I. I MATH 1 CALCfJL-fA$ I Sec fr ''''tomicheli/MATH1750F15/LECTURES/Math... · I. I MATH 1750 1...
147
I. I MATH 1750 1 INTEGRAL CALCfJL-fA$ ADVANctD SECTION I 3/3/15 Sec SYLLABUS fr ''''to 0", f-k00 °t Cou,x'S<' ;, w e., k gevuofL twJ- \Nv-rk two \ " i) II b wv-\-"- CJJv-.d 2) 1\ Aff \\ \ \N 'tV',,\l -L",-,ol- CD.;V'., be" VJY\s1-n}-o\.-<2d 0".$ -k at 0" of ('N:\\ £wi 00{" oj • Wlvrt \1 t ;S d 0"" o2L IR, 'I & 0 R *1\, r 0(" ) f(1)(, j == J I + xL. \NC0 CavY\.; f:..x Ov b (8Gvj; Ov Co - I J 't:::. l) Co\f\A. f\A--t b l. , . A = J J (1(,) d = J I -I
Transcript of I. I MATH 1 CALCfJL-fA$ I Sec fr ''''tomicheli/MATH1750F15/LECTURES/Math... · I. I MATH 1750 1...
I.
I MATH 1750 1 INTEGRAL CALCfJL-fA$ ADVANctD SECTION I
3/3/15
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'T~W'\-Wv\ 2.: L± f o~ G) det~V\wI O~ wV'v ~V\Jex-voL J Q'l{"
bo+~ OvV\ ovV\W.eY~vok~Y{, ot &) t~(MF (~) ::: G( ~) +- C Jar oflt ,?G lYl tv ~\Il+{~vJ,
r (if) Avv \I '"k,-"a£. 1\ \I\,QS \1.,1., {o'-I'VV [Q, b J) (Q) 10), [0-)0,)) (Q) b ] )
b \N+ otso [ Q,) ()O)) (Q) ()O ) '" (- CXJ / b J) (- 00 ) b )} Or (-co) co ~ =t lIZ.
~\oot (ot l"hxo'f".e'M Z) bo.s-uJ O'A- Th'N\. i j ,
::It f r ~ -f e<twJ G I .;;. t t\-c<'Nl !=" I - c; I = J -f ~ 0)
~"~, \=1 _ C;' ::: o. 'BlA+ t~ \Mfk(S (r -c-{) ::: O ..
B~ \~\('~ i J t=(~) - G(x,) = C (Q CoV'vst-ovWtJ) 30 f(-:;(.):: G(nvt) +- C. II
A§Vv'Mt ~ F is ON' QM,+\'d~vo1\,,,~ o{ -r ~,e, 1="' = J ' SO', \0 b
J {(-x.) d?6 " J F'C,><»(h:: F(!') - F(~) ~ t
So) \ P l.J\ov.'-\- d r~. Vd ~ I S e.-1 v'Vu.::M~.J
6;1, ~~ ~c,n:t jVUA<Lo-'Me;vJ~ t'(~ClIVv\ at ~k3
A.!v v....ow- ON\ OJ\ld~d .e, '\ ~v Q.;-K-V ~ 0+ S) .:1 0 \II.... ~\t\. Co\llA,?Jv l V\"~-v~ 'tAA,t, 0 VI 0" ~ l V\ -t-{,'rV at [ Q I lo ] I
21. ~: If (j0lA. lASe- o~ oJ1,cvr OJVvM~r;v",\,;v<- of -L Sc>.j
. G(x) ~ F[-x-) 1- L "'~"'LC ,J Q(PY\$tOMt of j0lA.C
ck-a('c-t/ t\0\,C d~;b ·I::tvv SOI\IV\e., '\eS'v..J~t. T V'L- ja.-cJ :
G(b) - G(a-) = (F(b)} c) - (F(a-)+c) "' F(b] - P(a)
N~~h-~",,', \t f \s OJvV OJV\.,t~,dtruv~vt- 0t t} \N~ c~cd~ bke-
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J t(x) ch == rex) + C
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Ku.p bl.Ms { \tV VV\-(,~d W~(;vv soev~~ f-w\e:,\.t\MS .
