Contents i i

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2 Contents i i

3 f() F'()=f() F() f() f() f() f() F() f() F()+C`(C) :`f() d=f()+c f()c f() F() G() F'()=f()G'()=f() {G()_F()}'=G'()_F'()=f()_f()=0 0 C G()_F()=C G()=F()+C f() F() f() F()+C`(C) C f() :`f() d= `+4+C f()(c) `f() F() F()=:`f() d F()=:`(4_8) d= `_8+C`(C) F()=0 D D =(_4) `_C=0 4 C=8 F()= `_8+8 F()=_8+8= : 미분한다. F() f() 적분한다. 부정적분 f() d= F()+C 피적분함수 미분 적분상수 f()=4_8 F() F()=0 F() 0 f() f '()= cos ` ; ;f(0)=f(p) ;4 ;+ ; ;+ ; ;+ p+ p+ `f '() f() f() f() f() f '()=3(_4) f() 5 0 _5 _6 _7 _3 q f()=:`sin (3p+) d f(p)=f '{;6 ;}f(0)= p+q p (pq) f '() f() f( ) f'( ) f()f '()=4_6f()=0 f()=0ab a `+b ` d [:`(+) ` d_:`(_) ` d]=a `+b+c a b c d a+b+c f()=:`(_tan ) cos df(0)=f {;4 ;} _/ _ / Structure 0 3 Level - Level - Level 3 3

4 0 f() F'()=f() F() f() f() f() 미분한다. F() f() 적분한다. f() F() f() F()+C`(C) :`f() d=f()+c 부정적분 : f() d= F()+C 미분피적분함수적분상수 f()c f() F() G() F'()=f()G'()=f() {G()_F()}'=G'()_F'()=f()_f()=0 0 C G()_F()=C G()=F()+C f() F() f() F()+C`(C) C f() :`f() d= `+4+C f()(c) f() :`(3 `_) d=f()+c f '()(C)

5 y=«`n C n+_:`«` d= n+ +C n+ n=_:` _ d=:` d=ln +C n+_{ n+ } ' = (n+) (n+)_ = n n+ n+ :` n d= n+ +C n+ n=_ (ln )'= :` ` d=:` d=ln +C = ` f()»(), :`kf()d=k:`f() d`(k) :`{ f()»()}d=:`f() d :`»() d`() f()»() F()G() F()=:`f() dg()=:`»() d F'()=f()G'()=»() {kf()}'=kf'()=kf() :`kf() d=kf()=k:`f() d {F() G()}'=F'() G'()=f()»() :`{ f()»()} d=f() G()=:`f() d :`»() d`() 3 :` ` / d :` d ` 4 :`(+)d+:`(+_ `)d :`(+) ` d_:`(_) ` d 5

6 0 C :`sin d=_cos +C :`sec ` d=tan +C :`sec tan d=sec +C :`cos d=sin +C :`cosec ` d=_cot +C :`cosec cot d=_cosec +C (cos )'=_sin :`sin d=_cos +C (sin )'=cos :`cos d=sin +C (tan )'=sec ` :`sec ` d=tan +C (cot )'=_cosec ` :`cosec ` d=_cot +C (sec )'=sec tan :`sec tan d=sec +C (cosec )'=_cosec cot :`cosec cot d=_cosec +C C :`e d=e +C`(e) :`a d= a ` ln a +C`(a>0a+) (e ` )'=e ` :`e ` d=e ` +C (a ` )' a ` (a ` )'=a ` ln a a ` = ={ } ' a ` :`a ` d= +C ln a ln a ln a 5 :`(4 sin +3 cos ) d :`{tan ` _ } d sin ` 6 :`e ` ± ` d (e) :`( ` +) ` d 6

7 f()=4_8 F() F()=0 F() 0 `f() F() F()=:`f() d F()=:`(4_8) d= `_8+C`(C) F()=0 D D 4 =(_4) `_C=0 C=8 F()= `_8+8 F()=_8+8= f() f '()=afi`+f(0)=3 f()=6a f() f '()= cos ` ; ;f(0)=f(p) ;4 ;+ ; ;+ ; ;+ p+ p+ 7

8 ` f()=:` d_:` d f(0)=3f(_)(+) : 6 : ;3&; ;%; ;3*; : 6 : :`{ f()»()} d=:`f() d :`»() d`() a `_b `=(a_b)(a `+ab+b `) ` ` `f()=:` d_:` d=:`{ _ } d `_ (_)( `++) `f()=:` d=:` d `f()=:`( `++) d `f()=;3!;æ `+;!;æ `++C`(C) f(0)=3 C=3 f()=;3!;æ `+;!;æ `++3 f(_)=_;3!;+;!;_+3=: 6 : 3 `+ ` f()=:` d+:` d f(0)=f() `++ `++ : 6 : ;3&; ;%; ;3*; : 6 :? ` 4 f()=:` d_:` d f()=f(3) _/ _/ ( ) ;#; ;!; _;4!; 8

9 f()=:`(_)(_) d : 6 :f() ;@#; ;@%; : 6 : ;4(; `f '()=(_)(_) f() == `f()=:`(_)(_) d=:`( `_3+) d=;3!;æ `_;#;æ `++C`(C) f '()=(_)(_) f() f '() f () y y y + 0 _ 0 + f() == `f() : 6 : `f()=;3*;_6+4+c=: 6 : C=;6&; f()=;3!;æ `_;#;æ `++;6&; f() `f()=;3!;_;#;++;6&;= 5 f() y=f '() (_0)(0) y=f() y=4_5f() y=f()»()=:`f() d y _»() y=f() O 9

10 f() f()=0 y=f() ( f()) 6 `_ f() (k)k _5 _4 _3 y=f() (f()) f '() `f()=:`f '() d y=f() ( f()) 6 `_ `f '()=6 `_ `f()=:`(6 `_) d= `_+C`(C) `f()=6_4+c=0 C=_ f()= ` k=f()= =_ 7 y=f() ( f()) _3 `+ y=f() (k)k y=f() +0 ( f()) +;K: y=f() A()B(ee `+)k(e) 3 0

11 f() f '()=_3+ cos {+; ;} f(0)=3f {;3 ;} _p _p 3_p +;3 ; +; ; cos {+; ;}=_sin :`sin d=_cos +C`(C) `f '()=_3+ cos {+; ;} `f()=:`[_3+ cos {+; ;}] d `f()=:`(_3_ sin ) d `f()=_3+ cos +C`(C) `f(0)=0++c=3 C= f()=_3+ cos + `f {;3 ;}=_3 ;3 ;+ ;!;+=_p e ` ` 9 f()=:` d_:` d (0)f() e ` ` +e ` + e ` ` +e ` + (e) e_ e_ e+ e `_ e `+ 0 sin ` f()=:` d f(0)=3f {; ;} +cos ( n +(n+)p) p+ p+ p+3 p+ 3p+

12 `f '() f() f() f() f() f '()=3(_4) f() 5 0 _5 _6 _7 _3 f()=:`sin (3p+) d f(p)=f '{;6 ;}f(0)= (pq) q p p+q 3 f '() f() y=f() +f '() ` ( <) f '()=g _ ( >) y=f() =_ f()=f(_) `f(0)=0 f()>0

13 f()f '()=4_6f()=0 f()=0ab a `+b ` d d a+b+c(abc) [:`(+) ` d_:`(_) ` d]=a `+b+c f()=:`(_tan ) cos df(0)=f {;4 ;} _/ _ / 4 4 ` _ f()f '()= f(0)= f() ` + ln ln ln ln ln ln 3

14 f()=:(_)(+)( `+) d f(0)=lim ` ` f()_f() `_ ;5!; ;5@; ;5#; ;5$; f() F() F()=f()+ `f() 3 f() `f()_ lim =;%; ` `_ :`( `+a+) d=f() a f()=:`k{sin ; ;+cos ; ;}` d y=f() (p f(p)) y=+3f(0)(k) 0 4

15 f() f '()= ` ++k`(k) `f() lim = ` 0 k+f() ln ln ln ln ln f() y=f() (y)6(_k) y=f()y=0y=8 f()(k k< f()f '() e ` (>0) `f '()=g 6+k (<0) f(ln )=4 f(_)=_6k(e) 4 f() y f(+y)=f()+f(y)_y_ f(_)=f(+) abf(/ )=a+b/ a+b 5

16 0»(t) =»(t) :`f() d=:`f(»(t))»'(t) dt F()=:`f() d` yy t=»(t) F()=F(»(t)) t d d d F()= F() dt d dt F()=f()»'(t) F()=f(»(t))» '(t) t F()=:`f(»(t))»'(t) dtyy :`f()d=:`f(»(t))»'(t) dt dt :`f(»())»'() d»()=t =»'() d :`f(»())»'() d=:`f(t)dt :`f(t)dtt»() :`(+5) ` d :`sin (_) d :`sin ` cos d :`4e ` d`(e) 6

17 f '() f() `f '() :` d=ln f() +C`(C) `f() dt f()=t =f '() d `f '() :` d=:` f '() d `f() f() :` d=:` dt t :` d=ln t +C :` d=ln f() +C `f '() :` d=ln f() +C`(C) `f() ` :` d=:`{_+ } d + + :` d=;!;æ `_+ln + +C`(C) 3 :` d :`tan d ` :` d :` d (+) `_ 7

18 0 f()»() :`f()»'() d=f()»()_:`f '()»() d f()»() { f()»()}'=f '()»()+f()»'() `f()»()=:`{ f '()»()+f()»'()} d `f()»()=:`f '()»() d+:`f()»'()d :`f()»'() d=f()»()_:`f '()»() d f()»'() ln :,,, : sin, cos : e f()`k h`» '() 그대로미분 : f()`» '()d=f()»()_: f '()»()d 적분 그대로 5 :`ln d :`e ` d`(e) 6 h()=:` sin d h{; ;}=; ;4h{;4 ;} _+; ; _+p +; ; p +p 8

19 f()=(a+) `»()=:`f() d»(0)=»(_)f() (aa>0) :`(a+b) n dt d a+b=t =a d :`(a+b) n d=:`;a!;æt n dt=;a!; t n+ +C=;a!; (a+b) n+ +C`(C) n+ n+ dt a+=t =a d»()=:`(a+) ` d=:`;a!;æt ` dt= t `+C`(C) 4a»()= (a+) `+C 4a»(0)=»(_) (0+) `+C= (_a+) `+C 4a 4a `=(_a+) ` _a+= a=4 ( a>0) f()=(4+) ` f()=6 `=6 6 q f()=:`6 cos {+;6 ;} d f {;6 ;}=5 f(0)= p+q p (pq) f() f '()=4? `+f(0)=;3%;æf() 9

20 `f '() :` d `f() f()=:` cos + sin df {;6 ;} _ ln _ln ln / +ln / +ln `f '() `f() `f '() :` d=ln f() +C`(C) `f() (+ sin )'= cos `f()=:` `f()=;!;æ:` cos + sin d cos + sin d `f()=;!;æln + sin +C`(C) f() `f(0)=0+c=0 C=0 f()=;!;æln + sin `f {;6 ;}=;!;æln + =ln / 3 +3 f()=:` d f()_f(0) `+3+ ln ln 3 ln ln 5 ln 3 4 f()=:` d f(e)=f(e `)`(e) ln _+ln _+ln 3 +ln +ln 3 0

21 + q f()=:` d f(3)=ln f(5)=ln p+q (_)(_) p (pq) `f '() `f() p+q A B = + AB (+a)(+b) +a +b A B :`{ + } d +a +b + A B + = + = (_)(_) (_)(_) A+B=A+B=_ + 3 A=_B=3 =_ + (_)(_) 3 `f()=:`{_ + } d `f()=_`:` d+3:` d `f()=_ ln _ +3 ln _ +C`(C) f(3)=_ ln +C=ln C=3 ln f()=_ ln _ +3 ln _ +3 ln `f(5)=_ ln 4+3 ln 3+3 ln =3 ln 3_ln =ln : : (A+B)_(A+B) (_)(_) p+q=+7= :` d(c) `+ ln +ln + +C ln + ln + +C ln _ln + +C ln _ln + +C ln _ ln + +C 6 `_5+8 f()=:` d f(0)=3 ln f(3) `_6+8

22 f()=:`cos ` d f(0)=;3$;æf {; ;} n cos«` sin«` dt sin =t =cos d f()=:`cos ` d f()=:`(_sin ` ) cos d f()=:`(_t `) dt f()=t_;3!;æt `+C`(C) f()=sin _;3!;æsin ` +C `f(0)=;3$;æ C=;3$; f()=sin _;3!;æsin ` +;3$; `f {; ;}=_;3!;+;3$;= 7 f()=:`sin ` (_) d f(0)=;3$;æf (p) f()=:`(+sin ) ` cos d f(0)=3f {; ;}

23 f() f '()=(_)e ` f() 0f(0) (e) e_ e_ e e+ e+ :`f()» '() d=f()»()_:`f '()»() d `f '()=(_)e ` =0 =`( e ` >0) <f '()<0 >f '()>0 f() = 0 f()=0 f()=:`(_)e ` d f()=(_)e ` _:`e ` d f()=(_)e ` _e ` +C`(C) f()=(_)e ` +C f()=_e+c=0 C=e f()=(_)e ` +e `f(0)=e_ 9 f()=:`(_)cos d f(0)=f() sin cos +sin +cos 0 :`e ` sin d(c e) ;!;æe ` (sin _cos )+C e ` (sin _cos )+C e ` (sin +cos )+C ;!;æe ` (cos +sin )+C e ` (cos _sin )+C 3

24 f()=:`? ln d f(e)=f(e `)(e) 5 : 3 : 6 : 3ª: 7 (` sin ` (>0) f()f '()= ` f(p)=f(_p)= 9 k`cos {; ;_} (<0) k ;3@; ;3$; ;3%; 3 f() `f(+h)_f(_h) lim =(+)e ` lim f()=e h` 0 h ` f(0)(e) e e 3e 4

25 f()=:`(_)( `_4+3) ` d f(0)=0f() <<pf()f '()=cos f {; ;}=0 f()=;;;4;;; /3` ; ; ;4#;p p ;4%;p ;#;p 3 f() f '()= 3 3 `+ f(0)=3f(4) ln 6+3 ln ln +3 3 ln +4 ln f()=:`4 ` ln d f()=0 f(e)=ae `+b aba+b (e)

26 f()=:(_sin ) ` cos d f {; ;}=0 f(h)=_;3!;æ h(_p<h<p) _;3 ; _;6 ; 0 ;6 ; ;3 ; f()=:` d y=f() (_)? `+3 ABAB ` 3 f()=:`e ;!; d f()=e n= `f(n) (e) e (e `_) (e `_) e e (e_) e_ (e_) 4 >0 f() F() F()=;!;æe F()=f()_;!;æ ` e ` f(3)(e) e e ` e ` e ` e ` 6

27 >0 f() `f() +f '()={+;!;} ln, f()=_;#; f(e)(e) ;4E; ;E; ;4#;e e ;4%;e f()»()=:`f()e d»(0)=»(3)(e) (_)f()=:`f() d `f()+3 lim =3 ` _ 3 >0 f() `f()= f '()_f()+3=0 ( f '()+0) `f '()= `f "()= `f(/3 )=3 7

28 03 n n n` y= `= S 0 n n y y= y y= O ;n!;;n@;;n#; n_ :;;n;;: O ;n!;;n@;;n#; n_ :;;n;;: S«T«n_ S«= ;n!;{;nk;} (n_)n(n_) n = T«= ;n!;{;nk;} = k=0 6n ` k= SS«<S<T«lim S«S lim T«lim S«= lim T«=;3!;S=;3!; n` n` n` n` n(n+)(n+) 6n ` rh n n n h r 8

29 f() ab ab n º(=a) y«(=b) lim ;Kn+!`f( )D {D= n` b_a n =a+kd} y y=f() f( ) O º «a Δ b = = f() a b :Ab``f()d :Ab``f()d= lim ;Kn+!`f( )D n` ab y=f()=a=b`(a<b) S ab f()>0 :Ab``f() d=s ab f()<0 :Ab``f() d=_s 3 ab f()æ0 (a c)f() 0 (c b) y=f() =a S y=f() =b S :Ab``f() d=s _S y O a y=f() S b y O a S b y=f() y O a y=f() S b c S 3 k lim ;Kn+! {+ n } n =:!à ` da n` :)3``(e `+) d= lim ;Kn+!`{e ;A;nK; +};Å;n; a(e) n`

30 03 f() ab F()f() :Ab``f() d=[f()]ba=f(b)_f(a) f(t)æ0y=f(t)t=at=`(a b) t S() y y=f(t) S()=:A f(t) dt S() D>0f(t) +D M m O a b t DS=S(+D)_S() DS m D DS M Dm D M DS lim m lim lim M D` 0 D` 0 D D` 0 D ` `0 m ` `f()m ` `f() S'()=f() d :A f(t) dt=f() d S()f() f() F() y O M ΔS m a +Δ y=f(t) b t S()=:A f(t) dt=f()+c`(c) yy S(a)=0C=_F(a) =bt :Ab``f() d=f(b)_f(a) yy f(t) 0D<0 F(b)_F(a) [F()]bA :Ab``f() df()ab :Ab``f() d=:ab``f(y) dy=:ab``f(t)dt=y a=b:aà `f() d=0a>b:ab``f() d=_:bà `f() d 4 :)` (6 `+3/ß ) d :) ; ; cos d+: ; ; ) cos d 3 30

31 f()»() ab :Ab``k f() d=k:ab``f() d (k) :Ab``{ f()»()} d=:ab``f() d :Ab``»() d () f()abc :Ab``f() d=:ac``f() d+:cb``f() d f()»()f()g() :Ab``{ f()»()} d=[f() G()]bA={F(b) G(b)}_{F(a) G(a)} :Ab``{ f()+»()} d={f(b)_f(a)} {G(b)_G(a)}=[F()]bA [G()]bA :Ab``{ f()+»()} d=:ab``f() d :Ab``»() d () f()f() :Ac``f() d+:cb``f() d=[f()]ca+[f()]bc={f(c)_f(a)}+{f(b)_f(c)} :Ac``f() d+:cd``f() d=f(b)_f(a)=[f()]ba=:ab``f() d y y축에대하여대칭 y y=f() 원점에대하여대칭 y=f() y f(_)=f() :_aa`f() d=:)à `f() d _a f(_)=_f() :_aa`f() d=0 _a O a O a 6 0:!` ` d :)` (+) ` d_:)` (_) ` d 7 :)` `_ d :)``sin d+:! ; ;`sin d 3

32 ah n (n_) k h ;n; (ka) ` ;nh; (n_) a `h V«V«= V V= limv«=;3!;a `h 6n ` n` a f(n)»(n)»() `f() n(n+) n(n+)(n+) n(n+) ;Kn+! k= ;Kn+! k `= ;Kn+! k `=[ ] 6 a a 3a (n_)a y n n n n ka k{ } =(ka) ` n n ` n_ ka V«= { } h a `h n_ a `h = k `= k= n n n ` k= n ` a `h V«= (n_)(n_) 6n ` n(n_)(n_) 6 a `h V= lim V«= lim [ (n_)(n_)]=;3!;a `h n` n` 6n ` f(n)=»(n)=(n_)(n_)»()=4 3= n ` `f() a h y=3 `= S 0 n y 3 y=3 n S= lim a ;Kn+! k `a+s n` n ` (a) O ;n!; ;n#; ;n@; n_ :;;n;;: 3

33 :!à `(+a) d=4a ;%; 3 ;&; 4 f() ab F'()=f():Ab``f() d=[f()]ba=f(b)_f(a) :`(+a) d= `+a+c`(c) :!à `(+a) d=[ `+a]a!=(a `+a `)_(+a)=a `_a_=4 a `_a_5=0(a+5)(a_3)=0 a=3`( a>0) (/ß_a) ` ab :!4`` d=_+b ln a+b ab :_ba`d=: : :)``(3ay_)(y_b)dy=0 ab ;%; 3 ;&; 4 33

34 ` `+ ( ) f()=g :_!`4 f() d 3_ ` (æ)»() (a c) `f()=g :Ab``f() d=:ac``»() d+:cb``h() d h() (c b) _ f()= `+ f()=3_ ` :_!`4 f() d=:_!`4 f() d+:!``4 f() d :_!`4 f() d=:_!`4( `+) d+:!``4(3_ `) d :_!`4 f() d=4:_!`( `+) d+4:!``(3 `_ `) d :_!`4 f() d=0+4[ `_;4!; `]! { f(_)=_f() :_aa` f() d=0} :_!`4 f() d=4[(8_4)_{_;4!;}]=3 3 4 :_4! e ` _ d=a ;e!;+be `+cabc a+b+c (e) _5 _4 _3 5 y=f():)3``(+) f() d y y=f() 8 : 3 : : 3 : 9 : 3 : O 3 34

35 f() :_!` f() d=:_!`(+) f() d=: 3º: f() f() f(_)=f():_!` f() d=:)``f() d f(_)=_f():_!` f() d=0 `f()=a+b`(a+0) :_!` f() d=:_!`(a `+b) d=a:)`` ` d=a[;3!; `])=a ;3!;=;3@; a= a=3 :_!`(+)f() d=:_!` ` f() d+:_!` f() d=b:)`` ` d+=b[;3!; `])+ :_!`(+)f() d=b ;3!;+=;3@; b+=: 3º: b= f()=3+f()=3+=5 5 6 : ; ;` (sin +cos +) ` d _; ; p+ p+ p+ p+ p+4 7 f() f(_)=_f() :_!` f() d=8 :_!`(+) ` f() d 35

36 f()=6 `+a :)``f() d=f()a _4 _ 0 4 f '() y=f()f()=f(3)=_3 y y=f() f(0)=_3:)3`` f '() d O _3 3 f() :)``f() d f(0)=f '(0)= 0<a<b<f '(a) f '(b) (0)f "()=e ` ;!;e_ ;#;e_ ;%;e_ ;&;e_ ;(;e_ 36

37 :)3`` ` d= lim ;Kn+! {:Ånapple:} 4 ;na;a n` :)``(6_3 `) d :)` ( + _? 4 ` ) d 3 5 ln ln ln ln ln 4 ` :_0!` d+:)-``` dy _ y_ ;3!; ;!; ;3@; ;6%; 5 :_5@ (4 `+3 `++sin ) d+:%` (4 `+3 `++sin ) d 37

38 :`p (6 sin _) d+: _; ; ;3 ; (3 sin _) d+: 3 sin d _; ; p p _;#; _;!; 0 ;!; ;#; :)` {;!;+;3!; `+;4!;fi`+y+; ; 9 } d ;5@; ; ; ; ; ;5$; ;!); 3 p : sin d 0 :) ; ; cos d : ;3@;p sin d _;3 ; : p p cos d 4 :_! (_) `(+) ` d=:)``{fl`_afi`+(a+3) `}da

39 ab :Ab` (_) ` d=a:ab` (_) ` d=b :Ab` (_)(_7)(_) ` da B A_4B A_9B 4A_9B A+4B 4A+9B f()f()+:)`` ` f(t) dt= `+:_@` f(t) dt :)``f() d _8 _6 _4 3 f()f(+3)=f() :)3``f() d=0 :_@ 0`f() d 4 f() f() f '() :_@ f() d=;pq;p+q y (p q) _ O _ y=f '() 39

40 04 ab f(t) ab t=»()» '() ab»(a)=a»(b)=b b :Ab``f(»())» '() d=: f(t) dt a ab f(t) F(t) b : f(t) dt=[f(t)] =F(b)_F(a) ab a a b t=»()» '() a b»(a)=a»(b)=b {F(»())}'=F'(»())» '()=f(»())» '() :Ab``f(»())» '() d=[f(»())]ba=f(»(b))_f(»(a)) :Ab``f(»())» '() d=f(b)_f(a) b :Ab``f(»())» '() d=: f(t) dt a :Ab``? k `_ ``d=k sin h`{_; ; h ; ;}(k) :Ab`` k `+ ` `d=k tan h`{_; ;<h<; ;}(k) :)``e ` d(e) e_ e e+ e_ e+ :)`` 4+ ` d ;8 ; ;6 ; ;4 ; ; ; p 40

41 f()»()f '()» '() ab :Ab``f()» '() d=[f()»()]ba_:ab``f '()»() d f()»() f()»() {f()»()}'=f '()»()+ f()» '() f()»()f '()»()+ f()» '() :Ab` {f '()»()+f()» '()} d=[ f()»()]ba :Ab``f '()»() d+:ab``f()» '() d=[ f()»()]ba :Ab``f()» '() d=[ f()»()]ba_:ab``f '()»() d :Ab``f()» '() d=[ f()»()]ba_:ab``f '()»() d f()» '()f() f() 3 :)``e d(e) ;4!; ;!; ;4#; ;4%; 4 :) ; ;` sin d ;!; ;#; ;%; 4

