i i

Transcription

1

2 i i

3 Level - Level - Level 3

4 0. F(x) f(x) F'(x)=f(x)F(x) f(x) : f(x)dx : f(x)dx=f(x)+c F'(x)=f(x) : f(x)dx=f(x)+c C f(x) F(x) G(x) F'(x)=f(x)G'(x)=f(x) {G(x)-F(x)}'=G'(x)-F'(x)=f(x)-f(x)=0 0 G(x)-F(x)=C C G(x)=F(x)+C f(x) F(x) f(x) F(x)+C C C 0 : f(x)dx=x +x +x+c f(x) C 0 : (x+a)dx=bx +3x+C aba+b C 03 f(x) : (3x -x+)dx=f(x)+c f'()c

5 y=x«+- : x«dx= x«± +C C + =- : x dx=: ;[!; dx=l x +C C +-{ x«± } ' = (+)x (+)- =x«+ + : x«dx= x«± +C C + =- (l x )'=;[!;=x : x dx=: ;[!; dx=l x +C C : x dx=;3!;x +C: x ;!; dx= 3 x ;!;+ +C=;3@;x ;#; +C C ;!;+ : dx : dx 04 : x dx : x 0 dx 05 : dx : dx x 06 : "çx dx : 'x dx 5

6 0 3. f(x) g(x) : kf(x)dx=k: f(x)dx k : { f(x)+g(x)} dx=: f(x)dx+: g(x)dx : { f(x)-g(x)} dx=: f(x)dx-: g(x)dx f(x) g(x) F(x)G(x) F(x)=: f(x)dxg(x)=: g(x)dx F'(x)=f(x)G'(x)=g(x) {kf(x)}'=kf'(x)=kf(x)k : kf(x)dx=kf(x)=k: f(x)dx k {F(x)+G(x)}'=F'(x)+G'(x)=f(x)+g(x) : { f(x)+g(x)} dx=f(x)+g(x)=: f(x)dx+: g(x)dx {F(x)-G(x)}'=F'(x)-G'(x)=f(x)-g(x) : { f(x)-g(x)} dx=f(x)-g(x)=: f(x)dx-: g(x)dx : 3x dx=3: x dx=3{;3!;x +C }=x +3C 3C C : 3x dx=x +CC 07 x + : (3x +x+)dx : dx x+ 08 x +x+3 : (x-)(x +x+)dx : dx x 09 x : (x+) dx-: (x-) dx : dx-: dx 'x- 'x- 6

7 4. C : si x dx=-cos x+c : sec xdx=ta x+c : sec x ta xdx=sec x+c : cos xdx=si x+c : cosec x dx=-cot x+c : cosec x cot x dx=-cosec x+c (-cos x)'=si x (si x)'=cos x (ta x)'=sec x (-cot x)'=cosec x (sec x)'=sec x ta x (-cosec x)'=cosec x cot x 5. C : e dx=e +C e : a dx= a l a +C a>0a+ (e )'=e e ) (a )'=a l a a>0a+) 0 : ( si x+3 cos x)dx : ta xdx si x : dx : dx si x cos x e e - : (e x+ + x+ )dx : dx e - 7

8 x : (3x+a)dx=0 a -3 -;#; -;3@; ;3@; 3 +- : x«dx= + x«± +C C : (3x+a)dx=;#;x +ax+c C ;#;x +ax+c=0ab a a+b=- 3=- ;#; a 3 a a+b= - = a= f(x)=4x+3 F(x) F(x)=0-F() f(x)=x+ F(x) x F(x)>0 F(0) 0<F(0)<; 6; ; 6;<F(0)<;8!; 0<F(0)<;4!; ;8!;<F(0)<;4!; F(0)>;4!; 8

9 f(x)g(x) ;dî[; { f(x)g(x)}=3x f()=3g()=0 f() f(x)g(x) : [;dî[; f(x)]dx=f(x)+c C ;dî[; { f(x)g(x)}=3x f(x)g(x)=: 3x dx=x +C C x= f()g()=+c f()=3g()=0 3 0=+C C=- f(x)g(x)=x -=(x-)(x +x+) f(x)=x +x+g(x)=x- f()=3g()=0 f()=4++=7 03 f(x)g(x) ;dî[;{ f(x)+g(x)}=6x+7f(0)=g(0)=4 f()+g() f(x)g(x) ;dî[; { f(x)+g(x)}=x+ ;dî[; { f(x)g(x)}=3x -4x+ f(0)=g(0)=-f()

10 3 y=f(x) (xy) x+;[!; y=f(x){;#;}f(e) x>0 e ;!;e + ;!;e + e + e + e + y=f(x) (xy) f'(x) f(x) f(x)=: f'(x)dx y=f(x) (xy) x+;[!; f'(x)=x+;[!; f(x)=: f'(x)dx=: {x+;[!;} dx f(x)=;!;x +l x +C C ) f(x)=;!;x +l x+c x>0) y=f(x){;#;} f()=;!;+c=;#; C= f(x)=;!;x +l x+ f(e)=;!;e +l e+=;!;e + 05 y=f(x) (xy) 'x+ y=f(x){;3@;} 'x f(9)x>0) y=f(x) (xy) ;[K; k 0 y=f(x) () (e3)f('e)x>0 e

11 4 f(x) y=f'(x) f(x) : 3 : f(x) y y=f'(x) ;6!; ;3!; ;!; - O x ;3@; ;6%; f(x) y=f'(x) y=f'(x) f'(x)=a(x+)(x-)a<0 y=f'(x) (0) =a (-) a=- f'(x)=-(x+)(x-) f'(x)=0 x=- x= y=f(x) f(x) x=- x= x f'(x) f(x) y y + 0 y - f(x)=: f'(x)dx=: {-(x+)(x-)}dx=: (-x +x+)dx=-;3!;x +;!;x +x+c C f()=: 3 : f()=-;3*;++4+c=: 3 : C= f(x)=-;3!;x +;!;x +x+ f(x) f(-)=;3!;+;!;-+=;6%; 07 f(x) f'(x)=3(x -)f(x) f(x) 08 f(x) y=f'(x) f(x) 4 0f() y y=f'(x) O x

12 5 x f(x) cos x (x>0) f'(x)=g e (x<0) f(-)=;e!; f{; ;}e : cos xdx=si x+c : e dx=e +C cos x (x>0) f'(x)=g : cos xdx=si x+c : e dx=e +C C C e (x<0) si x+c (x>0) f(x)=g e +C (x<0) f(-)=;e!; e +C =;e!; C =0 f(x) x=0 lim (si x+c )= lim (e +C )C =+C C = x +0 x -0 si x+ (xæ0) f(x)=g f{; ;}=si ; ;+= e (x<0) 09 x f(x) x (x>0) f'(x)=g f()= (x<0) f(-) x f(x) si x (x>0) f'(x)=g f{; ;}=f{-; ;}=- a cos x (x<0) a

13 6 f(x) f(p) a f(x) f'(x)= +a cos x lim =3 x 0 x p + ( p +)l ( p -)l ( p +) ( p -) l l f(x) lim x 0 f(x) x =aa f(0)=0 f'(0)=a lim x 0 f(x) x =3 f(0)=0f'(0)=3 f'(x)= +a cos x f'(0)=+a=3 a= f(x)=: f'(x)dx=: ( + cos x)dx= + si x+c C l f(0)= +C=0 C=l l p f(x)= + si x- f(p)= + si p- = ( p -) l l l l l f(x) f'(x)=ke lim x f(x) x- = f(0) k e ;e!;- ;e@;- 0 -;e!; -;e@; f(x) a+f(0)a f(x) f'(x)=a{si ;{;-cos ;{;} lim =a+ x p x-p p-3 p- p- p p+ 3

14 f '(x) f(a)=b f(x)=: f'(x)dx 0 f(x)x : f(x)dx=x -ax +axf()=0 f(3) a 0 y=f(x)x x (x<) f'(x)=[ f(0)=0 - (x>) f(x) ;!; ;3!; ;4!; ;5!; 03 f(x)x f(x)-: e f(x)dx=f(0)= f"(0)e f"(x) f(x)

15 Level 0 f(x)=: e cos pxdxf'()e ) -e - 0 e 0 f(x)=: (+3x +5x +7xfl +9x )dx f(0)=0f(-) cos x-x f(x)=: dxf{;4 ;}-f{-;4 ;} x cos x x>0 f(x) : xf'(x)dx=x +'x+c C f()= f(4) : : : : : : : : : ª: 5

16 Level 0 f(x)g(x) f()+g() f(x)g'(x)+f'(x)g(x)=3x +0x+8 f(x)=(x+)g(x) g(-)> f'(x)=+cos x f(x) f(0)= y=f(x) {; ;f{; ;}} y=ax+b ab a+b -p p 03 x>0 f(x) F(x) F(x)=xf(x)+si x-x cos xf(p)= f(p) -p - 0 p 04 x>0 f(x)x f(x)+xf'(x)=;[@;+3'xf()=4 f{;e!;} e ;e!; ;e@; e 'e 'e 6

17 Level f(x)=ta ;{;+cot ;{; g(x)=: dxf{; ;}=g{; ;} f(x) g{;3 ;} 0<x<p 0 f(x) y=f'(x) f(x) f(-) y y=f'(x) O x 03 f(x) f'(x)=e + e -e f(0)= f() f(-)e -e -e 0 +e +e 04 f(x)xy f(x+y)=f(x)+f(y)+xy f'(0)= f(0)=0 f'()= f(x) - 7

18 0. g(t) x=g(t) : f(x)dx=: f(g(t))g'(t)dt F(x)=: f(x)dx yy` xt x=g(t) F(x)=F(g(t)) t d d F(x)= F(x) dt dx F(x)=f(x)g'(t) F(x)=f(g(t))g'(t) dx dt t F(x)=: f(g(t))g'(t)dt yy` : f(x)dx=: f(g(t))g'(t)dt x xt t x x : f(g(x))g'(x)dx dt g(x)=tx =g'(x) dx : f(g(x))g'(x)dx=: f(t)dt 0 : 3x (+x ) dx : (x +x) (x+)dx 0 : xe x dx e 8

19 f'(x). : dx f(x) f'(x) : dx=l f(x) +C C f(x) dt f(x)=t x =f'(x) dx f'(x) : dx=: f'(x)dx=: ;t!; dt=l t +C=l f(x) +C C f(x) f(x) 3. : f'(x) f(x) dx=l f(x) +C C = { - }a+b (x+a)(x+b) b-a x+a x+b px+q A B = + AB (x+a)(x+b) x+a x+b px +qx+r A Bx+C = + ABC (x+a)(x +bx+c) x+a x +bx+c 03 x x : dx : dx x + (x +) 04 3x+ : dx : dx x(x+) x - 9

20 0 4. : si«xdx: cos«xdx( : siμ x cos«x dx(m m m : cos xdx=: +cos x dx=;!;x+;4!; si x+c C : cos xdx=: cos x (-si x)dx si x=t x dt =cos x dx : cos xdx=: cos x (-si x)dx : cos xdx=: (-t )dt : cos xdx=-;3!;t +t+c C : cos xdx=-;3!; si x+si x+c : si x cos xdx si x=t x dt =cos x dx : si x cos xdx=: t dt : si x cos xdx=;3!;t +C C : si x cos xdx=;3!; si x+c 05 : si x cos xdx : 8 cos xdx 06 : si xdx 0

21 5. f(x)g(x) : f(x)g'(x)dx=f(x)g(x)-: f'(x)g(x)dx f(x) g(x) { f(x)g(x)}'=f'(x)g(x)+f(x)g'(x) x f(x)g(x)=: { f'(x)g(x)+f(x)g'(x)} dx f(x)g(x)=: f'(x)g(x)dx+: f(x)g'(x)dx : f(x)g'(x)dx=f(x)g(x)-: f'(x)g(x)dx : f(x)g'(x)dx=f(x)g(x)-: f'(x)g(x)dx f(x) g'(x) f(x) g(x) g'(x) e a jk si xcos x jk xx y jk l xlogå x 07 : x cos xdx : xe x dx e 08 : (x l x+)dx

22 f(x)=: x " x + dx f(0)=3f(') dt : f(g(x))g'(x)dx g(x)=t x =g'(x) dx : f(g(x))g'(x)dx=: f(t)dt t g(x) dt x +=t x =x dx x f(x)=: dx " x + f(x) =: dt 't f(x) =t ;!; +C C f(x) =" x ++C f(0)=+c=3 C= f(x)=" x ++ f(')=7 0 f(x)=: x'ƒx+ dx f(0)=; 5;f(3) 0 f(x) d x- f(x)= f(0)=-: 3º: dx 'ƒx+ f(x)=

23 f'(x) : dx f(x) f(x) x f'(x)= f(0)=0 x +3 f(3) ;!; l ;!; l 3 l l 3 f'(x) : dx=l f(x) +C C f(x) x f'(x)= x +3 x x f(x)=: dx=;!;: dx=;!; l x +3 +C C x +3 x +3 x +3>0 f(x)=;!; l(x +3)+C f(0)=;!; l 3+C=0 C=-;!; l 3 f(x)=;!; l(x +3)-;!; l 3 f(3)=;!; l -;!; l 3=l 03 f(x) f'(x) (e +)f'(x)=e f(0)=l f()e l(e+) l(e -) l(e +) 04 0 x<; ; f(x)=: -ta x +ta x dx f{;4 ;}-f(0) -l -;!; l 0 ;!; l l 3

24 3 : 3x-4 x -3x+ dx C l x- - l x- +C l x- +l x- +C l x- + l x- +C l x- -l x- +C l x- +l x- +C px+q A B = + AB (x+a)(x+b) x+a x+b 3x-4 3x-4 a b = = + x -3x+ (x-)(x-) x- x- 3x-4 (a+b)x-a-b = (x-)(x-) (x-)(x-) 3x-4=(a+b)x-a-b a+b=3-a-b=-4 a=b= 3x-4 = + x -3x+ x- x- : 3x-4 dx=: { + } dx x -3x+ x- x- =l x- + l x- +C C 05 f(x)=: x+7 x +x-3 dx f(0)=0f(3) l l 3 l l 5 l 6 x +5x+8 06 f(x)=: dx f(-)=-f(0) x +4x+3 --l 3 -l 3 -+l 3 -l 3 l 3 4

25 4 f(x)=: (-3 cos x)si xdx f{; ;}=0 f(x)=0 p p 3p 4p 5p 0 x<p cos x=t dt cos x=t x =-si x dx y f(x)=: (-3 cos x)si xdx=: (-3t)(-dt) f(x)=: (3t-)dt=;#;t -t+c C O - f(x)=;#; cos x-cos x+c 3 - a p - p y=cos x 3 - p p p-a y= - 3 x f{; ;}=C=0 f(x)=;#; cos x-cos x=cos x{;#;cos x-} f(x)=0 cos x=0 cos x=;3@; cos x=0 x=; ; x=;#;p cos x=;3@; x=a x=p-a`{0<a<; ;} ; ;+;#;p+a+(p-a)=4p 07 f(x)=: (si x+cos x) dx f{; ;}-f(0) ;4!;(p+) ;4!;(p+) ;!;(p+) ;!;(p+) ;!;(p+3) 08 f(x) f'(x)=si x cos xf(0)= f{;3 ;}=;pq;p+q pq 5

26 5 y=f(x) (xy) xe x f() e ;4!;(e+) ;!;(e+) ;4!;(e +) ;!;(e +) e+ y=f(x) (xy) f'(x) : f(x)g'(x)dx=f(x)g(x)-: f'(x)g(x)dx y=f(x) f(0)=0 (xy) xe x f'(x)=xe x f(x)=: xe x dx f(x)=;!;xe x -: ;!;e x dx f(x)=;!;xe x -;4!;e x +C C f(0)=0 f(0)=-;4!;+c=0 C=;4!; f(x)=;!;xe x -;4!;e x +;4!; f()=;4!;(e +) 09 y=f(x) (xy) xe -x (-0) f()e 3 -;e#; - -;e!; - 0 e e 0 f(x) f(x)+xf'(x)=(x+3)e f()=3e f(3) e e ;3%;e ;3&;e ;%;e ;&;e 6

27 6 f(x) f'(x)=(x-)l x ;4#;f(x) -l - l - l -l -l : f(x)g'(x)dx=f(x)g(x)-: f'(x)g(x)dx f'(x)=(x-)l x=0 x= x= f(x) x (0) y y y f'(x) f(x) ;4#; f()=;4#; f(x)=: (x-)l xdx={;!;x -x} l x-: {;!;x -x} ;[!; dx f(x)={;!;x -x} l x-: {;!;x-} dx={;!;x -x} l x-;4!;x +x+c C f()=-;4!;++c=;4#; C=- f(x)={;!;x -x} l x-;4!;x +x- f()=- l -+4-=- l f(x) f'(x)=x cos x0 x p f(x) ; ;f(x) f(x)=: e si xdx f(0)=-;!; f(x)=;!;e 0 x<p e ) p ;#;p p ;%;p 3p 7

28 0 0<x<p f(x)=: cos x(+si x) si x dx f{; ;}=f{;6 ;} - l -l 0 l l 0 -; ;<x<; ; f(x) f'(x)=ta x+ta xf(0)=0 f{;3 ;} '3 '3 ;!; 4 ;#; 03 f(x) f(x+h)-f(x) lim =x e -x f(-)=-e h 0 h f(0)e

29 Level 0 3x f(x) f'(x) f'(x)= f(4)-f(0) 3x + -l 7 -l 6 l l 6 l 7 0 f(x) xf'(x)= l xf()=- f(e )e ) ;!; e e 03 f(x) d {e f(x) }=e f(x) l xf()=3 dx f(e) e 04 f(x) F(x) F(x)=xf(x)-x e F()=0 f()e e -e 3e -e 3e -e 3e + e +e 9

30 Level 0 f(x) g(x) f()e f(x)>0g(x)>0 f(3)= : f(x){ g(x)+g'(x)}dx=f(x)g(x) ;e!; e e e 0 f(x) f(x)=: (x+)" x +x+3 dxf(-)='3 f() 03 xe (xæ0) x f(x) f'(x)=g f{-; ;}=0 f(3) si x (x<0) e ) e -3 e - e e -3 e - 04 f(x) f(x)-f'(x)=x-3f(0)=- g(x)=e -x f(x) g() e ) ;e!; e e e 30

31 Level f(x)g(x) x f(x)>g(x)>0 f'(x)=-g(x)g'(x)=-f(x) f(0)=eg(0)=e e ) f(x)+g(x)=e -x+ { f(x)} -{ g(x)} =3e f()=;!;(e +3) 0 f(x)=: cosfi xdx f(0)= f {; ;}=;pq;p+q pq 03 f(x) f(x)- f'(x)=axe (x-) lim = x x- f()a e ;!;(3e -) ;!;(e -) e ;!;(e +) ;!;(3e +) 04 f(x)=: x si xdx g(x)=: ef(x) x si xdx. f(0)=g(0)=0g{; ;} e e- e- e e+ e+ 3

32 03. y=x xx= [0] [ ][ ] y y=x y y=x SS«<S<T«lim S«S lim T«y O y - x O y - [ ] [ ] [ ][ ] S«T«- (-)(-) (+)(+) S«= ;!;{;K;} = T«= ;!;{;K;} = k= 6 k= 6 y x lim S«= lim T«=;3!; S=;3!; y=x lim S«lim T«0 BC =aab =hb=90 ABCAB BC ABC ABC A h B y a C 0 r h h y r 3

33 f(x) [ab] [ab] x xº(=a)x x yx yx«(=b) b-a [x x ] (k=y) Dx k(b-a) lim ;K+! f(x )Dx {x =a+kdx=a+ } y=f(x) a b :Ab f(x)dx y O x 0 = a y y Dx f(xk ) y=f(x) x x yx k- x k y x = b x :Ab f(x)dx= lim ;K+! ; f(x )Dx a b y=f(x) xx=ax=ba<b S f(x)æ0:ab f(x)dx=sf(x)<0:ab f(x)dx=-s :Ab f(x)dx=:ab f(y)dy=:ab f(s)ds=:ab f(t)dt ak 03 :) xdx= lim ;K+! { };A; a k 04 lim ;K+! {+ } ;@;=:!a x dx a a>

34 03 3. f(x) [ab] F'(x)=f(x) :Ab f(x)dx=[f(x)]ba=f(b)-f(a) f(t)æ0 y=f(t) t=at=x(a x b) t S(x) y y=f(t) S(x)=:A/ f(t)dt Dx>0 f(t) [xx+dx] M m DS(x)=S(x+Dx)-S(x) DS(x) m Dx DS(x) M Dxm M Dx DS(x) lim m lim lim M Dx 0 Dx 0 Dx Dx 0 Dx 0m f(x)m f(x) S'(x)=f(x) d dx :A/ f(t)dt=f(x) S(x) f(x) f(x) F(x) S(x)=:A/ f(t)dt=f(x)+c C x=a C=-F(a) x=b O y O a S(x) x b t y=f(t) DS(x) m M a xx+dx b t :Ab f(t)dt=f(b)-f(a) f(t) 0Dx<0 :Aa f(x)dx=0:ab f(x)dx=-:ba f(x)dx 05 :) (x+'x)dx ;3@; ;6%; ;6&; ;3$; 06 :)» si xdx 07 :) l e dx e 34

35 4. f(x)g(x) [ab] :Ab { f(x)+g(x)} dx=:ab f(x)dx+:ab g(x)dx :Ab { f(x)-g(x)} dx=:ab f(x)dx-:ab g(x)dx :Ab kf(x)dx=k:ab f(x)dx k f(x) abc :Ab f(x)dx=:ac f(x)dx+:cb f(x)dx f(x)g(x) F(x)G(x) {F(x)+G(x)}'=f(x)+g(x) :Ab { f(x)+g(x)} dx=[f(x)+g(x)]ba={f(b)+g(b)}-{f(a)+g(a)}={f(b)-f(a)}+{g(b)-g(a)} :Ab { f(x)+g(x)} dx=:ab f(x)dx+:ab g(x)dx :Ab { f(x)-g(x)} dx=:ab f(x)dx-:ab g(x)dx {kf(x)}'=kf(x) :Ab kf(x)dx=[kf(x)]ba=k{f(b)-f(a)}=k:ab f(x)dx :Ac f(x)dx+:cb f(x)dx=[f(x)]ca+[f(x)]bc={f(c)-f(a)}+{f(b)-f(c)}=f(b)-f(a)=:ab f(x)dx 08 e :) (3x +4 'x)dx :_! (si x-e )dx 09 :! {;[!;+ } dx+:! {;[!;- } dx :) (cos x+ )dx-:) (cos x- )dx x x 0 :_! x dx :) ' ; ; cos xdx+: cos xdx ' 35

36 ABC AB A BC (-) '3 S«S«-lim S«< 50 y B C (+) (+)(+) (+) ;K+! k= ;K+! k = ;K+! k =[ ] 6 '3 '3 BC M AM = _;!; - ;!;;@;y - - '3 k '3 '3(-) '3(-) S«= { _ }= k= = k= k= 4 4 B '3 '3 - '3 '3 lim S«= S«-lim S«= - = _;!;< >: : A y M C 0 BCD S A H AH =h ABCD A kh A k=y AH S V«h h Sh V«=;K+! {S }=;K+! { S }= B y H C D V V=;3!;Sh =0k=5 ;qp;p+q pq 36

37 :!3 (x -4)dx :!3 (x -4)dx= lim ;K+! [{+ k } -4] 4(+)(+) 4(+) = lim [ = ] :Ab f(x)dx= lim ;K+! f(x )Dx :!3 (x -4)dx= lim ;K+! [{+ k } -4] 4k 4k :!3 (x -4) dx= lim ;K+! { + -3} :!3 (x -4) dx= lim { ;K+! k + ;K+! k- ;K+! } 8 (+)(+) 8 (+) 6 :!3 (x -4)dx= lim [ _ + _ - _] 6 4(+)(+) 4(+) :!3 (x -4) dx= lim [ ] 3 :!3 (x -4) dx=;3*;+4-6= ;3@; _6_;3@;=8 8 0 y=f(x) :N «± f(x)dx<0 (x<x>5f(x)>0) y y=f(x) O 3 5 x 03 f(x)=x +x+p a<bab b-a b-a lim ;K+! f{a+ k} >0 p 37

38 3 f(x)=(x +) ax+ :) f'(x)dx=:! f(x)dxa ;3*; 3 : 3º: : 3 : 4 f(x) [ab] F'(x)=f(x) :Ab f(x)dx=[f(x)]ba=f(b)-f(a) f'(x) f(x) :) f'(x)dx=[f(x)]) =[(x +) ax+ ]) = a+ -= a=0 f(x)=x + x :! f(x)dx=:! (x +)dx=[ +x]!={;3*;+}-{;3!;+} =: 3º: 3 04 ; ; :) {cos ;{;+si ;{;}{cos ;{;-si ;{;} dx ;!; ;#; ;%; 05 :) ' 4 dx l l l l l 38

