유형편(수Ⅰ)-해설-01-OK

0 0 64 0 04 4 05 8 06 8 0 0 0 0 04 05 06 65 07 08 09 0 8 4 5 6 7 8 7 9 0 0 "ç' 7_æ 6_ ' 56 "ç' 7_æ 6_ ' 56={( )! }! ' _ 45_ " 4 " 4 9~ ' "ç' 7_æ 6_ ' 56=' 4= 4 0 # 8 -# # 8 -# = # ( ) -# = # -( # 8 -# = #-{-(} =fl =64 64 0 8 5 a 0 log ' a a=fi'8= 5# log a=5# log _log 4_log 5_y_log 8 log log 4 log 5 log 8 = 4_ 4_ 4_y_ 4 log log log 4 log 7 =log 8= =logå =4! a 4! = a= =8 8 a>0, a+, b>0, b+, c>0, c+ logå b log a= logå b log c logç a= 06 log 50!0 n a 5555555555 0 na log 50!0=log 500 =-log(00_5) log 50!0=-log 00-log 5=--log 5 log 50!0=--log : º:=--(-log ) log 50!0=-+log log 50!0 - log n=-, a=log 44 =0 -na =0 log =0 log =0 log 8 =8 8 0 na 0 log ' a=0 log a=0_5#= 04 a log +log 4+log 5 =600 a log +log 4+log 5=log 60 a log 60 =60 log a =600=60 log a= a= =4 4 approach 0 a=fi 'ƒ-5 afi =-5 a a 05 logå (log )+logå (log 4)+logå (log 5)+y+logå (log 8)=4! a logå(log )+logå(log 4)+logå(log 5)+y+logå(log 8) =logå(log _log 4_log 5_y_log 8) a=fi ' -å5-5 5 afi =-5 afi =-5 a a a<0 a >0 a 00

a n a>0 n jk «'a n a-«'a n a a>0 n jk «'a n a a<0 n jk «'a n a a<0 n (') ' {(') ' } ' =(') =, 05 (x- '6)(x- '8)<0 g (x- '6)(x-fl 'ß)<0 0 a, b, c afl =, bfi =5, c =7(abc)«n afl = a= 6!, bfi =5 b=5 5!, c =7 c=7! (abc)«=( 6! 5 5! 7! )«yy n 6, 5, 0 0 0 ABCA BC DAB = '9, AC ='å7, BD =' CD CD =x AB :AC =BD :CD '9:'å7=':x '9 x='å7 ' @ x= #! x= #! 5555444 = #+!-@ = $ = '8å @ B '9 ' A D '7 C '6= " 6, '8= " 8, fl 'å= "ç 6 = _8, 8 = = _, = _9 fl 'å< '8< '6 (x- '6)(x- '8)<0 '8<x< '6 (x- '6)(x-fl'å)<0 fl'å<x< '6 '8<x< '6 x, y, z 06 a, b c a b=a _ 'a, a c=logå cå (6 8) 6 8=6 _ 'å6=( ) _ 8$ = _! = +! = : : p 'x, q 'y, r 'z (x>0, y>0, z>0) p, q, r p, q, r nk ø a mk = n ø a m k ABCA BC D AB :AC =BD :CD B D A C (6 8)= : : =log ( : : ) =log fl fi =65 65 07 OO log x x A, B, 6AB O log x x 04 a = a, b (') ' (') ' (') ' {(') ' } ' =, (') ' (') ' OA =log, OB =log 6 AB =log 6-log =log ^=log AB M O M =OA +! AB =log +! log =log ' M ' 08 800 a, a, a, y, a«log a +log a +log a +y+log a«log log 6 O A M B 00

800=fi _5 800 (5+)(+)=8() n=8 a _a =a _a =y=aª_a º=800 a a a ya =800 log a +log a +log a +y+log a =log (a a a ya ) =log 800 =9(log 00+log 8)=9(+log ) 0 Review N N N=p a p a p a yp«a«(p a i=,,, y, n N (a +)(a +)(a +)y(a«+) N (p +p +p +y+p a )(p +p +p +y+p a )y(p«+p«+p«+y+p«a«) log A=n+a`n 0{a< x -x log 500+k=0 n, a log 500 n+a= 5555, n a=k log 500=+log 5 log 5 n+a=+ 5555 log 5 0< 5555< log 5 n=, a= 5555 log 5 na= 5555=K k=log 5 0 Review x ax +bx+c=0 a, b a+b=-ab, ab=ac 09 a, b a b=a log b a b=b a a b=a log b =a log b =(b ) log a =b log a =b a e a e=e a=a abn f(a, b) a«{b<a«± f(a, b)=n f(, 0)=f(0, 00) a+b f(a, b)_f(b, a)=0 f(a, b)< f(b, a)> a log e =a log e=, log e=! e=! =' a a '=' a=a ' ' x x=x =' log x =!, log x=! x= 4! = ' x => x> {0< f(, 0)= 0 {00<0 f(0, 00)= f(, 0)+f(0, 00) f(, ) {< f(, )=0< f(, ) {< f(, )= a<b b {a<b f(b, a)=0 a>b a {b<a f(a, b)=0 a+b f(a, b)_f(b, a)=0 0 Review X X a a e=e a=a e(e<x) a x=x a=e x(x<x) a 0 A log A na x -x log 500+k=0 na k a, b ab= 6 a log b = ' (log a) -(log b) ab= 6 log ab=log 6 log b-log a=-4 a log b = ' log a log b =log ' (log a)(log b)=! (log a) -(log b) =(log a-log b) +(log a)(log b)(log a-log b) =4 +! 4 =64+4=68 004

0 Review a +b =(a+b) -ab=(a-b) +ab a +b =(a+b) -ab(a+b) a -b =(a-b) +ab(a-b) n log n f(n) n g(009)<g {0 09} Tip f(x)+g(x)=log x f(x)g(x) f(n)= f(n )=f(n)+ f(n)= 0{n<00 0{n<00, 00{n <0000 logn logn logn =logn+ logn logn 0{n<00 000{n <0000 0{n<50 {n<00 {n<50 n 50-=8 8 4 f(x), g(x) f(x) 0{g(x)< 0 f(x)+g(x) =x 5 x x=a_0«({a<0) n f(x) p, q f(pq)+f(p)+f(q) f(pq)= 0'å0<p<00 q<0'å0 x=a_0«({a<0)log x=n+log a(0{log a<) f(x) log x p, q f(p)=f(q)= log p=+a`(0{a<)log q=+b`(0{b<) log pq=log p+log q=+(a+b) (0{a+b<) f(pq)+f(p)+f(q)= p<0'å0 0'å0<q<00 {a+b< yy` log pq=+(a+b) {a+b< log pq f(pq)= log 0'å0=log 0 # =+! p<0'å0 log p<+! f(009)+g(009)=log 009 g(009)>g {0 09} f(009)+f{0 09}=0 0{a<!!<b< +!<log q< 0'å0<q<00 0 f(x)+g(x) =x f(x)+g(x)=log x x=009 f(009)+g(009)=log 009 log x=f(x)+g(x) f(x) 0{g(x)< f(x) log x g(x) log x 0 <009<0 f(009)= 0 <0 09<0 f {0 09}=-4 f(009)+f {0 09}=-4=- g(x) log x g(009)=log 009-=log.009 g {0 09}=log 0 09-f {0 09}=-log 009-(-4) g {0 09}=4-(+log.009)=-log.009 g(009)-g {0 09}=log.009-(-log.009) g(009)-g {0 09}= log.009-.009 g(009)-g {0 09}=log 555544 0 g(009)-g {0 09}<log =0 p=q=50 f(p)=f(q)=, f(pq)=f(500)= f(pq)+f(p)+f(q) 50>0'å0 p=q>0'å0 6 x log x f(x) 0 f(0)-f(0.0)=0 f(x)=f { 44 } x f(x )=0 f(x)=0 f(x)=! 0 0.0 f(0)-f(0.0)=0 0 x= =5 x 5 log log 5 0 f(x)+f { } x 005

f(x )=0 log x =n n logx=n log x=n k n=k f(x)=0 n=k+ f(x)=! n=k+ f(x)=@,, f(x )=0 f(x) 0! @ Tip x«0 x 8 000 00 r% 00 4 0r log =0.00, log.07=0.00 r% 0 _{+0R0} 0 =4_0, {+0R0} 0 =4 +0R0=4 0 log {+0R0}= 0log 4= 0log =0.00 log.07=0.00 +0R0=.07 r=7. 0r=7 7 9 [H±] phph=log [H±] ph 7.4 5! A ph 5.4 4 B A B ph a 0a log =0. 7 9 9 =a_0«(<a<0, n) a, n [ 4 ] 0«[x] xlog =0.00, log =0.477 9 =a_0«log 9 =log(a_0«) log a+n=0 log 9=0 log =0_0.477=9.54 <a<0 0<log a< n=9, log a=0.54 a=0 0.54 9 =0 0.54 9 _0, 5555 =0 0.54 =a 0 9 [ 5555 ]=[a] a 0 log =0.477, log 4=log =0.600 log <log a<log 4 <a<4 9 a [ 5555 ]= 0 N=a_0«{a<0 n N N a[ 5555 ]=[a] 0«N log N log N a log, log, log 4, y, log 9 log {a<log N log {a<log 4 N log 9{a N 9 ph 7.4, 5.4 p, q 7.4=log p!, 5.4=log q! A B pha, b a=log p%=log 5+log p!=(-log )+7.4 a=(-0.)+7.4=8. b=log 4 q=log 4!+log q!=-log +5.4 a=-_0.+5.4=4.8 ph a a=a-b=8.-4.8=. 0a= 0 M m r(pc) M-m=a+blog r (a, b) 0(pc) 0 5 8 5 ab r=0 M=m 0=a+blog 0 a+b=0 yy` M-(m+5)=a+blog 0r (M-m)-5=(a+blog r)+blog 0 (a+blog r)-5=(a+blog r)+b b=-5 b=-5 a=5 ab=-5 006

5~9 0-9 0 8 0 04 6 05-06 - 0 880 0 0 04 05 06 07 08 09 0 4 5 6 7 8 9 0 A =AA, A =A A, y A«a «an 0 = «0 = 0 0 b bn a 0 «a«0 = 0 b 0 b«0 0 0 A=, B= X-A=(X+B) X (, ) - - X-A=(X+B)=X+B X=-A-B -0 0 - X=- - - - -6 0-0 -4-6 -4 X= + = - -6-4 -9-7 X (, ) -9-9 04 a- 4 A= a 7 a- 4 A= 7 (a-)-4 7=0 a-4-8=0, a= a=6 6 0 - A=, B= (A-B)(A+B)=A -B a +b - a b (A-B)(A+B)=A +AB-BA-B =A -B AB-BA=O AB=BA - - = - a b a b - -a -b 7-4 = -a 6-b a+b -a-b -a=7, -b=-4, -a=a+b, 6-b=-a-b a=-, b= a +b =9+9=8 8 05 4 - a b A=, B= (AB ) = a+d - c d (AB ) =BA - - - - A = = 4- - -4 - -4 - - - - -7-9 BA = = - - - -4-5 -6 a+d=-7-6=- - 06 a- - x 0 x, y = x=0, y=0 a - a+ y 0 p, q p, q pq 0 0 A= A 0 0 0 A =AA= = 4 0 0 0 A =A A= = 4 6 0 0 0 A =A A= = 6 8 0 A«= n 0 A = 0 A a- x=0, y=0 - (a-)(a+)-=a -=0 a= ' 0 n 0 A«A«= (n=,,, y) 0 n A A +A A +A A +y+a ºA - a+ pq=' (-')=- - approach 007

