Lecture 3
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Leciqre
T \e
fwerrno6lbnq,'.^c probqbigi
T I'e t ct"n b e.
anic,,ro o*qte-b
a+
ttb tker-mo d6nl r^ i. Cona,d er
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? 15 crfted a fii,,,<,n r^a0@6t-c4+e
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p.obr
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lv\
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.
t
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t
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ftL
r h e t\ernod4 xarwis
probrlai0lt6
I
, b Aive-nng
NJ 1 tN i .!
fv-
r-t
i--o arPPrb.
o.6J5tern
d;ot<.Uwtuon. o pttz. vrr r hr rhD r -''
T \..e,rs I .5
N * uA, Fo " ;;a tec t
c N - - 1 Al --l-
t v
t
>
,
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vlot
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Qctltc'cte
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t
orf,e.r c ( el.-ice'
1z3L(
tka
€i*-ct, S ecundz\[*oA, -,-. 4.e-te-31(ste[:-
i,,tt\^c
1v-\
e-r'efril
b6 t|te nt't*bg,(5'
\6 desc*Ued
pa*ttc(.5
ttor ts- Tlte,rs q,(L ct prrrtre(c-s i.. tv.€- €t
1uppo;€
r \. e
' r\e 5 Q"enbe"U(" ) w'tfr Atr1>otffre(e
gd{ftLctc65
t v''r'-porfq'^"t fov r'wt'*-p[
i5 -t '.,^-e--
t\
g
U)e'r {ot- <-r"eqy /.-"e-( it v, ^75 d+ ckoosi^3 t1,.e ^\rvn
r.0 trrt- I ) (Al-'t) tN 2 4t
@ r r z . u . dA S 5 Q w t e . t \Y"*- t u"'o t[.f
v artcI
Ir5 ' otrt
tobe ptrc.J-"i^
e-^rry) ,b..1 eL P* ^n t1 . y4,"'a),*)lN-\)
5e-a,r,l 1
Vqrl-tc Ics r Le-
{-l--:
t
V^nf."cIe
S <-c.:nd
t
tue v. Po"e r
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(41-k) wa)5 C*"b1
e
9
t.rr (J ? q r t t c t c s
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r
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I
a^L tf,e tr$q( dot/s 4o" Se["nig
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w e
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tuoq(d, U> e_
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p l
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fl ni
h,\h"\ hsl --
e.re-r'"-r\suppoby tr.^ tLa porlt-leSfuW L \" et'Lahe^qJ-l'*l w) [el1 d^,2t-6-Lb,1fE-J , gqy one mol-ec-q( ' qryd).r vv) 0r J'rne Otr-We-!-ttn) ,",nybe- €"*Pry t t En*ryl.'.
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-z- N
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>>N
6y5*e.n5 A,13
we- \tg,v e
I uura l)te'fi"4d\Y\qw4
1_|.ermr,d4wtmc
q,n,dw,b &n-d r'
;Ura*o P0ohe,!D
probql,L,"lrV C,pr,'be b\.o'"'n \c b-
C-'
W*rl -- l{* ,u{8,
b "Lf enwo \o6 i I
4*"1 = a+* 6g ! vt Ab t, V<, .e:rrgr*?t a I
qnJ fu*lo
J
Wz-
pl.
p artTc le-z
F ^l
. r
c\rtd+bt ttJ.. ts[qb ie
b,tf {r.
rrotdiergdsf"b[
gest&
t^ri{[ k.'ya [nJ r I n a
degenon+leb5ste,r.lr (t
Cqqbe
u) e_
\{^.s{"tevayb
Ui;L(qL"-J
(P"" one,<-n6yJ e^e"3,\ " U Q-ew\ ConSiofo{ a bq,{le a((€-ve,l t!'efg c,tfP-5rrr"ur\
W:y-l
vre o[, d tkE wttv",b "t rsL !,nuJs i-.tke enu{9J-4u-(s trh i.th bk"d.tebg\; knvig ttu_ f_ne_rJ e-i ) hr ParTreia5 nL
i ,'"t\^"g Rt.5t b.ud.te o{ b,-te^,elr
,
s
*t
I
r
Seco,..d 5
\:aVr
5 S
Y O' r orH g- t J * r t cr r b t e p 1 r t r " I e t \ e
t( /t -1Qs
._ 2L c );
t-uith €nerg_y
5 lt i '{evg15
er
5
I
C -2
ur.trth
5
cv L ,
?