39 4 :_! x -4x dx : 4ª: : 4 : : : : 4 : : : :Ab f(x)dx=:ac f(x)dx+:cb f(x)dx y=x -4x y=x(x+)(x-) y :_! x -4x dx y=x 3-4x =:_0! (x -4x)dx+:) (-x +4x)dx - O x x x =[ -x ]0_!+[- +x ]) 4 4 =;4&;+4=: 4 : 06 ; ; :) cos x-si x dx '- ('-) '- 3('-) 3'- 07 'x+a (x>0) f(x)=g : a f(x)dx l ;!; e (x 0) a e ;4!; ;4#; ;4%; ;4(; : 4 : 39

40 0 f(x)=6x +ax :) f(x)dx=f() a :) x (x-) dx ;#; ;%; 3 ;&; 03 f'(x) y=f(x) f()= f(3)=-3 y y=f(x) f(0)=-3 :)3 f'(x) dx O 3 x

41 Level 0 :_! ( x -)(x + x +)dx -;#; -;4#; 0 ;4#; ;#; 0 :! 4 'x+ dx x +l +l (+l ) +3 l 3(+l ) 03 :)» si ;{;{si ;{;+cos ;{;} dx ; ; ; ;+ p ;#;p+ ;%;p 04 :_! e - dxe ) ;e!;+e- ;e!;+e- ;e!;+e ;e!;+e+ ;e!;+e+ 05 f(x)=e -ax :) f(x)dx=f() ae

42 Level 0 :)» si x+'3 cos x dx f(x) x (x<0) f'(x)=g si x (x>0) f{; ;}=0:_! ; ; f(x)dx - -;3%; -;3$; - -;3@; 03 [00] (00) f(x) f(x)>0f'(x)>0 f"(x)>0 0 A= f(k)b=;!; f{;k;}c=:) 0 f(x)dx k= 0 k= ABC A<B<C B<A<C B<C<A C<A<B C<B<A 04 f(a)=f(b)=f(c)=0 :Ab f(x)dx=3:ac f(x)dx=0 y y=f(x) y=f(x) f(x) F(x) F(x)=4 F(a) O a b c x a<b<c)

43 Level ;3 ; AOB A y B S«S _ lim S«p p 8p si 6p si 8p si p 6p si 8p si p 39 4 p 39 p - 3 O 0 :) e dx :) e dx= lim ;K+!e ;K; ;!; ;K+!e ;K; ;!;= lim (-e) (- ) = (- ) :) e dx= f() a b f()+a+be e- (e-) e+ e (e+) 03 f(x)=si x{0 x ; ;} y=g(x). [0] x x y=g(x) S«lim S«y p - y=g(x) ; ;- p- p... ; ;+ p+ O x 43

44 04. [ab] f(x) [ab] x=g(t) g'(t) [ab] a=g(a)b=g(b) :Ab f(x)dx=:ú f(g(t))g'(t)dt : f(x)dx x=g(t) : f(x)dx=: f(g(t))g'(t)dt F(x)=: f(x)dx :Ab f(x)dx=f(b)-f(a) yy x=g(t) a=g(a)b=g(b) :Ú f(g(t))g'(t)dt=[f(g(t))] Ú =F(g(b))-F(g(a)) =F(b)-F(a) yy :Ab f(x)dx=:ú f(g(t))g'(t)dt :Ab " c -x dx x=c si h {-; ; h ; ;} :Ab " c +x dx x=c ta h {-; ;<h<; ;} 0 e :) (x+) dx :) e- dx x+ 0 :) 'p x si x dx :) ;4 ; ta xdx :) " -x dx 03 e :) "çl 3 xe x l x dx :!e dx x 44

45 f(x)g(x) f '(x)g'(x) :Ab f(x)g'(x)dx=[ f(x)g(x)]ba-:ab f '(x)g(x)dx f(x) g(x) { f(x)g(x)}'=f '(x)g(x)+f(x)g'(x) :Ab { f(x)g(x)}' dx=:ab { f '(x)g(x)+f(x)g'(x)} dx [ f(x)g(x)]ba=:ab f '(x)g(x)dx+:ab f(x)g'(x)dx :Ab f(x)g'(x)dx=[ f(x)g(x)]ba-:ab f '(x)g(x)dx f(x) g'(x) 04 :) ; ; x si xdx :)» x cos xdx 05 :) xe dx e 06 e :!e l xdx :!e x l xdx 45

46 04 3. f(x) d d :A/ f(t)dt=f(x) :? x+a f(t)dt=f(x+a)-f(x) dx dx f(x) :A x+a lim f(t)dt=f(a) lim :A/ f(t)dt=f(a) x 0 x x a x-a F'(x)=f(x) d d :A/ f(t)dt= [F(t)]/A dx dx d = {F(x)-F(a)}=f(x) dx d :? x+a d f(t)dt= [F(t)]? x+a dx dx d = {F(x+a)-F(x)}=f(x+a)-f(x) dx F'(x)=f(x) :A x+a lim f(t)dt= [F(t)]A x+a x 0 lim x x 0 x F(x+a)-F(a) = lim x 0 =F'(a)=f(a) x lim x a :A/ f(t)dt= lim x a [F(t)]/A x-a x-a F(x)-F(a) = lim x a =F'(a)=f(a) x-a 07 F(x)=:)/ (t -)dtf'() x+; ; F(x)=:? si tdtf'(0) 08 e :! x+ lim " t +3 dt lim :E/ l tdt x 0 x x e x-e 46

47 4. f(x) [ab] lim k= b-a b-a f {a+ k} =:Ab f(x)dx =:) b-a f(x+a)dx =(b-a):) f((b-a)x+a)dx b-a b-a x =a+ kdx= lim k= b-a b-a f {a+ k} = lim f(x )Dx k= =:Ab f(x)dx 5. f(x)x x f(-x)=f(x) :_aa f(x)dx=:)a f(x)dx x f(-x)=-f(x) :_aa f(x)dx=0 :_aa f(x)dx=:_0a f(x)dx+:)a f(x)dx :_0A f(x)dx -x=t dt =-x=-at=ax=0t=0 f(-x)=f(x) dx :_0A f(x)dx=-:a0 f(-t)dt=:)a f(-t)dt=:)a f(-x)dx=:)a f(x)dx :_aa f(x)dx=:_0a f(x)dx+:)a f(x)dx=:)a f(x)dx 09 k 3k lim { }7 lim æ + 3 k= k= 0 :_!(x +x +x)dx : ;6 ; -;6 ; (si x+cos x+ta x)dx 47

48 :) ' x x - fi dx ;6!; ;3!; ;!; ;3@; ;6%; :Ab f(g(x))g'(x)dx g(x)=t g'(x) dx dt =g(a)=ag(b)=b :Ab f(g(x))g'(x)dx=:ú f(t)dt dx x -=t x = x=0t=-x='t= dt :) ' x x - fi dx=:_! ;!; t fi dt :) ' x x - fi dx=:_0! {-;!;tfi } dt+:) ;!;tfi dt :) ' x x - fi dx=[-; ;tfl ]0_!+[; ;tfl ]) :) ' x x - fi dx=; ;+; ;=;6!; 0 :) ; ; cos x ' si xdx ;3!; ;!; ;3@; ;3$; ;#; 0 :!e (l x) x dx e ;!; ;3!; ;4!; ;5!; ;6!; 48

49 :)» x si x cos xdx -; ; -;4 ; 0 ;4 ; ; ; :Ab f(x)g'(x)dx=[ f(x)g(x)]ba-:ab f'(x)g(x)dx :)» x si x cos xdx= :)» x si x dx f(x)=xg'(x)=si xf '(x)=g(x)=- :)» x si x cos xdx= :)» x si x dx cos x :)» x si x cos xdx=;!;[x {-;!; cos x}]»)-;!;:)» {-;!; cos x} dx :)» x si x cos xdx=;!;[{-; ;}-0]-;!;[-;4!; si x]») :)» x si x cos xdx=-;4 ; 03 :) x "çe dxe e :! 'x l xdx e ;9!;(e +) ;9!;(e +) ;9$;(e +) ;9$;(e +) ;9*;(e +) 49

50 3 f(x)x :!/ f(t)dt=(x-)fi +ax f(0)a :Aa f(x)dx=0 d dx :A/ f(t)dt=f(x) :!/ f(t)dt=(x-)fi +ax yy x= 0=+a a=- x f(x)=5(x-) (x-)'- f(x)=0(x-) - f(0)=0(-) -=9 05 f(x) f(x)=:!/ si(t -4)dt+ f()=a f"(a) a 06 lim x x - :!/ e cos pt dte 3 3e e e e e 50

51 4 k 5 lim { + }4 k= lim pk p f {a+ } =:A k= a+p f(x)dx lim k 5 5 k { + }4 = lim { + }4 k= 5 = :!3 x dx 5 xfi = [ ]3! 5 5 3fi = { - } 5 5 k= = lim k 5 k {+ }4 =5 lim {+ }4 =5:) (+x) dx k= +x=t dx dt k= =x=0t=x=t=3 5:) (+x) dx=5:!3 ;!;t dt=;%;:!3 t dt 07 lim p ta k= pk 4 l l 3 l 4 l 5 l 08 lim («'e+«"çe +«"çe +y+«"çe«)e e- e- e e+ e+ 5

52 0 :E e 3(l x) x dx : p 3p x si xdx p p 3p 4p 5p 03 f(x)x :!/ f(t)dt=x -ax +ax f(3) a 04 k f(x)=x +x lim f{+ } k= 5

53 Level 0 :) ;4 ; cos x si x+cos x dx+: ;4 ; 0 si x dx cos x+si x ;!; l l ;#; l l ;%; l 0 :) x l(x +)dx l - l -;!; l l +;!; l + 03 lim :)/ cos tdt x 0 si x ;3!; ;!; 3 04 f(x) f(x)=:)/ f(t) cos tdt+ f "(0) lim { + + +y+ } l l 3 l 3 l l 3 53

54 Level 0 f(x) f(x)= x x + f(0) +:) f(t) dt - l -l 0 l l 0 {a«} a«a«=:) x - e x«dx 0 = a«e 03 y=f(x) x x=ax=bx=c y F(x)=:B/ f(t)dt y=f(x) a<b<c) O a b c x F(a)>0 F(x) x=b F(x)=0 04 k k lim [{ } +]5 k= 0 : : : : 54

55 Level 3 f (x) 0 f(x)=x +x :) dx f'(f (x)) ;!; ;#; ;%; 0 f "(x) f(x) f(x)- f(x)-3 lim x =3 lim x =4 x- x- :! x f "(x)dx 03 f '(x) y=f(x) x= ex= f(3)=e :!3 f '(x) l f(x)dx y e y=f(x) e + e + e + e + e +3 e e O 3 x 04 x +y =(xæ0yæ0) Pº(0)P P yp - P«(0) y P (0, ) P - P - P k P x H OH OP OP k- S lim - k= S O O y y P k- P P 0 (, 0) H - H k H x ; ; ;4 ; ;6 ; ;8 ; p 0 55

56 05. y=f(x)[ab] y=f(x) x y x=ax=b S y=f(x) S=:Ab y dx=:ab f(x) dx O a c b x x=g(y) [cd] x=g(y) y y y=cy=d S x=g(y) d S=:Cd x dy=:cd g(y) dy y=f(x) [ac] f(x)æ0 [cb] f(x) 0 S c O x S=:Ac f(x)dx+:cb {-f(x)} dx S=:Ac f(x) dx+:cb f(x) dx S=:Ab f(x) dx 0 y=x -xx x=y yy= 0 y=si x {0 x ; ;}x=; ; x 03 e y=e xx=0x= y=l xyy=0y= 56

57 y=f(x)y=g(x) [ab] y=f(x) y=g(x) x=ax=b S y y=f(x) y=g(x) S=:Ab f(x)-g(x) dx O a b x x=f(y)x=g(y)[cd] x=f(y) x=g(y) y=cy=d S y d x=f(y) x=g(y) S=:Cd f(y)-g(y) dy c O x [ab] 0 g(x) f(x) S S=:Ab f(x)dx-:ab g(x)dx=:ab { f(x)-g(x)}dx y y=f(x) y=g(x) O a b x [ab] g(x) f(x) g(x) f(x) 0 g(x)+k f(x)+k y k S S=:Ab { f(x)+k} dx-:ab { g(x)+k} dx=:ab { f(x)-g(x)} dx y O a b y=f(x)+k y=g(x)+k y=f(x) x y=g(x) 04 y=x y=x x=y x=y+ 05 y=si x(0 x p)y=;!; y=ta x {0 x<; ;}y=x=0 06 e y=e y=e x= y=l xy= l xy= 57

58 05 3. [ab] x x S(x) V S(x) V=:Ab S(x)dx S(x) [ab] a x b x y=f(x)[ab] y=f(x) x x=ax=b x V x V x =p:ab y dx=p:ab { f(x)} dx y O a y y=f(x) b x x=g(y) [cd] x=g(y) y y=cy=d y y d x=g(y) V y x V y =p:cd x dy=p:cd { g(y)} dy c O x x[ab] x xº(=a)x x yx«(=b) x x S(x ) k V V b-a b-a V= lim V = lim S(x )Dx=:Ab S(x)dx {Dx= x =a+ k} k= k= 07 [3] x x x 08 x y=x x=y=0 y=si x(0 x p)y=0 09 y y=x y= y=l xxyy= 58

59 4. Pt(a t b) v(t) x(t) t=a t=b P :Ab v(t)dt t P x(a)+:at v(t)dt t=a t=b P :Ab v(t) dt Pt (xy) x=f(t)y=g(t) P t=a t=b s dx dy s=:ab æ { } +{ } dt=:ab " { f '(t)} +{ g'(t)} dt dt dt y=f(x)(a x b)l l=:ab æ +{ dy dx } dx=:ab " +{ f '(x)} dx 0 P t v(t) v(t)=3(t -t) t= t=3 P t=3 P t= t=3 P Pt (xy) t x=t y= -t 3 P t=0 t=3 f(x)=;3@;x'x (0 x 3) 59

60 y=x -x x x ;!; ;3!; ;4!; ;6!; ;8!; y=f(x) x y=f(x) f(x)æ0 f(x) 0 [ac] f(x)æ0 [cb] f(x) 0 S S=:Ab y dx=:ab f(x) dx=:ac f(x)dx+:cb {-f(x)} dx xæ0y=x -x y=x (x-) x<0y=x +x y=x (x+) y=x -x x S S S S=S +S S=:_0!(x +x )dx+:) (-x +x )dx x x x x S=[ + ]0_!+[- + ]) S=[(0+0)-{;4!;-;3!;}]+[{-;4!;+;3!;}-(0+0)] y y=x 3 -x (xæ0) S - O S y=x 3 +x (x<0) x S=; ;+; ;=;6!; 0 [0p] y=si x cos x x y=xe y=e y e e- e- e e+ e+ 60

61 y=x () y ;3!; ;!; 3 y=f(x)y=g(x) y=f(x)y=g(x) [ac] f(x)æg(x) [cb] f(x) g(x) S S=:Ab f(x)-g(x) dx=:ac { f(x)-g(x)} dx+:cb { g(x)-f(x)} dx y=x () y'=x y-=(x-) y=x- y y=x y=x- S S=:) {x -(x-)} dx O - x S=:) (x -x+) dx S=[ x 3 -x +x]) S={;3!;-+}-(0-0+0) S=;3!; 03 f(x)=si x [0; ;] y= f(x) f(x) y= f '(x) y '- '- '- 3'- 3'- 04 y=l(x+)y=l x x l -;!; l - l -;!; 3 l -;!; 3 l - 6

62 3 y=x (xæ0) x ; 0; ;5 ; ; 0;p ;5@;p ; ; [ab] y=f(x)y=g(x)(f(x)æg(x)æ0) x=ax=b x V V=p:Ab { f(x)} dx-p:ab { g(x)} dx=p:ab [{ f(x)} -{ g(x)} ] dx y O a y=f(x) y=g(x) b x y=x (xæ0) x=y y='x ( yæ0) y=x y='x y y=x V V=p:) ('x) dx-p:) (x ) dx y='x V=p:) xdx-p:) x dx x xfi V=p[ ])-p[ ]) 5 O x V=p {;!;- }-p {;5!;- } V=; ;-;5 ;=; 0;p 05 p p [ ] x x 6 3 ta x+cot x y= l x y=l y a b pa+b ab 6

63 4 4t (0 t ) Pt(0 t 0) v(t) v(t)=[ -4t+8 (<t 0) t=0 P P t v(t) P t=a t=b P :Ab v(t)dt P :Ab v(t) dt P -4t=a -5 0 t v(t)æ0 a> :)a v(t)dt=:) 4t dt+:!a (-4t+8)dt=[t ])+[-t +8t]a! =(-0)+{(-a +8a)-(-+8)}=-a +8a-5=-5 a=4 ( a>) P :)4 v(t) dt=:) 4t dt+:!4-4t+8 dt=:) 4t dt+:! (-4t+8)dt+:@4 (4t-8)dt =[t ])+[-t +8t]!+[t -8t]4@ =(-0)+{(-8+6)-(-+8)}+{(3-3)-(8-6)}=++8= 07 P t (x y) x=cos ty=si t P t=0 t=; ; ;!; ;#; ;%; 08 x e y=;!; {;!;x -l x} e ;4!;(e +) ;!;(e +) e + (e +) 4(e +) 63

64 0 y=e y=xe y A a y=e y=xe x= B bb-a y y=e x y=xe x ;#; e- A B ;%; e O x 0 y='xy='ƒ-x+0 x x ap a 03 Pt (xy) x=4(cos t+si t) [ (0 t p) y=cos t P t=0 t=p apa 64

65 Level 0 y='x- xx=0x= y=e x+ (e ) y e ;!;(e -e ) ;!;(e -e) ;!;(e -) ;#;(e -e ) ;#;(e -e) 03 [] x x "çl x ;6 ; l -;!; l +;!; l -;!; l +;!; l + 04 x +y =5(xæ0yæ0)y='ƒx- xy x : 6 :p : 3ª:p : :p : 3º:p : 6 :p 05 Pt (xy) x=;#;t y=;3$;t t=0 t= P ;qp;p+q pq 65

66 Level 0 y=;[!; xx=ax=b y x=k k ab 0<a<k<b) y= - x ' ab ' ab a+b a+b 3 a+b ab O a k b x 0 x= x=3 x= 3 y=f(x) f (x) y=f '(x) x y 3 y=f(x) O 3 x 03 y= y=f(x)y=g(x) x= y=f(x)y=g(x) x= y y=g(x) x=3 5x apa y= (g()>0f(3)>0) y=f(x) O 3 x 04 Pt(0 t c) v(t) P t s(t) s(0)=s(c)=s(b)=0 v(t) O a b c t :)c v(t)dt=0 :Ab v(t)dt=-:)a v(t)dt- s(t)= t(0c) 66

67 Level x + +si ; ;x xæ0 f(x) f(x)= lim 3 y=f(x) yy=4 x«+ p r - p+q+r pqr pq q p 0 y=f(x)y=g(x) y=f '(x)y=g'(x) y=f(x)y=g(x) x=ax=c y y=f'(x) S y=g'(x) (a<b<c) a O b c x f(a)=g(a) S=:Ac { f(x)-g(x)} dx f(b)=g(b) S=:Ac { g(x)-f(x)} dx :Ac {g(x)-f(x)} dx <S f(b)>g(b) 03 y=x y= y y y ; ;- ; ;+ p- p+ ;#;p- O x 04 P A(0) P y AP Q t t=0 y Q P t=p Q 3 - O A x

68 06.!_;!;=(-)! r(r ) «C _(r-)! ABCD 4 ABCD 4 4!=4 ABCDBCDACDABDABC A B C D B D C A D B A C C D A B 4 4! =(4-)!=3! 4! ;!;! =(-)! (-)

69 r «P «P = y_= r _4_4=4 =64 4 _ 4 _ P P =4_4_4=4 =64 «P ær «P <r abcde 69

70 06 3. pqyr! p+q+y+r= p!q!yr! 6 aaabbc xaaabbc. 3 a a a a 3! b b b! 3 a b (3!_!) a a a b b ca a a b b ca a a b b ca a a b b ca a a b b ca a a b b c a a a b b ca a a b b ca a a b b ca a a b b ca a a b b ca a a b b c x (3!_!) 3 a b x_3!_! 6 a a a b b c 6! x_3!_!=6! x= 6! 3!!! 06 5 aaabc

71 4. a ba B 7 aaaab bb abaabab A B 7! A B = =35 4!3! 3 A B 5 abcde a e. a ex 5 xxbcd 5 xbxcd x a x e a e 5!!! r r! 08 A P B P B A 09 6 abcdef b dd f

72 abcdef3 () C = =0 3 3 abc (3-)!=!= 3 def 3!=6 0 6=

73 3 AB CDE 5 3 5ABCDE r «P = X Y Z 5ABCDE 3 XYZ 3 5 P =3fi =43 5ABCDE Y Z 5 P =fi =3 5ABCDE X 4 Y Z C _ P =5_ = = X={--0} X f X x f(-x)=-f(x) f 73

74 3 ABCD abcd 8 ABCDabcd A a B b C cd d C c!= yy` Cc C c C c 6 AaBbDd A a B b D d A axx B byy D dz 5! xxyyz =30!!! yy` D d!= yy` _30_= S={34567} 5 5 > > > = _0 + _0 + _0 + _0+ _ 74

75 4 P O P 6 A() y 3 A(, ) - O 3 x - a a' b b' O A() aabb 6 A 6! aabbaa' =60 3!! 6! aabbbb' =60 3!! 60+60= A B B A

76 0 5 ABC A B 4C 5 5 A B AB 6 AB 4 A B A B A B 76

77 Level

78 Level ab c abc (abc) A B B A 78

79 Level a a a a _4 A= a a a a a Δ (i=j=34) 0 a Δ=a Δ (j=34) A AA A B B A

80 07. r «H «H =«C r> 4 ()()()() 0 (3)(4)(34)(34) H = C = C «H =«C jk jk «P jk jk «P jk jk «C jk jk «H 0 H H 0 abc

81 x+y+z=3 (xyz) xyz 3 xxxxxyxxzxyyxyzxzzyyyyyzyzzzzz (300)(0)(0)(0)()(0)(030)(0)(0)(003) x+y+z=3 (xyz) 3 (xyz) H = C = C = C =0 (a+b+c) abc 3 aaaaabaacabbabcaccbbbbbcbccccc a a ba cab abcac b b cbc c (a+b+c) 3 H = C = C = C =0 03 x+y+z+w=5 (xyzw) 04 (a+b+c) 05 X={34} Y={56789} f X x x x <x f(x ) f(x ) f 8

82 07 3. (a+b)«(a+b)«=«cºa«+«c a«b+«c a«b +y+«c a«b +y+«c«b«(a+b)«=;r+) «C a«b (a+b)««c a«b «Cº«C y«c«(a+b) (a+b) =a +3a b+3ab +b (a+b )(a+b )(a+b ) (a+b )(a+b )(a+b )=a +a (b +b +b )+a(b b +b b +b b )+b b b yy` a b b b 3 C =3 a b b b 3 C =3 b =b =b =b (a+b) =a + C a b+ C ab +b yy` a Cºb C (a+b) = Cºa + C a b+ C ab + C b (a+b )(a+b )(a+b )y(a+b«)=a«+a«b +a«b +y+a«b +y+ab«+b«yy` B =b +b +b +y+b«b =b b +b b +y+b«b«yb«=b b yb«b b b yb«r «C b =b =y=b«=b B =«C b (a+b)«=«cºa«+«c a«b+«c a«b +y+«c a«b +y+«c«ab«+«c«b««cº= 06 (x+3)fi x {x -;[!;}

83 4. «Cº+«C +«C +«C +y+«c«=««cº-«c +«C -«C +y+(-)««c«=0 «C +«C +3«C +y+«c«= «(+x)«(+x)«=«cº+«c x+«c x +y+«c x +y+«c«x«yy` x x= «=«Cº+«C +«C +«C +y+«c«x=- 0=«Cº-«C +«C -«C +y+(-)««c«x (+x)«=«c +«C x+y+r«c x +y+«c«x«yy` x= «=«C +«C +3«C +y+«c«=345y(a+b)«=(a+b) Cº C =(a+b) =3(a+b) =4(a+b) =5(a+b)fi «C «C «C «C +«C =«C Cº Cº Cº Cº C C C C C C C C C C C C C C log ( ºCº+ ºC + ºC +y+ ºC º) Cº+ C + C + C +y+ ºC ºC C C Cª Cª 83

84 abcde r «H =«C abcde 7 0 abcde H = C = C = C = = ae 0 bcd 765 Hº_ H + H _ H + H _ H = Cº_ªC + C _ C + C _ C = = =

85 x+y+z=0 (xyz) xyz (xyz) 66 xyz (xyz) 36 xyz (xyz) 6 x x yx«x +x +y+x«=rr «H =«C xyz x+y+z=0 (xyz) H º= C º= C = =66 xyz x=a+y=b+z=c+x+y+z=0 (xyz) a+b+c=7(abc) 9 8 H =ªC =ªC = =36 xyz x=(a+)y=(b+)z=(c+)x+y+z=0 (xyz) a+b+c=(abc) 4 3 H = C = =6 03 x+y+z+w=0 xyzw (xyzw) 04 (x+y+z) xyz

86 3 {x -;[#;}5 x px qp+q (a+b)«=«cºa«+«c a«b+«c a«b +y+«c a«b +y+«c«b«(a+b)«=;r+)«c a«b (a+b)««c a«b {x -;[#;}5 a=x b=-;[#; (a+b)fi C afi b = C (x )fi {-;[#;}r C afi b = C fi (-3) x x =x 0-3r= r=3 x p= C (-3) =-080 x =x 0-3r=4 r= x q= C (-3) =70 p+q= = {x - x }4 x {ax-;[!;}7 xfi a 86

87 4 log ( C + C +3 C +y+6 C ) (+x)«=«cº+«c x+«c x +y+«c x +y+«c«x«x (+x)«=«c +«C x+y+r«c x +y+«c«x«(+x)«(+x)«=«cº+«c x+«c x +y+«c x +y+«c«x«yy` x (+x)«=«c +«C x+y+r«c x +y+«c«x«yy` x==6 6 fl = C + C +3 C +y+6 C log ( C + C +3 C +y+6 C ) =log (6 fl ) =log ( fi ) =log =9 07 log ( Cº+ C + C +y+ fl C ) 08 log ( Cº+ C + C + C +y+ C )

88 (+x)« x (+x)«x ;K+!«C 3 88

89 Level USB 7 USB (x+y+z)fi (+ax)fi x (a+x)fl x a 04 «C + «C + «C +y+ «C «=5 89

90 Level 0 X={345} Y={3456} f f f(3)=3 X x x x <x f(x ) f(x ) x+y+z =7 xyz (xyz) (+i) log ( ºCº- ºC + ºC - ºC +y- ºC + ºC º) i=' - 04 (a+b+c) a bfi c a+b {(a+b)+c} c _(a+b) c (a+b) bfi _a bfi (a+b+c) a bfi c _ = 0! 3!5!!

