II
|
|
- えりか こやぎ
- 4 years ago
- Views:
Transcription
1 II
2 x α 16 x n 1 17 (x α) x P (x) P (x) = 3x 3 4x
3 Q(x) = x 4 ax + b ( ) 1 P (x) x Q(x) x P (x) x P (x) x = a P (a) P (x) = x 3 7x + 4 P (2) = = = 2 P ( 1) = ( 1) 3 7 ( 1) + 4 = = 10 P (a) = a 3 7a + 4 P (b + 1) = (b + 1) 3 7(b + 1) + 4 = b 3 + 3b 2 4b 2 3 ( ) a b + 1 a x a b P (x) = x 3 + 3x 2 x + 4 (1) P (1) (2) P ( 2) (3) P ( b) (4) P (a + 1) 1 P f polynomial function P, Q f, g, h 370
4 x α ( ) P (x) x α P (α) P (x) x α Q(x) R 4 P (x) = (x α)q(x) + R P (α) = (α α)q(α) + R = 0 Q(α) + R = R P (x) x α P (α) α x P α x + α x + α = x ( α) α P (x) = x 3 4x + 1 x 2 P (2) = = 1 1 P (x) x + 1 P ( 1) = ( 1) 3 4 ( 1) + 1 = 4 4 ( ) 2 α a a x α 0 0 R 371
5 96 P (x) x 2 x P (x) = x 3 3x 2 + 4x 2 x + 1, x 2, x Q(x) = x 3 + 2x 2 + ax + 4 x a Q(x) x + 1 Q( 1) a 12 a Q(x) x + 1 Q( 1) = a + 4 = a a + 5 = 12 a = 7 ( ) ( ) 173 P (x) = x 3 + x 2 + ax 2 x 2 8 a 56 P (x) x 2 4 x 3 6 P (x) (x 2)(x 3) 372
6 1 2 P (x) x 2 4 x 3 6 P (2) = 4, P (3) = 6 P (x) (x 2)(x 3) (x 2)(x 3) ax + b ( a, b ) a b 5 P (x) (x 2)(x 3) ( ) Q(x) P (x) = (x 2)(x 3)Q(x) + ax + b x 2 ( 2 x 2 2 ) P (2) = (2 2)(2 3)Q(2) + 2a + b (2 2)(2 3)Q(2) 2 2 = 0 0 ( 1) Q(2) = 0 P (x) x 2 4 P (2) = 4 4 = 2a + b (1) 6 x = 3 6 = 3a + b (2) a, b (1) (2) a, b 5 ax + b 1 a = 0 a = 0, b = 0 6 Q(x) 373
7 P (x) (x 2)(x 3) Q(x) ax + b P (x) = (x 2)(x 3)Q(x) + ax + b P (x) x 2 4 x 3 6 P (2) = 4, P (3) = 6 { 2a + b = 4 3a + b = 6 a = 2, b = 0 2x ( ) ( ) 174 x P (x) x 2 1 x P (x) (x 2)(x + 3) x α 1 7 ( ) P (x) 1 ax + b ( P b ) a S(x) = x 3 + 2x 2 x 3 2x 1 7 a, b 1, α 374
8 P (x) x α 0 8 ( ) P (x) x α P (α) = 0 P (α) = 0 P (x) x α P (x) x α P (α) P (α) = 0 0 P (x) x α 98 P (x) x α P (α) = 0 x α x α P (x) = x 3 3x 2 P (2) = = 0 P (x) x 2 P (1) = = 4 P (x) x 1 ( ) 176 x 1, x + 1, x + 2, x 3 P (x) = x 4 + 2x 3 x 2 57 P (x) = x 3 5x 2 + 2x
9 1 0 P (x) x α P (x) = (x α)(a 0 x n + + a n ) P (x) = a 0 x n+1 + αa n αa n 8 αa n = 8 α 8 9 α ±1, ±2, ±4, ± P ( 1) = ( 1) 3 5 ( 1) ( 1) + 8 = = 0 P (x) x a 0 x n + + a n α 0 376
10 P (x) x P (x) = (x + 1)(x 2 6x + 8) x 2 6x + 8 (x 2)(x 4) x 3 5x 2 + 2x + 8 = (x + 1)(x 2)(x 4) x 3 5x 2 + 2x + 8 = (x + 1)(x 2)(x 4) ( ) 177 (1) x 3 7x + 6 (2) x 3 2x 2 5x + 6 (3) x 3 6x x 8 (4) x 4 + 3x 3 5x 2 3x + 4 (5) x 4 x 3 7x 2 + x a 3 b 3 = (a b)(a 2 + ab + b 2 )
11 a 3 b 3 a P (a) = a 3 b a b P (b) = b 3 b 3 = 0 P (a) a b a 3 b 3 a b a 3 b ( ) A B A = BQ B A A B x 3 5x 2 + 2x + 8 = (x + 1)(x 2 6x + 8) x + 1 x 3 5x 2 + 2x + 8 x 3 5x 2 + 2x + 8 x + 1 ( ) P (x) x P (a) x a, b 378
