fòk mwen pi sistematik
sou kesyon limit yo.
Annou fè sa kounyea.
Mwen vle pale de limit
ak kontinuite.
E sa ap prepare nou
pou n derive tout lòt fòmil,
tout lòt fòmil nou pral bezwen
pou kalkile derive tout fonksyon nou konnen yo.
Sonje, se objektif nou sa, e se
yon semenn ase ki rete,
ki fè pito nou kòmanse.
Alò premyèman gen sa
mwen ta rele “limit fasil”.

French: 
Ki sa ki yon limit fasil?
Men yon egzanp limit fasil:
limit lè x ap pwoche 4 de (x + 3/ x^2 + 1).
limit lè x ap pwoche 4 de (x + 3/ x^2 + 1)
limit lè x ap pwoche 4 de (x + 3/ x^2 + 1)
Ak kalite limit sa a,
sèl sa mwen bezwen fè pou m evalye li,
se mete x = 4.
Kon sa, sa mwen vin jwenn la a, se 4 + 3/ (4^2 + 1).
se 4 + 3/ (4^2 + 1)
se 4 + 3/ (4^2 + 1)
E sa jis vin tounen 7/17.
Epi sa tou fini la.
Alò, sa se yon egzanp yon limit
ki fasil pou kalkile—yon limit fasil.
Men yon dezyèm kategori limit.
Bon, sa se pa sèl dezyèm kategori limit ki egziste.
Men, m jis vle pou nou note sa.
Sa enpòtan anpil.
Derive toujou pi difisil pase sa.

English: 
So what's an easy limit?
An easy limit is something like
the limit as x goes to 4 of x
plus 3 over x^2 + 1.
And with this kind of limit all
I have to do to evaluate it is
to plug in x = 4 because,
so what I get here is 4 + 3
divided by 4^2 + 1.
And that's just 7 / 17.
And that's the end of it.
So those are the easy limits.
The second kind of limit -
well so this isn't the only
second kind of limit but I
just want to point this out,
it's very important - is that:
derivatives are are always
harder than this.

French: 
Ou pa ka chape anba difikilte sa a.
Alò, pou ki sa?
Bon, lè ou pran yon derive,
ou pran
lè x ap pwoche x0 de f(x).
Alò, n a ekri li
nan tout bèlte li.
Men fòmil pou derive a.
Kounyea remake si nou mete
x = x0, sa toujou bay 0/0.
Ki fè tou senpleman
sa pa janm mache.
Ki fè nou pral toujou bezwen
yon anilasyon

English: 
You can't get away
with nothing here.
So, why is that?
Well, when you
take a derivative,
you're taking the limit
as x goes to x_0 of f(x),
well we'll write it all
out in all its glory.
Here's the formula
for the derivative.
Now notice that if you plug in
x = x:0, always gives 0 / 0.
So it just basically
never works.
So we always are going
to need some cancellation

French: 
pou limit la sa gen sans.
Kounyea, pou m ka rann bagay yo
yon jan pi fasil pou mwen,
pou m ka esplike ki sa k ap pase
ak limit yo, fòk mwen esplike
yon lòt ti notasyon.
Sa mwen pral esplike la a
se sa yo rele
limit a goch ak limit a dwat.
Si mwen pran limit
lè x ap pwoche x0
avèk la a yon siy plis la a pou yon fonksyon kèlkonk,
se sa yo rele
limit a dwat.
E mwen ka montre sa
ak yon ti shema.
Alò, sa sa vle di?
Sa vle di pratikman menm bagay
ak x ap pwoche x0
Men, gen yon lòt restriksyon sou x,
ki gen rapò
ak siy plis sa a, ki vle di
x ap soti sou kote pozitif x0.
Sa vle di x pi gwo pase x0.

English: 
to make sense out of the limit.
Now in order to make things
a little easier for myself
to explain what's
going on with limits
I need to introduce just
one more piece of notation.
What I'm gonna
introduce here is what's
known as a left-hand
and a right limit.
If I take the limit as x tends
to x_0 with a plus sign here
of some function, this is what's
known as the right-hand limit.
And I can display it visually.
So what does this mean?
It means practically
the same thing
as x tends to x_0 except there
is one more restriction which
has to do with this plus
sign, which is we're going
from the plus side of x_0.
That means x is bigger than x_0.

French: 
E lè mwen di a dwat,
sa vle di fòk gen yon tirè la a.
Limit la a dwat paske
sou liy chif yo, si x0 se la li ye,
x ap sou bò dwat x0.
Dakò?
Alò, se sa ki limit a dwat la.
E se sa ki se bò goch tablo a.
Epi sou bò dwat tablo a,
se la m ap mete limit a goch la,
jis pou m lage nou
nan konfizyon.
Alò sa a pral gen siy mwens lan la a.
Mwen yon ti jan disleksik
e mwen swete nou-menm nou pa disleksik.
Ki fè se posib
mwen fè fot nan sa.
Alò, se sa ki limit a goch la,
e m ap fè shema sa a.
Ki fè sa jis vle di
x ap pwoche x0,
men x ap sou bò goch x0.
Epi ankò, sou liy chif yo,
men x0 epi x li-menm
l ap sou lòt bò x0.
Bon, de notasyon sa yo
pral ede nou

English: 
And I say right-hand, so
there should be a hyphen here,
right-hand limit because
on the number line,
if x_0 is over here
the x is to the right.
All right?
So that's the right-hand limit.
And then this being the
left side of the board,
I'll put on the right side
of the board the left limit,
just to make things confusing.
So that one has the
minus sign here.
I'm just a little dyslexic
and I hope you're not.
So I may have gotten that wrong.
So this is the left-hand
limit, and I'll draw it.
So of course that just
means x goes to x_0 but x is
to the left of x_0 .
And again, on the number
line, here's the x_0
and the x is on the
other side of it.
Okay, so those two
notations are going

English: 
to help us to clarify
a bunch of things.
It's much more
convenient to have
this extra bit of
description of limits
than to just consider
limits from both sides.
Okay so I want to give
an example of this.
And also an example
of how you're going to
think about these
sorts of problems.
So I'll take a function which
has two different definitions.
Say it's x + 1, when x >
0 and -x + 2, when x < 0.
So maybe put commas there.
So when x > 0, it's x + 1.