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d! -t ( ~ (QL )) :0 J' ( ~ ( ?t )) . ~ lC 'A: )
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~ J Zx. . cas(X}) dx '" J iz ( SiNl.( ')62)) dJ(. = Sl"-( Q(,"l.) +- c
A ~ w~ +0 '\J1.Oceed Is ~ +Llo'N'~ (;1,- 'Noy\<.S '. ') :
• S'~t u,(?C-) ~ ')( .. ,:2. / 0'\ s~VV\Pto . VI = rx}
• t~ ~te,.,..~ 0t IN LS du ~ (?(.2) , ox. ~ 2x Jx
J 2x. easeX-) Ox. = J ens (%2) ~ '" j ease",,) dlA. = silo\. i.(.,
U au.
- s ~"" VL~~=?t'" <;;:i.'v\ . .(xT) + C. . (It WOr~~) M?w-J \+ \N(. 'JJoJW\- bo coVV\-pv\J-te. ~ d..tf~~te, ~vJ.~~
3 J Zx c:as(?t) dx
r l
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du.,.:: 2XdX. ?(, =. z. IL =- 4 ?<-=3 U.::3
Ex 0"'M p \.e.s,. J -taN', 'X. d-x. "" - k J CoS ?<-I + C
.. \ •
•
( L (3 \100 J ~ . QG 4-5) d'iG
J z-x! J?L (1 - Xl)2.
\
J SCYv' X d?G -\
• J Cf:JS2. X dx
@ ! Y\.+~ ~O~ bd ~ [ (,(,(x) vex j] dx
(-¥1 J ![U(~) v (x.) ]
t
,?a.,r1.s : ,
= u!(x,) v(x.j + u{o::..) v( 0::..) )
d'X - j IA'(OC)V(XJ OX ~ J u-(Oc-Jv(o::..)M
:r ~ J \ \'L (*) 1. eo_pwl-e- Ov DB 1= IIV lit lvd-~ Cot \ y,si-eo.,d ot 0Jvv ~v-vd{J~~t~ O\A,Q..,; \N L ~..Q.;t i
b J b . b
r J iJ u(-x-) V ('X-) J d-x, _ - j vG COL) v(x] J;t =j U(o<.) Vl(t)(,) dx OJ GL··· \;OJ.
r I \
V=rx- lL-=-} 7U
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· J ?G tw ')(,. d?0
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J CosL (A- dlA.
= J,J I - S l'lllL ~\- CDS(~c ') cI tA., ~ J V Gos"-IA COS U. dlA
= J I CoS It 1 . Cos u d IA
.::::
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f t ~,..---' J 1.::;- - S ~ y\.. \,\." Z. ! j ~' I - Co S (.,L.
~ = S(Vl cc
s ~~ tA. . Co> lit + J I· ellA ~ J Cos2 ilL ellA
S ~ Y\.- \A., , enS VL +- lA, -J GOs2 u- OIA
r ). J enS 2(N dlA- = ~ SC"" IA. . CoS lA.. +1 1-(, + C = (-.\-)
(-*) =
',$ <:-os ( Ov'rc. S iJ<v x, j 7 W'rvt-~
J wl"u ~ /1 -sivC l-\.,
-1 I -~-.~-------.--~
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I 1. s ,"-' ( (J,\'" C, ~ c V\, x.,)-I I -ls ~ 1\,' ( ~\- C. S Cve. 'J<.-) r +~ Clr-G s i "'-' x + C. 2 . Z
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VI.,(J.., 0.. b~~ oVU) 'M~+u:.c.. dlA.,bs+U\--u-+l.o\A.:
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dx = C.oS t..-L 1+%2-
d'X- _ z . '2..
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J dY\i- Cos2..(A., Cos2..u.
·OL-l - \ 1+ t:ov",z. tIl, 2.
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J dvL -I + $~V?u' Cos2.{A.
CDSz..lA,
COSLlA.- 4- S ~h}-lA. J i du ~
CDS2 Vl
II
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® I -t 1,s fr-~~~ s~Yv\-f\c., to ~e~ it Itf
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o
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. ~ ~ l/j~ }(x)
o a.- Io?(.. Q a.. h?L-
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ExO-'Mf''-: 'F,YvJ ~c, o,y",,- bd"'!<IA-; )d'(x);:; -?/+4?G "/ ~ (Qt.).~ rx: - ~% + 5
/
/
/ t 2-
% - G?C +-5 .=0
x = - b ± I ~ - ~ o __ c
-2<0
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to + J IOO-ito IO±~ x~ -~-__ _ - -
it lr b ~ 5+~"
v~: Z Z \:. .