42 04 f() d :A``f(t) dt=f() d :X +a`f(t) dt=f(+a)_f() d d f() lim :A +a`f(t) dt=f(a) lim :A `f(t) dt=f(a) ` `0 ` à _a f() F() d :A``f(t) dt= d [F(t)]A d d :A``f(t) dt= d {F()_F(a)}=f() d d :X +a`f(t) dt= d [F(t)]X d d :X +a`f(t) dt= d {F(+a)_F()}=f(+a)_f() d f() F() +a lim :A +a`f(t) dt= lim [F(t)]A ` 0 ` 0 :X +a`f(t) dt= F(+a)_F(a) lim =F '(a)=f(a) ` 0 lim :A``f(t) dt= lim [F(t)]A ` a _a ` a _a F()_F(a) :A``f(t) dt= lim =F '(a)=f(a) ` a _a +a 5 f() :#``f(t) dt= `+a+3f(0) (a) f()=sin +4 lim ` `0 ;!;:)``f(t) dt

43 f() ab b_a b_a lim ;Kn+!`f {a+ k} =:Ab``f() d n` n n b_a b_a =a+ kd= n n b_a b_a lim ;Kn+!`f {a+ k} = ;Kn+!`f( ) D=:Ab``f() d n` lim n n n` b_a b_a lim ;Kn+!`f {a+ k} =:Ab``f() d n` n n ;Kn+!`f {a+ k} =:) b_a`f(+a) d ;Kn+!`f {a+ k} =(b_a):)``f((b_a)+a) d lim ;Kn+!`f {;nk;} ;n!;=:)``f() d n` lim ;Kn+!`f {;np;k} ;np;=:)p``f() d n` lim ;Kn+!`f {a+;np;k} ;np;=:a a+p f() d=:)p``f(+a) d=p:)``f(p+a) d n` 7 lim ;Kn+! {+: napple:};n!; n` 8 lim n` / +/ +/3 +y+/ßn n/ßn` ;3!; ;3@; ;3$; ;3%; 43

44 e `+ :)`` d(e) e `+ e ` e ` e ` e ` 3e ` ln ln ln ln ln e+ e+ e+ e+ e+ :Ab`` f(»())» '() d»()=t dt =» '()»(a)=a»(b)=b d b :Ab`` f(»())» '() d=: f(t) dt a e `+=t =e ` =0t==t=e+ e ` + :)`` d=:@ e+ t+ { } dt=:@ e+ t+ dt e `+ t t_ t(t_) t+ A B t+ (A+B)t_A = + = A+B=_A= t(t_) t t_ t(t_) t(t_) A=_B= e ` + :)`` d=:@ e+ {_ + } dt e `+ t t_ :)`` dt d e+ d=[_ln t + ln t_ ]@` :)`` :)`` d=_ln (e+)++ln e ` d=ln e+ :! /7 d + ` ln / ln /3 ln ln 3 ln 4 (ln ) ` : d(e) e`e ` 44

45 (00)()y=f() y=f() = S:)``f '(/ß ) d y y=f() S S _S (_S) _S (+S) O S :Ab`` f()» '() d=[ f()»()]ba_:ab`` f '()»() d /ß=t dt = =0t=0=t= d /ß :)``f '(/ß ) d=:)``t f '(t) dt :)``t f '(t) dt u(t)=tv'(t)=f '(t)u'(t)=v(t)=f(t) :)``t f '(t) dt=[t f(t)])_:)``f(t) dt=f()_s :)``f '(/) d=:)``t f '(t) dt={ f()_s}=(_s)`( f()=) 3 :!`` `e ` d=;pq; e `p `+q ` (epq) 4 p : (3e _ sin 3+e _ cos 3) d(e) 0 e _p _ e _p _ e _p e _p + e _p + 45

46 0p f()=:)` ( sin t_) cos t dt=a=b a_b ;6 ; ;3 ; ; ; ;3@;p ;6%;p d :A``f(t) dt=f() d f()f '(a)=0 =af '()f() =a `f '()= d :)``( sin t_)cos t dt=( sin _) cos d `f '()=0 sin =;!; cos =0 0<<p=;6 ; =;6%;p =; ; f() f '() f() 0 y ;6 ; y ; ; y ;6%;p y p _ _ 0 + f() =;6 ;=;6%;p a_b = ;6 ;_;6%;p = _;3@;p =;3@;p 5 >0f() f()=+:!``f(t) dt f(e `)(e) 6 f()=:!``e sin pt dt y=f() (f())y=»()»(3)(e) /3` ;;;;;; e 3 e+3 46

47 lim ;n!;[cos {_; ;+;n ;}+cos {_; ;+: nδ:}+cos {_; ;+: nδ:}+y+cos {_; ;+:ñδ:}] n` 0 ;!; lim ;Kn+! f {a+;np;k} ;np;=:a a+p`f() d=:)p`` f(+a) d n` lim ;n!;[cos {_; ;+;n ;}+cos {_; ;+: nδ:}+cos {_; ;+: nδ:}+y+cos {_; ;+:ñδ:}] n` = lim ;n!; ;Kn+! cos {_; ;+:applenδ:} n` =;!; lim ;Kn+! [cos {_; ;+;n ;k}] ;n ; n` _; ;+p =;!;: ; ;``cos d _; ; cos d=;!;:_; ; ; ;``cos ]) ; ; 7 lim { + + +y+ } n` n+ n+4 n+6 n+n ln /7 ln /6 ln /5 ln ln /3 8 AB =AC =/CAB=; ; ABCBC C(=P«) n PºP P y P«P n_ P n lim ;Kn+! AP ` ;n!;=;pq;p+q n` (pq) / A / P P B(=Pº) 47

48 e f()=:)`` dtaf(a)=;!;:)à ` f() d +tfl` +fl` /e _` /e +` /e _ /e + f()f()= f() f '() f(a)=0: a : a a 4a { f()} ` ` f() d=k (a>00<k<) dk ;;4;; k ` ;;;; k ` k ` k k 3 f() :)``f(t) dt=e `+a+a f(ln )(a) e 3 e 48

49 :) ;4 ;`cos ` sin d ; 6; ;8!; ; 6; ;4!; ; 6; :!è `ln d(e) e_ e e+ 3 f() :!``f(t) dt=sin p+cos p+f {;4#;} _/ p /` /` _;;;;;; p 0 ;;;;;; p / p 4 lim h` 0 h ; ;+h : sin d ; ;_h ;4 ; ; ; p p 4p 5 lim ;n!;[{+;n!;} 3 +{+;n@;} 3 +{+;n#;} 3 +y+{+;nn;} 3 ]=;pq;p+q n` (pq) 49

50 :) ;6 ; cos ` d=;pq;p+q(pq) :_@ e ` d(e) e `_ 3 _ e ` e `_ 3 e ` e ` e ` e `_ 3 + e `_ + e ` e ` 3 y=f() O(a0) F()=:)``(_t) f(t) dt F '()=0 y y=f() (a>0) a a 3a O a 4a 5a 4 n k n k y=e `` A {;nk;, e ;nk; } l A l P lim ;n!; ;Kn+! OP n` (Oe) e `` e ` ;;;; e `` ;;3;; e `` ;;4;; e `` ;;5;; 50

51 F()=(_)(_e)e `+ F'()=f():)``e ` f(e `) d (e) ;!; e_ ;!;(e_) ;!; e ;!;(e+) ;!; e+ y=f() (a0)(b0) F()=:A``f(t)f '(t) dt y y=f() (0<a<b) F(a)=0 a+b F'{ }=0 F()=03 O a b 3 0 `f()= lim ;;; ;;; n` n ``{ n_ + n_ + n_3 +y+ n_n } ;5!; ;4!; ;3!; ;!; 4 f() f '()>0 f(0)=0f(5)=5 lim ;Kn+! [ f {: napple:}+f ` {: napple:}] ;n%;(f `()f()) n` 5

52 05 y=f() ab y=f() =a=b S y y=f() S=:Ab`~ y d=:ab`~ f() d y=f() ac f() 0 cb O a c b f()æ0 S S=:Ac`~{_f()} d+:cb`~ f() d S=:Ac`~ f() d+:cb`~ f() d S=:Ab`~ f() d =»(y) cd =»(y) y y=cy=d S S=:Cd`~ dy=:cd`~»(y) dy y d e =»(y) =»(y) ce»(y) 0 ed»(y)æ0 S S=:Cè ~{_»(y)} dy+:ed``»(y) dy O c S=:Cè ~»(y) dy+:ed``»(y) dy S=:Cd`~»(y) dy y=(_)(_4) y=sin `(0 p) ;&; 4 ;(; 5 : : 5

53 y=f()y=»() ab y=f()y=»() =a=b S S=:Ab`~ f()_»() d =f(y)=»(y) cd =f(y)=»(y) y=cy=d S S=:Cd`~ f(y)_»(y) dy ab 0»() f() S y y=f() S=:Ab`~f() d_:ab`~»() d=:ab`~{ f()_»()} d ab»() f() f()»() S y=»() y=f() y=»()y k O a b 0»()+k f()+k S y y=f()+k S=:Ab`~{ f()+k} d_:ab`~{»()+k} d S=:Ab`~{ f()_»()} d O a S S k y=»()+k y=f() b ac»() f() cb `f()»() S y y=»() S=:Ac`~{ f()_»()} d+:cb`~{»()_f()} d y=»() S=:Ac`~ f()_»() d+:cb`~ f()_»() d y=f() O a c b S=:Ab`~ f()_»() d 3 y= `y=/ ;4!; ;3!; ;!; ;3@; ;4#; 4 y=sin `(0 p)y=cos `(0 p)y=p / / 4 4/ 53

54 05 ab S() V S() V=:Ab``S() d`(s() ab ) a b ab n a=º y n_ n =b D k S( k ) S( k ) D S( k )D n V n V n = S( k )D V n V= lim V n = lim S( k )D=:Ab``S() d n` n k= n` k= a = S( ) b º +Δ = «y=f() ab y=f() =a=b V y O f() y=f() a b V =:Ab``py ` d=p:ab`~{ f()} ` d S() =»(y) cd =»(y) y y=cy=d y V y V y =:Cd``p ` dy=p:cd``{»(y)} ` dy y d S(y) y c O =»(y)»(y) 5 `cm (e +)`cm ` ln 3`cm(e) ln e ``cm ` ln 3e ``cm ` ln e ``cm ` ln 3e ``cm ` ln 4e ``cm ` 6 y=cos `{0 ; ;} y p ` p ` p ` p ` ;#;æp `

55 Pt (t) v(t)pt=a t=b P (b)=(a)+:ab``v(t) dt P :Ab``v(t) dt P :Ab`` v(t) dt Pt =f(t)y=»(t)t=a t=b P s d dy s=:ab`~ æ { }`+{ }` dt=:ab`~? { f '(t)} `+{» '(t)} ` dt dt dt t P =f(t)y=»(t)t=a t=b P l d dy l=:ab`~ æ { }`+{ }` dt=:ab`~? { f '(t)} `+{» '(t)} ` dt dt dt =a =b y=f()l dy l=:ab`~ æ +{ }` d=:ab`~? +{ f '()} ` d d 7 Pt v(t) v(t)=3t_t `t=0 t=4 P 6 : 3ª: : 3º: 7 : 3 : 8 y= e +e _ `(0 )(e) ;4!;æ{e_ } ;4!;æ{e+ } ;!;æ{e_ } ;!;æ{e+ } e_ e e e e e 55

56 y=f() S S S =6S =8 :)3``f() d y y=f() S (y=f()(00)(30)(60)) O S 3 6 y=f() =a=b S S=:Ab`~ y d=:ab`~ f() d S =:)3``{_f()} d=6 S =:#6`` f() d=8 dt :)3``f() d =t ==0t=0=3t=6 d :)3``f() d=;!;:)6``f(t)dt :)3``f() d=;!;æ[:)3``f(t) dt+:#6``f(t) dt] :)3``f() d=;!;æ _:)3``{_f(t)} dt+:#6``f(t) dt :)3``f() d=;!;æ(_6+8)=6 6 y=cos _`(0 p) ; ; p ;#;æp p ;%;æp ln f()= `(>0) =a by=f() q =a=bp+q p (p q) 56

57 y=e (0)(e `)(e) ;!; e e y=f()y=»() =a=b S S=:Ab`~ f()_»() d (0)(e `) y_= e ` 0 (_0) y=(e `_)+ y=e (0)(e `) S S=:)``{(e `_)+_e } d e `_ S=[ `+_;!;æe ]) e `_ e ` S= +_ +;!; S= y e y=(e _)+ y=e O 3 y `=y=_4 : : 7 : : 8 : : 4 y=ln (e) l y=ln l (e) ;E;_ ;E; e e+;!; e+ 57

58 y=ln =e (e) p(e_) p(e_) pe p(e+) p(e+) y=f() =a=b V V =:Ab``py ` d=p:ab``{ f()} ` d V V=p:!è `y ` d y y=ln V=p:!è `(ln ) ` d O e `f()=(ln ) `» '()= `f '()=(ln )»()= =e p:!è `(ln ) ` d=p[(ln ) `]e!_p:!è `ln d p:!è `(ln ) ` d=p[(ln ) `]e!_p[ ln _]e! p:!è `(ln ) ` d=pe_p=p(e_) 5 y=/ y= ;3@;æp p ;3$;æp ;3%;æp p 6 [;4 ;; ;] sin p+ p+ p+4 p+ p

59 PQt f(t)»(t) v=f(t)v=»(t) PQ t=a a v O v=f(t) t v=»(t) _ Pt (t)v(t)pt=a t=b P (b)=(a)+:ab``v(t) dt f(t)=t_»(t)=_;!;æt+ t=a PQ :)à `f(t) dt=:)à `»(t)dt :)à `(t_) dt=:)à `{_;!;æt+} dt t ` t ` a ` a ` [ _t]a)=[_ +t]a) _a=_ +a 4 4 3a{;4A;_}=0 a=0 a=4 PQ t=4 4 7 Pt v(t) v(t)=cos t_sin t t=; ;æp 0 8 P(y)t =e _t sin ty=e _t cos t t=0 t= P(e) _;e!; /`{_;e!;} +;e!; / / {+;e!;} 59

60 y= `_ `y=_ `+ y=a(_) a (0<a<) y y=_ + y=a(_) ;4!; ;8#; ;8%; ;4#; ;8&; O y= _ y=;!;æ ``(æ0) q pp+q p (p q) 3 P t(0 t d) v(t) :)à ` v(t) dt=:ad`` v(t) dt (0<a<b<c<d) P v(t) O a b c d t :)c``v(t) dt=:cd``v(t) dt :)b``v(t) dt=:bd`` v(t) dt 60

61 y=ln yy=(e) e_ e_ e e+ e+ y=;!;`(>0)y=_;@;`(>0)=;e!;=e (e) ;(; 5 : : 6 : : 3 _ q/3? _ ` p+q p (pq) 4 y=_/ y ; ; ;3 ; ;4 ; ;5 ; ;6 ; 5 Pt v(t) v(t)=cos`;4 ;æt t=0 t=6 P ; $; ; ^;' ; *;' : º:' : :' 6

62 f()=e»()=k sin a f(a)=»(a)f '(a)=»'(a) y=f()y=»()y æ0 y y=f() y=»() (k k>0e) O a (_/ )e ;4 ; _ (_/ )e ;4 ; _ (_/ )e ;4 ; (_/ )e ;4 ; + (_/ )e ;4 ; + y= `_k`(<k<) S = S y y= _k S +S k ;4%; / ;#; O S S k /3 ;4&; 3 ` y ` + = `+y `= 5 9 3V V p y 3 _5 _ O _ 5 _3 4 f() (03)(6) n lim æ +[ f '{ 6k }]` 3 n` n n k=

63 0 y=(_) n y y=(_) n+ S n q S n = p `+q ` n= p (np q) y=(_) «O y=(_) «± ;4!; ka(k0)b(kk) C(_kk)D(_k0) ABCD y=+ky=/ß ABCD y=/ß S y=+ky=/ß S y=+k S S S S C D _k y y=+k k B S S y=/ S A O k k 3 Pt`(0 t d) v(t)=(t_a)(t_b)(t_c)`(0<a<b<c<d) abc :)b``v(t) dt>0 :)à `v(t) dt+:cd``v(t) dt<0 P P :)b`` v(t) dt>:)d`` v(t) dt 63

64 9 \ 9 { \ \ { ( \ ( 06 n r «P «P =n\n\n\y\n=n ` r n r yr n 첫번째 두번째 세번째 r 번째 n 가지 n 가지 n 가지 n 가지 «P =n\n\n\y\n=n ` r «P nær «P n<r ( ) ABC 5 ()

65 n pqyr n n! (p+q+y+r=n) p!q! y r! a3 b aaabb 3 a a a a b b b aaabb 3!! 3! 가지 a a a b b a a a b b a a a b b a a a b b a a a b b a a a b b a a a b b a a a b b a a a b b a a a b b a a a b b a a a b b aaabb! 가지 3!! 3 a b 3!! 5! a a a b b 5! 3!!=5! = 3!! 3 6 COFFEE abcc3abacb3 65

66 06 A B B A 5 4 B ` A B ` ` ` ` ` ` 5 4 A B A 9! =6 5! 4! B A B C A C C B C A B C A B C A 5 A B C C B A 6 A B C B C A 66

67 n n! n =(n_)! n r(r n) «C (r_)! ABCD A B C D 4!=4 ABCD DABC CDAB BCDA A D C B B D A C D B C A C B A D 4! 4 4! ABCD =3!=6 4 n n n! =(n_)! n n 7 5 ABCDE () 67

68 9 \ { \ ( n r «P =n\n\n\y\n=n ` r P =5 `= P _5=5 `_5=0 5+0=45 abcdabc () A={abc}B={345} f`:`a` `B f(a)+ f 68

69 n pqyr n n! `(p+q+y+r=n) p!q! y r! ! =6!! 4 4 3! =3! 6 3= UNIVERSAL UIEA 9! 9! 9! 9! 9! 5! 4! 3!! 4 9 aabbbcccca mb nm_n 69

70 A B B 6+/ A / 6+/ 6+/ 7! 4 =05! 4! 7! 4 =05 4!! 6+/ 05+05=0 0 5 C A B C B A 6 A B () B A 70

71 ABC6A'B'C' 6 A A'B B'C C' A A' C C' B' B (6_)!=5!=0 A'B'C' 3 3!=6 0 6= () 7

72 5 5 ( ) A B A B ABCDEF A B ()

73 ABCD A B () B A

74 U={3456} AB A,UB,U A;B={} A={abcd}B={3} f`:`a` `B `f(a)+f(b)+f(c)+f(d)=0 f A B P A B P B A

75 X={34}Y={345678} f`:`x` `Y f f()f() f()<f() 0 aaabbccdef A P B C Q C C C C C A P C Q A 3 4 C PQ 8B 75

76 07 n r n r «H `«H = n+r_ C n r n r«c n(n_)(n_)y(n_r+) n! «C = = `(0 r n) r! r!(n_r)! «C 0 r n«h n<r ab 4 4 a b ab a a a b a a a b b b b b aaa aa `b a `bb bbb a b ab H 4! 3! 4! 4! H = = = C 3! 3!(4_3)! H = +3_ C H H + H

77 n +y+z=n yz (yz) H n = 3+n_ C n +y+z= yz (yz) (00)(0)(0)(00)(0)(00) yzyyyzzz (yz)yz H = C n (a+b+c) n H n = 3+n_ C n (a+b+c) ` a `b `c `abacbc aabbccabacbc abc H = C 3 +y+z=6 (yz) 4 (a+b+c)fi` 5 (+y)«` 8 n 77

78 07 n (a+b)«` (a+b)«` = n Cº a n + n C a«n_ b+ n C a n_ b `+y+ n C a n_r b r +y+ n C n b n (a+b)«` = n C a n_r b r n r=0 n (a+b) n n Cº n C n C y n C y n C n n C a n_r b r nc = n C n_r (a+b)«` a n_r b r a r b n_r (a+b)«` =(a+b)(a+b)(a+b) y(a+b)yy z3`n`3c (a+b)«` a n_r b r n (a+b) (n_r) a r b a n_r b r n (a+b) b r n C r (a+b)«` n C r a n_r b r`(r=0yn) n (a+b) n (a+b) (a+b) ` (a+b) ` (a+b) ` Cº C Cº C C Cº C C C Cº C C C C ` ` ` ` 3 ` 3 ` ` 4 ` 6 ` 4 ` n C r + n C r+ = n+ C r+ 6 {_;!;}6` ` 7 {+ }7` 8 C + C + C +ªC + ºC C C C C C 78

79 n n Cº+ n C + n C + n C +y+ n C n = n n Cº_ n C + n C _ n C +y+(_) n n C n =0 n Cº+ n C + n C +y= n C + n C + n C +y= n _ n C + n C +3 n C +y+n n C n =n n _ n Cº+;!; n C +;3!; n C +y+ n C n = n+ _ n+ n+ (+) n = n Cº+ n C + n C `+y+ n C n n yy = n = n Cº+ n C + n C + n C +y+ n C n yy =_ 0= n Cº_ n C + n C _ n C +y+(_) n nc n yy ;!;æ(+) n_ = n Cº+ n C + n C +y ;!;æ(_) n_ = n C + n C + n C +y n(+) n_ = n C + n C +3 n C `+y+n n C n n_ = yy n n_ = n C + n C +3 n C +y+n n C n (+) n+ +C= n Cº +;!; n C `+;3!; n C `+y+ nc n n+ (C) yy n+ n+ =0 C=_ n+ (+) n+ _ = n Cº +;!; n C `+;3!; n C `+y+ nc n n+ yy n+ n+ n+ _ = = n Cº+;!; n C +;3!; n C +y+ nc n n+ n+ 9 Cº+ C + C + C +y+ C 0 N= ºCº+ ºC + ` ºC +y+ ` ` ºC º N

80 3 4 ( 3 4) n r «H = n+r_ C r n r r H n = r+n_ C n 3 H = +3_ C 3 = 4 C 3 = 4 C = \5 H = 3+4_ C = C = C = =5 \ 4\5= ( )

81 +y+z+w=8 yzw (yzw) n +y+z=n yz (yz) H n = +n_ C n yz (yz) H n_ = n_ C n_ (næ3) = y+z+w=7 yzw (yzw) 9\8 H = 3+7_ C =ªC =ªC = =36 \ =3 y+z+w=5 yzw (yzw) 7\6 H = 3+5_ C = C = C = = \ =5 y+z+w=3 yzw (yzw) 5\4 H = 3+3_ C = C = C = =0 \ =7 y+z+w= yzw (yzw) H = 3+_ C = C =3 ~ = y+z=8 yz (yz) (+y+z) ` ` yz

82 a { + } 0 `40(a 0) a ` n (a+b) n n C r a n_r b r (a+b) n = n Cº a n + n C a n_ b+ n C a n_ b `+y+ n C r a n_r b r +y+ n C n b n (a+b) n =;Rn+) n C r a n_r b r a { + } 0 a ºC r { } 0_r a { } r = ºC r a r_0 0_r a 0_r= r=4 ` ºC a _ 0\9\8\7 = \ = 4\3\\ a ` 0 =40 a `=;!; a ` 0_r=_4 r=7 ` ºC a `= ºC a `= 0\9\8 3\\ 0 a ` \a `=0a `=0\{;!;}`=30 5 a {+ }8` ` 4a `a ` / / {_ }5` {+ }5` ` /ß /ß 8

83 ab a= Cº+ C + C +y+ C b= C + C + C +y+ C ab= m Cº+ m C + m C + m C +y+ m C m m (+) n = n Cº+ n C + n C `+y+ n C n n yy = n Cº+ n C + n C + n C +y+ n C n = n =_ n Cº_ n C + n C _ n C +y+(_) n n C n =0 n Cº+ n C + n C +y= n C + n C + n C +y= n_ = nc + n C +3 n C +y+n n C n =n n_ ncº+ n C + n C +y= n_ n= a= Cº+ C + C +y+ C = _ = nc + n C + n C +y= n_ n=6 b= C + C + C +y+ C = 6_ = 5 ab= \ 5 = 6 m Cº+ m C + m C + m C +y+ m C m = m ab= m Cº+ m C + m C + m C +y+ m C m 6 = m m= r C r r= log ( n C + n C +3 n C +y+n n C n )=0 n

84 r H r = 7 C 5 H r +y+z+w=4 (yzw) (+) n ` 45 n 84

85 7 7() (+a)(+b)(+c) ` 8 a bc (abc) (a+) ` ` ` ;!;æ0a ;4!; ;!; 4 4 n n a n = n C r r=0 n= a«;3!; ;3@; ;3$; ;3%; 85

86 ( ) (a+b+c+d) `(e+f+») ` a e y + z =7 0yz (yz) (cos np) `_(_)«` 0 n a n = a n ºC n n= `

87 8 ;5^;æp ;8&;æ 8 abc (abc)( ) n {+ } n+ n_3 0 a n n= a«;ª0; ; ; ;ª0; ;!); ;@); 3 북 B 서 남 동 A A B 87

88 08 S S={3456} {} {}{3}{4}{5}{6} AA={35} A B A B A'B AB A B A;B AB A;B=uA B A A A AÇ`` A B6 C A={35}B={46}C={36} A'C={356}A;C={3}CÇ``={45} A;B=u A B 3 A4 B A;B A'B B BÇ`` A B8 C (BÇ`` B) AC A;BB;C A'BBÇ`` 88

