수-적분2-1(126~153)eps교

.. 0 0 6

03 04 (+x)«05 7

06 84~93 «P ~5 94~96 «H 6~7 97 8 98~0 03 04~05 06 07 9 ~0 ~3 07 6~7/3 94~96 «H 3 8

08 8 64 6 Brahmagupta598670 n n(n-)(n-) y BhaskaraA485 n k n(n-)(n-) y (n-k+) k(k-)(k-) y 7 Pascal B6366 Leibniz G W64676BernoulliJ654705 8 3 NewtonI6477 (a+b)«a«b «C (a+b)«=;rn+)«c a«b n (a+b)«n=3y n= C 0 C n= C 0 C C n=3 3C 0 3 C 3 C 3 C 3 n=4 4C 0 4 C 4 C 4 C 3 4 C 4 n=5 5C 0 5 C 5 C 5 C 3 5 C 4 5 C 5 00 The Life of Isaac Newton SmithDE958History of mathematicsdover Publications 000 9

4 4 8 8 abcde dcaeb 84 85 86 87 88 44 5 0 m n m+n 35 P _ P C + C + C + C +ªC 0 5 30

n! p!q!yr! p!q!yr! «P 4 4 4 4!=4 4 4 4 4 4! =3!=6 4 4! = 4_3! 4 4 4 4 4 jk jk jk 4 4 P =4! 4 P 4! = =(4-)!=3!=6 4 4 n n «P«=n! n ABC ABC 3!=6 A B CC A BB C A A C BB A CC B A A C B 3 C B B A A C 3 A B C B C C A A B 3! ;3!; = 3 3

n «P«=n! n n «P«n! = =(n-)! n n n (n-) (n-)! n «P«n! = =(n-)! n n n «P«n! = =(n-)! n n 7 (7-)!=6!=6_5_4_3 =70 5 6 6 (6-)!=5!=5_4_3 =0 yy` 5 (5-)!=4!=4 9 (9-)!=8! 3 9 8 7 3 8 7 6 9 7 6 5 6 5 4 5 4 3 4 3 3_8! 8 8 7 3 4 7 6 3 6 5 5 4 _(8-)! 8 9 3 0 9 8 9 0 8 3 7 7 4 6 3 6 5 5 4 3 _(0-)! 4 4 3 8 8 4 7 3 7 6 5 6 5 4 7 8 6 7 8 6 5 5 4 3 4 3 4 4_(8-)! 3

64 65 5 (5-)!=4!=4 3 4 (4-)!=3!=6 3 3!=6 (4-)!_3!=36 3 (3-)!=!=6 3 3!=6 (3-)!_3!= (5-)!_=4_=48 yy` 3 3 0 9 0 9 0 0 0 0 0 0 0 0 0 0 9 0 0_0_0_0=0 =0000 «P n r n r 5 ABC DE 3 5 P =3fi =43 5 3 P =5 =5 5 5 5 5_5_5=5 =5 5 3 P =5 =5 33

4 0 0_0_0_0=0 =0000 3 P P P «P P Product P n r n r n r «P «P n r n n yr n r n n n n n r «P =n_n_y_n=n r ( M { M 9 034 0 34 4 034 4_ P =4_5 =500 yy` 4 5 5 5 0 34 4 04 3 03 4 4_ P _3=4_5 _3 4 =300 yy` 5 5 3 p.6 n r «P =n 4 0 P =4 =64 P =3 =8 «P nær «P n>rn=rn<r 5 0000()00() 34 34 34 4_4_4=4 =64 4 3 P =4 =64 3 3 P = =8 P =5 =5 P = =8 P =3 =7 64 65 0 3 ºP =0 =000 64 4 65 4 0 _ P =fi =3 34