A =:, f [- -x,2- -I- 1; 'JG - (?(,z - (,x. +- 5)] d?G
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Yov..- w~~l $ t Q.; -lv..(S ~
3.3.
(I LL~ 'N~\-h. \l\.,u,\l\A...b(l{s, EV(10)( EVE}...) =:, 1::V€,J (I)
ODD X ODD = oDD (2.)
EVtIJ .x Ot:>D ~ O.DD (3)
1='0, ~xQ.'M,p\e,.J ;t & ~S ~v~ ~ <\ is odol) ~ W(...,
dp1~~<G: "" (?<,) =: f('X.J ~(x) j W~ 't....a,,-G \u(--x,) .::; ~(-%) ~(-?C.) =: -f('X-)~(?V) = - h.(x)) s~ k is odJ.
e I~~ 0'1\ \j Ju.vvv-k,Q\A &0.+ C~ SlVv\.,\AJ~-\.O,,'A-e.ov.3 ~ eVJ..Jv... ~ odal '-.) t"'-'tt ~Cro 1lA.\Auh.QI~ (~. e ~ tk ~ IJ-N\,tX""Oy\; \:,~ ; s 2. rc YO JO, 0""j v k '0 1'?l.-') .
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• \~ o.-V\- od J 1 u.", ck~"", ls d 4~Md oJ-:2: e ,-0) .
t;""~ 1 ( 0 ) = o.
wW h<Lpp-tVlJ if d<OV\. ~Vt,., ted'"(,ok<. <Mtv .ev~ Ov- odd
r 0'1\, 0-- S¥f"'''''vtv ('' !io'M,OvCI4..? (1. "', of bke -to F-e.
iJ 1 t J is ev~",--: 'cr Ov (Ov
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2) if j is ~I 0... .
J f?<-) ~x = 0
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I:J"'-c., ~OJ~;> -L "'-e Y"(. Y''''-~ 6\u-\Yv-t~~ IS ~e.'\o.
I
9\1\; __ b~~ __ ~~~~ ___ ot ___ tI d \,l \M \M j VlL~~~~ ~~_~_~: _ I ?G == I
t= ; 'f'i -\- 'M>-\-~ l:,\,u,k J ex: Z J?(, ~ ~ [ x~ J = %3 o 3 'X.=<O
Nour };~\IJ.t, took o~ ~\\-e; e-y f"e ~S~ 0\1\, : I
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o
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1 ~ cov\ vI ~~~QV\. ) 1L\JJ..Jvv ~ t ~ Ovij'Uvrs 0'1\,; ti~ L~S 4 (*.J, J ?(}- o'X.- \ S t-JOT 0".g \.LV\, ~'v\ 4?G· o
A 'J\oi-~'("" 'wQ~ to see i, \- IS ~ .gaLLa \}J~\A-;~ :
f ')62 dx. =- l' t)2 d ~ = J>:,2 d C .) It)
6~CO-.ANs(. 'oJA o.x- e, e,9~.!oo ~. }JoV\ e, Q} ~% IS
o~ £ \A., V\. Gt00\A 0+ b"'-~ VCM-~aJ.o \~ lo~a;\- OvFf~o.-r~ ~'vv -t~ ~M~"Ul,f.
l~\ ~ t"~SoV\.-) ~ VO.)x--~oJo \-e. aX l.Yt-,-\-e~ LO . .;\-~ Gil\; 0..-
Dtfl}Jl1E \vvte~Lo.t ~S coJle.J \(DLAKMY" - It \-S ho+
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j ~1- 0%:: ~ ~ + C
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W~\-\A ~s ~V\; ~~) Co\t1S~tl("
d Q 3 e Iit-€. r ~ C. f v..-vcd-t. 0""
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3'.