89 A P(A) S n(a) n(s)a n(a)a n(s) n(a) ( A) P(A)= = n(s) ( ) n Ar«n r«ppa n r«p(a)= lim =p n` n ;!; ;!; (n) (r«) y y {;;n r«`} ; y ;!; ;!; n n r«n 3 8 p80p ;3!5&; ;3!5*; ;3!5(; ;7$; ;5#; 89

90 08 S A u 0 P(A) P(S)= P(u)=0 S A 0 n(a) n(s) n(s) n(a) 0 ` 0 P(A) n(s) S u n(s) n(u) 0 P(S)= =P(u)= = =0 n(s) n(s) n(s) S AB A B P(A'B)=P(A)+P(B)_P(A;B) AB P(A'B)=P(A)+P(B) S AB n(a'b)=n(a)+n(b)_n(a;b) n(s) n(a'b) n(a) n(b) = + _ n(s) n(s) n(s) P(A'B)=P(A)+P(B)_P(A;B) AB A;B=u P(A;B)=0 P(A'B)=P(A)+P(B) n(a;b) n(s) ;6!; ;5!; ;4!; ;3!; ;!; 6 AB P(A'B)=;5$;P(A)=3P(B)P(A) ;5!; ;5@; ;!; ;5#; ;4#; 90

91 A AÇ`` P(AÇ``)=_P(A) P(AÇ``;BÇ``)=P((A'B)Ç``)=_P(A'B) P(AÇ``'BÇ``)=P((A;B)Ç``)=_P(A;B) S A AÇ`` A;AÇ``=u AAÇ`` S A P(A'AÇ``)=P(A)+P(AÇ``) AÇ A'AÇ``=S P(A'AÇ``)= =P(A)+P(AÇ``) P(AÇ``)=_P(A) ~ ~` ~` ;!4#; ;7^; ;!4!; ;7%; ;ª4; 8 y _y > ;3!; ; 8; ;9$; ;!; ;9%; ;5$; ;6%; ;7^; ;8&; ;9*; 9

92 345 4 ; 0; ; 0; ;5!; ;4!; ; 0; n r n r «P n! «P =n(n_)(n_)y(n_r+)= (0 r n) (n_r)! S n(s)a n(a) A P(A) P(A)= n(a) n(s) = ( A) ( ) P 35 P 3 P P P P P 3 3 = =; 0; P ;6!; ;5!; ;4!; ;3!; ;!; 03 ; 8; ;9!; ;6!; ;9@; ; 8; 9

93 ;3!; ;9$; ;9%; ;3@; ;9&; AB A B P(A'B)=P(A)+P(B)_P(A;B) 6 A B 6 A'B 6(36) 4 0=40 P(A)=;9$0);=;9$; (357) 4 9=36 P(B)=;9#0^;=;5@; 6 4 4=6 P(A;B)=;9!0^;=;4 5; ~ P(A'B)=P(A)+P(B)_P(A;B)=;9$;+;5@;_;4 5;=;3@; ;4!; ; 8; ;3!6!; ;3!; ;3!6#; 4 X={3}Y={345678}f()< f()< f(3) f`:`x` `Y f()=3 f(3)=7 ;pq;p+q(pq) 93

94 34 ;ª8; ; 4; ;!8!; ;7#; ;!8#; AB A;B=u P(A;B)=0 A B P(A'B)=P(A)+P(B) C = =8 A BA'B C = C =3 P(A)=; 8; 4 3 C = =6 P(B)=; 8;=; 4; AB P(A'B)=P(A)+P(B)=; 8;+; 4;=;ª8; () ; 0; ;5!; ; 0; ;5@; ;!; 6 ababab 3 ;9@; ;4!; ; 8; ;3!6!; ;3!; 94

95 AB8AB ;!8#; ;!; ;!8%; ;7$; ;!8&; ~ ~ ~ ~ P(AÇ``)=_P(A) 88! AB XX XÇ`` AB AB AB AB77! AB 7! AB AB C =6CACB 66! AB 6 6! = 6! 7!+ 6! 6 6! P(XÇ``)= = =;!8#; 8! 8! P(X)=_P(XÇ``)=_;!8#;=;!8%; 7 A3B4 7 A ;7%; ;3@; ;5#; ;!; ;3!; ;7^; ;4#&; ;!(; ;!4#; ;@); 95

96 mni μ`` (_i)«` ;pq;p+q(i=/ _pq) 44 4 ;7#; ;3!5*; ;5#; ;3@5$; ;3@5&; 3 A B P(A)=P(B)P(A)P(B)=;9!; P(A'B) ;6!; ;3!; ;!; ;3@; ;6%; 96

97 ab `_a+b _ ;9@; ; 8; ;3!; ; 8; ;9$; () ; 0; ; 0; ; 0; ;5!; ;4!; 3 AB 73 4 AB ;5!; ;3!; ;7#; ;!; ;9%; 4 abc 4a ;4!; ;3!; ;!; ;3@; ;4#; 97

98 8 ; 4; ;7!; ; 4; ;7@; ; 4; ab a+? 4 b ;3!; ;3!6#; ; 8; ; ; ;9$; 3 abc a_b = b_c ; 8; ;4!; ;9@; ;3 6; ;6!; 4 6 ;3!; ;5@; ; 5; ; 5; ;5#; 98

99 ;5#0; ;50&0; ;@5; ;50(0; ;5 0; ;3 0; ; 8; ;9 0; ; 0; ;9!0!; 3 S={ } ;pq;p+q (pq) 99

100 09 SAB A B A B P(B A) A B P(B A)= P(A;B) P(A) (P(A)>0) P(B A)A A;B n(a;b) P(B A)= n(a) S n(s) `n(a;b)` P(B A)= n(s) P(A;B) = `n(a)` P(A) (P(A)>0) n(s) P(A)=P(A;B)+P(A;BÇ``)P(B A)= P(A;B) P(A;B)+P(A;BÇ``) AB P(A)=;!;P(B)=;3!;P(B A)+P(A B)=;4%;P(A;B) ;8!; ;6!; ; 4; ;4!; ; 4; ;7!; ;7@; ;7#; ;7$; ;7%; 00

101 P(A)>0P(B)>0AB P(A;B)=P(A)P(B A)=P(B)P(A B) A B P(B A)= P(A) P(A;B)=P(A)P(B A) (P(A)>0) B A P(A B)= P(A;B) P(A) P(A;B) P(B) P(B) P(A;B)=P(B)P(A B) (P(B)>0) P(A;B)=P(A)P(B A)=P(B)P(A B) 3 AB P(AÇ``)=;3!;P(B A)=;4!;P(A;B) (AÇ`` A) ; ; ;6!; ;4!; ;3!; ; ; `% 0`% 3 ; 5; ; 5; ; 5; ; 5; ;5!; () ;8!; ;4!; ;8#; ;!; ;8%; 0

102 09 AB P(B A)=P(B AÇ``)=P(B) P(A B)=P(A BÇ``)=P(A) AB AB P(A;B)=P(A)P(B) (P(A)>0P(B)>0) AB P(A;B)+P(A)P(B)AB AB P(B A)=P(B) P(A;B)=P(A)P(B A)=P(A)P(B) P(A;B) P(A)P(B) P(A;B)=P(A)P(B) P(B A)= = =P(B) AB P(A) P(A) AB A BÇ``AÇ`` BAÇ`` BÇ``(0<P(A)<0<P(B)<) P(A;BÇ``)=P(A)_P(A;B)=P(A)_P(A)P(B) P(A;BÇ``)=P(A){_P(B)}=P(A)P(BÇ``) P(AÇ``;BÇ``)=_P(A'B)=_{P(A)+P(B)_P(A;B)} =_P(A)_P(B)+P(A)P(B) ={_P(A)}{_P(B)} =P(AÇ``)P(BÇ``) 6 AB P(A)=;3@;P(B AÇ``)=;5@;P(A'B) (AÇ`` A) ;!; ;3@; ;4#; ;5$; ;6%; 7 ; ; ;6!; ;4!; ;3!; ; ; 0

103 A p Aq n Ar «C p r q n_r (p+q=r=0yn) 3 ;6!;\;6!;\;6%;={;6!;} ;6%; ;6!;\;6%;\;6!;={;6!;} ;6%; ;6%;\;6!;\;6!;={;6!;} ;6%; 3 C =3() ;6!;;6%; ;6!; ;6!; ;6%;={;6!;} ;6%; C (){;6!;} ;6%; C {;6!;} ;6%; 8 5 3p3fi`p 9 4 ;!; ;ª6; ;8%; ;!6!; ;4#; 03

104 ;3!; ( ) ; ; ;6!; ;4!; ;3!; ; ; A B P(B A)= P(A;B) P(A) (P(A)>0) A B P(A)=P(B) ;3!; P(A;B)=;3!;P(B) P(A;B) ;3!;P(B) P(B A)= = =;6!; P(A) P(B) ;4!; ;8#; ;!; ;8%; ;4#; ;8#; ;!; ;8%; ;4#; ;8&; 04

105 A B 345 A BB () ; 5; ;5!; ; 5; ; 5; ; 5; P(A)>0P(B)>0AB P(A;B)=P(A)P(B A)=P(B)P(A B) A X B Y A B B C P(X;Y)=P(X)P(Y X)=;5@; =;5@; ;5!;=; 5; C A B B P(XÇ``;Y)=P(XÇ``)P(Y XÇ``)=;5#; C C =;5#; ;5@;=; 5; P(Y)=P(X;Y)+P(XÇ``;Y)=; 5;+; 5;=; 5; () ;4!5^; ;4!5&; ;5@; ;4!5(; ;9$; 4 80`% 40`% 60`% ;ª; ; ; ; ; ; ; ; ; 05

106 B ;5$; AB () ;!5*; ;!5(; ;5$; ;@5!; ;@5@; P(A)>0P(B)>0 AB P(A;B)=P(A)P(B) AB A BÇ``AÇ`` BAÇ`` BÇ`` A X B Y P(X)=;5@;P(Y)=;5$; AB ZZ ZÇ`` AB ZÇ``=XÇ``;YÇ`` X Y XÇ``YÇ`` P(ZÇ``)=P(XÇ``;YÇ``)=P(XÇ``)P(YÇ``)={_;5@;}{_;5$;}=; 5; P(Z)=_P(ZÇ``)=_; 5;=;@5@; ;3!; ; 8; ;9$; ;!; ;9%; 6 A A ;3!;B B ;5!; A B ( A B) ;5@; ; 5; ; 5; ;5#; ;3@; 06

107 ;@7#; ;@4)3*; ;@4)3(; ;8&); ;@4!3!; A p A qn A r «C p r q n_r (p+q=r=0yn) ;6$;=;3@;;6@;=;3!; 5 AA AÇ``5 P(AÇ``)= C {;3@;} 5 {;3!;} 0 =; 4 3; P(A)=_P(AÇ``)=_; 4 3;=;@4!3!; ;3 ; ; 6; ;ª6; ;!6%; ;3#!; 8 4 p 3 3 p p p ;3@; ;3$; ;3%; 07

108 `:`3 70`% K 30`% K ;5!; K ;4!; ;3!; ; ; ;!; ; ; 3 34 ; 7; ;5!4!; ;9@; ;5!4#; ; 7; 3 AB 시계방향 A 시계반대방향 B A 5B ; 7; ;9@; ; 7; ;!7); ;9$; 08

109 ; 4; ;7@0(; ;7#0!; X={34} X Xf f()=f()= ;5!; ;9@; ;4!; ;7@; ;3!; 3 3 () ; 5; ; 5; ; ª5; ; 5; ; 5; ` ` `

110 ;!3!; ;!3@; ;!3#; ;!3$; ;!3%; ;3!7&; ;3!7*; ;3!7(; ;3@7); ;3@7!; 3 aaabb 3 a 3 b 4 a ;9@; ;3!; ;9$; ;9%; ;3@; 4 6p6 `p ;4!; ;!; 4 0

111 ;9!; ;6!; ;9@; ; 8; ;3!; A B AB A B B A A B () ;8!^; ;8!&; ;9@; ;8!(; ;8@); 3 53 m3 n A= ` 0 A m_n =E ;pq;p+q (Epq)

112 0 S R X`:`S` `R X X X P(X=) X y«xx p p p yp«x P(X= )=p `(i=3yn) X X X P(X= ) p p p y y «p«p(x) p p«p p O «X X P(X= )=p (i=3yn) 0 P(X= ) ;In+! P(X= )=;In+! p = P( X Δ)=;Kj+I p (ij=3yni j) XX P(<X 3) X 0 3 P(X=) a ;8#; b c ;8!; ;6!; ;4!; ;8#; ;!; 3456 XP(X=)= k+ 3 k _;7^; _;7%; _;7$; _;7#; _;7@;

113 X X X P(X= ) p p p y y «p«x E(X)= p + p + p +y+«p«=;in+! p =m V(X)=E((X_m) `)=;In+! ( _m) ` p V(X)=;In+! ` p _m `=E(X `)_{E(X)} ` XV(X) V(X)=;In+! ( _m) ` p =;In+! ( `_m +m `)p V(X)=;In+! ` p _m ;In+! p +m ` ;In+! p V(X)=;In+! ` p _m m+m ` V(X)=;In+! ` p _m ` V(X)=E(X `)_{E(X)} ` r(x)=? V(X) 3 XE(X) X 0 3 P(X=) ;6!; ;6!; a a : 6 : ;#8%; ;#8&; : 6 : ;$8!; 4 3 Xr(X)() /` ;4!; ;;;4;;; /` ;!; ;;;;;; 3

114 0 ax+b Xab(a+0) E(aX+b)=aE(X)+b V(aX+b)=a ` V(X) r(ax+b)= a r(x) X Y=aX+b X P(X= ) p p p y y «p«y a +ba +ba +bya«+b P(Y=a +b)=p(x= )=p `(i=3yn) Y Y P(Y=a +b) a +b p a +b p a +b p y y a«+b p«y E(Y)=;In+! (a +b)p =a ;In+! p +b ;In+! p =ae(x)+b E(X)=m E(Y)=am+b V(Y)=;In+! {(a +b)_e(y)} ` p V(Y)=;In+! {(a +b)_(am+b)} ` p V(Y)=a ` ;In+! ( _m) ` p =a ` V(X) r(y)=? V(Y)=? a ` V(X)= a r(x) 5 X 5 3Y=X+3 E(Y)+V(Y) X E(X)=3E(X `)=0r(5X+)

115 A pn A XX P(X=)=«C p q n_`(=0ynq=_p) X X P(X=) 0 «Cº p `q«` «C p `q«` ` «C p `q«` ` y y «C p `q«` ` y y n «C«p«`q ` X B(np) XB(np) E(X)=np V(X)=npq (p+q=) r(x)=/ƒnpq (p+q=) A pn A X h X lim P{ n` n _p <h}= A X A P(A) n 7 XB(np) E(X)=0r(X)=4n XE(X+)

116 XP( X )(a) X 0 3 P(X=) a a ` a a ` ;9@; ;3!; ;9$; ;9%; ;3@; a P( X Δ)=P(X= )+P(X= )+P(X= )+y+p(x=δ) (ij= 3 y n i j) a+a `+a+a `=3a `+a_=0 (3a_)(a+)=0 ` a=;3!; ( aæ0) X X 0 3 P(X=) ;3!; ;9!; ;3!; ;9@; P( X )=P(X=)+P(X=) P( X )=;9!;+;3!;=;9$; 3 3 XP(Xæ)() ;5#; ;3@; ;!5!; ;4#; ;5$; 33 5 XP(X `_6X+8>0) () ; 0; ;5!; ; 0; 3 3 ;5@; ;!; 6

117 X X 3 4 X : 6 :ab 30ab P(X=) a ;6!; b X X P(X= )=p `(i=3yn)x E(X)= p + p + p +y+«p«a+;6!;+b= ` a+b=;6%; yy X : 6 :E(X)= a+3 ;6!;+4 b=: 6 : a+4b=: 6 : ` a+b=;6&; yy a=;!;b=;3!; 30ab=30 ;!; ;3!;=5 3 X X 4 a b P(X=) ;3!; ;6!; ;4!; ;4!; E(X)=6V(X)=9abab XE(X) () ;4(; ;%; : 4 : 3 7

118 ax+b X Y=aX+b Y 3600aba+b(a>0) X 3 4 P(X=) ;5@; ; 0; ;5!; ; 0; X a (a+0)b E(aX+b)=aE(X)+b V(aX+b)=a ` V(X) r(ax+b)= a r(x) E(X)= ;5@;+ ; 0;+3 ;5!;+4 ; 0;= V(X)= ` ;5@;+ ` ; 0;+3 ` ;5!;+4 ` ; 0;_ `= E(Y)=E(aX+b)=aE(X)+b=a+b=36 V(Y)=V(aX+b)=a ` V(X)=a `=00 a=0b=6 ( a>0) a+b=6 yy yy 5 X E(X+3)=V(X+3)=36E(X `) XV(3X+4) ;3%; 5 : 3 : 7 9 8

119 X E(3X_) X B(np) E(X)=np V(X)=npq (p+q=) r(x)=? V(X)=/ npq (p+q=) C {;!;} {;!;}=;8#; X B{80 ;8#;} E(X)=80 ;8#;=30 E(3X_)=3E(X)_=3 30_= XE(X `) X B{80;4!;} (_0) ` P(X=) =0 9

120 X ax+b X X X _ 0 P(X=) 3_a 8 ;8!; 3+a 8 ;8!; P(0 X )=;8&; XE(X) ;4!; ;8#; ;!; ;8%; ;4#; ax+b X X 0 P(X=) ;4!; a a E(4X+0) X XB(00p) X 40X

121 XP(X=)= E(X) `_+ k (=03k 0) ;3&; ;3*; 3 : 3º: Xr(X) / /3 /6 3 X Y=X_E(Y)=9V(Y)=E(X `) X X 4 P(X=) a ;4!; b X V{;a!;X+b}(ab) ;8#; 3 : 4 : 6 : 4 : 5 XB{9;4!;}r(X+5) 4 6 7

122 X P(X=)= k /ƒ++/ƒ+ (=345y3) P(8 X 4)(k) ;6!; ;5!; ;3 0; ; 5; ;3!; AB ab `+a+b_=0 6 XE(X) 3 XX b=k M km(ab+0) X P(X=) a ;4!; ;4!; b ;4%; ;#; ;4&; 4 XB(80p)7P(X=38)=3P(X=39) E(6X_0)(0<p<)

123 X P(X=)= C C 0 (=03) ;X3+! {(4+0) P(X=)} 5 66 XE(X) ;%7%; : 8 º: : 8 : : 9º: : 8 : 3 A B PQR S R B S XE(X) ;!0(; ;@0!; A P Q : 5 : ;@0#; 4 0`% 90`%

124 XX ab X f() f() X f()æ0 (a b) :Ab``f() d= P(a X b) y=f() b P(a X b)=: f() d (a a b b) a a a b b () X () ( 상대도수 ) ( 계급의크기 ) ( 상대도수 ) ( 계급의크기 ) y=f() O O X0P(X=a)=P(X=b)=0 P(a<X<b)=P(a X<b)=P(a<X b)=p(a X b) 4 Xf()=kk ; 5; ; ; ; 5; ;6!; ;5!; _ Xf()=a `+;3!; P{0 X ;!;}(a) ; 6; ;8!; ; 6; ;4!; ; 6; 4

125 X X ab Xf() E(X)=m=:Ab`` f() d V(X)=E((X_m) `)=:Ab``(_m) ` f() d=:ab`` ` f() d_m `=E(X `)_{E(X)} ` r(x)=? V(X) V(X)=E((X_m) `)=:Ab``(_m) ` f() d V(X)=:Ab``{ ` f()_m f()+m ` f()} d V(X)=:Ab`` ` f() d_m:ab`` f() d+m `:Ab``f() d V(X)=:Ab`` ` f() d_m m+m ` V(X)=:Ab`` ` f() d_m `=E(X `)_{E(X)} ` ab X (aba+0) b E(aX+b)=: (a+b) f() d=a: f() d+b: f() d=ae(x)+b ab ab a b V(aX+b)=: [(a+b)_{ae(x)+b}] ` f() d=a `: {_E(X)} ` f() d=a ` V(X) ab a r(ax+b)= a r(x) 3 0 Xf()f()=;8#; `E(X) ;#; ;%; 3 ;&; 4 0 Xf()f()=_+ V(3X+) ; 8; ;6!; ;!; : 6 : ;%; 5

126 Xf() `f()= e _ (_m) ` r ` (mre_ << ) / p r X mrf() mr N(mr `) =m =m / p r mrr rm r r m m m mr <r X N(mr `) Z= _m _m P( X )=P{ Z } r r r m <m X_m r 0 ` f(z)= e _ z ` (_ <z< ) / p P(0 Z a)p(0 Z a)=:)à `f(z) dz N(0) _;""; f(z)= e z /ßp 0 a z 5 X N(03 `)P(X 6)=P(Xæa) a 6 XN(5000)P(30 X 60) (ZP(0 Z )=0.343P(0 Z )=0.477)

127 X B(np)n X N(npnpq) (q=_p) X_np Z= N(0) /ƒnpq a_np b_np P(a X b)=p{ Z } /ƒnpq /ƒnpq n XX B{n;6!;} n n= n y n=0 n=0 n=30 n=50 y n=0 n=0 n=30 n=50 O O npæ5nqæ5 n 7 X B{600;5@;}P(8 X 5) (ZP(0 Z )=0.343P(0 Z )=0.477) (ZP(0 Z )=0.343P(0 Z )=0.477)

128 0 Xf() ( ;4A;(+) (0 ) `f()={ ;A; ( ) 9 P{0 X ;#;}(a) ;7#; ;!; ;7$; ;ª4; ;7%; 0 f() :)``f() d= :)``f() d= :)``f() d=:)``;4a;(+) d+:!``;a; d :)``f() d=;4a;[;;;; ``+])+[;A;]!=;8#;a+;A;=;8&;a= a=;7*; P{0 X ;#;}=_P{;#; X }=_: ;#; ;7$; d=_[;7$;] ;#; =_;7@;=;7%; 0 Xf()f()=a(_) P{;!; X ;#;}(a) ;ª6; ;8%; ;!6!; ;4#; ;!6#; 03 Xf() y P( X )=P( X 3)ab b ab ;5!; ; 5; ; 5; y=f() a O 3 ; 5; ;ª5; 8

129 _ Xf() k+k (_ 0) `f()=g _k+k (0 ) V(X)(k) ;6!; ;3!; ;!; ;3@; Xf()(a b) b E(X)=m=: f() d a b V(X)=: (_m) ` f() d=: ` f() d_m `=E(X `)_{E(X)} ` ab a :_!` f() d= ;!; k= k= y=f()ye(x)=0 V(X)=:_!` ` f() d_0 `=:_0!` `(+) d+:)`` `(_+) d V(X)=[;4!; `+;3!; `]0_!+[_;4!; `+;3!; `]) y k y=f() _ O V(X)=0_{;4!;_;3!;}+{_;4!;+;3!;}_0=;6!; 3 0a Xf()f()=k X ;3!;P{;4!; X ;3!;}(ak) ;6!; ;3 6; ;9@; ;4!; ; 8; 4 0p Xf()f()=a sin E(4X_p)(a) ;4 ; ; ; p p 3p 9

130 80`g 4`g 76`g 8`g z P(0 Z z) XX N(804 `) X_80 Z= N(0) 4 76_80 8_80 P(76 X 8)=P{ Z } 4 4 P(76 X 8)=P(_ Z 0.5) P(76 X 8)=P(0 Z )+P(0 Z 0.5) P(76 X 8)= = Z N(0)P(_3 Z 3)=aP( Z 3)=b P( Z æ)ab ;!;_a+b ;!;_a+b _a+b _a+b _a+b z P(0 Z z)

131 43 X P(Xæ99) z P(0 Z z) X B(np)n X N(npnpq)`(p+q=) ;!;\;!;=;4!; X B{43;4!;} E(X)=43\;4!;=08 V(X)=43\;4!;\;4#;=8 n=43 X N(089 `) 99_08 P(Xæ99)=P{Zæ } 9 P(Xæ99)=P(Zæ_)=0.5+P(0 Z ) P(Xæ99)= = `% 508 n 93`% n z P(0 Z z) `% p00p z P(0 Z z)

132 X 0 X 4 X 00P(0 X ) y 3a a O 4 0 Xf()X;4!; :)``(a+5) f() d=0a 3 740`m 500`m 000`m 5`%000`m 5`% 000`m (ZP(0 Z 0.5)=0.) ;8#; ; 6; ;!; ;ª6; ;8%; 3

133 0 Xf()f()=;6K;(6_) P{;6%; ;;k;x; ;3%;}(k) ;4!; ;8@0!; ;4!0!; ;8@0#; ; 0; 0 a X `f() X 8 y y=f() E(X+3)(a>0) O a 3 XYN(05 `)N(00 `) ab P(a X 5)=P(0 Y b)a+b(a<0b>0) A 0`% 600 A z P(0 Z z)