0 0 0 A A = = =E 0 0 0 0 0 6 0 A A = = =6E 0 0 0 6 0 0 0 0 0 A ºA = = =0E 0 0 0 0 0 A«A«=n(n+)E A A +A A +A A +y+a ºA 440 0 =(+6++y+0)E=440E= 0 440 A A +A A +y+a ºA 880 880 n 0 A«= =ne A«A«=nE (n+)e=(n +n)e 0 n A A +A A +A A +y+a ºA = 0 n= (n +n)e 0 0 ={ 44 + 44444 }E=(85+55)E 6 440 0 =440E= 0 440 A A +A A +y+a ºA 880 0 A= B =A, B«=A«B«(n=,, 4, y) B º 0 B =A, B«=A«B«B =A B =A ± B =A B =A A ± =A ± ± B«=A +++y+n B º=Afi fi A= 0 A =AA= = 0 0 0 A =A A= = 0 0 0 n A«= 0 55 B º=Afi fi = 0 0 0 0 A = B= A«=A«B (n=,,, y) 0 0 A A +A A +A A +y+a ªªA ºº 0 0 A = B= A«=A«B 0 0 0 0 0 A =A B= = 0 0 0 0 0 0 A =A B= = =A 0 0 0 A =A B=A B=A A =A B=A B=A A «=A, A «=A 0 0 0 A A = = =A 0 0 0 A A =A A =y=a ªªA ºº=A 0 00 A A +A A +A A +y+a ªªA ºº=00A = 0 00 A A +A A +y+a ªªA ºº 00 Tip A«A A A =A A =A A =y=a A A 04 A, B P, Q a b p q X= Y= c d r s A A B P, Q A P, Q a, b q, s aq+bs X, Y XY (, ) 05 AB A+B=EAB=OA +A B+A B +y+ab +B E A+B=E B=E-A AB=A(E-A)=A-A =O A =A A=E-B AB=(E-B)B=B-B =O B =B A +A B+A B +y+ab +B =A+AB+AB+y+AB+B =A+B ( AB=O) =E P Q P a c p r Q b d q s 008

06 0 A B= E= AB 0 0A -A-E=OA a b A= c d a b a+b a+b AB= = c d c+d c+d AB 0 (a+b+c+d)=0 a+b+c+d=5 A -A-E=O a+d=, ad-bc=- -d -b -d -b A = 555555555555555555 = ad-bc -c -a -c -a A -d+b+c-a=(a+b+c+d)-(a+d)=5-4= 09 A E+A E-A A =ka k A +A +A E a b A= E+A E-A c d +a b -a -b E+A= E-A= c +d -c -d (+a)(+d)-bc=0 ad+a+d-bc+=0 yy (-a)(-d)-bc=0 ad-a-d-bc+=0 yy a=-d, ad-bc=- -d -b -a -b A = 555555555555555555 = 555555555 ad-bc -c -a - -c -d a b A = =A c d k= A =A A =E A 00 +A 009 +A 008 =E+A+E=A+E 07-7 A= {A +(A ) } E - -7-7 - 7 A = = - - - - 7 (A ) =(A ) = - -7 = - - - 7-7 - -0 A +(A ) = + = =-E - - -0 - {A +(A ) } =(-E) =E 08 a - A= (A -E)(A-E)=O A-E 4 b a, b a +b E (A -E)(A-E)=O (A-E)(A+E)(A-E)=O A-E (A-E) (A+E)(A-E)=O A -A-E=O yy a - A= 4 -b A -(a+b)a+(ab+8)e=o yy = a+b=, ab+8=- a +b =(a+b) -ab(a+b) = - (-) =8+66=74 0 x y- x +y = x, y k k x y- kx-(y-)+0 y+kx+ k x, y x +y = y+kx+ x +y = y=kx+ x +y = y=kx+ x +(kx+) = (k +)x +6kx+8=0 D 555555=(k) -8(k +)<0 4 k -8<0, (k+')(k-')<0 -'<k<' k 0 Review a, b, c x ax +bx+c=0 D=b -4ac D>0 HjK D=0 HjK D<0 HjK E A, E (A+E)(A-E)=A -E (A E) =A A+E (A E)(A A+E)=A E a b P(a, b), Q(c, d) PQ c d 6 6 6 6-0 + 0 009

6 6-555555555555= - 6 (6-) 55555555555555555555 = (-) 6 4 8 + = 5 8-5 555555555555= 4-6 - 0-6 = -0-6- 55555555555555555555555555 =- --(-) 0 Review HjK HjK y HjK y HjK - a b A= A+B=5E, AB=E B c a +b +c Ea, b, c A+B=5E B=5E-A AB=E A(5E-A)=E A -5A+E=O a+c=5, ac-b= a +b +c =(a+c) -ac+(ac-) a +b +c =5-ac+(ac) -ac+ a +b +c ={(ac) -4ac+4}+ a +b +c =(ac-) + a +b +c A 5 A +A-E=O A = 4 a -5 (A-E) = a, ba+be b -4 A +A-E=O A +A-E=(A-E)(A+E)=-E A-E -A-E a -5 (A-E) = -A-E b -4 a -5 5 5 =(-A-E) =A +E b -4 4 4 0 = + = 8 a+b=+= 4 a b b a A a, b, c, d A =, A =- c d d c E a a A =- ad-bc+0 A =E c c ad-bc+0 A =-A a a b a A =AA =A =- c c d c a a b a A =AA =A =- c c d c b b a b A =AA =-A =- d d c d a b a b A =- c d c d a b ad-bc+0 c d a b - A =-E c d A =-E A(-A)=E A =-A 5 a b d -b A= B= c d -c a AB=BA a+d+0 (A+B) =A +B C a+d+0 AC=CA BC=CB a b -d -b ad-bc 0 AB= = c d -c -a 0 ad-bc -d -b a b ad-bc 0 BA= = -c -a c d 0 ad-bc AB=BA a b -d -b a+d 0 A+B= + = c d -c -a 0 a+d A+B=(a+d)E (A+B) = 55555555555555 E a+d -d -b a b A +B = 555555555555555555 + 555555555555555555 ad-bc -c -a ad-bc c d A +B = 555555555555555555 B+ 555555555555555555 A ad-bc ad-bc a+d A +B = 555555555555555555 (A+B)= 555555555555555555 E ad-bc ad-bc (A+B) +A +B EC=CE (A+B)C=C(A+B) ( A+B=(a+d)E) AC+BC=CA+CB AC=CA BC=CB 00

6 a b a+ b A= B= c d c d+ a, b, c, d E AB=BA AB=A A A +B=O A =-E a+ b a b 0 B= = + B=A+E c d+ c d 0 AB=A(A+E)=A +A BA=(A+E)A=A +A AB=BA AB=A +A AB=A A=A +A A =O A B=A+E A +B=A +A+E=O A-E (A-E)(A +A+E)=A -E=O A =E A+B=E A +AB+BA+B =E AB=E ABA=A A (ABA)=A A BA=E A +B =-E A +B =-E A(A +B )=A(-E), A +AB =-A A +ABB=-A, A +B=-A ( AB=E) A =-(A+B)=-E B =-E A =B =-E 9 x y A= A =O 0 A =O A =x A 0 0 A= 0 7 AB AB=BA A -B =(A-B)(A+B) A«B=BA«n=,,, y AB (ABA) =(A ) B (A-B)(A+B)=A +AB-BA-B =A -B A«B=A«AB=A«BA=A«ABA=A«BA =y A«B=ABA«=BAA«=BA«(ABA) =(BA ) =(A ) B Tip AB=BA 8 A, B A+B=AB=E (A-E) =B-E A +B =-E A =B =E A+B=AB AB-A-B=O, (A-E)(B-E)=E (A-E) =B-E E x y A= A =O A 0 x 0- y=0 y=0 x 0 A= A =O A =O 0 x 0 A = =x A x 0 A =x A=O x=0 A=O A x=0 0 0 A= 0 0 a- -b x x-y x, y = x=y=0 b a+ y x+y a, b (a, b) a- -b x - x = b a+ y - y a- -b+ x 0 = b- -a+ y 0 a- -b+ x=y=0 b- -a+ (a-)(a+)-(-b+)(b-)=0 a +(b-) =4 (a, b) 4p 0

0 80 0 ' 0 8 04 75 05 87 06 55 0 0 0 04 05 06 07 08 09 0 90 7 4 5 6 7 8 9 487 0 90 4 77 5 6 7 8 7 9 0 4 0 {a«} a º=0, a -a =4 a º a d a º=a+9d=0 yy a -a =(a+d)-(a+7d)=4d=4 d=6 d=6 a+54=0 a=-4 a º=-4+9 6=-4+4=80 80 a d a -a =4d=4 d=6 a º=a º+0d=0+0 6=80 0 {a«} a =, a =6a a, a, a a =a a = 6= ~9 a =' a >0 ' 0 {a«} n S«S«=6n +kn{a«} k a a =S =6+k a«=s«-s«=(6n +kn)-{6(n-) +k(n-)} a«=n-6+k (n=,, 4, y) a«=n-6+k (n=,,, y) {a«} 6+k k= a =6+=8 8 04 0 k= 0 k= (k+)(k-) (k+)(k-)= (k -) 0 k= 0 (k+)(k-)= 444-0 6 (k+)(k-)=85-0=75 75 05 {a«} a = n a«-a«=4n a º a«-a«=4n n n+ a«-a«=4(n+) {a«} {b«} b«=4(n+) 9 9 0 a º=a + 4(k+)=+4{ 4+9} a º=+4 09=87 87 k= a«=a«+4n n=,, 4, y a =a +4 =+4 a =a +4 =+4 +4 a =a +4 4=+4 +4 +4 4 a º=+4 +4 +y+4 0 a º=4(+++y+0)- 0 a º=4 4 -=840-=87 06 a = a«=a«+(n=,,, y) {a«} a a«=a«+ a«+=(a«+) {a«+} a += a«+=_«=«a«=«- a = -=55 55 a = a =a +=+ a =a +=(+)+= ++ a =a +=( ++)+= + ++ («-) a«=«+«+y++= 444444=«- - a = -=56-=55 approach 0 {a«}, {b«} {a«} : 6, 4,, 0, y {b«} : 8, 7, 6, 5, y a =b k {a«}, {b«} a«=6+(n-)(-)=-n+8 b«=8+(n-)(-)=-n+9 a =b -k+8=(-k+9) k=9 0

