1
00:00:01,001 --> 00:00:02,000
Materyèl ou pral gade a se yon
materyèl nou ofri gras a

2
00:00:02,001 --> 00:00:04,000
Lisans Kreyativite non-komèsyal
pou pataje

3
00:00:04,001 --> 00:00:06,000
Sipò ou ap ede Klas Ouvè M.I.T.
kontinye ofri

4
00:00:06,001 --> 00:00:09,000
bon resous edikasyon gratis pou
tout moun.

5
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Pou bay yon kontribisyon pa w, oubyen
pou w gade lòt materyèl

6
00:00:13,001 --> 00:00:15,000
nan plis pase yon santenn kou,
al vizite Klas Ouvè M.I.T.

7
00:00:15,001 --> 00:00:22,000
nan sit entènèt: ocw.mit.edu

8
00:00:22,001 --> 00:00:27,000
Tou dabò, pou dezyèm leson sa a
mwen ta renmen fè nou sonje

9
00:00:27,001 --> 00:00:30,000
sa nou te fè nan dènye leson an.

10
00:00:30,001 --> 00:00:54,000
Nan dènye leson an nou te defini
derive kòm

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00:00:54,001 --> 00:01:04,000
pant liy tanjant.

12
00:01:04,001 --> 00:01:08,000
Sa se te aspè jeyometrik la.
Epi tou,

13
00:01:08,001 --> 00:01:10,000
nou te fè kèk kalkil.

14
00:01:10,001 --> 00:01:14,000
Nou te rive montre ki jan derive 1/x

15
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se te -1/ x^2.

16
00:01:20,001 --> 00:01:27,000
Epi tou nou te kalkile
derive x nan n-yèm puisans

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lè n=1, 2, elatriye. Se te bay x,
eskize m, nx^(n-1).

18
00:01:36,001 --> 00:01:46,000
Alò, se sa nou te fè nan dènye leson an.
Jodi a, mwen vle

19
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fin gade kèk lòt aspè
sou sa yon derive ye.

20
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Sa enpòtan anpil.
Se ka bagay pi enpòtan

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mwen di nan leson sa a.

22
00:01:58,001 --> 00:02:01,000
Men, n ap oblije refleshi sou sa
ankò lè nou rekòmanse

23
00:02:01,001 --> 00:02:04,000
epi lè nou kòmanse sèvi ak kalkil diferansyèl
pou nou rezoud pwoblèm pratik nan lavi.

24
00:02:04,001 --> 00:02:14,000
Alò, n ap kontinye pale sou
ki sa yon derive ye,

25
00:02:14,001 --> 00:02:19,000
e sa se suit esplikasyon
nou te bay nan dènye leson an.

26
00:02:19,001 --> 00:02:23,000
Nan dènye leson an, nou te pale
de entèpretasyonjeyometrik.

27
00:02:23,001 --> 00:02:28,000
Jodi a nou pral pale de

28
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to varyasyon kòm
entèpretasyon derive.

29
00:02:40,001 --> 00:02:47,000
Sonje nou te trase graf
fonksyon y=f(x).

30
00:02:47,001 --> 00:02:51,000
Epi nou te swiv varyasyon nan x,
epi bò isit la

31
00:02:51,001 --> 00:02:56,000
nou te suiv varyasyon nan y.

32
00:02:56,001 --> 00:03:02,000
Nou pral sèvi ak nouvo pèspektiv sa a
pou nou suiv,

33
00:03:02,001 --> 00:03:05,000
to varyasyon nan x
ak to varyasyon nan y.

34
00:03:05,001 --> 00:03:08,000
Se to varyasyon relatif sa a
ki enterese nou.

35
00:03:08,001 --> 00:03:14,000
Sa se delta y / delta x
e sa gen yon lòt

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entèpretasyon.

37
00:03:16,001 --> 00:03:21,000
Sa se to varyasyon mwayenn lan.

38
00:03:21,001 --> 00:03:26,000
Dabitid, nou ta panse kon sa:
Si x t ap mezire yon peryòd tan,

39
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alò mwayenn lan lè sa a,
li tounen yon to,

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e mwayenn lan kouvri
yon entèval tan delta x.

41
00:03:35,001 --> 00:03:45,000
Lè sa a, pou limit la,
ou ekri dy/dx, e sa se

42
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mwayenn to varyasyon an,
e sa se to ki

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kalkile vitès enstantane a.

44
00:03:59,001 --> 00:04:02,000
Bon, sa se lide mwen vle
diskite kounyea.

45
00:04:02,001 --> 00:04:06,000
Mwen pral ban nou kèk egzanp.

46
00:04:06,001 --> 00:04:12,000
Ann gade pou nou wè.

47
00:04:12,001 --> 00:04:19,000
Tou dabò, kèk egzanp
Nan syans fizik.

48
00:04:19,001 --> 00:04:31,000
Dabitid, Q se non pou yon chaj,
epi dq/dt

49
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se sa ki kouran.

50
00:04:33,001 --> 00:04:38,000
Alò, sa se yon egzanp
nan syans fizik.

51
00:04:38,001 --> 00:04:45,000
Yon lòt egzanp, youn ki pi
fasil pou n konprann,

52
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se pou n sèvi ak lèt “s” pou n reprezante distans/
Nan ka sa a, to varyasyon an,

53
00:04:51,001 --> 00:04:58,000
se sa nou rele vitès.

54
00:04:58,001 --> 00:05:03,000
Sa se 2 egzanp ki sèvi anpil,
e mwen ta renmen

55
00:05:03,001 --> 00:05:08,000
dekri dezyèm egzanp lan
ak plis detay

56
00:05:08,001 --> 00:05:10,000
paske mwen panse li enpòtan
pou nou konprann tout bon vre

57
00:05:10,001 --> 00:05:16,000
zafè vitès enstantane a.

58
00:05:16,001 --> 00:05:22,000
E mwen ka sèvi ak bilding kote
nou ye la a pou mwen fè sa.

59
00:05:22,001 --> 00:05:28,000
Nou ka deja konnen, petèt tou nou ka pa
konnen, chak ane pou fèt Latousen (Alowin),

60
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gen yon bagay ki fèt nan
tèt bilding sa a.

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00:05:32,001 --> 00:05:34,000
Se sa yo rele “lage joumou.”

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Yo lage yon joumou

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soti nan tèt bilding lan rive atè anba a.
Nou pral esplike lide to varyasyon sa a

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ak egzanp joumou k ap tonbe a.

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Men sa ki rive nan bilding sa a.
Ann gade bilding lan.

66
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Men yon pwen. Sa se bèl
gazon devan bilding lan.

67
00:06:03,001 --> 00:06:07,000
Epi men yon moun ak yon objè
nan men li. Objè a genlè piti.

68
00:06:07,001 --> 00:06:12,000
Men, obè pa vrèman piti
lè ou pre li.

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Yo jete objè a

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soti depi sou arebò a la a.

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Epi li tonbe.

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Nou konnen tout bagay nan M.I.T,
tout bagay oswa prèske tout bagay,

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se pwojè nan syans fizik.

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Se sa ki menmen zafè
lage joumou an.

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Alò bilding lan apeprè 300 pye

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nan wotè. Epi nou-menm nou bò isit la
nan premye etaj ki ka sèvi a.

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Olye de 300 pye, nap sèvi
ak 80 mèt

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pou kalkil yo

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ka pi fasil.

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00:06:55,001 --> 00:07:05,000
Alò nou gen wotè a ki kòmanse
a 80 mèt a lè t = 0.

81
00:07:05,001 --> 00:07:10,000
Epi, pou nou kalkile
akselerasyon akoz pezantè nou gen

82
00:07:10,001 --> 00:07:13,000
fòmil sa a pou h. h se wotè a.

83
00:07:13,001 --> 00:07:21,000
Alò, lè t=0, nou anwo nèt,
h se 80 mèt,

84
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nou mezire sa an mèt.

85
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Lè t=4, nou wè (5 * 4^2)
se 80.

86
00:07:32,001 --> 00:07:34,000
Mwen te chwazi chif sa yo
espre pou

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nou ka rive anba nèt.

88
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Alò, lide varyasyon mwayenn sa a,

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ki fè varyasyon mwayenn lan, oubyen
vitès mwayenn la, petèt nou ka rele l

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vitès mwayenn, se tan sa a
joumou an ap pran

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pou l tonbe. Se pral

92
00:08:06,001 --> 00:08:10,000
varyasyon nan h / varyasyon nan t

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E sa kòmanse kòm…
ak ki sa li kòmanse?

94
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Li kòmanse a 80, pa vre?

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E li fini a 0.

96
00:08:23,001 --> 00:08:26,000
Men, an reyalite, fòk nou fè sa al lanvè.

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Fòk nou konsidere 0-80 paske
premye chif la se dènye pozisyon an,

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epi dezyèm chif la,
se premye pozisyon an.

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E sa dwe divize pa 4 – 0;

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4 segonn mwens 0 segonn.

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Alò, natirèlman sa se -20 mèt
pa segonn.

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Alò vitès mwayenn nèg sa a
se 20 mèt pa segonn.

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Pou ki sa mwen chwazi egzanp sa a?

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00:09:00,001 --> 00:09:04,000
Paske, se nòmal, mwayenn lan,
menm si sa enteresan,

105
00:09:04,001 --> 00:09:06,000
sa pa vreman enteresan pou moun

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00:09:06,001 --> 00:09:08,000
k ap gade joumou sa a.