91 Level (x-) (x+) x x+y+z<5 xyz (xyz) r(>r) «C +«C =«C ;K+) «H æ «C««C««C««C««C« ABC A B C

92 08. S S={3456} A A={35} {}{}{3}{4}{5}{6}. A B A B A'B A B A B A;B A B A;B=uA B A A A AÇ A4 B5 CA={35}B={4}C={56} AÇ={46}A'B={345}A;B={} B;C=u B C 0 AA AA AÇ 9

93 3. S S (S)A (A) A P(A) (A) A P(A)= = (S) A 4. A r«r«p r«p(a)= lim =p pa y y y 0.5 A P(A)=0.5 r« ;8#; ; ; ;!4!; ;!; ;!4#; ;!!; ;7$; ;!#; ;3@; ;7%; 93

94 08 5. A 0 P(A) S P(S)= u P(u)=0 S A S (S) A (A) 0 (A) (S) (S) 0 (A) (S) 0 P(A) (S) (S) (S) S u (S) (u) P(S)= =P(u)= =0 (S) (S) 6. A B P(A'B)=P(A)+P(B)-P(A;B) A B P(A'B)=P(A)+P(B) S AB (S) AB (A)(B) (A'B)=(A)+(B)-(A;B) (S) (A'B) (A) (B) (A;B) = + - (S) (S) (S) (S) A B P(A'B)=P(A)+P(B)-P(A;B) A B A;B=u P(A;B)=0 P(A'B)=P(A)+P(B) ;3 6; ; ; ;3 6; ;3 6; ;4!; 06 AB P(A'B)=;3@;P(A)=;!;P(B) 94

95 7. A AÇ P(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) A AÇ A;AÇ=u A AÇ P(A;AÇ )=0 P(A'AÇ )=P(A)+P(AÇ ) P(A'AÇ )= =P(A)+P(AÇ ) A AÇ P(AÇ ) P(AÇ )=-P(A) AA AÇ P(AÇ )=;8!; P(A)=-P(AÇ )=-;8!;=;8&; 07 ;!; ; ; ;3@; ;4#; ;6%; ;!4!; ;!4#; ;8%; ;!4&; ;!4(; 09 ;3!; ;5@; ; 5; ; 5; ;5#; 95

96 aby=x+ax +y =b ;4!; ;3!6!; ;3!6#; ; ; ;3!6&; A P(A) (A) A P(A)= = (S) (x y ) ax+by+c=0 ax +by +c " a +b (ab) ()()y(66) 36 y=x+ax +y =b (00) x-y+a=0 'b a 'b a b ' ()()(3)(4)(5)(6)()(3)(4)(5)(6)(35)(36) 3 ;3!6#; 0 3 ; 6; ;4!; ; 6; ;8#; ; 6;

97 ;3!; ;9$; ;9%; ;3@; ;9&; A B P(A'B) P(A'B)=P(A)+P(B)-P(A;B) A B 36-4 P(A)=;3@6);=;9%;P(B)=;3@6*;=;9&;P(A'B)= =;9*; 36 P(A;B)=P(A)+P(B)-P(A'B)=;9%;+;9&;-;9*;=;9$; T ( : ) S T ;!5@; ;!5$; ;!5^; ;!5*; ;5$; ;7@; ;3!; ; ; ;7#; ;!); 97

98 ;7@; ;3!; ; ; ;7#; ;!); A B A;B=u P(A;B)=0 P(A'B)=P(A)+P(B) 7 C A B C C P(A)= =; ;=;7@; C C C P(B)= =; ;=;7!; C AB ;7@;+;7!;=;7#; ;pq;p+q pq ; 4; ;4!; ; 4; ;3!; ;8#; 98

99 4 AB P(A)+P(B)=3P(AÇ;BÇ ) P(A'B)AÇ A ; ; ;!; ; ; ;3@; ;4#; A AÇ P(AÇ ) P(AÇ )=-P(A) AB P(A;B)=0 P(A'B)=P(A)+P(B)-P(A;B) P(A'B)=P(A)+P(B) P(A)+P(B)=3P(AÇ;BÇ ) P(A'B)=3P((A'B)Ç ) P(A'B)=3(-P(A'B)) 4P(A'B)=3 P(A'B)=;4#; 07 AB P(AÇ;BÇ )=;6!;P(A;BÇ )=;!;P(BÇ ) AÇ A ;6!; ;3!; ;!; ;3@; ;6%; ;4@0&; ;4@0( ;4#0!; ;4#0#; ;8&; 99

100 0 3 ;5@; ;!; ;5#; ; 0; ;5$; 0 A B P(A)=P(B)P(A)P(B)=;9!; P(A'B) ;6!; ;3!; ;!; ;3@; ;6%; 03 X={3}Y={34}Z={0} f:x Y g :Y Z g Á f:x Z Z ;pq;p+q pq X x x x +x f(x )+f(x ) g Z 00

101 Level ; 0; ;5!; ; 0; ;5@; ;!; 0 JH007 6 ;3!; ; ; ;!; ; ; ;3@; ( : ) ;4@(; ;4#!; ;!4!; ;6%; ;4#&; 04 6 ABCDEF 3 AB ;5@; ;!; ;5#; ; 0; ;5$; 0

102 Level 0 4 ; ; ;6!; ;4!; ;3!; ; ; 0 X={34}Y={3456} X Y f ;pq;p+q pq X x x x <x f(x ) f(x ) f()=3 03 ;3@; ;!; ;3!; ;6!; ;3!; ;!; ;3@; ;6%; 04 3«a«.6 a a ya 3 3 ;pq;p+q pq 0

103 Level ABC A B C ;!; ;ª6; ;8%; ;!6!; ;4#; 0 ABC ; 5; ;5@; ; 5; ;3@; ;5$; 03 S 3 T S T ;6!; ;3!; ;!; ;3@; ;6%; ;6!; ;3 0; ; 0; ;3!0!; ;3!0#; 03

104 09. A B P(B A)= P(A;B) P(A) P(A)>0) SAB A B A B P(B A) P(B A) A A A;B (A;B) (A;B) (S) P(A;B) P(B A)= = = P(A)>0) (A) (A) P(A) (S) P(A;B) P(A;B) P(B A)= = P(A) P(A;B)+P(A;BÇ ) P(A;B)P(A;BÇ ) S A A;B C A;B B ;3ª7; ;5@; ;5#; ;ª9; ;!9); 0 AB P(A)=;!;P(B)=;3!;P(A B)=;4!;P(B A) ; 8; ; ; ;9!; ;6!; ;4!; 04

105 P(A)>0P(B)>0AB P(A;B)=P(A)P(B A)=P(B)P(A B) P(A)>0P(B A)= P(A;B)=P(A)P(B A) P(B)>0P(A B)= P(A;B)=P(B)P(A B) P(A;B) P(A) P(A;B) P(B) P(A) P(B) 3 4 A B P(A)=;7#;P(B A)=;6@; P(A;B) P(A;B)=P(A)P(B A)=;7#;_;6@;=;7!; ; ; ; ; ;7@; ;3!; ; ; AB P(A)=;3@;P(B A)=;4!;P(A;B) ; ; ;6!; ;4!; ;3!; ; ; 05

106 09 3. AB A B AB P(B A)=P(B AÇ )=P(B) A B AB AB A B AB P(A;B)=P(A)P(B) P(A)>0P(B)>0) A B A BÇ AÇ BAÇ BÇ 0<P(A)<0<P(B)<) 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 P(A;B)=P(A)P(B) P(A;BÇ )=P(A)-P(A;B)=P(A)-P(A)P(B) =P(A){-P(B)} =P(A)P(BÇ )( P(BÇ )=-P(B)) A BÇ P(AÇ ;BÇ )=P((A'B)Ç )=-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 )( P(AÇ )=-P(A)P(BÇ )=-P(B)) AÇ BÇ AB P(A B)=;3@;P(BÇ A)=;!;P(A;B) AÇ A 06

107 4. A p A q A r «C p q«p+q=r=0y) C =3() 3 ;6!;;6%; ;6!;_;6%;_;6%;=;6!; {;6%;} C ;6!;_;6%;_;6%; ;6%;_;6!;_;6%; 3 ;6%;_;6%;_;6!; C ;6!; {;6%;} 08 ;!; ;ª6; ;8%; ;!6!; ;4#; ;9$; ;!7$; ;!7^; ;3@; ;@7); 07

108 X Y X Y 40 0 X Y A B P(B A)= P(A;B) P(A) P(A)>0) X 050 Y 0 0 X Y (0+0)-300=30 X Y (50+0)-00=70 X Y A B P(A;B) P(A)= =;5!;P(A;B)=;5 0º0;=;5 0; P(B A)= =; 0; ; 0; 500 P(A) k 0 80 ;3@; k 0 ST S S T

109 A3 4 B4 3 A B ; 8; ;5!6#; ;7@; ;5!6(; ;!8!; AB A B AB A ;7#; A 4 B 5 3 ;7#;_;6@;_;8%;=;5 6; A B A ;7$; A 3 3 B 4 4 ;7$;_;6#;_;8$;=;7!; ;5 6;+;7!;=;5!6#; 03 A B AB ; 5; ;5!; ; 5; ; 5; ; 5; A334 B A B 09

110 3 A B P(A)-P(B)=;6!;P(A'B)=;6%; P(AÇ B)AÇ A ;6!; ;3!; ;!; ;3@; ;6%; A B P(A;B)=P(A)P(B) A B P(A;B)=P(A)P(B) P(A'B)=P(A)+P(B)-P(A;B) ;6%;=P(A)+P(A)-;6!;-P(A)[P(A)-;6!;] 6 6{P(A)} -3P(A)+6=0{P(A)-3}{3P(A)-}=0 P(A)=;3@; ( 0<P(A)<) AÇ B P(AÇ B)=P(AÇ )=-P(A)=;3!; 05 AB P(A;BÇ )=;3!;P(A;B)=;6!;P(B) BÇ B ; 8; ;9!; ;6!; ;3!; ;!; 06 AB P(A;B)=;6!;P(A B)+P(B A)=;6%; P(A'B) ;6!; ;3!; ;!; ;3@; ;6%; 0

111 ; ª5; ;ª 5; ;!5(; ;ª 5; ;!)5!; A p A q A r «C p q«p+q=r=0y) 3 AA AÇ =;5#; P(AÇ )= C {;5#;}3=; 5; P(A)=-P(AÇ )=-; 5;=;ª 5; ; 6; ;8!; ; 6; ;4!; ; 6; 08 AB A B 3 A 4 B 5 3 ;8!; ; 6; ;4!; ; 6; ;8#;

112 0 0 % 50 % 0 % ; 3; ; 3; ; 3; ; 3; ;ª3; 0 A B P(A'B)=;!;P(A B)=;8#;P(A;BÇ ) BÇ B ; 0; ; 0; ;5!; ;4!; ; 0; 03 3 A;!; B ;3!; C A B C ;6!; 70 ;3!; ;3!6!; ; 8; ;4!; ;9@;

113 Level 0 AB P(A)=P(B)P(A'B)=;3@;P(A B)=;3!; P(B) ; ; ;6!; ;4!; ;3!; ; ; 0 ; 8; ;9!; ;6!; ;3!; ;!; 03 ABCDEF 6 ABCDEF A C 04 3 ; 7; ; 7; ;9!; ; 7; ; 7; 3

114 Level ;pq; p+q pq 0 3 A3 B B AB A BB A ;9@; ;3!; ;9$; ;9%; ;3@; ;8!#; ; 7; ;8!&; ;8!(; ; 7; 4

115 Level ab 36 [ ]45 [ ] a a G(a, b) L(a, b) b b [ ] [ ] 44 G(ab) ab L(ab) ab 0 8 ; 8; ;9!; ;6!; ;3!; ;!; A B P(A)=;9$; P(B A)=;5@; A B ; 4; ; ; ;8!; ;6!; ;4!; 5

116 0. Xx P(X=x) X X H T S S={HHHTTHTT} HHHTTHTT HX X0 X 0 X P(X=0)=;4!;P(X=)=;4@;=;!;P(X=)=;4!; X X 0 P(X=x) ;4!; ;!; ;4!;. X X X x x x yx«xx p p p yp«p(x=x )=p (i=3y) X X P(X=x )=p (i=3y) 0 P(X=x ) P(X=x )= P(a X b)= P(X=x) i= b x=a 0 3 X X X P(X=x) 0 ( ;!; (x=) \ 3 X P(X=x)= { a (x=) a \ 9 ;3!; (x=3) 6

117 3. X X P(X=x )=p (i=3y) E(X) E(X)=x p +x p +y+x«p«= x p =m X P(X=x )=p (i=3y) X m V(X) V(X)= E((X-m) ) =(x -m) p +(x -m) p +y+(x«-m) p«= (x -m) p i= i= X mx V(X)= (x -m) p V(X)= x p -m x p +m p V(X)= x p -m m+m V(X)= x p -m V(X)=E(X )-{E(X)} X r(x) r(x)=" V(X) i= i= i= i= i= i= 03 X E(X) X 0 P(X=x) ;!; a ;4!; ;4!; ;!; ;4#; ;4%; 04 3 X P(X=x)=ar(X) '6 '7 ' 'å

118 0 4. ax+b Xab E(aX+b)=aE(X)+b V(aX+b)=a V(X) r(ax+b)= a r(x) X X Y=aX+b Y y =ax +b P(Y=y )=P(X=x )=p Y X x x y x«p(x=x ) p p y p«y y y y y«p(y=y ) p p y p«y E(Y)= y p = (ax +b)p i= E(Y)=a x p +b p =ae(x)+b E(X)=m E(Y)=am+b V(Y)= {y -E(Y)} p i= i= V(Y)= {(ax +b)-(am+b)} p i= V(Y)=a (x -m) p =a V(X) i= i= i= r(y)=" V(Y)=" a V(X)= a " V(X)= a r(x) 05 X E(X)=V(X)=4E(-3X+6)+r(-3X+6) X r(x)=3v(x-4) X E(3X-)=5r(3X-)=3E(X)_r(X)

119 5. A p A XX P(X=x)=«C x p q -x x=0yq=-p) B(p) X X 0 y x y P(X=x) «Cºq««C p q - «C p q - y «C x p q -x y «C«p (p+q)«(p+q)«=«cºq +«C p q - +«C p q - +y+«c«p«x B(p) E(X)=p V(X)=pq q=-p) r(x)=' pq q=-p) 6. A X A p h X lim P{ -p <h}= A A P(A) X 08 X B{00; 0;}X XX

120 k X P(X=x)= (x=y6)p(3 X 5) x(x+) k ;6!; ;3 6; ;9@; ;4!; ; 8; = { - } (A+B) AB B-A A B k k k k P(X=)+P(X=)+P(X=3)+y+P(X=6)= + + +y+ 3 3_4 6_7 P(X=)+P(X=)+P(X=3)+y+P(X=6)=k{ + + +y+ } 3 3_4 6_7 P(X=)+P(X=)+P(X=3)+y+P(X=6)=k[{-;!;}+{;!;-;3!;}+{;3!;-;4!;}+y+{;6!;-;7!;}] P(X=)+P(X=)+P(X=3)+y+P(X=6)=k{-;7!;}=;7^;k= k=;6&; P(3 X 5)=P(X=3)+P(X=4)+P(X=5) ;6&; ;6&; ;6&; P(3 X 5)= + + =;6&;{ + + } 3_4 4_5 5_6 3_4 4_5 5_6 P(3 X 5)=;6&;[{;3!;-;4!;}+{;4!;-;5!;}+{;5!;-;6!;}]=;6&;{;3!;-;6!;}=;6&;_;6!;=;3 6; 0 X P(X=x)=ax (x=345)p(x 4) a ; ; ; ; ; ; ;ª; ;!); XP(Xæ) 0

121 X X X 0 3 P(X=x) ;4!; a ;!;-a a ;!6&; ;8(; ;!6(; ;4%; X X (X )=(X )-(X ) ;4!;+a+{;!;-a}+a =a =;4!; a=;!; ( 0 P(X=x) ) E(X)=0_;4!;+_;!;+_0+3_;4!;=;4%; E(X )=0 _;4!;+ _;!;+ _0+3 _;4!;=: 4 : V(X)=E(X )-{E(X)} =: 4 :-{;4%;}=: 4 :-;@6%;=;!6(; 03 X X a 6 X : 4 :X P(X=x) ;4!; ;3!; b : 4 : 8 : 4 : : : : 4 : 04 XX ' '3 ;!; '5

122 3 ax+b X X+ X - 0 P(X=x) a a 3a 4a X P(X=x )=p X Y=aX+b Y y =ax +b P(Y=y )=P(X=x )=p a+a+3a+4a=0a= a=; 0; E(X)=-_; 0;+0_; 0;+_; 0;+_; 0;= = 0 E(X+)=E(X)+=3 05 ( ; 0; \ (x=0) 03 X P(X=x)= { ; 0; (x=) \ 9 ; 0; (x=3) Y=0X X 50X

123 X X A AÇ P(AÇ )=-P(A) 6 6_6 =;6!; -;6!;=;6%; X B{60;6%;} E(X)=60_;6%;=50 E(X-)=E(X)-=_50-= X X : 8 : ;#; : 8 : ;4&; : 8 : 08 p00 X X 6X {0<p<;!;}

124 0 X X - 0 P(X=x) 3-a 8 ;8!; 3+a 8 ;8!; P(0 X )=;8&; XE(X) ;4!; ;8#; ;!; ;8%; ;4#; 0 X X 0 P(X=x) ;7@; ;7#; ;7@; 7XV(7X) ax+b 0 X 4X+V(4X+) 4

125 Level 0 X 034 i=0 P(X=i)+P(X=4-i)=;4!; P(X=) ;!; ;3!; ;4!; ;5!; ;6!; ;3!; (x=) 0 34 X P(X=x)=[ X a (x=34) ;3&; ;3*; 3 : 3º: 03 X 4 Y Y=3X+4 Y 7 X ( +) 04 X B{0;5!;}P(X=)+P(X=8)= ºC _ ab 5å a+b

126 Level 0 3 X P(X<) ;!; ;3@; ;4#; ;5$; ;6%; X X V(X) V(X)= q p pqp+q 03 X X+3V(X+3) X 4 P(X=x) ;4!; ;3@; a 04 X P(X=x)= ºC x {;3!;} {;3@;} 30-x x=0y30) 30 (x-0) P(X=x) x=0 : 3 : : 3 : 6 : 3ª: : 3º: 6

127 Level X 35X C C x 3 y 456 x -y '3 C C X 6X-3V(6X-3) X X ' '3 '5 ;!; ' % 50 % 00 X X 7

128 . X X y=f(x) O x O x [ab] X f(x) f(x) X f(x)æ0 :Ú f(x)dx= P(a X b)=:ab f(x)dx a a b b) X 0P(x=a)=0 P(a X b)=p(a X<b)=P(a<X b)=p(a<x<b) 0 [0] X f(x)=kx k [0] X f(x)=;!;xp{ X ;#;} ; 6; ;8!; ; 6; ;4!; ; 6; 8

129 X [ab] X f(x)x E(X)=m=:Ú xf(x)dx V(X)=E((X-m) )=:Ú (x-m) f(x)dx r(x)=" V(X) V(X) V(X)=E((X-m) ) V(X)=:Ú (x-m) f(x)dx V(X)=:Ú {x f(x)-mxf(x)+m f(x)}dx V(X)=:Ú x f(x)dx-m:ú xf(x)dx+m :Ú f(x)dx V(X)=:Ú x f(x)dx-m m+m V(X)=:Ú x f(x)dx-m V(X)=E(X )-{E(X)} ab X E(aX+b)=aE(X)+bV(aX+b)=a V(X)r(aX+b)= a r(x) 03 [03] X f(x) f(x)=kx X k 4 ;4(; : 9 : ;@6%; ;#5^; 04 X f(x) f(x)=;3@;x( x )X ; 6 ; ; 7; ; 6 ; ;8 ; ;5 4; 9

130 3. X f(x) f(x)= e - (x-m) r (- <x< ) ' pr Xm r N(mr ) x=m x x x=m r m m r r r m m r m <m <m 3 r <r <r 3 m m m 3 x r 3 r r m x 4. X N(mr ) Z= x -m x -m P(x X x )=P{ Z } r r X-m r N(0) P( Z )=P(0 Z )-P(0 Z ) P(- Z )=P(0 Z )+P(0 Z ) P(Zæ)=0.5-P(0 Z ) P(- Z )=P(0 Z ) 05 Z N(0)P(0 Z.)=0.3665P(0 Z.67)=0.496 P(-. Z.67) X N(6 )P(4 X 8) Z P(0 Z )=0.343 )

131 5. X B(p) X N(ppq) a-p b-p P(a X b)=p{ Z } q=-p) ' pq ' pq Z N(0) X B(p) p B(p) XE(X) XV(X) B(p) p ' pq N(ppq)q=-p y O =0 =5 = x pæ5 qæ5 07 X B{450;3!;}P(40 X 50) Z P(0 Z )=0.343 ) Z P(0 Z )=0.477P(0 Z 3)= )

132 - x X f(x)=k(x-)(x+) P{-;!; X ;!;}k ;!6!; ;4#; ;!6#; ;8&; ;!6%; - x f(x) :_! f(x)dx= :_! f(x)dx= :_! k(x-)(x+) dx=k:_! (x -) dx=k:) (x -) dx :_! k(x-)(x+) dx=k[;3!; x -x])=-;3$; k= k=-;4#; P{-;!; X ;!;}=: ;!; [-;4#;(x-)(x+)]dx=-;4#; : ;!; (x -)dx -;!; -;!; P{-;!; X ;!;}=-;#; :) ;!; (x -)dx=-;#;[;3!;x -x]) ;!; =;!6!; 0 [0] X f(x)=kx(-x) k 0 [e] X f(x)=k l xp( X ) k e l -;4#; l -;5$; l -;7%; l -;7^; l - 3