0 0 n a, a, y, a«, a, a, y, a«, 0 n! @ 4# ª0 d (n+) 0 9 0=+(n+-)d d= yy n+ n,,, y 06 {a«} a +a +a +y+a =0, a +a +a =4 {a«} {a«} a r a +a +a =4 a(+r +r )=4 a +a +a =6 ar(+r +r )=6 yy yy r 4=6 r=# d=(, (, 4(, y =9(,!=ª8, 4#=ª, ª0 9 @= n n+ 0 " -6', a, b, c, " +6' a+c " 6'=" 'å8= ' " -6', a, b, c, " +6' d " +6'-" -6' =(+')-(-')='=4d ' d= a c b b=" -6' +d b=(-')+'= a+c=b= =6 0 Review " a+b+'aåb='a+'b a>0, b>0 " a+b-'aåb='a-'b a>b>0 04 0 {a«}, {b«} a +b =0, a + b =50 {a«}, {b«} då, d 0 k= 0 0(a +9då) 0(b +9d ) a + b = 4 + 4 k= a + b =5{(a +b )+9(då+d )} a + b =5{0+9(då+d )}=50 då+d =: 9º: 05 {a«} a a a =8 a a k= 0 k= 07 5 ' {a«} n S«a º-a ª S º-S 444444444 + 44444444 S º-S a º-aª a º a º - + a º-a ª S º-S a º-a ª a º+aª a ª aª 444444+ 4444= 4444+ 4444= 4444 + 4444 S º-S a º-aª a º+a ª a º-aª a º a º + a ª - aª '- '+ = 444+ 444=('-) +('+) '+ '- =(-')+(+')=6 08 {a«} n S«S =, S«=n +(n}) a +a º a =S = a =S -S =( +)-=4 a º=S º-Sª=(0 +)-(9 +)=9 a +a º=4+9= 09 9 +- N N= 4 y 8 9 0 7+ - N=00 N N=0 0 0 +++y+9+0= k= 4 =0 k= N=00 0-00=0 :!!:) +- - 55 0 0 n+- 0-n N a, a, a a a =a a a a =a =8 a = a, a, a a a=a = =4 :@!:)-+ +++y+4-5-6-7-8-9-0=0 0

0 00 k= sin Kp [ 4444 ] [x] x k k=,,, y sin Kp, 0, -, 0,, 0, -, 0, y sin Kp 4444 k, 0, -!, 0, 5!, 0, -7!, y sin Kp [ 4444 ], 0, -, 0, 0, 0, -, 0, 0, 0, -, y k sin Kp [ 4444] 4n- (n=,,, y) k - 0 4_5=00-5 {a«} {b«} {a«} :, 7, 9, 7, y {b«} : 6,, 8, y {b«} b«=6n a = a«=+ 6k 9 9 0 a º=+ 6k=+6 =7 7 a = a = - a = - a =4 - n- k= k= a º=0-9 =000-79=7 00 sin Kp 4444 k [ ] =+5_(-)=-4 k= Tip 0 0 sin x p sin Kp, 0, -, 0 {a«} {a«}, 5,, 5, 4, y {a«} 99 99 00-99 00+ 99 00-99 00+ {a«} {b«} {a«} :, 5,, 5, 4, y {b«} : 4, 8,, 6, y {b«} b«=4n 4 7 6 5 8 7 6 9 8 4 0 4 9 5 5 0 6 8 7 a +a +a +y+a«-6 4 {a«} {b«} b«= 444444 n 0 k= (a -b ) -6 4 {a«} a«=-6+(n-)4=4n-0 n n(n+) a +a +a +y+a«= (4k-0)=4 4444444-0n=n -8n n -8n b«= 44444 =n-8 n a«-b«=(4n-0)-(n-8)=n- 0 0 0 (a -b )= (k-)= 4-0=90 90 k= k= k= n a«a º n- a«=+ b =+ 4k=+n(n-) k= n- k= {a«} {a«} 4-0 A=, E= A«=nA-(n-)E (n=,,, y) - 0 n= A = A-0 E n=k A =ka-(k-)e A = A ± =k( )-(k-)a=, n a = a =7 a =9 a =7 n= A = A-0 E n=k A =ka-(k-)e yy A 04

A ± =ka -(k-)a A -A+E=O A = A-E A ± =k( A-E )-(k-)a= (k+)a-ke n A-E (k+)a-ke aº=!_000(+0.05) a ={!} _000(+0.05) a ={!} _000(+0.05) a«={!} «± _000(+0.05)«± 5 6 k= K + -K =p+q' p, q p-q 05! a ={!} fl _000(+0.05)fl =6 4_000_.4 a =0.975? { N }! ='! =' { -N } -! = -! = ' ' 6 ( K + -K 6 )= K 6 + -K [-{} fl ] ( K + -K '{(')fl -} ' ' )= 444 + 44 '- - ' ( K + -K 7' 8& )= 44+ 44 '- '- k= k= k= ( K + -K )=7(+')+8&('+) ( K + -K )=:!8!:(+: 8 :' p-q=:!8!:(-: 8 :=: 8 :=7 r a n n S«a(+r){(+r)«-} S«=a(+r)«+a(+r)«+y+a(+r)= 444 r r a n n S«a{(+r)«-} S«=a(+r)«+a(+r)«+y+a= r 8 7 0 %.0 =.4 a Å0 6 {a«} a =, a +a +a +y+a«=a«(n=,, 4, y) a º {a«} n S«S«=a +a +a +y+a«=a«(n=,, 4, y) S«-S«=a«S«=(S«-S«) S«=$S«(n=,, 4, y) % n 7 Å0(+0.0) fl Å0(+0.0) fi Å0(+0.0) 7 Å0 S«=$S«(n=,,, y) {S«} S =a = $ S«={$} «(n=,,, y) a º=S º-Sª={$} -{$} =! {$} 7 S S=Å0+Å0(+0.0)+Å0(+0.0) +y+å0(+0.0) fl Å0(.0 -) Å0(.4-) S= = 4 }00000.0-0.0 Å0_0.4}000 a}50000 50000 7 00000 5%!05!.05fl =.4 00+n! a«(n=0,,, y) Tip 7 7 Å0 Å0(+0.0) fl 05

9 «-,,, y, ««- a««- a =, a =6, a =9 a -,, y, fl ( -)-, (fl -)-, y, ( -)- 7 a= {( -)-}= ( ± -) k= 7 «k= ( -) a= -=487 487 - a = a =6 a =9 0(8+00) 44444=90 90 (), (, 4), (5, 7, 9), (0,, 4, 6), y n {a«} {a«} :,, 5, y {a«} {b«} {b«} :,, 5, y b«=n- a«=+ (k-) 0 a º=+ 9 k= n- k= (k-)=+8=8 0 0 8 0( 8+9 ) 444444444=90 0 n a«a =, a =6 a º n (n-) (n+) n 4 5 6 7 y a«6 0 5 8 6 y {a«} {b«} {b«} :, 4, 5, y {b«} b«=n+ n- n(n-) a«=a + (k+)=+ 4444444+(n-) k= a = n +n+ (n+)(n+) a«= 4444444444 = 44444444444444444 (0+)(0+) a º= 4444444 = 4=66 a =6 (n+) n += += 8 4 4 5 5 4 n a«a == + a =a +(a -)=a -=4= + a =a +(a -)=a -=0= + a«=a«-=«+ (n=,, 4, y) a = +,,, y 500 0 4 5 6 7 4 8 9 0 4 5 6 {a«} a«=n 9 8 0 8 00 0 0 4 5 6 7 8 9 0 4 5,,, y () (, ) (4, 5, 6) 4(7, 8, 9, 0) 06

n a«{a«},, 4, 7, y a«=+ n- k= k (n}) 0 0 a =+ k=+ 4 =466 a =+ k=+ 4 =497 a =+ k=+ 4 =59 500 4 500 4 4 5 (466, 467, 468, 469, y, 496) (497, 498, 499, 500, y, 58) (59, 50, 5, 5, 5, 54, y, 56) 500 468+469+5+5=00 4 0.00 0 0 x x=0.00000y x n a«00x n b«{c«} (a«=b«) c«= g 0 (a«+b«) 00 n= c«x=0.00000y n n+ c«= c«=0 {c«} 0, 0,, 0,, 0,, 0,,, 0,, 0,,,, 0,, 0, y (0, 0,, 0), (, 0,, 0), (,, 0,, 0), (,,, 0,, 0), y 4 k(k=,, 4, y) k+ k k k+ (k+)(k+) 4+4+5+6+y+(k+)= i-= - k= k= k= k= 4 5 4+4+5+6+y+4= 4-=0 c ºº 4 {c«} +++y+= 4=78 78 0 00 n= c«=78-=77 77 {c«} 0 +++y+= +=5 {c«} 0-5=78 i= c º =0, c º =, c º =0 00 c«=78-=77 n= 5 {a«} a«=g an n- (-) (n) (n) a ººº= n a«=a«m, n aμ«=aμ_a«{a«} n 4 5 6 7 8 9 0 4 5 6 y a«- - - - - - - y a ººº=a ºº=a º=a =(-)fl = a =, a =- a +a n=, 5, 9, y n=4k-(k=,,, y) a = n=, 7,, y n=4k- a =- m=4k-, n=4k'- k, k' mn=4(4kk'-k-k'+)- aμ=, a«=, aμ«= m=4k-, n=4k'- k, k' mn=4(4kk'-k-k'+)- aμ=-, a«=-, aμ«= m=4k-, n=4k'- k, k' mn=4(4kk'-k-k'+)- aμ=, a«=-, aμ«=-,, aμ«=aμ_a«6 {a«} a =, a«a«-a«+a«=0 (n=,,, y) a a +a a +a a +y+a ºa a«+0 a«a«-a«+a«=0 yy a«+0 a«a«- 4 + 4444=0 4444-4 =- a«a«a«a«[ 4 ] 4 - a«a 4 = 4 +(n-) (-)=-n+ a«a a«= 444444444444 -n+ a«a«=!(a«-a«) a a +a a +a a +y+a ºa =!{(a -a )+(a -a )+(a -a )+y+(a -a º)} =!(a -a )=!{- 9-} =-!9) 07

7 n x A«x l : y=x+ B«B«l x A«A (0, 0) A x y B B B y=x+ c n- a«a«=a«+a«+a«=a«+a«(n=, 4, 5, y) a =a+a =7 A O A A x A«x a«b«(a«, a«+) B«l y=-(x-a«)+a«+ y=0 x=a«+ a«=a«+ a«+=(a«+) {a«+} a + a =0 a«+= «a«=«- (n=,,, y) a =fl -=6 9 n A 000 n 0 n n+ A A-(n-) A<0? n Tip n x A A, A, y n A n A 0 000 000-000-(+) 000-(++5) n 8 A A a, b, c,, A A n a«b a A c A=000-{++5+y+(n-)}<0 000<++5+7+y+(n-) 000<n =96, =04 n a (a) a = (a, a), (b), (c) a = (a, a, a), (a, b), (b, a), (a, c), (c, a) a =5 4 (a, a, a, a), (a, a, b), (a, b, a), (b, a, a), (a, a, c), (a, c, a), (c, a, a), (b, b), (b, c), (c, b), (c, c) a = a, a, a, a, y a =, a = a =_+=_a +a =5 a =_+5=_a +a = a =_a +a =_5+= a =_a +a =_+=4 a=_a +a =_+4=85 a =_a +a=_4+85=7 7 n n a n- a«b n- a«0 c b c a b 0 c 0 n 0 n n+ c a+b n? n=9? n, a, b, c c n 0 4 5 6 7 8 9 a 5 5 b 0 0 0 8 8 c 0 5 8 4 n=9 c 4 4 a c 08