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00:09:08,001 --> 00:09:12,000
Sèl sa ki enterese nou vreman,
se vitès enstantane

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00:09:12,001 --> 00:09:22,000
lè joumou a frape atè.
Alò, nou ka kalkile vitès sa a

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lè li rive anba nèt.

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00:09:23,001 --> 00:09:25,000
Alò, ki sa vitès enstantane a ye?

111
00:09:25,001 --> 00:09:30,000
Se derive a, oubyen pou mwen kenbe menm
notasyon mwen te konmanse sèvi avè l deja a,

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se d/dt de h.

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Dakò?

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00:09:37,001 --> 00:09:39,000
Alò se d/dt de h.

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00:09:39,001 --> 00:09:42,000
Sonje nou gen fòmil pou
bagay sa yo.

116
00:09:42,001 --> 00:09:43,000
Nou ka kalkile derive.

117
00:09:43,001 --> 00:09:47,000
fonksyon sa a kounyea.
Nou te deja fè sa yè.

118
00:09:47,001 --> 00:09:51,000
Alò nou pral pran to varyasyon an
Epi si ou gade sa byen,

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to varyasyon 80 se 0,

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00:09:57,001 --> 00:10:02,000
mwens to varyasyon pou sa a
-5t^2, sa se -10t.

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00:10:02,001 --> 00:10:09,000
Alò, nou ka sèvi ak fòmil sa yo:
d/dt de 80 egal 0,

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00:10:09,001 --> 00:10:12,000
epi d/dt de t^2 egal 2t.

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00:10:12,001 --> 00:10:14,000
Ka sa a espesyal.

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00:10:14,001 --> 00:10:17,000
Men, m ap fè koken la a,
Gen yon ka espesyal

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00:10:17,001 --> 00:10:18,000
tout moun ka wè.

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00:10:18,001 --> 00:10:19,000
Mwen pa t mete li bò isit la.

127
00:10:19,001 --> 00:10:23,000
Ka n=2 se dezyèm ka ki la a.

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00:10:23,001 --> 00:10:30,000
Men, lè n=0 nou ka kalkile derive a tou.

129
00:10:30,001 --> 00:10:31,000
Paske sa se yon konstant.

130
00:10:31,001 --> 00:10:32,000
Derive yon konstan se 0.

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00:10:32,001 --> 00:10:36,000
Apre sa, faktè n lan la a
se 0, e nou toujou wè sa.

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00:10:36,001 --> 00:10:39,000
Kounyea, si ou gade fòmil ki sou tèt li
w ap wè

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00:10:39,001 --> 00:10:44,000
se ka kote n=-1.

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00:10:44,001 --> 00:10:49,000
Alò, nap gen yon pi gwo echantiyon tale,
lè nou rive nan kalkil ak puisans.

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00:10:49,001 --> 00:10:50,000
Oke.

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00:10:50,001 --> 00:10:54,000
Annou retounen bò isit la kote
nou gen to varyasyon nou an.

137
00:10:54,001 --> 00:10:55,000
Men sa li ye.

138
00:10:55,001 --> 00:11:00,000
Anba a, nan pwen kontak la,
nou gen

139
00:11:00,001 --> 00:11:07,000
t=4 epi h’, ki se derive a,

140
00:11:07,001 --> 00:11:12,000
egal -40 mèt pa segonn.

141
00:11:12,001 --> 00:11:19,000
Derive a 2 fwa pi vit ke vitès
mwayenn ki la a Si ou bezwen

142
00:11:19,001 --> 00:11:22,000
konvèti l, se apeprè
90 mil a lè.

143
00:11:22,001 --> 00:11:29,000
Se pou sa polis yo la a minui
lavèy fèt Latousen (“Alowin”)

144
00:11:29,001 --> 00:11:33,000
pou yo asire tout moun.
Epi tou, se pou sa

145
00:11:33,001 --> 00:11:37,000
fòk ou pare pou benyen
apre sa.

146
00:11:37,001 --> 00:11:40,000
An tou ka, se sa k pase.
Joumou a frape 90 mil a lè.

147
00:11:40,001 --> 00:11:42,000
An reyalite, bilding la
yon ti jan pi wo,

148
00:11:42,001 --> 00:11:46,000
gen reziztans van epi mwen sèten
nou ka fè yon rechèch

149
00:11:46,001 --> 00:11:50,000
ki pi konplè sou egzanp sa a.

150
00:11:50,001 --> 00:11:54,000
Bon. Kounyea mwen vle ban nou
de twa lòt egzanp

151
00:11:54,001 --> 00:11:58,000
paske tan ak kalite paramèt ak
varyab sa yo

152
00:11:58,001 --> 00:12:02,000
se pa sèl sa ki enpòtan nan
kalkil diferansyèl ak entegral.

153
00:12:02,001 --> 00:12:06,000
Si se te sèl kalite syans
fizik sa yo ki te ladan l,

154
00:12:06,001 --> 00:12:08,000
sijè sa a t ap pi
espesyalize anpil.

155
00:12:08,001 --> 00:12:13,000
Alò mwen vle ban nou kèk egzanp

156
00:12:13,001 --> 00:12:16,000
ki pa gen tan kòm varyab.

157
00:12:16,001 --> 00:12:21,000
Konsa twazyèm egzanp mwen
pral bay la se...

158
00:12:21,001 --> 00:12:31,000
Lèt t repranzate tanperati,
dt/dx reprezante

159
00:12:31,001 --> 00:12:38,000
sa ki rele gradyan tanperati.

160
00:12:38,001 --> 00:12:43,000
E sa enpòtan anpil lè
n ap prevwa meteyo

161
00:12:43,001 --> 00:12:46,000
paske se diferans tanperati sa yo

162
00:12:46,001 --> 00:12:52,000
ki lakoz van ak lòt chanjman nan atmosfè a.

163
00:12:52,001 --> 00:12:59,000
E gen yon lòt tèm enpòtan
ki nan tout syans ak jeni

164
00:12:59,001 --> 00:13:02,000
Mwen pral pale nan diskisyon

165
00:13:02,001 --> 00:13:15,000
sou sansibilite mezi.

166
00:13:15,001 --> 00:13:18,000
Ban m esplike sa.

167
00:13:18,001 --> 00:13:22,000
Mwen pa vle rete twò lontan sou sa
paske m ap montre nou sa

168
00:13:22,001 --> 00:13:26,000
jis pou mwen ka entwodui
kèk lide ki nan devwa nou yo.

169
00:13:26,001 --> 00:13:29,000
Nan premye pwoblèm nan devwa a,

170
00:13:29,001 --> 00:13:35,000
gen yon egzanp ki baze sou
yon sistèm pozisyònman global (“GPS”)

171
00:13:35,001 --> 00:13:39,000
ki senplifye nèt. Se yon GPS senp
kote latè tou plat.

172
00:13:39,001 --> 00:13:42,000
Epi nan sitiyasyon sa a,
si latè tou plat,

173
00:13:42,001 --> 00:13:45,000
se yon liy orizontal konsa.

174
00:13:45,001 --> 00:13:54,000
Apre sa, ou gen yon satelit,
ki bò isit la.

175
00:13:54,001 --> 00:14:03,000
Li ta pi bon si satelit la sou tèt late,
e si satelit la ka lokalize

176
00:14:03,001 --> 00:14:05,000
egazkteman pwen ki dirèkteman anba

177
00:14:05,001 --> 00:14:06,000
satelit la.

178
00:14:06,001 --> 00:14:12,000
Alò, pwen sa a se yon pwen
nou ka lokalize.

179
00:14:12,001 --> 00:14:24,000
E mwen chita la, avèk ti sistèm
pozisyonman global (“GPS”) mwen,

180
00:14:24,001 --> 00:14:26,000
e mwen vle konnen ki kote mwen ye.

181
00:14:26,001 --> 00:14:30,000
E jan pou mwen konnen ki kote mwen ye,
se pou m kominike

182
00:14:30,001 --> 00:14:36,000
ak satelit sa a gras ak siyal radyo.
Se kon sa mwen ka mezire distans sa a

183
00:14:36,001 --> 00:14:38,000
ki rele h.

184
00:14:38,001 --> 00:14:42,000
Epi sistem la ap kalkile
distans orizontal la

185
00:14:42,001 --> 00:14:47,000
ki se L.

186
00:14:47,001 --> 00:14:58,000
Alò, sa nou ka mezire se:

187
00:14:58,001 --> 00:15:04,000
h gras a siyal radio ak yon mont
oubyen plizyè mont.

188
00:15:04,001 --> 00:15:13,000
Apre sa, L depann de h.

189
00:15:13,001 --> 00:15:17,000
Epi, men sa ki enpòtan nan tout
sistèm sa yo:

190
00:15:17,001 --> 00:15:20,000
nou pa fouti konnen
ki sa h ye egzakteman.

191
00:15:20,001 --> 00:15:26,000
Gen yon erè nan h
nou pral reprezante ak delta h.

192
00:15:26,001 --> 00:15:31,000
Nou pa fouti konnen
ki sa h ye egzakteman.

193
00:15:31,001 --> 00:15:35,000
Sa ki bay pi gwo dout nan GPS
se nan iyonosfè a sa soti.

194
00:15:35,001 --> 00:15:38,000
Men, gen anpil koreksyon

195
00:15:38,001 --> 00:15:41,000
tout kalite ki fèt.