4 3 3 ( )!_3! 3 x 5 5! x_(!_3!)=5! 5! x= =0!_3! yy( ) 3! 3! 3 p.63 n pqrys n! p!q!r!ys! (p+q+r+y+s=n) 4 4 4 P =4!=4 6 aaab 3 a a a a a a a b a a a b a a a b a a a b a a a b a a a b a a a b 3 a a ba a a ba a a ba a a ba a a ba a a ba a ba a a ba a a ba a a ba a a ba a a ba a ba a a ba a a ba a a ba a a ba a a ba a a a =a =a =a a a a a aaab aaba abaa baaa aaab x 4! x= =4 3! 35

7 4 3 7! =35 4!3! 345 35 3 5 3 5 35 35a aaa4 a 3 5! =0 3! A B B a b A 4 3 A B aaaabbb 7! =35 4!3! 64 5 8 a b l 8! =5040!!! b 6 a l 6! =80!! 6 3 p.69 3 6! 6_5_4_3 = =60 3!!! (3 )_(_)_ baseball 8 b A B B A C B C A a b A B a B 4 a 4 b a b b a b C 8! b =70 yy` a 4!4! A A C aab C B aabbb 3! _ 5! =3_0=30!!!3! yy` p.63 n pqrys n! (p+q+r+y+s=n) p!q!r!ys! «Cπ_«πCœ_«π œ Æ C _y_ßcß 33 3 3 7! =0 3!!! 3 7! 4! C _ C _ C = 3!4!!! =0 36

65 5 a b A B aaaaabbbb 9! =6 5!4! 7 B C A A B A C B A B A C B A C B 4! 5! _ =60!! 3!! A C B 6-60=66 345 3 7 aabbbcc 5 (5-)!=4!=4 3 5 P =3fi =43 5 9 messenger ee egmnrss 7! =50! 3 7 a b 3c 7! =0!3!! ss eeegmnr 7! 3! =840 3360 37

3 3 3 «H r p.66 xyz 4 x+y+z=4 xyz 4 xyz x y z 3 3 3 P =3 =9 3 6 p.66 3 4 H (6=3+4-) xyz 4 ()()(3)(4)() (3)(4)(33)(34)(44) ()(3)(4)(33)(34) (44)(333)(334)(344)(444) 0 0 () () (3) (4) () (3) (4) (5) (6) (34) yy (44) (333) (334) (344) (444) (56) (345) (346) (356) (456) 3456 3 H = C = C =0 38

( ) 3y ( ) 4y 4 ( -y C «H H Homogeneus Product 6! 6! =5 = C =3+4-C4 4!! 4!! yy( ) n x x yx«r x y z 4 0 0 0 4 0 x +x +y+x«=r r 0 0 4 3 0 (n-) 3 0 0 3- (n+r-)! 4 =«C r!(n-)! n r «H n r «H (n+r-) r «C (a+b+c) (a+b+c) =(a+b+c)(a+b+c)(a+b+c)(a+b+c) (a+b+c) a a ba bcab cb c y 4 abc 4 (aaaa)(aaab)(aabc)(abbc)(bbcc)y (a+b+c) abc 4 H = C = C =5 yy` 5 5 n r «H =«C H = C = C =0 H = C = ºC =0 «C nær «H n>rn=rn<r «H =0 n (a+b+c+d)fi H H H ABC ABC A B C 3 P = =8 3 H = C = C =4 39

H = C = C = H = C = ºC =0 H = C = C =35 «H =«H +«H «H =«H +«H +«H «H =«H +3«H +3«H +«H «H =«H +4«H +6«H +4 «H +«H yy 3 3 4 6 4 yy 66 «H =«H +«H «H +«H =«C +«C =«C +«C =«C =«H «H =«H +«H +«H «H +«H +«H =«C +«C +«C =«C +«C +«C =(«C +«C )+(«C +«C ) =«C +«C =«C =«H 67 3 68 5 H = C = C = «H = n+3- C =«C (n+)! «H = 3!(n-)! (n+)(n+)n «H = 6 «H =0 (n+)(n+)n=6 0 =6_5_4 n=4 abcd 5 H = C = C =56 A={}B={34}fA B ij<ai<j f(i)<f(j) ij<ai<j f(i) f(j) x+y+z<5 xyz x+y+z=kk=034 xyz k H 4 k=0 4 H = 3+k- C k = C k=0 6 = Cº+ C + C + C + C =+3+6+0+5 =35 4 k=0 0 40