134 Xf() a `f()= ( e `) E(X)(ae) ;4!;(e `_) ;4!;e ` ;!;(e `_) ;!;e ` e `_ X X y=»() f()»()f()»() y=f() E(3X +5)=E(X _5) r(x )<r(x ) P(0 X 0)<P(0 X 30) 3 XN(mr `) z P(0 Z z) V(X+)=00P(X 90)=P(Xæ36) P(58 X 73) z P(0 Z z)

135 04 Xf()f()=;4!;t 4t `+4Xt+X+=0 X ;8!; ;4!; ;8#; ;!; ;8%; z P(0 Z z) ;3 ; ;8!; ;3 ; ;5#; ;!4&; 3 00`g 0`g 90`g 0`g 4 z P(0 Z z) (0.06) ` (0.6) ` (0.9) ` (0.3) ` (0.6) ` 4.4 0fl` 5 0fi` fl` 458 z P(0 Z z)

136 X mr `r nx X X yx n X S `S X = (X +X +X +y+x n ) n S `= {(X _X ) `+(X _X ) `+(X _X ) `+y+(x n _X ) `}S=? S ` n_ n_ m r n X r ` E(X )=mv(x )= r(x )= r n /ßn X 4 X X P(X=) 3 ;4!; ;!; ;4!; ;3 ; ; 6; ;8!; ;4!; ;!; X E(X )V(X )E(X )+00V(X )

137 X m r n X r ` X N{m } n n X N{m r ` n } N(mr `) n m95 %99 % r r r r 95`%[ _ ] 99`%[ _ ] /ßn /ßn /ßn /ßn N(mr `)n X r ` X _m N{m } Z= N(0) n ``r`` /ßn P(_.96 Z.96)=0.95 X _m r r P _ =P{X _.96 m X +.96 }=0.95 ª ``r`` º /ßn /ßn /ßn r r [X _.96 X +.96 ] m 0.95 /ßn /ßn r n r s X P(38 X 4) z P(0 Z z) m95 % (ZP( Z.96)=0.95)

138 p p^ n X p^ X p^= n X p^= X p n n B(np) B(np) X E(X)=npV(X)=npq`(q=_p) p^ X E(p^)=E{ }= E(X)= np=p n n n X pq pq V(p^)=V{ }= V(X)= npq= r(p^)=? V(p^)= æ n n ` n ` n n p^ pq p n p^ N{p n p^_p Z= N(0)(q=_p) ````pq` æ ```n npæ5n(_p)æ5 n } 5 80 % A 400A p^v(p^) z P(0 Z z) p^ p^

139 n p^ n p (q^=_p^) p^q^ p^q^ 95`%[p^_.96 æ p^+.96 æ ] n n p^q^ p^q^ 99`%[p^_.58 æ p^+.58 æ ] n n pn p^ n pq p^_p N{p }`(q=_p) Z= N(0) n ````pq` æ ``ǹ pq n p^pq p^ q^ (q^=_p^) n p^_p Z= N(0) ```p^q^` æ ` ` n p^_p P(_.96 Z.96)=P _ =0.95 ª ```p^q^` º æ ` ` n p^q^ p^q^ P{p^_.96 æ p p^+.96 æ }=0.95 n n p^q^ p^q^ [p^_.96 æ p^+.96 æ ] p 95 % n n 7 00 p^ ; 0;æ p 95`% ab b_a (ZP( Z.96)=0.95) p^ 0. p 99`%(ZP( Z.58)=0.99)

140 X X P(X=k)= k+ a (k=03) 3 X V(X )(a) ; ; ;6!; ;4!; ;3!; ;!; m r n X r ` E(X )=m V(X )= r(x )= r n /ßn = = a a a a a a=0 E(X)=0\; 0;+\; 0;+\; 0;+3\; 0;= E(X `)=0 `\; 0;+ `\; 0;+ `\; 0;+3 `\; 0;=5 V(X)=E(X `)_{E(X)} `=5_4= V(X) V(X )= =;3!; 3 X 40 r 40 X X r(x )=;#;æv(x) X X 4 P(X=) X E(0X +3) _ 0 ;5!; 3a 5a

141 X A z P(0 Z z) N(mr `) n X X N{m r ` n } A XX N(580 `) 5 X 0 E(X )=58r(X )= =4 / 5 X N(584 `) 60_58 66_58 P(60 X 66)=P{ Z } 4 4 P(60 X 66)=P(0.5 Z ) P(60 X 66)=P(0 Z )_P(0 Z 0.5) P(60 X 66)=0.477_0.95= `g 5`g 00 67`g z P(0 Z z) X N(004 `) n X P(X 88)=P(X æ0)n

142 m95`% ( kgz P( Z.96)=0.95) N(mr `) n m r r 95`% [ _ ] /ßn /ßn =8.r?s=0.8n=400 m 95`% [8._ ] / 400 / _ m m 99`% a m b m (gz P(0 Z.58)=0.4950) m 00 n m 99`% ab b_a0 n (ZP( Z.58)=0.99)

143 0`% A 00 A z P(0 Z z) p^ N{p pq n } 00 A p^ P(0.3 p^ 0.6) pq E(p^)=p=0.V(p^)= = =0.0009n=00 p^ N( ) n 00 p^_p p^_0. p^_0. Z= = = N(0) ```pq /ƒ æ n 0.3_0. 0.6_0. P(0.3 p^ 0.6)=P{ Z } P(0.3 p^ 0.6)=P( Z )=P(0 Z )_P(0 Z ) P(0.3 p^ 0.6)=0.477 _0.343= `% 600 p^ P(0.58 p^ 0.64) z P(0 Z z) `% (ZP(0 Z )=0.475)

144 X X p^ p^ X m 4P(m X a)= a_ (a cm) z P(0 Z z) X m 30 9 X G(k)H(k) G(k)=P(X m+30k)h(k)=p(x æm_30k) G(0)=H(0) G(3)=H() G()+H(_)= 3 m r 6.34 m 95`%.36 m aa+r(z P(0 Z.96)= mg)

145 n X 0.5 n X N(00 `) 5 X E(X `) 3 N(884 `) n X P(87 X 89)æ0.97 n z P(0 Z z) X m 5 n m95 % n (ZP( Z.96)=0.95)

146 334 6 X V(6X _) () z P(0 Z z) N(m5 `) n X m 99`% m X 0.75 k k (ZP( Z 3)=0.99) `% ab 000(b_a)(ZP(0 Z )=0.475)

147 X X P(X=) ;8!; ;8!; ;4!; ;!; X P(3 X 6) ;6#4!; ;3!%; ;6@4(; ; 6; ;6@4&; 60`g 4`g 5 550`g z P(0 Z z) `mL 4`mL 4 384`mL 46`mL z P(0 Z z) A A p^ 0.64 A p 99`% p^_p (ZP( Z.58)=0.99)

148 y=- y m y=f() y=f() A m> y y= y=f() C B O A y= y=f() B B y C OA =BC m ' 4 4' 8 8' m=5 A y y= y p p p p p ln ln ln ln ln 48

149 f()=cos ` +cos sin +;!;æ f() M m M `_m ` /6 / 3 /3 3/ lim ` `;4 ; 6 6 `_p ` : ;4 ;``f(t) dt ;!; ; #; ; $; ; %; 49

150 4 y= `( 4)y=4y= AB 4 y y=4 A(,`4) 4 y=;:; y=;4; O B(4,`) 4 a+ y= `( 4) P(ab) ` a+ SS b b+ ;3!; ;3@; ;3$; ;3%; 4 y= `( 4)y=4y= 4 ln 4 ln 6 ln 8 ln 0 ln 50

151 a af()=+»()=+ ` a=_lim ` ` `f()»() _ { f()} ` {»()} ` 08 a ` 3 5

152 ABCDEF (O AD A(0)D(_0)) y C B D O A E F _ 0 /3 f» ` _;!;æ ` 0 /3» _ Á f _ B( f ` f) A C D E F 6 ABCDEF XE(0X+/3 ) 8/3 9/3 0/3 /3 /3 5

153 0 3 ;7@;æ ` /+C 4 _;!;+C 4 ;#;æ `++C `+C 5 _4 cos +3 sin +C tan _+cot +C e + +C + ++C ln 4 ln ;3!;pr `h ; 0;(+5)fi`+C _;!;æcos (_)+C ;3!;æsin ` +C e +C 3 ;!;æln( `+)+C _ln cos +C 4 ln _ln + +C ln _ _ln + +C 5 ln _+C e _e +C

154

155

156

157 0 3 ;7@;æ ` /+C 4 _;!;+C 4 ;#;æ `++C `+C 5 _4 cos +3 sin +C tan _+cot +C e + +C + ++C ln 4 ln :`f() d= `+4+C =:`{( `+)+(+_ `)} d =:`(3+) d =;#;æ `++C`(C) :`(+) d+:`(+_ `) d =:`( `+) d+:`(+_ `) d =;3@;æ `+;!;æ `+C +{+;!;æ `_;3!;æ `+C } (C C ) ` f()=3 `+4 f()=3+4=7 =;3@;æ `+;!;æ `+C ++ `_;3@;æ `+C =;#;æ `++C`(C=C +C ) :`(3 `_) d=f()+c 3 `_=f '() f '()=3_= 3 :` `/ d=:` ;%; d 3 :` `/ d=;7@;æ ;&; +C`(C) 3 :` `/ d=;7@;æ `/+C :` ` d=:` _ d :` d= _+ +C`(C) _+ :` d= +C :` d=_ +C 4 :`(+) d+:`(+_ `) d =:`(+) d+:`(+_ `) d ;7@;æ `/+C _ +C :`(+) ` d_:`(_) ` d =:`{(+) `_(_) `} d =:`4 d = `+C`(C) ;#;æ `++C `+C 5 :`(4 sin +3 cos ) d=_4 cos +3 sin +C `(C) :`{tan ` _ } d=:`{(sec ` _)_cosec ` } d sin ` :`{tan ` _ } d=tan _+cot +C` (C) _4 cos +3 sin +C tan _+cot +C 6 :`e + d=:`e ` e d 6 :`e + d=e `:`e d 6 :`e + d=e ` e +C`(C) 6 :`e + d=e + +C 5

158 :`( +) ` d=:`{( ) `+ +} d :`( +) ` d=:`(4 + +) d :`( +) ` d= + ++C` ln 4 ln :`( +) ` d= + ++C ln 4 ln (C) 4 + e + +C + ++C ln 4 ln ( `+) `_ ` 3 f()=:` d `++ ( `++)( `_+) 3 f()=:` d `++ 3 f()=:`( `_+) d 3 f()=;3!;æ `_;!;æ `++C`(C) f(0)= f(0)=c= f()=;3!;æ `_;!;æ ` f '()=afi`+ f()=:`(afi`+) d=;6a;æfl`++c`(c) `f(0)=3 C=3 `f()=6 ;6A;++3=6 a=6 f '()= cos ` ; ;=+cos f()=;3!;_;!;++=: 6 :? ` 4 f()=:` d_:` d _/ß _/ß /ß_ 4 f()=:` d`( >0) _/ß _(_/ß) 4 f()=:` d _/ß 4 f()=:`(_) d 4 f()=_;!;æ `+C`(C) `f()= _+C= ` C=3 f()=_;!; `+3 f(3)=_;(;+3=_;#; `f()=:`(+cos ) d ` `f()=+sin +C`(C) `f(0)= f(0)=c= f()=+sin + `f(p)=p+ `+ ` 3 f()=:` d+:` d `++ `++ `+ `+ 3 f()=:` d `++ 5 f '()=+ `f()=:`(+) d= `++C`(C) y=f() y=4_5 `++C=4_5 `_+C+5=0 D D 4 =(_) `_(C+5)=0 C=_4 f()= `+_4 6

159 `f()=4+4_4=4 6 y=f() (00)(0) `f()=a(_)`(a>0)»()=:`a(_) d»()=a{;3!;æ `_;!;æ `}+C (C) `» '()=f()»() y 0 y y» '() + 0 _ 0 +» ()»() =0 = _»(0)=C=»()=_;6!;æa+=_ a=4»()=8 `_ `+»()=64_48+=8 8 7 ( f()) _3 `+ `f '()=_3 `+ k f()=:`{+ } d f()= `+k ln +C (C) y=f()a() `f()=+0+c= C= f()= `+k ln + y=f() B(ee `+) `f(e)=e `+k+=e `+ k= e 9 3 f()=:` d_:` d e +e + e +e + e 3 _ f()=:` d e +e + f()=:` f()=:`(e _) d d f()=e _+C`(C) f() (0) `f(0)=_0+c= C= f()=e _+ `f()=e `_+ =e `_ (e _)(e +e +) e +e + f()=:`(_3 `+) d f()=_ `++C`(C) y=f()(00) `f(0)=c=0 f()=_ `+ k=f()=_+= 8 +0 ( f()) k + ` f '()=+ k sin ` (_cos ` ) 0 f()=:` d=:` d +cos +cos (_cos )(+cos ) 0 f()=:` d +cos 0 f()=:`(_ cos ) d 0 f()=_ sin +C`(C) `f(0)=0_0+c=3 C=3 f()=_ sin +3 `f {; ;}= ; ;_ +3=p+ 7

160 5 3 f '()=3(_4) f '()=0 =0 =4 f() y 0 y 4 y f '() + 0 _ 0 + f () f() =0=4 `f()=:`3(_4) d (`_+C (<_)» 3 f()=:`f '() d={`;3!;æ `+C (_<<)» 9`_+C (>) (C C C ) f() lim ` 0 lim ` _0 f()= lim ` _+0 +C =_;3!;+C C =C _;3$; f() f()= lim f() ` +0 ;3!;+C =_+C C =C +;3$; `f()=:`(3 `_) d `f()= `_6 `+C`(C) f(0)=5 C=5 f()= `_6 `+5 f() `f(4)=64_6 6+5=_7 f()=:`sin(3p+) d f()=:`(_sin ) d f()=cos +C`(C) f '()=_sin `f(p)=f ' {;6 ;} _+C=_;!; C=;!; f()=cos +;!; y=f() lim f '()= lim (_)=_<0, ` 0 lim ` _+0 f '()= `=>0 f() =_ f() =_`() y=f() y f()=f(_)`() f(0)=0 C =0 f()= lim ` lim _ ` 0 ` _+0 f()=;3!;>0`() y O y=f() f(0)=+;!;=;#; p+q=+3=

161 f '()=4_6 f()=:`(4_6) d= `_6+C`(C) `f()=0 _6+C=0 C=4 f()= `_6+4 f()=0a b a+b=3ab= a `+b `=(a+b) `_ab=3 `_ =5 d [:`(+) ` d_:`(_) ` d] d d = :`{(+) `_(_) `} d d d = [:`(6 `+) d] d =6 `+ a=6b=0c= a+b+c=8 f()=:`( _) d f()= f(0)= ln ln ln 4 +C= _+C`(C) ln 4 C= _ =_ ln ln ln f()= ln ln 3 `f()= = _ ln ln ln `f()=:`(_tan ) cos d f()=:`(_)(+)( `+) d `f()=:`(cos _tan cos ) d `f()=:`(cos _sin ) d `f()=sin +cos +C`(C) f(0)= f(0)=+c= C=0 f()=sin +cos f {;4 ;}=;;;;;; /``+;;;;;; /``=/ 4 4 _ f '()= + 4 _ f()=:` d + ( _)( +) f()=:` d + f()=:`( `_) d f()=;5!;æfi`_+c`(c) f(0)= C= f()=;5!;æfi`_+ `f '()= `_ lim ` = lim ` f()_f() `_ { f()_f()}+{ f()_f()} (_)(+) (_)f() f()_f() = lim + ` lim (_)(+) ` (_)(+) `f() `f()_f() ` = lim + lim [ \ ] ` + ` _ + =;!;`f()+;!;`f '() 9

162 =;!; ;5^;+;!; 0 =;5#;+0 =;5#; f '(p)=k= f()=:`{sin +cos }` d f()=:`{+ sin cos } d f()=:`(+sin ) d`( sin h cos h=sin h) F'()=f() { f()}+={f()} f() f()=a `+b+c`(a+0) yy F()=:`f() d=;3a;æ `+;B;æ `+c+c (C)yy ;3A;æ `+;B;æ `+c+c= `+a `+b+c ;3A;=;B;=ac=bC=c f()=(_cos )+C`(C) y=f() (p f(p)) y_{(p+)+c}=(_p) y=_p+{(p+)+c}=++c y=+3 +C=3 C= f()=(_cos )+ `f(0)= (_)+=_ a=6b=c=c= f()=6 `++ `f()=6++= lim f()=f()= lim ` f '()=5 f()_f() ` = lim [ \ ] ` _ + =;!;`f '()=;%; `f '()= `+a+ = `f '()=+a+=5 a= f()_ `_ ` 4 f '()=k{sin +cos }` y=f() (p, f(p)) `f(0)=0 f '(0)= `f '(0)=+k k=`( f '(0)=) f()=:`f '() d f()=:`( ++) d f()= f(0)=0 `f(0)= ln ln C=_ ln +;!;æ `++C`(C) +C=0 0

163 f()= +;!;æ `+_ ln ln 4 3 `f()= ++_ = +4 ln ln ln 3 3 k+f()=+{ +4}= +5 ln ln f '()=6(_k) `f()=:`(6 `_6k) d= `_3k `+C` (C) f '()=0 =0 =k f()80 k<0f() =k=08 0 `f(k)=k `_3k `+C=8f(0)=C=0 k=_c=0 f()= `+6 ` `f()=8 3 >0 `f()=e +C `(C ) f(ln )=4 +C =4 C = <0 `f()=3 `+k+c `(C ) f(_)=_6 3_k+C =_6 C =k_9yy f() lim f()= lim (e +C )=+C =3 ` +0 lim f()= lim (3 `+k+c )=C ` _0 ` +0 ` _0 C =3 yy k_9=3 k= 4 y=0 `f()=f()+f(0)_ f(0)= f '()= f '()= f '()= f '()= lim h` 0 lim h` 0 lim h` 0 lim h` 0 yy f '()=f '(0)_ f()=_ `+f '(0)+C`(C) `f '(0) `f '(0) f()=_[_ ]`+[ ]`+C `f() =0 f(0)=c=`() y=f() = `f '(0) = f '(0)=4 f()=_ `+4+ f(/ )=_+4/ +=_+4/ a=_, b=4 a+b=_+4=3 `f(+h)_f() h `f()+f(h)_h f() h `f(h) h `f(h)_f(0) _`() h 3

164 0 ; 0;(+5)fi`+C _;!;æcos (_)+C ;3!;æsin ` +C e +C 3 ;!;æln( `+)+C _ln cos +C 4 ln _ln + +C ln _ _ln + +C 5 ln _+C e _e +C 6 +5=t dt = d :`(+5) ` d=:`;!;æt ` dt :`(+5) ` d=; 0;ætfi`+C (C) :`(+5) ` d=; 0;(+5)fi`+C dt _=t = d :`sin (_) d=:`;!;æsin t dt :`sin (_) d=_;!;æcos t+c (C) :`sin (_) d=_;!;æcos (_)+C ; 0;(+5)fi`+C _;!;æcos (_)+C sin =t dt =cos d :`sin ` cos d=:`t ` dt :`sin ` cos d=;3!;æt `+C (C) :`sin ` cos d=;3!;æsin ` +C dt `=t = d :`4e d=:`e dt :`4e d=e +C (C) :`4e d=e +C 3 ( `+)'= `+>0 :` d=;!;æ:` d `+ `+ :` d=;!;æln `+ +C`(C) :` d=;!;æln( `+)+C ;3!;æsin ` +C e +C sin tan = (cos )'=_sin cos sin :`tan d=:` d cos sin :`tan d=_:`{_ } d cos :`tan d=_ln cos +C`(C) ;!;æln( `+)+C _ln cos +C 4 = _ (+) + :` d=:`{ _ } d (+) + :` d=ln _ln + +C` (C) A B = = + `_ (_)(+) _ + +3 (A+B)+(A_B) = (_)(+) (_)(+) A+B=A_B=3 A=B=_ +3 = _ ` + +3 :` d=:`{ _ } d ` + :` d=:` d_:` d _ +

165 :` d= ln _ _ln + +C` (C) ln _ln + +C```````` ln _ _ln + +C ln = ln f()=ln» '()= :`ln d=:` ln d :`ln d= ln _:` d :`ln d= ln _+C`(C) :`f()d=f()_:`f '() d f()=» '()=e :`e d=e _:` e d :`e d=e _e +C`(C) ln _+C e _e +C +;6 ;=t dt = d f()=:`6 cos {+;6 ;} d f()=:`3 cos t dt f()=3 sin t+c`(c) f()=3 sin {+;6 ;}+C f {;6 ;}=5 3+C=5 C= f()=3 sin {+;6 ;}+ `f(0)=3 sin ;6 ;+=;&; p=q=7 p+q=+7=9 9 6 f()=» '()=sin h()=:` sin d h()= {_;!;æcos }_:` {_;!;cosæ} d h()=_;!;' cos +;!;:`cos d h()=_;!;` cos +;4!;'sin +C`(C) h{; ;}=; ;æ ;4 ;+C=; ; C=;4 ; h()=_;!;æ cos +;4!;æsin +;4 ; 4h{;4 ;}=4{0+;4!;+;4 ;}=+p dt `+=t =4 d `f()=:`4? `+ d `f()=:`/t`dt `f()=;3@;æt/t`+c`(c) `f()=;3@;æ( `+)? `++C `f(0)=;3%;æ ;3@;æ+C=;3%; C= f()=;3@;æ( `+)? `++ `f()=;3@;æ 9 3+=9 9 3

166 3 ( `+3+)'=+3 +3 `f()=:` d `+3+ `f()=ln `+3+ +C`(C) f()_f(0)=(ln 6+C)_(ln +C) =ln 6_ln =ln 3 `+3+=t dt =+3 d +3 f()=:` d `+3+ f()=:` dt f()=ln t +C`(C) f()=ln `+3+ +C t +3 = + ` f()=:` d `+3+ f()=:`{ + } d + + f()=ln + +ln + +C`(C) f()=ln (+)(+) +C f()=ln `+3+ +C dt ln =t = d `f()=:` d ln `f()=:` dt t `f()=ln t +C`(C) `f()=ln ln +C A B 5 = = + `+ (+) + +4 (A+B)+A = (+) (+) A+B=A=4 A=B=_ +4 = _ (+) + +4 :` d `+ =:`{ _ } d + =:` d_:` d + = ln _ln + +C`(C) 4 (ln )'= `f()=:` d ln `f()=:` d ln `f()=ln ln +C`(C) `f(e)=ln ln e +C =C= f()=ln ln + `f(e `)=ln ln e ` + =+ln `_5+8 6 =+ `_6+8 (_)(_4) A B =+ + 4 (A+B)_(A+B) = (_)(_4) (_)(_4) A+B=A+B=0 A=_B= =_ + (_)(_4) 4 4

167 f()=:`{_ + } d 4 f()=:` d_:` d+:` d 4 f()=_ln _ + ln _4 +C` (C) `f(0)=_ln + ln 4+C =3 ln +C=3 ln C=0 f()=_ln _ + ln _4 f(3)=3_0+0=3 3 `f()=;3!;(+sin ) `+C `f(0)= ;3!;+C= C=;3@; f()=;3!;(+sin ) `+;3@; 3f {; ;}=3{;3*;+;3@;}=0 9 f()=:`(_) cos d 0 dt 7 cos =t =_sin d sin (_)=_sin `f()=:`sin ` (_) d `f()=:`(_sin ` ) d `f()=:`{_(_cos ` ) sin } d 9 f()=(_) sin _:`sin d 9 f()=(_) sin +cos +C`(C) `f(0)=+c= C= f()=(_) sin +cos + `f()=+cos `f()=:`(_t `) dt `f()=t_;3!;æt `+C`(C) `f()=cos _;3!;æcos ` +C `f(0)=;3$;æ _;3!;+C=;3$; C=;3@; f()=cos _;3!;æcos ` +;3@;æ `f(p)=_+;3!;+;3@;=0 0 :`e sin d=e sin _:`e cos d 0 :`e sin d=e sin _{e cos +:`e sin d} 0 :`e sin d=e sin _e cos _:`e sin d :`e sin d=e (sin _cos ) :`e sin d=;!;æe (sin _cos )+C `(C) dt 8 +sin =t =cos d `f()=:`(+sin ) ` cos d `f()=:`t ` dt `f()=;3!;æt `+C`(C) 3 5