I (l.e. r'=j)
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eo.. \I\J o,-vt- u...o.. {;\j W '\ J e, b
f( 6) -1=(a.) :: J t(~) dt (]v
-~--~ ~~ -----~-------- ------~---- ------
Now; V\A) ~tk~ \,w~ I 5cx Q" T CavIA.. ot:\trQ3s 1~vv:l o--Vv o..,~+~~voJ-V\l-(, F 0t t S'lA-ck t;~/t r(a.,):: 0
~,e, sv..ck b~ t:"'--fL ~ 'Lo.-p\.-v 0t F d oes -G\'-.:tolA~\.v bk~ f.0\ivv\i ~Q.,)o). I~ ("*-¥--lE) \N.L ~os{., t= bo be:, Skck.. ovw/-cOIt,,~v~\I~) o~ ~+- Io<"-CO'N\!-s ~
b
F(b) = j t(t) dt Ov
37.
\Nt, 'vvJ..VL jOIl--~ Ov SO"'M-U-~ fa.- iHE AI-JTllYE£.lVI\"iIVe:
01" t THAT IS EQuAL TO ZERo"7 fa' x.. = 0..
(To VL,,;b t~, ¥LoU ~t- F(av) = Jo-;"'5(tJdt =0).
1\ ~ s e.,c.o vuo\ fVv\;\,d O.NV\ -{, y\Jot l~eo I{ ~ \M.. O} Co ... (('N...iu..s:
( Q . .k.. (N • llw H CoV\ s h-\A..c.oh.o V\ 'T\w,,, t """ {ou- A ",J.:.d ~ ~ V(}.,t\,v tS " :
It t \5 CoV\.1-\,Y\AA_O\A..-S C\!v Ovvv lNv+~Y-vvt) OvvuJ ~ Cs Ov
'c\. Vv YV\ \0 -Or ~ V\, t VvoJ- \ Vv \ ,<-,\y.oJ... ) 'X.
t'"'-"", l\c (, i "'-11\ cho",- F (-x) '" L t ( t) d -t )
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r
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J?(, C [CJt=?G f(x,) :; e - db == . - e-- = I - e. - x o t=O
Not~ b\wJt ; OJ F'(~) = e-x
e f(o).:: I
\s '('e...5e.'(yR"O CD OvS ~"'e- it lJ.JTE61<AL FLA NCTIOi-J ot.J "
(tW ts 0 for 1)(.= 0-J,
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t~ ~\A.~"" j (%) '" ~-~ o~s 0 k vvo \Iv \tV a.s 6\.A.e." G ~S~:'" -:f ":''I\-~''''- (ct \uJcs . Ovfft;.coLVt0 ItVv ?LobQ..b~~4-d ~ stQh.sKcs) cio($ YUot- ~\J~ ()\JVv
€-41M ~ t-evy-~ OM.,\-\,d ~ ,,()J\..:.v~, 1 '.' e. J i { .:I 0 "'"
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. J e -x d')(.. ~O\).. 'No""" IT I 2. lMoN~ \t ~
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% 2
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0....
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(". J iJ 7G 2. -l-lj
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(J,,) \ASL~ F {,OvlA- r<>-<-i- (a))
(c.) G"""ruke, ~ G ( ?(. ) .
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3
( <;. ) ~ G (f)(,) ~ ,] in, ( tvv ?<. ) • _1-=
d% ( £VV x )2 -I- ~ ')G )
b ~(,WA.S<:'... 6~ ~ ~"'-' rlA-e~)
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II
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CL
so.,:j s l\.w-k t~{, ~ V\,~ <-~ \A,~ 1 t "'-~ d.~ \J ~\J~
at ()., 3Vv\llv~~ ~s -t~~ j Vv~OVv l+,Saf-, /I) "TIve.- S~Co\l\,~ 1UJvula.NV\t.vv-tot -l~eD'\~ at eotCN...~ £,0,,::).$
lb~ F(x.) = J: gCt! ~t is (Lvv OM,M~\foh,,~ "13 ) I. c bbt * J: g(~) dt = j (-x. ) .
So -t \.e.. dv~ ~ \J O/\-\-" ~ oJ is J t+S~Q.j,
lh.e,·-~"-+~-wt\g~o"" ) at !
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F('X-) = j J(t) <1-t. Ov
How GQ,YV we ~Yv-te-"fY~{- t~t, jo...CJt-
t6t d F(",,) = ~ J"'-§(t) dt ~ -f(?<') ~ d'X. O?<..
