18
1 \'fD'2£""t~ t>E- 'K'Ol.l. ~ 'lA.1l'" fw...d'::"" -e'l. C\M.1r\1~ "'cJ~ j ":>.Q.~'\" [q 1",,1 (~kci",-,-<&''k & ~ ~)ID) ~~ ... .r(·~\.>-~(101 ::"D/ '2l ..;. -a.!l. ch~~~ ...a;.. U ~~ "'-~o, (~$i'--- ~ ~~ (),V>\ (s OJ.>, duiI.J.a.. 96~~ ~ ~""'\. it "'\:,/ ~". .\' (0."-') -e.,.l~ 0- fC'o-,) ~t~ '). ~ .:>~h/ ~ ( .... \ ')<.. ~~ ~ ~ eo.. ~s -ee~o!> .JJ 'fu.~~.. ----=- \ ~ (~"...f Cb) c: 0) -,μ.~'7 cL..... ~ J.Q. ~~+o L '-'(61; 't''Ad-- ~ k ~()5 "'-~ ~\-o .... ",-.c:.) 0. -<. Q <.. \0 I tJL~ μ &.M~ ~ ((Ie} ~ ~~ /,} ~, '-- r I .Q ,. ,\1 (~) =0 .. _ [ C'IJ - .n.'- , (.)<." 1.<.C........,.v-Q Co -:....)LJl ~.... ....' * '-"'<. •""'.. , t -- L «,.\ ~I ( ... 'l -= 0, ..... ~\n.t ~.,-fu ~.9-r ~.1~ .(' (,,'),,~ o.....f.."'k.t.u...... =- to. ,Ie] cLla.n- ;/.. ~ .... c~'t-a.'(' ~ 'IUI\ '!. f'M...iS,/ ,- l) 6 w... ~ ~ "'-" I IN. , ':::. ,.:{..:.,

Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

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Page 1: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

1\'fD'2£""t~ t>E- 'K'Ol.l€. ~ 'lA.1l'" fw...d'::"" -e'l. C\M.1r\1~ "'cJ~

j ":>.Q.~'\" [q 1",,1 (~kci",-,-<&''k & ~ ~)ID) ~~ ...

.r(·~\.>-~(101::"D/'2l ..;. -a.!l. ch~~~ ...a;.. U ~~"'-~o, (~$i'---

~ ~~ (),V>\ (s OJ.>, duiI.J.a.. 96~~ ~ ~""'\.it "'\:,/~". .\' (0."-') -e.,.l~ 0- fC'o-,) ~t~ '). ~ .:>~h/

~ ( ....\ ')<.. ~~ ~ ~ eo.. ~s -ee~o!> .JJ 'fu.~~..----=- \

~ (~"...fCb) c: 0)-€,µ.~'7 cL.....~ J.Q. ~~+o L'-'(61; 't''Ad-- ~ k~()5 "'-~ ~\-o ....",-.c:.) 0. -<. Q <.. \0 I tJL~ µ &.M~

~ ((Ie} ~ ~~ /,} ~,

'-- r I.Q ,. ,\1 (~) =0 .._

[

C'IJ • - .n.'- ,(.)<." 1.<.C........,.v-Q Co -:....)LJl ~.... ....' * '-"'<. • ""'.. , t--L«,.\ ~I (...'l -= 0, .......:

~\n.t ~.,-fu ~.9-r ~.1~ .('(,,'),,~ o.....f.."'k.t.u...... =- to. ,Ie]cLla.n- ;/.. ~ ....c~'t-a.'(' ~ 'IUI\ '!. f'M...iS,/ ,- l) 6 w... ~ ~ "'-" IIN. ,

':::. ,.:{..:.,

Page 2: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

~ """" l-'\ I ~,.J1M.Q.Q ') : {('" ') -::: "M." k::: C'tu.,,~<' ~<Ix~5 ,~I(v:) 0::.. 0 ,

a e. "'~ ~L ~ ~ ['lIb}.

"')I \tV... <- I-"\. '! --w...~ "" ~ :> 'tV-- =\: 0'; k ~0'"

~...... \{~r M.t"''''/ ~FrOllJ ~ I--'\. >0"

I

0"<"I

~(lC) s<.f(Q)~ )C "- C+},,)<,

~ (c -1:-1.)<.) s +'(c) )' v f,~ {J€~O\.c.

h:b ta-c/b~c1.

t (c -\ ~"') - ~(e) ~ ()

Page 3: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

\~ !./< ~0 i: 0- C''''''''-<> ~ c:Gv,.\~ ~~ ~

~r (c') ~ 0

~(((') ,?-o

Et ~'" ~ ~ S on.fo!'~

Seu-dlle.