168 dt ln =t = d `f()=:` `f()=:`/t dt d `f()=;3@;æt ;#; +C`(C) `f()=;3@;æ(ln ) ;#; +C f(e)= ;3@;+C= C=;3!;?çln f()=;3@;æ(ln ) ;#; +;3!; `f(e `)=;3@; 4 ;#; +;3!;=: 3 :+;3!;=: 3 : `f()=_k cos +C `(C ) f(_p)= k+c = C =_k ( ;3!;æcos ` _cos +;3!; (>0) `f()= { 9 _k cos +_k (<0) f() =0 `f(0)= lim f()= lim f()yy lim ` +0 lim ` _0 ` +0 f()= lim {;3!;æcos ` _cos +;3!;} ` +0 f()=;3!;_+;3!;=_;3!; f()= lim (_k cos +_k) ` _0 f()=_k+ ` _0 >0 sin ` =sin ` sin =(_cos ` ) sin cos =t dt =_sin d _;3!;=_k+ k=;3@; f()=:`sin ` d f()=:`(_cos ` ) sin d f()=:`(t `_) dt f()=;3!;æt `_t+c `(C ) f()=;3!;æcos ` _cos +C f(p)= ;3@;+C = C =;3!; <0 cos {; ;_}=sin `f()=:`k cos {; ;_} d `f()=:`k sin d 3 f() lim h` 0 f(+h)_f(_h) h f(+h)_f() = lim + h` 0 h =f '()+f '() = f '() f '()=(+)e f '()=(+)e f()=:`(+)e d f()=(+)e _:`e d f()=(+)e _e +C`(C) f()=e +C f() f() lim h` 0 lim f()=f()=ee+c=e ` `f(+(_h))_f() (_h) 6

169 C=e f()=e +e `f(0)=e =;3 ; =;3@;p =;6 ; =;3 ; ;6 ;+;3 ;=; ; 3 4 `_4+3=t dt =_4=(_) d f()=:`(_)( `_4+3) ` d f()=:`t ` dt f()=;3!;æt `+C`(C) f()=;3!;æ( `_4+3) `+C f(0)=9+c=0 C= f()=;3!;æ( `_4+3) `+ 3 (3 `+)'=6 3 `f()=:` d 3 `+ 6 `f()=;!;:` d 3 `+ `f()=;!;æln (3 `+)+C`(C) f(0)=0+c=3 C=3 f()=;!;æln (3 `+)+3 f(4)=;!;æln 49+3=ln 7+3 `f()=0+= 4 f()=:`4 ` ln d dt =t = d f()=:`cos d f()=:`;!; cos t dt f()=;!; sin t+c`(c) f()=;!; sin +C f {; ;}=0+C=0 C=0 f()=;!; sin 0<<p f()=;;;4;;; /3` f()= ` ln _:` ` d f()= ` ln _;4!;æ `+C`(C) f()=_;4!;+c=0 C=;4!; f()= ` ln _;4!; `+;4!;æ `f(e)=;4#;æe `+;4!; ab a=;4#;b=;4!;æ a+b=;4#;+;4!;= 7

170 dt 3 ;!;æ=t =;!;æ d `f()=:`e ;!; d dt _sin =t =_cos d f()=:`(_t `) dt `f()=:`e t dt `f()=e t +C`(C) `f()=e ;!; +C f()=_;3!;æt `+C`(C) f()=_;3!;(_sin ) `+C f {; ;}=0+C=0 C=0 f()=_;3!;æ(_sin ) ` f(h)=_;3!;æ sin h=0 `f()=e+c=e C=0 f()=e ;!; = n= f(n) n= e n ;e!; =;!; _;e!; = (e_) h=0`( _p<h<p) dt `+3=t = d f()=:` f()=:`? `+3 /t d ;!;ædt 4 `f()=f()+ f '()_e _;!; `e f '()={+;!;æ}e `( >0) f()={+;!;æ}e _:`;!;æe d f()=;!; t ;!; +C`(C) f()=/t +C f()=? `+3+C y=? `+3+C (_) _=+C C=_3 y=? `+3_3 y 0 0=? `+3_3? `+3=3 `=6 = /6 (_/6 0)(/6 0) AB `= /6 _(_/6 ) ` AB `=(/6 ) `=4 4 f()={+;!;æ}e _;!;æe +C`(C) f()=;!;æ(+)e +C `yy F()=;!;æe F()=f()_;!;æe `f()=e `f()=;!;æ(+)e+c=e C=0 f()=;!;æ(+)e `f(3)=e ` 8

171 3 `f() + f '()={+;!;} ln `f()+ f '()=(+) ln d `f()+ f '()= { f()} d f()=:`(+) ln d f()=( `+) ln _:`( `+);!; d f()=( `+) ln _:`(+) d f()=( `+) ln _{;!; `++C}`(C) f()=( `+) ln _;!; ` C = f()=_;#;_c f()=_;#; (_) f '()=f() yy `f()=a n +a n_ +a n_ +y+a«+a«(na a a ya +0) na =a ` n=`( a +0) f()f()=a+b`(a+0) a=3b=_6 f()=3_6»()=:`(3_6)e d»()=(3_6)e _:`3e d»()=(3_6)e _3e +C`(C)»(0)= _6_3+C= C=»()=(3_6)e _3e +»(3)= C=0 f()=( `+) ln _;!; `_ =e e f(e)= f(e)=;e; 0 lim{ f()+3}=0 ` lim ` e ` f()=_3 `f() f() = f()= lim f() ` f()=_3 yy `f()+3 `f()_f() lim = =3 ` lim _ ` _ f '()=3 yy `f()+(_) f '()= f() 3 = f '()_ f()+3=0 f()= f '()=`() f '()+ f "()_ f '()=0 f "()=f '() yy = f "()=f '()=`() `f "() = `f '() `f "() :` d=:` d `f '() ln f '() =ln +C `(C ) yy f '()= = ln f '() =ln +C 0=0+C C =0 f '() = 9

172 f '()f '()>0 f '()=yy `f()=;!;æ `+C `(C ) 03 f()= ;!;+C = ;3!;pr `h C =;#; f()=;!;æ `+;#; n;nh; ;nh; ;nr; f(/3 )=;#;+;#;=3`() r r 3r y nr n n n n nv«h r V«=;Kn+! [p { kr } h ] n n V«= pr `h n ` ;Kn+! k ` V«= pr `h n ` V«=pr `h n(n+)(n+) 6 (n+)(n+) 6n ` (n+)(n+) lim V«= lim [pr `h ] n` n` 6n ` V«=;3!;pr `h ;3!;pr `h =+: napple:d=;n@; º=«=+: n :=3 lim ;Kn+! {+: napple:};n@;=:!3`` d n` a=3 3 f()=e ` +f() 03 D=;n#; =kd=: napple: 0

173 :)3``(e ` +) d= lim ;Kn+! (e +) D :)3``(e ` +) d= lim ;Kn+! (e : napple: +) ;n#; :)3``(e ` +) d= lim ;Kn+! (e ;^nk; +) ; n; a=6 n` n` n` 7 0 y `_ =_( `_) `_ = ` O :)`` `_ d _ =:)` {_( `_)} d+:!``( `_) d y= _ 4 :)``(6 `+3/ß) d=[ `+ ;#; ]) =_[;;3;; ``_])+[;;3;; ``_]! :)``(6 `+3/ß) d=4_0 :)``(6 `+3/ß) d=4 =_{;3!;_}+[{;3*;_}_{;3!;_}] = :)``sin d+:! ; ;`sin d=:) ; ;`sin d 5 : ; ; 0`cos d=_:) ; ;`cos d :)``sin d+:! ; ;`sin d=[_cos ]) ; ; 4:) ; ;`cos d+: ; ; 0`cos d :)``sin d+:! ; ;`sin d=0_(_) =4:) ; ;`cos d_:) ; ;`cos d = =3:) ; ;`cos d ; ; =3[sin ]) =3(_0) =3 6 0:!`` ``d=:!``0 ``d 0:!`` ``d=[fi`]! 0:!`` ``d=64_ 0:!`` ``d=6 :)``(+) ` d_:)``(_) ` d n n 0;n!;;n@;y;nN; n n S«S«=;n!; 3{;n!;} +;n!; 3{;n@;} +y+;n!; 3{;nN;} =:)``{(+) `_(_) `} d =:)``8 d =[4 `]) =4_0 S«=;;; ;;; n `` ;Kn+! k ` S«=;;; ;;; n `` S S= lim S«= n` a=3,s= n(n+)(n+) 6 =4 a+s=3+=

174 (/_a) ` _a/ß+a ` :` d=:` d :` d=:`{_a _;!; + a ` } d :` d=_4a/ß +a ` ln +C (, C) (/_a) ` :!4`` d=[_4a/ß +a ` ln ]4! :!4`` d=(4_8a+a ` ln )_(_4a) :!4`` d=(3_4a)+a ` ln :!4`` d=_+b ln 3_4a=_a `=b a=b= a+b=3 =[_e ` +]0_!+[e ` _]4) =;e;!;+e `_5 a=b=c=_5 a+b+c=++(_5)=_3 5 0 f()= 3f()= :)3``(+)f() d =:)``(+) d+:!3``(+) d =:)``( `+4) d+:!3``(+4) d =[;3@; `+ `])+[ `+4]3! ={;3@;+}+{(9+)_(+4)} 3 :_ba`d=[]b_a=b_(_a)=b+a =: 3 : a+b=: : yy :)``(3ay_)(y_b)dy=:)``{3ay `_(3ab+)y+b}dy :)``(3ay_)(y_b)dy=[ay `_ :)``(3ay_)(y_b)dy=a+b_ 3ab+ a+b_ =0 yy ab=4 3ab+ 3ab+ y `+by]) ; ; 6 : sin d=0: sin d=0 _; ; _; ; : ; ;` (sin +cos +) ` d _; ; =: _; ; =: _; ; =: _; ; ; ; ; ;` {sin `+cos ` + +(sin cos +cos +sin )}d ; ;` (+ sin cos + cos + sin ) d ; ;` d+:_; ; ; ;` sin d+:_; ; ; ;` cos d +: _; ; ; ;` sin d ``e ` _ (æ0) 4 e ` _ =g _e ` + (<0) y y= e _ =4:) ; ; d+0+4:) ; ; cos d+0 ; ; ; ; =4[]) +4[sin ]) :_4! e ` _ d =:_0!`(_e ` +) d+:)4``(e ` _) d _ O 4 =4{; ;_0}+4(_0) =p+4

175 7»()= ` f()»(_)=(_) ` f(_)»(_)=_ ` f()`()»(_)=_»() :_!`»() d=:_!` ` f() d=0 :)3`` f '() d={ f()_f(0)}+{_f(3)+f()} :)3`` f '() d={_(_3)}+(3+) :)3`` f '() d=4+4 :)3`` f '() d=8 :_!`f() d=0 :_!`(+) ` f() d =:_!`( `++) f() d 3 (0,)f "()=e ` `f '()=:`f "() d=:`e ` d=e ` +C ` =:_!` ` f() d+:_!` f() d+:_!`f() d = =6 6 f '(0)= +C = ` C =0 f '()=e ` (C ) 3 f()=6 `+a :)``f() d=:)` (6 `+a) d :)``f() d=[ `+a `]) :)``f() d=+a f()=6+a :)` f() d=f() +a=6+a a=_4 `f()=:`f '() d=:`e ` d=e ` +C ` (C ) f(0)= +C = ` C =0 f()=e ``(0<<) f()= f '() f()=ef '()=e f()=e ``(0 ) <<f '() f '() e f '()`( f '()=e) :!` e dt :!` f '(t) dt [et]! [ f(t)]! e_e f()_f() e_e f()_e`( f()=e) e f()`( ) 0 f '()æ0 3f '() 0 :)3`` f '() d=:)``f '() d+:!3``{_f '()} d :!``e d :!``f() d [;!;e `]! :!``f() d :)3`` f '() d=[ f()])+[_f()]3! ;#;e :!``f() d 3

176 :)``f() d=:)``f() d+:!``f() d :)``f() dæ:)``e ` d+;#;e :)``f() d=[e ` ])+;#;e :)` ( + _? 4 ` ) d= :)` ( + _? 4 ` ) d= ln ln _ ln :)``f() d=(e_)+;#;e :)``f() d=;%;e_ ;%;e_ ` 4 :_0!` d+:)-` `` dy _ y_ ` =:_0!` d_:_0!` d =:_0!` =:_0!` ` d (_)( `++) _ d =:_0!`( `++) d f()= `f() 03 =[;3!; `+;!; `+]0_! =;3!;_;!;+ =;6%; D=;n#; =0+kD=: napple: :)3`` ``d= lim ;Kn+! `D :)3`` ``d= lim ;Kn+! {: napple:} 4 ;n#; a=3 n` n` 5 :_5@`(4 `+3 `++sin ) d +:%``(4 `+3 `++sin ) d =:_@`(4 `+3 `++sin ) d :)``(6_3 `) d=[3 `_ `]) =:_@`3 ` d :)``(6_3 `) d=_8 :)``(6_3 `) d=4 =:)``3 ` d =[ `])=6 6 3 :)` ( + _? 4 ` ) d=:)` ( ` _? ( ` ) ` ) d :)` ( + _? 4 ` ) d=:)` ( ` _ ` ) d :)` ( + _? 4 ` ) d=:)`` ` d :)` ( + ` _? 4 ` ) d=[ ]) ln 3 4 4

177 p : (6 sin _) d+: _; ; p ;3 ; _; ;`(3 sin _) d+: 3 sin d p : _;3 ; ;3@;p sin d=:_;3 ; 0`` (_sin ) d+:) ;3@;p sin d =: _; ; p p (3 sin _) d (6 sin _) d_:_; ; : _;3 ; ;3@;p sin d=[cos ]0 _;3 ; +[_cos ]) ;3@;p ;3 ; +: 3 sin d p : _;3 ; ;3@;p sin d={_;!;}+[_{_;!;}_(_)] =: _; ; {(6 sin _)_(3 sin _)}d+: p : _;3 ; p ;3 ; 3 sin d ;3@;p sin d=;!;+;#; p =: _; ; ;3 ; =: _; ; 3 sin d+: p 3 sin d ;3 ; 3 sin d : _;3 ; ;3@;p sin d= p ;#;p cos 0 y y= cos =[_3 cos ] ;3 ; =_;#; _; ; ;#;p p cos æ0 : cos d=: pp p;#;p p cos d (_cos ) d+:;#;p O p p ;#;p :)``{;!;+;3!; `+;4!;fi`+y+; ; 9 } d =[ `+ `+ fl`+y+ 0 ]) = + + +y =;!;{ + + +y+ } =;!;[{_;!;}+{;!;_;3!;}+{;3!;_;4!;}+y+{; 0;_; ;}] =;!;{_; ;} ;#;p p : cos d=[_sin ] pp p +[sin ] ;#;p p : cos d=+= p p : sin d 0 4 +=t=t_d=dt :_! (_) `(+) ` d=:)``(t_3) ` t ` dt :_! (_) `(+) ` d=:)``(_3) ` ` d :_! (_) `(+) ` d=:)``( `_6+9) ` d =; ; p p 3 : sin d=[_cos ])=_(_)= 0 :) ; ; cos d=:) ; ; cos d :) ; ; ; ; cos d=[sin ]) :) ; ; cos d=(_0) :) ; ; cos d= _;3 ; 0sin 00 ;3@;p sin æ0 :_! (_) `(+) ` d=:)``(fl`_6fi`+9 `) d :)` (fl`_6fi`+9 `) d_:)` {fl`_afi`+(a+3) `}d =:)` {(a_6)fi`_(a_6) `}d =(a_6):)` (fi`_ `) d =;^5$;(a_6)=0 a_6=0 a=6 5

178 f() d=:!3``f() d :Ab` (_)(_7)(_) ` d :(`0``f() d=:)``f() d :_0@`f() d+:(`0``f() d=:!3``f() d+:)``f() d =:Ab` {(_)+3}{(_)_3}(_) ` d :_0@`f() d+:(`0``f() d=:)3``f() d yy =:Ab` {4(_) `_9}(_) ` d =:Ab` {4(_) `_9(_) `}d =4:Ab` (_) ` d_9:ab` (_) ` d =4A_9B f()+:)`` ` f(t) dt= `+:_@` f(t) dt f()+ `:)``f(t) dt= `+:_@` f(t) dt f()={_:)``f(t) dt} `+:_@` f(t) dt :)``f() d=k yy :_@ 0``f() d=:_0@`f() d+:)9``f() d+:(`0``f() d :_@ 0``f() d={:_0@`f() d+:(`0``f() d}+:)9``f() d :_@ 0``f() d=:)3``f() d+3:)3``f() d :_@ 0``f() d=4:)3``f() d :_@ 0``f() d=4 0=40 f()p `f(+p)=f() 40 y=f() y y=f() :_@` f() d=:)``f() d=k`(f(_)=f()) f()=(_k) `+k yy :)``{(_k) `+k}d=k _k [ `+k])=k 3 ;3*;(_k)+4k=k ab a+pb+p, :Ab``f() d=: b+p`f() d a+p O a b b+p a+p k=_8 :A a+p f() d=:b b+p f() d 3 f(+3)=f()f() :)3``f() d=:#6``f() d 4 f '() ``_ (<_) `f '()=g`_ (>_) :)3``f() d=:^9``f() d yy `f()=:`f '() d 6

179 _+C (<_) `f()=g (C C )`` _ `+C (>_) f() `f(0)=c =0 f() _ (_)+C =_(_) ` C =_3 3 ( _) f()=g _ ` (æ_) y=f() :_@` f() d=:_@` _;#;`( 3) d+:)`` ` d -;#; :_@` f() d=[_ `_3]_@ +[;3!; `]) :_@` f() d=[{_;4(;+;(;}_(_4+6)]+;3@; :_@` f() d=;!;+;3@;=;6&; y O _ `=t dt ==0t=0= d t= :)``e ` d=:)``;!;e t dt=;!;[e t ])=;!;(e_) = tan h {_; ;<h<; ;} d = sec ` h dh =0h=0=h=;4 ; :)`` d=:) ;4 ; sec ` h dh 4+ ` 4+4 tan ` h :)`` d=:) ;4 ; sec ` h dh 4(+tan ` h) :)`` d=:) ;4 ; ;!; dh p=6, q=7 p+q=3 3 :)`` ;4 ; d=[;π;]) =;8 ; 3 f()=»'()=e ` f '()=»()=e ` :)``e ``d=[e `])_:)``e ` d :)``e ``d=[e `])_[e `])= 4 f()=»'()=sin f '()=»()=_cos :) ; ; ; ; ; ; sin d=[_ cos ]) +:) cos d :) ; ; ; ; ; ; sin d=[_ cos ]) +[sin ]) = 7

180 5 d :#``f(t)dt= d ( `+a+3) d d `f()=+a =3 0=9+3a+3 a=_4 f()=_4 f(0)= 0_4= `=t dt = d =t==/7t=8 :! /7 d=;!;:@8``;;t;!; dt + ` 6 f() F() lim ` 0 ;!;:)``f(t) dt= lim ;!;[F(t)]) ` 0 ;!;:)``f(t) dt= lim ` 0 ;!;:)``f(t) dt=f'(0) ;!;:)``f(t) dt=f(0) ;!;:)``f(t) dt=4 F()_F(0) _0 :! /7 d=;!;[ln t ]8@ :! /7 d=;!;(ln 8_ln ) :! /7 d=;!; ln 4 :! /7 d=ln :! /7 d=;!;:! /7 d + ` + ` 7 lim ;Kn+! {+: napple:};n!;= lim ;4!; ;Kn+! {+: napple:};n$; n` n` ;Kn+! {+: napple:};n$;=;4!;:!5`` d ;Kn+! {+: napple:};n$;=;4!;[;;;; ``]5! :! /7 d=;!;:! /7 (+ `)' d + ` :! /7 /7 d=;!; [ln + ` ]! :! /7 d=;!; (ln 8_ln ) :! /7 d=ln 8 lim n` ;Kn+! {+: napple:};n$;=;4!;{: :_;!;} ;Kn+! {+: napple:};n$;=3 /+/+/3+y+/n n/ßn 3 ln =t dt =;!; d =et==e `t=3 (ln ) ` : e ` d=:!3` t ` dt e : e ` d=[;;4;; t ``]3!=0 e = lim ;n!;{ ;n!;+ ;n@;+ ;n#;+y+ ;nn; } n` 0 = lim ;Kn+! ;n!; ;nk; n` =:)` /ß`d =[;3@; ;#; ]) =;3@; 3 `=t dt = d =t==t=4 :!`` `e ` d=;!;:!4``te t dt :!4``te t dt `f(t)=t»'(t)=e t f '(t)=»(t)=e t 8

181 :!4``te t dt=[te t ]4!_:!4``e t dt :!4``te t dt=[te t ]4!_[e t ]4! :!4``te t dt=3e ` :!`` `e ` d=;!;:!4``te t dt=;#;e ` p=q=3 p `+q `=3 3 6 f()=:!``e sin pt dt=0 f '()= d :!``e sin pt dt=e sin p f '()= d y=f() (f()) (0) y_0= (_)y=_»()=_»(3)= p 4 : e _ sin 3 d f()=e _»'()=sin 3 0 f '()=_e _»()=_;3!; cos 3 p : e _ sin 3 d 0 =[_;3!;e _ cos 3] p p )_;3@;: e _ cos 3 d 0 p : e _ p sin 3 d+;3@;: e _ cos 3 d 0 0 =[_;3!;e _ cos 3] p ) 3 p 3: e _ p sin 3 d+: e _ cos 3 d 0 0 =3 [_;3!;e _ cos 3] p ) 7 lim { + + +y+ } n` n+ n+4 n+6 n+n n n n n = lim ;n!;{ + + +y+ } n` n+ n+4 n+6 n+n = lim ;n!; ª + + +y+ n` º +;n@; +;n$; +;n^; +: n : = lim ;n!;;kn+! n` +: napple: =;!; lim ;n@;;kn+! n` +: napple: =;!;:!3``;!; d =;!;[ln ]3! =;!; ln 3 p : (3e _ sin 3+e _ cos 3) d 0 =[_e _ cos 3] p ) =ln /3 =e _p + 5 f()+ f '()=+f() >0 f '()=;!; f()=ln +C`(C) =f()= `f()=ln +C= C= `f()=ln +f(e `)=3 3 8 A C P / 4k :n: ;4; p ABP AB =/BP =: napple:abp =;4 ; B AP `=(/ ) `+{: napple:} _ / : napple: cos ;4 ; AP `=8+6{;nK;} _6 ;nk; 9

182 lim ;Kn+! AP ` ;n!;= lim ;Kn+! [8+6{;nK;} _6 ;nk;] ;n!; n` n` ;Kn+! AP ` ;n!;=:)``(8+6 `_6) d ;Kn+! AP ` ;n!;=8:)``( `_+) d ;Kn+! AP ` ;n!;=8[;3@; `_ `+])=: 3 : p=3q=6 p+q=9 9 f(a)=0f()= f() f '() `f(a)= f(a) f '(a)=0 :A a { f()} ` d=:a a ;!; f() f '() d ` :A a d=:a a f() d =t dt = d =at=a=at=4a :A a { f()} ` d=:a a f() d ` 4a :A a d=: a f(t) t dt=k 3 f()=:)`` dt =0f(0)=0 +tfl` f()=:)`` dt +tfl` `f '()= +fl` e f() :)à ` d=:)à `e f() f '() d +fl` e f() =t dt =e f() f '() d =0t==at=/e :)à ` e f() +fl` :A a { f()} ` d ` d=:! /e` dt=/e_ u()={ f()} `v'()=;;; ;;; `` 3 :)``f(t) dt=e `+a+a `yy `f()=e `+a =0 0=e `+a=+a a=_ `f()=e `_f(ln )=e ln _=_= cos =t dt =_sin d =0t==;4 ;t=;;;; ;;;; ` / :) ;4 ; cos ` sin d=:! ;;;; ;;;; / ` t `(_) dt u'()= f() f '()v()=_;!; :A a { f()} ` d ` a a =[_;!;{ f()} `]A +:A ;!; f() f '() d =_; ;a;{ f(a)} `+;a!;{ f(a)} `+:A a ;!; f() f '() d :) ;4 ; cos ` sin d=: ;;;; ;;;; ``t ` dt ` / :) ;4 ; cos ` sin d=[;;4;; `` ` t ``];;;; ;;;; / :) ;4 ; cos ` sin d=;4!;_;4!;{;;;; ;;;; `}4 =; 6; / 30