133 ( ;!; (0 x ) q X f(x)= { X p 9 -;4!;x+k ( x 3) pqp+q k :)3 f(x)dx= :)3 f(x)dx=:) ;!; dx+:!3 {-;4!;x+k}dx :)3 f(x)dx=[;!;x])+[-;8!;x +kx]3! :)3 f(x)dx=;!;+(-+k)= k=;4#; E(X)=:)3 xf(x)dx=:) ;!;xdx+:!3 {-;4!;x +;4#;x}dx E(X)=[;4!; x ])+[-; ; x +;8#;x ]3! E(X)=;4!;+{-: 6 :+ }=;!#; E(X )=:)3 x f(x)dx=:) ;!;x dx+:!3 {-;4!;x +;4#;x }dx E(X )=[;6!;x ])+[-; 6;x +;4!;x ]3! E(X )=;6!;+{-5+: :}=;3%; V(X)=E(X )-{E(X)} =;3%;-{;!#;} =;3%;-;!4^4(;=; 4 4; p+q=44+7= X f(x)= x- (0 x ) X

134 3 A g 4 g g z P(0 Z z) X X N(3054 ) X-305 Z= N(0) 4 P(Xæ33)=P{Zæ }=P(Zæ)=0.5-P(0 Z ) P(Xæ33)= = z P(0 Z z) z P(0 Z z)

135 z P(0 Z z) X B(p) X N(ppq)(q=-p) 3 {;!;}3 +{;!;}3 =;8!;+;8!;=;4!; XX B{08;4!;} E(X)=08_;4!;=7V(X)=08_;4!;_;4#;=: 4 : 08 X N{7{;(;} } 8-7 P(Xæ8)=P ª Zæ 3 ;(; =P(Zæ-)=P(0 Z )+0.5= =0.977 º 06 ;3!; 6 60 Z P( Z )=0.686 ) % z P(0 Z z) 00 4 p0000p

136 0 [0] X f(x) a(-x) (0 x<) f(x)=g b(x-) ( x ) P( X )= a 6 a-b ;!; ;3!; ;4!; ;5!; 0 [0] X f(x)x ;4!; :) (ax+5)f(x)dx=0 a m 500 m 000 m 5 %000 m 5% 000 m Z P(0 Z 0.5)=0. ;8#; ; 6; ;!; ;ª6; ;8%; 36

137 Level 0 [0] X f(x)=ax P(0 X )=P{;!; X b} abab ' '3 ;4!; ;!; ' [ab] X f(x) :Ab x f(x)dx=k :Ab x f(x)dx=4k X '3 k 03 X N(mr ) P( X-m.5r) r> z P(0 Z z) X X B{00;4!;}P{ -;4!; <;4 0;} z P(0 Z z)

138 Level 0 [04] X f(x) f(-x)=f(+x) :)3 f(x)dx=;4#; P( X 3) ;!; ;3!; ;4!; ;5!; ;6!; 0 [04] X y=f(x) X '6 '7 3 3 ' 3 y k O y=f(x) 4 x 'å g 0. g g g z P(0 Z z) X z P(0 Z z) P(X=x)= ººCÆ{;5!;} x {;5$;} 00-x (x=0y00) P(8 X 8)

139 Level A XX y=f(x) A y k y=f(x) 4 (0 X 6) O - 6 x ;!); ;ª; ; ; ; ; ; ; 0 [-] X f(x) f(-x)=f(x) :) x f(x)dx=4 [04] X g(x) g(x)=f(x-) E(X )+V(X ) % A A Z P(0 Z.05)=0.48 ) A75 cm B7 cm3 C78 cm ABC ABC ACB BAC BCA CAB 39

140 . X X m r r X X X yx«x S S X +X +X +y+x«x = S = {(X -X ) +(X -X ) +y+(x«-x ) }S="çS - m r X E(X )=mv(x )= r(x )= X r N(mr ) X N{m } X N{m r } r r ' X 50 5 X P(X 49) (Z P(0 Z.5)= )

141 N(mr ) x r r 95% [x -.96 m x +.96 ] ' ' r r 99% [x -.58 m x +.58 ] ' ' r N(mr ) X N{m X -m Z= N(0) r ' P(-.96 Z.96)=0.95 X -m P =0.95 ª r º ' r r r r P{-.96 X -m.96 }=0.95P{X -.96 m X +.96 }=0.95 ' ' ' ' r r [X -.96 X +.96 ] 95% ' ' r xæ r r [xæ-.96 xæ+.96 ] m 95% ' ' (æ30) r s P( Z k) [ab] b-a=_k_ r ' } m 95% Z P( Z.96)=0.95 ) [8.6.84] [8..88] [8.08.9] [ ] [ ] m 99 % [ab]00(b-a) Z P( Z.58)=0.99 ) 4

142 3. p ^p X X ^p ^p= X ^p= X p B(p) B(p) X ^p X E(^p)=E{ }= E(X)= p=p X pq V(^p)=V{ }= V(X)= pq= q=-p) pq r(^p)=" V(^p)=æ q=-p) pq ^p-p ^p N{p } Z= pq æ N(0) q=-p) pæ5qæ % 00 ^p r(^p) '6 '7 ' ;5 0; 'å ; 0; 8 ^p P(^p 0.) Z P(0 Z 3)= )

143 4. ^p p ^q=-^p 95 % [^p-.96 æ ^p^q ^p+.96 æ ^p^q ] 99 % [^p-.58 æ ^p^q ^p+.58 æ ^p^q ] ^p^q(=-^p) ^p-p Z= æ ^p^q N(0) P(-.96 Z.96)=0.95 ^p-p P =0.95 ª ^p^q º æ P ^p-.96æ ^p^q p ^p+.96æ ^p^q =0.95 [^p-.96æ ^p^q ^p+.96æ ^p^q ] p 95% 95% 95% p ^p=0. p95% Z P( Z.96)=0.95 ) [ ] [ ] [ ] [ ] [ ] ^p=0.5 p99% [ab]00(b-a) Z P( Z.6)=0.99 )

144 X X 3 X X X 0 P(X=x) ;4!; a a ;4!8!; ;4!; ;4!8#; ; 4; ; 6; m r X E(X )=mv(x )= r r(x )= r ' ;4!;+a+a=;4!;+3a=3a=;4#; a=;4!; X X E(X)=0_;4!;+_;4!;+_;!;=;4%;E(X )=0 _;4!;+ _;4!;+ _;!;=;4(; X V(X)=E(X )-{E(X)} =;4(;-{;4%;}=;4(;-;@6%;=;!6!; ;!6!; V(X) V(X )= = =;4!8!; 3 0 X m r 6 X r(x )=V(X-3) 0 X X P(X ) X 3 P(X=x) ;!; ;3!; ;6!; ;3#6!; ;9*; ;!!; ;!8&; ;3#6%; 44

145 X 6000 kg/cm 00 kg/cm kg/cm z P(0 Z z) N(mr ) X N{m r X -m } Z= r ' N(0) X X N( ) 5 X 00 E(X )=6000r(X )= =0 'å5 X N(60000 ) Z= X Z N(0) P(X 5960)=P{Z } 0 P(X 5960)=P(Z -) P(X 5960)=P(Zæ) P(X 5960)=0.5-P(0 Z ) P(X 5960)= = lm 0.05 lm lm lm z P(0 Z z)

146 m 95% [ab]bz P( Z.96)=0.95 ) r r s =00xÆ=0.4r?s= % [ _ _ ] ' 00 ' 00 [ _ _0.005] [ ] b= % [ab] b-a Z P( Z.58)=0.99 ) N(m ) X m 95 % [ab]b-a Z P( Z.96)=0.95 ) 46

147 4 0% 64 5% z P(0 Z z) pæ5(-p)æ5 pq 0._ E(^p)=p=0.V(^p)= = = =0.005 = ^p N( ) ^p-p ^p-0. ^p-0. Z= 3 = 333 = 3 N(0) pq 'ƒ æ 3 P(^pæ0.5)=P{Zæ P(^pæ0.5)=0.5-P(0 Z ) P(^pæ0.5)= =0.587 }=P(Zæ) 06 A 0% 5% 0.07 Z P( Z.5)=0.86 ) 07 AB 5 A 0 A 95 % Z P( Z )=0.95 ) [0.7, 0.88] [0.7, 0.9] [0.68, 0.9] [0.66, 0.94] [0.64, 0.96] 47

148 X X 0 X z P(0 Z z) % mmz P(0 Z.96)= p ^p ^p p 95 % [ ] Z P( Z.96)=

149 Level X X ;#; ;%; X 4X r 6 X r r % [a-.58ba+.58b] ab ;ba; Z P( Z.58)=0.99 )

150 Level X X ;7 ; ;3 6; ; 4; ; 8; ;7 ; 0 X 4 X N(008 ) P(84 X 3) z P(0 Z z) z P(0 Z z) x x yx ºº x +x +y+x ºº=800 (x -8) +(x -8) +y+(x ºº-8) =99 m 95 % Z P( Z.96)=0.95 ) [7.804, 8.96] [7.75, 8.49] [7.58, 8.49] [7.395, 8.605] [7.88, 8.8] 50

151 Level a 3 X E(X )=P(X ) ;!7^; ;!7&; ;3@; ;!7(; ;@7); 0 (PVC) cm 0.06 cm PVC 4 60 cm z P(0 Z z) p % 00 40% p 0 p z P(0 Z z) m k X f(m)=p{x.96_ } X E(X )V(X ) ' k k Z P(0 Z.96)= E(X )=mv(x )=;k!; f(0)= m m f(m) 5

152 memo

153 0 0 f(x)=3x +x ;4!;x +C ;0 3;x +C 05 -;[!;+C x+c 06 ;5@;x ;%; +C ;4#;x ;3$; +C 07 x +x +x+c ;3!;x -;!;x +x+c 08 ;4!;x -x+c ;!;x +x+3 l x +C 09 4x +C ;3@;x ;#; +x+c 0 - cos x+3 si x+c ta x-x+c -cot x+c sec x+c x+ e x+ + +C e +x+c l l x -l x+ +C l x- +l x+ +C 05 ;8!;x-;3 ; si 4x+C 3x+ si x+;4!; si 4x+C 06 ;3!; cos x-cos x+c 07 x si x+cos x+c xe x -e x +C 08 ;!;x l x-;4!;x +x+c Level Level Level Level Level Level 3 0 ;4&; ah 0 0 ;3!;pr h e+e 09 l 0 ;%; l 0 0 ;5!;(+x )fi +C ;4!;(x +x) +C 0 e x +C 03 ;4!; l(x +)+C - +C 4(x +) Level

154 Level Level : 3 : 08 ;5 ; 09 ; ; ; ;(e -) 0 6 : 3 : p ;!; l ;4 ; 03 ;!; ;4!;(e +) : 3 : 0 ;3@; Level Level Level Level Level Level ;6!; ;3!; 0 03 e- e- 04 ;6!; ;(; 05 '3-;3 ; ;4 ;-;!; l 06 e+;e!;- e-'e+ Level Level Level

155 Level ;!; 04 5 Level Level AÇ={()()()} ;6!; Level Level Level ;7#; ;3!; ;3!; ;4!; Level ;5!; 04 Level Level ;!!; ;!!; 0 ;6!; ;3@; Level Level Level

156 Level Level Level Level Level 3 Level

157 0 f'()=3 - += 0 f(x)=3x +x ;4!;x +C ;0 3;x +C 05 -;[!;+C x+c 06 ;5@;x ;%; +C ;4#;x ;3$; +C 07 x +x +x+c ;3!;x -;!;x +x+c 08 ;4!;x -x+c ;!;x +x+3 l x +C 09 4x +C ;3@;x ;#; +x+c 0 - cos x+3 si x+c ta x-x+c -cot x+c sec x+c e x+ + x+ +C e +x+c l 04 : x dx= x + +C 3+ : x dx=;4!;x +C : x dx= x + +C 0+ : x dx=;0 3;x +C ;4!;x +C ;0 3;x +C 05 : x dx=: x - dx : dx= x -+ +C -+ : dx=-x +C : dx=-;[!;+c : dx=: dx=: x dx 0 : f(x)dx=x +x +x+c : dx= x + +C 0+ =x+c (x +x +x+c)'=3x +x+ f(x)=3x +x+ f(x)=3x +x+ -;[!;+C x+c 0 : (x+a)dx=bx +3x+C (bx +3x+C)'=bx+3 x+a=bx+3 =ba=3 b=a=3 a+b=3+= : "çx dx=: x ;#; dx : "çx dx= 3 x ;#;+ +C ;#;+ : "çx dx=;5@;x ;%; +C : 'x dx=: x ;3!; dx : 'x dx= 3 x ;3!;+ +C ;3!;+ 03 : (3x -x+)dx=f(x)+c : 'x dx=;4#;x ;3$; +C { f(x)+c}'=f'(x) f'(x)=3x -x+ ;5@;x ;%; +C ;4#;x ;3$; +C 5

158 07 : (3x +x+)dx =: 3x dx+: x dx+: dx =3: x dx+: xdx+: dx =3{;3!;x +C }+{;!;x +C }+(x+c ) C C C =x +3C +x +C +x+c =x +x +x+c C=3C +C +C : x + (x+)(x -x+) dx=: x+ x+ dx : dx=: (x -x+)dx : dx=;3!;x -;!;x +x+c =: 8x dx =8 ;!;x +C =4x +C x : dx-: dx 'x- 'x- x =: { - } dx 'x- 'x- =: =: x- 'x- dx ('x-)('x+) 'x- =: ('x+)dx =: {x ;!; +} dx = 34 x ;!;+ +x+c ;!;+ dx x +x +x+c ;3!;x -;!;x +x+c =;3@;x ;#; +x+c 08 : (x-)(x +x+)dx=: (x -)dx 4x +C ;3@;x ;#; +x+c : =;4!;x -x+c dx=: {x++3 ;[!;} dx =;!;x +x+3 l x +C ;4!;x -x+c ;!;x +x+3 l x +C : f(x)g(x)dx+: f(x)dx : g(x)dx : x +x+3 x : f(x)dx f(x) dx+ g(x) : g(x)dx 09 : (x+) dx-: (x-) dx =: {(x+) -(x-) } dx =: {(x +4x+4)-(x -4x+4)} dx 0 : ( si x+3 cos x)dx=- cos x+3 si x+c +ta x=sec x ta x=sec x- : ta xdx=: (sec x-)dx =ta x-x+c - cos x+3 si x+c ta x-x+c si x+cos x= +ta x=sec x +cot x=cosec x =cosec x si x : dx=: { } dx si x si x =: cosec xdx =-cot x+c 6

159 si x =sec x =ta x cos x cos x si x si x : dx=: dx cos x cos x cos x F(x)=x +3x-4 F()=+3-4= =: sec x ta xdx =sec x+c -cot x+c sec x+c si x cos x ta x= cot x= cos x si x =cosec x =sec x si x cos x =cot x ta x 0 F(x)=: (x+) dx=x +x+c x F(x)>0 F(x)=0 D<0 D=-4C<0 C>;4!; F(0)=C>;4!; : (e )dx=: (e e + )dx 03 ;dî[; { f(x)+g(x)}=6x+7 : (e )dx=e e + +C l : (e )dx=e + ± + +C l e - (e -)(e +) : dx=: dx e - e - =: (e +)dx =e +x+c e + ± + +C e +x+c l f(x)+g(x)=: (6x+7) dx f(x)+g(x)=3x +7x+C yy x=0 f(0)+g(0)=c f(0)=g(0)=4 +4=C C=5 f(x)+g(x)=3x +7x+5 f()+g()=3+7+5=5 04 ;dî[; { f(x)+g(x)}=x+ 0 F(x)=: (4x+3) dx=x +3x+C x +3x+C=0ab C ab= C =- C= f(x)+g(x)=: (x+) dx f(x)+g(x)=x +x+c C ) yy` x=0 f(0)+g(0)=c f(0)=g(0)=- +(-)=C C =- f(x)+g(x)=x +x- yy` ;dî[;{ f(x)g(x)}=3x -4x+ f(x)g(x)=: (3x -4x+) dx f(x)g(x)=x -x +x+c C y` 7

160 x=0 f(0)g(0)=c C = (-)=- f(x)g(x)=x -x +x- =(x-)(x +) yy` f(0)=g(0)=- f(x)g(x) f(x)=x +g(x)=x- f()=+= 05 y=f(x) (xy) 'x+ 'x f'(x)='x+ f(x)=: f'(x)dx f(x)=: {'x+ } dx f(x)=: {x ;!; +x ;!; } dx f(x)=;3@;x ;#; +x ;!; +C f(x)=;3@;x'x+'x+c y=f(x){;3@;} f()=;3@;++c=;3@; C=- 'x f(x)=;3@;x'x+'x- f(9)=;3@; = 06 y=f(x) (xy) ;[K; f'(x)=;[k; k 0 f(x)=: f'(x)dx f(x)=: ;[K; dx 'x f(x)=k l x +C f(x)=k l x+c ( x>0) y=f(x)()(e3) f()=k l +C= f(e)=k l e+=3 C= k= f(x)= l x+ f('e )= l 'e+= ;!;+= 07 f'(x)=3(x -)=3(x+)(x-) f'(x)=0 x=- x= y=f(x) x y - y y f'(x) f(x) f(x) x=- x= f(x)=: f'(x) dx f(x)=: 3(x -)dx f(x)=: (3x -3)dx f(x)=x -3x+C f()= f()=-3+c= C=3 f(x)=x -3x+3 f(x) f(-)=-+3+3=5 08 y=f '(x) f'(x)=ax(x-) (a>0) f'(x)=0 x=0 x= y=f(x) x y 0 y y f'(x) f(x) 5 8

161 f(x) x=0 x= f{-; ;}=- f(x)=: f'(x) dx=: ax(x-) dx a si {-; ;}+C =- C =a- f(x)=a: (x -x) dx=a{;3!;x -x }+C f(0)=4 f(0)=c=4 C=4 f()=0 f()=a{;3*;-4}+c=0-;3$;a+4=0 a=3 f(x) x=0 lim (-cos x+c )= lim (a si x+c ) x +0 x -0 -+C =C C =0 ( C =) C =a- 0=a- a= f(x)=3{;3!;x -x }+4=x -3x +4 f()=-3+4= f(x) lim x = f()=0f'()= x- f'(x)=ke - f'()=ke =k= k= x (x>0) 09 f'(x)=g (x<0) : x dx=x +C : dx=x+c C C x +C (x>0) f(x)=g x+c (x<0) f()= +C = C = f(x) x=0 lim (x +C )= lim (x+c ) x +0 x -0 C =C C = x + (xæ0) f(x)=g x+ (x<0) f(-)=(-)+=0 si x (x>0) 0 f'(x)=g a cos x (x<0) : si x dx=-cos x+c f(x)=: f'(x)dx=: e - dx f(x)=: ;e@; e dx=;e@; e +C f()=+c=0 C=- f(x)=e - - f(0)=;e@;- f(x) lim x p =a+ x-p f(p)=0f'(p)=a+ f'(x)=a{si ;{;-cos ;{;} f'(x)=a{si ;{;- si ;{; cos ;{;+cos ;{;} f'(x)=a(-si x) f'(p)=a(-si p)=a=a+ a=- f(x)=: f'(x) dx : a cos x dx=a si x+c C C f(x)=: (-) (-si x) dx -cos x+c (x>0) f(x)=g a si x+c (x<0) f {; ;}= -cos ; ;+C = C = f(x)=: (si x-) dx f(x)=-cos x-x+c f(p)=-cos p-p+c=-p+c=0 C=p- f(x)=-cos x-x+p- 9

162 f(0)=-cos 0+p-=p- x> f'(x)<0 a+f(0)=-+(p-)=p-3 f(x) x= f()=;3!; : f(x)dx=x -ax +ax x ;dî[;: f(x)dx=;dî[;(x -ax +ax) f(x)=3x -4ax+a f()=0 f()=3-4a+a=3-3a=0 a= f(x)=3x -4x+ f(3)=7-+= f(x)-: e f(x)dx= x f'(x)-e f(x)=0 yy` x=0 f'(0)-f(0)=0 f'(0)-=0 f(0)= f'(0)= x f"(x)-{e f(x)+e f'(x)}=0 x=0 f"(0)-{ f(0)+f'(0)}=0 f"(0)={ f(0)+f'(0)} =(+) =6 x (x<) 0 f'(x)=g - (x>) ;3!;x +C (x<) f(x)= { C C ª -x+c (x>) f(x)x lim {;3!;x +C }= lim (-x+c ) x -0 x +0 ;3!;+C =-+C C =C +;3$; yy` f(0)=0 Level f(x)=: e cos px dx x f'(x)=;dî[;: e cos px dx=e cos px f'()=e cos p=-e ;3!; 0+C =0 C =0 C =;3$; ;3!;x (x ) f(x)= { ª -x+;3$; (x>) 0<x< f'(x)>0 0 f(x)=: (+3x +5x +7xfl +9x )dx f(x)=x+x +xfi +x +x +C f(0)=c=0 C=0 f(x)=x+x +xfi +x +x f(-)=-5 0

163 cos x-x 03 f(x)=: dx x cos x f(x)=: {;[!;- } dx cos x 0 f(x)g'(x)+f'(x)g(x)=3x +0x+8 f(x)g'(x)+f'(x)g(x)={ f(x)g(x)}' f(x)g(x)=: (3x +0x+8) dx=x +5x +8x+C f(x)=: {;[!;-sec x} dx f(x)=l x -ta x+c f{;4 ;}=l ;4 ; -ta ;4 ;+C=l ;4 ;-+C f{-;4 ;}=l -;4 ; -ta {-;4 ;}+C f{-;4 ;}=l ;4 ;++C f(x)=(x+)g(x) f(-)=0 f(-)g(-)=-+5-8+c=c-4=0 C=4 f(x)g(x)=x +5x +8x+4=(x+)(x+)(x+) (x+)g(x)g(x)=(x+)(x+)(x+) g(x)=x+ ( g(-)>0) f(x)=(x+)(x+) f()+g()= 3+3=9 f{;4 ;}-f{-;4 ;}=- 0 f(x) f'(x)=+cos x f(x)=: f'(x) dx=: (+cos x) dx=x+si x+c 04 : xf'(x)dx=x +'x+c x f(0)=c= C= f(x)=x+si x+ xf'(x)=x+ 'x f'(x)=+ ( x>0) x'x f(x)=: {+ } dx x'x f(x)=: {+;!;x -;#; } dx f{; ;}=; ;+si ; ;+=; ;+ f'{; ;}=+cos ; ;= y=f(x) {; ;; ;+} y-{; ;+}=x-; ; f(x)=x-x ;!; +C C f(x)=x- +C 'x f()=-+c =+C = f(x)=x- 'x f(4)=8-;!;+=: : + C = y=x+ a+b=+=3 03 f(x) F(x) F'(x)=f(x) F(x)=xf(x)+si x-x cos x F'(x)=f(x)+xf'(x)+cos x-cos x+x si x f(x)=f(x)+xf'(x)+cos x-cos x+x si x xf'(x)=-x si x f'(x)=-si x(x>0) f(x)=: f'(x)dx=: (-si x)dx Level f(x)=cos x+c f(p)=cos p+c=+c= f(x)=cos x+ f(p)=cos p+=(-)+=0 C=

164 04 g{;3 ;}=-;!; cos ;3 ;+=-;4!;+=;4&; xf(x)=: dx ;4&; xf(x)= l x +x'x+c f()=4 f()=+c=4 C= xf(x)= l x +x'x+ f(x)=;[!;( l x+x'x+) x>0 0 y=f'(x) f'(x)=ax(x-)(x-) (a>0) f'(x)=0 x=0 x= x= y=f(x) f{;e!;}=e{ l ;e!;+;e@;æ;e!; +}= 'e x y 0 y y y f'(x) f(x) f(x) x= x=0x= Level 3 f(x)=: f'(x) dx 0 ;4&; f(x)=: ax(x-)(x-) dx f(x)=a: (x -3x +x)dx si ;{; cos ;{; 0 f(x)=ta ;{;+cot ;{;= cos ;{; si ;{; si ;{;+cos ;{; f(x)= 343 = cos ;{; si ;{; f(x)= 333 = si x cos ;{; si ;{; si x g(x)=: dx=: dx f(x) g(x)=-;!; cos x+c f{; ;}= 33 = si ; ; g{; ;}=-;!; cos ; ;+C=C f{; ;}=g{; ;} C= g(x)=-;!; cos x+ 33 cos ;{; si ;{; f(x)=a {;4!;x -x +x }+C f(0)=cf()=c f(0)=f()=c= C= f()= f()=a{;4!;-+}+c=;4!;a+= a=4 f(x)=4{;4!;x -x +x }+=x -4x +4x + f(-)=+4+4+=0 03 f'(x)=e + e -e e -e=0 x e =e x= `xæ `f'(x)=e +(e -e)=e -e `x< `f'(x)=e -(e -e)=e 0

