a a apier sin 0; 000; 000 = 0 7 sin 0 0; 000; 000 a = 0 7 ;r = 0: = 0 7 a n =0 7 ( 0 7 ) n n =0; ; 2; 3; n =0; ; 2; 3; ; 00 a n+ =0 7 ( 0 7 ) n

Size: px
Start display at page:

Download "a a apier sin 0; 000; 000 = 0 7 sin 0 0; 000; 000 a = 0 7 ;r = 0: = 0 7 a n =0 7 ( 0 7 ) n n =0; ; 2; 3; n =0; ; 2; 3; ; 00 a n+ =0 7 ( 0 7 ) n"

Transcription

1 apier John apier(550-67) = =7 2 n n 2 n 2 m n + m a 0 ;a ;a 2 ;a 3 ; a = a 0 ; r = a =a 0 = a 2 =a = a 3 =a 2 = n a n a n = ar n a r 2 a m = ar m ;a n = ar n a m a n = a(ar n+m ) a 0 ;a ;a 2 ;a 3 ;, m; n m + n a a a 0 a =0 p a p a p p a p 2 a r a a r

2 a a apier sin 0; 000; 000 = 0 7 sin 0 0; 000; 000 a = 0 7 ;r = 0: = 0 7 a n =0 7 ( 0 7 ) n n =0; ; 2; 3; n =0; ; 2; 3; ; 00 a n+ =0 7 ( 0 7 ) n ( 0 7 )=a n 0 7 a n ; ; 2; 3; ; 00 logarithms(= ratio numbers) : logarithm 00 log : = 00 x =0 7 ( 0 7 ) y log x = y 0; 000; 000 5; 000; 000 6; 900; 000 logarithms x = ar m ;y = ar n y=x = r n m logarithms : ' = 0 5 a =0 7 ;r = ( 0 5 ) r ; r =0; ; 2; 3; ; 50 2

3 log : ' = 5; 000: : ' = = ( =2000) p ;p = 0; ; 2; 3; ; 20 p = :5780 p = :34 log :5780 ' = 00000: :5789 ' = , 0 7 ( =00) q ;q =0; ; 2; 3; 68; a pq =0 7 p q ; p =0; ; 2 20; q =0; ; p q a pq : : : : : : : : : : : :4034 5; 000; 000 a pq log a pq = plog qlog logarithms log ' 500:24506; log ' 05026:5: 3

4 apier log ' :2 log = :8: apier ar n n b ar n bn apier Jost Burgi b =0;a =0 8 ;r =+0 4 apier apier Burgi ( 0 7 ) ( 0 4 ) ( 0 7 ) ( 0 4 ) ( 0 7 ) ( 0 4 ) 3 n 0 7 ( 0 7 ) n 0 n 0 8 ( 0 4 ) n ( ) 23;027 ' 0 Burgi n =23; 027 a; b 0 Burgi a =;b=0 4 Bog x = n 0 4 () x =( 0 4 ) n Bog x x ( ) ( ) 2 n 0 4 ( ) n n 0 4 = m Bog x = m () x = h (+0 4 ) 04i m : 4

5 0 4 0 p Bog x = m () x = h ( + 0 p ) 0pi m : ( ) 04 =2:78: ( + =k) k =[(k +)=k] k ;k =; 2; 3; ; ; ; ; ; ; Euler e e = lim + k =2:782882::: k! k log x = y () x = e y log x x a log a x = y () x = a y log a x a x Leonhard Euler( ) Henry Briggs apier 0, 0 log ,000 90,000 00,000 (624) Adian Vlacq 00, apier P; L P 0 7 P 0 O P 0 O L L 0 L 0 P t OP x L 0 7 t L 0 L y x y og x = y x =0 7 ( 0 7 ) n og 0 7 ( 0 7 ) n ' n apier dx dt = x; y =07 t: 5

6 y = og x =0 7 log 07 x : apier og 0 7 ( 0 7 ) n = n ψ ! + ' : n: Gregory St Vincent 647 x [a; b] y = =x S(a; b) t>0 S(ta; tb) =S(a; b): [a; b] n a = x 0 <x <x 2 < <x n = b; x i x i = b a ; i =; 2; n n [x i ;x i ] =x i S n (a; b) ta = tx 0 <tx <tx 2 < <tx n = tb [ta; tb] n S n (ta; tb) nx tb ta nx b a S n (ta; tb) = = = S n (a; b): tx i n x i n i= n S n (a; b) S(a; b) S n (ta; tb) S(ta; tb) i= S(ta; tb) = lim n! S n(ta; tb) = lim n! S n(a; b) =S(a; b): Vincent AA de Sarasa ( S(;x) x L(x) = S(x; ) 0 <x< <x<y L(xy) =L(x)+L(y) L(xy) =S(; xy)=s(; x)+s(x; xy) =S(; x)+s(; y)=l(x)+l(y): 6