French: 
klarifye plizyè bagay.
Li pi fasil vre pou
nou gen tout deskripsyon sa yo
sou afè limit yo, olye pou
nou jis konsidere an menm tan
limit sou tou 2 bò yo.
Dakò, konsa m vle bay
yon egzanp sou sa.
Epi tou, yon egzanp
sou ki jan pou nou kalkile
kalite pwoblèm sa yo.
Ki fè m ap pran yon fonksyon
ki gen 2 definisyon diferan.
Ann di x + 1,
x + 1, lè x > 0,
epi -x + 2, lè x < 0.
Alò, petèt fòk nou mete yon vigil la.
Ki fè lè x>0,
se x + 1.

French: 
Kounyea mwen ka desinen yon graf pou sa.
Li pral yon jan piti paske
m bezwen
foure li anba la a.
Men, petèt m a mete aks la anba a.
Ki fè nan wotè 1, mwen gen
sou bò dwat la yon liy ki gen pant 1,
ki fè li monte konsa.
Dakò?
Apre sa, sou bò goch la,
mwen genyen yon liy ki gen pant -1.
Men, li vin kontre ak aks la nan 2
ki fè li anlè la a.
Ki fè mwen gen
yon fòm antèn ki dwòl,
Men graf la.
Petèt m ta dwe ekri sa nan yon
lòt koulè pou sa parèt pi klè.
Apre sa, si mwen kalkile 2 limit
sa yo la a, men sa mwen vin wè:

English: 
Now I can draw a
picture of this.
It's gonna be kind
of a little small
because I'm gonna try
to fit it down in here,
but maybe I'll put
the axis down below.
So at height 1, I have to
the right something of slope
1 so it goes up like this.
All right?
And then to the left of 0 I have
something which has slope -1,
but it hits the axis
at 2 so it's up here.
So I had this sort of
strange antenna figure here,
which is my graph.
Maybe I should draw these in
another color to depict that.
And then if I calculate
these two limits here,
what I see is that
the limit as x

French: 
limit lè x ap pwoche 0 sou bò dwat
de f(x),
se menm jan ak limit lè x ap pwoche x0
de fòmil ki la a,
x + 1.
Sa vin tounen 1.
E si mwen pran limit sa a,
ki fè se limit a goch la.
Eskize m, m te di nou
mwen disleksik.
Sa a se dwat la.
Alò se men dwat la.
Ann re-derape.
Kounyea m pral soti a goch,
e se f(x) ankò. Men, kounyea
poutèt se sou bò sa a mwen ye,
mwen pral sèvi
ak lòt fòmil lan,
- x + 2, e sa a li menm ap ban nou 2.
Kounyea, annou gade limit a goch
ak limit a dwat la, e sa
se yon ti nyans, e se prèske
sèl bagay mwen bezwen
pou nou rive suiv
ak anpil atansyon kounyea.

English: 
goes to 0 from above of f(x),
that's the same as the limit
as x goes to 0 of the
formula here, x + 1.
Which turns out to be 1.
And if I take the limit, so
that's the left-hand limit.
Sorry, I told you
I was dyslexic.
This is the right, so
it's that right-hand.
Here we go.
So now I'm going from the
left, and it's f(x) again,
but now because I'm on that
side the thing I need to plug
is the other formula, -x + 2,
and that's gonna give us 2.
Now, notice that the left
and right limits, and this
is one little tiny subtlety
and it's almost the only thing
that I need you to really
pay attention to a little bit

English: 
right now, is that this, we
did not need x = 0 value.
In fact I never even told
you what f(0) was here.
If we stick it in we
could stick it in.
Okay let's say we stick
it in on this side.
Let's make it be that
it's on this side.
So that means that this point
is in and this point is out.
So that's a typical notation:
this little open circle
and this closed dot for
when you include the.
So in that case
the value of f(x)
happens to be the same
as its right-hand limit,
namely the value is
1 here and not 2.

French: 
Nou pa t bezwen valè x=0.
Avrèdi m pa t menm janm di nou
sa f(0) te ye la a.
Si nou foure l ladan l...
OK, nou ka foure l ladan l.
Dakò, annou foure l
nan bò sa a.
Annou fè pou l vin
sou bò sa a.
Sa vle di pwen sa a anndan,
pwen sa a li-menm, li deyò.
Sa se yon notasyon ki popilè:
ti sèk ki ouvri sa a
epi ti pwen ki fèmen sa a
pou lè ou vle mete valè sa a anndan...
Ki fè nan ka sa a,
valè f(x) vin menm
ak limit a dwat la.
La a, valè a se 1. Se pa 2.