196
00:15:41,001 --> 00:15:43,000
Epi tou si ou nan
bilding lan,

197
00:15:43,001 --> 00:15:44,000
li difisil pou mezire l.

198
00:15:44,001 --> 00:15:48,000
Men se yon sijè ki
enpòtan anpil.

199
00:15:48,001 --> 00:15:49,000
Mwen pral eksplike sa talè.

200
00:15:49,001 --> 00:16:01,000
Lide a se pou nou konsidere
delta L.

201
00:16:01,001 --> 00:16:06,000
Nou estime l gras a konsiderasyon rapò
delta L / delta h ki

202
00:16:06,001 --> 00:16:11,000
apeprè menm ak derive L

203
00:16:11,001 --> 00:16:13,000
pa rapò a h.

204
00:16:13,001 --> 00:16:16,000
Alò sa byen fasil paske

205
00:16:16,001 --> 00:16:18,000
se kalkil diferansyèl
ak entegral.

206
00:16:18,001 --> 00:16:22,000
Kalkil diferansyèl ak entegral
se pati ki pi fasil la e li pèmèt nou

207
00:16:22,001 --> 00:16:28,000
konprann plizyè pwoblèm pratik
ki tou pre nou la a.

208
00:16:28,001 --> 00:16:31,000
Konsa, rezon ki fè sa dwe enterese nou anpil,
se paske

209
00:16:31,001 --> 00:16:34,000
sa sèvi tout tan pou
fè avyon ateri.

210
00:16:34,001 --> 00:16:38,000
Se pou sa fòk yo konnen kote avyon w lan ye,
ak ki wotè li ye, elatriye.

211
00:16:38,001 --> 00:16:42,000
E fòk erè nan kalkil distans sa yo

212
00:16:42,001 --> 00:16:48,000
pa depase kèk mèt.

213
00:16:48,001 --> 00:16:49,000
Dakò. Kounyea, nou fini

214
00:16:49,001 --> 00:16:51,000
ak entwodiksyon jeneral

215
00:16:51,001 --> 00:16:52,000
sou ki sa derive ye.

216
00:16:52,001 --> 00:16:54,000
Mwen sèten w ap fin abitye
ak sa

217
00:16:54,001 --> 00:16:56,000
nan anpil lòt diskisyon nan kou sa a.

218
00:16:56,001 --> 00:17:04,000
E kounyea fòk nou retounen
nan bon jan detay.

219
00:17:04,001 --> 00:17:09,000
Èske tout moun satisfè ak sa nou
rive fè jouk kounyea?

220
00:17:09,001 --> 00:17:10,000
Wi?

221
00:17:10,001 --> 00:17:13,000
Etidyan: Kòman ou te rive kalkile ekwazyon wotè a?

222
00:17:13,001 --> 00:17:14,000
Pwofesè: Aaa... Bèl kesyon.

223
00:17:14,001 --> 00:17:18,000
Kesyon an se koman mwen rive
kalkile ekwasyon wotè sa a?

224
00:17:18,001 --> 00:17:25,000
Mwen envante l paske se fòmil
nan syans fizik

225
00:17:25,001 --> 00:17:30,000
ou pral aprann lè ou pran klas 8.01.
Sa gen rapò ak vitès la.

226
00:17:30,001 --> 00:17:35,000
Si ou kalkile derive vitès,
sa ba w akselerasyon.

227
00:17:35,001 --> 00:17:39,000
Epi akselerasyon akoz pezantè,
se 10 mèt pa segonn.

228
00:17:39,001 --> 00:17:42,000
E sa se dezyèm derive sa a.

229
00:17:42,001 --> 00:17:44,000
An tou ka,

230
00:17:44,001 --> 00:17:46,000
mwen annik rale li

231
00:17:46,001 --> 00:17:48,000
nan kou syans fizik nou.

232
00:17:48,001 --> 00:17:55,000
Alò ou ka annik di:
al gade 8.01.

233
00:17:55,001 --> 00:18:02,000
Oke. Lòt kesyon?

234
00:18:02,001 --> 00:18:04,000
Dakò, ann kontinye.

235
00:18:04,001 --> 00:18:09,000
Kounyea fòk mwen pi sistematik
sou kesyon limit yo.

236
00:18:09,001 --> 00:18:20,000
Annou fè sa kounyea.

237
00:18:20,001 --> 00:18:30,000
Mwen vle pale de limit
ak kontinuite.

238
00:18:30,001 --> 00:18:34,000
E sa ap prepare nou
pou n derive tout lòt fòmil,

239
00:18:34,001 --> 00:18:38,000
tout lòt fòmil nou pral bezwen

240
00:18:38,001 --> 00:18:41,000
pou kalkile derive tout fonksyon nou konnen yo.

241
00:18:41,001 --> 00:18:44,000
Sonje, se objektif nou sa, e se
yon semenn ase ki rete,

242
00:18:44,001 --> 00:18:47,000
ki fè pito nou kòmanse.

243
00:18:47,001 --> 00:18:58,000
Alò premyèman gen sa
mwen ta rele “limit fasil”.

244
00:18:58,001 --> 00:19:00,000
Ki sa ki yon limit fasil?

245
00:19:00,001 --> 00:19:07,000
Men yon egzanp limit fasil:
limit lè x ap pwoche 4 de (x + 3/ x^2 + 1).

246
00:19:07,001 --> 00:19:09,000
limit lè x ap pwoche 4 de (x + 3/ x^2 + 1)

247
00:19:09,001 --> 00:19:11,000
limit lè x ap pwoche 4 de (x + 3/ x^2 + 1)

248
00:19:11,001 --> 00:19:16,000
Ak kalite limit sa a,
sèl sa mwen bezwen fè pou m evalye li,

249
00:19:16,001 --> 00:19:22,000
se mete x = 4.
Kon sa, sa mwen vin jwenn la a, se 4 + 3/ (4^2 + 1).

250
00:19:23,001 --> 00:19:24,000
se 4 + 3/ (4^2 + 1)

251
00:19:24,001 --> 00:19:27,000
se 4 + 3/ (4^2 + 1)

252
00:19:27,001 --> 00:19:31,000
E sa jis vin tounen 7/17.

253
00:19:31,001 --> 00:19:33,000
Epi sa tou fini la.

254
00:19:33,001 --> 00:19:38,000
Alò, sa se yon egzanp yon limit
ki fasil pou kalkile—yon limit fasil.

255
00:19:38,001 --> 00:19:43,000
Men yon dezyèm kategori limit.
Bon, sa se pa sèl dezyèm kategori limit ki egziste.

256
00:19:43,001 --> 00:19:45,000
Men, m jis vle pou nou note sa.

257
00:19:45,001 --> 00:19:55,000
Sa enpòtan anpil.

258
00:19:55,001 --> 00:19:59,000
Derive toujou pi difisil pase sa.

259
00:19:59,001 --> 00:20:03,000
Ou pa ka chape anba difikilte sa a.

260
00:20:03,001 --> 00:20:05,000
Alò, pou ki sa?

261
00:20:05,001 --> 00:20:08,000
Bon, lè ou pran yon derive,
ou pran

262
00:20:08,001 --> 00:20:20,000
lè x ap pwoche x0 de f(x).
Alò, n a ekri li

263
00:20:20,001 --> 00:20:24,000
nan tout bèlte li.

264
00:20:24,001 --> 00:20:28,000
Men fòmil pou derive a.

265
00:20:28,001 --> 00:20:39,000
Kounyea remake si nou mete
x = x0, sa toujou bay 0/0.

266
00:20:39,001 --> 00:20:42,000
Ki fè tou senpleman
sa pa janm mache.

267
00:20:42,001 --> 00:20:51,000
Ki fè nou pral toujou bezwen
yon anilasyon

268
00:20:51,001 --> 00:21:05,000
pou limit la sa gen sans.

269
00:21:05,001 --> 00:21:12,000
Kounyea, pou m ka rann bagay yo
yon jan pi fasil pou mwen,

270
00:21:12,001 --> 00:21:16,000
pou m ka esplike ki sa k ap pase
ak limit yo, fòk mwen esplike

271
00:21:16,001 --> 00:21:18,000
yon lòt ti notasyon.

272
00:21:18,001 --> 00:21:21,000
Sa mwen pral esplike la a
se sa yo rele

273
00:21:21,001 --> 00:21:23,000
limit a goch ak limit a dwat.

274
00:21:23,001 --> 00:21:27,000
Si mwen pran limit
lè x ap pwoche x0

275
00:21:27,001 --> 00:21:32,000
avèk la a yon siy plis la a pou yon fonksyon kèlkonk,
se sa yo rele

276
00:21:32,001 --> 00:21:42,000
limit a dwat.

277
00:21:42,001 --> 00:21:44,000
E mwen ka montre sa
ak yon ti shema.

278
00:21:44,001 --> 00:21:45,000
Alò, sa sa vle di?

279
00:21:45,001 --> 00:21:49,000
Sa vle di pratikman menm bagay
ak x ap pwoche x0

280
00:21:49,001 --> 00:21:51,000
Men, gen yon lòt restriksyon sou x,
ki gen rapò

281
00:21:51,001 --> 00:21:55,000
ak siy plis sa a, ki vle di
x ap soti sou kote pozitif x0.

282
00:21:55,001 --> 00:21:58,000
Sa vle di x pi gwo pase x0.