0 lim 0-0! 0 0 04 05 06 4 0 0 0 04 0 05 06 05 07 08 09 0 9 4 5 6 7 8 9 0 4 5 -n-n 5555 n + n ~7 55-n!- -n-n n lim 55 = lim 5 =- - n + +55 n a«{ -}= n= a«lim { -}=0,! lim a«-=0 lim a«= 06 n= n= «+(-)«55555555 5««+(-)«- 55555555 = {5@}n + { 5555555}n 5«n= n= 5 5@ <, -5# < 5@ -5# - {5@}n + { 5555555}n = 5555555555555 + 555555555555555555555 n= n= 5-5@ -{-5#} {5@}n + { }n =@-8#=4 4 0 - lim (" n + n-n) (" n +n -n)(" n +n +n) lim (" n +n -n)= lim 555555555 " n +n +n n (" n +n -n)= lim 55555 = lim 555555555 " n +n +n Æ +n! + (" n +n -n)= 5555 =!! + 0 lim [{!}n +{-!}n ]! <, -! < lim [{!}n +{-!}n ]= lim {!}n + lim {-!}n =0+0=0 0 approach 0 a«-b«{a«}, {b«} lim a«=, lim (a«-b«)= lim a«+b«lim a«=, lim a«-b«555555 a«=0 lim (a«-b«)= a«-b«lim (a«-b«)+a«lim = 555555555555 5555555555 a«+b«lim a«-(a«-b«) (a«-b«) 555555555+ a«= lim 55555555555555 a«-b«-55555555555 a«=! 04 555 + 555 + 555 +y 4 S«= 55 + 55 +y+ 55555555 n(n+) S«={-!}+{!-!}+y+{n!- 55 }=- 55 n+ n+ S= lim S«= lim {- 55 }=- lim 55 =-0= n+ n+ 05 a«{a«} { 5 -}= lim a«n= 0 {a«}, {b«} {a«} lim (a«-b«)=0 lim a«= lim b«lim a«=a`a, a«-b«=c«lim c«=0 lim b«= lim (a«-c«)= lim a«- lim c«=a-0=a {a«}, {b«} {a«b«} {a«}, {b«} {a«+b«} a«=(-)«, b«=(-)«{a«}, {b«} a«b«=(-) «= {a«b«} a«=n, b«=-n {a«}, {b«} a«+b«=n+(-n)=0 {a«+b«} 0 09

0 {a«} n S«S«=n 4«(n=,,, y) S«=n 4«n} a«=s«-s«=n 4«-(n-) 4«=(n+) 4«n= a =S =4 a =4 n= a«=(n+) 4«(n}) yy` lim S«a«S«n 4«4n lim 555 = lim 5555555555555555 = lim 5555555555=$ a«(n+) 4«n+ 04 - {a«}, {b«} lim 0a«n a«+b«=n a«b«=n- a«<b«a«, b«x -nx+(n-)=0 x=n " n - n+ a«<b«a«=n-" n - n+ n- lim 0a«=0 lim (n-" n - n+)=0 lim 55555555555555555555555555 n+" n - n+ -n# 0a«=0 lim 5555555555555=0 5555555=0 0 + +æ -n@ +5 n 0 Review x ax +bx+c=0 a, b a+b=-ab, ab=ac a =" n +kn-n (k=,,, y, 0) kn k lim a = lim (" n +kn -n)= lim 555555555 = lim 5555555 =K " n +kn+n Æ +nk + 0!+@+y+: º:=!(++y+0)=! 5555555 =05 05 07 x+ f(x)=x+- y=x f(x) f «± (x)=( fωf «)(x), f (x)=f(x) (n=,,, y) lim f «(-4) lim f «(-) lim f «() ( x+ 5555555 (x}-) x+ f(x)=x+-55555555555 f(x)= { (x+) 9 555555555555555 (x<-) y=f(x) x x -y=x x -, y y=x ( - (x<-) y=f(x) lim f «(x)={ x (x=-, ) -4- - O x 9 (-<x<, x>) lim f «(-4)=- lim f «(-)= lim f «()= y - O y=x y=f(x) x Tip 05 - n "4 n + n a«lim a«"ç4nω <"ç4nω ç+ n<"ç4nω ç+ 4 n+ " (n) <"ç4nω ç+ n <" (n +) n<"ç4nω ç+ n<n+ 06 - lim (" n +n +" n +n +y+" n +0n -0n) lim (" n + n +" n +n +y+" n +0 n -0n) = lim {(" n + n -n)+(" n +n -n)+y+(" n +0 n -n)} "ç4nω ç+ n n a«a«="ç4nω ç+ n -n lim a«= lim ("ç4nω ç+ n -n) n a«= lim 555 = lim 55555 =4! "ç4nω ç+ n +n Æ 4+n! + f (x) a y=x y=f(x) x a y a y=f(x) x a y a y=f(x) f «(x) 08 4 x -x-=0 a, b lim x -x-=0 x= 'ƒ+ = ' a«± +b«± 555 a«+b«a=-', b=+' ƒ < lim { ƒ}n =0 00

b«a{ ƒ}n +b a«± +b«± lim 555555555555555 = lim 555555555555555 =b=+' a«+b«{ ƒ}n + 09 4 A«n 6«A«lim 6«6«=««A«A«=(++ + +y+«)(++ + +y+«) «± - «± - A«= 55555 _ 55555 - - A«=!(«± -)(«± -) {a«} a a a (a -)+{ 55 -}+{ 55 -}+{ 55 -}+y 5 7 n+a«lim 555 n-+a«a«{ 55 -} n= n- a«a«lim { 55 -}=0 lim 55 = n- n- n a«n+a«55555+ 55555 lim 55555555= lim n- n- n-+a«555555555555555555 a«+55555555555 n-!+ = 55555 =4& + A«(«± -)(«± -) lim 555 = lim 55555555 6«6«6 6«-«± -«± + = lim 555555555 6«a«lim a«=0 n= 6- {6@}n - {6#}n +{6!}n = lim 555555555 =^= 0 Review A A=aπ bœ c a, b, c p, q, r A S S=(a +a +y+aπ )(b +b +y+bœ )(c +c +y+c ) a, r n S«a(-r«) a(r«-) r+s«= 555555555 = 555555555 r=s«=na -r r- 0 {a«} a =5, na«-(n+)a«=4 (n=,,, y) a«lim n na«-(n+)a«=4 n(n+) a«a«4 55-55 = 55555 n+ n n(n+) a«a 4 [ 555555 ] 555555 =5 [ 55555] n n(n+) a«n- 4 n- 555555 =5+ 55555=5+4 {k!- 555555} n k= k(k+) k= k+ =5+4[{!-!}+{!-!}+y+{ 5555555 -n!}] n- =5+4{-n!}=9-n$ a«lim 555555 = lim {9-n$}=9 9 n n x(n, 0) C«6 C«a«n= a«c«(n, 0) a«=(n) - =4n - 6 6 5= 55555555 n= a«n= 4n - 6 = 555555555555555 n= (n-)(n+) n 6 n = lim 555555555555555 = lim ^{ 555555555-555555555 } k= (k-)(k+) k= k- k+ = lim [{-!}+{!-5!}+y+{ 555555555-555555555 }] n- n+ = lim {- 55 }= n+ y O a«n C«n <k (k) =a n= n n= n n= n S= + + +y+ +y 5 (n-) n < (n+) n(n+) = + < + n= n n= (n+) n= n(n+) n = + lim { - }= k k+ a= + + +y+ +y=s+[ + +y+ +y]=s+ _a n 4 (n) S= _a k= a, b, c, d ad+bc x 0

n 5555555555555 < 555555555555555 (n+) n(n+) n= 55 = + 5555555555555 < + n n= (n+) n = + lim {k!- 5555555}=+ lim {- 5555555 }= k= k+ n+ 5 + 5 +y+ 55555555 +y= 55555555 =4! 55 =4!a 4 (n) n= (n) n= n a= 5 + 5 + 5+y+ 5 +y n a=s+[ 5 + 5 +y+ 55555555 +y]=s+ 4! _a 4 (n) S= 4# _a n= 555555555555555 n(n+) 6 r«a lim a «(sin «a+) +a «a < a=0 a > a= a < sin a < n a «0, sin «a 0 a = sin a < n a «=, sin «a 0 =a a = a=! a=0 a=! a a=, b=, c=4!, d=4# ad+bc=4#+4@=4% a=kp+ (k=0,,, y) n a «, sin «a= a «(sin «a+) sin «a+ + lim = lim 44 = 4= +a «0+ 45+ a «4 a«, na«a«=a na«=b n (a«-a«) n= n= ab a, b n n= n= n (a«-a«)= (a -a )+ (a -a )+y+n (a«-a«)+y n (a«-a«)= a +( - )a +( - )a +y n (a«-a«)= {n -(n-) }a«= (n-)a«n (a«-a«)= n= n= n= na«- a«=b-a n= Tip + (n-)a«n= n= n= 7 r«0 r [ r «± - r «+r ] -@ 0! ` r >lim r «= lim 55 =0 r «r-55 r «± - lim 5555= lim r «55555555555=r r<- r> r «+r +55555 r «` r < (r+0)lim r «= lim r «± =0 r «± - lim 5555=- 555 r «+r r -<r<0, 0<r< - 555 >! - 555 <-! r r r «± - - r= lim 5555= 555444=0 r «+r + r «± - -- r=- lim 5555= 5554444 = r «+r -! 5 (x+a)«, 4(ax+)«x a a<a<ba+b n= n= -<x+a<, -<ax+< --a<x<-a, -a@<x<0 a>0 8 n A«A«a«b«a =(, b =6 lim a«a ={!+} a ={!+@+} a«b«n A A n} x -a@<-a a -a-<0, (a+)(a-)<0 -<a< a 0<a< a=0, b= a+b= n(n+) 5 a«={n!+n@+y+nn}= 555555555 n =#(n+) b«0

b =++ b =+++4 (n+)(n+) b«=++y+n+(n+)= 555555 a«b«(n+) (n+) lim 55555 = lim 555555555 =4# n 4n a«=b«a«c«=!(p-a«) a«= -!a«a«- =-!{a«- } a«= +{a - }{-!} «lim a«= lim [ +{a - }{-!} «]= 9 lm x=klm P Q S +S +y+s«op Q S lim n O y O l P S Q x=k m x a«=pa«+q (p+0, p+, q+0)a«-a=p(a«-a) {a«-a} a -a p a«=a+(a -a)p«l, m x 60, 0 l, m ' y='x, y= 555x P (k, 'k), Q {k, ' 555 ' ' S =!{'k-555k} k= 555k k} n S +S +y+s«= n S = ' 555k k= k= ' n(n+)(n+) 'n(n+)(n+) S +S +y+s«= 555 555555555 = 555555555 6 8 S +S +y+s«'n(n+)(n+) ' lim 5555555 = lim 555555555 = 555 n 8n 9 Tip y O P S 60 0 Q x=k A AB :AC =BD :DC B D C ' k :O P =: k S =@_{!_'k_k}= k 0 A B C A B C B C, A C, A B A, B, C A B C A B C B C, A C, A B A, B, C A«B«C«B«A«C«=a«a«lim A«B«C«O A«C«OB«A«C«O=A«B«O= B«A«C«=a«C«OB«=p-a«C«C A«a«l B«m a«b«c«a«o B A A C B x B C A a =, a =Ik+!a + lim Kn+! a a =, a =Ik+! a + k}a -a ={ a +}-{ a +}=a a =a (k}) a =, a =a += a = (k}) n n n- 5 = 5 + 5 =+!{!} k= a a k= a k= k i=![-{!} «] =+ 55 =+@[-{!} «] -! k- i= n lim 5 = lim +@[-{!} n- ] =+@=% k= a n «0 S«+ + +y S S S S««0 S«= 5 {(++y+«)-(++y+«)} ««(«+) «(«+) S«= 5 [ - ] ««+ «+ S«= - =«5 + 5 + 5 +y= 5 = = 55 =# S S S n= S«n= «-! {a«} a «a +a +a +y+a «+y=: 5 : {a«} {a«} a, r a«=ar«n= a +a +aª+y+a «+y=@9$ {a «}, {a «}, {a «} r, r, r 0