French: 
Dakò, sa a se te premye
kalite egzanp lan.
Kesyon?
Dakò. Kounyea, pwochen
travay nou pral fè,
se defini kontinuite
Sa se te lòt sijè a.
Se kon sa nou pral defini kontinuite:
f kontini sou x0, sa vle di
limit f(x) lè x ap pwoche x0
se f(x0).
Pa vre?
Alò rezon ki fè m
pase tout tan sa a ap okipe
bò goch la ak bò dwat la, elatriye,
se paske m vle
pou nou fè atansyon pou yon moman

English: 
Okay, so that's the
first kind of example.
Questions?
Okay, so now our next
job is to introduce
the definition of continuity.
So that was the
other topic here.
So we're going to define.
So f is continuous at x_0 means
that the limit of f(x) as x
tends to x_0 is
equal to f(x_0) .
Right?
So the reason why I spend
all this time paying
attention to the left and the
right and so on and so forth
and focusing is that I want you
to pay attention for one moment

French: 
ak sa definisyon sa a
gen ladan l.
Sa li vle di a se sa:
kontinuite nan x0
gen divès engredyan.
Men premye engredyan an:
fòk limit sa a egziste.
E sa, sa vle di fòk limit la gen valè
ni sou bò goch,
ni sou bò dwat.
Epi tou fòk 2 valè sa yo menm.
Dakò, ki fè se sa
k ap pase la a.
Epi, men dezyèm pwopriyete a:
fòk f(x0) defini.
Ki fè mwen pa ka nan yon sitiyasyon
kote mwen pa

English: 
to what the content
of this definition is.
What it's saying is the
following: continuous at x_0
has various ingredients here.
So the first one is
that this limit exists.
And what that means
is that there's
an honest limiting value
both from the left and right.
And they also have
to be the same.
All right, so that's
what's going on here.
And the second property
is that f(x_0) is defined.
So I can't be in one
of these situations
where I haven't
even specified what

English: 
f(x_0) is and they're equal.
Okay, so that's the situation.
Now again let me
emphasize a tricky part
of the definition of a limit.
This side, the left-hand side
is completely independent,
is evaluated by a
procedure which does not
involve the right-hand side.
These are separate things.
This one is, to evaluate it, you
always avoid the limit point.
So that's if you like a
paradox, because it's exactly
the question: is it true
that if you plug in x_0
you get the same answer as
if you move in the limit?
That's the issue that
we're considering here.
We have to make that
distinction in order
to say that these
are two, otherwise
this is just tautological.
It doesn't have any meaning.
But in fact it
does have a meaning

French: 
menm montre ki sa f(x0) ye.
Epi, fòk yo egal.
Dakò, se sitiyasyon an sa.
Kounyea, ban mwen reprann yon pati
ki gendwa lakòz ti konfizyon nan
definisyon limit.
Bò sa a, bò a goch la,
endepandan nèt,
li sèvi ak yon definisyon
ki pa rantre
nan definisyon bò dwat la.
Sa se 2 bagay ki separe.
Sa a li menm se, pou evalye li,
fòk ou toujou evite
pwen limit la.
Alò, sa se yon paradòks.
Sa se egzakteman
kesyon an sa: èske se vre si
ou mete x0 pou x, w ap jwenn
menm repons kòmsi ou te kalkile
limit lè x ap pwoche x0?
Se kesyon sa a
n ap konsidere la a.
Fòk nou fè distenksyon an
pou nou ka di
sa yo se 2 bagay separe. Si se pa sa,
sa ap jis yon totoloji.
Li pa gen okenn sans.
Men, avrèdi li gen sans paske
youn nan bagay yo

French: 
nou ka evalye li
pa rapò ak tout lòt pwen yo
esepte x0. Epi lòt la,
nou ka evalye li dirèk-dirèk
sou pwen x0 an menm.
E vrèman, sa bagay sa yo ye,
se egzakteman
egzanp limit ki fasil pou kalkile.
Se egzakteman
sa n ap pale la a.
Se yo nou ka evalye
nan fason sa a.
Alò, fòk nou fè distenksyon an.
E lòt sa yo se yo
nou pa p ka evalye nan fason sa a.
Ki fè se limit sa yo ki byennelve.
Se poutèt sa nou okipe yo.
Se rezon sa a ki fè nou gen
tout definisyon sa yo pou yo.
Dakò?
Kounyea, ki sa nou pral fè?
Bon, fòk nou fè nou fè yon ti vizit,
yon ti vizit byen kout,
nan pak an dezòd kote sa nou rele
fonksyon diskonti yo rete.
Alò sa se preske tout lòt fonksyon
ki pa kontini yo.

English: 
because one thing is evaluated
separately with reference
to all the other
points and the other
is evaluated right at
the point in question.
And indeed what
these things are,
are exactly the easy limits.
That's exactly what
we're talking about here.
They're the ones you
can evaluate this way.
So we have to make
the distinction.
And these other ones are
gonna be the ones which
we can't evaluate that way.
So these are the
nice ones and that's
why we care about them, why
we have a whole definition
associated with them.
All right?
So now what's next?
Well, I need to give you a a
little tour, very brief tour,
of the zoo of what are known
as discontinuous functions.
So sort of everything else
that's not continuous.

French: 
Ki fè, premye egzanp lan la a,
ban mwen jis ekri l anba la a.
Sa se egzanp diskontinuite ki sote.
Ki sa yon diskontinuite ki sote ta ka ye?
A vrè di, nou deja wè sa.
Diskontinuite ki sote,
nou te wè sa nan egzanp
nou te genyen la a.
Se lè limit a goch ak limit a dwat,
tou lè 2 egziste.
Men, yo pa egal.
Dakò, ki fè sa te nan
egzanp lan.
Pa vre?
Nan egzanp sa a, 2 limit yo,
youn ladan yo te 1,
epi lòt la te 2.

English: 
So, the first example here,
let me just write it down here.
It's jump discontinuities.
So what would a jump
discontinuity be?
Well we've actually
already seen it.
The jump discontinuity
is the example
that we had right there.
This is when the limit
from the left and right
exist, but are not equal.
Okay, so that's
as in the example.
Right?
In this example, the
two limits, one of them
was 1 and of them was 2.