283
00:21:58,001 --> 00:22:01,000
E lè mwen di a dwat,
sa vle di fòk gen yon tirè la a.

284
00:22:01,001 --> 00:22:07,000
Limit la a dwat paske
sou liy chif yo, si x0 se la li ye,

285
00:22:07,001 --> 00:22:14,000
x ap sou bò dwat x0.

286
00:22:14,001 --> 00:22:15,000
Dakò?

287
00:22:15,001 --> 00:22:16,000
Alò, se sa ki limit a dwat la.

288
00:22:16,001 --> 00:22:20,000
E se sa ki se bò goch tablo a.
Epi sou bò dwat tablo a,

289
00:22:20,001 --> 00:22:23,000
se la m ap mete limit a goch la,
jis pou m lage nou

290
00:22:23,001 --> 00:22:24,000
nan konfizyon.

291
00:22:24,001 --> 00:22:30,000
Alò sa a pral gen siy mwens lan la a.
Mwen yon ti jan disleksik

292
00:22:30,001 --> 00:22:33,000
e mwen swete nou-menm nou pa disleksik.

293
00:22:33,001 --> 00:22:38,000
Ki fè se posib
mwen fè fot nan sa.

294
00:22:38,001 --> 00:22:41,000
Alò, se sa ki limit a goch la,
e m ap fè shema sa a.

295
00:22:41,001 --> 00:22:45,000
Ki fè sa jis vle di
x ap pwoche x0,

296
00:22:45,001 --> 00:22:48,000
men x ap sou bò goch x0.

297
00:22:48,001 --> 00:22:54,000
Epi ankò, sou liy chif yo,
men x0 epi x li-menm

298
00:22:54,001 --> 00:22:56,000
l ap sou lòt bò x0.

299
00:22:56,001 --> 00:22:59,000
Bon, de notasyon sa yo
pral ede nou

300
00:22:59,001 --> 00:23:01,000
klarifye plizyè bagay.

301
00:23:01,001 --> 00:23:06,000
Li pi fasil vre pou
nou gen tout deskripsyon sa yo

302
00:23:06,001 --> 00:23:10,000
sou afè limit yo, olye pou
nou jis konsidere an menm tan

303
00:23:10,001 --> 00:23:15,000
limit sou tou 2 bò yo.

304
00:23:15,001 --> 00:23:25,000
Dakò, konsa m vle bay
yon egzanp sou sa.

305
00:23:25,001 --> 00:23:30,000
Epi tou, yon egzanp
sou ki jan pou nou kalkile

306
00:23:30,001 --> 00:23:32,000
kalite pwoblèm sa yo.

307
00:23:32,001 --> 00:23:38,000
Ki fè m ap pran yon fonksyon
ki gen 2 definisyon diferan.

308
00:23:38,001 --> 00:23:39,000
Ann di x + 1,

309
00:23:39,001 --> 00:23:43,000
x + 1, lè x > 0,

310
00:23:43,001 --> 00:23:47,000
epi -x + 2, lè x < 0.

311
00:23:47,001 --> 00:23:51,000
Alò, petèt fòk nou mete yon vigil la.

312
00:23:51,001 --> 00:23:55,000
Ki fè lè x>0,

313
00:23:57,001 --> 00:23:58,000
se x + 1.

314
00:23:58,001 --> 00:24:01,000
Kounyea mwen ka desinen yon graf pou sa.

315
00:24:01,001 --> 00:24:04,000
Li pral yon jan piti paske
m bezwen

316
00:24:04,001 --> 00:24:07,000
foure li anba la a.
Men, petèt m a mete aks la anba a.

317
00:24:07,001 --> 00:24:14,000
Ki fè nan wotè 1, mwen gen
sou bò dwat la yon liy ki gen pant 1,

318
00:24:14,001 --> 00:24:16,000
ki fè li monte konsa.

319
00:24:16,001 --> 00:24:18,000
Dakò?

320
00:24:18,001 --> 00:24:24,000
Apre sa, sou bò goch la,
mwen genyen yon liy ki gen pant -1.

321
00:24:24,001 --> 00:24:30,000
Men, li vin kontre ak aks la nan 2
ki fè li anlè la a.

322
00:24:30,001 --> 00:24:33,000
Ki fè mwen gen
yon fòm antèn ki dwòl,

323
00:24:33,001 --> 00:24:35,000
Men graf la.

324
00:24:35,001 --> 00:24:43,000
Petèt m ta dwe ekri sa nan yon
lòt koulè pou sa parèt pi klè.

325
00:24:43,001 --> 00:24:50,000
Apre sa, si mwen kalkile 2 limit
sa yo la a, men sa mwen vin wè:

326
00:24:50,001 --> 00:25:00,000
limit lè x ap pwoche 0 sou bò dwat
de f(x),

327
00:25:00,001 --> 00:25:06,000
se menm jan ak limit lè x ap pwoche x0
de fòmil ki la a,

328
00:25:06,001 --> 00:25:07,000
x + 1.

329
00:25:07,001 --> 00:25:10,000
Sa vin tounen 1.

330
00:25:10,001 --> 00:25:15,000
E si mwen pran limit sa a,
ki fè se limit a goch la.

331
00:25:15,001 --> 00:25:20,000
Eskize m, m te di nou
mwen disleksik.

332
00:25:20,001 --> 00:25:23,000
Sa a se dwat la.
Alò se men dwat la.

333
00:25:23,001 --> 00:25:25,000
Ann re-derape.

334
00:25:25,001 --> 00:25:32,000
Kounyea m pral soti a goch,
e se f(x) ankò. Men, kounyea

335
00:25:32,001 --> 00:25:35,000
poutèt se sou bò sa a mwen ye,
mwen pral sèvi

336
00:25:35,001 --> 00:25:37,000
ak lòt fòmil lan,

337
00:25:37,001 --> 00:25:43,000
- x + 2, e sa a li menm ap ban nou 2.

338
00:25:43,001 --> 00:25:48,000
Kounyea, annou gade limit a goch
ak limit a dwat la, e sa

339
00:25:48,001 --> 00:25:51,000
se yon ti nyans, e se prèske
sèl bagay mwen bezwen

340
00:25:51,001 --> 00:25:54,000
pou nou rive suiv
ak anpil atansyon kounyea.

341
00:25:54,001 --> 00:26:06,000
Nou pa t bezwen valè x=0.

342
00:26:06,001 --> 00:26:11,000
Avrèdi m pa t menm janm di nou
sa f(0) te ye la a.

343
00:26:11,001 --> 00:26:14,000
Si nou foure l ladan l...
OK, nou ka foure l ladan l.

344
00:26:14,001 --> 00:26:20,000
Dakò, annou foure l
nan bò sa a.

345
00:26:20,001 --> 00:26:22,000
Annou fè pou l vin
sou bò sa a.

346
00:26:22,001 --> 00:26:32,000
Sa vle di pwen sa a anndan,
pwen sa a li-menm, li deyò.

347
00:26:32,001 --> 00:26:37,000
Sa se yon notasyon ki popilè:
ti sèk ki ouvri sa a

348
00:26:37,001 --> 00:26:41,000
epi ti pwen ki fèmen sa a
pou lè ou vle mete valè sa a anndan...

349
00:26:41,001 --> 00:26:46,000
Ki fè nan ka sa a,
valè f(x) vin menm

350
00:26:46,001 --> 00:26:56,000
ak limit a dwat la.
La a, valè a se 1. Se pa 2.

351
00:26:56,001 --> 00:27:01,000
Dakò, sa a se te premye
kalite egzanp lan.

352
00:27:01,001 --> 00:27:06,000
Kesyon?

353
00:27:06,001 --> 00:27:13,000
Dakò. Kounyea, pwochen
travay nou pral fè,

354
00:27:13,001 --> 00:27:17,000
se defini kontinuite

355
00:27:17,001 --> 00:27:20,000
Sa se te lòt sijè a.

356
00:27:20,001 --> 00:27:23,000
Se kon sa nou pral defini kontinuite:

357
00:27:23,001 --> 00:27:39,000
f kontini sou x0, sa vle di
limit f(x) lè x ap pwoche x0

358
00:27:39,001 --> 00:27:44,000
se f(x0).

359
00:27:44,001 --> 00:27:47,000
Pa vre?

360
00:27:47,001 --> 00:27:52,000
Alò rezon ki fè m
pase tout tan sa a ap okipe

361
00:27:52,001 --> 00:27:55,000
bò goch la ak bò dwat la, elatriye,
se paske m vle

362
00:27:55,001 --> 00:27:57,000
pou nou fè atansyon pou yon moman

363
00:27:57,001 --> 00:28:01,000
ak sa definisyon sa a
gen ladan l.

364
00:28:01,001 --> 00:28:12,000
Sa li vle di a se sa:
kontinuite nan x0

365
00:28:12,001 --> 00:28:17,000
gen divès engredyan.

366
00:28:17,001 --> 00:28:24,000
Men premye engredyan an:
fòk limit sa a egziste.

367
00:28:24,001 --> 00:28:28,000
E sa, sa vle di fòk limit la gen valè

368
00:28:28,001 --> 00:28:35,000
ni sou bò goch,
ni sou bò dwat.

369
00:28:35,001 --> 00:28:39,000
Epi tou fòk 2 valè sa yo menm.

370
00:28:39,001 --> 00:28:41,000
Dakò, ki fè se sa
k ap pase la a.