165 e -e (xæ) `f'(x)=g e (x<) 0 : (e -e)dx=e -ex+c C : e dx=ex+c C e -ex+c (xæ) f(x)=g ex+c (x<) f(0)=c = C = f(x) x= lim (e -ex+c )= lim (ex+c ) x +0 x -0 e-e+c =e+c C =C C = e -ex+ (xæ) f(x)=g ex+ (x<) f() f(-)=(e+)(-e+)=-e 0 ;5!;(+x )fi +C ;4!;(x +x) +C 0 e x +C 03 ;4!; l(x +)+C - +C 4(x +) 04 l x -l x+ +C l x- +l x+ +C 05 ;8!;x-;3 ; si 4x+C 3x+ si x+;4!; si 4x+C 04 f(x)xy f(x+y)=f(x)+f(y)+xyf'(0)= f(x+y)=f(x)+f(y)+xy x=0y=0 f(0)=f(0)+f(0) f(0)=0 f'(x)= f'(x)= f'(x)= lim h 0 lim h 0 lim h 0 f(h) f'(x)= lim h 0 +x h f'(x)=x+ f(x+h)-f(x) h f(h)+xh h f(h) f(0+h)-f(0) { lim = lim h 0 h 0 =f'(0)=} h h f'()=+=4 f(x)+f(h)+xh-f(x) h f(x)=: f'(x)dx=: (x+)dx=x +x+c 06 ;3!; cos x-cos x+c 07 x si x+cos x+c xe x -e x +C 08 ;!;x l x-;4!;x +x+c 0 +x =t x dt =3x dx : 3x (+x ) dx=: t dt : 3x (+x ) dx=;5!;tfi +C : 3x (+x ) dx=;5!;(+x )fi +C x +x=t x dt =x+ dx : (x +x) (x+)dx=: t dt f(0)=c=0 C=0 f(x)=x +x=(x+) - f(x) f(-)=- : (x +x) (x+)dx=;4!;t +C : (x +x) (x+)dx=;4!;(x +x) +C ;5!;(+x )fi +C ;4!;(x +x) +C 3

166 0 x =tx dt =x dx l x -l x+ +C l x- +l x+ +C : xe x dx=: e dt 05 : si x cos xdx=: ;4!; si x dx =e +C =e x +C e x +C : si x cos xdx=: ;8!;(-cos 4x)dx : si x cos xdx=;8!;x-;3 ; si 4x+C 03 x +=t x dt =4x dx x : dx=: x dx x + x + : : : dx=: ;t!; ;4!; dt=;4!;: ;t!; dt dx=;4!; l t +C dx=;4!; l(x +)+C ( x +>0) dt x +=t x =x dx x : dx=: ;!; dt (x +) t : dx=-;4!; +C t : dx=- +C 4(x +) 04 : dx=: {;[!;- } dx x(x+) x+ =l x -l x+ +C 3x+ 3x+ a b = = + x - (x-)(x+) x- x+ 3x+ (a+b)x+a-b = (x-)(x+) (x-)(x+) 3x+=(a+b)x+a-b a+b=3a-b= a=b= 3x+ : dx=: { + } dx x - x- x+ ;4!; l(x +)+C - +C 4(x +) = l x- +l x+ +C : 8 cos xdx=: 8{ } dx : 8 cos xdx=: (+ cos x+cos x)dx : 8 cos xdx=: (3+4 cos x+cos 4x)dx : 8 cos xdx=3x+ si x+;4!; si 4x+C ;8!;x-;3 ; si 4x+C 06 : si xdx=: si x(-cos x)dx 3x+ si x+;4!; si 4x+C cos x=t x dt =-si x dx : si xdx=: si x(-cos x)dx : si xdx=: (t -)dt : si xdx=;3!;t -t+c : si xdx=;3!; cos x-cos x+c 07 f(x)=xg'(x)=cos x f'(x)=g(x)=si x : x cos x dx=x si x-: si xdx +cos x =x si x+cos x+c f(x)=xg'(x)=e x f'(x)=g(x)=e x ;3!; cos x-cos x+c 4

167 : xe x dx=xe x -: e x dx =xe x -e x +C x si x+cos x+c xe x -e x +C 08 : (x l x+)dx =: x l xdx+: dx f(x)=l xg'(x)=x f'(x)=;[!;g(x)=;!;x : x l xdx+: dx =;!;x l x-: ;!;x dx+: dx =;!;x l x-;4!;x +x+c ;!;x l x-;4!;x +x+c f(3)=;5@; fi -;3@; +; 5;=8 x- 0 ;dî[;f(x)= 'ƒx+ x- f(x)=: dx 'ƒx+ 'ƒx+=t x+=t dt x = dx t f(x)=: f(x)=: x- 'ƒx+ t - t dx t dt f(x)=: (t -4) dt f(x)=;3@;t -4t+C f(x)=;3@;(x+)'ƒx+-4'ƒx++c f(0)=;3@;-4+c=-: 3º: C=0 f(x)=;3@;(x+)'ƒx+-4'ƒx+ f(x)=0 ;3@;(x+)'ƒx+-4'ƒx+=0 0 x+=t x dt = dx ;3@;'ƒx+ {(x+)-6}=0 x=5 x>- f(x)=: x'ƒx+ dx f(x)=: (t-)'t dt f(x)=: (t ;#; -t ;!; )dt f(x)=;5@;t ;%; -;3@;t ;#; +C f(x)=;5@;(x+) ;%; -;3@;(x+) ;#; +C f(0)=;5@;-;3@;+c=; 5; C=; 5; f(x)=;5@;(x+) ;%; -;3@;(x+) ;#; +; 5; e 03 f'(x)= e + f(x)=: e e + dx f(x)=l e + +C e +>0 f(x)=l(e +)+C f(0)=l +C=l C=0 f(x)=l(e +) f()=l(e +) 5

168 si x ta x cos x cos x-si x 04 = 343 = +ta x si x cos x+si x +343 cos x -ta x cos x-si x f(x)=: dx=: dx +ta x cos x+si x -si x+cos x f(x)=: dx cos x+si x f(x)=l cos x+si x +C 0 x<; ; si x+cos x>0 x+5 = - (x+)(x+3) x+ x+3 f(x)=: {+ - } dx x+ x+3 f(x)=x+ l x+ -l x+3 +C f(-)=-+c=- C=0 f(x)=x+ l x+ -l x+3 f(0)=-l 3 f(x)=l(cos x+si x)+c {0 x<; ;} 07 f(x)=: (si x+cos x) dx f{;4 ;}-f(0)=(l '+C)-(l +C) f(x)=: (si x+ si x cos x+cos x) dx =l ' =;!; l f(x)=: (+si x) dx f(x)=x-;!; cos x+c x+7 x+7 a b 05 = = + x +x-3 (x-)(x+3) x- x+3 x+7 (a+b)x+3a-b = (x-)(x+3) (x-)(x+3) a+b=3a-b=7 a=b=- x+7 - = + x +x-3 x- x+3 x+7 - f(x)=: dx=: { + } dx x +x-3 x- x+3 f(x)= l x- -l x+3 +C f(0)=-l 3+C=0 C=l 3 f(x)= l x- -l x+3 +l 3 f(3)= l -l 6+l 3=l f{; ;}-f(0)={; ;+;!;+C}-{-;!;+C} f{; ;}-f(0)=;!;(p+) 08 f(x)=: si x cos x dx=: si x( cos x-)dx cos x=t x dt =-si x dx f(x)=: (t -)(-dt) f(x)=: (-t )dt f(x)=t-;3@;t +C f(x)=-;3@; cos x+cos x+c x +5x+8 x+5 06 f(x)=: dx=: {+ } dx x +4x+3 x +4x+3 x+5 x+5 a b = = + x +4x+3 (x+)(x+3) x+ x+3 x+5 (a+b)x+3a+b = (x+)(x+3) (x+)(x+3) a+b=3a+b=5 a=b=- f(0)=-;3@;++c= C=;3@; f(x)=-;3@; cos x+cos x+;3@; f{;3 ;}=-; ;+;!;+;3@;=;!#; p=q=3 p+q=5 5 6

169 09 f'(x)=xe -x - f(x)=: xe -x dx f(x)=-xe -x -: (-e -x )dx f(x)=-xe -x -e -x +C (-0) f(-)=c=0 f(x)=-(x+)e -x f()=- 3 e 0 f(x)+xf'(x)=(x+3)e x {xf(x)}'=(x+3)e x xf(x)=: (x+3)e x dx f(x)=: e si xdx f(x)=e si x-: e cos xdx f(x)=e si x-[e cos x-: (-e si x)dx] f(x)=e (si x-cos x)-: e si xdx f(x)=e (si x-cos x)-f(x) f(x)=;!;e (si x-cos x)+c f(0)=-;!;+c=-;!; C=0 f(x)=;!;e (si x-cos x) xf(x)=(x+3)e x -: e x dx xf(x)=(x+)e x +C f()=3e+c=3e C=0 xf(x)=(x+)e x x=3 3f(3)=7e f(3)=;3&;e f(x)=;!;e ;!;e (si x-cos x)=;!;e si x-cos x= ( e >0) ' si {x-;4 ;}=si {x-;4 ;}= x-;4 ;=;4 ; x-;4 ;=;4#;p ' f'(x)=x cos x=0 (0 x p) x=0 x=; ; f(x) x=; ; x=p ;#;p x 0 y ; ; y p f'(x) p f(x) f(x) ; ; f{; ;}=; ; f(x)=: x cos xdx f(x)=x si x-: si xdx f(x)=x si x+cos x+c f{; ;}=; ;+C=; ; C=0 f(x)=x si x+cos x f(0)=f(p)=- f(x) si x=t x dt =cos x dx f(x)=: f(x)=: f(x)=: cos x(+si x) si x +si x si x +t t dt dx cos xdx 7

170 f(x)=: { +;t!;} dt f(x)=-;t!;+l t +C f(x)=- f(x)=- t si x si x +l si x +C +l(si x)+c f(x)=-x e -x -xe -x -e -x +C f(x)=-(x +x+)e -x +C f(-)=-e+c=-e C=0 f(x)=-(x +x+)e -x f(0)=- ( 0<x<psi x>0) f{; ;}= -+C= C= Level f(x)=- si x +l(si x) f{;6 ;}=-+l ;!;+=-l 3x 0 f(x)=: dx 3x + 3x +=t x 0 f(x)=: (ta x+ta x) dx f(x)=: ta x(+ta x) dx f(x)=: ta x sec xdx ta x=tx dt =sec x dx f(x)=: t dt=;!;t +C=;!; ta x+c dt =6x dx f(x)=: f(x)=: 3x 3x + t dt f(x)=;!; l t +C dx f(x)=;!; l 3x + +C f(0)=c=0 C=0 f(x)=;!; l(3x +)+C ( 3x +>0) f(x)=;!; ta x f(4)-f(0)={;!; l 49+C}-C=l 7 f{;3 ;}=;#; f(x+h)-f(x) 03 lim =f'(x)=x e -x h 0 h f(x)=: x e -x dx l x 0 xf'(x)= l x f'(x)= x l x f(x)=: dx x l x=t x dt =;[!; dx f(x)=-x e -x -: (-xe -x )dx f(x)=: t dt=t +C=(l x) +C f(x)=-x e -x +: xe -x dx f(x)=-x e -x -xe -x -: (-e -x )dx f()=- f()=c=- f(x)=(l x) - f(e )=(l e ) -= 8

171 03 ;dî[;{e f(x) }=e f(x) f'(x)=e f(x) l x f'(x)=l x f(x)=: l xdx f(x)=x l x-: x ;[!; dx f(x)=x l x-x+c f()=-+c=3 C=4 f(x)=x l x-x+4 f(e)=e-e+4=4 4 f'(x) f(x) = ( f(x)>0) x f'(x) : dx=: dx f(x) l f(x)=x+c ( f(x)>0) l f(3)=3+c f(3)= C=-3 l f(x)=x-3 f(x)=e -3 f()= e 04 F()=f()-e=0 f()=e F(x)=xf(x)-x e x f(x)=f(x)+xf'(x)-xe -x e xf'(x)=xe +x e f'(x)=e +xe x>0 0 f(x)=: (x+)" x +x+3 dx " x +x+3=t x +x+3=t x dt t =x+ dx f(x)=: (e +xe ) dx f(x)=: t t dt f(x)=e +xe -: e dx f(x)=;3@;t +C f(x)=e +xe -e +C f(x)=e +xe +C f()=e+e+c=e C=-e f(x)=(x+)e -e f()=3e -e f(x)=;3@;(x +x+3)" x +x+3+c f(-)='3+c='3 C=0 f(x)=;3@;(x +x+3)" x +x+3 f()=8 8 Level : f(x){ g(x)+g'(x)} dx=f(x)g(x) x f(x){ g(x)+g'(x)}=f'(x)g(x)+f(x)g'(x) f(x)g(x)=f'(x)g(x) f(x)=f'(x) ( g(x)>0) xe (xæ0) 03 f'(x)=g si x (x<0) : xe dx=xe -: e dx=xe -e +C : si x dx=- cos x+c C C (x-)e +C (xæ0) f(x)=g - cos x+c (x<0) f{-; ;}=C =0 (x-)e +C (xæ0) f(x)=g - cos x (x<0) 9

172 f(x)x lim x +0 {(x-)e +C }= lim (- cos x) x -0 -+C =- C =- (x-)e - (xæ0) f(x)=g - cos x (x<0) f(3)=e - 04 g(x)=e -x f(x) x=0 g(0)=f(0)=- g(x)=e -x f(x) x g'(x)=-e -x f(x)+e -x f'(x) =-e -x (x-3)( f(x)-f '(x)=x-3) g(x)=: {-e -x (x-3)}dx g(x)=e -x (x-3)-: e -x dx g(x)=e -x (x-3)+e -x +C g(x)=e -x (x-)+c g(0)=- g(0)=-+c=- C=0 g(x)=e -x (x-) g()=;e!; l { f(x)+g(x)}=-x+l 3e f(x)+g(x)=e -x+l 3e =3e -x+ f'(x)=-g(x)g'(x)=-f(x) f'(x)-g'(x)=f(x)-g(x) : f'(x)-g'(x) f(x)-g(x) = ( f(x)>g(x)>0) dx=: dx l { f(x)-g(x)}=x+c C l { f(0)-g(0)}=c f(0)=eg(0)=e C = f'(x)-g'(x) f(x)-g(x) l { f(x)-g(x)}=x+ f(x)-g(x)=e x+ f(x)+g(x)=3e -x+ f(x)-g(x)=e x+ { f(x)} -{ g(x)} =3e f(x)+g(x)=3e -x+ f(x)-g(x)=e x+ f(x)=3e -x+ +e x+ f()=3+e f()=;!;(e +3) 0 f(x)=: cosfi xdx Level 3 f(x)=: cos x cos x cos xdx f'(x)=-g(x) g'(x)=-f(x) f'(x)+g'(x)=-{ f(x)+g(x)} : f'(x)+g'(x) f(x)+g(x) f'(x)+g'(x) f(x)+g(x) =- ( f(x)>g(x)>0) dx=: (-) dx l { f(x)+g(x)}=-x+c ( f(x)+g(x)>0) l { f(0)+g(0)}=c f(0)=eg(0)=e C=l 3e f(x)=: cos x(-si x)(-si x) dx si x=t x dt =cos x dx f(x)=: (-t )(-t )dt f(x)=: (t -t +)dt f(x)=;5!;tfi -;3@;t +t+c f(x)=;5!; sifi x-;3@; si x+si x+c f(0)=c= 0

173 f(x)=;5!; sifi si x+si x+ f{; g(0)=e f(0) +C=0 C=- g(x)=e f(x) - p=5q=3 f(x)=: x si xdx p+q=38 38 f(x)=-x cos x-: (-cos x)dx f(x)- 03 lim x = x 0 x- 0 { f(x)-}=0 lim x f(x)= f(x) f(x) lim x lim x f(x)=f()= f()= f(x)- f(x)-f() lim = lim x =f'()= x- x x- f'(x)=axe (x-) f'()=a= (x-) f'(x)=xe f(x)=-x cos x+: cos xdx f(x)=-x cos x+si x+c C f(0)=c =0 f(x)=-x cos x+si x g(x)=e -x cos x+si x - g{; ;}=e- f(x)=: xe (x-) dx f(x)=xe (x-) -: e (x-) dx f(x)=xe (x-) -;!;e (x-) +C f()=-;!;+c= C=;!; f(x)=xe (x-) -;!;e (x-) +;!; f()=;#;e +;!;=;!;(3e +) 04 f(x)=: x si xdx x f'(x)=x si x g(x)=: e f(x) x si xdx g(x)=: e f(x) f'(x)dx g(x)=e f(x) +C

174 03 ah 0 0 ;3!;pr h e+e 09 l l V«= _ (+)(+) V«=pr h_ 6 lim pr h V«=;3!;pr h (+)(+) 6 (+)(+) V«= lim [pr h_ ] 6 (+)(+) V«=pr h lim 6 ;3!;pr h 0 ;%; 0 ;H; 03 :) xdx= lim ;K+!{ k } a= ( a>0) a 3a a ;A; y S«h ka S«=;K+!{ _ } S«= S«= S«= ah ah ;K+! k _ ah(+) (+) ABC ah(+) ah lim S«= lim = ah k 04 x =+ Dx=;@; xº=x«=+ =3 k lim ;K+!{+ } ;@; =:!3 x dx a=3 05 :) (x+'x)dx=[;!;x +;3@;x ;#; ]) :) (x+'x)dx={;!;+;3@;}-(0+0) :) (x+'x) dx=;6&; 0 ;H; r r 3r r y V«06 :)» si xdx=[-cos x]») =(-cos p)-(-cos 0) = kr V«=;K+! [;H;_p{ } ] pr h V«= ;K+! k 07 :) l l e dx=[e ]) =e l -e =-=

175 08 :) (3x +4 'x)dx =:) 3x dx+:) 4 'x dx =[x ])+[3x ;3$; ]) =+3 =4 :_! (si x-e )dx =:_! si xdx-:_! e dx =;!;+ =;%; :) ' ; ; cos xdx+: cos xdx ' ; ; =:) cos xdx ; ; =[si x]) =-0 = =[-cos x]_!-[e ]_! ;%; =0-(e-e ) =-e+e 4 -e+e 09 :! {;[!;+ } dx+:! {;[!;- } dx x x =:! ;[!; dx = [l x]! = l :) (cos x+ )dx-:) (cos x- )dx =:) dx =:) dx =[ ]) l ={ - } l l = l 0 :_! x dx =:_0! (-x)dx+:) xdx x x =[- ]0_!+[ ]) l l kh 0 A AH S kh S :S={ } :h k S :S={ } : k S ={ } S V«h V«=;K+!{S } k =;K+![ { } h S ] Sh = ;K+! k Sh (+)(+) = _ 6 Sh (+)(+) = _ 6 V= lim V«=;3!;Sh { k } 3

176 (+)(+) 6 k { } _ =0k=5 5 (0+)(0+) { } _ =: 8 : 0 6 p+q=77+8=85 0 [][35] f(x) 0 :N «± f(x)dx< b-a b-a 03 lim ;K+! f{a+ k} =:Ab f(x) dx>0 x f(x)æ0 f(x)=0 D D 4 =-p 0 pæ (+)(+) 6 p x ;4 ;cos xæsi x ;4 ;<x ; ;cos x<si x ; ; :) cos x-si x dx ;4 ; ; ; =:) (cos x-si x)dx+: (-cos x+si x)dx ;4 ; ;4 ; =[si x+cos x]) +[-si x-cos x] =('-)+{(-)-(-')} =('-) ; ; ;4 ; 07 f(x) x=0 lim f(x)= lim f(x)=f(0) lim x -0 x -0 a= e = lim ( 'x+a)= x +0 x +0 : a f(x)dx=: f(x)dx l ;!; -l : a f(x)dx=: 0 e dx+:) (x;3!; +)dx l ;!; -l : a f(x) dx=[e ]0 +[;4#;x ;3$; +x]) l ;!; -l ; ; 04 :) {cos ;{;+si ;{;}{cos ;{;-si ;{;} dx ; ; =:) {cos ;{;-si ;{;} dx ; ; =:) cos xdx : a f(x)dx=(-e -l )+{;4#;+-0} l ;!; : a l ;!; : a l ;!; f(x)dx={-e l ;!; }+;4&; f(x) dx=;4(; ; ; =[si x]) =-0 = :) "ç4 dx=:) dx :) "ç4 dx=[ ]) l :) "ç4 dx= - = l l l 0 :) f(x)dx=:) (6x +ax)dx =[x +ax ]) =+a f()=6+a 4

177 +a=6+a a=-4 x x =[- -x]0_!+[ -x]) 4 4 ={-;4#;}+{-;4#;} 0 [0] -x +x (0 x<) x (x-) =g x -x ( x ) =-;#; :) x (x-) dx =:) (-x +x )dx+:! (x -x )dx =[-;4!;x +;3!;x ])+[;4!;x -;3!;x ]! ={; ;-0}+[;3$;-{-; ;}] =;#; 03 0 x f'(x)æ0 x 3 f'(x) 0 'x+ 0 :!4 dx x =:!4 { +;[!;} dx 'x =:!4 dx+:!4 ;[!; dx 'x =[x ;!; ]4!+[l x]4! =+l 4 =(+l ) :)3 f'(x) dx=:) f'(x)dx+:!3 {-f'(x)}dx 03 :)» si ;{;{si ;{;+cos ;{;} dx =[f(x)])+[-f(x)]3! =:)» {si ;{;+si ;{; cos ;{;} dx ={ f()-f(0)}+{-f(3)+f()} ={-(-3)}+(3+) =4+4=8 -cos x =:)» { +;!; si x} dx =;!;:)» (-cos x+si x)dx =;!;[x-si x-cos x]») =;!;(p+) Level p = + 04 :_! e - dx 0 :_! ( x -)(x + x +) dx =:_0! (-e )dx+:) (e -) dx =:_! ( x -)( x + x +) dx =[x-e ]0_!+[e -x]) =:_! ( x -)dx ={(0-)-(--e )}+{(e-)-(-0)} =;e!;+e- =:_0! (-x -)dx+:) (x -)dx 5

178 05 :) f(x) dx=:) (e -ax) dx ax :) f(x)dx=[e - ]) :) f(x)dx={e-;a;}-(-0) :) f(x)dx=e-;a;- f()=e-a :)» si x+'3 cos x dx =:)» si {x+;3 ;} dx ;3@;p` =:) si {x+;3 ;} dx+: p``[-si {x+;3 ;}]dx ;3@;p ;3@;p =[-cos {x+;3 ;}]) +[cos {x+;3 ;}] p ;3@;p e-;a;-=e-a =[-{-;!;}]+[{-;!;}-(-)] ;A;= a= =3+ =4 x (x<0) 0 f'(x)=g si x (x>0) x +C (x<0) f(x)=g C C -cos x+c (x>0) Level f{; ;}= si x+'3 cos x '3 ={;!; si x+ cos x} = si {x+;3 ;} f(x)=si x+'3 cos x0 x ;3@;p f(x)æ0;3@;p<x p f(x)<0 :)» si x+'3 cos x dx ;3@;p` =:) (si x+'3 cos x)dx +: ;3@;p p``(-si x-'3 cos x)dx ;3@;p =[-cos x+'3 si x]) +[cos x-'3 si x] p ;3@;p =[{;!;+;#;}-(-+0)]+[(--0)-{-;!;-;#;}] f{; ;}=-cos ; ;+C =0 C =0 f(x) x=0 lim f(x)= lim f(x) lim (x +C )= lim (-cos x) x -0 C =- x - (x<0) f(x)=g -cos x(xæ0) ; ; ; ; :_! f(x) dx=:_0! f(x) dx+:) f(x) dx ; ; ; ; :_! f(x) dx=:_0! (x -) dx+:) (-cos x)dx ; ; :_! x -0 x +0 x +0 x f(x) dx=[ -x]0_!+[-si x] ) 3 ; ; :_! f(x)dx=-;3@;+(-) ; ; :_! f(x)dx=-;3%; ; ; =3+ =4 03 f(x)>0f'(x)>0f''(x)>0 y=f(x) 6