12 ( ) 2 ( ) 2 a 2 bx 2 y 3 a 2 b 3 xy a 2 bxy a 2 b 3 x 2 y 3 ( ) 58 x 3 3x 2 + 2x x 4 10x x 3 3x 2 + 2x = x(x 1)(x 2) x 4 10x = (x + 1)(x 1)(x + 3)(x 3) 379
13 x 1 x(x + 1)(x 1)(x 2)(x + 3)(x 3) ( ) ( ) 178 (1) b 2, b 3 c 2 (2) (x + 1)(x 2), (x + 2)(x 2) (3) (x + 1)(x 2) 2, (x 2) 3 (x 1) 2 (4) x 2 + 6x + 8, x 2 + 5x + 6 (5) x 2 + x 6, x 2 + 4x + 3, x 2 + 3x (6) x 3 + 4x 2 3x 18, x 3 + 4x 2 9x ( ) (0 ) 1 0 ( ) A, B A B Q R A B B R x 3 3x 2 + 2x x 4 10x
14 x 4 10x x 3 3x 2 + 2x 3x 2 6x + 9 x 3 3x 2 + 2x 3x 2 6x + 9 3x 2 x x 1 3 x x 15 3x 2 6x x 15 ) 15x 15 15x 15 = 15(x 1) 15 x 1 ( ) ( ) ( ) ( ) A, B D L AB = DL 381
15 ( ) A, B D A = A D, B = B D A B A B D x + 2 2x 3 + 3x 2 2x A, B A = A (x + 2), B = B (x + 2) A B A B (x + 2) x + 2 A, B A, B 2 A B 1 A, B A = A (x + 2), B = B (x + 2) A B 2x 3 + 3x 2 2x = x(x + 2)(2x 1) A, B 2 A, B 1 A = x, B = 2x 1 x(x + 2) (2x 1)(x + 2) ( ) ( ) A B A B A B A, B 179 P, Q P, Q x+1 x 4 +3x 3 +4x 2 +3x+1 P, Q 382
16 16.4 (1976) 1 (2000)
> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3
13 2 13.0 2 ( ) ( ) 2 13.1 ( ) ax 2 + bx + c > 0 ( a, b, c ) ( ) 275 > > 2 2 13.3 x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D >
More informationax 2 + bx + c = n 8 (n ) a n x n + a n 1 x n a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 4
20 20.0 ( ) 8 y = ax 2 + bx + c 443 ax 2 + bx + c = 0 20.1 20.1.1 n 8 (n ) a n x n + a n 1 x n 1 + + a 1 x + a 0 = 0 ( a n, a n 1,, a 1, a 0 a n 0) n n ( ) ( ) ax 3 + bx 2 + cx + d = 0 444 ( a, b, c, d
More information漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト
https://www.hmg-gen.com/tuusin.html https://www.hmg-gen.com/tuusin1.html 1 2 OK 3 4 {a n } (1) a 1 = 1, a n+1 a n = 2 (2) a 1 = 3, a n+1 a n = 2n a n a n+1 a n = ( ) a n+1 a n = ( ) a n+1 a n {a n } 1,
More information, 1. x 2 1 = (x 1)(x + 1) x 3 1 = (x 1)(x 2 + x + 1). a 2 b 2 = (a b)(a + b) a 3 b 3 = (a b)(a 2 + ab + b 2 ) 2 2, 2.. x a b b 2. b {( 2 a } b )2 1 =
x n 1 1.,,.,. 2..... 4 = 2 2 12 = 2 2 3 6 = 2 3 14 = 2 7 8 = 2 2 2 15 = 3 5 9 = 3 3 16 = 2 2 2 2 10 = 2 5 18 = 2 3 3 2, 3, 5, 7, 11, 13, 17, 19.,, 2,.,.,.,?.,,. 1 , 1. x 2 1 = (x 1)(x + 1) x 3 1 = (x 1)(x
More informationA A = a 41 a 42 a 43 a 44 A (7) 1 (3) A = M 12 = = a 41 (8) a 41 a 43 a 44 (3) n n A, B a i AB = A B ii aa
1 2 21 2 2 [ ] a 11 a 12 A = a 21 a 22 (1) A = a 11 a 22 a 12 a 21 (2) 3 3 n n A A = n ( 1) i+j a ij M ij i =1 n (3) j=1 M ij A i j (n 1) (n 1) 2-1 3 3 A A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33