ar ar a «= 5 =: 5 : yy` a «= 5 =@9$ yy` n= -r n= -r ar(-r ) +r+r 55555555555 =!0(, 55555555=!0( ar (-r ) r(+r) 9r +9r-0=0, (r+5)(r-)=0 r=@ ( r <) @a 555555 =: 5 : a= -9$ 7 ar a «= 5 = 555555 =6$5* n= -r -8!^ 4 0g 80% 0% n a«a =0_0. a =(0_0.+0)_0.=0_0. +0_0. a =(0_0. +0_0.+0)_0.=0_0. +0_0. +0_0. a«=0_0.«+0_0.«+y+0_0. n 0_0., 0. 0_0. lim a«= 55555 = =.5(g) -0. 0.8 5 Aº x! A A 0 {!} A, A 0 {!} A A, A, y, A«, y n A«A«x x«, y y«lim x«=[!-{!}!-{!}!]+[{!}4 -{!}5!-{!}6!]+y y O 0 A { } A A { }4 0!-{!}!-{!}!!-9!!- 7! 7 x«= 5555555555555555= 55555555 = 55555 =6 -{!} @7^ @7^ lim y«' ' ' ' =[0+{!} 555 -{!} 555 ]+[0+{!}5 555 -{!}6 555 ]+y ' ' ' ' ' {!} 555-{!} 555 9! 555-7 555 555 7 ' = 555555555555555555 = 5555555555555 = = 555 -{!} @7^ @7^ 6 Aº 0 A ' A«{6, 555 } 6 { } x 0-0 0 04 0 05 06 0 0 0 04 05 06 07 08 09 0 4 5 6 7 8 5 9 0 0 9~4 y= x y a, b y=4 + a+b y=4 += ± + y= x y -, a+b=(-)+=- - 0 y= -x -x y= -x -x -x -x -x -x=-(x+) + x=- y= -x -x x=- = 0 x + -x+ =4 ( ) -4 +=0 =t t -4t+=0, (t-)(t-)=0 t= t= x=0 x= 0+= 04 y= x- + y=logå (x+b)+c a, b, c a+b+c y= x- + x- =y- x=log (y-)+ x y y=log (x-)+ y= x- + y=log (x-)+ a=, b=-, c= a+b+c=0 0 05 {x{000 y=(log x) -log x log x=t 0{t{ log x = log x y=t -t=(t-) - t= -, t= -+= 06 log (x-4)<log (x-) x 04

log (x-4)<log (x-) log (x-4) <log (x-) (x-4) <x- x -9x+8<0, (x-)(x-6)<0 <x<6 x>4 4<x<6 x 5 approach x -x-p=0 D D 55555555 =+p<0 p<- yy 4 -<p<- 0 Review ax +bx+c=0 D=b -4ac x ax +bx+c>0 HjK a>0, D<0 x ax +bx+c<0 HjK a<0, D<0 0 y= A BA B y x C A, B x a, b ABC a+b C A y y= B 04 y=logå x (, ), (4, b) b y b y=logåx a O b x A, B AC =BC O 4 x ABC!_AC _BC = AC =BC = -å=b-a= yy y=logå x (, ) logå =, (4, b) logå 4=b b=logå 4=logå = logå = =6 b=a+ å± -å=, å= a=log @ b=log @+=log * a+b=log @+log *=log : 9 :=4- log 0 f(x)= + f(x-)=f(x+) x f(x-)=f(x+) x- + -(x-) = x+ + -(x+) 05 y=log 8(x-) y= log x x a y b a+b y=log 8(x-) =log 8+log (x-) =+ log (x-) y= log x x y a+b=+=4 x- + -x+ = x+ + -x-, x {9-!}= -x {-9!} x : := -x : 9 :, x = -x- x=-x- x=-! 06 y=log x y=-k y=k A BA B x a b a k! _b -k! k>0 y a O B b y=log x y=k x A y=-k 0 A={x x - px- >0}, B=[x {!} x -{!} x+p <0] AB A(a, log a), B(b, log b) log a=-k a=0, log b=k b=0 p a k! _b k! =(0 ) k! _(0 ) k! =0 _0 =0!0 A x > px- x >px-, x -px+>0 yy` B {!} x <{!} x+p x >x+p, x -x-p>0 yy` AB=R R A=B=R x -px+=0 D D =p -4<0, (p+)(p-)<0 -<p< yy 07 y=log x 4x +x-=0 n a n_0 a y 0.7 0.5 0.5 O a b c y=logx x 05

4x +x-=0 n, a n+a=-: 4 :=-4+4# n=-4, a=4# 0{a< n_0 a =(-4)_0 4# 0 4# =A log A=4#=0.75=0.5+0.5=log a+log b=log ab 0 logæ +logæ +logæ +logæ +y+logæ =n n n logæ+logæ +logæ +logæ +y+logæ =n 0 logæ +++4+y+0 4 =n, logæ =n, logæ 0 =n 0 4# =ab n_0 a =-4ab N log Nlog N=n+a n0{a< n log Na log N n{log N<n+ n [log N] 08 y=log (x-)+% k (a, b) a+b 0 (x-)= x=$ y=log {$-}+%=log +%=% k x«= 0 x= :@n!:) yy x n 0 0= _ _5 _7 n (+)(+)(+)(+)=6 ± + ± f(x)= 55555555 f(4.9), f(5), f(5.) + a>b a, b å± +å± ± + ± f(a)-f(b)= 555555555-555555555 å+å + (å± +å± )( + )-( ± + ± )(å+å) f(a)-f(b)= 555555555555555555555555555555555 (å+å)( + ) ( å+ å)( + )-( + )(å+å) f(a)-f(b)= 555555555555555555555555555555555555555 (å+å)( + ) å -å (å -å ) f(a)-f(b)= 5555555555555555555 = 5555555555555555555 >0 (å+å)( + ) (å+å)( + ) a>b f(a)>f(b) f(4.9)< f(5)< f(5.) {$, %} a+b=$+%= y= y=5 y=0 y=k x a, b, cab-bc-ca y=k y y=0 y=5 y= 09 (log x)(log y)=- g x, y logæy x>y log xy= O cb a x k> log xy=log x+log y=, (log x)(log y)=- log x, log y t '5 t -t-=0 t= 555 +'5 -'5 log x= 555, log y= 555 x>y -'5 555 log y -+'5 logæy= 5= 555555 = 5555555 log x +'5 555 Tip a =5 b =0 c a =5 b = c 5 c a-c =5 c yy b ab-bc =5 bc yy a =5 b c ca =5 cb yy ab-bc ca 55555555555555=, ab-bc-ca =, ab-bc-ca = ab-bc-ca=0 log x log y A, B A, B f(x)=@_ _ +#_ _ f(x)=6 f(x)=@_ _ +#_ _ f(x)= x- _ -(x-) + -(x-) _ x- 06

g(x)= _ + _ y=f(x) y=g(x) x g(x)= _ + _ g(-x)=g(x) y=g(x) y y=f(x) x= f(x)=6 a, b a+b 5 = a+b= Tip f(x) f(x) g(x)= _ + _ x g(x) y=g(x) y y=f(x) x= x= 4 N y=f(m, n) {(m, n) m<n, n<n} N f(m, n )=f(m, n ) m =m, n =n f(m, n)=μ _4«f(m, n)=μ _6«f(m, n)=5μ _0«6 a ABCD AC y=log x a= ABCD 0<a< ABCD a> ABCD A x t D y log t+a C y log (t+a) C D y log t+a=log (t+a), log t å=log (t+a), t å=t+a a t= 5 å- a= t= 5 = A (, 0) - ABCD a 0<a< å> t= 5 >0 å- ABCD a a> å> t= 5 >0 å- ABCD y O D A C B y=log x x f(4, )= 4 =fl, f(, )= 4 =fl f(4, )=f(, ) 4+, + f(m, n )=f(m, n ) m 6 n = m 6 n m (_) n = m (_) n, n m +n = n m +n n =n, m +n =m +n n =n, m =m f(m, n )=f(m, n ) 5 m 0 n =5 m 0 n 5 m (_5) n =5 m (_5) n, n 5 m +n = n 5 m +n 5 n =n, m +n =m +n n =n, m =m 5 ( -) +(4 -) +(8Ω -) =( -)(4 -)(8Ω -) 00 x, y, z (x, y, z) -=a, 4 -=b, 8Ω -=c a +b +c =abc a +b +c -abc=0 (a+b+c)(a +b +c -ab-bc-ca)=0!(a+b+c){(a-b) +(b-c) +(c-a) }=0 x, y, z a+b+c>0 (a-b) +(b-c) +(c-a) =0 -=4 -=8Ω - =4 =8Ω a=b=c = = Ω, y=z z y x, y, z 00 {x=z{00 z=, 4, 6, 8, y, 6 7 p, q x, y x=log (p+q), y=log p+log q (x, y) x=log (p+q) p+q= yy y=log p+log q=log pq pq= yy p, q t - t+ =0 D D=( ) -4 }0, - ± }0 } ± x}y+ y{x- y=x- p>0, q>0 p+q}' pq log (p+q)}log ' pq log (p+q)=+log ' pq=+! log pq log (p+q)=+!(log p+log q) x}+!y y{x- (x, y) y=x- 07

8 x -x- x log (x -x-)-log x=0, log 5555555 =0, log 5555555 =0 x x -x- a, b, c a+b+c log (x -x-)-log x=0 yy x -x- log 5555555555 x =0 yy x log 5555555555=0 yy x -x- log (x -x-)=log x x -x-=x, x -x-=0 x= ' x -x->0, x>0 x=+' a= x -x- 5555555555 x = x -x-=x x -x-=0 x= ' b= x= ' c= a+b+c=++=5 5 0 7 0 6 0 4 04 05 05 0 06 8 0 0 0 04 05 0 06 07 08 09 0 4 5 6 7 8 0 9 0 0 5 5 (, 4), (, ), (, ), (4, ) 4 0 (4, 6), (5, 5), (6, 4) 45~49, 4+=7 7 0 5 9 (log x) -plog x+q=0 a, b (a<b) 4 -p +q=0 c, d (c<d) a=4, c p, q (log x) -p log x+q=0 a, b log x=t t -pt+q=0 log a, log b 4 -p +q=0 c, d =s s -ps+q=0 ç, t -pt+q=0 s -ps+q=0 a<b, c<d log a<log b, ç< log a=ç yy a=4 ç=log 4= c= 8 P =8 7=56 5 P =5 4=0 56-0=6 6 0 4 5 (5-)!=4!=4 4 0 0<x<p log [&+sin x]{, [x] x 0<x<p -{sin x{ %{&+sin x{( [&+sin x]=,, 4 yy 04 A PB (A P B) =(A P)_(P B)! 7! = _ 4444=05 05! 4!! A P B log [&+sin x]{ [&+sin x]{ yy [&+sin x]= %{&+sin x< -{sin x<-! 0<x<p 6&p<x<: 6 :p a, b A B (a+b)! 5 a!b! A a B b 08