C _(4-)!=5_6 =90-6 4 4 4000 04 3 5 3_ P =3_5=375 375+=376 4000 376 034 4000 3 B A B P Q A A PQ B 4 x+y+z=6 xyz 3 5 abcd a b A B aaaaabbb 8! =56 5!3! A P Q B 4! 3! =8!!!! 5 P =4_3= P =4 =6 4! C = =6!! H = C = 5!!3! =0 4 3 6 H = C = 8! =8 6!! 4

6 ABCDE 6 6 6 8 8 4 4+4 4 +4 4+ +3 3+3 3+ A B 4 C 6 D 4 E A B 4 C 6 D 4 E A B C D E 6 ABCDE A={abcd} 6 (3x +x+5){x+ } x 3 (x+y+3)(x-y-3) {x+ } x 4 4x -y -6y-9 x +4x +6+ + x x n «C +«C +«C =4 4 4

(+x)«;!; a b aba ab (a+b) (a+b) =(a+b)(a+b)(a+b) =aaa+aab+aba+abb+baa+bab+bba+bbb =a +3a b+3ab +b ab 3 ab 3 a+b (a+b)(a+b)(a+b) b a a a a a = Cºa abb a a b 3! 3! = C =3 ab!!!! a b a 3a b= C a b C =3 a Cº= a b b a a C =3 b C = a b b b a b 3ab = C ab b b a (a+b) (a+b) = Cºa + C a b+ C ab + C b b b b b = C b (a+b) a b aaa aab aba baa abb bab bba bbb (a+b+c)«aπ bœ c (p+q+r=n) (a+b+c)«={(a+b)+c}«(a+b)««c (a+b) n-r c «Cœa n-r-q bœ (a+b+c)««c «Cœaπ bœ c aπ bœ c n! (n-r)! «C «Cœ= _ r!(n-r)! q!(n-r-q)! n! = p!q!r! (a+b+c)fl a b c 6! =60!3!! 43

n (a+b)«(a+b)«=(a+b)(a+b)y(a+b) n a+b a b a n-r b r a+b n r b (n-r) a a n-r b r n! =«C (n-r)!r! (a+b)«=«cºa«+«c a n- b+y+«c a n-r b r +y+«c«b«n ( M M { M M 9 fi {x+;[#;} x C x 5-r {;[#;} r = C 3 r x 5-r (a+b)«a=xb=;[#; 5-r= r= x C 3 =90 yy` (a+b)«=;rn+) «C a n-r b r «Cº«C y«c y«c«[ ] {x +;[@;} fl [x ] {x-;[!;} [xfi ] «C a n-r b r p.75 n (a+b)«=«cºa«+«c a n- b+y+«c a n-r b r +y+«c«b«(a+b)«=;rn+) «C a n-r b r «C (n-)! = (r-)!(n-r)! (n-)! «C = r!(n-r-)! «C +«C r(n-)!+(n-r)(n-)! = r!(n-r)! n(n-)! = r!(n-r)! n! = =«C r!(n-r)! n=3y(a+b)«n= n= n=3 n=4 C0 C C0 C C 3C0 3C 3C 3C3 4C0 4C 4C 4C3 4C4 3 3 4 6 4 n=5 5C0 5C 5C 5C3 5C4 5C5 5 0 0 5 «C +«C =«C «C =«C«a n-r b r a r b n-r p.78 (a+b) = Cº(a) + C (a) b+ C (a)b + C b (a+b) =8a +3 4a b+3 ab +b (a+b) =8a +a b+6ab +b 3 (a+b)fl (x-y)fi (3x+y) (x-) (a+b)fi (+x)«(+x)«=«cº+«c x+«c x +y+«c«x«75 (3x+y) =7x +54x y+36xy +8y (x-) =x -4x +6x -4x+ (a+b)fi =afi +5a b+0a b +0a b +5ab +bfi 76 77 C (x )fl { } = C x x x r=3 x C =60 C (x) {- } = C (-) x x xfi r= xfi C fl (-)=-448 (a+b)fl =afl +6afi b+5a b +0a b +5a b +6abfi +bfl (x-y)fi =xfi -0x y+40x y -80x y +80xy -3yfi 76 4 77 4 (+x) n x (+x) n = n Cº+ n C x+ n C x +y+ n C n x n x=- ncº- n C + n C -y+(-) n nc n =0 yy x= ncº+ n C + n C +y+ n C n = n yy +( n Cº+ n C + n C +y)= n n Cº+ n C + n C +y= n- -( n C + n C + n C +y)= n n C + n C + n C +y= n- 44