More informationx = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b)
2011 I 2 II III 17, 18, 19 7 7 1 2 2 2 1 2 1 1 1.1.............................. 2 1.2 : 1.................... 4 1.2.1 2............................... 5 1.3 : 2.................... 5 1.3.1 2.....................................
More information18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C
8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,
More informationad bc A A A = ad bc ( d ) b c a n A n A n A A det A A ( ) a b A = c d det A = ad bc σ {,,,, n} {,,, } {,,, } {,,, } ( ) σ = σ() = σ() = n sign σ sign(
I n n A AX = I, YA = I () n XY A () X = IX = (YA)X = Y(AX) = YI = Y X Y () XY A A AB AB BA (AB)(B A ) = A(BB )A = AA = I (BA)(A B ) = B(AA )B = BB = I (AB) = B A (BA) = A B A B A = B = 5 5 A B AB BA A
More informationa 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 informationuntitled
yoshi@image.med.osaka-u.ac.jp http://www.image.med.osaka-u.ac.jp/member/yoshi/ II Excel, Mathematica Mathematica Osaka Electro-Communication University (2007 Apr) 09849-31503-64015-30704-18799-390 http://www.image.med.osaka-u.ac.jp/member/yoshi/
More information17 ( ) II III A B C(100 ) 1, 2, 6, 7 II A B (100 ) 2, 5, 6 II A B (80 ) 8 10 I II III A B C(80 ) 1 a 1 = 1 2 a n+1 = a n + 2n + 1 (n = 1,
17 ( ) 17 5 1 4 II III A B C(1 ) 1,, 6, 7 II A B (1 ), 5, 6 II A B (8 ) 8 1 I II III A B C(8 ) 1 a 1 1 a n+1 a n + n + 1 (n 1,,, ) {a n+1 n } (1) a 4 () a n OA OB AOB 6 OAB AB : 1 P OB Q OP AQ R (1) PQ
More information13,825,228 3,707,995 26.8 4.9 25 3 8 9 1 50,000 0.29 1.59 70,000 0.29 1.74 12,500 0.39 1.69 12,500 0.55 10,000 20,000 0.13 1.58 30,000 0.00 1.26 5,000 0.13 1.58 25,000 40,000 0.13 1.58 50,000 0.00 1.26
More information平成20年5月 協会創立50年の歩み 海の安全と環境保全を目指して 友國八郎 海上保安庁 長官 岩崎貞二 日本船主協会 会長 前川弘幸 JF全国漁業協同組合連合会 代表理事会長 服部郁弘 日本船長協会 会長 森本靖之 日本船舶機関士協会 会長 大内博文 航海訓練所 練習船船長 竹本孝弘 第二管区海上保安本部長 梅田宜弘
More information
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
More informationuntitled
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 informationuntitled
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 informationx () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
More informationuntitled
351 351 351 351 13.0 0.0 25.8 1.0 0.0 6.3 92.9 0.0 80.5 0.0 1.5 15.9 0.0 3.5 13.1 0.0 30.0 54.8 18.0 0.0 27.5 1.0 0.0 2.5 94.7 0.0 91.7 0.0 1.3 14.7 0.0 3.8 14.4 0.0 25.0 50.5 16.0 0.0 27.5 2.0 0.0 2.5