,

-\'(tt) Oc\ (b)- 7"'"' ·t(('~t\-=:.a,

Page 4: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

~~\., " ~ 2e ~vo-~ ~I(l<')~ ~9~-k ~ -j.,~<D r6d..... LC<lk>31)~.(>,~" ¥ "'~~AC) ..M~~ \e~1M.../.J.. ~~~ '-'.0 "So. f"-'-cL. ~ "'" ..,

..("(1") -e:> o...AlUA,('".e.<.<. +.,t. '(Co, ( k,J '

[-1,{]~o..,:: -1~'0 "'- -1:L

~/()<.)~b VXEt~/bJ)

t.)c ~ ~+-z, a...... 'K '" 0 ,

J" (>< ') ~ 0 ~ ~"-~t.\ F+-c, ~ t C( Ik,1 '

. I .--~"". 'Ii>

rl'C-,c\ ~ ~ ';<y. . / O~x ~ i..l-)I'.+3 {J ~x~3

---7

\,I

'=-y2 I-t

/

Page 5: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

Ncs>~~ ':> dv-'- ~ \ (>0) ~o/ '..J)<. ~ [ ell ~ 11 ~"- ~+(lc) ~~ cu...+(IO(,'lt

.f(~) ".(C~)=o..o_

'"?-e'Y'O ,.(' '(Ie') "'\<, '" ~t.,,-k...... 'K ~ d , ?"'"~d".,J·...) ~ P~ ...-

"""te:.fe~ ~ t.A'Tt24"?(j6-- ..\ -EOe.t;olo-tA- 'til:L ~'j!.. Ke'P (0. tpl'.5I::.t.t\otcrv- ').. ,- . ...

~()'rtU.<. .., -A~~~_(\~,,~r:.~ V~ ~Q\"Sol. ,('(",,). \ll.l..o ~c:.~c....~(...')"-~ .. "::·~".. -tU<...\.tlW.'" ~O~

J. ~{.'r,,~o ['l' (1:>::11 ~ J,t-k>\-Q.o. t<~~ AI. (<\, h~ trr"'" h....i.C\u. 1..'(Q-4) 0.f I(~-) e.d~L'\ ~tsk: pd<" JA "'-'-l~"~

~'" (1'1'::+0 ): ~CI Q <. -C "' b/ j-..Q ~,- -.("((') "'- {o;)- +,(c,,?

I.a - C>..

~O~".. ~ ~ -fCb) - {(Q'2. J.0-<;.

""<,. ,*b..=t (I<) ;;: ..(' (I<) - fcCl. ') - t> b: -C{ ') ,

-

Page 6: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

-- - - - -

y

C...."""k ~ ea-.-c9.. ~s~ ~

Cc<JG)) I (bJ ~ (10')')

1/ /.' J., l"'--- ~ f"I- It~~...L + i{ d S oj"- >S'rI ........'

~~ILJ,""I~~ '(r= f(b\~t0') ('r:-o..')10 -Cl.

te'.,') _+Cu)I~P,-o...

-\- .f (0.)

\\.u.. ', "\(/()::o ~(I(:l- L.fca')+-&(,<-c...')]

"e~-\-." +~ct.\<;1M~b-t~ ~~)l~~(0:') ...........j ~ SR c......-I" ~.::: ~ ('( '"'"\ ..I. fc,),

~"-01. .\-6c ') 1 s ~J,'~ J ~-=- ~ (k-O'\ -+ -\ (4\ ~.s,-~ 1=""r,,) u C«A...\. i~ ......~,

Page 7: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

i-J.h'Q ~, -1(>0-) \~ c4'+~+l)1.A \"\A LCl,b:1",,1\>. <!.L ~ ("'\ 0 ()'::..& (..,-0.) -\-.(ca..') ~ so,

f..JO\U .'t-('o]:: ..{'(~ - ~(a-.l ~ & C6-t»J

>- -f CI!» - \-\(0.) "' 4 (b)-{ (<<\ (b-~)1L b-c.. ''j

-=- J C~) - ~ Ceo.} .-\: .\ (~') - ~(y.()

~ ,{(~I- [+llo)jl~

=F(~. =- O.,

- ~ ( 0-') ~ 'Q •OJ-\l(..))-- .rC...)