7 x 3 L(x) =logx Euler ewton, Mercator ewton ( ) 667 y = x + (x > ) [0;x] A( + x) S(; +x) x + x 2 +x ) +x x x x 2 x 2 x 2 + x 3 ewton A( + x) = = y = +x = x + x2 x 3 + : Z x 0 Z x 0 +x dx dx Z x 0 3 x dx + Z x = x x2 2 + x 3 x : 0 x 2 dx Z x A(( + x)( + y)) = A( + x)+a( + y) ψ! +x A = A( + x) A( + y): +y 0 x 3 dx + x = ±0:; ±0:2; A(0:8);A(0:9);A(:);A(:2) 57 2=(:2 :2)=(0:8 0:9) A(2) = 2A(:2) A(0:8) A(0:9) 7

8 A(2) A(3);A(5);A();A(0);A(00); ewton icolas Mercator ( ) 668 log( + x) =x x2 2 + x 3 3 x : Wallis(66-703) 668 Mercator, [0;x] n h = x=n y ==( + x) [0;x], =( + x) A( + x) ' h + h A( + x) n X j= h +jh +jh = jh +(jh)2 (jh) 3 + : ' nh h 2 [ (n )] + h 3 [ (n ) 2 ] = x x2 3 x [ (n )] + 2 n n 3 [ (n ) 2 ] : Wallis 656 n! lim k +2 k + + n k = n k+ k + n! Mercator Wallis k» 0 Wallis log( + x) x< y ==x [x ;x 2 ] log(x 2 =x ) Vincent, Sarasa 660 Mercator 668 Euler Leonhard Euler( ) John(=Jean=Johann) Bernoulli ( ) (727-74, ) (74-766) 8

9 Euler Euler, Euler a x log a x = y a y = x y Euler a 0 =, ffl a ffl =+kffl k ffl x = x=ffl a x = a ffl =(a ffl ) =(+kffl) = a x = lim n! ψ! + kx n n ψ! + kx : ψ! + kx ψ! ψ! 2 ψ! 3 kx ( ) kx ( )( 2) kx = ! 3! =+kx + 2! k 2 x 2 + 3! 2 k 3 x 3 + : 0= = 2 = ψ a x = n! lim + kx n! n =+ kx! + k 2! 2 x 2 + k3 x 3 3! + : a k x = ψ! a = n! lim + k n =+ k n! + k 2 2! + k 3 3! + : Euler e k = a e = n! lim + n =+ n! + 2! + 3! + 9

10 23 e ' 2: : e x = n! lim + x n x =+ n! + x 2 2! + x 3 3! + : Euler log e ( + x) =y, +x = e y : +x = e y = + y : ( + x) = =+(y=) y = ewton ( + x) = =+ x + h ( + x) = i : 2! x 2 + = =0 2 y = x + 2! x2 + ( )( 2) x 3 + : 3! y =log e ( + x) =x x2 2 + x 3 3 : 3! x 3 + e log e ( + x) y ==( + x) [0;x] A( + x) Euler R sin A 2A cos A Euler x sin x; cos x sin 2 x +cos 2 x = sin(x ± y) = sin x cos y ± cos x sin y; cos(x ± y) = cos x cos y sin x sin y De Moivre (cos z + i sin z) n = cos nz + i sin nz i = p 0

11 x x ffl = x= x = ffl: De Moivre z = ffl; n = cos x + i sin x =cosffl + i sin ffl = (cos ffl + i sin ffl) : ffl cos ffl =; sin ffl = ffl (cos ffl + i sin ffl) =(+iffl) = + ix = e ix : Euler cos x + i sin x = e ix e ix Euler x x z = x + iy e z = e x+iy = e x e iy = e x (cos y + i sin y): e z e x y w = e z log w = z w C H Edwards, Jr, "The Historical Development of the Calculus", Springer- Verlag, 979

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ

More information

( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED)

( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED) rational number p, p, (q ) q ratio 3.14 = 3 + 1 10 + 4 100 ( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED) ( a) ( b) a > b > 0 a < nb n A A B B A A, B B A =

More information

Microsoft Word - 触ってみよう、Maximaに2.doc

Microsoft Word - 触ってみよう、Maximaに2.doc i i e! ( x +1) 2 3 ( 2x + 3)! ( x + 1) 3 ( a + b) 5 2 2 2 2! 3! 5! 7 2 x! 3x! 1 = 0 ",! " >!!! # 2x + 4y = 30 "! x + y = 12 sin x lim x!0 x x n! # $ & 1 lim 1 + ('% " n 1 1 lim lim x!+0 x x"!0 x log x

More information

n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1)

n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1) D d dx 1 1.1 n d n y a 0 dx n + a d n 1 y 1 dx n 1 +... + a dy n 1 dx + a ny = f(x)...(1) dk y dx k = y (k) a 0 y (n) + a 1 y (n 1) +... + a n 1 y + a n y = f(x)...(2) (2) (2) f(x) 0 a 0 y (n) + a 1 y