More informationi
i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,
More informationEPSON エプソンプリンタ共通 取扱説明書 ネットワーク編
K L N K N N N N N N N N N N N N L A B C N N N A AB B C L D N N N N N L N N N A L B N N A B C N L N N N N L N A B C D N N A L N A L B C D N L N A L N B C N N D E F N K G H N A B C A L N N N N D D
More informationありがとうございました
- 1 - - 2 - - 3 - - 4 - - 5 - 1 2 AB C A B C - 6 - - 7 - - 8 - 10 1 3 1 10 400 8 9-9 - 2600 1 119 26.44 63 50 15 325.37 131.99 457.36-10 - 5 977 1688 1805 200 7 80-11 - - 12 - - 13 - - 14 - 2-1 - 15 -
More informationEPSON エプソンプリンタ共通 取扱説明書 ネットワーク編
K L N K N N N N N N N N N N N N L A B C N N N A AB B C L D N N N N N L N N N A L B N N A B C N L N N N N L N A B C D N N A L N A L B C D N L N A L N B C N N D E F N K G H N A B C A L N N N N D D
More information公務員人件費のシミュレーション分析
47 50 (a) (b) (c) (7) 11 10 2018 20 2028 16 17 18 19 20 21 22 20 90.1 9.9 20 87.2 12.8 2018 10 17 6.916.0 7.87.4 40.511.6 23 0.0% 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2.0% 4.0% 6.0% 8.0%
More informationQ1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 A B (A/B) 1 1,185 17,801 6.66% 2 943 26,598 3.55% 3 3,779 112,231 3.37% 4 8,174 246,350 3.32% 5 671 22,775 2.95% 6 2,606 89,705 2.91% 7 738 25,700 2.87% 8 1,134
More information橡hashik-f.PDF
1 1 1 11 12 13 2 2 21 22 3 3 3 4 4 8 22 10 23 10 11 11 24 12 12 13 25 14 15 16 18 19 20 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 144 142 140 140 29.7 70.0 0.7 22.1 16.4 13.6 9.3 5.0 2.9 0.0
More information198
197 198 199 200 201 202 A B C D E F G H I J K L 203 204 205 A B 206 A B C D E F 207 208 209 210 211 212 213 214 215 A B 216 217 218 219 220 221 222 223 224 225 226 227 228 229 A B C D 230 231 232 233 A
More informationネットショップ・オーナー2 ユーザーマニュアル
1 1-1 1-2 1-3 1-4 1 1-5 2 2-1 A C 2-2 A 2 C D E F G H I 2-3 2-4 2 C D E E A 3 3-1 A 3 A A 3 3 3 3-2 3-3 3-4 3 C 4 4-1 A A 4 B B C D C D E F G 4 H I J K L 4-2 4 C D E B D C A C B D 4 E F B E C 4-3 4
More information1
1 2 3 4 5 (2,433 ) 4,026 2710 243.3 2728 402.6 6 402.6 402.6 243.3 7 8 20.5 11.5 1.51 0.50.5 1.5 9 10 11 12 13 100 99 4 97 14 A AB A 12 14.615/100 1.096/1000 B B 1.096/1000 300 A1.5 B1.25 24 4,182,500
More information05[ ]戸田(責)村.indd
147 2 62 4 3.2.1.16 3.2.1.17 148 63 1 3.2.1.F 3.2.1.H 3.1.1.77 1.5.13 1 3.1.1.05 2 3 4 3.2.1.20 3.2.1.22 3.2.1.24 3.2.1.D 3.2.1.E 3.2.1.18 3.2.1.19 2 149 3.2.1.23 3.2.1.G 3.1.1.77 3.2.1.16 570 565 1 2
More information/9/ ) 1) 1 2 2) 4) ) ) 2x + y 42x + y + 1) 4) : 6 = x 5) : x 2) x ) x 2 8x + 10 = 0