French: 
Alò, se sa ki diskontinuite ki sote.
E kalite pwoblèm sa a,
kesyon si yon fonksyon kontini ou pa,
sa gen dwa parèt yon ti jan teknik.
Men, se yon pwoblèm ki te bay anpil moun anpil pwoblèm.
yon pwoblèm ki te bay anpil moun anpil pwoblèm.
Bob Merton, ki te yon pwofesè
nan MIT lè li te fè travay
ki te ba li pri Nobèl nan ekonomi,
te enterese nan
kesyon sa a menm:
Èske pri diferan kalite estòk kontini
sou bò goch ouswa sou bò dwat
nan divès fonksyon.
E sa se te yon bagay ki te
enpòtan nan devlopman modèl
ki deside pri nou peye
pou envestisman nou fè
tout tan kounyea.
Ki fè bò goch ak bò dwat ka vle di 2 bagay
trè diferan. Nan ka sa a, bò goch la reprezante

English: 
So that's a jump discontinuity.
And this kind of issue,
of whether something
is continuous or not, may
seem a little bit technical
but it is true that people
have worried about it a lot.
Bob Merton, who was a
professor at MIT when
he did his work for the
Nobel prize in economics,
was interested in
this very issue
of whether stock
prices of various kinds
are continuous from the left
or right in a certain model.
And that was a
very serious issue
in developing the model
that priced things
that our hedge funds
use all the time now.
So left and right can really
mean something very different.

French: 
sa ki te deja fèt nan lepase, epi bò dwat la
reprezante sa ki pral fèt nan lavni.
Se konsa vin gen yon gwo diferans
si fonksyon an kontini
a goch oswa a dwat.
Èske pwen an la a menm,
yon kote nan mitan,
yon lòt kote ?
Sa a se yon kesyon enpòtan.
Alò pwochen egzanp mwen vle
ban nou an
yon ti jan pi difisil.
Se sa yo rele
« diskontuinite ki pwolonjab ».
E sa sa vle di, sa vle di limit a goch
ak limit a dwat, tou lè 2 egal.
Ki fè si ou fè yon graf,
ou gen yon fonksyon
k ap vini konsa, epi li gen yon twou.
Ki kote? Sa k konnen.
Swa petèt fonksyon an pa defini
ouswa petèt li defini
anwo la a, epi apre li jis
kontinye ale.
Dakò?

English: 
In this case left is the
past and right is the future
and it makes a big
difference whether things
are continuous from the left
or continuous from the right.
Right, is it true that
the point is here,
here, somewhere in the
middle, somewhere else.
That's a serious issue.
So the next example
that I want to give you
is a little bit more subtle.
It's what's known as a
removable discontinuity.
And so what this means is that
the limit from left and right
are equal.
So a picture of
that would be, you
have a function which is
coming along like this
and there's a hole
maybe where, who knows
either the function is undefined
or maybe it's defined up here,
and then it just continues on.
All right?

French: 
Ki fè 2 limit yo se menm bagay.
Epi apre, sa tou natirèl, fonksyon an
se sipliye l ap sipliye pou nou redefini l
pou nou ka retire twou sa a.
E se pou sa nou rele sa
yon « diskontuinite ki pwolonjab ».
Kounyea, ban m ban nou
yon egzanp sou sa, ou pito
de twa egzanp.
Sa yo se egzanp ki
enpòtan. Nou pral
travay avèk yo toutalè.
Premye egzanp lan se fonksyon
g(x), ki se sinx / x.
Dezyèm egzanp lan se fonksyon
h(x), ki se 1-cosx / x.

English: 
So the two limits are the same.
And then of course the function
is begging to be redefined
so that we remove that hole.
And that's why it's called
a removable discontinuity.
Now let me give you
an example of this,
or actually a
couple of examples.
So these are quite
important examples
which you will be working
with in a few minutes.
So the first is the function
g(x), which is sin x / x,
and the second will be the
function h(x), which is 1 -
cos x over x.

English: 
So we have a problem at
g(0), g(0) is undefined.
On the other hand it turns
out this function has what's
called a removable singularity.
Namely the limit as x goes
to 0 of sin x / x does exist.
In fact it's equal to 1.
So that's a very important limit
that we will work out either
at the end of this lecture or
the beginning of next lecture.
And similarly, the
limit of 1 - cos x
divided by x, as
x goes to 0, is 0.
Maybe I'll put that
a little farther
away so you can read it.
Okay, so these are
very useful facts
that we're going
to need later on.
And what they say is
that these things have
removable singularities, sorry
removable discontinuity at x

French: 
Alò, nou gen yon pwoblèm nan g(0):
g(0) pa defini.
An tou ka, fonksyon sa a gen sa yo rele
yon sengilarite ki pwolonjab.
Ki fè limit lè x ap pwoche 0 de sinx / x,
limit sa a egziste.
Avrèdi, li egal a 1.
Alò, sa se yon limit ki vreman enpòtan.
Nou pral travay sou li
oswa nan fen leson sa a,
oswa nan kòmansman
pwochen leson an.
Epi tou, limit lè x a pwoche 0
de 1-cosx / x, limit sa a se 0.
Petèt m a mete sa
yon ti jan pi lwen
pou nou ka li l.
Dakò. Alò, sa yo se done
nou pral bezwen
pi devan.
E sa yo di sèke
fonksyon sa yo gen sengilarite

French: 
ki pwolonjab—eskize, diskontinuite
ki prolonjab—nan x=0.
Dakò. Alò, jan m di a,
n ap tounen sou sa toutalè.
Dakò. Ki fè èske gen okenn
kesyon anvan mwen kontinye?
Wi?
Etidyan: (Son pa klè)
Pwofesè: Kesyon an se:
Pou ki sa sa vre?
Se sa kesyon ou an?
Repons lan vreman vreman ta dwe
parèt li pa klè. Mwen poko montre nou sa.
E si nou pa t sezi wè sa,
sa twa
dwòl nèt.
Alò, nou poko esplike sa.
Fòk nou rete branche jisaske
nou esplike sa.
Dakò?
Nou poko montre sa.