More information

xyr x y r x y r u u

xyr x y r x y r u u xyr x y r x y r u u y a b u a b a b c d e f g u a b c d e g u u e e f yx a b a b a b c a b c a b a b c a b a b c a b c a b c a u xy a b u a b c d a b c d u ar ar a xy u a b c a b c a b p a b a b c a

More information

untitled

untitled 1 1 1. 2. 3. 2 2 1 (5/6) 4 =0.517... 5/6 (5/6) 4 1 (5/6) 4 1 (35/36) 24 =0.491... 0.5 2.7 3 1 n =rand() 0 1 = rand() () rand 6 0,1,2,3,4,5 1 1 6 6 *6 int() integer 1 6 = int(rand()*6)+1 1 4 3 500 260 52%

More information

i) M C F Richter : ) km 2800) A µm) M L = log 0 A ) ii) ph ph mol/l [H + ] ph = log 0 [H + ] = log 0 [H + ] 909 Søren Pete

i) M C F Richter : ) km 2800) A µm) M L = log 0 A ) ii) ph ph mol/l [H + ] ph = log 0 [H + ] = log 0 [H + ] 909 Søren Pete Pierre Simon Laplace : 749827) 2 Christopher Columbus : 45506) Vasco da Gama : 469524) 400 650 024 2 0 ) 048576 2 20 ) 024 048576 07374824 log 2 024 048576) 2 30 0 + 20 30 2 n!! log a MN = log a M + log

More information

2 log 3 9 log 0 0 a log 9 3 2 log 3 9 y 3 y = 9 3 2 = 9 y = 2 0 y = 0 a log 0 0 a = a 9 2 = 3 log 9 3 = 2 a 0 = a = a log a a = log a = 0 log a a =. l

2 log 3 9 log 0 0 a log 9 3 2 log 3 9 y 3 y = 9 3 2 = 9 y = 2 0 y = 0 a log 0 0 a = a 9 2 = 3 log 9 3 = 2 a 0 = a = a log a a = log a = 0 log a a =. l 202 7 8 logarithm a y = y a y log a a log a y = log a = ep a y a > 0, a > 0 log 5 25 log 5 25 y y = log 5 25 25 = 5 y 25 25 = 5 3 y = 3 log 5 25 = 3 2 log 3 9 log 0 0 a log 9 3 2 log 3 9 y 3 y = 9 3 2

More information

1 1 1 1 1 1 2 f z 2 C 1, C 2 f 2 C 1, C 2 f(c 2 ) C 2 f(c 1 ) z C 1 f f(z) xy uv ( u v ) = ( a b c d ) ( x y ) + ( p q ) (p + b, q + d) 1 (p + a, q + c) 1 (p, q) 1 1 (b, d) (a, c) 2 3 2 3 a = d, c = b

More information

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0 A c 2008 by Kuniaki Nakamitsu 1 1.1 t 2 sin t, cos t t ft t t vt t xt t + t xt + t xt + t xt t vt = xt + t xt t t t vt xt + t xt vt = lim t 0 t lim t 0 t 0 vt = dxt ft dft dft ft + t ft = lim t 0 t 1.1

More information

., a = < < < n < n = b, j = f j j =,,, n, C P,, P,,, P n n, n., P P P n = = n j= n j= j j + j j + { j j / j j } j j, j j / j j f j 3., n., Oa, b r > P

., a = < < < n < n = b, j = f j j =,,, n, C P,, P,,, P n n, n., P P P n = = n j= n j= j j + j j + { j j / j j } j j, j j / j j f j 3., n., Oa, b r > P . ϵριµϵτρoζ perimetros 76 Jones, Euler. =.,.,,,, C, C n+ P, P,, P n P, P n P n, P P P P n P n n P n,, C P, P j P j j =,,, n P n P., C.,, C. f [a, b], f. C = f a b, C l l = b a + f d P j P j a b j j j j

More information

http://know-star.com/ 3 1 7 1.1................................. 7 1.2................................ 8 1.3 x n.................................. 8 1.4 e x.................................. 10 1.5 sin

More information

untitled

untitled 47 48 10 49 2005.6.1 17 500 50 1988 1994.1.1 16 22 51 18 1989 2005 17 2006 18 4 12 18 2007 19 1 12 2007 19 H18.8. J.H. 20 19 52 53 42.9 54 50 50 3080 55 30 100 56 57 22 96 6.8 9.4 31.44 58 10 780 250 59

More information

A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B

A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B 9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A

More information

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t 6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]

More information

&

& & & & c & B & c & & aa c c a a nk a a & a aa aaa & A A A A & 7 & & c og og & c c k g a og ok c c & 7 7 n & aa & & & & & & & & & & & g & & a a & & 171 1 & 1 7 1 1 6 de 666 66 6 7 6 f 6 & & c & 1 1 1 1 1

More information

Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3. 39. 4.. 4.. 43. 46.. 46..

Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3. 39. 4.. 4.. 43. 46.. 46.. Cotets 6 6 : 6 6 6 6 6 6 7 7 7 Part. 8. 8.. 8.. 9..... 3. 3 3.. 3 3.. 7 3.3. 8 Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3.