183 f()=ln»'()= `f '()=;!;»()= :!è `ln d=[ ln ]e!_:!è ` d :!è `ln d=[ ln ]e!_[]e! :!è `ln d= 5 lim ;n!;[{+;n!;} 3 +{+;n@;} 3 +{+;n#;} 3 +y+{+;nn;} 3 ] n` = lim ;n!; ;Kn+! {+;nk;} 3 n` =:)``(+) ` d =:@3`` ` d =[;;4;; ``]3@=: 4 : p=4q=65 3 d :!``f(t) dt= d (sin p+cos p+) d d `f()=p cos p_p sin p p+q=69 69 f {;4#;}=p cos ;4#; p_p sin ;4#; p /` /` f {;4#;}=_;;;;;; p_;;;;;; p f {;4#;}=_/ p :) ;6 ; cos ` d=:) ;6 ; cos ` cos d 4 f()= sin f() F() ; ;+h lim : sin d h` 0 h ; ;_h ; ;+h = lim : f()d h` 0 h ; ;_h ; ;+h = lim [F()] h` 0 h ; ;_h F{; ;+h}_f{; ;_h} = lim h` 0 h F{; ;+h}_f{; ;} F{; ;_h}_f{; ;} = lim _ h` 0 h h :) ;6 ; cos ` d=:) ;6 ; (_sin `) cos d sin =t dt =cos d =0t=0=;6 ;t=;!; :) ;6 ; cos ` d=:) ;!; (_t `) dt :) ;6 ; ;!; cos ` d=[t_;;3;; t ``]) =;!4!; p=4q= p+q=35 35 F{; ;+h}_f{; ;} F{; ;_h}_f{; ;} = lim + lim h` 0 h h` 0 _h =F'{; ;} = f {; ;} = ; ; sin ; ;=p :_@` e ` d=_:_0@`e ` d+:)``e ` d `f()=»'()=e ` f '()=»()=e ` :_0@`e ` d=[e ` ]0_@_:_0@`e ` d :_0@`e ` d=[e ` ]0_@_[e ` ]0_@ :_0@`e ` d=;;; ;;; e `` _ 3

184 :)``e ` d=[e ` ])_:)``e ` d ;n!; ;Kn+! OP =[;!; `+;!;e ]) :)``e ` d=[e ` ])_[e ` ]) :)``e ` d=e `+ ;n!; ;Kn+! OP = e ` :_@` e ` d=_{;;; ;;; e `` _}+(e `+) :_@` e ` d=e `_;;; ;;; e `` y=f() f()=k(_a)`(k>0) F()=:)``f(t) dt_:)``t f(t) dt F'()= d {:)``f(t) dt}_ d :)``t f(t) dt d d F'()=:)``f(t) dt+ f()_ f() F'()=:)``f(t) dt F'()=k:)``t(t_a) dt e `=t dt =e ` d =0t==t=e :)``e ` f (e `) d=;!;:!è `f(t) dt :)``e ` f (e `) d=;!; [F(t)]e! :)``e ` f (e `) d=;!;{f(e)_f()} F(e)=eF()=;!;(e_) F'()=k[;;3;; t ``_at `]) F'()=k{;;3;; ``_a `} F'()=k `{;3 ;_a} F'()=0 =0 =3a 0+3a=3a 4 y=e ` y'=e ` A {;nk;e ;nk; } e ;nk; A {;nk;e ;nk; } y_e ;nk; =_ {_;nk;} y=0=;nk;+e : napple: OP =;nk;+e : napple: e ;nk; lim ;n!; ;Kn+! OP =:)``(+e ) d n` F()=:A``f(t) f '(t) dt =a F(a)=:Aà `f(t) f '(t) dt=0 () F()=:A``f(t) f '(t) dt F'()= d :A``f(t) f '(t) dt d F'()=f() f '() f()=k(_a)(_b) (k>0) a+b f '()=k{_ } F'()=f() f '() a+b F'()=k `(_a)(_b){_ } F'{ }=0 () ;!;{ f()} ` ' =;!; f() f '() ;!;{ f()} ` ' =f() f '()=F'() a+b 3

185 F()=;!;{ f()} `+C`(C) F()=;!; k `(_a) `(_b) `+C F(a)=0 C=0 F()=;;!; k `(_a) `(_b) ` 4 f() `f `(0)=0f `(5)=5 y=f()y=f `() y y=f() 5 y=f () y=f() =a=b F()=0() lim ;Kn+! [ f {: napple:}+f `{: napple:}];n%; n` O 5 F'()=f() f '() y=f()= a+b f '{ }=0 a+b a+b a+b F'{ }=f { } f '{ }=0 F()=;!;{ f()} `f()=0=a =b F()=;!;{ f()} `=0 () a+b =:)5` { f()+f `()} d =:)5` f() d+:)5` f `() d =:)5` f() d+{5_:)5` f() d} =5 5 3 f()= lim ;;; ;;; n ``{ n_ + n_ + n_3 +y n` + n_n } f()= lim ;n!;[ _;n!; + _;n@; + _;n#; +y n` f()= lim ;Kn+! _;nk; ;n!; n` f()=:)`` _t dt + _;nn; ] f()=:)``(_t) dt+:x``(t_) dt f()=[t_;;;; t ``])+[;;;; t ``_t]x f()= `_+;!; f()={_;!;} +;4!; f()=;!;;4!; 33

186 05 S=:) p p sin d+: (_sin ) d p p p S=[_cos ])` +[cos ] p S= y y=(_)(_4) 3 y y= y=/ O 4 O y= `y=/ß (00)() S S S=:)``(_)(_4) d+:@4` {_(_)(_4)} d S=:)``(/ß _ `) d S=[;3@;æ ;#; _;;3;; ``])=;3!; S=:)``( `_6 `+8) d+:@4``(_ `+6 `_8) d S=[;4!;æ `_ `+4 `])+[_;4!;æ `+ `_4 `]4@ S=8 y=(_)(_4) (0) S=:)``(_)(_4) d S=:)``( `_6 `+8) d S=[;4!;æ `_ `+4 `]) S= (4_6+6)=8 4 y y=sin sin =cos =;4 ;=;4%;æp ;4%;p O p ;4; p p y=cos =p 0 ;4 ;cos æsin ;4 ; ;4%;æp sin æcos ;4%;æp pcos æsin S ;4 ; S=:) (cos _sin ) d+:`;4%;æp `(sin _cos ) d ;4 ; p`(cos _sin ) d y O y=sin p p +: ;4%;p S=[sin +cos ]) ;4 ; ;4%;æp +[_cos _sin ] ;4 ;` p +[sin +cos ] ;4%;p S S=:) p sin d S=(/ _)+/ +(+/ ) S=4/ 34

187 5 V V=:) ln 3`(e +) d V=[e +])` ln 3 V=e ln 3 +ln 3_ V=+ln 3 e ++e _ +(y') `= 4 e +e _ +(y') `={ }`? +(y') ` = e +e _ l V=ln 3e `(cm `) 6 V V=p:) ; ;`y ` d y y=cos l=:)``? +(y') ` d e +e _ l=:)`` d e _e _ l=[ ]) l=;!;æ{e_;e;!;} V=p:) ; ;`cos ` d O p ;; V=p:) ; ;` +cos d ; ; V=p[; ;+;4!;æsin ])` V= p ` v(t)=t(3_t) 0 t 3væ03 t 4 v 0t=0t=4s y O p s=:)4`` v(t) dt s=:)4`` 3t_t ` dt s=:)3``(3t_t `) dt+:#4``(t `_3t) dt s=[;#;æt `_;;3;; t ``]3)+[;;3;; t ``_;#;æt `]4# s=: 3ª: y=cos _ S S=:)`p (_cos ) d p S=[_sin ])` =p e 8 _e _ y'= e _e _ +(y') `=+{ }` +(y') `=+ e _+e _ 4 ;!; _ln _ln f '()= = ` ` `f '()=0 =e {_;!;} `_(_ln ) _3+ ln `f "()= = ` ` f "()=0 =e ;#; 35

188 f() (0) y e y e ;#; y f '() + 0 f " () 0 + f() y ` S=:_4@`{y+4_ } dy y ` y ` S=[ +4y_ ]4_@=8 6 f() =e e ;#; y ln y=:::: O =e =e ;#; 4 y=ln y '= y=ln (e) l y_= (_e) y= e e y y=;e; y=ln S S=:E e;#;` ln d O e ln =t dt = =et==e ;#; d t=;#;æ S S=:)è ` e d_:!è `ln d S=:E e;#;` ln d S=:! ;#; t dt S=[ t ` ;#; ]!` =;8%; S=[ `]e)_[ ln _]e! e e e S= _(e_e+)= _ p=8q=5 p+q=3 ` 3 S=:)``(e y _ey) dy S=[e y _ e y `]) e S= _ 3 y `=y=_4 y y `=(y+4)y `_y_8=0 y=_ y=4 y 4 y=_4 y = 5 / = = ` (_)=0 =0 = y=/ y= =0 = y y= y=/ß O 4 O S V 36

189 V=p:)` {(/ ) `_ `} d V=p:)``(_ `) d V=p[ `_ ` 3 ])=;3$;æp 6 S() S()=sin ` V Pt=0 t= s d dy s=:)`` æ { }`+{ }` dt dt dt s=:)``/ e _t dt s=[_/ e _t ]) s=/ {_ } e V=: ;4 ; ; ; S() d V=: ;4 ; ; ; sin ` d V=: ;4 ; ; ; V=[ _cos d _;4!;æsin ] ; ; ;4 ; = p+ 8 7 t P (t)p (0)=0 7 3 :)``{(_ `+)_( `_ `)} d =:)``{(_ `+)_(a_a `)} d :)``(_ `+ `+) d =:)``{_ `+a `+(_a) }d t=; ;æp{; ;} d 8 =_e _t sin t+e _t cos t=_e _t (sin t_cos t) dt dy =_e _t cos t_e _t sin t=_e _t (cos t+sin t) dt {; ;}=(0)+:) ; ; v(t) dt {; ;}=0+:) ; ; (cos t_sin t) dt ; ; {; ;}=[;!;æsin t+cos t])` {; ;}=_ d dy æ { }`+{ }` dt dt =? (_e _t ) `{(sin t_co s t) `+(cos t+sin t ) `} =/ e _t :)``(_ `+ `+) d_:)``{_ `+a `+(_a)} d =0 :)``[(_ `+ `+)_{_ `+a `+(_a)}] d=0 :)``{ `_a `+(a_)} d=0 ` a_ [ _;3@;æa `+ `])=0 4 ;4!;_;3@;æa+a_;!;=0 a=;4#; y=;!;æ ` y =;!;æy ` y y= / { æ0} y=/ 37

190 y=;!;æ `(æ0) y= ;!;æ `= =0 = V V=p:)``(/ ) ` d_p:)``{;!;æ `}` d ` V=p:)``{_ } d 4 fi` V=p[ `_ ]) 0 y O y=;!; y=/ß a t cb c t dc A>B P 0<t d :)à `v(t) dt=:)à ` v(t) dt=a :Ac``v(t) dt=_:ac`` v(t) dt=_b :)c``v(t) dt=:)à `v(t) dt+:ac``v(t) dt=a_b :Cd``v(t) dt=:cd`` v(t) dt=c A_B=C :)c``v(t) dt=:cd``v(t) dt :Ab`` v(t) dt=s:bc`` v(t) dt=t V=: 5 :æp p=5q= p+q=7 7 :)b``v(t) dt=a_s:bd`` v(t) dt=t+c S+T=BA=B+C A_S=A_(B_T)=A_B+T=C+T :)b``v(t) dt=:bd`` v(t) dt 3 v(t) O A a b B C cd t 0a ac cd v(t) t ABC :)à ` v(t) dt=a:ac`` v(t) dt=b:cd`` v(t) dt=c :)à ` v(t) dt=:ad`` v(t) dt :)à ` v(t) dt=:ac`` v(t) dt+:cd`` v(t) dt y y=ln y= A=B+C `yy P 0 t aa O y=ln =e y S 38

191 S=:)``e y dy S=[e y ]) 4 V V=p:)``(_/ß) ` d y y=_/ S=e_ V=p:)``(_/ß+) d O ln = =e S V=p[_;3$;æ ;#; ` + ]) V=;6 ; S=e _:!è `ln d S=e_[ ln _]e! S=e_ 5 0 t v(t)æ0 t 6v(t) 0 Pt=0t=6 s s=:)6`` v(t) dt S S=:`è ` [ _{_ }] d ;e!; S=:`è ` ;e!; 3 d S=[3 ln ]è ` ;e!; S=6 y e O _e =e =;:; e y=;:; y=_;:; s=:)``cos`;4 ; t dt+:@6``{_cos`;4 ;æt} dt s=[; $;æsin ;4 ; t])+[_; $;æsin ;4 ; t]6@ s= p 3 S() /3 S()= (? _ ` ) ` 4 /3 S()= (_ `) 4 V V=:_!`S() d /3 V= :_!`(_ `) d `f(a)=»(a) eå`=k sin a f '(a)=» '(a) eå`=k cos a a=;4 ;k=/ e ;4 ; yy yy /3 V= :)``(_ `) d /3 ` V= [_ ]) 3 5/3 V= 6 p=6q=5 p+q= S S=:) ;4 ; (e _/ e ;4 ; sin ) d S=[e +/ e ;4 ; cos ])` ;4 ; S=e ;4 ; +e ;4 ; / e ;4 ; S=(_/ )e ;4 ; _ 39

192 S =:)k``(k_ `) d k ` S =[ `_ ]k) 3 k ` k ` k ` S = _ = 3 6 S =:K``( `_k) d ` k S =[ _ `]K 3 k ` k ` S =;3*;_k_ + 3 k ` S = _k+;3*; 6 S +S =S(k) k ` S(k)= _k+;3*; 3 S'(k)=k `_=0 k=/ `( <k<) S(k) V y V y V =;3$;æpab `V y =;3$;æpa `b n 6k 3 4 lim ' +[ f '{ }]` n` k= n n n 6k 6 =;!;æ lim ' +[ f '{ }]` n` k= n n =;!;:)6``? +{ f '()} ` d :)6``? +{ f '()} ` dy=f()`(0 6) (03)(6) :)6``? +{ f '()} ` d ;!;æ? (6_0) `+ (_3) ` =;!;æ/ƒ36+64=5 k () y / y () S'(k) _ 0 + S(k) S(k) k=/ k/ 3 ` y ` + = y `=9_;ª5;æ ` `+y `= 5 9 y `=_ ` 0 3 S n =:)``{(_) n _(_) n+ } d S n =:_0!~{(+) n _(+) n+ } d S =:_0!~( n n _ n+ ) d V=[p:)5``{9_;ª5;æ `} d_p:)``(_ `) d] V=p[[9_; 5;æ `]5)_[_;3!;æ `])] V=p[(45_5)_{_;3!;}] V=: ;3&; :æp 3V =76 p 76 ` y ` + =`(a>0b>0) a ` b ` S n =[ _ ]0_! n+ n+3 S n = n+ n+ _ S n = { _ } n= n= n+ n+3 n S n = lim { _ } n` k= k+ k+3 S n = lim [{;3!;_;5!;}+{;5!;_;7!;}+{;7!;_;9!;} n` n+3 n+3 +y+{ _ }] n+ n+3 40

193 S n = lim {;3!;_ } n` n+3 S n =;3!; p=3q= p `+q `=0 S S S S +S =S yy y=+k ABCD S +S =S yy S =S S =3S k ` S +S +S =6S =(k) ` S = 3 S =:)k``/ß d S =[;3@;æ ;#; ]k) 0 ABCD a+c b= B=C _A+B>0A<B _A+D<0A>D P D C B A B A C D O P B=CA>D :)b`` v(t) dt=(a+b) :)b`` v(t) dt=a+a+b+b :)b`` v(t) dt=a+b+a+b :)b`` v(t) dt>a+b+d+c :)b`` v(t) dt=:)d`` v(t) dt S =;3@;æk ;#; k ` 3 =;3@;æk ;#; k `=k ;#; k `=k ` k `(k_)=0 k>;4!; k= 3 v(t) v(t) O A D B c a b C d t t=at=bt=cp 0 P 4

194 P =3 `=7 ABC P =3fi`=43 3 C O F E 6! =80!! 4 3 abc 3 6! =0 3! 0 5 A C ` ` 5! =0! 3! C B `! =!! 0 =0 6 A B ` 4 5 9! =6 4! 5! A C 5! =0C B! 3! 4! =6 A C B!! 0 6=60 6_60= A B C D E (5_)!=4!=4 8 6 (6_)!=5!=0 0 =40 40 ABC ABC 4 P =3 `=8 4

195 AaB AbcB 4 P =4 5 `=00 3 UIEA XXXX X UIEA 9 XXXXNVRSL 9! 4! 00 f(a)= P _ P =5_5=00 5 A Q C P A B A P Q B A B 9! =84 6! 3! A P 5! =5 4! P Q Q B 3! =3! 84_5 3=84_5=69 B 69 4 a 7 7! =35 4! 3! 7 a C =8 m=35 8=980 b B 7 aabcccc 7! n= =05 4!! m_n=980_05= a 9! 8! m= _ =980! 3! 4! 3! 4! 6 A C D F E CD EF A B 9! =6 5! 4! CD A B 6! =40 3! 3! EF A B 5! 3! =30 3!!! B 43

196 CDEFA B 3! =! 6_(40+30)+=68 4! =6!! C B 4! =6!! 6 6= (4_)! =3! =6 = C = C =35 4 (4_)!=3!=6 C 3!=35 6= ! =56 3! 5! C A B 3 A B 5 (5_)!=4!=4 A B 4 = P _ P =4 `_4 ` =56_64 = P =4 `= =9 A C 4ABCD 44

197 4! =6!! 3 B C D E 3 A B B A AB 456 P =3 `=8 8 A A B A` `C` `B 4! =4 3! A` `D` `B 4! 3! =6 3=8!!! A` `E` `B 4! 3! { _} { _}=3 =6 3!! 4+8+6=8 8 4 C =6 5 (5_)!=4!=4 6 4= AB f(a)+f(b)+f(c)+f(d)=0 f(a)f(b) `f(c)f(d)3+3+3+= = ! 4! + =4+6=0 3!!! 3 P A ` `B 9! =6 5! 4! A ` `P ` `B 4! 5! =60!! 3!! P 6_60=66 P A Q R' P R Q' A ` `Q ` `P ` `Q' ` `B 3! 4! =8!!! A ` `R ` `P ` `R' ` `B 3! 4! =! 3! B 45

198 P 8+= = (5_)!!=48 (4_)!!!=4 48_4=4 a bcdef 4! C =0 3! abc 4! C =8!! abc def 4! C C =360! abcdef 4 C 4!= = f()=f() `f()=3f() `f()=5f() 68 `f()=7f()=8 f()f() 4+3++=0 f(3)f(4)y f(3)f(4) P =8 `=64 f 0 64= B 3 4 C A Q 35 7 P Q P 3 Q P P 4C Q P C P 5 Q P P 5! =0! 3! 0 46

199 H 6 = 3+6_ C 6 H 6 = H 6 =8 = 8 C 6 = 8 C 8\7 \ H + H = 4+3_ C + 3+4_ C = 6 C + 6 C = 6 C + 6 C 6\5\4 6\5 H + H = + 3\\ \ H + H =0+5= H = 3+4_ C H = 6 C = 6 C 6\5 H = \ H = y+z=6=3y=z= 3 y z 3 yz 6 (yz) H 6 = 3+6_ C 6 = 8 C 6 = 8 C 8\7 H = \ H = abc 5 H = 3+5_ C = 7 C = 7 C 7\6 H = = \ 5 (+y) n y n H n = +n_ C n = n+ C n = n+ C =n+=8 n=7 6 {_;!;}6` 6C r () 6_r {_;!;} r = 6 C r 6_r {_;!;} r 6_r 6_r=4 r= ` 6C ` {_;!;} 6\5 = \ `\{_;!;} =60 \ 7 {+ }7` 7C r 7_r { 7_r= r=3 }r`= 7 C r r 7_r 7\6\5 7C `= \ `=80 3\\ 8 nc r + n C r+ = n+ C r+ 7C + 7 C + 8 C + 9 C + 0 C 5 = 8 C + 8 C + 9 C + 0 C 5 = 9 C + 9 C + 0 C = 0 C + 0 C 5 = C 5 47

200 9 ncº+ n C + n C + n C +y+ n C n = n n=8 8Cº+ 8 C + 8 C + 8 C +y+ 8 C 8 = `= (+) ` `= ºCº+ ºC + ºC `+ ºC `+y+ ºC º ` ` = 3 ` `= ºCº+ ºC + ºC `+ ºC `+y+ ºC º ` ` N=3 ` ` N=3 ` ` 34 5 H = 3+5_ C H = 7 C = 7 C 7\6 H = = \ 56_= H = 4+4_ C = C = C 7\6\5 H = =35 3\\ H = 3+5_ C H = 7 C = 7 C 7\6 H = = \ 3 4 H = 3+4_ C H = 6 C = 6 C 6\5 H = =5 \ \5= H = 4+5_ C H = 8 C = 8 C 8\7\6 H = =56 3\\ 3 +y+z=8`(yz) ='+y=y'+z=z'+ ('+)+(y'+)+(z'+)=8 '+y'+z'=5 ('y'z') yy 'y'z' ('y'z') H = 3+5_ C H = 7 C = 7 C 7\6 H = = \ 4 (+y+z) ` ` yz a y b z c a+b+c= a bc (abc) a=a'+b=b'+c=c'+ (a'+)+(b'+)+(c'+)= a'+b'+c'=4 (a'b'c')yy a'b'c' (a'b'c') H = 3+4_ C H = 6 C = 6 C 6\5 H = =5 \ 48

201 a 5 {+ }8` 8C r 8_r { 8_r=_ r=5 ` a }r`= 8 C r a r 8_r 8C afi`= 8 C afi` 8\7\6 8C afi`= \afi`=56afi` 3\\ 8_r= r=3 ` 8\7\6 8C a `= \a `=56a ` 3\\ ` 4a ` ` 56afi`+56a `=4a ` 56afi`_68a `=056a `(a `_3)=0 a=/3``( a>0) 8 nc + n C +3 n C +y+n n C n =n n_ log ( n C + n C +3 n C +y+n n C n ) =log (n n_ ) =log n+log n_ =log n+n_ =0 log n+n=yy log n n= m (m=03y) m=3 n= `= {_ }5` {+ }5`=[{_ }{+ }]5` /ß /ß /ß /ß ={ `_ }5` { `_ }5` C r ( `) 5_r {_ 0_3r=4 r= ` 5\4 C (_) ` = \=0 \ 7 (+) 8 = 8 Cº+ 8 C + 8 C `+y+ 8 C r r +y+ 8 C 8 8 =3 4 8 = 8 Cº+ 8 C 3+ 8 C 3 `+y+ 8 C r 3 r +y+ 8 C = 3 r 8 C r r=0 }r`= C r (_) r fi 5_r 0_3r 0 H r = 3+r_ C r = +r C r = +r C +rc = 7 C +r=7 r=5 H r = H = 5+5_ C =ªC =ªC H r = H r =6 9\8\7\6 4\3\\ 6 +y+z+w=4 (yzw) 4 yzw 4 H = 4+4_ C = 7 C = 7 C 7\6\5 H = 3\\ H = r 8 C r =4 ` `= `fl` r=

202 +y+z=`(yz) 3 yz H = 3+_ C = C 4\3 H = =6 \ (yz) (3)(3)(3)()() () 5 (3)(3)(3) a+b=3 (ab) ab 3 H = +3_ C = C =4 ()()() =5 4 (+) n n C r r ` nc n(n_) nc = =45 n `_n_90=0 (n_0)(n+9)=0 n=0`( n) 0 H 7 = 4+7_ C 7 = 0 C 7 = 0 C 0\9\8 H = 3\\ H =0 (+a)(+b)(+c) ` a+b+c a+b+c=8 a+b+c=8`(abc)a=a'+ b=b'+c=c'+ (a'+)+(b'+)+(c'+)=8 a'+b'+c'=5`(a'b'c') a'+b'+c'=5 a'b'c' (a'b'c') H = 3+5_ C H = 7 C = 7 C 7\6 H = = \ 3 (a+) ` 7C r (a) 7_r r = 7 C r a 7_r r 7_r 7_r=3 r=4 ` 7C a ` `=6 C a ` 7_r=4 r=3 ` 7C a ` `=8 C a ` ` ` ;!; 6 C a `=;!; 8 C a ` 6=4a ( a+0) a= n 4 a n = n C r r=0 a n = n Cº+ n C + n C + n C +y+ n C n a n = n = n= a«n= n = {;4!;}ǹ n= 50

203 ;4!; = _;4!;` =;3!; H = 4+_ C = C 5\4 H = =0 \ 5\0= y + z =7 ='+ y =y'+ z =z'+ ('+)+(y'+)+(z'+)=7 '+y'+z'=4`('y'z') '+y'+z'=4 'y'z' ('y'z') H = 3+4_ C H = 6 C = 6 C 6\5 H = =5 \ ('y'z') (yz) `=8 (yz) 5\8=0 ('y'z') () =, y = z =3 (yz) (3)(_3)(_3)( 3) (_3)( 3)( 3) ( 3) 8 (a+b+c+d) ` a (b+c+d) ` H = C = C 6\5 H = \ H =5 (e+f+») ` e e H _ H = C _ C H _ H = C _ C 5\4 H _ H = _4 \ H _ H =0_4=6 5 6=90 4 m n=m_ cos np=cos (m_)p=_ (_) n =(_) m_ =_ (_) `_(_) a n = = n=m cos np=cos mp= (_) n =(_) m = `_ a n = =0 (n=35y) a n =g 0 (n=46y) 0 a n 0 C n = 0 C + 0 C + 0 C +y+ 0 C 9 n= a n 0 C n = 0_ = 9 5