English: 
= 0.
All right so as I say, we'll
get to that in a few minutes.
Okay so are there any
questions before I move on?
Yeah?
Student: [INAUDIBLE]
Professor: The question
is: why is this true?
Is that what your question is?
The answer is it's
very, very unobvious,
I haven't shown it to you
yet, and if you were not
surprised by it then that
would be very strange indeed.
So we haven't done it yet.
You have to stay
tuned until we do.
Okay?
We haven't shown it yet.

French: 
E an reyalite menm lòt pawòl sa a,
ki pral
kontinye parèt pi dwòl toujou,
nou poko esplike sa non plis.
Dakò. Ki fè nou pral rive sou sa,
jan mwen di a,
nan fen leson sa a oswa
nan kòmansman pwochen leson an.
Gen lòt kesyon?
Dakò. Annou kontinye vizit
pak diskontinuite yo.
E mwen kwè mwen vle sèvi
ak ni limit a dwat ni limit a goch.
Ki fè m ap kenbe sa ki sou tablo sa a
sou limit a dwat ak limit a goch.
Alò, yon twazyèm kategori diskontinuite,
se sa yo rele

English: 
And actually even
this other statement,
which maybe seems
stranger still,
is also not yet explained.
Okay, so we are going
to get there, as I said,
either at the end
of this lecture
or at the beginning of next.
Other questions?
All right, so let me
just continue my tour
of the zoo of discontinuities.
And, I guess, I want
to illustrate something
with the convenience of
right and left hand limits
so I'll save this board about
right and left-hand limits.
So a third type of
discontinuity is

French: 
« diskontinuite enfini ».
E nou deja kwaze
ak youn nan sa yo.
M pral desine yo
bò isit la.
Sonje fonksyon y se 1/x.
Se fonksyon sa a ki la a.
Men, kounyea mwen ta renmen
desinen lòt branch lòt branch ipèbòl la anba la a.
Epi mwen pral konsidere
valè negatif
pou x.
Alò, men graf pou 1/x.
E fasilite nou genyen pou nou distenge
ant limit a goch ak limit a dwat,
sa enpòtan vre paske la a
mwen ka ekri
limit lè x ap pwoche 0 de 1/x.
limit lè x ap pwoche 0 de 1/x.
Ki fè sa a li menm, l ap sòti
de bò dwat la epi l ap monte.

English: 
what's known as an
infinite discontinuity.
And we've already
encountered one of these.
I'm going to draw
them over here.
Remember the
function y is 1 / x.
That's this function here.
But now I'd like to draw
also the other branch
of the hyperbola down here
and allow myself to consider
negative values of x.
So here's the graph of 1 / x.
And the convenience here
of distinguishing the left
and the right hand limits is
very important because here I
can write down that the limit
as x goes to 0+ of 1 / x.
Well that's coming from the
right and it's going up.

French: 
Alò limit sa a se enfini.
Alòske, limit nan lòt direksyon an,
limit a goch la,
sa a li menm, l ap desann.
E sa a, li diferan nèt.
Sa se mwens enfini.
Kounyea gen de moun ki di
limit sa yo pa defini.
Men, tout bon vre, yo prale
nan yon direksyon ki byen defini.
Depi sa posib, fòk nou pote bon jan presizyon
sou sa limit sa yo ye.
An tou ka, pawòl ki di
limit lè x ap pwoche 0 de 1/x
se enfini, pawòl sa a pa kòrèk menm.
Oke, sa pa vle di
moun pa ekri sa.
Sa pa kòrèk menm
Sa pa vle di yo pa ekri l.
Siman ou dwe deja wè bagay kon sa.
Se paske moun sa yo annik konsidere
branch a dwat la sèlman.
Yo pa fè yon fot yo nesesèman
Men, an tou ka, gen neglijans.
E gen kèk neglijans nou ka sipòte.
Epi gen kèk lòt,
fòk nou eseye evite.

English: 
So this limit is infinity.
Whereas, the limit in
the other direction,
from the left, that
one is going down.
And so it's quite different,
it's minus infinity.
Now some people say that
these limits are undefined
but actually they're going in
some very definite direction.
So you should,
whenever possible,
specify what these limits are.
On the other hand, the
statement that the limit
as x goes to 0 of 1 / x is
infinity is simply wrong.
Okay, it's not that
people don't write this.
It's just that it's wrong.
It's not that they
don't write it down.
In fact you'll probably see it.
It's because people are just
thinking of the right hand
branch.
It's not that they're making
a mistake necessarily,
but anyway, it's sloppy.
And there's some sloppiness
that we'll endure
and others that
we'll try to avoid.

French: 
Konsa la a, ou vle di sa,
epi sa fè yon diferans.
Pa egzanp, plis enfini se ka yon kantite enfini
lajan ou gen nan pòch ou,
Men, mwens enfini se ka yon kantite enfini
lajan ou dwe.
Sa se yon gwo diferans.
Yo pa menm menm.
Konsa, sa a se neglijans,
e sa a vreman pi kòrèk.
Oke. Kounyea, an plis de sa,
m vle montre
yon lòt bagay.
Sonje nou te kalkile
derive a.
Se te -1/x^2.
Men, mwen vle trase graf sa
a epi m ap fè
kòmantè sou li.
Kidonk, mwen pral trase graf la
dirèkteman anba
graf fonksyon an.
Epi remake byen
ki sa graf sa a ye.
Li fèt konsa, li toujou negatif,
e li pwente sou anba.
Konsa, kounyea, sa ka parèt
yon jan dwòl
pou derive fonksyon sa a se nèg sa a.
Men se akòz
yon bagay ki enpòtan anpil.