More information

F8302D_1目次_160527.doc

F8302D_1目次_160527.doc N D F 830D.. 3. 4. 4. 4.. 4.. 4..3 4..4 4..5 4..6 3 4..7 3 4..8 3 4..9 3 4..0 3 4. 3 4.. 3 4.. 3 4.3 3 4.4 3 5. 3 5. 3 5. 3 5.3 3 5.4 3 5.5 4 6. 4 7. 4 7. 4 7. 4 8. 4 3. 3. 3. 3. 4.3 7.4 0 3. 3 3. 3 3.

More information

1 6 2011 3 2011 3 7 1 2 1.1....................................... 2 1.2................................. 3 1.3............................................. 4 6 2.1................................................

More information

II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1

II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1 II 2014 1 1 I 1.1 72 r 2 72 8 72/8 = 9 9 2 a 0 1 a 1 a 1 = a 0 (1+r/100) 2 a 2 a 2 = a 1 (1 + r/100) = a 0 (1 + r/100) 2 n a n = a 0 (1 + r/100) n a n a 0 2 n a 0 (1 + r/100) n = 2a 0 (1 + r/100) n = 2

More information

...3 1-1...3 1-1...6 1-3...16 2....17...21 3-1...21 3-2...21 3-2...22 3-3...23 3-4...24...25 4-1....25 4-2...27 4-3...28 4-4...33 4-5...36...37 5-1...

...3 1-1...3 1-1...6 1-3...16 2....17...21 3-1...21 3-2...21 3-2...22 3-3...23 3-4...24...25 4-1....25 4-2...27 4-3...28 4-4...33 4-5...36...37 5-1... DT-870/5100 &DT-5042RFB ...3 1-1...3 1-1...6 1-3...16 2....17...21 3-1...21 3-2...21 3-2...22 3-3...23 3-4...24...25 4-1....25 4-2...27 4-3...28 4-4...33 4-5...36...37 5-1....39 5-2...40 5-3...43...49

More information

z z x = y = /x lim y = + x + lim y = x (x a ) a (x a+) lim z z f(z) = A, lim z z g(z) = B () lim z z {f(z) ± g(z)} = A ± B (2) lim {f(z) g(z)} = AB z

z z x = y = /x lim y = + x + lim y = x (x a ) a (x a+) lim z z f(z) = A, lim z z g(z) = B () lim z z {f(z) ± g(z)} = A ± B (2) lim {f(z) g(z)} = AB z Tips KENZOU 28 6 29 sin 2 x + cos 2 x = cos 2 z + sin 2 z = OK... z < z z < R w = f(z) z z w w f(z) w lim z z f(z) = w x x 2 2 f(x) x = a lim f(x) = lim f(x) x a+ x a z z x = y = /x lim y = + x + lim y

More information

13 0 1 1 4 11 4 12 5 13 6 2 10 21 10 22 14 3 20 31 20 32 25 33 28 4 31 41 32 42 34 43 38 5 41 51 41 52 43 53 54 6 57 61 57 62 60 70 0 Gauss a, b, c x, y f(x, y) = ax 2 + bxy + cy 2 = x y a b/2 b/2 c x

More information

2 1 17 1.1 1.1.1 1650

2 1 17 1.1 1.1.1 1650 1 3 5 1 1 2 0 0 1 2 I II III J. 2 1 17 1.1 1.1.1 1650 1.1 3 3 6 10 3 5 1 3/5 1 2 + 1 10 ( = 6 ) 10 1/10 2000 19 17 60 2 1 1 3 10 25 33221 73 13111 0. 31 11 11 60 11/60 2 111111 3 60 + 3 332221 27 x y xy

More information

untitled

untitled 20 7 1 22 7 1 1 2 3 7 8 9 10 11 13 14 15 17 18 19 21 22 - 1 - - 2 - - 3 - - 4 - 50 200 50 200-5 - 50 200 50 200 50 200 - 6 - - 7 - () - 8 - (XY) - 9 - 112-10 - - 11 - - 12 - - 13 - - 14 - - 15 - - 16 -

More information

untitled

untitled 19 1 19 19 3 8 1 19 1 61 2 479 1965 64 1237 148 1272 58 183 X 1 X 2 12 2 15 A B 5 18 B 29 X 1 12 10 31 A 1 58 Y B 14 1 25 3 31 1 5 5 15 Y B 1 232 Y B 1 4235 14 11 8 5350 2409 X 1 15 10 10 B Y Y 2 X 1 X

More information

lim lim lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d

lim lim lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d lim 5. 0 A B 5-5- A B lim 0 A B A 5. 5- 0 5-5- 0 0 lim lim 0 0 0 lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d 0 0 5- 5-3 0 5-3 5-3b 5-3c lim lim d 0 0 5-3b 5-3c lim lim lim d 0 0 0 3 3 3 3 3 3

More information

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C 0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,

More information

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9 1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),