204 =;!; n= a«n= n(n+) 0 =;!;æ { _ } n= n n+ p ` 8=3p ;8&;æ ;8!;\3p=4p abc =;!;æ[{;!;_;!;}+{;!;_;3!;}+{;3!;_;4!;}+ y+{; 0;_; ;}] =;!;æ{_; ;} =;!;æ ;!);=; ; ;5^;æp(a+b+c) 4p a+b+c ;; 3º;; a+b+c : 3º: a+b+c= (abc) H = 3+_ C = C =3 a+b+c= (abc) H = 3+_ C = C 4\3 H = =6 \ a+b+c=3 (abc) H = 3+3_ C = C = C 5\4 H = =0 \ ~ (abc) 3+6+0= abcyij a b c d e f» h i A A B 0 0C + 0 C + 0 C +y+ 0 Cª= 0_ = 9 =5 5 bch A B a b c d e f» h i A adf»i A B a b c d e f» h i A j j j B B B {+ } n+ n+c r (n+)_r { } r = n+ C r r n+_r n+_r=n_3 r= n_3 n+c `= (n+)n a n =n(n+) \ `=n(n+) 5

205 p=80 ;4!;= S={3456} A={4} B={456} A;B={4} A'B={456} BÇ` ={3} C = = C C = 3=8 ;3!5*; S={ } A={369} B={357}C={48} A;C=u AC A;B={3}B;C={} (A;B);(B;C)=u A;B B;C A'B={ } BÇ` ={ } (A'B);BÇ``={69}+u A'B BÇ`` 3 6 6=36 ab 8 a+b= (ab)() a+b=4 (ab)(3)()(3) 3 a+b=8 (ab)(6)(35)(44)(53)(6) 5 ~ =9 p=;3ª6;=;4!; 5 3 A4 B 3 4 A'B P(A)=;4!0#; P(B)=;4!0);=;4!; P(A;B)=;4 0; P(A'B)=P(A)+P(B)_P(A;B) P(A'B)=;4!0#;+;4!;_;4 0; P(A'B)=;!; 6 ABP(A)=3P(B) P(A'B)=P(A)+P(B) P(A'B)=P(A)+;3!; P(A) P(A'B)=;3$; P(A)=;5$; 53

206 P(A)=;5#; C = = AA AÇ`` 4 54 C = C =5 P(AÇ``)=;7 0;=; 4; P(A)=_P(AÇ``) P(A)=_; 4; P(A)=;!4#; ºC = =0 3 AA AÇ`` C = =0 3 P(AÇ``)=; º0;=;6!; P(A)=_P(AÇ``) P(A)=_;6!; P(A)=;6%; 8 y (y) 6 6=36 (y) _y > A A AÇ`` (y) _y _y =0 (y) ()()(33)(44)(55)(66) 6 _y = (y) ()()(3)(3)(34)(43) (45)(54)(56)(65) 0 _y (y) 6 P(AÇ``)=;3!6^;=;9$; P(A)=_P(AÇ``) P(A)=_;9$; P(A)=;9%; P =4 3 = != !=6 36+6= ;!4@;=;!; 54

207 !=3 3 =8 C = C =4 4 =4 ; 8;=;9@; 3 A 6 B 6 A'B ªC = = C = =6 P(A)=;3 6;=;6!; C = =6 P(B)=;3 6;=;6!; 6 6 P(A;B)=;3 6; P(A'B)=P(A)+P(B)_P(A;B) P(A'B)=;6!;+;6!;_;3 6; P(A'B)=;3!6!; 4 f()< f()< f(3)f()=3 Af(3)=7 Bf()=3 `f(3)=7 A'B `f()<f()<f(3)f`:`x` `Y C = =56 3 3=f()< f()< f(3) C = =0 P(A)=;5!6);=; 8; f()< f()< f(3)=7 C = P(B)=;5!6%; =5 3=f()< f()< f(3)=7 C =3 P(A;B)=;5 6; P(A'B)=P(A)+P(B)_P(A;B) P(A'B)=; 8;+;5!6%;_;5 6; P(A'B)=;!8!; p+q=8+= ºC = = A 30 B330 A'B C = =0 3 P(A)=; º0;=;6!; C = C =4 P(B)=;$0;=;3 0; 55

208 AB P(A'B)=P(A)+P(B) P(A'B)=;6!;+;3 0;=;5!; 6 ab (ab) 6 6=36 ab A3 B 3 A'B ab ab3 5 (ab)()()(3) (3)(5)(5) ! A A AÇ`` 5 5! 6 4 P 5! P P(AÇ``)= 5! P 9! P(A)=_P(AÇ``) P(A)=_;4 ; P(A)=;4#&; =;4 ; P(A)=;3 6;=;6!; ab 3 ab 495 (ab) (4)() (4)(33)(55) 5 P(B)=;3 6; AB P(A'B)=P(A)+P(B) ! A XX XÇ``B B P =4 3= 55! B 5! P(A'B)=;6!;+;3 6; P(A'B)=;3!6!; P(XÇ``)= 5! 7! =;7@; P(X)=_P(XÇ``) P(X)=_;7@;=;7%; 6 6=36 i m (_i) n =(_) n i m+n nm+n48 (mn) ()(6)(44)(6)(66) 5 nm+n60 (mn) ()(5)(33)(5)(55) 5 p+q=8+5= =; 8; C C C C =05 4! 56

209 44 C C =7 ; 0 5;=;3@5$; 4 _a+b=5 b_a=4(ab) (5)(6) ~ =;3!; 3 AB P(A;B)=0 6 (6_)!=5! P(A)=P(B)P(A)P(B)=;9!; {P(A)} `=;9!; P(A)=;3!; ( P(A)æ0) P(A'B)=P(A)+P(B)_P(A;B) P(A'B)=;3!;+;3!;_0=;3@; 3 (3_)!=! 3 3!! 3!! 3! 5! =; 0; 3 4 ab (ab) 6 6=36 `_a+b_ `_a +b=_a+b _a+b _a+b= a=b(ab) ()()(33)(44)(55)(66) 6 _a+b=3 b_a=(ab) (3)(4)(35)(46) C C = =35 3 AB X YA B X'Y AB C C =5 P(X)=;3 5;=;7!; AB 5 4 C C = C = =0 P(Y)=;3!5);=;7@; 57

210 XY P(X'Y)=P(X)+P(Y) P(X'Y)=;7#; 4! P 4! P 4! = =; 4; 8! 8! 4 abc 4 H = C = C = C 6 5 H = H =5 a AA AÇ`` a bc 4 H = C = C = C =5 P(AÇ``)=; 5;=;3!; P(A)=_P(AÇ``) P(A)=_;3!; P(A)=;3@; ab (ab) 6 6=36 b b a+? 4 b =4 a+ = a+ b a+ a= b=3456 (ab) 6 a= b=36 (ab) a=3 b=46 (ab) 3 a=4 b=5 (ab) fi a=5 b=36 (ab) ~fi a+? 4 b (ab) =4 ;3!6$;=; 8; ! 4684! P 3 abc (abc) 6 6 6=6 ` a_b = b_c a_b=b_c a_b=_b+c a+c=4b a=c a+c=4b Xa=c Y a+c=4b b=a+c=4 (ac) (3)()(3)3 b=a+c=8 (ac) 58

211 (6)(35)(44)(53)(6)5 b=3a+c= (ac) (66) a+c=4b (abc) 3+5+=9 9 P(X)= =; 4; 6 ` a=c a=c (ac) ()()(33) (44)(55)(66) 6b a=c (abc) 6 6=36 36 P(Y)= =;6!; 6 ` a+c=4ba=c a=b (ab) ()(4) (63) 3 a+c=4b a=c (abc) 3 3 P(X;Y)= =;7 ; 6 ` P(X'Y) P(X'Y)=P(X)+P(Y)_P(X;Y) P(X'Y)=; 4;+;6!;_;7 ;=;3 6; 4 W BR G WWBBRG 6 6! =80!! WW XBB Y WW 5! =60! P(X)=; 8º0;=;3!; BB 5! =60! P(Y)=; 8º0;=;3!; WWBB 4!=4 P(X;Y)=; 8 0;=; 5; ~ P(X'Y)=P(X)+P(Y)_P(X;Y) P(X'Y)=;3!;+;3!;_; 5;=; 5; XÇ``;YÇ`` P(XÇ``;YÇ``)=P((X'Y)Ç``) P(XÇ``;YÇ``)=_P(X'Y) P(XÇ``;YÇ``)=_; 5;=; 5; a bcd 7 a+b+c+d=7 (aæbcd) a=a'+ (a'+)+b+c+d=7 a'+b+c+d=6 (a'bcd) yy a'bcd (a'bcd) H = C =ªC =ªC H = =84 3 ;60*0$0;=;50&0; 6! =80!! 59

212 != != !=6 ~ =;9 0; A B P(A)=; 0º0'=; 0; P(B A)+P(A B) P(A;B) P(A;B) = + P(A) P(B) P(A;B) P(A;B) = + ;!; ;3!; =P(A;B)+3P(A;B) =5P(A;B) =;4%; P(A;B)=;4!; 3 ªC +ªC +ªC +y+ªcª= `_(ªCº+ªC ) = `_(+9) =50 A AÇ`` {3579} C + C + C + C =fi`_( Cº+ C )=6 P(AÇ``)=;5 0 ;=; 5 ; P(A;B)= 70_30 00 =;5@; ;5@; P(A;B) P(B A)= = =;7$; P(A) ; 0; 3 P(A)=_P(AÇ``)=_;3!;=;3@; P(A;B)=P(A)P(B A) P(A;B)=;3@;\;4!;=;6!; P(A)=_P(AÇ``)=_; 5 ;=;@5#*; p+q=5+38= A B 3 60`% 60

213 P(A)=; 0º0;=;5#; 0`% P(B A)=; 0º0;=;5!; P(A;B)=P(A)P(B) P(A;B)=;!;\;6%;=; ; A;B P(A;B) P(A;B)=P(A)P(B A) P(A;B)=;5#;\;5!;=; 5; 5 A B P(A;B)=P(A)P(B A) P(A;B)=;8%;\;7$;=; 4; P(AÇ``;B)=P(AÇ``)P(B AÇ``) P(AÇ``;B)=;8#;\;7%;=;5!6%; P(B)=P(A;B)+P(AÇ``;B) P(B)=; 4;+;5!6%;=;5#6%;=;8%; 6 AB P(B AÇ``)=P(B)=;5@; P(A;B)=P(A)P(B) P(A'B)=P(A)+P(B)_P(A;B) P(A'B)=P(A)+P(B)_P(A)P(B) P(A'B)=;3@;+;5@;_;3@; ;5@; P(A'B)=;!5@;=;5$; 8 ;6@;=;3!; ;3@; 5 3p p= C {;3!;} 3 {;3@;} 4 40 =0 = 3fi` 3fi` 40 3fi`p=3fi` =40 3fi` 40 9 ;!; ;!; 4 AA AÇ`` 4 0 P(AÇ``)= Cº {;!;} 0 {;!;} 4 + C {;!;} {;!;} 3 P(AÇ``)=; 6;+; 6;=; 6; P(A)=_P(AÇ``)=_; 6;=;!6!; A B P(A)=;4@;=;!; P(B)=;3#6);=;6%; N A B (N) 6

214 A B ;n^;; P(A;B) P(B A)= = =; 6;=;8#; P(A) : n : A B P(A)=0.4 P(A;B)=0. P(A;BÇ``)=P(A)_P(A;B) =0.4_0.=0.3 P(A;BÇ``) 0.3 P(BÇ`` A)= = =;4#; P(A) A B 6 P(A;B)=P(A)P(B A) P(A;B)=;6$;\ P(A;B)=;3@;\;5@;=; 5; 6 P(AÇ``;B)=P(AÇ``)P(B AÇ``) P(AÇ``;B)=;6@;\ C + C C P(AÇ``;B)=;3!;\; 5;=;4 5; P(B)=P(A;B)+P(AÇ``;B) P(B)=; 5;+;4 5;=;4!5(; C + C C 4 A B P(A;B)=P(B)P(A B) P(A;B)=; 0º0;\; 0º0;=; 5; P(A;BÇ``)=P(BÇ``)P(A BÇ``) P(A;BÇ``)={_; 0º0;}\; 0º0; P(A;BÇ``)=; 5; P(A)=P(A;B)+P(A;BÇ``) P(A)=; 5;+; 5;=;!5!; P(B A)= P(A;B) P(A) ; 5; P(B A)= =; ; ;!5!; B BÇ`` (A) (AÇ``) A B P(A)=P(B)=;6$;=;3@; P(AÇ``)=P(BÇ``)=_;3@;=;3!; A B A BÇ``AÇ`` B P(A;BÇ``)+P(AÇ``;B) =P(A)P(BÇ``)+P(AÇ``)P(B) =;3@;\;3!;+;3!;\;3@; =;9$; 6

215 6 A A B B P(A)=;3!;P(B)=;5!; A B XX XÇ``A B XÇ``=AÇ``;BÇ`` A B AÇ`` BÇ`` P(XÇ``)=P(AÇ``;BÇ``) P(XÇ``)=P(AÇ``)P(BÇ``) P(XÇ``)={_;3!;}{_;5!;} P(XÇ``)=; 5; P(X)=_P(XÇ``)=_; 5;=; 5; p = C {;3!;} {;3@;} =; 7; 43 3p p = C {;3@;} 3 {;3!;} =;8#@; p ;8#@; = =;3$; p ; 7; 3 K A B K ;5!; P(A;B)=;5!; P(A'B)=P(A)+P(B)_P(A;B) P(A'B)=; 0º0;+;5@;_;5!;=;ª0; 7 3 ;6#;=;!; 3 ;6#;=;!; 5 3 AA AÇ`` P(AÇ``)= C {;!;} 5 {;!;} 0 =;3 ; P(A)=_P(AÇ``) P(A)=_;3 ;=;3#!; 8 ;6@;=;3!; 3 ;6$;=;3@; 4 p K P(BÇ`` AÇ``)= P(BÇ`` AÇ``)= P(AÇ``;BÇ``) P(AÇ``) _P(A'B) _P(A) _;ª0; P(BÇ`` AÇ``)= =;3!; _; 0; 4 A BAB A P(A)= =; 8; 6 6 P(B)=;6$;=;3@; 63

216 P(A;B)=P(A)P(B) P(A;B)=; 7; 3 3 a3 b 5Ba 3b b5 C {;3!;} 3 {;3@;} + Cº {;3!;} 0 {;3@;} 5 =; 4º3;+; 4 3; C {;3!;} 3 {;3@;} + Cº {;3!;} 0 {;3@;} 5 =; 4 3;=; 7; 3!! P(A)= =;4!; P(A;B)= =; ; 4! 4! P(B A)= ; ; P(B A)= ;4!; P(B A)=;3!; P(A;B) P(A) 3 C + C 3+ = =;5@; C 0 _;5@;=;5#; 3 4 ;5#;\;5#;\;5@;=; 5; A B P(A)=;5#0%0);=; 0; P(A;B)=;5!0%0%;=; 0 0; P(B A)= P(A;B) P(A) C {;!;} {;!;} = = fi` ` 3 C {;!;} 3 {;!;} 0 5 = = fi` ` 5 ` 5 \ = ` 5 ` ; 0 0; P(B A)= =;7#0!; ; 0; X Xf4! f f()= Af()= B 3 4 A B 64

217 0.6 P(B)=; 0º0; n(b)=00 ; 0º0;= P(AÇ``;BÇ``)=; 0 0; n(aç``;bç``)=; 0 0; 00=8 n(a;b)=4 n(bç``)=00_n(b) =00_60=40 n(a;bç``)=n(bç``)_n(aç``;bç``) =40_8= n(a)=n(a;b)+n(a;bç``) =4+=46 P(BÇ`` A)= ; 0 0; P(BÇ`` A)= ; 0 0; P(BÇ`` A)=;4@6@;=;!3!; B BÇ`` 3 A B A AÇ`` 3 P(AÇ``)= P(A;BÇ``) P(A) (A) 4 46 C C C 4 ` =;6@4&; (AÇ``) P(A)=_P(AÇ``)=_;6@4&;=;6#4&; A;B 3 3! =3! 3! =3! P = 3+3+ P(A;B)= =;3ª; 4 ` P(B A)= ;3ª; P(B A)= ;6#4&; P(B A)=;3!7*; P(A;B) P(A) 3 3 aabb 3 ;6@;=;3!; aabb a 3!! =;6#;=;!; `` 4! ``!! ;3!;\;!;=;6!; 3 aaab 3 ;6$;=;3@; aaab a 3!! =;4#; 4! 3! ;3@;\;4#;=;!; 65

218 ~ =;4!5);=;9@; 4 ;6!; ;6#;=;!; p p= C {;6!;} 5 {;!;} p= C {;6!;} 5 {;!;} p= 6 ` 6 `p=6 ` =;!; 6 ` A B a B A b a=b= A B ;3@;\;3!;=;9@; A B ;3@;\;3!;=;9@; ;9@;\;9@;=;8 ; a=b= A B 3 37 ;3@;\;3@;=;9$; A B ;3!;\;3@;=;9@; ` 0abc a+b+c=5b+c=6 abc c=3b=0a= aaccc 5! =0!3! c=b=a= abbcc 5! =30!! c=b=4a=0 bbbbc 5! =5 4! ;9$;\;9@;=;8 ; a=b= A B A B a=b= A B ;3!;\;3@;=;9@; A B ;3@;\;3!;=;9@; ` ;9@;\;9@;=;8 ; 66

219 ~ ;8 ;+;8 ;+;8 ;=;8!^; 0 _0 _ 3 A `= ` ` = ` 0 0 A `=A `A _0 _ A `= ` ` 0 0 A `= ` 0 A `=E A m_n =Emn m+n=5 m_n =3 m=n=4 m=4n= m=n=4 C {;3!;} {;3@;} 4 =; 4º3; m=4n= C {;3!;} 4 {;3@;} =; 4º3; ; 4º3;+; 4º3;=;ª4º3;=;!7); p+q=7+0= P(<X 3)=P(X=)+P(X=3) P(<X 3)= C {;!;} {;!;} + C {;!;} 3 {;!;} 0 P(<X 3)=;8#;+;8!;=;!; k+ ( )= 3 k+=;7!; ` k=_;7^; 3 ;6!;+;6!;+a+a= 3a=;3@; ` a=;9@; X X 0 3 P(X=) ;6!; ;6!; ;9@; ;9$; E(X)=0 ;6!;+ ;6!;+ ;9@;+3 ;9$; E(X)=;6!;+;9$;+: 9 :=;#8%; 4 XX=X= C C P(X=)= =;6#;=;!; C C Cº P(X=)= =;6#;=;!; C 67

220 E(X)= ;!;+ ;!;=;#; E(X `)= ` ;!;+ ` ;!;=;%; V(X)=E(X `)_{E(X)} ` V(X)=;%;_;4(;=;4!; r(x)=? V(X)= ;4!; =;!; 5 Y=X+3E(X)=5V(X)=3 E(Y)=E(X+3)=E(X)+3 = 5+3=3 V(Y)=V(X+3)= ` V(X) =4 3= E(Y)+V(Y)=3+=5 6 V(X)=E(X `)_{E(X)} ` =0_9= r(x)=? V(X)= r(5x+)= 5 r(x)= X0X C P(X=0)= =;5!; C C C P(X=)= =;5#; C C P(X=)= =;5!; C X X P(X=) 0 ;5!; P(Xæ)=P(X=)+P(X=) P(Xæ)=;5#;+;5!;=;5$; 5 ;5#; C =0 X3456 X ;5!; 7 E(X)=np=0 r(x)=? np(_p)=4 0(_p)=6 p=;5!; np=;5!;n=0 X P(X=) ; 0; ;5!; ;5@; ;5!; X `_6X+8>0 (X_)(X_4)>0 X< X>4 6 ; 0; n=00 P(X `_6X+8>0)=P(X< X>4) P(X `_6X+8>0)=P(X<)+P(X>4) P(X `_6X+8>0)=P(X=5)+P(X=6) 8 XB{80;3!;} E(X)=80 ;3!;=60 P(X `_6X+8>0)=;5!;+; 0;=; 0; E(X+)=E(X)+ = 60+= 3 E(X)=6V(X)=9 E(X)= ;3!;+4 ;6!;+a ;4!;+b ;4!;=6 68

221 a+b=0 V(X)={ ` ;3!;+4 ` ;6!;+a ` ;4!;+b ` ;4!;}_6 `=9 a `+b `=08 (a+b) `_(a `+b `)=ab (a+b) `_(a `+b `) ab= 0 `_08 ab= ab=96 E(X)= ;!;+ ;3!;+3 ;6!;=;3%; E(X `)= ` ;!;+ ` ;3!;+3 ` ;6!;=: 3º: V(X)=E(X `)_{E(X)} ` V(X)=: 3º:_: 9 :=;9%; V(3X+4)=3 ` V(X) V(3X+4)=9 ;9%;=5 4 X= 3 P(X=)= =; 0; C X= C C = C C =3 C C + C C = P(X=)= =;!0#; C X=3 C C C P(X=3)= =; 0; C ;3@; XB{90;3@;} E(X)=90 ;3@;=60 V(X)=90 ;3@; ;3!;=0 E(X `)=V(X)+{E(X)} ` E(X `)=0+3600=360 8 E(X)=80 ;4!;=0=m V(X)=80 ;4!; ;4#;=5 E(X)= ; 0;+ ;!0#;+3 ; 0; E(X)=;$0%;=;4(; 5 E(X+3)=E(X)+3=E(X)=4 V(X+3)= ` V(X)=36V(X)=9 E(X `)=V(X)+{E(X)} ` =9+4 `=5 80 (_0) ` P(X=) =0 80 = (_m) ` P(X=) =0 =V(X)=5 3 6 X X 3 P(X=) ;!; ;3!; ;6!; P(0 X )=P(X=0)+P(X=)+P(X=) 3+a P(0 X )=;8!;+ +;8!; 8 P(0 X )= 5+a 8 =;8&; 69

222 a= X X P(X=) _ 0 ;8!; ;8!; ;8%; ;8!; E(X)=_ ;8!;+0 ;8!;+ ;8%;+ ;8!; E(X)=;4#; X X P(X=) 0 3 ; ; ; ; ;4!; ; ; XE(X) E(X)=0 ; ;+ ; ;+ ;4!;+3 ; ; E(X)=;@*;=;3&; ;4!;+a+a=3a=;4#; X X 4 5 a=;4!; P(X=) ;3!; ;6!; ;6!; ;3!; E(X)=0 ;4!;+ ;4!;+ ;4@;=;4%; E(4X+0)=4E(X)+0 E(4X+0)=4 ;4%;+0=5 3 X B(00p)E(X)=40 00 p=40 p=;5!; V(X)=00 ;5!; ;5$;=3 E(X)= ;3!;+ ;6!;+4 ;6!;+5 ;3!;=3 E(X `)= ` ;3!;+ ` ;6!;+4 ` ;6!;+5 ` ;3!;= V(X)=E(X `)_{E(X)} `=_9=3 r(x)=? V(X)=/3 3 E(Y)=E(X_)=E(X)_=9 E(X)=5 V(Y)=V(X_)=4V(X)= V(X)=3 E(X `)=V(X)+{E(X)} ` =3+5 `=8 4 a+;4!;+b= a+b=;4#; yy P(X=0)+P(X=)+P(X=)+P(X=3) X E(X)=a+ ;4!;+4b= =;k!;+;k!;+;k#;+;k&; a+4b=;#; yy =: k := k= a=;!;b=;4!; 70

223 V(X)= ` ;!;+ ` ;4!;+4 ` ;4!;_ `=;#; V{;a!;X+b}=V{X+;4!;} V{;a!;X+b}= ` V(X) V{;a!;X+b}=4 ;#;=6 5 XB{9;4!;} r(x)= 9 ;4!; ;4#; =6 r(x+5)=r(x)= `+a+b_=0 D=a `_4(b_)æ0 a `æ4(b_) (ab) ()()(3)(4)(5)(6) 6 ()(3)(4)(5)(6) 5 (33)(43)(53)(63) 4 (44)(54)(64) 3 (45)(55)(65) 3 (56)(66) 3 `+a+b_=0 ;3@6#; 6 X B{6;3@6#;} k P(X=)= /ƒ+ +/ƒ+ k(/ƒ+ _/ƒ+ )` P(X=)= (/ƒ+ +/ƒ+ )(/ƒ+ _/ƒ+ ) P(X=)=k(/ƒ+ _/ƒ+ ) P(X=3)+P(X=4)+P(X=5)+y+P(X=3) =k(/5 _/4 )+k(/6 _/5 )+k(/7 _/6 ) +y+k(/ 5 _/ 4 ) =k(/ 5 _/4 )=k(5_) =3k= k=;3!; P(8 X 4) =P(X=8)+P(X=9)+y+P(X=4) =;3!;(/ 0 _/9 )+;3!;(/ _/ 0 )+y =;3!;(/ 6 _/9 ) +;3!;(/ 6 _/ 5 ) E(X)=6 ;3@6#;=38 3 a+;4!;+;4!;+b= a=_b+;!; `yy a>00<b<;!; E(X)=0 a+ ;4!;+4 ;4!;+6 b=6b+;#; E(X `)=0 ` a+ ` ;4!;+4 ` ;4!;+6 ` b=36b+5 V(X)=E(X `)_{E(X)} ` V(X)=36b+5_{6b+;#;} V(X)=_36b `+8b+: 4 : V(X)=_36{b_;4!;} +5 V(X)b=;4!; 5 k=;4!;m=5 38 =;3!; km=;4%; 7