English: 
So here, you want to say this,
and it does make a difference.
You know, plus infinity is
an infinite number of dollars
and minus infinity is and
infinite amount of debt.
They're actually different.
They're not the same.
So, you know, this is sloppy and
this is actually more correct.
Okay, so now in
addition, I just want
to point out one more thing.
Remember, we calculated
the derivative,
and that was -1/x^2.
But, I want to draw
the graph of that
and make a few
comments about it.
So I'm going to draw
the graph directly
underneath the graph
of the function.
And notice what this graphs is.
It goes like this, it's always
negative, and it points down.
So now this may look
a little strange,
that the derivative of
this thing is this guy,
but that's because of
something very important.

English: 
And you should always remember
this about derivatives.
The derivative function looks
nothing like the function,
necessarily.
So you should just forget
about that as being an idea.
Some people feel like
if one thing goes down,
the other thing has to go down.
Just forget that intuition.
It's wrong.
What we're dealing with here,
if you remember, is the slope.
So if you have a slope
here, that corresponds
to just a place over
here and as the slope
gets a little bit
less steep, that's
why we're approaching
the horizontal axis.
The number is getting a
little smaller as we close in.
Now over here, the
slope is also negative.
It is going down and
as we get down here
it's getting more
and more negative.
As we go here the slope,
this function is going up,
but its slope is going down.
All right, so the slope is down
on both sides and the notation

French: 
E nou dwe toujou sonje sa
lè n ap pale de derive.
Derive yon fonksyon a pa oblije sanble ak
fonksyon an ditou.
Konsa, ou dwe retire
lide sa a nan tèt ou.
Gen kèk moun ki panse
si yon fonksyon desann, derive a
dwe desann tou.
Annik bliye entuisyon sa a.
Li pa kòrèk.
Sa n ap travay sou li la a,
si ou sonje, se pant lan.
Konsa si ou gen yon pant la a,
ki koresponn senpleman a yon kote la a,
e pandan pant lan
ap vin yon ti jan mwen rèd,
se pou sa n ap pwoche
aks orizontal la.
Kantite a ap vin pi piti
plis n ap pwoche.
Kounyea, bò isit la,
pant lan negatif tou.
L ap desann, e pandan n ap
desann la a, l ap vin
pi negatif toujou.
Pandan n a prale la a nan pant lan
fonksyon sa a ap monte.
Men, pant li ap desann.
Dakò, konsa pant lan desann
sou de bò yo. Epi notasyon

French: 
nou sèvi pou sa byen koresponn
ak afè bò goch ak bò dwat.
Sètadi, limit lè x ap pwoche 0
de -1/ x^2, sa pral egal
mwens enfini.
E sa se vre ni pou x lè l ap pwoche 0+
ni pou x lè l ap pwoche 0-.
Konsa tou lè 2 limit yo
gen pwopriyete sa a.
Finalman, m ap fè yon dènye kòmantè
sou 2 graf sa yo.
Fonksyon sa a se yon fonksyon enpè.
Lè ou kalkile derive
yon fonksyon enpè,
sa toujou ba wou
yon fonksyon pè.
Sa byen koresponn ak afè 1/x
se yon puisans enpè

English: 
that we use for that is
well suited to this left
and right business.
Namely, the limit as x
goes to 0 of -1 / x^2,
that's going to be
equal to minus infinity.
And that applies to x going
to 0+ and x going to 0-.
So both have this property.
Finally let me just
make one last comment
about these two graphs.
This function here
is an odd function
and when you take the
derivative of an odd function
you always get an even function.
That's closely related to the
fact that this 1 / x is an odd

French: 
epi x^1 se yon puisans enpè
epi x^2 se yon puisans pè.
Ak tou sa, entuisyon nou ta dwe
ranfòse lide nou genyen
ki di graf sa yo kòrèk.
Kounyea, gen yon dènye
kategori diskontinuite
m vle mansyone rapidman
e m ap rele yo
« lòt diskontinuite ki lèd ».
E yo anpil anpil.
Men yon egzanp :
y = sin 1/x
y = sin 1/x lè x ap pwoche 0.
E sa sanble yon bagay prèske konsa.

English: 
power and-- x^1 is an odd
power and x^2 is an even power.
So all of this your intuition
should be reinforcing the fact
that these pictures look right.
Okay, now there's one
last kind of discontinuity
that I want to mention
briefly, which I will call
other ugly discontinuities.
And there are lots
and lots of them.
So one example would
be the function y = sin
1 / x, as x goes to 0.
And that looks a
little bit like this.

English: 
Back and forth and
back and forth.
It oscillates infinitely
often as we tend to 0.
There's no left or right
limit in this case.
So there is a very large
quantity of things like that.
Fortunately we're not gonna
deal with them in this course.
A lot of times in
real life there
are things that oscillate
as time goes to infinity,
but we're not going to
worry about that right now.
Okay, so that's our final
mention of a discontinuity,
and now I need to do just
one more piece of groundwork
for our formulas next time.