More information

2 1 2 3 27 2 6 2 5 19 50 1 2

2 1 2 3 27 2 6 2 5 19 50 1 2 1 2 1 2 3 27 2 6 2 5 19 50 1 2 2 17 1 5 6 5 6 3 5 5 20 5 5 5 4 1 5 18 18 6 6 7 8 TA 1 2 9 36 36 19 36 1 2 3 4 9 5 10 10 11 2 27 12 17 13 6 30 16 15 14 15 16 17 18 19 28 34 20 50 50 5 6 3 21 40 1 22 23

More information

f (x) f (x) f (x) f (x) f (x) 2 f (x) f (x) f (x) f (x) 2 n f (x) n f (n) (x) dn f f (x) dx n dn dx n D n f (x) n C n C f (x) x = a 1 f (x) x = a x >

f (x) f (x) f (x) f (x) f (x) 2 f (x) f (x) f (x) f (x) 2 n f (x) n f (n) (x) dn f f (x) dx n dn dx n D n f (x) n C n C f (x) x = a 1 f (x) x = a x > 5.1 1. x = a f (x) a x h f (a + h) f (a) h (5.1) h 0 f (x) x = a f +(a) f (a + h) f (a) = lim h +0 h (5.2) x h h 0 f (a) f (a + h) f (a) f (a h) f (a) = lim = lim h 0 h h 0 h (5.3) f (x) x = a f (a) =

More information

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................ 5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)

More information

Z: Q: R: C: 3. Green Cauchy

Z: Q: R: C: 3. Green Cauchy 7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................

More information

1: *2 W, L 2 1 (WWL) 4 5 (WWL) W (WWL) L W (WWL) L L 1 2, 1 4, , 1 4 (cf. [4]) 2: 2 3 * , , = , 1

1: *2 W, L 2 1 (WWL) 4 5 (WWL) W (WWL) L W (WWL) L L 1 2, 1 4, , 1 4 (cf. [4]) 2: 2 3 * , , = , 1 I, A 25 8 24 1 1.1 ( 3 ) 3 9 10 3 9 : (1,2,6), (1,3,5), (1,4,4), (2,2,5), (2,3,4), (3,3,3) 10 : (1,3,6), (1,4,5), (2,2,6), (2,3,5), (2,4,4), (3,3,4) 6 3 9 10 3 9 : 6 3 + 3 2 + 1 = 25 25 10 : 6 3 + 3 3

More information

平成27年度三菱重工グループ保険 フルガードくん(シニア)

平成27年度三菱重工グループ保険 フルガードくん(シニア) TEL 0120-004-443 TEL 045-200-6560 TEL 042-761-2328 TEL 0120-539-022 TEL 042-762-0535 TEL 052-565-5211 TEL 077-552-9161 TEL 0120-430-372 TEL 0120-45-9898 TEL 0120-63-0051 TEL 0120-252-892 TEL 083-266-8041

More information

a q q y y a xp p q y a xp y a xp y a x p p y a xp q y x yaxp x y a xp q x p y q p x y a x p p p p x p

a q q y y a xp p q y a xp y a xp y a x p p y a xp q y x yaxp x y a xp q x p y q p x y a x p p p p x p a a a a y y ax q y ax q q y ax y ax a a a q q y y a xp p q y a xp y a xp y a x p p y a xp q y x yaxp x y a xp q x p y q p x y a x p p p p x p y a xp q y a x p q p p x p p q p q y a x xy xy a a a y a x

More information

IV.dvi

IV.dvi IV 1 IV ] shib@mth.hiroshim-u.c.jp [] 1. z 0 ε δ := ε z 0 z

More information

A B 5 C 9 3.4 7 mm, 89 mm 7/89 = 3.4. π 3 6 π 6 6 = 6 π > 6, π > 3 : π > 3

A B 5 C 9 3.4 7 mm, 89 mm 7/89 = 3.4. π 3 6 π 6 6 = 6 π > 6, π > 3 : π > 3 π 9 3 7 4. π 3................................................. 3.3........................ 3.4 π.................... 4.5..................... 4 7...................... 7..................... 9 3 3. p

More information

0 (1 ) 0 (1 ) 01 Excel Excel ( ) = Excel Excel =5+ 5 + 7 =5-5 3 =5* 5 10 =5/ 5 5 =5^ 5 5 ( ), 0, Excel, Excel 13E+05 13 10 5 13000 13E-05 13 10 5 0000

0 (1 ) 0 (1 ) 01 Excel Excel ( ) = Excel Excel =5+ 5 + 7 =5-5 3 =5* 5 10 =5/ 5 5 =5^ 5 5 ( ), 0, Excel, Excel 13E+05 13 10 5 13000 13E-05 13 10 5 0000 1 ( S/E) 006 7 30 0 (1 ) 01 Excel 0 7 3 1 (-4 ) 5 11 5 1 6 13 7 (5-7 ) 9 1 1 9 11 3 Simplex 1 4 (shadow price) 14 5 (reduced cost) 14 3 (8-10 ) 17 31 17 3 18 33 19 34 35 36 Excel 3 4 (11-13 ) 5 41 5 4

More information

CSE2LEC2

CSE2LEC2 " dt = "r(t "T s dt = "r(t "T s T T s dt T "T s = "r ln(t "T s = "rt + rt 0 T = T s + Ae "rt T(0 = T 0 T(0 = T s + A A = T 0 "T s T(t = T s + (T 0 "T s e "rt dy dx = f (x, y (Euler dy dx = f (x, y y y(x

More information

1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.