371
00:28:41,001 --> 00:28:50,000
Epi, men dezyèm pwopriyete a:
fòk f(x0) defini.

372
00:28:50,001 --> 00:28:53,000
Ki fè mwen pa ka nan yon sitiyasyon
kote mwen pa

373
00:28:53,001 --> 00:29:05,000
menm montre ki sa f(x0) ye.
Epi, fòk yo egal.

374
00:29:05,001 --> 00:29:09,000
Dakò, se sitiyasyon an sa.

375
00:29:09,001 --> 00:29:13,000
Kounyea, ban mwen reprann yon pati
ki gendwa lakòz ti konfizyon nan

376
00:29:13,001 --> 00:29:15,000
definisyon limit.

377
00:29:15,001 --> 00:29:20,000
Bò sa a, bò a goch la,
endepandan nèt,

378
00:29:20,001 --> 00:29:24,000
li sèvi ak yon definisyon
ki pa rantre

379
00:29:24,001 --> 00:29:25,000
nan definisyon bò dwat la.

380
00:29:25,001 --> 00:29:26,000
Sa se 2 bagay ki separe.

381
00:29:26,001 --> 00:29:32,000
Sa a li menm se, pou evalye li,
fòk ou toujou evite

382
00:29:32,001 --> 00:29:34,000
pwen limit la.

383
00:29:34,001 --> 00:29:37,000
Alò, sa se yon paradòks.
Sa se egzakteman

384
00:29:37,001 --> 00:29:41,000
kesyon an sa: èske se vre si
ou mete x0 pou x, w ap jwenn

385
00:29:41,001 --> 00:29:44,000
menm repons kòmsi ou te kalkile
limit lè x ap pwoche x0?

386
00:29:44,001 --> 00:29:46,000
Se kesyon sa a
n ap konsidere la a.

387
00:29:46,001 --> 00:29:49,000
Fòk nou fè distenksyon an
pou nou ka di

388
00:29:49,001 --> 00:29:55,000
sa yo se 2 bagay separe. Si se pa sa,
sa ap jis yon totoloji.

389
00:29:55,001 --> 00:29:56,000
Li pa gen okenn sans.

390
00:29:56,001 --> 00:29:58,000
Men, avrèdi li gen sans paske
youn nan bagay yo

391
00:29:58,001 --> 00:30:01,000
nou ka evalye li
pa rapò ak tout lòt pwen yo

392
00:30:01,001 --> 00:30:05,000
esepte x0. Epi lòt la,
nou ka evalye li dirèk-dirèk

393
00:30:05,001 --> 00:30:06,000
sou pwen x0 an menm.

394
00:30:06,001 --> 00:30:12,000
E vrèman, sa bagay sa yo ye,
se egzakteman

395
00:30:12,001 --> 00:30:18,000
egzanp limit ki fasil pou kalkile.
Se egzakteman

396
00:30:18,001 --> 00:30:19,000
sa n ap pale la a.

397
00:30:19,001 --> 00:30:24,000
Se yo nou ka evalye
nan fason sa a.

398
00:30:24,001 --> 00:30:25,000
Alò, fòk nou fè distenksyon an.

399
00:30:25,001 --> 00:30:27,000
E lòt sa yo se yo

400
00:30:27,001 --> 00:30:29,000
nou pa p ka evalye nan fason sa a.

401
00:30:29,001 --> 00:30:32,000
Ki fè se limit sa yo ki byennelve.
Se poutèt sa nou okipe yo.

402
00:30:32,001 --> 00:30:36,000
Se rezon sa a ki fè nou gen
tout definisyon sa yo pou yo.

403
00:30:36,001 --> 00:30:38,000
Dakò?

404
00:30:38,001 --> 00:30:40,000
Kounyea, ki sa nou pral fè?

405
00:30:40,001 --> 00:30:48,000
Bon, fòk nou fè nou fè yon ti vizit,
yon ti vizit byen kout,

406
00:30:48,001 --> 00:30:54,000
nan pak an dezòd kote sa nou rele
fonksyon diskonti yo rete.

407
00:30:54,001 --> 00:30:57,000
Alò sa se preske tout lòt fonksyon
ki pa kontini yo.

408
00:30:57,001 --> 00:31:04,000
Ki fè, premye egzanp lan la a,
ban mwen jis ekri l anba la a.

409
00:31:04,001 --> 00:31:13,000
Sa se egzanp diskontinuite ki sote.

410
00:31:13,001 --> 00:31:15,000
Ki sa yon diskontinuite ki sote ta ka ye?

411
00:31:15,001 --> 00:31:18,000
A vrè di, nou deja wè sa.

412
00:31:18,001 --> 00:31:21,000
Diskontinuite ki sote,
nou te wè sa nan egzanp

413
00:31:21,001 --> 00:31:23,000
nou te genyen la a.

414
00:31:23,001 --> 00:31:35,000
Se lè limit a goch ak limit a dwat,
tou lè 2 egziste.

415
00:31:35,001 --> 00:31:42,000
Men, yo pa egal.

416
00:31:42,001 --> 00:31:51,000
Dakò, ki fè sa te nan
egzanp lan.

417
00:31:51,001 --> 00:31:52,000
Pa vre?

418
00:31:52,001 --> 00:31:53,000
Nan egzanp sa a, 2 limit yo,

419
00:31:53,001 --> 00:31:57,000
youn ladan yo te 1,
epi lòt la te 2.

420
00:31:57,001 --> 00:32:02,000
Alò, se sa ki diskontinuite ki sote.

421
00:32:02,001 --> 00:32:09,000
E kalite pwoblèm sa a,
kesyon si yon fonksyon kontini ou pa,

422
00:32:09,001 --> 00:32:21,000
sa gen dwa parèt yon ti jan teknik.
Men, se yon pwoblèm ki te bay anpil moun anpil pwoblèm.

423
00:32:21,001 --> 00:32:26,000
yon pwoblèm ki te bay anpil moun anpil pwoblèm.

424
00:32:26,001 --> 00:32:30,000
Bob Merton, ki te yon pwofesè
nan MIT lè li te fè travay

425
00:32:30,001 --> 00:32:35,000
ki te ba li pri Nobèl nan ekonomi,
te enterese nan

426
00:32:35,001 --> 00:32:39,000
kesyon sa a menm:
Èske pri diferan kalite estòk kontini

427
00:32:39,001 --> 00:32:42,000
sou bò goch ouswa sou bò dwat
nan divès fonksyon.

428
00:32:42,001 --> 00:32:46,000
E sa se te yon bagay ki te
enpòtan nan devlopman modèl

429
00:32:46,001 --> 00:32:50,000
ki deside pri nou peye
pou envestisman nou fè

430
00:32:50,001 --> 00:32:51,000
tout tan kounyea.

431
00:32:51,001 --> 00:32:57,000
Ki fè bò goch ak bò dwat ka vle di 2 bagay
trè diferan. Nan ka sa a, bò goch la reprezante

432
00:32:57,001 --> 00:33:01,000
sa ki te deja fèt nan lepase, epi bò dwat la
reprezante sa ki pral fèt nan lavni.

433
00:33:01,001 --> 00:33:04,000
Se konsa vin gen yon gwo diferans
si fonksyon an kontini

434
00:33:04,001 --> 00:33:06,000
a goch oswa a dwat.

435
00:33:06,001 --> 00:33:09,000
Èske pwen an la a menm,
yon kote nan mitan,

436
00:33:09,001 --> 00:33:11,000
yon lòt kote ?

437
00:33:11,001 --> 00:33:13,000
Sa a se yon kesyon enpòtan.

438
00:33:13,001 --> 00:33:18,000
Alò pwochen egzanp mwen vle
ban nou an

439
00:33:18,001 --> 00:33:22,000
yon ti jan pi difisil.

440
00:33:22,001 --> 00:33:32,000
Se sa yo rele
« diskontuinite ki pwolonjab ».

441
00:33:32,001 --> 00:33:39,000
E sa sa vle di, sa vle di limit a goch

442
00:33:39,001 --> 00:33:46,000
ak limit a dwat, tou lè 2 egal.

443
00:33:46,001 --> 00:33:48,000
Ki fè si ou fè yon graf,
ou gen yon fonksyon

444
00:33:48,001 --> 00:33:52,000
k ap vini konsa, epi li gen yon twou.
Ki kote? Sa k konnen.

445
00:33:52,001 --> 00:33:55,000
Swa petèt fonksyon an pa defini
ouswa petèt li defini

446
00:33:55,001 --> 00:33:58,000
anwo la a, epi apre li jis
kontinye ale.

447
00:33:58,001 --> 00:33:59,000
Dakò?

448
00:33:59,001 --> 00:34:01,000
Ki fè 2 limit yo se menm bagay.

449
00:34:01,001 --> 00:34:05,000
Epi apre, sa tou natirèl, fonksyon an
se sipliye l ap sipliye pou nou redefini l

450
00:34:05,001 --> 00:34:07,000
pou nou ka retire twou sa a.

451
00:34:07,001 --> 00:34:14,000
E se pou sa nou rele sa
yon « diskontuinite ki pwolonjab ».

452
00:34:14,001 --> 00:34:18,000
Kounyea, ban m ban nou
yon egzanp sou sa, ou pito

453
00:34:18,001 --> 00:34:22,000
de twa egzanp.