05 0 0 0! 0_9_8 ºC = 4444 = 444444444444=0 0!7! 06 (x+) x (x+) C x x C =4 =8 8 04 (-)!=!=!= =6 _6= approach 0 0 0 5 A5 B0 AB A, B, AB 04 0+4-= 05 X={,,, 4} f X X f()+f()+f()+f(4)=7,,, 4 7 +++4, +++, +++ 4! 4! 4! =4, =, =4!!! f(), f(), f(), f(4) 4++4=0 0 0 06 8 6 0 6_0=60 7 7! 4 5 6 7 6 4 5 7 5 6 4 7 7! 4 0,,, 4, 5 0 5 55,,, 4, 5 P =5_4_=60 Tip 8!, 8 8! 4!, 8_ 4_ 6! 6_4 09

07 9 9 9_8_7 ªC = 44444444444=84 8 84-8=76 08 0 0 0,, 6 7 7 6_5 C = 44445 =5 _, C _C =7_7=49, 5+49=64 09 n A={a, a, y, a«}, B={b, b, y, b«} AB=u U=A'B U A B A B «C n A, B A, B «C «C _«C =«C A B A, B «C «C +«C =«C U «C «C =«C +«C 5 8 n a«n (n-) a«(n-) a«, a«a«=a«+a«(n=, 4, 5, y) a =, a =, a =5 a =8, a =, a =, a=4, a =55 55 A, B A B A A 4A A A _4 =48 B =54 A, B =4 (48-4)+(54-4)=54 6 A, B, C, D, E, F 8 [ ][ ] 8 [ ]5 6 A B A B 5 0 ('+x)fi ('+x)fi C (')fi x 5-r=0,, 4 r=5,, C (') + C (') + C (') =+0+45=76 C D E F C D E F [ ] [ ] 5 6 4 C, D, E, F A, B C, D, E, F A, B 5 6 A, B 4 00

4 = 6 4 RNA RNA A, C, G, U A, C, G, U 4 A-C-C-G G-C-C-A A, C, G, U 4 P =4 =56 P =4 =6 56-6=40 : $º:+6=6 P =fi =4 5 P =fi = 4 C _ P =5_ =5_6=80 4--80= 5 A={,,, 4, 5} X'Y=A, XY+u X, Y (X, Y) X'Y=A A k (k=,,, 4, 5) k<x-y, k<y-x, k<xy k (k=,,, 4, 5) X-Y, Y-X, XY (X, Y) (X, Y) 5 fi =4 X'Y=A, XY=u k (k=,,, 4, 5) X-Y, Y-X (X, Y) (X, Y) 5 fi = (X, Y) 4-= Tip XY ~5 A A 5 fi XY=ufi X Y A 4 5 f(a) X-Y XY Y-X XY=u A f(a) X 4 Y 5 7 8 abc a, b, c a b, b c, c a A={,, 5, 7}, B={, 4, 6, 8} a, b, c A A C =4 A B C _ C =6_4=4, 6 A C = 4-=, 5 5_!=5_6=50 8 4 6 5 C _ C =4 4 C _ C =5 4_5=60 C _ C _ 44=! C _ C =0 _0=60, 60+60=0 0 0

0 5 6 5 0 5 ºC _ C _ 4 =6! 5 C =6, 6-6=0 9 40MB, 50MB, 60MB, 70MB, 80MB, 90MB 56MB (4), (), () (4) 4 56MB {40, 50, 60, 70}, {40, 50, 60, 80}, {40, 50, 60, 90}, {40, 50, 70, 80}, {40, 50, 70, 90}, {40, 60, 70, 80} 6 () 40MB 56MB C _ C _ C =0 =60 () 56MB C _ C _ C _ 44=5_6 6!=5!,, 6+60+5=8 0 6 0 5 0 5@5& 04! 05 4! 06 6 0 0 0 04 05 06 4 07 6 08 09 0 6 4 5 6 7 8 9 0 587 0 5 5~55 6_6_6=6 5 (,, ), (,, )!! + =6!! 6 = 6 6 6 0 4 A B C C C C P(A)= 5555555555, P(B)= 5555555555, P(C)= 5555555555 ºC ºC ºC A, B, C P(A'B'C)=P(A)+P(B)+P(C) P(A'B'C)=4 5+4 5+4 5= 5 5 0 +(+x)+(+x) +(+x) +y+(+x) x n} (+x)«x «C x C + C + C +y+ ºC = C + C + C +y+ ºC ( C = C =) C + C + C +y+ ºC = C + C +y+ ºC ( «C +«C =«C ) C + C + C +y+ ºC = C +y+ ºC =y C + C + C +y+ ºC = ºC + ºC C + C + C +y+ ºC = C «Cº=«C«= «C =«C«0{r{n «C =«C +«C {r{n- 0 8 C - 5555555555=-5@5*=5@5& 5@5& C 04 A B P(B A) P(A)=!, P(AB)=6! 6! P(AB) P(B A)= 55555555555555555555555= 55555555=!! P(A)! 0

05 A, B P(B)=! P(ABÇ )=!P(AB) A, B P(ABÇ )=P(A)P(BÇ ) P(BÇ )=-P(B)=-!=@, P(ABÇ )=!!=@P(A) P(A)=4# P(AB)=P(A) P(B)=4#!=4! 4! 0 A B C 4 A 7 B C 4 9 ABC C _ C _ C 4 55555555555555555555555555555555 = 55555555555555555555 =7@ ªC 9_8_7 55555555555555555555 A, B HjK A, B C HjK A C, B HjK A C, B C 0 8 6 06 5! 5 C {!} {!}= 6 6 6 6 6! 8 6! C! 4 4 4! C! 4! 5555555555555555555555555555 =5! 6! approach 04 A, B P(A)=!, P(B)=!, P(AÇ BÇ )=4! P(AB) AÇ A 0 A F P(A'B)=P(A)+P(B)-P(AB) P(A)=!, P(B)=! B E P(A'B)=!+!-P(AB)=6%-P(AB) yy C D P(AÇ BÇ )=P((A'B)Ç )=-P(A'B)=4! 6! 6_5_4 C = 5555 = 555555555555 =0!! ABC, BCD, CDE, DEF, EFA, FAB, ACE, BDF 8 B A C F D E P(A'B)=4# yy 4#=6%-P(AB) P(AB)= 0=5@ n «C n «C 05 A, B, CA ab b, C c a+b+c=6 X, a=b=c Y P(Y X) a+b+c=6(,, 4), (,, ), (,, ) (,, 4) a+b+c=6! =! (,, ) a+b+c=6!=6 0

(,, ) a+b+c=6 a+b+c=6 a=b=c (,, ) P(XY) P(Y X)= 55555555555555555= 55555555555= 0 P(X) +6+ C 8!0 8!0=6 0 a+b=6 6 Tip P(YX) P(Y X)= 54444 P(X) P(Y X) X Y X 0 Y a, b, c 0 08,, k (k=,, ) k -k A k B (k=,, ) A=(B A)'(B A)'(B A) 06 P A B P(B A)=P(B )P(A B )=! K (k=,, ) B A, B A, B A P(A)=P(B A)+P(B A)+P(B A) P(A)=!!+! @+! #=@ 4 5 6 5A P B pq p+q, p, q!! 6! 5 A P B 5! _{!}_!_6!= 6! p+q=6+5=4 4 07 A, B, C,, A B B C C ab a+b A B C, a, b 09 A, B P(AÇ B)=-P(A) P(B)=P(A)P(B)+P(AÇ )P(B) A B P(AB)=P(A)P(B) P(AÇ B) P(AÇ B)= 5555555555555555555 P(B) P(AÇ )P(B) P(AÇ B)= 5555555555555555555555 =P(AÇ ) P(B) P(AÇ B)=-P(A) AB=u P(A'B)=P(A)+P(B) AB+u P(A'B)=P(A)+P(B)-P(AB) <P(A)+P(B) P(A'B){P(A)+P(B) P(B)=P(AB)+P(AÇ B) P(B)=P(A)P(B)+P(AÇ )P(B) P(A'B){P(A)+P(B) C A B C C A B B C C C C C 555555555 555555555 555555555=! 6! 0=8!0 C C C P(A)>0, P(B)>0 A, BHjK P(B A)=P(B A C ) HjK P(B A)=P(B) HjK P(AB)=P(A)P(B) 04

0 A, B 7 4 A A A A 4, 5, 6 {!}=8! {!} A 4, 5, 6 7 C {!} {!}!= 4!!!= 6, A 8!+ 6= 6 999 999 pq p+q, p, q 999= 7 999 999 7 999 7 7 7 9 999 999-(+7-9)=999-5=648 9^9$9*=@7$ A={,,, 4, 5, 6} f:a A K={x f(x)=x, x<a} f K={, 5} f K={, 5} f()=, f(5)=5 f(k)+k (k=,, 4, 6) f()= f()= f(4)=6, f(6)=4 f()=4 f(4)=6, f(6)= f()=6 f(4)=, f(6)=4 f()=4, f()=6 f()= K={, 5} f _=9 A A f 6! 9 =8 0 6! Tip «C n! 5 4 9 p+q=7+4=6 6 6 5 n 4 m+n m, m, n 5 5 5!=0!=6 P =6!=6 6_6=6 P _=6!=6 6_6=6,, 6+6+6=78 0=!0# m+n=0+= 4 6 6,,,, fi, fl i (i=,, ) f(i), f()<f()<f() f()<f()<f() +fi <fl fl 5 fl C fl!( C _! 4 C _!! 6 6! ( C _!)_( C _!)_! 5555555555555555555555555555555555555555555555555555=6! 6! 05

5 S A, B 7 6, P(A)+0, P(B)+0 A, B, C 6 A, B P(A)+P(B){ P(A)+P(B)=P(AB)=P(AÇ )P(BÇ ) A, B 5 A, B P(B A)+P(A B)= 4 P(A'B)=P(A)+P(B)-P(AB){ A, B P(AB)=0 P(A'B)=P(A)+P(B){ P(AB)=P(AÇ )P(BÇ ) P(AB)={-P(A)}{-P(B)} P(AB)=-P(A)-P(B)+P(A)P(B) P(AB)=P(A)P(B) P(A)+P(B)= A, B S S={,, } A={,, }, B={} A B 4 C,, 5, 6 B,, 5, 6 C4 B 6 5 4 B C 5 A,, 4, 5, 6 6_6!_{6!_6$+6$_6@}=!6@=! P(A)=, P(B)=!, P(AB)=! P(AB)=P(A)P(B) A, B P(B A)=!, P(A B)= P(B A)+P(A B)+ _4= _!=_6=7 6 7 =! 6 6 A, B, C C C A B C 5!! 4! 5#!! 5! 6! 4! 8 A, B, C, D, E, F 6 A B A B ABAB C A, B 4!_5!_6!=!0 C A, B AB A 6$=@!_5!_!= 0 B 5@ C A, B 4!_5#_!= 0,, C!0+ 0+ 0= 0 C 0 0 = 0 A!_!=4!B! A, B @_5@_4!_!= 0 B A 0 A B 0+ 0= 5 06