1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1. 1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N

More information

PowerPoint プレゼンテーション

PowerPoint プレゼンテーション 0 1 2 3 4 5 6 1964 1978 7 0.0015+0.013 8 1 π 2 2 2 1 2 2 ( r 1 + r3 ) + π ( r2 + r3 ) 2 = +1,2100 9 10 11 1.9m 3 0.64m 3 12 13 14 15 16 17 () 0.095% 0.019% 1.29% (0.348%) 0.024% 0.0048% 0.32% (0.0864%)

More information

http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n

http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 1 1 1.1 ɛ-n 1 ɛ-n lim n a n = α n a n α 2 lim a n = 1 n a k n n k=1 1.1.7 ɛ-n 1.1.1 a n α a n n α lim n a n = α ɛ N(ɛ) n > N(ɛ) a n α < ɛ

More information

34 2 2 h = h/2π 3 V (x) E 4 2 1 ψ = sin kxk = 2π/λ λ = h/p p = h/λ = kh/2π = k h 5 2 ψ = e ax2 ガウス 型 関 数 1.2 1 関 数 値 0.8 0.6 0.4 0.2 0 15 10 5 0 5 10

34 2 2 h = h/2π 3 V (x) E 4 2 1 ψ = sin kxk = 2π/λ λ = h/p p = h/λ = kh/2π = k h 5 2 ψ = e ax2 ガウス 型 関 数 1.2 1 関 数 値 0.8 0.6 0.4 0.2 0 15 10 5 0 5 10 33 2 2.1 2.1.1 x 1 T x T 0 F = ma T ψ) 1 x ψ(x) 2.1.2 1 1 h2 d 2 ψ(x) + V (x)ψ(x) = Eψ(x) (2.1) 2m dx 2 1 34 2 2 h = h/2π 3 V (x) E 4 2 1 ψ = sin kxk = 2π/λ λ = h/p p = h/λ = kh/2π = k h 5 2 ψ = e ax2

More information

タ 縺29135 タ 縺5 [ y 1 x i R 8 x j 1 7,5 2 x , チ7192, (2) チ41299 f 675

タ 縺29135 タ 縺5 [ y 1 x i R 8 x j 1 7,5 2 x , チ7192, (2) チ41299 f 675 139ィ 48 1995 3. 753 165, 2 6 86 タ7 9 998917619 4381 縺48 縺55 317832645 タ5 縺4273 971927, 95652539358195 45 チ5197 9 4527259495 2 7545953471 129175253471 9557991 3.9. タ52917652 縺1874ィ 989 95652539358195 45

More information

example2_time.eps

example2_time.eps Google (20/08/2 ) ( ) Random Walk & Google Page Rank Agora on Aug. 20 / 67 Introduction ( ) Random Walk & Google Page Rank Agora on Aug. 20 2 / 67 Introduction Google ( ) Random Walk & Google Page Rank

More information

7) ẋt) =iaω expiωt) ibω exp iωt) 9) ẋ0) = iωa b) = 0 0) a = b a = b = A/ xt) = A expiωt) + exp iωt)) = A cosωt) ) ) vt) = Aω sinωt) ) ) 9) ) 9) E = mv

7) ẋt) =iaω expiωt) ibω exp iωt) 9) ẋ0) = iωa b) = 0 0) a = b a = b = A/ xt) = A expiωt) + exp iωt)) = A cosωt) ) ) vt) = Aω sinωt) ) ) 9) ) 9) E = mv - - m k F = kx ) kxt) =m d xt) dt ) ω = k/m ) ) d dt + ω xt) = 0 3) ) ) d d dt iω dt + iω xt) = 0 4) ω d/dt iω) d/dt + iω) 4) ) d dt iω xt) = 0 5) ) d dt + iω xt) = 0 6) 5) 6) a expiωt) b exp iωt) ) )

More information

案内(最終2).indd

案内(最終2).indd 1 2 3 4 5 6 7 8 9 Y01a K01a Q01a T01a N01a S01a Y02b - Y04b K02a Q02a T02a N02a S02a Y05b - Y07b K03a Q03a T03a N03a S03a A01r Y10a Y11a K04a K05a Q04a Q05a T04b - T06b T08a N04a N05a S04a S05a Y12b -

More information

無印良品のスキンケア

無印良品のスキンケア 2 3 4 5 P.22 P.10 P.18 P.14 P.24 Na 6 7 P.10 P.22 P.14 P.18 P.24 8 9 1701172 1,400 1701189 1,000 1081267 1,600 1701257 2,600 1125923 450 1081250 1,800 1125916 650 1081144 1,800 1081229 1,500 Na 1701240