1. 2018/9/ ) 1) 8 9) 2) 6 14) + 14 ) 1 4 8a 8b) 2 a + b) 4) 2 : 7 = x 8) : x ) x ) + 1 2 ) + 2 6) x + 1)x + ) 15 2. 2018/9/ ) 1) 1 2 2) 4) 2 + 6 5) ) 2x + y 42x + y + 1) 4) : 6 = x 5) : x 2) x 2 15 12
More informationL P y P y + ɛ, ɛ y P y I P y,, y P y + I P y, 3 ŷ β 0 β y β 0 β y β β 0, β y x x, x,, x, y y, y,, y x x y y x x, y y, x x y y {}}{,,, / / L P / / y, P
005 5 6 y β + ɛ {x, x,, x p } y, {x, x,, x p }, β, ɛ E ɛ 0 V ɛ σ I 3 rak p 4 ɛ i N 0, σ ɛ ɛ y β y β y y β y + β β, ɛ β y + β 0, β y β y ɛ ɛ β ɛ y β mi L y y ŷ β y β y β β L P y P y + ɛ, ɛ y P y I P y,,
More information, = = 7 6 = 42, =
http://www.ss.u-tokai.ac.jp/~mahoro/2016autumn/alg_intro/ 1 1 2016.9.26, http://www.ss.u-tokai.ac.jp/~mahoro/2016autumn/alg_intro/ 1.1 1 214 132 = 28258 2 + 1 + 4 1 + 3 + 2 = 7 6 = 42, 4 + 2 = 6 2 + 8
More information14 (x a x x a f(x x 3 + 2x 2 + 3x + 4 (x 1 1 y x 1 x y + 1 x 3 + 2x 2 + 3x + 4 (y (y (y y 3 + 3y 2 + 3y y 2 + 4y + 2 +
III 2005 1 6 1 1 ( 11 0 0, 0 deg (f(xg(x deg f(x + deg g(x 12 f(x, g(x ( g(x 0 f(x q(xg(x + r(x, r(x 0 deg r(x < deg g(x q(x, r(x q(x, r(x f(x g(x r(x 0 f(x g(x g(x f(x g(x f(x g(x f(x 13 f(x x a q(x,
More informationkoji07-01.dvi
2007 I II III 1, 2, 3, 4, 5, 6, 7 5 10 19 (!) 1938 70 21? 1 1 2 1 2 2 1! 4, 5 1? 50 1 2 1 1 2 2 1?? 2 1 1, 2 1, 2 1, 2, 3,... 3 1, 2 1, 3? 2 1 3 1 2 1 1, 2 2, 3? 2 1 3 2 3 2 k,l m, n k,l m, n kn > ml...?
More informationa n a n ( ) (1) a m a n = a m+n (2) (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 552
3 3.0 a n a n ( ) () a m a n = a m+n () (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 55 3. (n ) a n n a n a n 3 4 = 8 8 3 ( 3) 4 = 8 3 8 ( ) ( ) 3 = 8 8 ( ) 3 n n 4 n n
More information7 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 informationR R 16 ( 3 )
(017 ) 9 4 7 ( ) ( 3 ) ( 010 ) 1 (P3) 1 11 (P4) 1 1 (P4) 1 (P15) 1 (P16) (P0) 3 (P18) 3 4 (P3) 4 3 4 31 1 5 3 5 4 6 5 9 51 9 5 9 6 9 61 9 6 α β 9 63 û 11 64 R 1 65 13 66 14 7 14 71 15 7 R R 16 http://wwwecoosaka-uacjp/~tazak/class/017
More informationuntitled
i ii (1) (1) (2) (1) (3) (1) (1) (2) (1) (3) (1) (1) (2) (1) (3) (2) (3) (1) (2) (3) (1) (1) (1) (1) (2) (1) (3) (1) (2) (1) (3) (1) (1) (1) (2) (1) (3) (1) (1) (2) (1) (3)
More information平成18年度「商品先物取引に関する実態調査」報告書
... 1.... 5-1.... 6-2.... 9-3.... 10-4.... 12-5.... 13-6.... 15-7.... 16-8.... 17-9.... 20-10.... 22-11.... 24-12.... 27-13... 29-14.... 32-15... 37-16.... 39-17.... 41-18... 43-19... 45.... 49-1... 50-2...