Ov
/ .. I ;
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SO JD~ S'M.oU .6.)(,;
1= '(1)(,) ~ ,F" (Q(, -l-t6'X) - fC'x) ) or: F( 'X +6')<. ) - F(~) '::!. F (I)(;)Lbc ~'")<v '}
T Y\seY'-li..'v\,~ 1= '('7<.) = 6 ('X-) ) W (, ~~t:
t F(~Hh) - r(~) ~ j(~)~?C,1 1 . j
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(A,'lW- J(x) 6% Of 0- '\~CA.Cl.;'Nd~e. II
. /' (;OM, $'~f b\.-v.:..s r~<- l\i\.,. ctass. 4Z, ~. . ~
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\N~ WOMit bo 1'1-0'1(": '.iff'· G ts OIU.~ ~~vO-ttw{.,- 1- 1= L I ~J-t""""J, , b~~ J~ J(x) dx::; G(b) - <;(a) J Tkiol-t~ llS'~ fu jo..J. ~ t{x.) '= JXJ(t) db is ~·l 2,IIPiu.wl. G-M,ttd ~"olW~ 0 t 5. 0.. J 1\\\, .• .-.1.- .
. L(;+ G b(; ~"" ovv-b~trax~ ~-tv.:.v~".(, 1- 1) 8 0 b\t,o.Jt- G = ~ I W ~ o.1.s-oka".(., 'F' = J S' 0
G I - j: I :: 0 0vwI ( G - F)' = o. -n\.G~t.soV'{./ ~r(, i..s .. (J...; CoV\ S+o.Mi~" SlA.~ ~1-
?<"
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L f(av) = 0
b
z. j f(b)db '" F(b)
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o & 5( ~) ;; & (c) ::: 0
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j-o-c- 'X. ~ 0 .
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I) b.~J\N l ~ Ov O.J\A.d c /
t='(x) =- t(?C) >0 so' f IS i'M;,\w..~~~
e F( b) = A 2- - A I=:O
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t~~ \Nt eoQi ~~ f0V~_
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~ 'fu f Q\DThcli..Q.e...-r ) jD ("- ?(,::::; 0) S' IQ II \ s tk,e- CD...S.e., ~
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r'. . s ( does", 't- (\'''C<:>.N-.S-<- t ~~ hA>t odd ~'" .
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Zt(?L) =0
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it <J-, ~\AN\'vkoV\. \ s vvat tUV~ 'N Or VUJt oJ ~, ~ : -\-CO-M '-\- b-e- 'c"'-<t. :!corD t v.,w:.hJ.>", ,
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s+J1Q,'M ~V\+.s IS t'\\f..Z.
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it is both, \\n;rTOIf -SjVV\W\e,t-,,:c..
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Bowdoin Math 1750B, Fall 2015 – Extra Problem Set #1
Computing Volumes
Tuesday, October 27, 2015
Problem 1. Compute the volume of the following solids of revolution:
(a) The solid obtained by rotating the region simultaneously bounded by y = x2, the lines y = 0, andthe line x = 2, around the x-axis.
(b) The solid obtained by rotating the region simultaneously bounded by y = x2, the lines y = 0, andthe line x = 2, around the y-axis.
(c) The solid obtained by rotating the region simultaneously bounded by y = x2, the lines x = 0, andthe line y = 4, around the line y = 4.
(d) The solid obtained by rotating the region simultaneously bounded by y = x2, the lines x = 0, andthe line y = 4, around the line y = 6.
(e) The solid obtained by rotating the region simultaneously bounded by y = x2, the lines x = 2, andthe line y = 0, around the line x = 2.
(f) The solid obtained by rotating the region simultaneously bounded by y = x2, the lines x = 2, andthe line y = 0, around the line x = 3.
Problem 2. Find the volume of the solid obtained by rotating the region bounded by y = e−x, y = 1,and x = 2, around the line y = 1.
Problem 3. Find the volume obtained by rotating the graphs of y = 9− x2 and y = 12 for 0 ≤ x ≤ 3around the line y = 15.