More information

O157 6/23 7/4 6 25 1000 117,050 6 14:00~15:30 1 2 22 22 14:30~15:30 8 12 1 5 20 6 20 10 11 30 9 10 6 1 30 6 6 0 30 6 19 0 3 27 6 20 0 50 1 2 6 4 61 1 6 5 1 2 1 2 6 19 6 4 15 6 1 6 30 6 24 30 59

More information

24.15章.微分方程式

24.15章.微分方程式 m d y dt = F m d y = mg dt V y = dy dt d y dt = d dy dt dt = dv y dt dv y dt = g dv y dt = g dt dt dv y = g dt V y ( t) = gt + C V y ( ) = V y ( ) = C = V y t ( ) = gt V y ( t) = dy dt = gt dy = g t dt

More information

7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6

7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6 26 11 5 1 ( 2 2 2 3 5 3.1...................................... 5 3.2....................................... 5 3.3....................................... 6 3.4....................................... 7

More information

17 3 31 1 1 3 2 5 3 9 4 10 5 15 6 21 7 29 8 31 9 35 10 38 11 41 12 43 13 46 14 48 2 15 Radon CT 49 16 50 17 53 A 55 1 (oscillation phenomena) e iθ = cos θ + i sin θ, cos θ = eiθ + e iθ 2, sin θ = eiθ e

More information

86 6 r (6) y y d y = y 3 (64) y r y r y r ϕ(x, y, y,, y r ) n dy = f(x, y) (6) 6 Lipschitz 6 dy = y x c R y(x) y(x) = c exp(x) x x = x y(x ) = y (init

86 6 r (6) y y d y = y 3 (64) y r y r y r ϕ(x, y, y,, y r ) n dy = f(x, y) (6) 6 Lipschitz 6 dy = y x c R y(x) y(x) = c exp(x) x x = x y(x ) = y (init 8 6 ( ) ( ) 6 ( ϕ x, y, dy ), d y,, dr y r = (x R, y R n ) (6) n r y(x) (explicit) d r ( y r = ϕ x, y, dy ), d y,, dr y r y y y r (6) dy = f (x, y) (63) = y dy/ d r y/ r 86 6 r (6) y y d y = y 3 (64) y

More information

閨75, 縺5 [ ィ チ573, 縺 ィ ィ

閨75, 縺5 [ ィ チ573, 縺 ィ ィ 39ィ 8 998 3. 753 68, 7 86 タ7 9 9989769 438 縺48 縺55 3783645 タ5 縺473 タ7996495 ィ 59754 8554473 9 8984473 3553 7. 95457357, 4.3. 639745 5883597547 6755887 67996499 ィ 597545 4953473 9 857473 3553, 536583, 89573,

More information

13 21 13 3 10 2010 5 6 20 32 10 10 3 JR 14 3 1 8 2 15 6 ( ) 135 1 8 2 15 135 5 135 1 8 2 15 5 JR 135 1 8 2 15 JR 1 135 1 8 2 15 JR 135 135 135 JR 135 1 8 2 15 135 1 8

More information

untitled

untitled 2015 2004 6 22 (1) 2 3 4 50 550 2 80 10 3 100 10 165 50 1,000 2223 2 6,800 8,400 5001,000 23 () 5 1 1 02 04 5,400 3 46 80 90 70 7090 610 02 18.2 05 22.6 1120 3,000 30 5 90 15 90 22.8 1015 80 90 1120

More information

2 3 4 5 6 7 8 ( ) 9 10 11 12 13 14 S JR 16 22 23 24 25 27 29 30 31 32 33 34 35 36 37 38 39 40 41 JR JR 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72

More information

report_c.ai

report_c.ai 39 182646163205 40 41 42 43 44 45 46 47 48 49 ( 50 1 2 1 P.4 2 P.5 51 50 20 52 3 4 3 P.4 4 P.5 53 JR 10 54 5 6 5 P.4 6 P.5 55 20 20 510 26 46 56 26 57 ( 510 58 26 59 ( 40 = 510 60 26 26 61 ( ( ( ) 10 62

More information

広報えちぜん12月号_4校.indd

広報えちぜん12月号_4校.indd 24 7,140,000 24 3,129,000 24 20,685,000 24 9,292,500 24 8,925,000 24 13,492,500 24 NO. 5,670,000 24 4,470,900 24 8,494,500 24 2,341,500 24 4,021,500 Jr. [ ] 128 1214 845 855 1345 1355 2015 2025 121 127

More information

Fourier (a) C, (b) C, (c) f 2 (a), (b) (c) (L 2 ) (a) C x : f(x) = a 0 2 + (a n cos nx + b n sin nx). ( N ) a 0 f(x) = lim N 2 + (a n cos nx + b n sin

Fourier (a) C, (b) C, (c) f 2 (a), (b) (c) (L 2 ) (a) C x : f(x) = a 0 2 + (a n cos nx + b n sin nx). ( N ) a 0 f(x) = lim N 2 + (a n cos nx + b n sin ( ) 205 6 Fourier f : R C () (2) f(x) = a 0 2 + (a n cos nx + b n sin nx), n= a n = f(x) cos nx dx, b n = π π f(x) sin nx dx a n, b n f Fourier, (3) f Fourier or No. ) 5, Fourier (3) (4) f(x) = c n = n=

More information

* 1 2014 7 8 *1 iii 1. Newton 1 1.1 Newton........................... 1 1.2............................. 4 1.3................................. 5 2. 9 2.1......................... 9 2.2........................