179 y y=f(x) F(a)<4 F(a)+3>4 <F(a)<4 F(a)3 5 O 0 x A= f(k) [ ] 0 B=;!; f{;k;} [ ] k= 0 k= [ ] y y=f(x) y y=f(x) O 90x O 90 x [ ] [ ] C=:) 0 f(x)dxy=f(x) x x=0x=0 C<B<A Level p 0 3 S«p p S«=_;!; si = si 3 3 lim S«= lim si p 3 p si 3 p S«= lim ª º p 3 3 p si 3 S«=;3@;p lim 343 p 3 04 :Ab f(x) dx=[f(x)]ba=f(b)-f(a)=3 F(b)=F(a)+3 :Ac f(x) dx=[f(x)]ca=f(c)-f(a)=0 F(c)=F(a) F'(x)=f(x) y=f(x) y=f(x) F(x)=4 y=f(x) y=4 y y=f(x) F(b)=F(a)+3 4 y=4 F(a)=F(c) O a b c x S«=;3@;p p S _ lim S«=4_si _;3@;p=6p si 36 0 :) e dx= lim e ;K; ;!; k= :) e dx= lim ;!; e ;K; ) e ;!; (-e) :) e dx= lim { ;!;_ 3433 } ª -e ;!; 0 e ;!; (-e) ) :) e dx= lim 333 { } ª (- e ;!; ) 0 k= h=;!; h 0 p 36 7

180 lim 333= (- ) e ;!; lim h 0 =- lim h 0 = - h -e 3 e - 3 h 04 ) e ;!; (-e) :) e dx= lim { 3333 } ª (- e ;!; ) 0 :) e dx=(-e)_ lim e ;!; _ lim 3333 (- ) =(-e) (-) = e- f()=e ;!; a=-b=e- f()+a+b=e+(-)+(e-)=(e-) e ;!; ;!; l ;4 ; 03 ;!; ;4!;(e +) : 3 : 0 ;3@; 03 lim S«y=g(x)x= x y=f(x) y=g(x) y=x y=f(x) y= y y - p y=g(x) y=f(x) O - p x dx 0 x+=t = dt x=0t=x=t=3 :) (x+) dx=:!3 ;!;t dt =[;8!;t ]3! =: 8 :-;8!; =0 dx x+=t = dt x=0t=x=e-t=e e- :) x+ dx=:!e t dt =[l t]e! ; ; ; ; _; ;-:) si xdx=; ;-[-cos x]) =l e-l =; ;- = 0 dx 0 x =t x = dt x=0t=0x='pt=p :) 'p x si x dx=:)» ;!; si tdt 8

181 :) 'p x si x dx=[-;!; cos t]») :) 'p x si x dx=;!;+;!; :) 'p x si x dx= :) ;4 ; ta xdx=:) ;4 ; si x dx cos x cos x=t -si x x=0t=x=;4 ;t= :) ;4 ; si x dx=:! {- } dt cos x t =: dt t =[l t] =0-l =;!; l x=si h{-; ; h ; ;} = x=0h=0x=h=; ; :) " -x dx =:) ; ; cos h" -si h dh =:) ; ; cos h`dh =:) ; ; +cos h dh ; ; h si h =[ + ]) 4 ={;4 ;+ }-(0+0) =;4 ; ' ' ' ' dx dt ' dx dh =cos h l 3 =[;!;e ]) =;!;e l 3 -;!; = l x=t ;[!; = x=t=0x=et= :!e l x x dx=:) tdt =[;!;t ]) =;!; 04 f(x)=xg'(x)=si x f '(x)=g(x)=-cos x ;!; :) ; ; ; ; x si xdx=[-x cos x]) -:) ; ; (-cos x)dx ; ; =0-[-si x]) = dx dt f(x)=xg'(x)=cos x f '(x)=g(x)=si x :)» x cos xdx=[x si x]»)-:)» si xdx =(0-0)-[-cos x]») =-{-(-)} =- 05 f(x)=xg'(x)=e f '(x)=g(x)=e - ;!; l ;4 ; :) xe dx=[xe ])-:) e dx dx 03 x =t x = dt x=0t=0x="çl 3t=l 3 :) "çl 3 xe x l 3 dx=:) ;!;e dt =(e-0)-[e ]) =e-(e-) = 9

182 06 :!e l xdx=:!e l xdx f(x)=l xg'(x)= f '(x)=;[!;g(x)=x :!e l xdx=[x l x]e!-:!e dx =(e-0)-[x]e! =e-(e-) = f(x)=l xg'(x)=x f '(x)=;[!;g(x)=;!; x x x :!e x l xdx=[ `l x]e!-:!e dx e x = -[ ]e! 4 e e = -{ -;4!;} 4 =;4!;(e +) = lim x e [F(t)]/E x-e = lim x e =F'(e) =f(e) = k 09 lim { }7 =:) x dx k= x =[ ]) 8 =fi -0 =3 3k 3 lim æ + =:!4 'x dx k= =[;3@;x ;#; ]4! =: 3 :-;3@; F(x)-F(e) x-e ;4!;(e +) =: 3 : 07 F'(x)=x - F'()= -=3 F'(x)=si {x+; ;}-si x F'(0)=si ; ;-si 0=-0= 3 3 : 3 : 0 :_! (x +x +x)dx=:_! (x +x)dx+:_! x dx =0+:) x dx x =[ ]) 3 08 f(t)=" t +3 F'(t)=f(t) x+ x+ lim :! " t +3 dt= lim x 0 :! f(t)dt x x 0 x x+ = lim x 0 [F(t)]! x = lim =F'() =f() = f(t)=l tf'(t)=f(t) lim x e x-e x 0 :E/ l t dt= lim x e F(x+)-F() x x-e :E/ f(t) dt ;6 ; =;3@; : (si x+cos x+ta x)dx -;6 ; ;6 ; =: -;6 ; (si x+ta x)dx+: cos xdx -;6 ; ;6 ; =0+:) cos xdx ;6 ; =0+[si x]) = ;6 ; ;3@; 30

183 =;3$;e -[;9$;x ;#; e ]! =;3$;e -{;9$;e -;9$;} =;9*;e +;9$; dx 0 si x=t cos x = dt =;9$;(e +) x=0t=0x=; ;t= ; ; :) cos x 'ƒsi xdx=:) 't dt =[;3@;t ;#; ]) =;3@; dx 0 l x=t = x dt x=t=0x=et= (l x) :!e dx=:) t dt x t =[ ]) 4 05 f()=:! si (t -4)dt+=0+==a a= x f '(x)=si(x -4) x f"(x)={cos(x -4)}_x f"(a)=f"()=(cos 0)_4=4 06 f(t)=e cos pt F'(t)=f(t) lim :!/ e cos pt x dt x - = lim x :!/ f(t)dt x - 4 =;4!; = lim x [F(t)]/! x - F(x)-F() = lim x [ _ ] x- x+ 03 :) x "çe dx=:) xe ;{; dx F(x)-F() = lim x _ x- lim x x+ =[x e ;{; ]) -:) e ;{; dx =F'()_;!; =(4e-0)-[4e ;{; ]) =e _;!; =4e-(4e-4) =4 = e e e 04 :! 'x l xdx=:! x ;!; l xdx =[;3@;x ;#; e e l x]! -:! ;3@;x ;#; ;[!; dx e ={;3@;e -0}-:! ;3@;x ;!; dx p pk 07 lim ta k= 4 pk p =4 lim ta _ k= 4 4 ;4 ; =4:) ta xdx 3

184 ;4 ; =4:) ;4 ; =-4:) si x cos x dx (cos x)' cos x ;4 ; =-4[l (cos x)]) ' =-4{l -0} =-4l -;!; =l dx 03 :!/ f(t)dt=x -ax +ax x= 0=-a+a a= :!/ f(t)dt=x -ax +ax x f(x)=3x -4ax+a=3x -4x+ ( a=) f(3)=7-+= lim («'e+«"çe +«"çe +y+«"çe«) = lim «"çe k= k 04 lim f {+ } k= k = lim f{+ } k= =;!;:!3 f(x)dx = e ;K; lim k= =;!;:!3 (x +x)dx =:) e dx =;!;[;4!;x +;!;x ]3! =[e ])=e- =;!;{: 4 :+;(;-;4!;-;!;} =;!;(0+4) = dx 0 l x=t = x dt x=et=x=e t= e :E 3(l x) x 0 :@ 3p x si xdx dx=:! 3t dt=[t ]!=8-=7 =[x(-cos x)]3@» -:@ 3p (-cos x)dx ={3p-(-p)}-[-si x]3@» Level ;4 ; 0 :) ;4 ; =:) ;4 ; =:) cos x si x+cos x cos x si x+cos x cos x-si x si x+cos x dx+: ;4 ; 0 si x dx cos x+si x ;4 ; dx-:) dx si x+cos x=t si x cos x+si x dx =5p-(0-0) =5p (cos x-si x) dx dt = x=0t=x=;4 ;t=' 3

185 ;4 ; :) cos x-si x si x+cos x dx=:! ' dt t =[l t]! ' f(0)= f '(0)= f "(0)=f '(0)cos 0-f(0)si 0= =l '-0 =;!; l 05 lim { + + +y+ } = lim k= +k dx 0 x +=t x = dt = lim _ k= k + x=0t=x=t= :) x l(x +)dx =:! ;!; l tdt =;!; :! l tdt =:! dx x =[l x]! =l -l =l =;!;[[t l t]!-:! dt] =;!;[( l -0)-[t]!] Level =;!;{ l -(-)} =l -;!; :) f(t)dt k 03 lim x 0 :)/ cos tdt si x x = lim x 0 {;!; ;[!;:)/ cos tdt} si x f(x)= x x + k=:) f(t)dt +k(k) =;!; cos 0 =;!; 04 f(x)=:)/ f(t)cos t dt+ yy x f '(x)=f(x)cos x yy x f "(x)=f '(x)cos x-f(x)si x x=0 k=:) { k=:) k=:) +k} dt dt+:) k dt dt+[kt]) t k=:) dt+k yy t + t :) dt t +=s t + t dt ds t t + t t + t t + = t=0s=t=s= t :) dt=:! ds t + s 33

186 =;!; :! ;s!; ds =;!;[l s]! =;!; l F(b)=:Bb f(t)dt=0 F'(x)=f(x) y=f(x) y=f(x) y y=f(x) k=;!; l +k k=-;!; l O a b c x f(x)= f(0)=-l x x + -l y O a b c y=f(x) x 0 x«=t x - dx = dt x=0t=0x=t= a«=:) x - e x«dx F(x)=0 a«=:) x«x - e x«dx a«= a«= :) te dt {[te ])-:) e dt} k k 04 lim [{ } +]5 k= k k = lim [{ } +]5 _ k= =:) (x +)fi xdx a«= [(e-0)-[e ])] a«= a«= {e-(e-)} 0 0 0(0+) = = =55 = a«= 55 x +=t x = x=0t=x=t= :) (x +)fi x dx=:! tfi ;!; dt :) (x +)fi xdx=:! tfi dt :) (x +)fi xdx=[;6!;tfl ]! dx dt :) (x +)fi xdx=: 6 :-;6!; 03 F(a)=:Ba f(t)dt=-:ab f(t)dt :) (x +)fi xdx=: : :Ab f(t)dt>0 F(a)<0 d F'(x)= :B/ f(t)dt=f(x) dx F'(b)=f(b)=0 x=b F'(x) f(x) + - F(x) x=b Level

187 dx 0 f (x)=t { f (x)}' = dt f(f (x))=x x f '(f (x)) { f (x)}'= { f (x)}'= dx = yy f'(f (x)) dt f(x)=x +x y=x +xxy x=y +y x=0 y=f (0)=0x= y=f ()= x=0t=f (0)=0x=t=f ()= :) f (x) f'(f (x)) dx=:) t dt=[;!;t ]) =;!;-0=;!; f(x)- 0 lim x =3 x 0 x- 0 lim x { f(x)-}=f()-=0 f()= lim x lim x 0 lim x =f '()=3 =4 x 0 { f(x)-3}=f()-3=0 f()=3 lim x f(x)-f() x- f(x)-3 x- f(x)-f() x- f'(f (x)) =f '()=4 :! xf "(x)dx=[xf'(x)]!-:! f '(x)dx ={f '()-f '()}-[ f(x)]! 03 [] f(x) [3] f(x) :!3 f '(x) l f(x) dx =-:! f '(x)l f(x)dx+:@3 f '(x)l f(x)dx dx f(x)=t f '(x) = dt x=t=ex=t= x=3t=e -:! f '(x)l f(x)dx+:@3 f '(x)l f(x)dx e =-:E l t dt+:! l tdt e e =-{[t l t]e-:e dt}+{[t l t]! -:! dt} e =-[(0-e)-[t]E]+[(e -0)-[t]! ] =-{-e-(-e)}+{e -(e -)} =-(-)+(e +) =e + k p kp 04 PºOP = _ H {cos 0} cos S =;!;_cos p kp _ - - kp p lim S = lim { _cos _ } k= k= ; ; =:) ;!; cos xdx ; ; =;!; :) p +cos x dx ; ; =;4!; :) (+cos x) dx si x =;4!;[x+ ]) ; ; kp =(_4-3)-{ f()-f()} =5-(3-) =4 4 =;4!;[{; ;+ }-(0+0)] =;8 ; 35

188 05 S y S=:) y dy=[ ])=;3!;-0=;3!; 3 ;6!; ;3!; 0 ;6!; ;3!; 0 03 e- e- 04 ;6!; ;(; 05 '3-;3 ; ;4 ;-;!; l 06 e+;e!;- e-'e+ 07 : 3 : 08 ;5 ; p 0 S ; ; S=:) si xdx ; ; S=[-cos x]) S=0-(-) S= y O y=si x p - x 09 ; ; ; ;(e -) S y y=e x : 3 : S=:) e dx e 0 y y=x -x S=[e ]) S=e- O x= x O S x y=l x x=e S S=:) e dy y y=l x y= S=:) x -x dx S=[e ]) O x S=:) (-x +x)dx S=e- x x S=[- + 3 ]) S=[{-;3!;+;!;}-(0+0)] S=;6!; y x=y y= 04 y=x y=x x x =xx(x-)=0 x=0 x= S y e- e- y=x y=x O x O x S=:) (x-x )dx x x S=[ - ]) 3 36

189 S={;!;-;3!;}-(0-0)=;6!; x=y y x=y+ y y =y+ (y+)(y-)=0 O y=- y= - - S S=:_! (y+-y )dy x=y+ 4 x=y x ;4 ; S=:) si x {- } dx cos x ;4 ; S=[x+l(cos x)]) ' S={;4 ;+l }-(0+0) S=;4 ;-;!; l '3-;3 ; ;4 ;-;!; l y y S=[ +y- ]_! 3 06 S y=e -x y y=e x S={+4-;3*;}-{;!;-+;3!;} S=:) (e -e )dx S=;(; S=[e +e ]) ;6!; ;(; S=(e+e )-(+) O x= x S=e+;e!;- 05 y=si x y=;!; x si x=;!; - y y=si x y= - O p p p p 6 x y=l x x=e y=l x x=e ;}; S S=:) (e -e ;}; )dy y O y=lx y=l x y= x x=;6 ; x=;6%;p S ;6%;p S=: {si x-;!;} dx ;6 ; ;6%;p S=[-cos x-;!;x] ;6 ; S=[e -e ;}; ]) S=(e-e ;!; )-(-) S=e-'e+ e+;e!;- e-'e+ '3 '3 p S={ -; ;p}-{- - } S='3-;3 ; 07 S(x)=x V V=:!3 S(x)dx y=ta xy= x ta x= x=;4 ; S ;4 ; S=:) (-ta x)dx y O y=ta x p - 4 p - y= x V=:!3 x dx x V=[ ]3! 3 V=9-;3!;=: 3 : : 3 : 37

190 08 V V=p:) y dx V=p:) x dx xfi V=p[ ]) 5 V=;5 ; V V=p:)» y dx V=p:)» si xdx V=p:)» dx V=; ; :)» (-cos x)dx V=; ;[x-;!; si x]») V=; ;{(p-0)-(0-0)} V= p -cos x y O y O y=si x x= p - p y=x x x V=; ;(e -) ; ; ; ;(e -) 0 t= t=3 P :!3 v(t) dt=:!3 3(t -t) dt :!3 v(t)dt=[t -3t ]3! =(7-7)-(-3) = t=3 P +:)3 3(t -t) dt =+[t -3t ]3) =+{(7-7)-(0-0)} = t= t=3 P :!3 3(t -t) dt =:! (-3t +6t) dt+:@3 (3t -6t) dt ;5 ; p =[-t +3t ]!+[t -3t ]3@ 09 V V=p:) x dy V=p:) ydy y V=p[ ]) y O y=x y= x ={(-8+)-(-+3)}+{(7-7)-(8-)} =+4=6 6 P s dx dy s=:)3 æ { } +{ } dt dt dt V=; ; s=:)3 " (t) +(t -) dt y=l x x=e V V=p:) x dy y y=l x y= s=:)3 " t +t + dt s=:)3 (t +) dt V=p:) e dy O x s=[;3!;t +t]3) V=p[;!;e ]) V=p {;!;e -;!;} s=(9+3)-(0+0) s= 38

191 ;#; f '(x)=x ;!; l l=:)3 " +{ f '(x)} dx 0 (00) (0)(e)(0e) y e y=xe x l=:)3 æ +{x ;!; } dx l=:)3 'ƒ+x dx y=xe x= x O x l=[;3@;(+x) ;#; ]3) S l=: 3 :-;3@;=: 3 : S=_e-:) xe dx : 3 : S=e-{[xe ])-:) e dx} S=e-[(e-0)-[e ])] S=e-{e-(e-)} S=e y=si x cos x=;!; si x [0p] y=;!; si x x y 03 f(x)=si x f '(x)=cos x y=si xy=cos x x si x=cos x x=;4 ; y O y=si x p - 4 p - y=cos x x O y= -si x p - 4 p p p x S ;4 ; S=:) (cos x-si x)dx ;4 ; S=[si x+cos x]) S ; ; S=:) ;!; si x dx ; ; S=:) si x dx ' ' S={ + }-(0+) S='- ; ; S=[-;!; cos x]) S=;!;-{-;!;} S= 04 y=l(x+)y=l x x l (x+)=l x x+=x x= 39

192 y l y=l x y=l(x+) 06 y= l x xæy=l x x=e 0<x<y=-l x x=e y O - y=l xy=l (x+) x l y=-l x y=l x y=l x=;!;e x=e - S O x S=:) l [;!;e -(e -)]dy=:) l {-;!;e } dy V S=[y-;!;e ]) l V=p:) l (e ) dy-p:) l (e ) dy S=(l -)-{0-;!;} V=p:) l e dy-p:) l e dy S=l -!; V=p[;!;e ]) l -p[-;!;e - ]) l V=p{;!;e l -;!;}-p{-;!;e - l +;!;} x V=p{;!;e l 4 -;!;}-p{-;!;e l ;4!; +;!;} ;!; S=:) l(x+)dx+: {l(x+)-l x} dx ;!; V=;#;p-;8#;p 05 S(x) S(x)=;!;(ta x+cot x) si ;3 ; V=;8(;p a+b=7 7 '3 S(x)= (+ta x+cot x) 4 V ;3 ; V=: ;6 ; '3 V= : 4 '3 4 ;6 ; ;3 ; (+ta x+cot x) dx (+ta x+cot x) dx ;3 ; '3 V= : (sec x+cosec x) dx 4 ;6 ; ;3 ; '3 V= [ta x-cot x] 4 ;6 ; '3 V= [{'3- }-{ -'3}] 4 '3 '3 '3 V= _{'3- }= 4 '3 07 P t=0 t=; ; s ; ; s=:) dx dy æ { } +{ } dt dt dt ; ; s=:) " (-3 cos t si t) +(3 si t cos t) dt ; ; s=:) " 9 cos t si t+ 9 si t cos tdt ; ; s=:) 3" (cos t si t) (cos t+si t)dt ; ; s=:) 3 si t cos tdt ; ; s=;#; :) si t dt ; ; s=;#; [-;!; cos t]) 40

193 s=;#;[;!;-{-;!;}] s=;#; dy 08 =;!;{x-;[!;} l dx l=:!e æ +{ dy dx } dx l=:!e æ +;4!;{x-;[!;} dx b-a=e-(e-)= 0 y='x y='ƒ-x+0 x 'x='ƒ-x+0 x=-x+0 x=5 V y 0 y= -x+0 y= x 5 O 5 0 x l=:!e æ ;4!;{x+;[!;} dx V=p:)5 xdx+p:% 0 (-x+0)dx l=;!; :!e {x+;[!;} dx x x V=p[ ]5)+p[- +0x]%0 l=;!; [;!;x +l x]e! V=: :p+p [{-50+00}-{-: :+50}] l=;!; [{;!;e +}-{;!;+0}] l=;4!;(e +) V=5p a= P t=0 t=p :)» dx dy æ { } +{ } dt dt dt =:)» " {4(-si t+cos t)} +(- si t) dt =:)» " 6(-si t) +4 si t dt 0 y=e y=xe x e =xe x=( e +0) =:)» " 4(-si t) +si t dt a=:) (e -xe )dx =:)» " (-si t) dt a=:) (-x)e dx =:)» -si t dt a=[(-x)e ]) +:) e dx =:)» (-si t)dt ( - si t ) a=e- b=:! (xe -e )dx a=:! (x-)e dx a=[(x-)e ]! -:! e dx a=e =[t+;!; cos t])» =_4p =8p a=8 a =

194 Level S(x) S(x)=(' l x) _si ;6 ;=;!; l x V V=:! ;!; l xdx 0 y y='x- V=;!; :! l xdx V=;!;{[x l x]!-:! dx} O - 4 x=4 x V=;!;[( l -0)-[x]!] V=;!;{ l -(-)} y='x- S V=l -;!; S=:) (-'x)dx+:!4 ('x-)dx S=[x-;3@;x ;#; ])+[;3@;x ;#; -x]4! S=[{ -;3@;}-(0-0)]+[{: 3 :- }-{;3@;- }] 04 x +y =5(xæ0yæ0)y='ƒx- xy y S=;3!;+;3%; '5 x +y =5 y= x- S= -'5 O '5 x 0 y'=e x+ x= e y=e x+ y y-e =e (x-) y=e x-e S S=:) (e x+ -e x+e )dx S=[;!;e x+ -e x +e x]) e O x y=e 3 x-e 3 -e 3 S={;!;e -e +e }-{;!;e-0+0} S=;!;(e -e) 03 ' l x ;6 ; e 3 -'5 x x +('ƒx-) =5 x +x-=5 x +x-6=0 (x-)(x+3)=0 x= ( xæ) V V=p:) (5-x )dx-p:! ('ƒx-) dx x x V=p[5x- ])-p[ -x]! 3 V=p[{0-;3*;}-(0-0)]-p[(-)-{;!;- }] V=: 3 :p-;!;p=: 6 :p 4

195 05 P s dx dy s=:) æ { } +{ } dt dt dt k =ab k>0 k='aåb s=:) " (3t) +(4t ) dt s=:) " 9t +6t dt 0 y=f '(x) y y=f'(x) s=:) t" 6t +9 dt 6t +9=u 3t = t=0u=9t=u=5 s=:( 5 ;3 ; 'udu s=;3 ; :( 5 'udu s=;3 ;[;3@;u ;#; ](5 dt du O 3 S S=:!3 f '(x) dx S=:! f'(x)dx+:@3 {- f'(x)} dx x s=;3 ;{;3@;_5 -;3@;_3 } S=[ f(x)]!+[-f(x)]3@ s=;3 ;_;3@;(5-3 ) s=;3 ;_;3@;_98 s=;$4(; S={ f()-f()}+{-f(3)+f()} S=-f()+ f()-f(3) S=-+_3- S=4 p+q=49+4=73 73 Level :Ak dx=:kb dx x x [l x]ka=[l x]bk l k-l a=l b-l k l k=l a+l b l k =l ab 03 y=f(x)y=g(x) y= f(x)+g(x) = f(x)+g(x)=4 y=f(x)y=g(x) x=x=3 5 :!3 f(x)-g(x) dx=5 V V=p:! [{ f(x)} -{g(x)} ] dx +p:@3 [{ g(x)} -{ f(x)} ] dx V=p:! { f(x)-g(x)}{ f(x)+g(x)} dx +p:@3 { g(x)-f(x)}{ f(x)+g(x)} dx V=p:! 4{ f(x)-g(x)} dx+p:@3 4{g(x)-f(x)} dx 43