,q

-r(rt,\ -::. o·T~ 1--.lll!.!5> +t:....~ ",-J.h!-.x-·' "P(lc ')',

I . . . ~

\~ =/.,-6 ~ '(.'" c", '~ .u~t" "fICe) =0 I ~ < c .:..b,

'1~, .., +- I I. . T (~y-::=1 (1<-) -~ ,

o ~ ;f:' (Q') =:. -r I(c'l -&:::') (!) ~..f (-C) - &.

.a '3 ~

,10••

Page 8: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

r@ ""~ (c)

+' C b) - +((.) :::0 ~ Ire),-~~.D.

~CMu. k'~ (-kM'~~ l4~ ~~~. -\'.:It..,...'"j"l::\;, 6 eo __ (So C'....~ d..-,.. ~ '>R" a...-l ~ I ~ • ((l '-'\"-j a ,,",s:. ~ ~ ('0.; )"""-~lc, } CJ,~') L:r4 ~ '!. ~ s,.Q., ~ '/

& ""A(I,')-.(Cc..')b~'".-

-r~ ~ '-.)~~~ "\N..t.~9 ~+-"µ"'N...'<. X~<;.\-s ':>. ~~ ~'I.u... <t> ~ '(f'rR- ..~ ~. kcl-l4...t"2~fl"\ ",4 ... f'Oc,A. I< ","0) 9-u-ck -\4,,-\- ~ s-Q.~

c.\- R..,f.- ~ 9-s. +4 ~ t). ~ sQgf'-" 0+ ~

,\

(I

c Jo

Page 9: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

~ ( ~ 't'~ -a G E'fK I 0 < (9 <. I ~J...-\-1..)-11

•'"- ~ :::. S (1.,-"'\ --\~.~~l~ ~j .

..\(b')..:t(Cll~ '"'"-r' ( (j. -I fI( k,)),l? -c...--------------------------------------

'§?_ -r~~ o'U....feu. 'j1A~~ s 0 ~ +OH> t...ckcas 'P\i~.

( C'MClllo(r~T~H~) (Y'f',::> k..R I.<4I\J-\ •

~ONµ." ~{ ~ (V.\ -a.J t(lr1o') ..(.wo e>cu.+lIA.UlN!. ~C+teu. Cl'v.

~ ,...-\.v~ Cat ~'1 f II I'\J Jd~W-l~""\..-6..... Cl-\. (~\ b), 4Rsod ":>~ +Cv.-~ ~/(l<)~ 0 C<.I.. ((I lb). T\..o....., ~~ e.\Crd~(

((()~~....,t CIoN.,') c €:- Co. ('rJ ') I ~ ~i'.-\(10 'I ~;(:(0.) =- -\ (')

__ lf1 ('oj - ~(G.) tt>'(C)

T.,'R"ClC <rk~! ~ = ~(b)-+ro.~{J(Io~-l£ (a.)~f'S ..

-

Page 10: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

~t>{~ ~w l(' (\0')- lQ( ct.) 4:00' ~\".d:!...o.!'f (b ') -~ (0.-') ~C) I "N-J ~ ~ (~ +l...~-<-I

~ ~ol.Jcl t:\<Ot c,{ So C ~ (Q (b") I %c1 {J...,r.;,{-

~ ((('\"-0; ~..t- ~~ ~,,+ b-<- ~ &((or-.~

+(> \-t..L ~\""~~~ ",f ~ rr-s...,,J +l.o~, +~~\0! n~J-~(6\ 4C),

~~ I 'JS ~ r'-'{~ ct.'cf( ck.~"

TC)C) ~ +(\<') ~ -\'Co.\ - ~ t'-\l ("') -l(J(a.\1 'No\.to!' 4t",~:.~C"'') Is o..u.V\<.u.Qh.uS u...... [Q(\o3, 'miC'Q

-KJc') ~& cQ (A.) ~ -",c.

• i=-(k) p"" Jp( ~ \{i,u.. "'" 1(Q ( b) /"'""""-<1>.\'C)c\ ~~ qJ(~\ 'he 30,

~ f'(Cl\ ~ -fCet) --\Co.) - & [f(o.) _ ~("Jj::::.0 .

'" tn») ~ ..f(~)-+(C() -<Si'[cf(t:l) -C{l(i~

~ ((10') - fra.) - (fm-{r4J) L'f(~)-1~ce(+>\-ct' Co..) .