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.
ic
n
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9
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T he- r^,^'qthe.n.ot$r"Qr. t- prub qbifrf6-" t? fI'e toe{
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W
tr2
t
/ tJ t II
fi N,r"
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fb"L) P i > t e s s \ h q ^ \ ) ( u e { , r w wqs 9'e"oY-n b et 'r <-€n b\a ti:\
C o , tot.t
a5 a "r d tL. er^",ro {V
wric b
P h ) s i c , r l a i 9 " i ! - i c ^ n c co L I Z ( M , V , t ) Ctrno,dert-c,
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t
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vr'.rt ba
nstfuf"'e-
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,, lr tL.
4o-
n !t' 9i {'n^B+uo
Thel dePu"lon t!'<-
btc,tqs€
+ lke aJrl^
L t- .il.e wqlts Nt ,N
'rr,.r,*tr-*tic^\
A,'A'
ere 3(-l7Yde-d bY a n3iJ u W4\
fLe-nn =- Cp € )
V ,, V . o ^ l tq/ffi
5 evt\eryJoL i*erqetuon, rs \oaig ngl ee-t=&.C6 b e cqctt c- N, >il,
rV, rv,
otrc
corvb\qn-*5
"
i b t[ee"ro\) ol-U- ?!'te>Ysle'; 4o=At+/z
n
in'e.^ S) ,Wr ru,',Lr\ ,!,tn,
J 2 .t E , ) , 9 , C t - y . ) -.'Wt nq.yr\ber o L
is girrn l"y
-O'LL,)
"L6") -cz-
W( Lr) =
a(ceu"orble 6\te5
= 52.rcL')-C),tt'-
a',J
I -cr"cerl E2_
Ez
"WLEr) -- C Q* L E, ) (), I L r) w orb w',ln
Ez)
c -
= c-QoLet)
t s>"(Ez)
t-kt^^e EL
;1s'te'* Ao ,& tlte ConTpobit<
vre ywugf ^o'lLi'^a$c
Wlbz\
Fo, ft*a^^6ho,ot probubte *un C-onwAqiet;t
v)<-
er
,!-v,1oridrr", 6,
'dF, J L.v_
, t-^ * (il --OE' _.e:*) \€u
J*#)
=o ' = - - >
b6r
*yo--' €r^trcPJ Qor botl" ito{-ot1"e;[
rt" A-
frre 3t{tl' 'V''E)'W'yt
lvcrcrob\rtte-S' de-te-r,*'i.^.dlc$ tLe^"r rttP eef tve txtck'n"'e eLe"ry] 4n I *^*"* whgn t g d',^'n& i'.,i' t['r*^"\ '-{- "*Y"::*peruof, A**%-'tlv'.b id,ex \Corvb /v'vLl i-.. lYr' 6ets A ,tb -n n $-L b"tb tJao* 4, totLe' ^ ^ 4 a ^ Frrcn € n e "J) qrst trqh5 (nt- O t\.qt
6 t.: t\.
5wb S'1:i..n 4z ,tt\e*5 p\otc( , t *tne_(pyre_b d^^1 pun e-lvye
i^ ttttn- entuP)
d{_ n € (ornposita 3yste"^ Ao wouldbe-
'U]-r Aso: (os)Ar (e>r )A.: f\ b E r J [_tr) - ft.. ) t-.-^ -tbE ) ,N,r( ,,tv77v-, ztVz =,>t[(P-t -r?jir ] L\ b€,./r,,C t dq lu,,,q\ occordrg {u €'rtrup)-
S"
Sec..,",A \ot'',-rcl{te-c'^o
@*
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svu'o. ?b--9, tN
tov \o\{q
t\e n
h,r^^bf\ ^ c re'.,.S€j
rbs, \ \ ! ) b - ), N , , V , [ - -
dilt'ottt'tcs \\e
N.,V,
|
-:>._Tr=;T'
5 )o = .: -:-5 T. =-T2-
---<-
i
T \e
fwerrno6lbnq,'.^c probqbigi
T I'e t ct"n b e.
anic,,ro o*qte-b
a+
ttb tker-mo d6nl r^ i. Cona,d er
A/
? 15 crfted a fii,,,<,n r^a0@6t-c4+e
Frt
p.obr
parf..o-le-3 i-, r 4i
\aofit-t d.ignbwte J
f.rr b o {
A,
No
toY +
A,
lv\
-t-l
\TJ\
-
.
t
!