196 V=4p:!3 f(x)-g(x) dx V=4p_5=0p a= s(0)=s(c)= :)c v(t)dt=0 s(b)=0s(c)= :Bc v(t) dt= :)c v(t) dt=:)a v(t) dt+:ab v(t) dt+:bc v(t) dt =:)a v(t) dt+:ab v(t) dt+ =0 :Ab v(t)dt=-:)a v(t)dt- :)a v(t)dt=as(t) s(t) a+ f(x)=si ; ; x x= + +si ; ; f()= lim 3 = + x> f(x)= lim x +0 f(x)= +0 f(x)=x x +3 si ; ;x x«3 +3 x«si ; ;x (0 x ) f(x)=[ x (x>) y=f(x) (00) (0)(4)(04) y=f(x) xx= y 4 y=f(x) O a b c t s(t)= t(0c) y=x y=si p -x O S S=_4-:) f(x)dx x Level S=8-{ :) si ; ;x dx+:! x dx} x S=8-{[-;ç@;cos ; ;x])+[ ]!} 3 S=8-[{0+;ç@;}+{;3*;-;3!;}] 0 0 x< x + +si ; ;x f(x)= lim x + 0+si ; ;x f(x)= 3 0+ S=: 3 :-;ç@; p+q+r=7+3+= 0 y=f(x)-g(x) 44

197 x y a y b y c y f '(x)-g'(x) f(x)-g(x) f(a)=g(a) f(a)-g(a)=0 y y=f(x)-g(x) a O b c x f(c)-g(c)<0 S=:Ac f(x)-g(x) dx 't 't cos ;4 ;='t_ =' t S(t) S(t)=p('t) -('åt) =pt-t V V=:) (py-y) dy V=[; ;y -y ]) V=; ;- ' 't p - 4 t 't S+:Ac { f(x)-g(x)} dx f(b)=g(b) f(b)-g(b)=0 y=f(x)-g(x) y y=f(x)-g(x) a O c b x S=:Ac f(x)-g(x) dx S=:Ac {g(x)-f(x)} dx :Ac { f(x)-g(x)} dx <S [ac] y=f(x)-g(x) x=b y=f(x)-g(x) f(b)-g(b) f(b)>g(b) y 04 P tp μap=t POA=t P(cos tsi t)q(cos tsi t+t) Q p s dx dy s=:)» æ { } +{ } dt dt dt s=:)» " (-si t) +(cos t+) dt s=:)» " si t+cos t + cos t+ dt s=:)» " (+cos t)dt s=:)» æ 4 cos ;T; dt s=:)» cos ;T; dt s=:)» cos ;T; dt { 0 t pcos ;T;æ0} s=[4 si ;T; ]»)s a O b y=f(x)-g(x) c x s=4-0=4 03 y=t(0 t ) y 45

198 0 5 5!_;5!;=(5-)!=4!=4 3 = C = C = =5 4 (4-)!=3!=3 =6 5_6= _ 3 _ 3 _ 3 _ 3 P =3_3_3_3_3= _ 3 _ 3 _ 3 P - P =3-3 =8-7= = aaabc 5! 3! =5_4= abc (ab)(bc)(ca) 3 (ab) aaab 4! =4 3!! aabb 4! =6!! abbb 4! =4!3! (bc)(ca) 3_(4+6+4)=4 08 a b A P aabbb 5! =0!3! P B aab 3! =3! 46

199 0_3= abcdef b d d f bdf x xxxace 6! 3! =6 5 4= _5_3=60 04 X x f(-x)=-f(x) f(0)=0 f() f(-)=-f() f() f(-)=-f() f() 5 f() 5 f 5_5= abcde C =5 a bcde (4-)!=3!=6 5_6= ! =30!! ! =60! 30+60=90 4 (4-)!=3!=6 4 4!=4 6_4=44 06 S={34567} C = C = = > > > aaa bbaaabb 5! =0 3!! 47

200 07 _0=0 0 P B Q R A A B PQR PQR A P B _= A Q B 3! 5! _ =30!!3! A R B 5! _ =5 4! +30+5= ! =30!! ABBBC 5! =0 3! ABBBB 5! =5 4! = ABCDEFCD EF C = =6 CD AB A BXXX XCD 4! =! 6_=7 08 abc xyz. xyzd6 6 abcddd 6! 3! =6 5 4=0 03 A B A B ABBCC A Q P B A B A B 48

201 A P B 4! 4! { -}_ =5_4=0!! 3! A Q B 4! 4! _{ -}=4_5=0 3!!! 0+0=40 Level AaBbCcDd (4-)!=3!=6 =6 6_6=96 0 P = P = P = P = fi P =fi (fi -) fi = = abcdefg 40 C =7 abcdefg a (6-)!=5!=0 7_0= a b 7 6 C = C = = aaabb 5! 5 4 = =0 3!! _0=0 0 Level (8)(7)(36)(45) (8) (7)(36)(45) 3!=6 (7)(36)(45) =8 6_8=48 0 abc (abc) 3 49

202 P = = a bb ccc 6 c c c abb 3! =3! c c c abb 3! =3! c c b c ab!= c b c c ab!= 3+3++= !=4 6_4=44 a a a a 0 A= a a a a 0 4 P = = a b ca B aabbcc 6!!!! =90 Level PQ Q B A P A B 7! =35 3!4! A P B 5! _ =5 4! A Q B 50

203 5! _=5 4! = ! 3! - =9!! 03 4!-3!=8 0 4! -=3 3! 9+8+3= H = C = C = =5 H = C = C = C = abc H = C = C = C = =5 03 x+y+z+w=5 (xyzw) xyzw 5 H = C = C = C H = = (a+b+c) abc 4 H = C = C = C 6 5 H = = X={34} Y={56789} f Xx x x <x f(x ) f(x ) f

204 H = C = C H = 4 3 H =70 06 (x+3)fi a=xb=3 (x+3)fi «C a«b = C (x)fi 3 = C fi 3 xfi xfi =x 5-r=3 r= C fi 3 = C C fi 3 = 8 9 C fi 3 =70 07 {x -;[!;}6 a=x b=-;[!; {x -;[!;}6 «C a«b = C (x )fl {-;[!;}r «C a«b = C (-) x x =x -3r=0 r=4 C (-) = C (-) = C 6 5 C (-) = =5 70 Cº+ C + C + C +y+ ºC = Cº+ C + C + C +y+ ºC ( Cº= Cº) = C + C + C +y+ ºC ( Cº+ C = C ) = C + C +y+ ºC ( C + C = C ) = ºC + ºC = C H = C = C = C H = = H = C = C = C =6 56-6= H = C =ªC =ªC = = ºCº+ ºC + ºC +y+ ºC º= log ( ºCº+ ºC + ºC +y+ ºC º) =log =0 09 «C +«C =«C 03 x+y+z+w=0 xyzw (xyzw) x=a+y=b+z=c+w=d+ a+b+c+d=3 (abcd) H = C = =

205 04 (x+y+z) xyz xπ yœ z p+q+r=7 pqr (pqr) p=a+q=b+r=c+ a+b+c=4 abc (abc) 6 5 H = C = C = =5 fl = Cº+ C + C +y+ C yy` x=-=6 0= Cº- C + C -y+ C yy` + fl =( Cº+ C + C +y+ C ) Cº+ C + C +y+ C = fi log ( Cº+ C + C +y+ C )=log fi =5 05 {x - }4 x C (x ) {- }r = C (-) x fi x x fi =x 8-5r=3 r= x C (-) =-8 06 {ax-;[!;}7 C (ax) {-;[!;}r = C a (-) x x =x fi 7-r=-5 r=6 xfi C a(-)fl =7a= a=3 07 (+x)«=«cº+«c x+«c x +y+«c«x«x==6 3 fl = Cº+ C + C +y+ fl C log ( Cº+ C + C +y+ fl C )=log 3 fl =6 08 (+x)«=«cº+«c x+«c x +y+«c«x«yy` x== x+y+z= xyz 3 H = C = C =6 (3)(3)(3)()()() 5 (3)(3) (3) a+b=3 ab 3 H = C = C =4 ()()() =5 0 (+x)«= «C x x «C «C = k=0 (-) = =0(-0)(+9)=0 =0 ( >0) 0 53

206 03 ;K+!«C =«C +«C +y+«c«=«- «=«Cº+«C +«C +y+«c««- 3 =468y Level ;!; xyz x+y+z=7 xyz (xyz) 9 8 H =ªC =ªC = =36 a=;!; 04 (+x) = «Cº+ «C x+ «C x + «C x +y+ «C «x - + «C «x yy` x= = «Cº+ «C + «C + «C +y+ «C «+ «C «yy` x=- 0= «Cº- «C + «C - «C +y- «C «+ «C «yy` - =( «C + «C + «C +y+ «C «) «C + «C + «C +y+ «C «= - - =5 - = -=9=0 =5 ;!; 5 0 (x+y+z)fi xπ yœ z (xy)(yz)(zx) 3 xy p+q=5 pq (pq) H = C = C =4 3_4= 03 (+ax)fi C (ax) = C a x x C a (a+x)fl C a 6-r x x C a C a = C a C = C a ( a>0) = a 3 0=0a Level X Y3 X f H _ H = C _ C =6_0=60 0 x+y+z =7 xyz (xyz) z=0z=z= z=0 x+y=7 xy (xy) H = C = C =8 54

207 z= x+y=6 xy (xy) H = C = C =7 z= x+y=3 xy (xy) H = C = C =4 (xyz) 8+7+4=9 03 (+i) (+i) =+i+i =i (+i) =(i) =4i =-4 (+i) =(-4) = yy` (+i) = ºCº+ ºC i- ºC - ºC i+ ºC + ºC i - ºC - ºC i+y+ ºC + ºC i- ºC - ºC ªi+ ºC º yy` ºCº- ºC +y- ºC + ºC º= log ( ºCº- ºC +y- ºC + ºC º)=log = a+b {(a+b)+c} c ºC _(a+b) c (a+b) bfi C _a bfi (a+b+c) a bfi c 0! 8! 0! ºC _ C = _ =!8! 5!3! 3!5!! += ºC + C += ºC + C = + 3 +=0 (a+b+c)«aπ bœ c a+b {(a+b)+c}«c «C (a+b)«c (a+b)«bœ «Cœa«œ bœ =«Cœaπ bœ ( p+q+r=) (a+b+c)«aπ bœ c «C _«Cœ! (-r)! = _ r!(-r)! q!(-r-q)!! (-r)! = _ ( p+q+r=) r!(-r)! q!p! =! p!q!r! Level (x-) C (x) (-) r=03 (x+) Cßx ßß s=034 (x-) (x+) C (x) (-) _ Cßx ßß = C (-) Cßß x ß x 7-r-s=3 r+s=4 (rs)=(04)(3)()(3) (rs)=(04) Cº (-) C =8 (rs)=(3) C (-) C =-384 (rs)=() C (-) C =44 (rs)=(3) C (-) C =-8 x 8+(-384)+44+(-8)=-0 0 x+y+z<5xyz (xyz) 55

208 x+y+z=0 Hº= Cº= x+y+z= H = C =3 x+y+z= 4 3 H = C = =6 x+y+z=3 5 4 H = C = C = =0 fi x+y+z=4 6 5 H = C = C = =5 (xyz) =35 a+b+c= H _ H = C _ C =30 3 x+y+z= a+b+c=3 H _ H = C _ C =30 fi x+y+z= a+b+c= H _ H = C _ C = =6 03 ;K+) «H =«Hº+«H +«H +y+«h«=«cº+«c +«C +y+ «C«=«Cº+«C +«C +y+ «C«( «Cº=«Cº) =«C +«C +y+ «C«( «Cº+«C =«C ) =«C +y+ «C«( «C +«C =«C ) = «C«+ «C«= «C«04 4 A B C x yz abc 4 x+y+z=4 6 5 H = C = C = =5 4 a+b+c=4 6 5 H = C = C = =5 3 x+y+z=3 56

209 AÇ ={()()()} ;6!; S S={3456}A={35} A u{}{4}{6}{4}{6}{46}{46} 8 {46} =8 0 (ab) A AÇ 3 AÇ ={()()()} AÇ ={()()()} _ P = = =4 3 3!=6 34 3!= =0 ;4@8);=; ; C C _ C C _ C C =;7$; 05 (ab) ()()y (66) 36 3 ()() 4 (3)()(3) 3 ;3 6;+;3 6;=;3 6; 06 AB P(A;B)=0 P(A'B)=P(A)+P(B)-P(A;B) ;3@;=;!;+P(B)-0 P(B)=;6!; 07 AA AÇ 3_3 P(AÇ )= =;4!; 36 ;6!; 57

210 P(A)=-P(AÇ )=-;4!;=;4#; AA AÇ 3 C P(AÇ )= =; 4; ºC P(A)=-P(AÇ )=-; 4;=;!4&; 09 C A A AÇ 6 P(AÇ )= 6 C =;5@; P(A)=-P(AÇ )=-;5@;=;5#; 0 (ab) ()()()(3) ()()()(3) ()()()(3) (3)(3)(3)(33) 6 ()()()()(3)(3) 6 ; 6;=;8#; a b a-b = a=b= a=b= a=b=3 a=3b= 3 ;!;;4!;;4!; ;!;_;4!;+;4!;_;!;+;4!;_;4!;+;4!;_;4!;=;8#; 0 () 0-() 0 3 ºC =0 A B (-) «C º «C = (0-) (-) (0-) (-)(0-) P(A)= = 0 40 (0-)(9-) «C º «C = (0-)(9-) (0-)(9-) P(B)= = 0 40 P(A)=3P(B) (-)(0-) (0-)(9-) =3_ =3(9-) =7 03 T A B 400 P(A)= =;5@;

211 P(B)= = P(A;B)= =;4!; 000 P(A'B)=P(A)+P(B)-P(A;B) 49 P(A'B)=;5@;+ -;4!; 00 P(A'B)=;!5^; 3 C C P(C)= C A BB CC A P(A'B'C)=P(A)+P(B)+P(C) P(A'B'C)= P(A'B'C)= P(A'B'C)=;6!6(; C + C + C C ªC 3 A 8 B C _ C C _ C P(A)= =;8!4%;=; 8; ªC C _ C C _ C P(B)= =;8@4!;=;4!; ªC C _ C _ C C _ C _ C P(A;B)= =;8 4;=; ; ªC P(A'B)=; 8;+;4!;-; ;=; ; p=66q=9 p+q= !_4! A B 4!_ 4!_ P(A)= =; 4; 4!_4! 4!_ C _ 4!_ C _ P(B)= =;4!; 4!_4! AB P(A)+P(B)=; 4;+;4!;=; 4; C A B3 C C C P(A)= C C P(B)= C C 07 P(AÇ ;BÇ )=;6!; P((A'B)Ç )=;6!; -P(A'B)=;6!; P(A'B)=;6%; P(A;BÇ )=;!; 59

212 P(B)=P(A'B)-P(A;BÇ ) P(B)=;6%;-;!; P(B)=;3!; P(BÇ )=-P(B)=-;3!;=;3@; 08 C + C + C + C + C + C = -( Cº+ C )=0 3 AA AÇ C P(AÇ )= =;4 0; 0 P(A)=-P(AÇ )=-;4 0;=;4#0#; «Cº+«C +«C +y+«c«=«p(a)=;3!; ( P(A)æ0) P(A'B)=P(A)+P(B)-P(A;B) P(A'B)=;3!;+;3!;-0=;3@; 03 g -=4 f g Á f Z 0 -; 4;=;7^; X 3 X 3 f f Y 3 4 Y 3 4 g g Z 0 Z 0 p+q=7+6= C C _ C _ C C =; 0;=;5@; 0 A B P(A;B)=0 P(A)=P(B)P(A)P(B)=;9!; {P(A)} =;9!; Level ! 3 4!! 4!_! 4!_! =;5@; 5! 60

213 0 JH007 6! 6! 007 4!! JH P 4! 3_ P! P 343 = =;3@; 6! 6_5 3! _ P AA AÇ 3 C P(AÇ )= =;4 ; ªC P(A)=-P(AÇ )=-;4 ;=;4#&; 04 6 ABCDEF 3 C _ C!! AB M M MÇ AB C P(MÇ )= 33 =;5!; C _ C _3_!! P(M)=-P(MÇ )=-;5!;=;5$; 4!! 0 4 4!!!!_!!_! =;6!; 4! 0 X Y f P X Y 3 C X 34 Y3456 H C _ H C _ H 3_ C 5 ;pq;= = = P 6 6 p=6q=5 p+q= 03 A B A;B (A'B)Ç P(A)=;3@;P(B)=;!;P(A;B)=;3!; P(A'B)=P(A)+P(B)-P(A;B) P(A'B)=;3@;+;!;-;3!; P(A'B)=;6%; P((A'B)Ç )=-P(A'B)=-;6%;=;6!; Level a =3a =9a =7a =a =3y AA AÇ

214 C + C _ C + C _ C P(AÇ )= =; 4 0; C P(A)=-P(AÇ )=-; 4 0;=;!4@0&; p=40q=7 p+q=67 67 Level C _ C _ C _ 3! AB C _ C AB C! C _ C _! C _ C _! 33 =; 5; C _ C _ C _3 3! C - C _3 (++++)_=7_= C - C _3 =;3!*;=;ª6; !5 S AT B A S! S 3!!_3!!_3! P(A)= =; 0; 5! B T P T 3! P _3! P _3! P(B)= =; 0; 5! A;B TS ST C _ C + C _ C! ( C _ C + C _ C )_! ( C _ C + C _ C )_! P(A;B)= =; 5; 5! P(A'B)=P(A)+P(B)-P(A;B) P(A'B)=; 0;+; 0;-; 5; P(A'B)=;3!; 6

215 C 6 3k3k-3k-k 3k k k k+3k3k-+3k- C + C _ C =0+30=40 0 C + C _ C =3+9= 40- C =;3 0; 09 0 A B P(A)=;3!7(;P(A;B)=;3ª7; P(B A)= P(A;B) P(A) ;7#; ;3!; ;3ª7; P(B A)= =;ª9; ;3!7(; ;ª9; P(A;B) P(A;B) 0 P(A B)= = =;4!; P(B) ;3!; P(A;B)=; ; P(B A)= P(A;B) P(A) ; ; P(B A)= =;6!; ;!; 03 ;7$; 63

216 3 3 ;6#; ;7$;_;6#;=;7@; P(AÇ )=0.6_0.7=0.4 P(A) P(A)=-P(AÇ ) =-0.4 = A B P(A;B)=P(A)P(B A) P(A;B)=;7#;_;6@; P(A;B)=;7!; P(AÇ ;B)=P(AÇ )P(B AÇ ) P(AÇ ;B)=;7$;_;6#; P(AÇ ;B)=;7@; P(B)=P(A;B)+P(AÇ ;B) P(B)=;7!;+;7@; P(B)=;7#; 07 AB P(A B)=P(A)=;3@; ABABÇ P(BÇ A)=P(BÇ )=;!; P(B)=;!; P(A;B)=P(A)P(B)=;3@;_;!;=;3!; ;3!; 08 C {;!;} {;!;} =6{;!;}4 C {;!;}3 {;!;} =4{;!;}4 ;7#; 6{;!;}4 +4{;!;}4 =0{;!;}4 =;8%; 05 P(A;B)=P(A)P(B A) P(A;B)=;3@;_;4!; P(A;B)=;6!; 09 ;3@; 4 3 C {;3@;}3 {;3!;} =;8#@; 06 AA AÇ 4 4 C {;3@;}4 =;8!^; P(AÇ ) ;8#@;+;8!^;=;8$*;=;!7^; 64

217 ;3!; ;4!; =;3!; 0 A B P(A;B)= P(B)= P(A B)=;3@; k 3 P(A;B) 80 P(A B)= = 3 P(B) k k P(A B)= =;3@; k+0 k=40 40 k=40 k k+0 k 80 k+0 80 =;3@; 03 A B ;5!;_;5#;=; 5; A B ;5#;_;5!;=; 5; ; 5;+; 5;=; 5; 04 A 4 B 3 ;6!;_;6%;=;3 6; A 3 B ;6!;_;6#;=;3 6; A B ;6!;_;6!;=;3 6; S A B 00 P(A;B)= =;5!; P(B)= =;5#; 000 P(A B)= P(A;B) P(B) ;5!; P(A B)= =;3!; ;5#; ;3 6;+;3 6;+;3 6;=;3ª6;=;4!; 05 P(A;BÇ )=;3!;P(A;B)=;6!; P(A)=;3!;+;6!;=;!; A B P(A;B)=P(A)P(B) ;4!; ;3!; ;6!;=;!; P(B) 65

218 P(B)=;3!; 06 A B P(A B)=P(A) P(B A)=P(B) P(A)+P(B)=;6%; P(A'B)=P(A)+P(B)-P(A;B) P(A;B)=;6%;-;6!; P(A;B)=;3@; 07 ;!; 4 C {;!;} {;!;} C ;!; ;!; C {;!;} {;!;} _ C ;!; ;!;= 6; 08 3 x y x+y=3 yy A B 5 x+5y= yy x=y= 3 C {;!;} {;!;} =;8#; A B P(B A)= P(B A)= P(B A)= P(B A)=; 3; 0 A B P(A B)=P(A)=;8#; P(A'B)=P(A)+P(B)-P(A;B) =P(A)+P(B)-P(A)P(B) B A ; 0;_;!; ;ª0;_;5!; AÇ ; 0;_;!; ;ª0;_;5$; P(B;A) P(A) P(B;A) P(B;A)+P(BÇ ;A) ; 0;_;!; ; 0;_;!;+;ª0;_;5!; ;!;=;8#;+P(B)-;8#;P(B) P(B)=;5!; A BA BÇ P(A;BÇ )=P(A)P(BÇ )=;8#;_{ -;5!;}=; 0; A B C C B A ;!;_;6!;=; ; ;3!;_;3!;=;9!; ;6!;_;!:=; ; BÇ 66

219 ; ;+;9!;+; ;=; 8; B ;5$;_;4!;=;5!; ;5!; Level ;5!; P(A;B) 0 P(A B)= = ;3!; P(B) P(A;B)=;3!;P(B) P(A'B)=P(A)+P(B)-P(A;B) ;3@;=P(B)+P(B)-;3!; P(B) ;3@;=;3*;P(B) P(B)=;4!; C _;3!;_;3@;_;3!;=; 7; A B P(A)=;3!; P(B;A) = + 0 A B A 3_3+3_3 P(A)= =;!; 6 A;B 3_3 P(A;B)= =;4!; 6 =;3!;_;3!;_;3@;_;3!;+;3!;_;3@;_;3!;_;3!; =;8 ; P(B A)= ;8 ; = ;3!; =; 7; P(B;A) P(A) ;4!; P(B A)= =;!; ;!; Level 03 A 5 C

220 0 000 A B (A)= =40 (A;B)=80+90=70 ; 0 0º0; P(A;B) P(B A)= = =;4!&; P(A) ; 0 0º0; p=4, q=7 p+q=58 0 A B B A B A B ;3!; ;3!; ;3!; 4 ;3!;_;3!;_;3!;_;3!;=;8 ; 4! _;3!;_;3!;_{;3!;} =;8!@;! 4! _{;3!;} _{;3!;} =;8 ;!_! ;8 ;+;8!@;+;8 ;=;8!(; ;3@;_;3!;+;3!;_;3@;=;9$; 03 ) x 00-x x 0-x 300 Level A B 0 [ ] ;3!;[ ] P(A)=;3!0)0);=;3!; ;3@; P(B)= 80+x 300 P(A;B)= x 300 AB P(A;B)=P(A)P(B) 4 A 4 B P(A)=P(A;B)+P(A;BÇ ) P(A)=;3!;_;3!;+;3!;_;3@;=;3!; P(A;B)=;3!;_;3!;=;9!; x =;3!;_ 300 3x=80+x x=40 80+x ;9!; P(A;B) P(B A)= = =;3!; P(A) ;3!; 68

221 0 0 ºC + + C + C =0 0 P(A)= =;9$; ºC C + C = P(B)= = 5; ºC C + C + C + C =8 P(A;B)= P(A)=;9$; 8 ºC =;4 5; ;4 5; P(A;B) P(B A)= = =;5@; P(A) ;9$; P(B)+P(B A) A B 0 0 ;!!; ;!!; 0 ;6!; X C _ C 3_ P(X=)= = =;!; C 4_3 C 3 P(X=)= = =;!; C 4_3 X X P(X=x) ;!; ;!; ;!; ;!; 03 4 a b c a+b+c=4 yy a+c=3 yy b+c= yy a=b=c= ;!;+a+;3!;= a=-;!;-;3!;=;6!; 03 ;!;+a+;4!;= a=-;!;-;4!;=;4!; ;6!; 4!! _{;!;} _;3!;_;6!;=;6!; E(X)=0_;!;+_;4!;+_;4!;=;4!;+;!;=;4#; 04 69