9 A, B A B B A A 0 n p«log @ (pª+p º) A B B A p =!!+!!=! A A B! @+! @=@ 9 pª={@}! 0 p º={@}!+{@} ={@} pª+p º={@}!+{@} 57~6 0!& 0 0 0 04 6 05 0.64 0 0 0 04 90 05 06 07 08 09 0 4 5 6 7 6 8 9 44 0 0 X E(X)=! V(X) X - 0 P(X) 4! a b! 4!+a+b+!= a+b= yy E(X)=! pª+p º={@} {!+@}={@} E(X)=(-) 4!+0 a+ b+!=! b= log @ (pª+p º)=log @ {@} =8 a>0, a+ M>0, N>0 logå =0, logå a= logå MN=logå M+logå N M logå 44 =logå M-logå N N logå M =k logå M k b= a=! E(X )=(-) 4!+0!+ +!=% V(X)=E(X )-{E(X)} =%-4!=!&!& 0 ax+b 0 X (X+00) X B {0,!} E(X)=0_!=5 0 4 pq p+q, p, q 4 C {!} {!} =6 4! 4!=8# -8#=8% 8%_8%_8#=5 p+q=5+75=587 587 E(X+00)=E(X)+00 = 5+00=0 0 0 X f(x) f(x)=kx (0{x{) k f(x) k>0 f(x)=kx! k= k= y k f(x)=kx O x 04 X N(, )P({X{5) Y N(0, ) P(0{Y{k) k 07

X N(, ) - 5- P({X{5)=P { 444{Z{ 444}=P(0{Z{) Y N(0, ) 0-0 k-0 P(0{Y{k)=P { 444{Z{ 444}=P {0{Z{K} P(0{Z{)=P {0{Z{K} K= k=6 6 0 ax+b,,,,, 4 X 5X+ X E(X)=_!+_!+4_6!= X 4 P(X)!! 6! E(5X+)=5E(X)+=5 += a, b P(a{Z{b)=P(0{Z{b)-P(0{Z{a) P(-a{Z{b)=P(-a{Z{0)+P(0{Z{b) =P(0{Z{a)+P(0{Z{b) P(Z}a)=0.5-P(0{Z{a) P(Z{a)=0.5+P(0{Z{a) 05 X00 0 0 X V(X ) X 0 4 6 8 4 6 4 0 E(X)= 0(0_+_4+4_6+6_4+8_)=*0)=4 E(X )= 0(0 _+ _4+4 _6+6 _4+8 _)=: 0 :=.4 V(X)=E(X )-{E(X)} =.4-6=6.4 0 V(X) 6.4 V(X )= 44 = 4=0.64 0.64 0 0 0 ax+b m r X T X-m T=0_ 4444444 +00 r T 0 0m E(X)=m, r(x)=r T= X- 444+00 r r 0 0m 0m 0m E(T)= E(X)- 444+00= 444-444+00=00 r r r r 0 0 r(t)= r(x)= _r=0 r r 04 90 X X X B {90,!} E(X)=90_!=0, V(X)=90_!_@=0 V(X)=E(X )-{E(X)} E(X )=V(X)+{E(X)} =0+0 =90 90 05 X f(x) f(x)=kx (0{x{)P(0{X{) k approach 0 k- X P(X=k)= 4444 (k=,, 4) X m r 6 mr f(x) k>0 f(x)=kx P(0{X{)=! k= k=9@ P(0{X{)=! k=k=9$ y k k O f(x)=kx x X X 4 k- P(X=k)= (k=,, 4) 6 X P(X) 6! 6@ 6# E(X)=_6!+_6@+4_6#=: º:=m V(X)= _6!+ _6@+4 _6#-{: º:} =: :-:!9):)=9% '5 r(x)=" V(X)= =r '5 0'5 mr=: º:_ = 444 9 06 X, Y N(0, ), N(0, ) P(8{X{4)=P(4{Y{k) k X, Y N(0, ), N(0, ) X-0 Y-0 Z X = 444, Z Y = 444 Z X, Z Y N(0, ) 8-0 4-0 P(8{X{4)=P { 4{Z X { 444} P(8{X{4)=P(-{Z X {)=P(-{Z X {) 08

4-0 k-0 k-0 P(4{Y{k)=P { 444{Z Y { 4}=P {-{Z Y { 4} P(8{X{4)=P(4{Y{k) k-0 4= k= Tip X P(-{Z{), k-0 Y P {-{Z{ 44 } k k-0 Z 0 44 = 07 X 50 0 4 X E(X ) E(X)=50, V(X)=0, n=4 E(X )=50, V(X )=: 4º:=5 V(X )=E(X )-{E(X )} E(X )=V(X )+{E(X )} =5+50 =505 08 N(0, 4) 8 Y Y 8 X Y=8X V(X )=8$=! V(Y)=V(8X )=8 V(X )=64_!= 09 50 40 00 X P(46{X {58) 40 X N {50, 55 } N(50, 4 ) 00 X -50 Z= 44444 Z N(0, ) 4 46-50 58-50 P(46{X {58)=P { 44444444 {Z{ 44444444 } 4 4 P(46{X{58)=P(-{Z{) P(46{X{58)=P(0{Z{)+P(0{Z{) P(46{X{58)=0.4+0.477 P(46{X{58)=0.885 z.0.5.0.5 P(0{Z{z) 0.4 0.4 0.477 0.498 0 0 cm 95% P( Z {.96)=0.95 m X, n 95% r X -m {.96_ 'n 0.96_ {, 'n}9.6 n}9.6 =84.6 'n n 85,, 0 X X 0.6,,,, x, y, z X x+y+z=0 yy x+y+z E(X)= 44 = x+y+z=0 yy 0 (-) x+(-) y+(-) z x+z V(X)= 44444 = 4444=0.6 0 0 x+z=6 yy x=, y=4, z=,, 4 4 XX _ C 444= X B {, } C E(X)=_ = V(X)=_ _ =4 P(X=) X 6 X=r P(X=r)4444 P(X=4) X B(n, p) E(X)=np=6 V(X)=np(-p)= 6 (-p)= p=@ p=@ n=9 yy yy X P(X) 0 0 0 09

P(X=r)=ªC {@} {!} P(X=) 4444 P(X=4) ªC {@} {!} fl = 4 =! ªC {@} {!} fi 4 70 cm 5cm X 0 cm A, B, C a, b, c a, b, c x A 70 B 68 C 74 A, B, C P(65{X{75)>P(6{X{7)>P(69{X{79) a>b>c A:65{X{75, B:6{X{7, C:69{X{79 5 A, B, C A, B, C N A (80, 5 ), N B (74, 6 ), N C (70, 8 ) A-80 B-74 C-70 Z A = 44, Z B = 44, Z C = 44 Z A, Z B, Z C 5 6 8 N(0, ) 86, 88, 84 86-80 88-74 84-70 Z A = 444=5@, Z B = 444=8&, Z C = 444=9& 5 6 8 Z B >Z C >Z A 86 80 5 88 74 6 84 70 8 X-0 Z= 44444 Z N(0, ) 8 40-0 P(X{40)=P{Z{ 44444444 }=P(Z{.5) 8 P(X{40)=0.5+P(0{Z{.5)=0.5+0.498 P(X{40)=0.998 7 00 X P(7{X{k)=0.885 k 5 X B {00, 0} E(X)=00_ 0=0, V(X)=00_ 0_ª0=9 C 4 = 0 C n=00 X N(0, ) X-0 Z= 4444444 Z N(0, ) k-0 P(7{X{k)=P {-{Z{ 444444 } k-0 P(7{X{k)=P(0{Z{)+P {0{Z{ 444444 }=0.885 k-0 0.4+P {0{Z{ 444444 }=0.885 k-0 P {0{Z{ 444444 }=0.477 k-0 444444 = k=6 6 z.0.5.0.5 P(0{Z{z) 0.4 0.4 0.477 0.498 6 08 40 400 X X B {400, 5$} E(X)=400_5$=0, V(X)=400_5$_5!=8 n=400 X N(0, 8 ) z....4.5 P(0{Z{z) 0.48 0.486 0.489 0.498 0.498 8 64 x, x, x, y, x 64 k= x =0, x =856 m 95% P( Z {.96)=0.95 X s x k= X = 4444 =: 6 4º:=5 64 64 64 x k= s = 444444 -X =:!6*4%:^-5 =9-5=4 s= 64 m 95% 5-.96_ 44{m{5+.96_ 44 4.5{m{5.49 '6å4 '6å4 64 k= 040

9 6 6~65 0 0 0 04 05 06 07 08 09 0 4 5 6 9 7 6 8 88 9 7 0 N(0, ) Z P( Z {k)=0ƒ0 r n r a% m k_ 'n 6 k_ 44 = k= 'å6 n { 'n} n}44 'n 44 44 Tip r = 'n 4 0 a, b " afi = 4 logå b b " afi = 4 HjK a % =b HjK a -6% =b b logå b=logå a -6% =-6% 0 lim " n -0-n 44444444 n-" n +0 " n -0-n -0(n+" n +0) n+" n +0 lim 4444= lim 4444444 =lim 4444 n-" n +0-0(" n -0+n) " n -0+n 0 0 +æ + n +' = lim 44 = 44 = 0 '+ æ - + n 0, 5#,, % 5 A, B, C, D, E {5#} -% A B C D E 0 5 5 x 0 r n m a%x a n X -m a n m an y={5#} {5#} -% >{5#} =% {5#} -% E y 5 - O - 5 x P( Z {k)=0ƒ0 r X -m {k_ a k X -m 'n n X -m a n X -m r r a% X -k_ {m{x +k_ 'n 'n a% m (00-a)% m r k_ a k 'n n 04 {a«} a =, a +a =a +6 a º a +a =a a =a +6 {a«} d a =a +d=+d=6 a º=a +9d=+9 %=6 d=% a =6 05 a, b a b=a+b-ab log 5 04

e a e=e a=a+e-ae=a e(-a)=0 a e=0 log 5 x (log 5) x=log 5+x-x log 5=0 x(-log 5)=-log 5, x log =-log 5 -log 5 x= 444=-log 5 log 06 {a«}, {b«} lim a«= lim b«=- lim (a«-b«)= {a«+b«} lim (a«+b«)=0 a«a«[ 4 ] lim 4 <0 b«b«lim (a«-b«)= lim a«-lim b«=-(-)= a«=n+, b«=-n lim a«= lim b«=- lim (a«+b«)= lim {(n+)-n}= lim =+0 a«=n, b«=-n lim a«= lim b«=- a«n lim = lim 4444 = lim {-n!}=0 b«-n 07 {a«}, {b«} n A -A-E=O (A-E) =4E (A-E) =(4E) =6E (A-E) 6E 09 n a x=«_{!} å x log =0.00, log =0.477 log =00 log =00_0.00=0.0 0 0.0 log =0<0.0<log n=, a= x= _{!} = log =0 log =0_0.477=4. 4 5 log N n N n+ log N -n N n 0 N log N a log, log, log, y log {a<log N log {a<log N log {a<log 4 N a =b = a«+b«=a«b«+a«=b«5 k= b b«=a«-a«{b«} {a«} a«=b«-b«{a«} {b«} a =a + 5 k= b {a«} :,, 5,, 4, 89, y {b«} :,, 8,, 55, y a =89 5 b =a -a =89-=88 k= 08 A A -A-E=O A A+E (A-E) A -A-E=O A(A-E)=E E A =!(A-E) A=E A -A-E=O A+E=4E A+E 0 A B P(B A) P(AB)=5# 4@= 0 P(AB C )=5@ 4#= 0 P(A)=P(AB)+P(AB C )= 0+ 0= 0=5# P(AB) 0 P(B A)= 44444 = 44 =! P(A) 5# Tip 04