454
00:34:22,001 --> 00:34:28,000
Sa yo se egzanp ki
enpòtan. Nou pral

455
00:34:28,001 --> 00:34:34,000
travay avèk yo toutalè.

456
00:34:34,001 --> 00:34:41,000
Premye egzanp lan se fonksyon
g(x), ki se sinx / x.

457
00:34:41,001 --> 00:34:50,000
Dezyèm egzanp lan se fonksyon
h(x), ki se 1-cosx / x.

458
00:34:50,001 --> 00:35:00,000
Alò, nou gen yon pwoblèm nan g(0):
g(0) pa defini.

459
00:35:00,001 --> 00:35:03,000
An tou ka, fonksyon sa a gen sa yo rele
yon sengilarite ki pwolonjab.

460
00:35:03,001 --> 00:35:14,000
Ki fè limit lè x ap pwoche 0 de sinx / x,
limit sa a egziste.

461
00:35:14,001 --> 00:35:17,000
Avrèdi, li egal a 1.

462
00:35:17,001 --> 00:35:19,000
Alò, sa se yon limit ki vreman enpòtan.
Nou pral travay sou li

463
00:35:19,001 --> 00:35:22,000
oswa nan fen leson sa a,
oswa nan kòmansman

464
00:35:22,001 --> 00:35:23,000
pwochen leson an.

465
00:35:23,001 --> 00:35:35,000
Epi tou, limit lè x a pwoche 0
de 1-cosx / x, limit sa a se 0.

466
00:35:35,001 --> 00:35:38,000
Petèt m a mete sa
yon ti jan pi lwen

467
00:35:38,001 --> 00:35:40,000
pou nou ka li l.

468
00:35:40,001 --> 00:35:45,000
Dakò. Alò, sa yo se done
nou pral bezwen

469
00:35:45,001 --> 00:35:47,000
pi devan.

470
00:35:47,001 --> 00:35:51,000
E sa yo di sèke
fonksyon sa yo gen sengilarite

471
00:35:51,001 --> 00:36:04,000
ki pwolonjab—eskize, diskontinuite
ki prolonjab—nan x=0.

472
00:36:04,001 --> 00:36:13,000
Dakò. Alò, jan m di a,
n ap tounen sou sa toutalè.

473
00:36:13,001 --> 00:36:16,000
Dakò. Ki fè èske gen okenn
kesyon anvan mwen kontinye?

474
00:36:16,001 --> 00:36:30,000
Wi?
Etidyan: (Son pa klè)

475
00:36:30,001 --> 00:36:38,000
Pwofesè: Kesyon an se:
Pou ki sa sa vre?

476
00:36:38,001 --> 00:36:40,000
Se sa kesyon ou an?

477
00:36:40,001 --> 00:36:44,000
Repons lan vreman vreman ta dwe
parèt li pa klè. Mwen poko montre nou sa.

478
00:36:44,001 --> 00:36:49,000
E si nou pa t sezi wè sa,
sa twa

479
00:36:49,001 --> 00:36:51,000
dwòl nèt.

480
00:36:51,001 --> 00:36:53,000
Alò, nou poko esplike sa.

481
00:36:53,001 --> 00:36:55,000
Fòk nou rete branche jisaske
nou esplike sa.

482
00:36:55,001 --> 00:36:57,000
Dakò?

483
00:36:57,001 --> 00:36:59,000
Nou poko montre sa.

484
00:36:59,001 --> 00:37:02,000
E an reyalite menm lòt pawòl sa a,
ki pral

485
00:37:02,001 --> 00:37:05,000
kontinye parèt pi dwòl toujou,
nou poko esplike sa non plis.

486
00:37:05,001 --> 00:37:09,000
Dakò. Ki fè nou pral rive sou sa,
jan mwen di a,

487
00:37:09,001 --> 00:37:15,000
nan fen leson sa a oswa
nan kòmansman pwochen leson an.

488
00:37:15,001 --> 00:37:22,000
Gen lòt kesyon?

489
00:37:22,001 --> 00:37:29,000
Dakò. Annou kontinye vizit

490
00:37:29,001 --> 00:37:34,000
pak diskontinuite yo.

491
00:37:34,001 --> 00:37:38,000
E mwen kwè mwen vle sèvi

492
00:37:38,001 --> 00:37:42,000
ak ni limit a dwat ni limit a goch.

493
00:37:42,001 --> 00:37:52,000
Ki fè m ap kenbe sa ki sou tablo sa a
sou limit a dwat ak limit a goch.

494
00:37:52,001 --> 00:37:55,000
Alò, yon twazyèm kategori diskontinuite,
se sa yo rele

495
00:37:55,001 --> 00:38:07,000
« diskontinuite enfini ».

496
00:38:07,001 --> 00:38:11,000
E nou deja kwaze
ak youn nan sa yo.

497
00:38:11,001 --> 00:38:14,000
M pral desine yo
bò isit la.

498
00:38:14,001 --> 00:38:19,000
Sonje fonksyon y se 1/x.

499
00:38:19,001 --> 00:38:22,000
Se fonksyon sa a ki la a.

500
00:38:22,001 --> 00:38:26,000
Men, kounyea mwen ta renmen
desinen lòt branch lòt branch ipèbòl la anba la a.

501
00:38:26,001 --> 00:38:31,000
Epi mwen pral konsidere
valè negatif

502
00:38:31,001 --> 00:38:32,000
pou x.

503
00:38:32,001 --> 00:38:35,000
Alò, men graf pou 1/x.

504
00:38:35,001 --> 00:38:42,000
E fasilite nou genyen pou nou distenge
ant limit a goch ak limit a dwat,

505
00:38:42,001 --> 00:38:46,000
sa enpòtan vre paske la a

506
00:38:46,001 --> 00:38:49,000
mwen ka ekri
limit lè x ap pwoche 0 de 1/x.

507
00:38:49,001 --> 00:38:51,000
limit lè x ap pwoche 0 de 1/x.

508
00:38:51,001 --> 00:38:57,000
Ki fè sa a li menm, l ap sòti
de bò dwat la epi l ap monte.

509
00:38:57,001 --> 00:39:00,000
Alò limit sa a se enfini.

510
00:39:00,001 --> 00:39:07,000
Alòske, limit nan lòt direksyon an,
limit a goch la,

511
00:39:07,001 --> 00:39:10,000
sa a li menm, l ap desann.

512
00:39:10,001 --> 00:39:16,000
E sa a, li diferan nèt.
Sa se mwens enfini.

513
00:39:16,001 --> 00:39:19,000
Kounyea gen de moun ki di
limit sa yo pa defini.

514
00:39:19,001 --> 00:39:22,000
Men, tout bon vre, yo prale
nan yon direksyon ki byen defini.

515
00:39:22,001 --> 00:39:26,000
Depi sa posib, fòk nou pote bon jan presizyon

516
00:39:25,966 --> 00:39:25,966
sou sa limit sa yo ye.

517
00:39:26,001 --> 00:39:30,000
An tou ka, pawòl ki di
limit lè x ap pwoche 0 de 1/x

518
00:39:30,001 --> 00:39:37,000
se enfini, pawòl sa a pa kòrèk menm.

519
00:39:37,001 --> 00:39:40,000
Oke, sa pa vle di
moun pa ekri sa.

520
00:39:40,001 --> 00:39:41,000
Sa pa kòrèk menm

521
00:39:41,001 --> 00:39:43,000
Sa pa vle di yo pa ekri l.

522
00:39:43,001 --> 00:39:45,000
Siman ou dwe deja wè bagay kon sa.

523
00:39:45,001 --> 00:39:47,000
Se paske moun sa yo annik konsidere

524
00:39:47,001 --> 00:39:48,000
branch a dwat la sèlman.

525
00:39:48,001 --> 00:39:51,000
Yo pa fè yon fot yo nesesèman

526
00:39:51,001 --> 00:39:53,000
Men, an tou ka, gen neglijans.

527
00:39:53,001 --> 00:39:56,000
E gen kèk neglijans nou ka sipòte.
Epi gen kèk lòt,

528
00:39:56,001 --> 00:39:57,000
fòk nou eseye evite.

529
00:39:57,001 --> 00:40:00,000
Konsa la a, ou vle di sa,
epi sa fè yon diferans.

530
00:40:00,001 --> 00:40:04,000
Pa egzanp, plis enfini se ka yon kantite enfini
lajan ou gen nan pòch ou,

531
00:40:04,001 --> 00:40:07,000
Men, mwens enfini se ka yon kantite enfini
lajan ou dwe.

532
00:40:07,001 --> 00:40:08,000
Sa se yon gwo diferans.

533
00:40:08,001 --> 00:40:09,000
Yo pa menm menm.

534
00:40:09,001 --> 00:40:13,000
Konsa, sa a se neglijans,

535
00:40:13,001 --> 00:40:15,000
e sa a vreman pi kòrèk.

536
00:40:15,001 --> 00:40:18,000
Oke. Kounyea, an plis de sa,
m vle montre

537
00:40:18,001 --> 00:40:21,000
yon lòt bagay.

538
00:40:21,001 --> 00:40:24,000
Sonje nou te kalkile
derive a.

539
00:40:24,001 --> 00:40:25,000
Se te -1/x^2.

540
00:40:26,001 --> 00:40:31,000
Men, mwen vle trase graf sa
a epi m ap fè

541
00:40:31,001 --> 00:40:32,000
kòmantè sou li.