Problem 4. The base of a solid is the region simultaneously bounded by y = x2, y = 1, and the y-axis.Cross sections perpendicular to the x-axis are squares. Find the volume of the solid.
Problem 5. The base of a solid is the region simultaneously bounded by y = x2, y = 1, and the y-axis.Cross sections perpendicular to the y-axis are squares. Find the volume of the solid.
Problem 6. The base of a solid is the region simultaneously bounded by y = x2, y = 1, and the y-axis.Cross sections perpendicular to the y-axis are equilateral triangles (with one of the two shorter sideslying on the xy-plane, and the other ‘standing’ on the y-axis, in a perpendicular fashion) . Find thevolume of the solid.
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i _ e, . b~~ .. <- (M-<., k.: oM-d H s tA,~ ~ K. ~ s Y) ~ M)
oJvvd S VI i 5 Yv\ 0 Y\,O +0 'f\ t. ~1JvC 'I ea...s ~ ~ :
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I I -.f---lI--"---:t--.-1~I---f~"-' --4-1 j -- ~ ~
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VI \S S9lA.,ee,t ~ b{+\NH~ (LVI o~ en,.)
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S' ERIES; -t~~ s ~'M ot Ov'itv ~'At~w.+(. n~\iV\b er of V\-u"lMbers '
( ~Vv'("f';'~~\t\~ ~j J H~ eov",", be ~ -fC~k \lW...W\be,,!) G ~\J-WV 0", S ~u.,(M,cc., a 1) G\. Z ) Q 3 ) a... Lj I ,.,
Wt ~t~'I\.(' Ov V\.tw- se1~c{,/ CQ,GW 6~e rA~i'AL SUM;
S\ == Ov,
S2. :; Q, +- 0.,2-
S = Q, -\- (1l + a.3 3
\
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D~~~~~~., e 1 t t~~ S~\A.<Mce. 0+ f~ot S\J..V\AS: CoYv.Je,,,~~s b
Ov . '(\AlJW\ b~r 5 ~ . e I \Nt, ho.,\J~ ,I
~'M S'V1 .: S 'fI~0l0
t~~ \N{' sO-.j t"'-;-t \1 \::,~(, Se,'C~e-s
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• I ~ S \'\- J;" \t ~X' ~ e S ) \N t, S (J.;~ ·\:\L<'.Jt 0()
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Co'IU\I (,'("~ vwt s(, .... ~e:; > 6-"1 bl.0:.,,--~ <:Iv S"''b ,,",ewv1- 0+ 4~",""k 2.Lv-.'},:"," . ~ s-lux-t IN'~) o-MJ b,-<a..k::~;'+, \,cf' ''''-to
OM \ "t" 'Iv.. tt V\.\!-'M \?<.,r 0.\- f "tt<. 5 , I 'n. t 11'\ t' 'AA ~ <- S "'-1M 04-t"'-t f~W.S IN,'v\' ~~" (., :)0"'- bo..cM.. 1:,,,,-,, ~~ -\-Lol.. S"d 1M ~
( !~~h, ~~~~J' fo, ~o.,'<v\pk I k\r~ Ov S~'Mw:\ .. ~'I\~+~ 1 0Jvv=J 'b'C<-<:L~ ~\- ~vvto two f"'U,(s) t;k:~
~ ~. Stw"-d ho.tf OY\C~ Cl% oJ-\\; ~V\;+o two f~ues J o..Jvvd. ~o Orv ~ >:0 -f0r~. 'Nt W I.Lf' srt;,~~ t"'-", s~ \vl<MJI- ''''+0 O-m \ VI t~ t<- "0vVv'M b't~ at f ~u:.t.s.
1-1< >1
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11 ~ ~ I ~L! I
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E04 p~t~ W k~l-h a,~ = (l)V!-) Y\ ~ I) 2.,3, ...
S ~~ct. -'G"-<.. ~tot ~~ +-"'- \s i) 'vJ ~ 'vv:L-J.t
~ (~)\, _ J. + J- + 1. 4- -' + I" ::: i. L= I l - l Lt 8 I b •
L(jt's YVOW- b~ o-b 0.. vY\O'\~ ~ '<.f'.tve,~ co-U.