(+x) n = n C 0 + n C x+ n C x +y+ n C n x n yy x=- n C 0 - n C + n C -y+(-) n nc n =0 yy x= n C 0 + n C + n C +y+ n C n = n yy nc 0 + n C + n C 4 +y= n C + n C 3 + n C 5 +y= n- n(+x) n- = n C + n C x+3 n C 3 x +y+n n C n x n- x= yy «Cº+«C +«C +y+«c«=«(+x)«=«cº+«c x+«c x +y+«c«x«(a+b)«x= a=b=x «=«Cº+«C +«C +y+«c«4 «Cº-«C +«C -y+(-)««c«=0 «Cº+«C +«C +y=«c +«C +«C +y= n- 5 A={ 3 y 8} n n- = n C + n C +3 n C 3 +y+n n C n (+x) n+ nc nc nc n = n C 0 x+ x + x +y+ x n+ n+ 3 n+ yy x= n+ nc nc = n C 0 + + +y+ n C n n+ 3 n+ (x-y)fi (a+b) {x+;[@;} fl x 3 ºCº+ ºC + ºC +y+ ºC º (+x) n n Cº+ n C + n C +y= n C + n C + n C +y= n- C + C + C + C = =8 3 (+x) n n Cº+ n C + n C +y+ n C n = n 0Cº+ 0 C + 0 C +y+ 0 C 0 = 0 =04 (x-y)fi =xfi -5x y+0x y -0x y +5xy -yfi (a+b) =a +8a b+4a b +3ab +6b C r x 6-r { } r = C r r x x 6-r x r= x C =60 (x+y) 0 x y r n C r =n n- C r- nc + n C +3 n C 3 +y+n n C n =n n- n C + n C +3 n C 3 +y+n n C n 0 =n n- C 0 +n n- C +n n- C +y+n n- C n- =n( n- C 0 + n- C + n- C +y+ n- C n- ) =n n- 45

C (3a) 5-r (-b) r = C 3 5-r (-) r a 5-r b r a b r= a b C 3 (-) =70 a b r= a b C 3 (-)=-405 - (3a-b)fi a b a b {3x +;[!;} fl x C (3x ) 6-r {;[!;} r = C 3 6-r x -3r -3r=3r=3 x C 3 =540-3r=0r=4 C 3 =35 3 (+x) (+x) x 4 n 500< «C + «C + «C +y+ «C «<000 5 Cº+ C + C + C +y+ ºC 3 (+x) C 3-r x r (+x) Cß 4-s (x) s = Cß s x s (+x) (+x) ( C x r )( Cß s x s )= C Cß s x r+s x r+s= r=0s= Cº C =8 r=s=0 C Cº =3 x 8+3= 5 Cº= Cº Cº+ C + C + C +y+ ºC = Cº+ C + C + C +y+ ºC = C + C + C +y+ ºC = C + C +y+ ºC = = ºC + ºC = C 4 n «C + «C + «C +y+ «C «= n- 500< n- <000 =5 =04 n-=9 n=5 46