= f( ~\-(r",\ - (t(b)~t-C\\~. ~~

t:- (10) :::.. OJ

Page 11: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

\,'"• •

,I~ -

,t'£..

~l~(> \Lo~; :3 <: I 0... < c <1) 'rwJ..1= I(e~ !:::..(!J (

f(c) ::o~~I(C) ...,

..(' C( (' \ _ -f (l;,') - .( (a.\, -

t - •C{ (C) f' (I:l) '- 4:(CJ..) •

tuA- olo'(""~ ~ dt~'M>oI :~.(..(~, ~e:D.•

NG~~ ~~ Cor-.<.I*- f'J"1l~ C''4-U~ ($ -tt..e~ ~{~~ (~-J~ ~t1~ ... 1,,-&..u09·,

j c~ ~ (Q ( 1,,') I ~ .{~i "

..t 'e (,,)::- -CCIo) -,{' ("':.)~

~ ('z. t- (Qr!o) , 'I::> -Q..

~((C"Z) ~lf(b)-.l(p(~\··\,;)~"'-

Page 12: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

a~\l?'c1~ ~ .6.cQ0"YaetM.. clJ QM~ to da.. h~. c j,.bTetcG)\1\)O~*

~,- 01 ~.cl()~ ct. ks E'd'So ~\, --l-?-e. -f\i'k.2(M[

~~> ~ ')\<') es. &!{~ ,,,,\...e...

-e...... (q,'o) (6 c.......~[~r;, ~. [«1101).\~"'". /(~~CQlb),t'u.--kCQ.C,i ~~"'-~~\ F .((~)e\«d'l'. fg eke, 'f""

J.. ~;~:

eyc(").\.or d g ~ct.'b ~\k So. k ~':>

4!...y; ') ~.'F -:=c '!=-.o I

~(y.c*t'l,x) ~~(~\-(::,..,..

f", ~ ""l!\~l \-tc'::> ~:' ,"", .

L \~(Y;"-l~~)-J!~) ~~I(~;lc::.0rhl<~o l b.1-- "J

{>~.~':'b7. 'i;t~,.;Qf(kl:'fki:'(}clt~~&lJO-...• -\ ( ~ t> -I.!.I<') '- ~ (¥.0 _ .\ f Il'c,')

~

~~ 'd \\w'o ~ t::, l< ---"'} 0.. ~b{.u.u".s. 'tv-<- e.'l..hdt>~"-<..~~{fe.v..o&o" ~~'" c£k ~-kj. )C., ~ 6.",

Page 13: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

Q, ~....""\O So ..... -6...09$.-rJ('Ko,AF) ~ f,(---o~A...\ - -\ (~!> - -\ 1(>--0").

Ar-r

'd "::>\ '

----~!) (,

Q,..~~.....~ .('[-rca ~IC\ o\'~ ....0 ., ,- ,l~("d &'1-) ::,., !(IC~ -+ ~ (x;) box

C\<,J.. !l{ ("bI Ac.') - ,::", 0,1:>.,. -')0

t::..~( ("""0 b./<. ~ ~ -::. 'f..t,/ <e.u\-w..(!Q,S Co":.:i!!-::" ~ _~~f",_

,of-" r..fC<) ~.\ r",') ,{'("" \"-"<0,) + ;;("0, xl ~x_~\]._----- --,-

Page 14: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

B< ""..-.vr e..... ~ ':>Fo,<.~~4~·,1I~ ~ w'oQ."., l..~'E. 'P o"C" ~ <iI (:.~c id : .f(X) - \: ("') / 1"'-< ~J,.., ~~

..C C ~'>'Ub ~~~~ J. ~ r-

&tw't»Jc ~ c.<UA~QQ.<..u 0~ )(, )(,e / ~S r.r

(Ovt..c> ~" <:> ros b...2r< 61.. ('I<., "0 . ~..RJ..() I \.x~~e..,

~Se U ~~."~~ V-4JJ.,'- C:> dJ> ~~ \.

0>- 9r..- (-kj'~4'"C. J. t:J. ('y:., Y-. ") "

6L()CrlCo) ~ ~ ('I",) -:_~(~ - -\(('1'0) ,)C - )Co

~o ~ {'()<: ') - \ ('r-o') 'PC>"(' ~ -+~ aOJ ..r.J&, U.t(~- ...... /

)<. -'>Co .

~ lc t'<rt -Q..C, EO C~ ) N) hi- "r-v,.[ ('f')~.r'(~~c::~) (' ') .