1
f GEOMETRIC S' E"1<.Ie.S 1
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't \>\ +00
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I I I I + Z + it -+- g 1- "I
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'PA~I\AL SUM (*~~)
L C ,'-I == t;:::; I v\~oo 1=1
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----_._-_ .•.. _-----
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00
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is 9~ bo:
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II~ lj) \fV\.e,i-~c, \\ o .. :\~ O-.W\ tNv+ .ro..r~'\ ..
'R~o.x·'v,,: ,,,-<- r("'t'Nu.Ro.. (1'0~ oik~.J CoJvv be.. \.(,Wv:.b\;CN\., ~\AJ Ov
S~M~ CoY\.\J~~+- f'rVV\.:
\, - \ ioo •
'( = ,=\
I
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C
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c = \ .
00 •
~ t- I L r =
\ \" \ < I L-_____ , ___ ._~ .. _________ J
L0AI~ 1150] L e,cJvvrt \9:
N ~oo n=o
T'vI<- 'r<A.Son:,"d u..".+ ~ to (-If) 'S ; ""Po.. . .tQI'lkJ so:
N-l
'iioof O} (*) (~aA,'I\) 'PoxKot S('c'M: SN = nf rV'
2 1\.1-2 /-..1- \ ( to}o..,{:) So +- r + '(' +- '" + r- + r N t:.e<M>
blH-e~~cQ.,~ SN - ,SN::; \- r N
(\-\)SN ::: \- r N
So 1- r
No\kl'. 00 N
. t ,V\:::. ~yv\ t.- ,\V\ = ~W\ S'N = ~W\ 1- ,IV V\=O N"-700 n=o N~OO N~oo 1- r t 1- r •
\it1rl<1
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1.31.
V_iS_\A_~_·_~_Q.i_~_oy\ __ 0f: s~~~ce.3 ~ se..~ CoYlS~dey- ~e, S'9~ce:
a 2 = (;J ) Oc3 = (~ y (}M~ c.~~ C'D'\~e.spov\"~6 S9\MM~ 0+ fo.x-+Lo1 SU-'McS:
S f r 113 S 7 L = Q., I::: ~ ) ~2.. = Ov L + 0.2. .= - +- T = ~}I 3 = ~ 1 + IX 2. +- a..,;3 == T) -'-, _ .z '1
It \N~ Q,~'Q.., iyd~t·n:.s+w tV\; d{,+e,VV\.~"i w~d-~(.r cA.,
5el("~t.s (:.a~\J e./"~ e.,.s Or" 'vvot) it 'Is covvv ~(M..i+ to v ~s \A.oJ.;.~e, t'k~ v a.fJv . .J..,~s Ot ~~(., S ~\A.<W\ce.. <JvS Ov ,I b'tr ~ 'laM \\ j W ~t-h.
~ bOv,<""s G't w~:}l i 0J1M:l ~~itt <L ¥\, :
ell ------ t~s Wo...~ b~ Su"IJv\ O~ ~
Se,Y-~ ~$ ~wa.t to b""~ Sv...VV'I O~_ t\t,~ ~I(""eo..s 0t ~k '1~d-~~~ ee.$ .
00.