"t
Parrctr s t
t
lvt
A'
ftL
r h e t\ernod4 xarwis
probrlai0lt6
I
, b Aive-nng
NJ 1 tN i .!
fv-
r-t
i--o arPPrb.
o.6J5tern
d;ot<.Uwtuon. o pttz. vrr r hr rhD r -''
T \..e,rs I .5
N * uA, Fo " ;;a tec t
c N - - 1 Al --l-
t v
t
>
,
' ) 9 5
I
, \1,3\
,\zEtt,
*
€i.st
s
t\'.v I
(
€orT\
{'y es otvY
vlot
l..e.45:J)
Qctltc'cte
J €
t
orf,e.r c ( el.-ice'
1z3L(
tka
€i*-ct, S ecundz\[*oA, -,-. 4.e-te-31(ste[:-
i,,tt\^c
1v-\
e-r'efril
b6 t|te nt't*bg,(5'
\6 desc*Ued
pa*ttc(.5
ttor ts- Tlte,rs q,(L ct prrrtre(c-s i.. tv.€- €t
1uppo;€
r \. e
' r\e 5 Q"enbe"U(" ) w'tfr Atr1>otffre(e
gd{ftLctc65
t v''r'-porfq'^"t fov r'wt'*-p[
i5 -t '.,^-e--
t\
g
U)e'r {ot- <-r"eqy /.-"e-( it v, ^75 d+ ckoosi^3 t1,.e ^\rvn
r.0 trrt- I ) (Al-'t) tN 2 4t
@ r r z . u . dA S 5 Q w t e . t \Y"*- t u"'o t[.f
v artcI
Ir5 ' otrt
tobe ptrc.J-"i^
e-^rry) ,b..1 eL P* ^n t1 . y4,"'a),*)lN-\)
5e-a,r,l 1
Vqrl-tc Ics r Le-
{-l--:
t
V^nf."cIe
S <-c.:nd
t
tue v. Po"e r
\'.
(41-k) wa)5 C*"b1
e
9
t.rr (J ? q r t t c t c s
tt\-e
(ru_t I -, i
Fot'
L
r
5
tLe- ge<:o+I !-*l
i
\\rothbe ChnS€n [-"
(tr- '4) (N -2)
I
a^L tf,e tr$q( dot/s 4o" Se["nig
l,'eThrJFor
l*o
farnac[a \q-qbou€
,l-eunl,L u^tgb
i:
)w 4> LN-' ) LN-5) {vtrv-l) trt.l-z t\ F "o ,,,., q.bo\/€-
w e
CL\, Ccrn
Coqct"J. fxruv
.I 'rbqb\.0. ts- )
tuoq(d, U> e_
Nl
p l
Vv.
*\'e-
*\.erDruodJ*ti..
fl ni
h,\h"\ hsl --
e.re-r'"-r\suppoby tr.^ tLa porlt-leSfuW L \" et'Lahe^qJ-l'*l w) [el1 d^,2t-6-Lb,1fE-J , gqy one mol-ec-q( ' qryd).r vv) 0r J'rne Otr-We-!-ttn) ,",nybe- €"*Pry t t En*ryl.'.
;:#;;
-'"r ":,:{
\ 9ryll€f,in poPtVfionqg-i"
e ^ )4
,'l')ii, .t?JJ) r ? q '
3 5 * p p o b e atl..t\e
dnd
1".,Lec,r-(r5 ? L C e p f
WzJ-NI
f he n
0" e
Qre.
i.^_(2weW
-z- N
( n / - t lt1l o l o !
w h c -n
N i:
Fo, two
lq"g(, l\l !
>>N
6y5*e.n5 A,13
we- \tg,v e
I uura l)te'fi"4d\Y\qw4
1_|.ermr,d4wtmc
q,n,dw,b &n-d r'
;Ura*o P0ohe,!D
probql,L,"lrV C,pr,'be b\.o'"'n \c b-
C-'
W*rl -- l{* ,u{8,
b "Lf enwo \o6 i I
4*"1 = a+* 6g ! vt Ab t, V<, .e:rrgr*?t a I
qnJ fu*lo
J
Wz-
pl.
p artTc le-z
F ^l
. r
c\rtd+bt ttJ.. ts[qb ie
b,tf {r.
rrotdiergdsf"b[
gest&
t^ri{[ k.'ya [nJ r I n a
degenon+leb5ste,r.lr (t
Cqqbe
u) e_
\{^.s{"tevayb
Ui;L(qL"-J
(P"" one,<-n6yJ e^e"3,\ " U Q-ew\ ConSiofo{ a bq,{le a((€-ve,l t!'efg c,tfP-5rrr"ur\
W:y-l
vre o[, d tkE wttv",b "t rsL !,nuJs i-.tke enu{9J-4u-(s trh i.th bk"d.tebg\; knvig ttu_ f_ne_rJ e-i ) hr ParTreia5 nL
i ,'"t\^"g Rt.5t b.ud.te o{ b,-te^,elr
,
s
*t
I
r
Seco,..d 5
\:aVr
5 S
Y O' r orH g- t J * r t cr r b t e p 1 r t r " I e t \ e
t( /t -1Qs
._ 2L c );
t-uith €nerg_y
5 lt i '{evg15
er
5
I
C -2
ur.trth
5
cv L ,
?