6 5 x+y+z=8 5 03456 5 6 {ax + }fi x 70 a x 3 A B B 7 P 0 P A A(8) 4 X={3}Y={345} f:x3 Y i<j f(i) f(j) (i<xj<x) f:x 3 Y 8 (+x)«3 300 _(6-)!=40 4_ P _=4_5_=40 3 A P B P B A Q B4 A R B36 A S B7 +4+36+7=86 4 P =5 =5 H = C =35 A Q R S 5 H = ºC =45 H = C = 6 C a 5-r x 0-4r x r= C a =70 a=3 7 x 0-x x-(0-x)=8 x=6 0! =0 6!4! 8 3 0 300 (+x) = ºCº+ ºC x+ ºC x +y+ ºC ºx 3 = ºCº+ ºC 30+ ºC 30 +y+ ºC º 30 3 300 47

.. 5 4 4. (4-)!=6 _5_6=30.. ( ) ( ) ( )4 ( ). C = 6_5 =5. ( ) 4 (4-)!=6 5_6=90.. ( ) ( ) ( 3)( 3) (). C = 6_5 =5 4 (4-)!_= 5_=80 3. 6. 5.. 5_80_ =450!.. 5 A A 6 30 A A 5_30_ =75!.. 5_90_ =5! 48

ABCD 7 3 36 64 8 4 H + 3 H 9 + 5 H 7 + 7 H 5 + 9 H 3 + H 5 3 00 80 440 650 960 5 A B B Q P Q P B A 3 3335 6 (x+) 49

3 4 P =3 =8 5 3 5 (5-)!=4! 3 P =60 4!_ P =440 3 ` 0 4! `3335 =4 3! ` 4! `333 =4 3! 4! `335 =! ` 4! `33 =6!! 4! `35 =! 4+(4+)+(6+)=38 4 H + 3 H 9 + 5 H 7 + 7 H 5 + 9 H 3 + H = C + C 9 + C 7 + C 5 + C 3 + C = 0 =04 5 A P 4! =6 yy!! P B 5! =0!3! P Q 3! _=6!! P Q 0-6=4 yy 6_4=4 yy 6 8 C r x 8-r r 8 C r r yy 8C r r (9-r) æ æ r 6 8C r- r- r 8C r r r+ æ æ ræ5 8C r+ r+ (8-r) 5 r 6 r=5 r=6 yy 8C 5 fi = 8 C 6 6 =79 79x 79x A P P Q B 8C r x 8-r r 8 C r r r 30 % 50 % 0 % yy 0 % 60 % 0 % 50

6 0 40 360 480 600 4 A={a, b, c, d}, B={,, 3, 4} f A B f(a)+f(b)+f(c)+f(d)=7 f 6 0 4 8 3 A a b c d f B 3 4 P =«P n 5 a +b+c+d= abcd 3 ABBccd 6 {x+ } «70 n x 5

6 (6-)!=5!=0 3 6 5 4 3 5 0_3=360 6 4 3 4 5 3 6 4 f(a)+f(b)+f(c)+f(d)=7 (f(a) f(b) f(c) f(d)) 3 (4)(3)() 4! (4) =4 3! 4! (3) =! 4! () =4 3! f 4++4=0 P =3fl «P =n 3fl =(3 ) =n n=3 =9 n>0 3 A BBccd 5! =30!! B ABccd 5! =60! 30+60=90 yy yy yy 5 a=0 b+c+d= H = C =9 a= b+c+d= H = C =78 a= b+c+d=8 H = ºC =45 a=3 b+c+d=3 H = C =0 9+78+45+0=4 6 «C x n-r { } =«C x n-r yy x n-r=0 n=r yy n 4 6 8 y r 3 4 y C =70 n=8 yy A 40 % 30 % B 40 % n r 30 % 0 % 70 n 40 % 5

5 (5 0 0) (4 0) (3 0) (3 ) ( ) 5 5 P =3fi =43 5 H = C = C = 5 (500) yy C = (40) yy C _ C =5 (30) yy C _ C =0 (3) yy C _ C _ =0! () yy C _ C _ =5! +5+0+0+5=4 4 n r «P =n «H =«C 53