More information1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ( ) 24 25 26 27 28 29 30 ( ) ( ) ( ) 31 32 ( ) ( ) 33 34 35 36 37 38 39 40 41 42 43 44 ) i ii i ii 45 46 47 2 48 49 50 51 52 53 54 55 56 57 58
More information23 15961615 1659 1657 14 1701 1711 1715 11 15 22 15 35 18 22 35 23 17 17 106 1.25 21 27 12 17 420,845 23 32 58.7 32 17 11.4 71.3 17.3 32 13.3 66.4 20.3 17 10,657 k 23 20 12 17 23 17 490,708 420,845 23
More information1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA appointment Cafe D
1W II K200 : October 6, 2004 Version : 1.2, kawahira@math.nagoa-u.ac.jp, http://www.math.nagoa-u.ac.jp/~kawahira/courses.htm TA M1, m0418c@math.nagoa-u.ac.jp TA Talor Jacobian 4 45 25 30 20 K2-1W04-00
More information) 9 81
4 4.0 2000 ) 9 81 10 4.1 natural numbers 1, 2, 3, 4, 4.2, 3, 2, 1, 0, 1, 2, 3, integral numbers integers 1, 2, 3,, 3, 2, 1 1 4.3 4.3.1 ( ) m, n m 0 n m 82 rational numbers m 1 ( ) 3 = 3 1 4.3.2 3 5 = 2
More information2012 A, N, Z, Q, R, C
2012 A, N, Z, Q, R, C 1 2009 9 2 2011 2 3 2012 9 1 2 2 5 3 11 4 16 5 22 6 25 7 29 8 32 1 1 1.1 3 1 1 1 1 1 1? 3 3 3 3 3 3 3 1 1, 1 1 + 1 1 1+1 2 2 1 2+1 3 2 N 1.2 N (i) 2 a b a 1 b a < b a b b a a b (ii)
More informationMicrosoft 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 information1990 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 informationuntitled
1 ( 12 11 44 7 20 10 10 1 1 ( ( 2 10 46 11 10 10 5 8 3 2 6 9 47 2 3 48 4 2 2 ( 97 12 ) 97 12 -Spencer modulus moduli (modulus of elasticity) modulus (le) module modulus module 4 b θ a q φ p 1: 3 (le) module
More informationA_chapter3.dvi
: a b c d 2: x x y y 3: x y w 3.. 3.2 2. 3.3 3. 3.4 (x, y,, w) = (,,, )xy w (,,, )xȳ w (,,, ) xy w (,,, )xy w (,,, )xȳ w (,,, ) xy w (,,, )xy w (,,, ) xȳw (,,, )xȳw (,,, ) xyw, F F = xy w x w xy w xy w
More informationa x x x x 1 x 2 Ý; x. x = x 1 + x 2 + Ý + x = 10 1; 1; 3; 3; 4; 5; 8; 8; 8; 9 1 + 1 + 3 + 3 + 4 + 5 + 8 + 8 + 8 + 9 10 = 50 10 = 5 . 1 1 Ý Ý # 2 2 Ý Ý & 7 7; 9; 15; 21; 33; 44; 56 21 8 7; 9; 15; 20; 22;
More information04年度LS民法Ⅰ教材改訂版.PDF
?? A AB A B C AB A B A B A B A A B A 98 A B A B A B A B B A A B AB AB A B A BB A B A B A B A B A B A AB A B B A B AB A A C AB A C A A B A B B A B A B B A B A B B A B A B A B A B A B A B A B
More informationPart y mx + n mt + n m 1 mt n + n t m 2 t + mn 0 t m 0 n 18 y n n a 7 3 ; x α α 1 7α +t t 3 4α + 3t t x α x α y mx + n
Part2 47 Example 161 93 1 T a a 2 M 1 a 1 T a 2 a Point 1 T L L L T T L L T L L L T T L L T detm a 1 aa 2 a 1 2 + 1 > 0 11 T T x x M λ 12 y y x y λ 2 a + 1λ + a 2 2a + 2 0 13 D D a + 1 2 4a 2 2a + 2 a
More informationx, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
More information記号と準備
tbasic.org * 1 [2017 6 ] 1 2 1.1................................................ 2 1.2................................................ 2 1.3.............................................. 3 2 5 2.1............................................