542
00:40:32,001 --> 00:40:36,000
Kidonk, mwen pral trase graf la
dirèkteman anba

543
00:40:36,001 --> 00:40:38,000
graf fonksyon an.

544
00:40:38,001 --> 00:40:41,000
Epi remake byen
ki sa graf sa a ye.

545
00:40:41,001 --> 00:40:48,000
Li fèt konsa, li toujou negatif,
e li pwente sou anba.

546
00:40:48,001 --> 00:40:52,000
Konsa, kounyea, sa ka parèt
yon jan dwòl

547
00:40:52,001 --> 00:40:57,000
pou derive fonksyon sa a se nèg sa a.
Men se akòz

548
00:40:57,001 --> 00:40:58,000
yon bagay ki enpòtan anpil.

549
00:40:58,001 --> 00:41:01,000
E nou dwe toujou sonje sa
lè n ap pale de derive.

550
00:41:01,001 --> 00:41:03,000
Derive yon fonksyon a pa oblije sanble ak

551
00:41:03,001 --> 00:41:04,000
fonksyon an ditou.

552
00:41:04,001 --> 00:41:07,000
Konsa, ou dwe retire
lide sa a nan tèt ou.

553
00:41:07,001 --> 00:41:10,000
Gen kèk moun ki panse
si yon fonksyon desann, derive a

554
00:41:10,001 --> 00:41:11,000
dwe desann tou.

555
00:41:11,001 --> 00:41:13,000
Annik bliye entuisyon sa a.

556
00:41:13,001 --> 00:41:14,000
Li pa kòrèk.

557
00:41:14,001 --> 00:41:20,000
Sa n ap travay sou li la a,
si ou sonje, se pant lan.

558
00:41:20,001 --> 00:41:24,000
Konsa si ou gen yon pant la a,
ki koresponn senpleman a yon kote la a,

559
00:41:24,001 --> 00:41:29,000
e pandan pant lan
ap vin yon ti jan mwen rèd,

560
00:41:29,001 --> 00:41:31,000
se pou sa n ap pwoche

561
00:41:31,001 --> 00:41:33,000
aks orizontal la.

562
00:41:33,001 --> 00:41:36,000
Kantite a ap vin pi piti
plis n ap pwoche.

563
00:41:36,001 --> 00:41:41,000
Kounyea, bò isit la,
pant lan negatif tou.

564
00:41:41,001 --> 00:41:43,000
L ap desann, e pandan n ap
desann la a, l ap vin

565
00:41:43,001 --> 00:41:44,000
pi negatif toujou.

566
00:41:44,001 --> 00:41:48,000
Pandan n a prale la a nan pant lan
fonksyon sa a ap monte.

567
00:41:48,001 --> 00:41:50,000
Men, pant li ap desann.

568
00:41:50,001 --> 00:41:55,000
Dakò, konsa pant lan desann
sou de bò yo. Epi notasyon

569
00:41:55,001 --> 00:42:03,000
nou sèvi pou sa byen koresponn

570
00:42:03,001 --> 00:42:09,000
ak afè bò goch ak bò dwat.

571
00:42:09,001 --> 00:42:16,000
Sètadi, limit lè x ap pwoche 0
de -1/ x^2, sa pral egal

572
00:42:16,001 --> 00:42:18,000
mwens enfini.

573
00:42:18,001 --> 00:42:21,000
E sa se vre ni pou x lè l ap pwoche 0+

574
00:42:21,001 --> 00:42:24,000
ni pou x lè l ap pwoche 0-.

575
00:42:24,001 --> 00:42:31,000
Konsa tou lè 2 limit yo
gen pwopriyete sa a.

576
00:42:31,001 --> 00:42:34,000
Finalman, m ap fè yon dènye kòmantè

577
00:42:34,001 --> 00:42:37,000
sou 2 graf sa yo.

578
00:42:37,001 --> 00:42:43,000
Fonksyon sa a se yon fonksyon enpè.
Lè ou kalkile derive

579
00:42:43,001 --> 00:42:45,000
yon fonksyon enpè,
sa toujou ba wou

580
00:42:45,001 --> 00:42:50,000
yon fonksyon pè.

581
00:42:50,001 --> 00:42:54,000
Sa byen koresponn ak afè 1/x
se yon puisans enpè

582
00:42:54,001 --> 00:43:01,000
epi x^1 se yon puisans enpè
epi x^2 se yon puisans pè.

583
00:43:01,001 --> 00:43:05,000
Ak tou sa, entuisyon nou ta dwe
ranfòse lide nou genyen

584
00:43:05,001 --> 00:43:11,000
ki di graf sa yo kòrèk.

585
00:43:11,001 --> 00:43:16,000
Kounyea, gen yon dènye
kategori diskontinuite

586
00:43:16,001 --> 00:43:27,000
m vle mansyone rapidman
e m ap rele yo

587
00:43:27,001 --> 00:43:33,000
« lòt diskontinuite ki lèd ».

588
00:43:33,001 --> 00:43:39,000
E yo anpil anpil.

589
00:43:39,001 --> 00:43:44,000
Men yon egzanp :
y = sin 1/x

590
00:43:44,001 --> 00:43:50,000
y = sin 1/x lè x ap pwoche 0.

591
00:43:50,001 --> 00:43:59,000
E sa sanble yon bagay prèske konsa.

592
00:43:59,001 --> 00:44:00,000
Ale vini, ale vini.

593
00:44:00,001 --> 00:44:06,000
Li varye a lenfini lè x ap pwoche 0.

594
00:44:06,001 --> 00:44:19,000
Pa gen ni limit a goch
ni limit a dwat nan ka sa a.

595
00:44:19,001 --> 00:44:25,000
Se konsa gen youn voum ak yon pakèt
fonksyon ki nan ka sa a.

596
00:44:25,001 --> 00:44:29,000
Erezman, nou pa pral okipe yo
nan kou sa a.

597
00:44:29,001 --> 00:44:32,000
Nan lavi nou, anpil fwa,
gen fonksyon k ap pede monte-desann

598
00:44:32,001 --> 00:44:35,000
lè tan ap pwoche enfini.
Men, nou pa p fatige tèt nou

599
00:44:35,001 --> 00:44:40,000
ak sa kounyea.

600
00:44:40,001 --> 00:44:49,000
Oke, sa se dènye pale
n ap fè sou diskontinuite.

601
00:44:49,001 --> 00:44:56,000
Kounyea mwen bezwen tabli
baz pou fòmil

602
00:44:56,001 --> 00:44:59,000
pou pwochen leson an.

603
00:44:59,001 --> 00:45:09,000
Mwen vle gade avè nou
yon lide enpòtan,

604
00:45:09,001 --> 00:45:10,000
yon zouti pou limit.

605
00:45:10,001 --> 00:45:12,000
Konsa, sa pral tounen yon teyorèm.

606
00:45:12,001 --> 00:45:17,000
Erezman se yon teyorèm tou kout.

607
00:45:17,001 --> 00:45:19,000
E li gen yon prèv tou kout.

608
00:45:19,001 --> 00:45:22,000
Teyorèm lan rele

609
00:45:22,001 --> 00:45:28,000
«si yon fonksyon gen derive,
fonksyon an gen kontinuite ».

610
00:45:28,001 --> 00:45:33,000
E men sa teyorèm lan di :
Si f gen derive,

611
00:45:33,001 --> 00:45:41,000
sa vle di, si derive a egziste nan x0,

612
00:45:41,001 --> 00:45:59,000
, f kontini nan x0.

613
00:45:59,001 --> 00:46:02,000
Konsa, nou pral bezwen teyorèm sa a
tankou yon zouti,

614
00:46:02,001 --> 00:46:05,000
se yon etap enpòtan
nan règ pwodui ak kosyan.

615
00:46:05,001 --> 00:46:12,000
Alò, mwen ta renmen
bay prèv teyorèm la kounyea.

616
00:46:12,001 --> 00:46:16,000
Bon, men prèv la.

617
00:46:16,001 --> 00:46:20,000
Erezman, prèv la
se sèlman yon grenn liy.

618
00:46:20,001 --> 00:46:24,000
Dabò, m vle byen ekri sa nou bezwen

619
00:46:24,001 --> 00:46:27,000
tcheke a. Men sa nou bezwen

620
00:46:27,001 --> 00:46:31,000
tcheke a : èske limit,

621
00:46:31,001 --> 00:46:41,000
èske limit lè x ap pwoche x0, de f(x)
- f(x0), sa egal 0 ?

622
00:46:41,001 --> 00:46:42,000
Wi, se sa nou vle tcheke.

623
00:46:42,001 --> 00:46:45,000
Nou poko konn si se vre.
Men, n ap eseye tcheke

624
00:46:45,001 --> 00:46:47,000
si sa se vre ou si se pa vre.

625
00:46:47,001 --> 00:46:50,000
Sa se menm ak pawòl ki di
fonksyon an kontini

626
00:46:50,001 --> 00:46:54,000
paske limit f(x) la
sipoze egal f(x0),

627
00:46:54,001 --> 00:46:59,000
e konsa diferans sa a
dwe gen limit 0.

628
00:46:59,001 --> 00:47:03,000
E kounyea, jan pou nou pwouve sa, se annik reekri li

629
00:47:03,001 --> 00:47:09,000
apre nou miltipliye
epi divize li pa (x-x0).

630
00:47:09,001 --> 00:47:18,000
Konsa, m ap re-ekri limit lè x ap pwoche 0
de f(x ) – f (x0 ) divize pa x-x0 miltipliye pa x-x0.