224 4 B(80p) P(X=38)= ºC p 38 (_p) 4 P(X=39)= ºC ª p 39 (_p) 4 7P(X=38)=3 P(X=39) 7 ºC p 38 (_p) 4 =3 ºC ª p 39 (_p) 4 80! 80! 7 (_p)=3 p 38!4! 39!4! ;X3+!{(4+0) P(X=)} =4 P(X=)+8 P(X=)+3 P(X=3) =4 ;ª0;+8 ;ª0;+3 ; 0; =: º0º:=5 ;6!;(_p)=;3!;p p=;3!; E(X)=80 ;3!;=: 3º: E(6X_0)=6E(X)_0 E(6X_0)=6 : 3º:_0 E(6X_0)= m6 n (mn) (05)(4)(3)(3)(4)(50) X= m_n X P(X=)= C {;3@;} {;3!;} 3 + C {;3@;} 3 {;3!;} P(X=)=; 4º3;+; 4º3;=;8$); P(X=3)= C {;3@;} {;3!;} 4 + C {;3@;} 4 {;3!;} P(X=3)=; 4º3;+; 4º3;=;!7); P(X=5)= Cº {;3@;} 0 {;3!;} 5 + C {;3@;} 5 {;3!;} 0 P(X=5)=;4!3;+; 4 3;=;8!!; X P(X=)= C C 0 (=03) X X P(X=) 0 ; 0; ;ª0; ;ª0; 3 ; 0; E(X)= ;8$);+3 ;!7);+5 ;8!!; E(X)= E(X)=: 8 : ;X3+!{(4+0) P(X=)} =;X3+){(4+0) P(X=)}_0 P(X=0) =4 ;X3+) P(X=)+0 ;X3+) P(X=)_0 ; 0; 3 A B 6! =0 3!3! X=0 A B =4{0 ; 0;+ ;ª0;+ ;ª0;+3 ; 0;}+0_ P(X=0)=; 0;=; 0; =6+0_=5 { ;X3+) P(X=)=} 5 X= SQ P(X=)=; 0;=; 0; 7

225 X= PQPSQRSR! 4 P(X=)= =; 0;=;5@; 0 X=3 A` `P` `Q` `R` `B A` `P` `S` `R` `B!! P(X=3)= =; 0;=;5@; 0 E(X)=0 ; 0;+ ; 0;+ ;5@;+3 ;5@; E(X)=;@0!; :!4``f() d= :!4``f() d=:!4``k d=[;k; `]4! :!3``f() d=8k_;!;k :!3``f() d=: : k= = = =; 3; 50 XX B{50; 3;} E(X)=50 ; 3;=60 60 k=; 5; :_!` f() d= :_!`{a `+;3!;} d=:)``{a `+;3!;} d :_!`{a `+;3!;} d=[;3a; `+;3 ;]) :_!`{a `+;3!;} d=: 3;Å:+;3@;= a=;!; P{0 X ;!;}=:) ;!;`{;!; `+;3!;} d ;!; P{0 X ;!;}=[;6!; `+;3 ;]) P{0 X ;!;}=;4 8;+;6!;=; 6; 3 E(X)=:)`` f() d E(X)=:)``{ ;8#; `} d E(X)=:)``;8#; ` d E(X)=[;3 ; `])=;#; 73

226 4 E(X)=:)`` f() d E(X)=:)``(_+) d E(X)=:)``(_ `+) d E(X)=[_;3@; `+ `])=;3!; E(X `)=:)`` ` f() d E(X `)=:)`` `(_+) d E(X `)=:)``(_ `+ `) d E(X `)=[_;!; `+;3@; `]) E(X `)=_;!;+;3@;=;6!; V(X)=E(X `)_{E(X)} ` V(X)=;6!;_{;3!;} =;6!;_;9!;=; 8; V(3X+)=3 ` V(X) V(3X+)=9 ; 8;=;!; 5 X N(03 `) X y=f()=0 7 XB{600;5@;} E(X)=m=600 ;5@;=40 r `=600 ;5@; ;5#;=44 n=600 X N(40 `) 8_40 5_40 P(8 X 5)=P{ Z } P(8 X 5)=P(_ Z ) P(8 X 5)=P(0 Z ) P(8 X 5)= 0.343= XX B{00;!;} m=00 ;!;=50r `=00 ;!; ;!;=5 n=00 X N(505 `) 45_50 P(X 45)=P{Z } 5 P(Xæ45)=P(Z _) P(Xæ45)=0.5_P(0 Z ) P(Xæ45)=0.5_0.343 P(Xæ45)= P(X 6)=P(Xæ4) a= _50 X_50 60_50 6 P(30 X 60)=P{ } P(30 X 60)=P(_ Z ) P(30 X 60)=P(0 Z )+P(0 Z ) P(30 X 60)= P(30 X 60)=0.885 :)``f() d= :)``a(_) d=:)``(a `_a) d :)``a(_) d=[;3a; `_a `]) :)``a(_) d=;3*;a_4a=_;3$;a= 74

227 a=_;4#; ka `= yy P{;!; X ;#;}=: ;!; ;#; f() d P{;!; X ;#;}=: ;!; ;#; [_;4#;(_)] d P{;!; X ;#;}=_;4#;: ;!; ;#; ( `_) d a=;!;k=8 P{;4!; X ;3!;}=: ;4!; ;3!; f() d P{;4!; X ;3!;}=: ;4!; ;3!; 8 d P{;!; X ;#;}=_;4#;[;3!; `_ `] ;!; ;#; P{;4!; X ;3!;}=[4 `] ;4!; ;3!; P{;!; X ;#;}=_;4#;[;8(;_;4(;_{; 4;_;4!;}] P{;!; X ;#;}=_;4#; {_;!!;}=;!6!; P{;4!; X ;3!;}=4{;9!;_; 6;} P{;4!; X ;3!;}=;3 6; _;B; +b (0 ) `f()= a_a ( 3) 4 p : f() d=: a sin d 0p 0 p p : f() d=[_a cos ])=a= 0 :)3``f() d= a=;!; ;!; b+;!; a=b+;!;a= a+b= yy P( X )=P( X 3) p E(X)=: ;!; sin d 0 E(X)=;!;[[_ cos ] p p )_: (_cos ) d ] 0 ;!; ;B; =;!;a E(X)=;!;[[_ cos ] p )+[sin ] p ) ] b=a a=;5@;b=;5$; yy E(X)=; ; E(4X_p)=4E(X)_p E(4X_p)=4 ; ;_p=p ab=; 5; f()=;!; sin (0 p) 3 :)à `f() d=:)à `k d=[;k; `]a)=;k;a `= ka `= yy E(X)=:)à ` f() d=:)à `k ` d E(X)=[;3K; `]a)=;3!; ka `=;3!; =; ; E(X)=; ; 5 P(_3 Z 3)=a P(0 Z 3)=;A; P( Z 3)=P(0 Z 3)_P(0 Z ) P( Z 3)=;A;_P(0 Z ) P( Z 3)=b 75

228 6 XX N(4030 `) k k_40 P(Xæk)=P{Zæ } 30 P(0 Z )=;A;_b P( Z æ)=p(z _)+P(Zæ) P( Z æ)=p(zæ) P( Z æ)={0.5_p(0 Z )} P( Z æ)=[0.5_{;a;_b}] P( Z æ)=_a+b P(Xæk)=; 5º0º0;=0.6 k_40 P{0 Z }= P(0 Z )=0.34 k_40 30 k=70 = ` XX B{50;5#;} E(X)=50 ;5#;= XX B{400;5!;} E(X)=400 ;5!;=80 V(X)=400 ;5!; ;5$;=64 400X N(808 `) 68_80 P(Xæ68)=P{Zæ } 8 P(Xæ68)=P(Zæ_.5) P(Xæ68)=0.5+P(0 Z.5) P(Xæ68)= =0.93 p= p= ;!; a+;!; 3 3a= a=;5!; (0){4;5#;} V(X)=50 ;5#; ;5@;=36 50 X N(906 `) P(Xæn)=P{Zæ P(0 Z.5)=0.43 P(Zæ_.5)=0.93 n_90 6 n 8 _.5 n_90 6 n8 }æ0.93 y=;5!;(_)=;5!; P(0 X )=;!; ;5!;+;!; ;5!;=;5!; 00P(0 X )=00 ;5!;=0 :)``f() d= yy :)`` f() d=;4!; yy 0=:)``(a+5) f() d 0 76

229 0=:)``a f() d+:)``5 f() d 0=a:)`` f() d+5:)``f() d 0=;4!;a+5 () ;4!;a=5 a=0 :)``(a+5) f() d=e(ax+5) 0 :)``;6K;(6_) d=;6k;[6_;!; `]) :)``;6K;(6_) d=;3%;k= k=;5#; f()=; 0;(6_) X P{;6%; ;3%;}=P{;6%;k X ;3%;k} k P{;6%; ;3%;}=P{;!; X } P{;6%; ;3%;}=: ;!;` ``; 0;(6_) d :)``(a+5) f() d=ae(x)+5 ;4A;+5=0`{ E(X)=;4!;} a=0 P{;6%; P{;6%; P{;6%; ;3%;}=; 0;[6_;!; `] ;!; ;3%;}=; 0;[{6_;!;}_{3_;8!;}] ;3%;}=;8@0!; 3 XX N( `) 000_740 P(Xæ000)=P{Zæ } 500 P(Xæ000)=P(Zæ0.5) =0.3 f()f()=m`(m>0) :)à `f() d= ma ` :)à `m d=[;;;m; `]a)= = ma `= `yy 000`m 000`m =;8#0%;=; 6; E(X)=:)à ` f() d E(X)=:)à `m ` d E(X)=[;;3;M; `]a)= E(X)=;3@;a () V(X)=:)à ` ` f() d_{e(x)} ` V(X)=:)à `m ` d_{;3@;a} V(X)=[ V(X)= m ` 4 ma ` 4 ]a)_;9$;a ` _;9$; a ` ma ` 3 V(X)=;!;a `_;9$;a ` () :)``f() d= V(X)=; 8; a `=8 77

230 a= ( a>0) E(X)=;3@;a=8 E(X+3)=E(X)+3 = 8+3 =9 =P(0 Z )+P(0 Z 0.5) = = X Y N(05 `)N(00 `) a_0 5_0 P(a X 5)=P{ Z } 5 5 a_0 P(a X 5)=P{ Z } 5 0_a P(a X 5)=P{0 Z }+P(0 Z ) 5 ( a<0) yy 0_0 b_0 P(0 Y b)=p{ Z } 0 0 b_0 P(0 Y b)=p{_ Z } 0 b_0 P(0 Y b)=p{0 Z }+P(0 Z ) 0 ( b>0) yy 0_a 5 = 0_a=b_0 a+b=40 b_ A XX B{600; 0;} E(X)=600 ; 0;=60 V(X)=600 ; 0; ;ª0;= X N(60 `) P(48 X 66) 48_60 66_60 =P{ Z } =P(_ Z 0.5) 3 4 :! e ` f() d= :! e ` f() d=:! e ` a d :! e ` f() d=;a;[ln ] ` :! e ` f() d=a= E(X)=:! e ` f() d E(X)=:! e ` { } d e ` E(X)=[;!;] ` E(X)=;!;(e `_) e ` E(X )=0E(X )=0 E(3X +5)=3E(X )+5=35 E(X _5)=E(X )_5=35 E(3X +5)=E(X _5) () r r y=f()y=»() r(x )<r(x ) () X X 0_0 0_0 P(0 X 0)=P{ Z } r(x ) r(x ) 0 P(0 X 0)=P{0 Z } yy r(x ) 78

231 0_0 30_0 P(0 X 30)=P{ Z } r(x ) r(x ) 0 P(0 X 30)=P{0 Z } yy r(x ) r(x )<r(x ) 0 0 < r(x ) r(x ) P(0 X 0)<P(0 X 30) () 3 V(X+)=4V(X)=00 V(X)=5 ` r=5 =m m= =63 58_63 73_63 P(58 X 73)=P{ Z } 5 5 P(58 X 73)=P(_ Z ) P(58 X 73)=P(0 Z )+P(0 Z ) P(58 X 73)= = XX N(480 `) 60_48 P(Xæ60)=P{Zæ } 0 P(Xæ60)=P(Zæ0.6) P(Xæ60)=0.5_P(0 Z 0.6) P(Xæ60)=0.5_0.3 = =648() 3 4 4t `+4Xt+X+=0 D ;;4;D;=4X `_4(X+)æ0 X `_X_=(X_)(X+)æ0 X _ Xæ 0 X 4 P( X 4)=:@4``f() d P( X 4)=:@4``;4!; d P( X 4)=[;4!; ]4@ P( X 4)=;4!;(4_) P( X 4)=;!; XX N(78 `) P(Xæ88 Xæ80)= `P{Zæ54} 88_7 P(Xæ88 Xæ80)= 8 `P{Zæ54} 80_7 8 P(Zæ) P(Xæ88 Xæ80)= P(Zæ) 0.5_P(0 Z ) P(Xæ88 Xæ80)= 0.5_P(0 Z ) 0.5_0.48 P(Xæ88 Xæ80)= 0.5_ P(Xæ88 Xæ80)= 0.6 P(Xæ88 Xæ80)=;8!; P(Xæ88) P(Xæ80) 3 XX N(000 `) X<90 X>0 P(X<90)+P(X>0) 90_00 0_00 =P{Z< }+P{Z> } 0 0 =P(Z<_)+P(Z>) ={0.5_P(0 Z )} 79

232 =(0.5_0.34) =0.3 C (0.3) `=(0.3) ` 4 X.04 0fl`_.4 0fl` P(X<.04 0fl`)=P{Z< } 5 0fi` P(X<.04 0fl`)=P(Z<.8) P(X<.04 0fl`)=0.5+P(0 Z<.8) P(X<.04 0fl`)= = fl` Y Y B(6000.9) E(Y)= =440 V(Y)= =44 Y N(440 `) 458_440 P(Y 458)=P{Z } P(Y 458)=P(Z.5) P(Y 458)=0.5+P(0 Z.5) P(Y 458)= = m=e(x)=\;4!;+\;!;+3\;4!;= r `=V(X)= `\;4!;+ `\;!;+3 `\;4!;_ `=;!; ;!; r ` E(X )=m=v(x )= = =;8!; n 4 E(X )\V(X )=\;8!;=;4!; mr E(X )=m=40 r ` 0 ` V(X )= = = n 00 E(X )+00V(X )=40+00\=40 3 mrm=40r=0 n=00 r 0 E(X )=m=40r(x )= = = /ßn / 00 X N(40 `). 38_40 4_40 P(38 X 4)=P{ Z } P(38 X 4)=P(_ Z ) =P(_ Z 0)+P(0 Z ) =P(0 Z )+P(0 Z ) = = =4r=n=64 P( Z.96)=0.95 m 95`% [4_.96\ 4+.96\ ] / 64 / 64 80

233 3.5, p^ V(p^) 0.8\0. V(p^)= = _ p=0.5 n=400 E(p^)= \0.5 V(p^)= =;6 00; 400 n=400 p^ N{0.5 }. 40 ` p^_0.5 Z= Z ;4 0; N(0) p^ _0.5 P(p^æ0.45)=P Zæ ª º ;4 0; P(p^æ0.45)=P(Zæ_) P(p^æ0.45)=0.5+P(0 Z ) P(p^æ0.45)= = n=00p^=0. P( Z.96)=0.95 p 95`% 0.\0.9 0.\0.9 [0._ ] _ a=0.04b=0.588 b_a= n=56p^=0. P( Z.58)=0.99 p 99`% 0.\0.8 0.\0.8 [0._ ] r r(x )= =;#; / 40 r=3/ 0 V(X)=r `=90 V(X)= ` V(X) =4\90 =360 ;5!;+3a+5a= a=; 0; E(X)=_\;5!;+0\; 0;+\; 0;=; 0; E(X )=; 0; E(0X +3)=0 E(X )+3=6 3 mrm=68r=5 n=00 E(X )=m=68 r 5 r(x )= = =0.5 /ßn / 00 X N(680.5 `) 67_68 P(X æ67)=p{zæ } 0.5 P(X æ67)=p(zæ_) P(X æ67)=0.5+p(0 Z ) 8

234 = = E(X )=00V(X )= X n 6 X _00 N{00 }Z= Z n ``4`` /n N(0,) P(X 88)=P{Z P(X æ0)=p ª Zæ }=P(Z _3) º =P{Zæ /n } P(X 88)=P(X æ0) /n 3= /n=6 n=36 88_00 4 0_00 ``4`` /n 5 =050r=40n=00 m 99`% _.58 m / 00 / m m04004y060 6 m99`% [ _ ] /ßn /ßn 00 b_a=\.58\ 0 /ßn /ßn æ5.8 næ n 666 pq 0.6\0.4 7 E(p^)=p=0.6V(p^)= = = n 600 n=600 p^ N( ) p^_p p^_0.6 p^_0.6 Z= = = `` pq `` 0.6\ ` ` `` n `` 600 N(0) P(0.58 p^ 0.64) 0.58_ _0.6 =P{ Z } =P(_ Z ) =P(0 Z )+P(0 Z ) = = p^=;3@0@0%;=0.75 p 95`% 0.75\ \0.5 [0.75_ ] _ X N(m4 `). a_m P(m X a)=p{0 Z }= a_m = 4 a=m+4yy 6 X N(m `) P(X æa_) a m =P{Zæ } =P(Zæ)`() =0.5_0.477 =0.08 8

235 X N(m30 `) 9 X 30 E(X )=mr(x )= =0 /ß9 X N(m0 `) G(k)=P(X m+30k) m+30k_m G(k)=P{Z } 30 G(k)=P(Z k) H(k)=P(X æm_30k) m_30k_m H(k)=P{Zæ } 0 H(k)=P(Zæ_3k) G(0)=P(Z 0)=0.5 H(0)=P(Zæ0)=0.5 G(0)=H(0) G(3)=P(Z 3)=0.5+P(0 Z 3) H()=P(Zæ_3) =P(_3 Z 0)+0.5 =P(0 Z 3)+0.5 G(3)=H() G()=P(Z )H(_)=P(Zæ3) G()+H(_)=P(Z )+P(Zæ3) =_P( Z 3) P( Z 3)>0 G()+H(_)<. 3 XX N(mr `) 6 =.34 95`%m r r.34_.96 m /ß 6 /ß 6.36 m a r.96 =.34_.36 /ß rr=0 X r 0 r(x )= = /ßn /ßn /ßn 5 /n 0 5 n 400 n 400_5+=376 E(X )=E(X)=0 V(X) V(X )= =: º5º:=4 n E(X `)=V(X )+{E(X )} =4+400= X N{88 } n P(87 X 89)æ0.97 P ª P{0 Z Z }æ0.97 º æ0.97 P(0 Z z)=0.485 z.7 /n 4 87_88 ``4`` /ßn æ.7 /ßn æ8.68 /n 4 næ _88 ``4`` /ßn n r=a= ;4@;=3.3 a+r=3.3+=5.3 4 r=5 n P( Z.96)=0.95 m 95`% 83

236 5 5 [ _.96\ +.96\ ] /ßn /ßn 5 \.96\ =5.98_4.0=.96 /ßn /ßn=0 n= X X 3 4 P(X=) ;3!; ;6!; ;3!; ;6!; m r ` m=\;3!;+\;6!;+3\;3!;+4\;6!;=;3&; r `= `\;3!;+ `\;6!;+3 `\;3!;+4 `\;6!;_{;3&;}` r `=: 3º:_: 9ª:=: 9 : r ` V(X )= =;!;\: 9 :=;!8!; V(6X _)=36V(X ) =0.5_0.343 = n 5 5 P{X _3\ m X +3\ }=0.99 /ßn /ßn 5 5 P{_3\ m_x 3\ }=0.99 /ßn /ßn 5 P{ m_x 3\ }=0.99 /ßn m X \ 0.75=;4#; ;4!; /ßn /ßn /n æ0 næ400 k n=900p^=;9#0@0$;=0.36q^=_p^=0.64 p 95`% 0.36\ \0.64 [0.36_ ] _ a=0.38b= (b_a)=000\0.064=64 V(6X _)=36 ;!8!;= X X N( `) 8 X 4.5 E(X )=4.5r(X )= =0.5 / 8 X N( `) P(X æ5)=p{zæ 5_ P(X æ5)=0.5_p(0 Z ) }=P(Zæ) 3 4 X X X X X

237 X X P(X=) ;8!; ;8!; ;4!; ;!; P(3 X 6) =_{P(X =)+P(X =7)+P(X =8)} =_[;8!;\;8!;+{;4!;\;!;+;!;\;4!;}+;!;\;!;] =_{;6 4;+;4!;+;4!;} =_;6#4#;=;6#4!; X X N(604 `) 5 X 4 ` E(X )=60V(X )= 5 4 ` X N{60 } `g 550 X =6 5 6_60 P(X 6)=P Z ª º ;5$; P(X 6)=P(Z.5) =0.5+P(0 Z.5) = = r(y)=r(4x )=4r(X )=8 Y N(4008 `) P(Y<384)+P(Y>46) 384_400 46_400 =P{Z< }+P(Z> } 8 8 =P(Z<_)+P(Z>) =P(Z>) ={0.5_P(0 Z )} =(0.5_0.48) = = \0.04=000() 4 p 99`% p^q^ p^q^ [p^_.58 æ p^+.58 æ ] n n n=400p^=0.64q^= \0.36 p^_p \0.6 p^_p =.58\ 0 p^_p =0.069 p^_p XX N(004 `) 4 X 4 ` E(X )=00V(X )= =4 4 X N(00 `) 4X Y=4X E(Y)=E(4X )=4E(X )=400 85

238 `f()=cos ` +cos sin +;!; `f()= f() F() : ;4 ;` f(t) dt=[f(t)];4 ;` =F()_F{;4 ;} 6 lim ` `;4 ; 6 `_p ` : f(t) dt ;4 ;` F()_F{;4 ;} = lim ` `;4 ; `_{;4 ;}` +;!; sin +;!; `f()=;!;æ(sin +cos )+ `f()= +cos / sin {+;4 ;}+ _ sin {+;4 ;} / / _ + f() + / / M= +m=_ + M `_m `=(M+m)(M_m)=/ ( F()_F{;4 ;} ) = lim { } ` `;4 ; _;4 ; +;4 ; 9 0 a+ 3 A= ` a+ 4 AB A'B' A(4)B(4) S 4 S=OAA'+:!4`` d _OBB' S=;!; 4+:!4`` S=:!4`` 4 d S=[4 ln ]4! S=8 ln b A b+ (a+)(b+)_b(a+)=a+b++0 ( a 4 b 4) A A ` b+ A `= ` a+b+ _a_ S= a+b+ P(ab)y= a 4 b 4 ab=4 4 _b S a+ d_;!; 4 `( 4) a+b æ/ ab= ( a=b=) S= =;3!; a+b+ 4+ S ;3!;æ 4 y y=4 A OA' 4 y=;:; B B' y=;4; =F'{;4 ;} =f {;4 ;} ={;!;+;!;+;!;} #; 5 a=_ h()=f()»()=(+){_ } ` h()=0 lim ` `f()»() _ h()_h() = lim =h'() ` _ 86

239 h'()={_ }+(+){+ } ` ` h'()=0+(+)=6 `f()»() lim =6 ` _ 8 6 6\5\4 C = =0 3\\ a 6 { f()} ` {»()} `=(+) ` {+ }4` ` a { f()} ` {»()} `=( `++){+ }4` yy ` a {+ }4` ` C r 4_r a { }r``= C r a r 4_3r yy ` r=3 ` ` fi` C a `=4a ` 4a ` ` 4a `=08a `=7 a=3 _ 0 7 ` 0 ;!;æ ` _/3 _;!;æ ` _/3 _;!;æ ` /3 /3 ` _;!; _:"": /3 = /3 ` :"": _;!; cos 0 = ` sin 0 _sin 0 cos 0 0 f f _ BE»» _ EC» _ Á`f _ B C 3 /3 ;!;\\\sin 0 = 4 6 /3 P{X= }=; 0;=; 0; 4 /3 ;!;\\\sin 60 = /3 P{X= }=;!0@;=;5#; 3 /3 4 (/3 ) `= 3/3 4 3/3 P{X= }=; 0;=; 0; 4 /3 /3 3/3 E(X)= \; 0;+ \;5#;+ \; 0; 4 4 8/3 9/3 E(X)= = 40 0 E(0X+/3 )=0E(X)+/3 9/3 E(0X_/3 )=0\ +/3 0 E(0X_/3 )=0/3 87

240 memo