More information20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33
More informationn ( (
1 2 27 6 1 1 m-mat@mathscihiroshima-uacjp 2 http://wwwmathscihiroshima-uacjp/~m-mat/teach/teachhtml 2 1 3 11 3 111 3 112 4 113 n 4 114 5 115 5 12 7 121 7 122 9 123 11 124 11 125 12 126 2 2 13 127 15 128
More informationii-03.dvi
2005 II 3 I 18, 19 1. A, B AB BA 0 1 0 0 0 0 (1) A = 0 0 1,B= 1 0 0 0 0 0 0 1 0 (2) A = 3 1 1 2 6 4 1 2 5,B= 12 11 12 22 46 46 12 23 34 5 25 2. 3 A AB = BA 3 B 2 0 1 A = 0 3 0 1 0 2 3. 2 A (1) A 2 = O,
More information) Binary Cubic Forms / 25
2016 5 2 ) Binary Cubic Forms 2016 5 2 1 / 25 1 2 2 2 3 2 3 ) Binary Cubic Forms 2016 5 2 2 / 25 1.1 ( ) 4 2 12 = 5+7, 16 = 5+11, 36 = 7+29, 1.2 ( ) p p+2 3 5 5 7 11 13 17 19, 29 31 41 43 ) Binary Cubic
More informationA(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6
1 1 1.1 64 A6, 1) B1, 1) 65 C A, 1) B, ) C 66 + 1 = 0 A1, 1) B, 0) P 67 A, ) B1, ) C4, 0) 1) ABC G ) A B C P 64 A 1, 1) B, ) AB AB = 1) + 1) A 1, 1) 1 B, ) 1 65 66 65 C0, k) 66 1 p, p) 1 1 A B AB A 67
More information( 28 ) ( ) ( ) 0 This note is c 2016, 2017 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercial purp
( 28) ( ) ( 28 9 22 ) 0 This ote is c 2016, 2017 by Setsuo Taiguchi. It may be used for persoal or classroom purposes, but ot for commercial purposes. i (http://www.stat.go.jp/teacher/c2epi1.htm ) = statistics
More information1 1 n 0, 1, 2,, n n 2 a, b a n b n a, b n a b (mod n) 1 1. n = (mod 10) 2. n = (mod 9) n II Z n := {0, 1, 2,, n 1} 1.
1 1 n 0, 1, 2,, n 1 1.1 n 2 a, b a n b n a, b n a b (mod n) 1 1. n = 10 1567 237 (mod 10) 2. n = 9 1567 1826578 (mod 9) n II Z n := {0, 1, 2,, n 1} 1.2 a b a = bq + r (0 r < b) q, r q a b r 2 1. a = 456,
More informationさくらの個別指導 ( さくら教育研究所 ) A 2 P Q 3 R S T R S T P Q ( ) ( ) m n m n m n n n
1 1.1 1.1.1 A 2 P Q 3 R S T R S T P 80 50 60 Q 90 40 70 80 50 60 90 40 70 8 5 6 1 1 2 9 4 7 2 1 2 3 1 2 m n m n m n n n n 1.1 8 5 6 9 4 7 2 6 0 8 2 3 2 2 2 1 2 1 1.1 2 4 7 1 1 3 7 5 2 3 5 0 3 4 1 6 9 1
More information1 θ i (1) A B θ ( ) A = B = sin 3θ = sin θ (A B sin 2 θ) ( ) 1 2 π 3 < = θ < = 2 π 3 Ax Bx3 = 1 2 θ = π sin θ (2) a b c θ sin 5θ = sin θ f(sin 2 θ) 2
θ i ) AB θ ) A = B = sin θ = sin θ A B sin θ) ) < = θ < = Ax Bx = θ = sin θ ) abc θ sin 5θ = sin θ fsin θ) fx) = ax bx c ) cos 5 i sin 5 ) 5 ) αβ α iβ) 5 α 4 β α β β 5 ) a = b = c = ) fx) = 0 x x = x =
More information13 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 informationx x x 2, A 4 2 Ax.4 A A A A λ λ 4 λ 2 A λe λ λ2 5λ + 6 0,...λ 2, λ 2 3 E 0 E 0 p p Ap λp λ 2 p 4 2 p p 2 p { 4p 2 2p p + 2 p, p 2 λ {