U={,,, y, 9} A, B n(a)=4, n(b)=5, n(a'b)=6 A-B B-A (A, B) n(a'b)=n(a)+n(b)-n(ab) 6=4+5-n(AB) n(ab)= AB a, b, c U AB 6 p, q, r (p<q<r) A={a, b, c, p}, B={a, b, c, q, r} (A, B) 9_8_7 6_5_4 ªC _ C = 4_ 4 ªC _ C =84_0=680 A B AB O 0. 0. 0. 0.4 00 B 5 00 B X X B{00, 5!} E(X)=00_5!=0, V(X)=00_5!_5$=6 z 0.75.00.5 00 X N(0, 4 ) X-0 5-0 P(X}5)=P{ 444 } 444 }=P(Z}.5) 4 4 P(X}5)=0.5-P(0{Z{.5)=0.5-0.944=0.056 P(0{Z{z) 0.74 0.4 0.944 x, y a [{] 0 ª º= -4 [}] 0 x, y a, b a, b (a, b) 6 [x] x a=-6 ab a+b 0 4 A t y y=a (k) 0 6 x n<x<n+ nlog =0.0, log =0.48 0 A =A, = 0k=log k= 0 log x 6 a [{] 0 [{]=0, [}]=0 ª º= -4 [}] 0 x, y [{]=0, [}]=0 x, y a, b A 0 log =6A, 0 log =6 0 log =log 6 log 0.0 x=0 log 6=0{+ }=0{+ 44 } log 0.48 x=0(+0.65)=48.75 a=0, b=0,, n<x<n+ n 48 a, b (a, b) 6 [{]=0, [}]=0 a -4 --a+0 a+-6 a=0, b=0 a+b 0 a=, b= ab 5 A, B, C, D, E, F 6 A B E F C D a=-6[{]=n, [}]=n (n) A, B _!= 4 C, D E, F A B a, b (a, b) [{]=0, [}]=0 _!_!=8 _8=6 04

6 8 pqp+q p, q 8 8 7 6 C = 44 =56 C _ C _ C =8 8-6= 4 +4=6( 5!6^=7@ p+q=7+=9 9 7 X N(00, 4 ) 4 X P( X -E(X ) <8)=P( X-E(X) <k) k X E(X)=00, V(X)=6 n=4 X V(X) E(X )=E(X)=00, V(X )= 44 =4 4 X N(00, ) P( X -E(X ) <8)=P( X -00 <8) X -00 P( X -E(X ) <8)=P { 44444 <*}=P( Z <4) X-00 P( X-E(X) <k)=p{ 44444 <4K}=P{ Z <4K} 4 P( X -E(X ) <8)=P( X-E(X) <k) 4K=4 k=6 6 9 n S«S 5 4 6 7 8 4 9 0 5 6 7 8 6 7 8 9 5 0 9 0 4 S S S S 0 S S º 4 5 S k S Sª 9 9 0 9 k = 44 =85 k= 6 S º 0_4-4=6 S º 8_4-4=8 S º 6_4-4=0 S º 4 5 S º 85+6+8+0+=7 7 Tip 0 n A«P«A«x n P«A«A«n} 8 x {x{00 x, y log y =(log x) 44 x=a M y a+log M A ºP =5#, A «P «=5#_P «A«n} lim P «A«=pQ p+q p, q log y = log y=(log x) log y=!(log x) P P P P n P n+ x x log =4 log x-log y=4 log x--log y y log =-!(log x) +4 log x- log x=t {x{00 0{t{log 00 x log =-!t +4t-=-!(t-4) +7 y x x log t=4 7 x=8 y y a=8, M= a+log M=8+7=88 88 Aº A A A A«- A«A«+ r=5# P A =-A ºP =-r P A =-A P =-r P A =-r(-r)=-r+r P A =-A P =-r P A =-r(-r+r )=-r+r -r P A P «A«=-r+r -r +y+(-)«r«lim P «A«= 444 = 4 =8% p+q=8+5= -(-r) +5# 044

66~69 0 0 0 04 05 06 07 08 09 0 4 5 6 5 7 6 8 5 9 98 0 05 N(6, 5 ) n X P(4{X {8)}0.9 n z.8.64.96 P(0{Z{z) 0.400 0.450 0.475 0 'å_ '4 ø " 8 'å_ '4 ø "Ω8 =" fi _ " fl" ( ) = %+4@-6( = # =' X N{6, 5 4} n 4-6 8-6 P(4{X {8)=P 4{Z{ 4 ª 5 5 4 4 º 'n 'n 0 - - A=, B=, C= AB+AC 4 - - 5 AB+AC=A(B+C) - - AB+AC= [ + ] 4 - - 5-0 -4 AB+AC= = 4 0-6 8 AB+AC -4+-6+8=0 0 lim {'ƒ+4ƒ+yƒ+n-" + +y +( n-)} lim {" +4 +y +n-" + +y +( n-)} = lim {" n(n+)-" n }= lim (" n +n-n) n = lim 44 = lim 44 =! " n +n+n æ +n!+ 04 4 5 C _ C _ C _ =6! C _ C _ C =0 6_0=80 'n 'n P(4{X {8)=P{- 4 {Z{ 4 }}0.9 5 5 P(-.64{Z{.64)=0.9 'n 4 }.64 n}4. =6.8 5 n 7 06 0 C _C 444444 = ºC 5! _5!= 07 a b A= A = ax+by=k, cx+dy=-, x+y=4 c d 4 k ax+by=k a b x k g = cx+dy=- c d y - x a b = k k k-4 = = y c d - 4-4k-6 ax+by=k, cx+dy=- (k-4, 4k-6) x+y=4(k-4, 4k-6) (k-4)+(4k-6)=4, 6k-4=4 k= n p q r(p+q+r=n) p, q, r «Cπ_«πCœ_ C p, q, r «Cπ_«πCœ_ C _ 44! p, q, r «Cπ_«πCœ_ C _ 44! 08 y logæy <x x, y (x, y) x x>0, x+, y y>0, y+ x> y logæy <x x logæy logæy <logæx, (logæy) <, (logæy+)(logæy-)<0 -<logæy<0 0<logÆ y< yy 045

logæ [!<logæy<logæ logæ <logæy<logæ x [!<y< <y<x 0<x< y logæy <x x logæy logæy >logæx, (logæy) >, (logæ y+)(logæ y-)>0 logæ y<- logæ y> yy logæy<logæ [! logæy>logæx y>[! 0<y<x Tip 09 y O a b M=[ a, b] -b a y=x A, B<M AB<M A, B<M AB=BA A<M A A A <M y= x x a b p q A= <M, B= <M -b a -q p a b p q ap-bq aq+bp AB= = <M -b a -q p -(aq+bp) ap-bq p q a b ap-bq aq+bp BA= = -q p -b a -(aq+bp) ap-bq AB=BA a b A= <M a +b +0 A -b a a -b a (-b) A = 4444 = 5554 <M a +b b a a +b -(-b) a 0 A={,,, y, } 50 606 606 log =0.00 «, 4, 8, 6,, 4, 8, 6, y 00=4_50+ 50 A log =00 log =00_0.00=605.0 605 606 log «m a log «± =log «+log =m+a+0.00 log «± m log «± a+0.00}0.00 «± log «± m+ log «± (a-)+0.00<0.00 «± log =0.00, log =605.0 log «± log «n 605 A 605 n E n E log E=a+bn (a, b) 6 00 0 4 log =0. 6 E, E log E =a+b yy log E =a+6b yy E =00E - E log E -log E =-5b, log 44 =-5b E log 00=-5b b=-0.4 0 Eº log Eº=a 4Eº n log 4Eº=log Eº-0.4n, log 4+log Eº=log Eº-0.4n log =-0.4n, _0.=-0.4n n=-.5 -.5 n n a, a, y, a«b, b, y, b«(a +a +y+a«)(b +b +y+b«)}(a b +a b +y+a«b«) n==a b = n=k A=a +a +y+a, B=b +b +y+b, C=a b +y+a b AB-C 0 (A+a )(B+b )-(C+a b ) =(AB-C )+( ) +("çab-c)a b }0 n=k+ n n==a b =(a b ) = a b 046

n=k A=a +a +y+a, B=b +b +y+b, C=a b +a b +y+a b (a +a +y+a )(b +b +y+b )}(a b +a b +y+a b ) AB}C AB-C } 0 (A+a )(B+b )-(C+a b ) =AB+Ab +Ba +a b -C -Ca b -a b =(AB-C )+( " Ab -" Ba ) +("çab-c)a b }0 n=k+ n a b } " Ab -" Ba B, C, D A B, C, D!= =6 6_{!_!_!_!}= 7 7+ 7+ 7=7 Tip A n A P«P«A AA B, C, D! A P«=0_P«+!(-P«) y=logå(x+5)+6(a>0, a+) a (p, q) p +q y=logå x(a>0, a+) a (, 0) y=logå(x+5)+6 y=logå x x -5 y 6 a (-5, 0+6)=(-4, 6) p +q =6+6=5 5 [ ] [ ][ ] [ ][ ] [ n] «a«a«n= 4! A 4A A 4 A B, C, D A B A B A, A C A C A, A D A D A! _{!_!_!_!}= 7 B, C, D A A A A A A A B, C, D P =_=6 A A A B, C, D P =_=6 B, C, D _{!_!_!_!}= 7 B A C D n+ ø r«r«o, O, O r«o C, C, C r«r«r«a O =ø πo Oπ π-πa O r«o A O =" 4r«-r«='r«O r«r«o A O O O O ' O O =@ A O = 444r«O O +r«=r«' 444r«+r«=r«, 'r«+r«=r«r«=('-)r«n+ n {a«} p ('-) p a«= p{('-) }«= 4 n= n= -('-) p +8' a«= 444 = p (-+8') 6 y=8, y= y=k A, BAB =6k k> y O y=8 x y= x A B y=k x 047

y=8 y=k A 8 =k x=log k A(log k, k) y= y=k B =k x=log k B(log k, k) AB =6 log k-log k=6, log k-! log k=6 @ log k=6, log k=9 k= =5 5 7 O ABCDABDC B 78 C n n ABDC a, a+d, a+d, a+db=78 a+d=78 yy 80 B=78 D=80 -B=0 a+d=0 yy a=54, d=4 C=a+d=54 +_4 =6 n=6 6 8 B A O C D 9 - ++ +y+ = 4 =0 - + Cº+ Cº+ Cº+y+ªCº =++y+=0 C + C + C +y+ªc =+++y+9=45 Cº+ C + C +y+ªc =+++y+9=45 + C + C + C +y+ªcª=++y+=0 Cº C C 0-(0+45+45+0)+(+ Cº+ C + C )=98 98 n}4 «Cº, «C, «C, y, «C«n k=4 ( C + C +y+ C )= ( C + C +y+ C )= Cº C Cº C C Cº C C C Cº C C C C Cº C C C C C ªCº ªC ªC ªC ªC ªCª n k=4 n k=4 { -( Cº+ C + C + C )} { -(k+)} Cº C Cº C C Cº C C C Cº C C C C Cº C C C C C ªCº ªC ªC ªC ªC ªCª 9 (fl -) 6(5+0) { -(k+)}= 444-444=04-6-90=98 k=4 - n+ n n+ n n+ 0 n} n a«a = n+ n n+ a«n n+ a«=a«+ a«+=(a«+) {a«+} a º+=(aª+)= (a +)=y= (a +)= a = a º= -=5 5 {a«+} a + a«+=(a +)«= «=«a«=«- a º= -=5 0 9 X 0 X= E(7X) X,, X= C _ C _ C P(X=)= 444 =ª8 ªC X= (_ C )_(_ C ) P(X=)= 44444 =!8* ªC X= _ C P(X=)= = 8 ªC X X P(X=x) ª8!8* 8 E(X)=_ª8+_!8*+_ 8=: 7 E(7X)=7E(X)= 048