More information

: 1: 3:

: 1: 3: 1 2013 10 11 google maps engine : : : : : : : : : : 1 4.6.1 2: 1: 3: 4: 6: 5: 7: 2 2.1 2.1.1 2.1.2 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.3 2.3.1 18 2.3.2 2.3.3 2.3.4 2.3.5 2000 SA 2.3.6 2.4 2.4.1 2.4.2 2.4.3

More information

I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin. sin. sin + π si

I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin. sin. sin + π si I 8 No. : No. : No. : No.4 : No.5 : No.6 : No.7 : No.8 : No.9 : No. : I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin.

More information

1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v =

1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v = 1. 2. 3 3. 4. 5. 6. 7. 8. 9. I http://risu.lowtem.hokudai.ac.jp/ hidekazu/class.html 1 1.1 1 a = g, (1) v = g t + v 0, (2) z = 1 2 g t2 + v 0 t + z 0. (3) 1.2 v-t. z-t. z 1 z 0 = dz = v, t1 dv v(t), v

More information

36 16 16 26 1 8 9 30 70 15 12 20 6 21 7 23 12 25 6 23 7 23 9 24 8 24 10 25 3 9 26 11 JR 2 9 80 10 S.A 3600 PTA 4300 10 19 10 20 10 16 16 16 16 No.010101 No.011701 Ho To No.020101 No.033401

More information

96 7 1m =2 10 7 N 1A 7.1 7.2 a C (1) I (2) A C I A A a A a A A a C C C 7.2: C A C A = = µ 0 2π (1) A C 7.2 AC C A 3 3 µ0 I 2 = 2πa. (2) A C C 7.2 A A

96 7 1m =2 10 7 N 1A 7.1 7.2 a C (1) I (2) A C I A A a A a A A a C C C 7.2: C A C A = = µ 0 2π (1) A C 7.2 AC C A 3 3 µ0 I 2 = 2πa. (2) A C C 7.2 A A 7 Lorentz 7.1 Ampère I 1 I 2 I 2 I 1 L I 1 I 2 21 12 L r 21 = 12 = µ 0 2π I 1 I 2 r L. (7.1) 7.1 µ 0 =4π 10 7 N A 2 (7.2) magnetic permiability I 1 I 2 I 1 I 2 12 21 12 21 7.1: 1m 95 96 7 1m =2 10 7 N

More information

-

- - - v vt t y r y W0W9WwWq c zx t - -4 ud d dr y r y x dx id d d d d x d d r Wq Wq d Uu Xd Xd -5 x dt r o Tx Ii Xd XdXd v c z x d t r o Ii Xd XdXd -6 -7 o y v vt t y W0W9WwWq -8 cc zx t d d v z r d y -9

More information

d dt A B C = A B C d dt x = Ax, A 0 B 0 C 0 = mm 0 mm 0 mm AP = PΛ P AP = Λ P A = ΛP P d dt x = P Ax d dt (P x) = Λ(P x) d dt P x =

d dt A B C = A B C d dt x = Ax, A 0 B 0 C 0 = mm 0 mm 0 mm AP = PΛ P AP = Λ P A = ΛP P d dt x = P Ax d dt (P x) = Λ(P x) d dt P x = 3 MATLAB Runge-Kutta Butcher 3. Taylor Taylor y(x 0 + h) = y(x 0 ) + h y (x 0 ) + h! y (x 0 ) + Taylor 3. Euler, Runge-Kutta Adams Implicit Euler, Implicit Runge-Kutta Gear y n+ y n (n+ ) y n+ y n+ y n+

More information

36.fx82MS_Dtype_J-c_SA0311C.p65

36.fx82MS_Dtype_J-c_SA0311C.p65 P fx-82ms fx-83ms fx-85ms fx-270ms fx-300ms fx-350ms J http://www.casio.co.jp/edu/ AB2Mode =... COMP... Deg... Norm 1... a b /c... Dot 1 2...1...2 1 2 u u u 3 5 fx-82ms... 23 fx-83ms85ms270ms300ms 350MS...

More information

閨 [

閨 [ 1303000709 000 03. 070503 170, 0 3 0806 タ07 09 090908090107060109 04030801 縺0408 縺0505 03010708030060405 タ05 縺0400703 060504050ィ 03090405080050400909 03.03. 030007030000908 060005090809 0501080507 080500705030504040701

More information