631
00:47:18,001 --> 00:47:25,000
limit lè x ap pwoche 0 de f(x ) – f (x0 )
divize pa x-x0 miltipliye pa x-x0.

632
00:47:25,001 --> 00:47:29,000
Oke, m te ekri menm espresyon
mwen te genyen la a.

633
00:47:29,001 --> 00:47:31,000
Se menm limit la.

634
00:47:31,001 --> 00:47:38,000
Men, mwen te ni miltipliye l
ni divize l pa (x-x0).

635
00:47:38,001 --> 00:47:45,000
Epi kounyea, lè m kalkile limit la, men sa
k rive: limit premye faktè a se f’(x0).

636
00:47:45,001 --> 00:47:47,000
limit premye faktè a se f’(x0).

637
00:47:48,001 --> 00:47:53,000
Se sa nou konnen ki egziste
an patan, dapre sipozisyon nou.

638
00:47:53,001 --> 00:48:00,000
Epi limit dezyèm faktè a
se 0 paske

639
00:48:00,001 --> 00:48:06,000
limit lè x ap pwoche x0
de (x-x0) se 0.

640
00:48:06,001 --> 00:48:09,000
Se byen sa.

641
00:48:09,001 --> 00:48:12,000
Repons lan se 0,
e se byen sa nou te vle.

642
00:48:12,001 --> 00:48:14,000
Kidonk se prèv la.

643
00:48:14,001 --> 00:48:19,000
Kounyea, gen yon bagay
ki pa klè ditou nan prèv sa a.

644
00:48:19,001 --> 00:48:26,000
Kite m montre nou sa
anvan nou kontinye.

645
00:48:26,001 --> 00:48:33,000
Sa vle di, nou abitye
ak limit kote x egal x0.

646
00:48:33,001 --> 00:48:35,000
E sa sanble n ap miltipliye,
epi pa 0.

647
00:48:35,001 --> 00:48:40,000
Sa se yon operasyon ki gate prèv

648
00:48:40,001 --> 00:48:43,000
nan nenpòt ki sitiyasyon aljebrik.

649
00:48:43,001 --> 00:48:45,000
Yo te montre nou
sa pa ta ka janm mache.

650
00:48:45,001 --> 00:48:47,000
Dakò.

651
00:48:47,001 --> 00:48:52,000
Men, kalkil limit sa yo jwenn yon jan
pou yo kabre pwoblèm sa a,

652
00:48:52,001 --> 00:48:55,000
e mwen pral byen esplike
ki jan sa fèt.

653
00:48:55,001 --> 00:49:03,000
Nan kalkil limit sa a,
nou pa janm sèvi ak x=x0.

654
00:49:03,001 --> 00:49:06,000
Se egzakteman valè x nou pa konsidere

655
00:49:06,001 --> 00:49:09,000
ditou nan limit sa a.

656
00:49:09,001 --> 00:49:11,000
Se konsa nou kalkile limit.

657
00:49:11,001 --> 00:49:15,000
E se sa ki te tèm lan
jodi a jouk kounyea :

658
00:49:15,001 --> 00:49:18,000
nou pa dwe konsidere x=x0,
e sa fè miltiplikasyon

659
00:49:18,001 --> 00:49:21,000
ak divizyon pa kantite sa a legal.

660
00:49:21,001 --> 00:49:23,000
Kantite sa a ka piti.
Men, li pa janm 0.

661
00:49:25,001 --> 00:49:28,000
Kidonk, sa mache tout bon vre.

662
00:49:28,001 --> 00:49:32,000
Epi nou fèk sot montre ki jan yon fonksyon
ki gen derive se yon fonksyon ki kontini.

663
00:49:32,001 --> 00:49:38,000
Konsa, m ap gen pou m kontinye
ak limit sa yo ki enteresan anpil,

664
00:49:38,001 --> 00:49:42,000
limit nan pwochèn leson an.

665
00:49:42,001 --> 00:49:46,000
Men, an nou fè yon kanpe pou
yon segonn pou n wè si gen kesyon

666
00:49:46,001 --> 00:49:47,000
anvan nou rete.

667
00:49:47,001 --> 00:49:48,000
Wi, gen kesyon.

668
00:49:48,001 --> 00:49:49,000
Etidyan (Son an pa klè)

669
00:49:49,001 --> 00:49:53,000
[Son pa klè]

670
00:49:53,001 --> 00:50:00,000
Pwofesè: Repete prèv sa a la a?

671
00:50:00,001 --> 00:50:02,000
Repete sa w di a.

672
00:50:02,001 --> 00:50:08,000
Etidyan: [Son pa klè]

673
00:50:08,001 --> 00:50:13,000
Pwofesè: Oke, konsa gen 2 etap
nan prèv la.

674
00:50:13,001 --> 00:50:17,000
Etap w ap poze kesyon
sou li a se premye etap la.

675
00:50:17,001 --> 00:50:18,000
Kòrèk?

676
00:50:18,001 --> 00:50:21,000
E mwen sa m ap te di:
si ou gen yon kantite epi

677
00:50:21,001 --> 00:50:24,000
ou miltipliye kantite sa a pa 10/10,
se menm kantite a.

678
00:50:24,001 --> 00:50:26,000
Si ou miltipliye l pa 3/3,
se menm menm kantite la.

679
00:50:26,001 --> 00:50:30,000
2/2, 1/1 epi ale nèt.

680
00:50:30,001 --> 00:50:32,000
Konsa, sa kòrèk si
ou chanje sa a pou sa a.

681
00:50:32,001 --> 00:50:34,000
Se menm bagay la egzakteman.

682
00:50:34,001 --> 00:50:36,000
Se premye etap la.

683
00:50:36,001 --> 00:50:41,000
Etidyan: ( Son pa klè)

684
00:50:41,001 --> 00:50:45,000
Pwofesè: Chhhhhhh.

685
00:50:45,001 --> 00:50:52,000
Kesyon an, se kouman prèv la,
kouman liy sa a ye?

686
00:50:52,001 --> 00:50:53,000
Kote pwen entewogasyon an ye a.

687
00:50:53,001 --> 00:50:56,000
Konsa, sa m te tcheke se si
kantite sa a ki sou bò goch la

688
00:50:56,001 --> 00:51:02,000
egal a kantite sa a ki long e ki
konplike a

689
00:51:02,001 --> 00:51:06,000
ki egal a kantite sa a ki egal a
kantite sa a.

690
00:51:06,001 --> 00:51:08,000
Kidonk mwen tcheke pou wè si
kantite sa a egal paske

691
00:51:08,001 --> 00:51:12,000
dènye bagay la se 0.

692
00:51:12,001 --> 00:51:16,000
Sa a egal sa a ki egal sa a
ki egal 0.

693
00:51:16,001 --> 00:51:17,000
E sa se prèv la.

694
00:51:17,001 --> 00:51:21,000
Etidyan: ( Son pa klè)

695
00:51:21,001 --> 00:51:30,000
Pwofesè: Sa se yon kesyon diferan.

696
00:51:30,001 --> 00:51:36,000
Oke. Mwen sèvi ak ipotèz
diferansyèl la

697
00:51:36,001 --> 00:51:39,000
paske limit sa a egal
a kantite sa a.

698
00:51:39,001 --> 00:51:40,000
Limit sa a egziste.

699
00:51:40,001 --> 00:51:44,000
Se konsa m sèvi ak
ipotèz teyorèm lan.

700
00:51:44,001 --> 00:51:47,000
Konklizyon teyorèm lan
se se menm ak sa a paske

701
00:51:47,001 --> 00:51:52,000
kontinuite se menm ak
limit lè x ap pwoche x0

702
00:51:52,001 --> 00:51:56,000
limit lè x ap pwoche x0 de f( x0)

703
00:51:56,001 --> 00:51:57,000
Se definisyon kontinuite.

704
00:51:57,001 --> 00:52:02,000
E m te retire f(x0) nan
tou le 2 bò yo pou m fè

705
00:52:02,001 --> 00:52:06,000
Konsa m di se kontinuite
e se menm ak

706
00:52:06,001 --> 00:52:10,000
kesyon sa a la a.

707
00:52:10,001 --> 00:52:11,000
Dènye kesyon.

708
00:52:11,001 --> 00:52:16,000
Etidyan: Kouman ou fè
jwenn 0 a? (Son an pa klè)

709
00:52:16,001 --> 00:52:18,000
Pwofesè: Kouman nou jwenn 0 a
nan sa a?

710
00:52:18,001 --> 00:52:20,000
Konsa, sa mwen di a,
se pou ki sa kantite sa a

711
00:52:20,001 --> 00:52:24,000
ap pwoche lòt kantite sa a.

712
00:52:24,001 --> 00:52:27,000
M ap bay yon egzanp.
M ap oblije efase yon bagay

713
00:52:27,001 --> 00:52:28,000
pou m esplike sa

714
00:52:28,001 --> 00:52:35,000
Men sa m di: limit lè x
ap pwoche x0 de x-x0 se 0.

715
00:52:35,001 --> 00:52:37,000
Se sa m di.

716
00:52:37,001 --> 00:52:39,000
Oke. Èske sa reponn
kesyon an?

717
00:52:39,001 --> 00:52:42,000
Dakò.

718
00:52:42,001 --> 00:52:45,000
Mande m lòt kesyon apre leson an.