French: 
Ale vini, ale vini.
Li varye a lenfini lè x ap pwoche 0.
Pa gen ni limit a goch
ni limit a dwat nan ka sa a.
Se konsa gen youn voum ak yon pakèt
fonksyon ki nan ka sa a.
Erezman, nou pa pral okipe yo
nan kou sa a.
Nan lavi nou, anpil fwa,
gen fonksyon k ap pede monte-desann
lè tan ap pwoche enfini.
Men, nou pa p fatige tèt nou
ak sa kounyea.
Oke, sa se dènye pale
n ap fè sou diskontinuite.
Kounyea mwen bezwen tabli
baz pou fòmil
pou pwochen leson an.

French: 
Mwen vle gade avè nou
yon lide enpòtan,
yon zouti pou limit.
Konsa, sa pral tounen yon teyorèm.
Erezman se yon teyorèm tou kout.
E li gen yon prèv tou kout.
Teyorèm lan rele
«si yon fonksyon gen derive,
fonksyon an gen kontinuite ».
E men sa teyorèm lan di :
Si f gen derive,
sa vle di, si derive a egziste nan x0,
, f kontini nan x0.

English: 
Namely, I want to check
for you one basic fact,
one limiting tool.
So this is going
to be a theorem.
Fortunately it's a
very short theorem
and has a very short proof.
So the theorem goes under
the name differentiable
implies continuous.
And what it says
is the following:
it says that if f is
differentiable, in other words
its-- the derivative
exists at x_0, then
f is continuous at x_0.

English: 
So, we're gonna need
this is as a tool,
it's a key step in the
product and quotient rules.
So I'd like to prove
it right now for you.
So here is the proof.
Fortunately the proof
is just one line.
So first of all, I want to
write in just the right way what
it is that we have to check.
So what we have to check is that
the limit, as x goes to x_0,
of f(x) - f(x_0) is equal to 0.
So this is what we want to know.
We don't know it
yet, but we're trying
to check whether
this is true or not.
So that's the same
as the statement
that the function is continuous
because the limit of f(x)
is supposed to be f(x_0) and
so this difference should
have limit 0.

French: 
Konsa, nou pral bezwen teyorèm sa a
tankou yon zouti,
se yon etap enpòtan
nan règ pwodui ak kosyan.
Alò, mwen ta renmen
bay prèv teyorèm la kounyea.
Bon, men prèv la.
Erezman, prèv la
se sèlman yon grenn liy.
Dabò, m vle byen ekri sa nou bezwen
tcheke a. Men sa nou bezwen
tcheke a : èske limit,
èske limit lè x ap pwoche x0, de f(x)
- f(x0), sa egal 0 ?
Wi, se sa nou vle tcheke.
Nou poko konn si se vre.
Men, n ap eseye tcheke
si sa se vre ou si se pa vre.
Sa se menm ak pawòl ki di
fonksyon an kontini
paske limit f(x) la
sipoze egal f(x0),
e konsa diferans sa a
dwe gen limit 0.

French: 
E kounyea, jan pou nou pwouve sa, se annik reekri li
apre nou miltipliye
epi divize li pa (x-x0).
Konsa, m ap re-ekri limit lè x ap pwoche 0
de f(x ) – f (x0 ) divize pa x-x0 miltipliye pa x-x0.
limit lè x ap pwoche 0 de f(x ) – f (x0 )
divize pa x-x0 miltipliye pa x-x0.
Oke, m te ekri menm espresyon
mwen te genyen la a.
Se menm limit la.
Men, mwen te ni miltipliye l
ni divize l pa (x-x0).
Epi kounyea, lè m kalkile limit la, men sa
k rive: limit premye faktè a se f’(x0).
limit premye faktè a se f’(x0).
Se sa nou konnen ki egziste
an patan, dapre sipozisyon nou.

English: 
And now, the way this
is proved is just
by rewriting it by multiplying
and dividing by (x - x_0).
So I'll rewrite the limit
as x goes to x_0 of f(x) -
f(x_0) divided by x
- x_0 times x - x_0.
Okay, so I wrote down the same
expression that I had here.
This is just the same limit,
but I multiplied and divided
by (x - x_0).
And now when I take the limit
what happens is the limit
of the first factor is f'(x_0).
That's the thing we know
exists by our assumption.

French: 
Epi limit dezyèm faktè a
se 0 paske
limit lè x ap pwoche x0
de (x-x0) se 0.
Se byen sa.
Repons lan se 0,
e se byen sa nou te vle.
Kidonk se prèv la.
Kounyea, gen yon bagay
ki pa klè ditou nan prèv sa a.
Kite m montre nou sa
anvan nou kontinye.
Sa vle di, nou abitye
ak limit kote x egal x0.
E sa sanble n ap miltipliye,
epi pa 0.
Sa se yon operasyon ki gate prèv
nan nenpòt ki sitiyasyon aljebrik.
Yo te montre nou
sa pa ta ka janm mache.
Dakò.
Men, kalkil limit sa yo jwenn yon jan
pou yo kabre pwoblèm sa a,
e mwen pral byen esplike
ki jan sa fèt.

English: 
And the limit of the second
factor is 0 because the limit
as x goes to x_0 of (x
- x_0) is clearly 0 .
So that's it.
The answer is 0, which
is what we wanted.
So that's the proof.
Now there's something
exceedingly fishy-looking
about this proof and let me just
point to it before we proceed.
Namely, you're used in limits
to setting x equal to 0.
And this looks like we're
multiplying, dividing by 0,
exactly the thing
which makes all proofs
wrong in all kinds of
algebraic situations
and so on and so forth.
You've been taught
that that never works.
Right?
But somehow these
limiting tricks
have found a way around
this and let me just
make explicit what it is.