f cty, == ~ I + aZ + 0..3 + '" o Vl=1
I
Not i, b\w.k fu-e. Slt-W\ 0t t"'-e.- Ser~S is j LL3.+ t-~ S\k,'M (Jt b"'-~ o..'\W-.S 0t ~t, \e,Gh~~. So.) wVt.e."W
(lS \A;,\I\,~ ~ O\lv't-s(.t\J-fl..~ w~{;-\-"'-tX' 01(- Yuot' b"'-lJ 5{,r~e..s Covv\l~,,~e~.) ~ou...''\~ '('eoA~ O-$K;,~ 30 \J.J'C"sef,v<.-s
w\vJr~ex tv...e.. v~ of b\ve S9lA.W-.Ct. Q,\) 0.2.) Q 3 "I
00 40 2e"D _!OJsi- ruv...,0\A.?o""- So tW t~ ~ O-~~ la'll- (t,I,...e, Sv.."", of -l:,'M, <l
13Z. t: x O.;Vvvp It. :
1+1 + 2
I , + - + - 4- II, 3 ~ 5 - 00
(t""-e.. g\AW\
d..,0v tr~ es t-o +- (0)
\ (0+ 0 •• 1 \'s . , ~ tk !~ _____ ~_ "GI/\A. IS NOT 0.., ~ QoW\ (,,+-,~ se;r;;?'S) ". e. or ~
10 ,,,,,,, 1. r"'. A ,\\110""'(,t";'(, stx-;es 'NOv.U b~, 40 <"
V) = 0 I I I ~)(OJVV\.f\.(, ) I + l + ~ + I .... \\ :Sw.+- tke.. ~c.r":e..s g)pov~
~.s OvLL .at t"'-,,- &~~\t\£ Ii iY\; bJ.w~e.Vl- ,\ \
No+~ b~ t:,\".~ :SU"IN\ o~ t~e.Ov'\WvS o~ tu '\(.J..Ov~ ~e.s t~ GR-eAl'8R. t~ tk~ Ux~tov G'vt~t,seJ b:1 -b\v~ ~ lofk o t j::s X 0Jw:! I~~ 'X - a. )<'\s -
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2,5
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lk'C"~ (I~-wt ICS+-): Sl.<fpos~. a" '" t(\'1.)) VI = I) 2, 3) ". ) wlLe,.<.. \:h" fl.<-~o"" ::i = 5(?<-) t$. fOS ~~\i t- Of'N1 de.c~e.a.$:~~. \N -t ~'-I-t &.oJ..:
· J~ 1'(0<.) dO(. CDVV'l«'"<'J(S' ~ l 0"", COvw4'~O
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fo(' e.,xOv'M,plt y S~nc~ wR.., ~"JL, ~+-:
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J ~r Govw ~r~ .e.s 60
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- -~f V\=l it ~yt~%~.s f~ I.
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(w'v-...f.K~ Q", =:: ~ (V\ ') ). ts \Nt fJOl' V\ ece,sso\JI(""~S ~o 6~ ~<lNV\e,
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k wt \ +- \ $ -t;~t. c.o.,sc. lc~
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.so, b e-s:·b .. ~ '''' \::I..cL 9.a.b W t, ~: Q VI = ( ± )'1\, J VI, ; 0) I ) l) ...
s == . 0
.l\trw;,;vqy;;tFn' 4 o I t.
Mo~{. i.w ~ e,""6raQ :
lJo-te ~. w~
Si = 1+ 1 ,z ~ I ( ~2. = I + -;;..j. J.
;<... 'i) I"
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o I ~ ~ 0 I Z. 1. 2-z.. .z. 4
SVv = ~o +- a.,\+ 0..2, "" '" t- Q,«'_I = C'.o.J\l\ ofLSo 'N ~~ t
I . Sl; =Z-g)m
w~(,k. is the, c1ose,d - (X~
Now) w (., cou,tJ k~ 0" 'More. ~ e\tU)ro..e o{-='p'roo..ctv
~ w'CJ\- <-) 40 (" ~'~ 'W\,~'\~ ""' E. nt) t""-l fa.,rt;"oL Su..V\\.',
Q'C" :
1+ r +
\N ~~ tw... ~ V\.<h:x of t~e ~ ~ \A..~c,{J (h V\
sio;y-ts '~'("'OW\ 0 J' &00-<; ~s )\.0 ~e\A~~ - Q.cc.,.er~e.ol (.;:)y\ V-W\,MV\.. Ow \N\uJA- 60 eo n OS ~ t'll" ,I F~ S\J.NV\ \\ :
Cia .J.. (L I'" Ov2. +, ,,, + Q.,J.(._I (*) (,{( te(''MS) (1.,0 + Ov \ +- Q., z. -I- HI+- Ov ok. (~ ) (I\(.J- I ~vV\.s..J
13b.
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S ~ == I +- r +- r 2..j.. .\1 4- r k j Y\,(Jw" m,u..\+~f't.:1 b~ r!
'\ S l(.:::J '\ i- ,2.1- \~ -) 'll + r~.} , j V'v(JW'" ~ b\""~ cUft <.k'-RMte. :
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o t JO.v~ 'fu,vVVl b {;I("' '\:/:- I· ('¥Vot-
FORMULA io{"': 2. -k
I + r t r +- \\\ + r .
~or OMj v~~ h(;('t,S~Q'1(,~ I r l < I)
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