222 a+a+a=3a= V(X)=pq=64_;!;_;!;=6 a=;3!; E(X)=_;3!;+_;3!;+3_;3!;= E(X )= _;3!;+ _;3!;+3 _;3!;=: 3 : V(X)=E(X )-{E(X)} =: 3 :- =;3@; 0 0 ;3@; r(x)=" V(X)=æ;3@;= ' E(-3X+6)=-3E(X)+6=-3_+6=0 V(X)=4 r(x)=" V(X)= r(-3x+6)= -3 r(x)=3_=6 E(-3X+6)+r(-3X+6)=6 0 P(X=)+P(X=)+y+P(X=5) =a+4a+y+5a =55a= a=;5 5; 06 r(x)=3 V(X)={r(X)} =3 =9 P(X 4)=-P(X=5) P(X 4)=-5_;5 5; V(X-4)= V(X)=4_9=36 P(X 4)=-; ; 07 E(3X-)=3E(X)-=5 E(X)= r(3x-)=3r(x)=3 r(x)= E(X)_r(X)=_= P(X 4)=; ; 0 X03 P(Xæ)=P(X=)+P(X=3) C _ C P(Xæ)= + ºC Cº_ C ºC 08 =00p=; 0; r(x)=' pq r(x)='ƒp(-p) 6_5 6_5_4 4 _ 3 P(Xæ)= + 0_9_8 0_9_8 3 3 r(x)=æ 00_; 0;_;ª0; P(Xæ)=; º0;+; º0; r(x)='9=3 P(Xæ)=;3@; 09 ;!; ;3@; 64 X B{64;!;} 03 ;4!;+;3!;+b= 70

223 b=-;3!;-;4!;= -4-3 =; ; V(X)=;#0#;-{;#;} X : 4 : V(X)=;#0#;-;4(; E(X)=_;4!;+a_;3!;+6_; ;=;3A;+: 4 :=: 4 : V(X)=;@0!; ;3A;= a=3 V(Y)=V(0X-0) =0 V(X) E(X )= _;4!;+3 _;3!;+6 _; ; V(Y)=00_;@0!; E(X )=;4!;+3+5=: 4 : V(Y)=05 04 X0 X=0 P(X=0)=;!;_;!;=;4!; X= P(X=)=;!;_;!;_=;!; X= P(X=)=;!;_;!;=;4!; 06 X0 8_7 C P(X=0)= = 3 =;4@5*; ºC 0_9 C _ C P(X=)= = _8 3 =;4!5^; ºC 0_9 C P(X=)= = 3 =;4 5; ºC 0_9 E(X)=0_;4!;+_;!;+_;4!;= E(X)=0_;4@5*;+_;4!5^;+_;4 5;=;5@; E(X )=0 _;4!;+ _;!;+ _;4!;=;#; E(X )=0 _;4@5*;+ _;4!5^;+ _;4 5;=;9$; V(X)=E(X )-{E(X)} =;#;-=;!; r(x)=æ;!;= ' V(X)=E(X )-{E(X)} V(X)=;9$;-{;5@;} V(X)=;9$;-; 5; 05 E(X)=0_; 0;+_; 0;+_; 0;+3_; 0; E(X)= E(X)=;#; E(X )=0 _; 0;+ _; 0;+ _; 0;+3 _; 0; E(X )= E(X )=;#0#; V(X)=E(X )-{E(X)} V(X)=; 5; r(x)=" V(X)=; 5; r(50x+5)=50r(x)=50_; 5;= =0.5 XB(00.5) V(X)=0_0.5_0.75 7

224 V(X)=0_;4!;_;4#; V(X)=: 8 : 08 X B(00p) 6 V(X)=00p(-p)=6 5p -5p+4=0 (5p-)(5p-4)=0 p=;5!; { 0<p<;!;} E(X)=00_;5!;=0 E(X )=V(X)+{E(X)} =6+0 =46 V(X)=E(X )-{E(X)} =: 7 :-=;7$; 7XV(7X) V(7X)=49V(X)=49_;7$;=8 03 ;!;_;!;=;4!; X B{0;4!;} V(X)=0_;4!;_;4#;=: 8 : V(4X+)=6V(X)= Level P(0 X )=P(X=0)+P(X=)+P(X=) 3+a P(0 X )=;8!;+ +;8!; 8 P(0 X )= a+5 8 a+5 8 =;8&; a= X 0 P(X=i)+P(X=4-i)=;4!; P(X=0)+P(X=4)=;4!; P(X=)+P(X=3)=;4!; P(X=) =-{P(X=0)+P(X=)+P(X=3)+P(X=4)} =-{;4!;+;4!;} X - 0 P(X=x) ;8!; ;8!; ;8%; ;8!; =;!; XE(X) E(X)=-_;8!;+0_;8!;+_;8%;+_;8!;=;4#; 0 E(X)=0_;7@;+_;7#;+_;7@;= E(X )=0 _;7@;+ _;7#;+ _;7@;=: 7 : 0 ;3!;+a+a+a=;3!;+3a= 3a=;3@; a=;9@; E(X)=_;3!;+_;9@;+3_;9@;+4_;9@; 7

225 E(X)=;3!;+: 9 : E(X)=;3&; P(X<)=-;5!; P(X<)=;5$; 03 V(X)=E(X )-{E(X)} =4 yy E(Y)=E(3X+4)=3E(X)+4=7 E(X)= E(X )- =4 E(X )=5 yy 04 X B{0;5!;} X P(X=x)= ºC x {;5!;} x {;5$;} 0-x (x=0y0) P(X=)+P(X=8) = ºC {;5!;} {;5$;}8 + ºC {;5!;}8 {;5$;} fl = ºC + ºC ( ºC = ºC ) 5 5 = ºC = ºC a=0b= a+b= fl + 5 ( +) =6 +3+5=9 X0 6_5 C _ P(X=0)= = 3 =;7!; C 5_4 3 _ C _ªC P(X=)= = 6_9 33 =;3!5*; C 5_4 _ 9_8 ªC _ P(X=)= = 3 =;3!5@; C 5_4 3 _ E(X)=0_;7!;+_;3!5*;+_;3!5@;=;5^; E(X )=0 _;7!;+ _;3!5*;+ _;3!5@;=;3^5^; V(X)=E(X )-{E(X)} V(X)=;3^5^;-{;5^;} V(X)=;3^5^;-;#5^; V(X)=; 7 5; p+q=75+78=53 Level X0 P(X<)=-P(X=) C P(X<)=- C P(X<)= _5 _ 03 ;4!;+;3@;+a=;!!;+a= a=; ; E(X)=_;4!;+_;3@;+4_; ;=;@#; E(X )= _;4!;+ _;3@;+4 _; ;=: 4 : V(X)=E(X )-{E(X)} 73

226 V(X)=: 4 :-{;@#;} V(X)=: 4 :-;%4@4(; V(X)=; 4 4; V(X+3)= V(X) V(X+3)=44_; 4 4; V(X+3)=83 04 X B{30;3!;} E(X)=30_;3!;=0 30 (x-0) P(X=x)V(X) x=0 30 (x-0) P(X=x)=V(X) x=0 (x-0) P(X=x)=30_;3!;_;3@; (x-0) P(X=x)=: 3º: P(X=5)= C _;7$;_;6#;_;5#;_;4@;_;3@; P(X=4)=+ C _;7#;_;6@;_;5$;_;4#;_;3!; P(X=4)=;3!5@;+;3 5; P(X=4)=;3!5*; E(X)=3_;7!;+4_;3!5@;+5_;3!5*; E(X)= =: 3 5 : E(35X-0)=35E(X)-0 E(35X-0)=35_: 3 5 :-0=43 W BX=4 3 3 (WWBW)(WBWW)(BWWW) (BBWB)(BWBB)(WBBB) 0 C C y C C Level 3 - O x X 345 P(X=3)=;7$;_;6#;_;5@;+;7#;_;6@;_;5!; P(X=3)=;3 5;+;3 5; P(X=3)=;7!; P(X=4)= C _;7$;_;6#;_;5#;_;4@; - C C 3 C C y 3 C C P(X=4)=+ C _;7#;_;6@;_;5$;_;4!; - O x P(X=4)=;3ª5;+;3 5;=;3!5@; - 74

227 C C 456 C (-'3) C C " (-) +('3) = C C C C P(X=)=; 4;=;6!; P(X=)=-{;6!;+;6!;}=;3@; E(X)=0_;6!;+_;3@;+_;6!;= C y E(X )=0 _;6!;+ _;3@;+ _;6!;=;3$; '3 V(X)=E(X )-{E(X)} V(X)=;3$;- =;3!; - O - C x r(x)=" V(X)= '3 3 C C X P(X=)=;3@; P(X=)=;3!; E(X)=_;3@;+_;3!;=;3$; 04 A B P(A)=0.6P(B A)=0.5 P(A;B)=P(A)P(B A) P(A;B)=0.6_0.5=0.3 XB(000.3) V(X)=00_0.3_0.7= E(X )= _;3@;+ _;3!;= V(X)=E(X )-{E(X)} V(X)=-{;3$;} V(X)=;9@; V(6X-3)=6 V(X) V(6X-3)=36_;9@; V(6X-3)= (5-)!=4 X=0!_!=4 P(X=0)=; 4;=;6!; X=!_!=4 75

228 04 E(X)=:! xf(x) dx E(X)=:! dx E(X)=[;9@;x ]! E(X)=: 9 : E(X )=:! x f(x) dx E(X )=:! ;3@;x dx 0 :) f(x)dx= E(X )=[;6!; x ]! k k :) kx dx=[ x ])= = 3 3 E(X )=;%; k=3 V(X)=E(X )-{E(X)} V(X)=;%;-{: 9 :} ;#; 0 P{ X ;#;}=:! ;!;x dx V(X)=;%;-: 8ª : ;#; P{ X ;#;}=[;4!;x ]! V(X)= P{ X ;#;}=;4!;{;4(;- } V(X)=; 6 ; P{ X ;#;}=; 6; 03 :)3 f(x)dx= k :)3 kx dx=[ x ]3)=9k= 3 05 P(-. Z.67) =P(-. Z 0)+P(0 Z.67) =P(0 Z.)+P(0 Z.67) = =0.867 k=;9!; f(x)=;9!;x E(X)=:)3 xf(x) dx E(X)=:)3 ;9!;x dx E(X)=[;3 6;x ]3) P(4 X 8)=P{ Z } P(4 X 8)=P(- Z ) P(4 X 8)=P(0 Z ) P(4 X 8)=_0.343 P(4 X 8)=0.686 E(X)=;4(; 07 X B{450;3!;} E(X)=450_;3!;=50 76

229 =450 X N(500 ) P(40 X 50) =P{ Z } 0 0 =P(- Z 0) =P(0 Z ) = ;6#;=;!; 00 X B{00;!;} E(X)=00_;!;=50 V(X)=00_;!;_;!;=5 =00 X N(505 ) P(60 X 65) =P{ Z } 5 5 =P( Z 3) =P(0 Z 3)-P(0 Z ) = =0.05 :) kx(-x)dx=k[;!;x -;3!;x ]) :) kx(-x) dx=;6k;= k=6 0 :!e f(x)dx= :!e k l x dx=k:!e l xdx =k[x l x-x]e! =k= P( X ) =:! l xdx =[x l x-x]! =(l -)-(-) =l - 03 E(X)=:) xf(x) dx E(X)=:) x x- dx E(X)=:) x(-x) dx+:! x(x-) dx E(X)=:) (x-x ) dx+:! (x -x) dx E(X)=[;!;x -;3!;x ])+[;3!;x -;!;x ]! 6 E(X)={;!;-;3!;}+{;3&;-;#;} 0 :) f(x)dx= :) kx(-x) dx=k:) (x-x ) dx E(X)=-+= E(X+)=E(X)+=_+=3 04 X X N(0050 ) P(50 X 300) =P{ Z } =P(- Z ) 77

230 =P(- Z 0)+P(0 Z ) =P(0 Z )+P(0 Z ) = = X X N(805 ) P(X<70)=P{Z< } 5 P(X<70)=P(Z<-) =P(Z>) =0.5-P(0 Z ) = = X V(X)=00_; 0;_;ª0;=9 =00 X N(03 ) 4-0 P(X 4)=P{Z } 3 P(X 4)=P(Z -) P(X 4)=P(Zæ) P(X 4)=0.5-P(0 Z ) P(X 4)= P(X 4)= p= X B{6;3!;} E(X)=6_;3!;=54 0 P( X )=:! f(x) dx V(X)=6_;3!;_;3@;=36 P( X )=:! b(x-)dx =6 X N(546 ) P(Xæ60)=P{Zæ } 6 P(Xæ60)=P(Zæ) P(Xæ60)=0.5-P(0 Z ) P(Xæ60)=0.5-;!;P( Z ) P( X )=b[;!; x -x]! P( X )=;B;=;6A; b=;3a; P(0 X )=:) f(x) dx P(Xæ60)=0.5-;!;_0.686 P(0 X )=:) a(-x)dx P(Xæ60)= P(0 X )=a[x-;!;x ]) P(Xæ60)=0.587 P(0 X )=;A; P(0 X )= ;A;+;6A;= X X B{00; 0;} E(X)=00_; 0;=0 a=;#;b=;!; a-b= 78

231 0 :) f(x)dx= yy E(X)=;4!; :) xf(x)dx=;4!; yy :) (ax+5)f(x)dx=a:) xf(x)dx+5:) f(x)dx :) (ax+5)f(x)dx=a_;4!;+5=0 a=0 03 XX N( ) P(Xæ000)=P{Zæ } 500 P(Xæ000)=P(Zæ0.5) P(Xæ000)=0.5-P(0 Z 0.5) P(Xæ000)= P(Xæ000)=0.3 P(X<000)=-P(Xæ000) =-0.3 = m A B P(A;B)=0.7_0.05=0.035 P(AÇ ;B)=0.3_0.5=0.045 P(B)=P(A;B)+P(AÇ ;B)=0.08 P(A;B) P(A B)= = =; 6; P(B) 0.08 Level a=;!; P(0 X )=P{;!; X b} :) ;!;xdx=: b ;!;xdx ;!; [;4!;x ])=[;4!;x ]b ;!; ;4!;=;4!;b -; 6;b =;4%; b= ab= '5 '5 4 { bæ;!;} 0 :Ab xf(x) dx=e(x)=k :Ab x f(x) dx=e(x )=4k V(X)={r(X)} =('3) =3 V(X)=E(X )-{E(X)} =4k-k =3 k -4k+3=0 (k-)(k-3)=0 k= k=3 k4 4 k -4k+3=0 k -4 - =4 X-m 03 P( X-m.5r)=P{.5} r P( X-m.5r)=P( Z.5) =P(-.5 Z.5) =P(0 Z.5) =_0.433 = :) f(x)dx= :) ax dx=[;a; x ])=a= 04 XB{00;4!;} E(X)=00_;4!;=300 79

232 V(X)=00_;4!;_;4#;=5 =00 X N(3005 ) P{ X 00 -;4!; <;4 0;}=P( X-300 <30) X-300 =P{ <} 5 =P( Z <) =P(-<Z<) =P(0<Z<) =_0.477 = f(x)=-;8!;x+;!; E(X)=:)4 x{-;8!;x+;!;}dx E(X)=:)4 {-;8!;x +;!;x}dx E(X)=[-; 4;x +;4!;x ]4) E(X)=-;3*;+4 E(X)=;3$; E(X )=:)4 x {-;8!;x+;!;}dx Level E(X )=:)4 {-;8!;x +;!;x }dx E(X )=[-;3 ;x +;6!;x ]4) E(X )=-8+: 3 : 0 P( X 3)=:@3 f(x) dx E(X )=;3*; V(X)=E(X )-{E(X)} P( X 3)=:)3 f(x)dx-:) f(x)dx yy V(X)=;3*;-{;3$;} y=f(x) x= :)4 f(x)dx= :) f(x)dx=;!; V(X)=;9*; r(x)=" V(X)= ' 3 P( X 3)=:)3 f(x) dx-:) f(x) dx P( X 3)=;4#;-;!; 03 X X N( ) P(45.07 X 45.93) P( X 3)=;4!; =P{ Z } =P(-.5 Z.5) =P(0 Z.5) 0 :)4 f(x)dx= =_0.484 = ;!;_4_k= k=;!; 0000_0.9684=

233 04 X B{00;5!;} E(X)=00_;5!;=0 V(X)=00_;5!;_;5$;=6 =00 X N(04 ) P(8 X 8)=P{ Z } 4 4 P(8 X 8)=P(-0.5 Z ) =P(-0.5 Z 0)+P(0 Z ) =P(0 Z 0.5)+P(0 Z ) = = Level :)6 f(x)dx= ;!;_: :_k=: 4 :k= k=; ; i(-x)=(-x) f(-x)=x f(x)=i(x) E(X )=:_@ x f(x) dx=:) x f(x) dx=8 V(X )=E(X )-{E(X )} =8 g(x)=f(x-) X =X + E(X )=E(X +)=E(X )+= V(X )=V(X +)=V(X )=8 E(X )+V(X )=0 03 X X N(500 ) % a a-50 P(Xæa)=P{Zæ } 0 a-50 P(Xæa)=0.5-P{0 Z } 0 P(Xæa)=0.0 a-50 P{0 Z }= P(0 Z.05)=0.48 a-50 =.05 0 a-50=0.5 a=70.5 A 70.5-(80+55)=35.5 P(X<4)=-P(4 X 6) P(X<4)=-;!;_(6-4)_{; ;_;!;} P(X<4)=-;!; ; ; P(X<4)=;ª; 0 E(X )=:_@ xf(x)dx h(x)=xf(x) h(-x)=-xf(-x)=-xf(x)=-h(x) E(X )=:_@ xf(x)dx=0 04 ABC z z z z = =0.75 z = =0.66 z = = z >z >z BAC E(X )=:_@ x f(x)dx i(x)=x f(x) 8

234 [ ] [ ] 00(b-a)=00( )= % 40% p=0.6q= _0.4 r(^p)=æ = 00 ' E(X )=m=0 r 4 r(x )= = = ' 'å6 E(X )_r(x )=0 0 X E(X )=m=50 r r(x )= = =;5@; ' 'å5 X N{50{;5@;} } 0 06 p=; 0; E(^p)=; 0; 9 _ 0 0 V(^p)= =;90!0; 8 =8 ^p N{; 0;;90!0;} P(^p 0.)=P Z ª º æ ;90!0; P(X 49)=P{Z } 5 P(X 49)=P(Z -.5) P(X 49)=P(Zæ.5) P(X 49)=0.5-P(0 Z.5) P(X 49)= P(X 49)= xæ=0r=4=6 P( Z.96)=0.95 m95 % 4 4 [0-.96_ 0+.96_ ] 'å6 'å6 [ ] 04 xæ=80 r=5 =00 P( Z.58)=0.99 m99 % 5 5 [80-.58_ _ ] ' 00 ' 00 P(^p 0.)=P(Z 3) P(^p 0.)=0.5+P(0 Z 3) P(^p 0.)= =00^p=0. P( Z.96)=0.95 p 95 % 0._0.8 0._0.8 [0.-.96æ æ ] [0.-.96_ _0.04] [ ] 08 =400^p=0.5 P( Z.6)=0.99 p 99% 0.5_ _0.5 [0.5-.6æ æ ] [0.5-.6_ _0.05] [ ] 00(b-a)=00_0.3=3 8

235 r(x )= r r = = 'å6 4 r=4 r =V(X)=6 V(X-3)=4V(X)=64 0 x x + x =x = X = = P(X =)=;!;_;!;=;4!; x =x = x =x = + X = =.5 P(X =.5)=;!;_;3!;+;3!;_;!;=;3!; x =x =3 x =x = x =3x = +3 + X = = = P(X =)=;!;_;6!;+;3!;_;3!;+;6!;_;!;=; 8; P(X )=;4!;+;3!;+; 8;=;3#6!; 03 X X N( ) 00 X 0.05 E(X )=0.5r(X )= =0.005 ' 00 X N( ) X -0.5 Z= Z N(0) P( X 0.500) =P{ Z } =P(-0.04 Z 0.04) =P(0 Z 0.04) =_0.06 = =5xÆ=00r?s=0 99 % 0 0 [00-.58_ _ ] 'å5 'å5 [ ] [ ] a=89.68b=0.3 b-a= b-a=_.58_ =0.64 'å5 05 r= 95% [xæ-.96_ xæ+.96_ ] ' ' a=xæ-.96_ b=xæ+.96_ ' ' b-a=_.96_ ' 'æ3.9 æ _ E(^p)=0.V(^p)= ^pn{0. P(^pæ0.5)=P ª Zæ P(^pæ0.5)=P Zæ ª 0.3 º 3 ' ' P(^pæ0.5)=P{Zæ } æ } º 6 83

236 P(^pæ0.5)=0.5-P{0 Z P{0 Z } 0.43 P( Z.5)=0.86 P(0 Z.5)=0.43 ' 6 8 ' 6.5' 9 8 ' 6 }æ r s s=0=00xæ=45 m 95% 0 0 [45-.96_ _ ] ' 00 ' 00 [ ] =5^p=;@5);=;5$; P( Z )=0.95 p 95 % [ ;5$;-» «4 _ «;5$;+» [0.8-_; 5;0.8+_; 5;] [ ] [ ] 4 _ ] 03 p95% ^p^q ^p^q [^p-.96æ ^p+.96æ ] [ ] ^p= ^p= X X^p= 300 X 300 =0.75 X= X X N(600 ) 5 X 0 N{60 } N(60 ) 5 P(5X æ450)=p(x æ58) P(5Xæ450)=P{Zæ } P(5Xæ450)=P(Zæ-) P(5Xæ450)=0.5+P(0 Z ) P(5Xæ450)= P(5Xæ450)=0.843 Level P«=5«=5 = r= r V(X )= = = 84

237 r 4 0 V(X )= = =4 =4 E(X +)=E(X +4) =E(X )+4 =V(X )+{E(X )} +4 =V(X )+{E(X)} +4 = =7 03 r 6 X r V(X )= yy 6 r r 6 <rr <6rr(r-6)<0 0<r<6 r y ^p=;ª0º0;=0.9 ^q=0. =00 P( Z.58)=0.99 p99 % 0.9_0. 0.9_0. [ _æ _æ ] [ _ _0.03] a=0.9b=0.03 a 0.9 = =30 b X X X 3 4 P(X=x) ;3!; ;3!; ;6!; ;6!; E(X)=_;3!;+_;3!;+3_;6!;+4_;6!;=: 6 : E(X )= _;3!;+ _;3!;+3 _;6!;+4 _;6!;=: 6 : V(X)=E(X )-{E(X)} V(X)=: 6 :-{: 6 :} V(X)=;3$6!; 4 V(X) 36 V(X )= = =;7 ; XN(mr ) r m=00 =8 4 r =56=6 X N(006 ) P(84 X 3) =P{ Z } 6 6 =P(- Z ) =P(- Z 0)+P(0 Z ) =P(0 Z )+P(0 Z ) = =0.885 Level X X N(458 ) 4 X N(454 ) P(4X 64)=P(X 4) 4-45 P(4X 0)=P{Z } 4 P(4X 0)=P(Z -) =P(Zæ) 85

238 =0.5-P(0 Z ) = =0.587 P(X )=-P(X >) 9+ P(X )=- 7 P(X )=;!7&; x +x +y+x ºº 04 x = x = =8 00 (x -8) +(x -8) +y+(x ºº-8) s = 00- s =;9(9(;= P( Z.96)=0.95 m 95% [8-.96_ 8+.96_ ] ' 00 ' 00 [ ] [ ] Level E(X)=_;3!;+_;3!;+a_;3!;= E(X )=E(X) 3+a = a= _3_3=7 P(X )=-P(X >) X > ! _3=9! a 3 0 PVC X X N(0.06 )4 PVC X N{ } N{{ } } 4 5 P(4 X æ60)=p{x æ } P(4 X 60)=PªZæ 0.5+Pª0 Z Pª0 Z æ0.05 ºæ P(0 Z.7)= æ.7-5 ºæ ºæ æ0 yy = = <0 æ ^p E(^p)=p p(-p) V(^p)= 00 =00 ^p N{p p(-p) }

239 0.4-p P(^pæ0.4)=PªZæ º 'ƒp(-p) 0 4-0p P(^pæ0.4)=P{Zæ }=0.977 'ƒp(-p) 0p P{0 Z }=0.977 'ƒp(-p) 0p-4 P{0 Z }=0.477 'ƒp(-p) P(0 Z )= p-4 = 'ƒp(-p) 5p-='ƒp(-p) yy` 5p -0p+4=p-p 6p -p+4=0 (p-)(3p-4)=0 p=;!; p=; 3; p=; 3; p=;!; 00p=50 04 N(m ) k X N{m{ E(X )=mv(x )=;k!; f(m)=p{x.96_ } 'k.96_-m 'k f(m)=pªz º 'k f(m)=p(z.96-m'k ) f(0)=p(z.96) 'k } } f(0)= = f(m)=p(z.96-m'k ) m f(m) 87

240 memo