French: 
Nan kalkil limit sa a,
nou pa janm sèvi ak x=x0.
Se egzakteman valè x nou pa konsidere
ditou nan limit sa a.
Se konsa nou kalkile limit.
E se sa ki te tèm lan
jodi a jouk kounyea :
nou pa dwe konsidere x=x0,
e sa fè miltiplikasyon
ak divizyon pa kantite sa a legal.
Kantite sa a ka piti.
Men, li pa janm 0.
Kidonk, sa mache tout bon vre.
Epi nou fèk sot montre ki jan yon fonksyon
ki gen derive se yon fonksyon ki kontini.
Konsa, m ap gen pou m kontinye
ak limit sa yo ki enteresan anpil,
limit nan pwochèn leson an.
Men, an nou fè yon kanpe pou
yon segonn pou n wè si gen kesyon
anvan nou rete.
Wi, gen kesyon.
Etidyan (Son an pa klè)
[Son pa klè]

English: 
In this limit we never
are using x = x_0.
That's exactly the
one value of x that we
don't consider in this limit.
That's how limits are cooked up.
And that's sort of been
the themes so far today,
is that we don't
have to consider that
and so this multiplication
and division by this number
is legal.
It may be small, this number,
but it's always non-zero.
So this really works,
and it's really true,
and we just checked that a
differentiable function is
continuous.
So I'm gonna have to carry
out these limits, which
are very interesting 0
/ 0 limits next time.
But let's hang on for one second
to see if there any questions
before we stop.
Yeah, there is a question.

English: 
Student: [INAUDIBLE] Professor:
Repeat this proof right here?
Just say again.
Student: [INAUDIBLE]
Professor: Okay, so there
are two steps to the proof
and the step that you're
asking about is the first step.
Right?
And what I'm saying is
if you have a number,
and you multiply it by 10
/ 10 it's the same number.
If you multiply it by 3
/ 3 it's the same number.
2 / 2, 1 / 1, and so on.
So it is okay to
change this to this,
it's exactly the same thing.
That's the first step.
Yes?
Student: [INAUDIBLE]
Professor: Shhhh...
The question was how does the
proof, how does this line,
yeah where the question mark is.
So what I checked was
that this number which
is on the left hand side
is equal to this very long

French: 
Pwofesè: Repete prèv sa a la a?
Repete sa w di a.
Etidyan: [Son pa klè]
Pwofesè: Oke, konsa gen 2 etap
nan prèv la.
Etap w ap poze kesyon
sou li a se premye etap la.
Kòrèk?
E mwen sa m ap te di:
si ou gen yon kantite epi
ou miltipliye kantite sa a pa 10/10,
se menm kantite a.
Si ou miltipliye l pa 3/3,
se menm menm kantite la.
2/2, 1/1 epi ale nèt.
Konsa, sa kòrèk si
ou chanje sa a pou sa a.
Se menm bagay la egzakteman.
Se premye etap la.
Etidyan: ( Son pa klè)
Pwofesè: Chhhhhhh.
Kesyon an, se kouman prèv la,
kouman liy sa a ye?
Kote pwen entewogasyon an ye a.
Konsa, sa m te tcheke se si
kantite sa a ki sou bò goch la

English: 
complicated number which is
equal to this number which
is equal to this number.
And so I've checked that
this number is equal to 0
because the last thing is 0.
This is equal to that is
equal to that is equal to 0.
And that's the proof.
Yes?
Student: [INAUDIBLE]
Professor: So that's
a different question.
Okay, so the hypothesis
of differentiability I
use because this limit
is equal to this number.
That that limit exits.
That's how I use the
hypothesis of the theorem.
The conclusion of the
theorem is the same
as this because being
continuous is the same as limit
as x goes to x_0 of
f(x) is equal to f(x_0).
That's the definition
of continuity.

French: 
egal a kantite sa a ki long e ki
konplike a
ki egal a kantite sa a ki egal a
kantite sa a.
Kidonk mwen tcheke pou wè si
kantite sa a egal paske
dènye bagay la se 0.
Sa a egal sa a ki egal sa a
ki egal 0.
E sa se prèv la.
Etidyan: ( Son pa klè)
Pwofesè: Sa se yon kesyon diferan.
Oke. Mwen sèvi ak ipotèz
diferansyèl la
paske limit sa a egal
a kantite sa a.
Limit sa a egziste.
Se konsa m sèvi ak
ipotèz teyorèm lan.
Konklizyon teyorèm lan
se se menm ak sa a paske
kontinuite se menm ak
limit lè x ap pwoche x0
limit lè x ap pwoche x0 de f( x0)
Se definisyon kontinuite.

French: 
E m te retire f(x0) nan
tou le 2 bò yo pou m fè
Konsa m di se kontinuite
e se menm ak
kesyon sa a la a.
Dènye kesyon.
Etidyan: Kouman ou fè
jwenn 0 a? (Son an pa klè)
Pwofesè: Kouman nou jwenn 0 a
nan sa a?
Konsa, sa mwen di a,
se pou ki sa kantite sa a
ap pwoche lòt kantite sa a.
M ap bay yon egzanp.
M ap oblije efase yon bagay
pou m esplike sa
Men sa m di: limit lè x
ap pwoche x0 de x-x0 se 0.
Se sa m di.
Oke. Èske sa reponn
kesyon an?
Dakò.
Mande m lòt kesyon apre leson an.

English: 
And I subtracted
f(x_0) from both sides
to get this as being
the same thing.
So this claim is continuity and
it's the same as this question
here.
Last question.
Student: How did you
get the 0 [INAUDIBLE]
Professor: How did we
get the 0 from this?
So the claim that is
being made, so the claim
is why is this tending to that.
So for example, I'm going
to have to erase something
to explain that.
So the claim is that the limit
as x goes to x_0 of x - x_0
is equal to 0.
That's what I'm claiming.
Okay, does that
answer your question?
Okay.
All right.
Ask me other stuff
after lecture.