~

~()C)- Y(,><) -;o,C~f (C) - \ 1('f'O~ ('( -K.')

$::""- '-'e> ~ ~ ~ t-es A~ <fC OWl>

-=So "'-

Page 15: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

~ ~~ '(' Co ~ ...J.... I"--W..~ vi.-{~ U ~~

( 1<-0 I ",.\ ~ So ~ C'c"o c&..,o ~

v-u. elI <> ""e <; ell = ~~ ""-~W><'O Q Q.\C (." f 0 ( ewv 'f;S lAD'" ~ CJ< ~u.f~-E~

~ ~u..£ ~ ~":>cto e s..

~~: ~~~«~ d.... ~ ~-cu:l~

oJ.. ~ ~ coL' c1.~ ? ~i"'-<. III ~ •

U~o ",,; ~(lC'\':'" ,f)<. a '(,F b </, J.~ ..tJ~..\{v:~"-~., "'-~ 0: ~ • I::'CM.o .. ~.. !> ", .......' 't. "- b'S:

U~ ~ ~'Il ~ ~l~"~~ A~.y ()()::::'~f(~o\('Y:. - ~~ -\- f (1<.0

Gu.." { ( ('oC'> ~.J. 1..iII.\.+"'" ~s: ~((lcc\\ ~ ~ ~..L,;:}~ .2vH :<t8\

- ±- G &,tn"co.s J '>L - ¥'o .;;- 6S -~4 ~.L/~ "'~

'i (65) ~ t o.i + SR "'- '2 -I ~ "- So. 0(92S;

~ ~ ~(&'S,\:::. '\ ((,5)-+ <i. ('(..,'1.0) (-. -~\,

~S~()6Z5 T rx['f..,Yco')(l<. -'><0\.~8'1""

Page 16: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

l \......Q~,\\. +to. ~\W'" 1)u..... q,\ 1

~ C ,Y.e.) ('i<-'fCo')

kIA9<-> .\t...,.\- (-" - ¥Oc'J :2-- b S - Go <I'; .::. 1 t +-I

oliX, ~), 13 tL-\- I g~ WoO f""'-\)b.~

d ("" 't.) ~ -\,1<1:') -- (( )<0)

0. ~ ('f..", "'') ,::: (GLJ1 b 51, T~ I

t:J. ('F I 'l<o') ~ --t . _ .-L 1.(..l ~ ~~~SC .:<.~ ~ 2 \fC -IRQ

- J ~ -- --rc. .- :2,

-< .".:;n \fQ: '~"': Yo :0 b 4 :

~{Yt (,4) ::. \ ~ - ~~ , •p ~-JC:"'

b.t...o>I'>r- ~ ~

~")'.c .J.. . "'..O!~

0({lCc64')'::;o -L -..L2U' ((j

7x" C l<,

Page 17: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

F"" ~. 1Md ..,." hw.",,'-(.

')( ";.--~ -f: ~ nl' ~'1 i

L.<-Lz.l;?'( ;?.J2 (6

..l <~<.-L1'6 2{'c.. Ib :

~c"'l>? It,,QY. C>i (1« 64) 0\.!.~" ""'-0 ~ 0 y' \.L'A,':>

\ ' c" ; --+-- -ll,,'311"-''''(~~11.<~o;( fY.'"("~ "l.\} C. / ~ ... 9 (~ ;. <>-::,i·

cf. ('f I b.f\ -;:.:. ..L. _ 1- -.\.-12J'c. if, - t~ fp

-- - -- -

O!f'?( b 4- ') <.. CJ •

Th,~ ~ 'h-L') J.Q..* a..clr 'd f' ~ I> ~ ~~ <t.o.

+I.t ... \L .J \a.. ~':;I c+ IP>-Lu.) ~+(.t- IS ~

~::: 0 c {)C) 6'1-1</. . __ .

-S3--,--0' --

j '!. 'J'r<' j1'>'y~\...'1'" ~".'

Page 18: Chapter 2 Differential Calculus A · ~ ( ~ 't'~ -a G E'fK I 0

Eo;, ~c"",, ~~J ~::.d., :>f' 'is

-L ::.. ~ ::: J. ~ O~0<5)'),.".2~ .2~'1 tB

~ +.J. -= 'g. () S5 _.,4. ~s2-Ji\

~+-.l..,g-i- - ..L =- O. dO 6,,\ -<14<\ , ..1'" \'3