K E N Z OU 2008 8. 4x 2x 2 2 2 x + x 2. x 2 2x 2, 2 2 d 2 x 2 2.2 2 3x 2... d 2 x 2 5 + 6x 0 2 2 d 2 x 2 + P t + P 2tx Qx x x, x 2 2 2 x 2 P 2 tx P tx 2 + Qx x, x 2. d x 4 2 x 2 x x 2.3 x x x 2, A 4 2
More information1: *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入試の軌跡
4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf
More informationfunction2.pdf
2... 1 2009, http://c-faculty.chuo-u.ac.jp/ nishioka/ 2 11 38 : 5) i) [], : 84 85 86 87 88 89 1000 ) 13 22 33 56 92 147 140 120 100 80 60 40 20 1 2 3 4 5 7.1 7 7.1 1. *1 e = 2.7182 ) fx) e x, x R : 7.1)
More information応用数学III-4.ppt
III f x ( ) = 1 f x ( ) = P( X = x) = f ( x) = P( X = x) =! x ( ) b! a, X! U a,b f ( x) =! " e #!x, X! Ex (!) n! ( n! x)!x! " x 1! " x! e"!, X! Po! ( ) n! x, X! B( n;" ) ( ) ! xf ( x) = = n n!! ( n
More information1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
More information0 (18) /12/13 (19) n Z (n Z ) 5 30 (5 30 ) (mod 5) (20) ( ) (12, 8) = 4
0 http://homepage3.nifty.com/yakuikei (18) 1 99 3 2014/12/13 (19) 1 100 3 n Z (n Z ) 5 30 (5 30 ) 37 22 (mod 5) (20) 201 300 3 (37 22 5 ) (12, 8) = 4 (21) 16! 2 (12 8 4) (22) (3 n )! 3 (23) 100! 0 1 (1)
More information行列代数2010A
a ij i j 1) i +j i, j) ij ij 1 j a i1 a ij a i a 1 a j a ij 1) i +j 1,j 1,j +1 a i1,1 a i1,j 1 a i1,j +1 a i1, a i +1,1 a i +1.j 1 a i +1,j +1 a i +1, a 1 a,j 1 a,j +1 a, ij i j 1,j 1,j +1 ij 1) i +j a
More informationD xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y
5 5. 2 D xy D (x, y z = f(x, y f D (2 (x, y, z f R 2 5.. z = x 2 y 2 {(x, y; x 2 +y 2 } x 2 +y 2 +z 2 = z 5.2. (x, y R 2 z = x 2 y + 3 (2,,, (, 3,, 3 (,, 5.3 (. (3 ( (a, b, c A : (x, y, z P : (x, y, x
More information1 I
1 I 3 1 1.1 R x, y R x + y R x y R x, y, z, a, b R (1.1) (x + y) + z = x + (y + z) (1.2) x + y = y + x (1.3) 0 R : 0 + x = x x R (1.4) x R, 1 ( x) R : x + ( x) = 0 (1.5) (x y) z = x (y z) (1.6) x y =
More informationii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.
24(2012) (1 C106) 4 11 (2 C206) 4 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 (). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5... 6.. 7.,,. 8.,. 1. (75%)
More information1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2
1 Abstract n 1 1.1 a ax + bx + c = 0 (a 0) (1) ( x + b ) = b 4ac a 4a D = b 4ac > 0 (1) D = 0 D < 0 x + b a = ± b 4ac a b ± b 4ac a b a b ± 4ac b i a D (1) ax + bx + c D 0 () () (015 8 1 ) 1. D = b 4ac
More information2002.N.x.h.L.......g9/20
1 2 3 4 5 6 1 2 3 4 5 8 9 1 11 11 12 13 k 14 l 16 m 17 n 18 o 19 k 2 l 2 m 21 n 21 o 22 p 23 q 23 r 24 24 25 26 27 28 k 28 l 29 m 29 3 31 34 42 44 1, 8, 6, 4, 2, 1,2 1, 8 6 4 2 1, 8, 6, 4, 2, 1,2 1, 8
More information