\-t\ leugl
.
ic
n
+
i
y \ r , t m b e " ,I hnaefs 5tq te qnd r.or*c robq Vrr\A
flo.
L qt\t ot
\^/
gr ..,.9:' n.t
.
h l l,
nr\
\
9
irL ^{n i !
=
L .
T he- r^,^'qthe.n.ot$r"Qr. t- prub qbifrf6-" t? fI'e toe{
rrq"'rrer o'Ldejcnc-rse
W
tr2
t
/ tJ t II
fi N,r"
F)"--
I
fb"L) P i > t e s s \ h q ^ \ ) ( u e { , r w wqs 9'e"oY-n b et 'r <-€n b\a ti:\
C o , tot.t
a5 a "r d tL. er^",ro {V
wric b
P h ) s i c , r l a i 9 " i ! - i c ^ n c co L I Z ( M , V , t ) Ctrno,dert-c,
?Voio.\ t5s1e*..
AL
N,,V,, Et
rt/yr\/, , L -
rn acfo..:il.qte
-Q, 1A/,r V, , bt)
u 4 -A ,
t fr.
t
-(),
til, tV, tL,)
,Qa.49.T\e xteJ
I
vr'.rt ba
nstfuf"'e-
-f'k- 6 a r n e
,, lr tL.
4o-
n !t' 9i {'n^B+uo
Thel dePu"lon t!'<-
btc,tqs€
+ lke aJrl^
L t- .il.e wqlts Nt ,N
'rr,.r,*tr-*tic^\
A,'A'
ere 3(-l7Yde-d bY a n3iJ u W4\
fLe-nn =- Cp € )
V ,, V . o ^ l tq/ffi
5 evt\eryJoL i*erqetuon, rs \oaig ngl ee-t=&.C6 b e cqctt c- N, >il,
rV, rv,
otrc
corvb\qn-*5
"
i b t[ee"ro\) ol-U- ?!'te>Ysle'; 4o=At+/z
n
in'e.^ S) ,Wr ru,',Lr\ ,!,tn,
J 2 .t E , ) , 9 , C t - y . ) -.'Wt nq.yr\ber o L
is girrn l"y
-O'LL,)
"L6") -cz-
W( Lr) =
a(ceu"orble 6\te5
= 52.rcL')-C),tt'-
a',J
I -cr"cerl E2_
Ez
"WLEr) -- C Q* L E, ) (), I L r) w orb w',ln
Ez)
c -
= c-QoLet)
t s>"(Ez)
t-kt^^e EL
;1s'te'* Ao ,& tlte ConTpobit<
vre ywugf ^o'lLi'^a$c
Wlbz\
Fo, ft*a^^6ho,ot probubte *un C-onwAqiet;t
v)<-
er
,!-v,1oridrr", 6,
'dF, J L.v_
, t-^ * (il --OE' _.e:*) \€u
J*#)
=o ' = - - >
b6r
*yo--' €r^trcPJ Qor botl" ito{-ot1"e;[
rt" A-
frre 3t{tl' 'V''E)'W'yt
lvcrcrob\rtte-S' de-te-r,*'i.^.dlc$ tLe^"r rttP eef tve txtck'n"'e eLe"ry] 4n I *^*"* whgn t g d',^'n& i'.,i' t['r*^"\ '-{- "*Y"::*peruof, A**%-'tlv'.b id,ex \Corvb /v'vLl i-.. lYr' 6ets A ,tb -n n $-L b"tb tJao* 4, totLe' ^ ^ 4 a ^ Frrcn € n e "J) qrst trqh5 (nt- O t\.qt
6 t.: t\.
5wb S'1:i..n 4z ,tt\e*5 p\otc( , t *tne_(pyre_b d^^1 pun e-lvye
i^ ttttn- entuP)
d{_ n € (ornposita 3yste"^ Ao wouldbe-
'U]-r Aso: (os)Ar (e>r )A.: f\ b E r J [_tr) - ft.. ) t-.-^ -tbE ) ,N,r( ,,tv77v-, ztVz =,>t[(P-t -r?jir ] L\ b€,./r,,C t dq lu,,,q\ occordrg {u €'rtrup)-
S"
Sec..,",A \ot'',-rcl{te-c'^o
@*
, @ I
* ^a
r?st 1
svu'o. ?b--9, tN
tov \o\{q
t\e n
h,r^^bf\ ^ c re'.,.S€j
rbs, \ \ ! ) b - ), N , , V , [ - -
dilt'ottt'tcs \\e
N.,V,
|
-:>._Tr=;T'
5 )o = .: -:-5 T. =-T2-
---<-
i
