example2_time.eps
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- やすはる はしかわ
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1 Google (20/08/2 ) ( ) Random Walk & Google Page Rank Agora on Aug. 20 / 67
2 Introduction ( ) Random Walk & Google Page Rank Agora on Aug / 67
3 Introduction Google ( ) Random Walk & Google Page Rank Agora on Aug / 67
4 Introduction Σ ( ) Random Walk & Google Page Rank Agora on Aug / 67
5 Introduction Plan of Talk Google (Section 2) (Section ) (Section ) (Section ) (Section ) Google (Section 5) ( ) Random Walk & Google Page Rank Agora on Aug / 67
6 Introduction Google ( ) Random Walk & Google Page Rank Agora on Aug / 67
7 Google Google ( ) Random Walk & Google Page Rank Agora on Aug / 67
8 Google Google Google (Section.) Google ( ) Random Walk & Google Page Rank Agora on Aug / 67
9 Google Google Google (Section 2) Google S.Brin L.Page (999 ) L.Page ( ) Random Walk & Google Page Rank Agora on Aug / 67
10 Google (Section 2.)...,, ( ) Random Walk & Google Page Rank Agora on Aug / 67
11 Google (Section 2.) HyperText Mark up Language HTML Hyper Link... A B A B ( ) Random Walk & Google Page Rank Agora on Aug. 20 / 67
12 Google (Section 2.) v, v 2, v 3, v v 2, v 3, v 2 v 3, v 3 v. v (Example 2.., P.5) v 2 v 3. (Definition 2..2, P. 5) ( ) Random Walk & Google Page Rank Agora on Aug / 67
13 Google (Section 2.) (Definition 2..3, P. 6) v i r(v i ) r(v i ) = r(v j ) v j v j B vi (2.) B vi v i {r(v i )} (2.). ( ) Random Walk & Google Page Rank Agora on Aug / 67
14 Google (Section 2.) Example 2..6 r = r 3, r 2 = 2 r, (2.2) (Example 2..6, P. 7) r 3 = 2 r + r 2, {r,r 2,r 3 } (2.2) r = k, r 2 = k/2, r 3 = k ( ) Random Walk & Google Page Rank Agora on Aug / 67
15 Google (Section 2.) Example (2.2) r r 3 = 0, 2 r + r 2 = 0, 2 r r 2 + r 3 = 0 (2.3) (0,0,0) (0,0,0) ( ) Random Walk & Google Page Rank Agora on Aug / 67
16 Google (Section 2.) (2.) ( ) Random Walk & Google Page Rank Agora on Aug / 67
17 Google (Section 2.2) t = n v i p i (n) v i t = n + ( ) Random Walk & Google Page Rank Agora on Aug / 67
18 Google (Section 2.2), p(v i ) = p i = lim n p i(n) v i. {p i (n)} {p i (n + )}. p i (n + ) = v j B vi v j p j(n) (2.4) ( ) Random Walk & Google Page Rank Agora on Aug / 67
19 Google (Section 2.2) Example 2.. p (n + ) = p 3 (n), p 2 (n + ) = 2 p (n), p 3 (n + ) = 2 p (n) + p 2 (n) (2.5) p (0) =, p 2 (0) = p 3 (0) = 0 t p 0 /2 /2 /4 /2 3/8 3/8 7/6 3/8 2/5 p 2 /2 0 /4 /2 /8 /4 3/6 3/6 7/32 /5 p 3 /2 /2 /4 /2 3/8 3/8 7/6 3/8 3/32 2/5 (p,p 2,p 3 ) = (2/5,/5,2/5) Example 2..6 ( ) Random Walk & Google Page Rank Agora on Aug / 67
20 Google (Section 2.2) Theorem (P. 9) {p i (n)} (2.4). i p i (n) n p i, {p i } p i = v j B vi v j p j (2.6), (2.6), (2.) ( ) Random Walk & Google Page Rank Agora on Aug / 67
21 Google (Section 2.3) ( ) Random Walk & Google Page Rank Agora on Aug / 67
22 Google (Section 2.3),,,, t = 0, ( ) Random Walk & Google Page Rank Agora on Aug / 67
23 ( ) Random Walk & Google Page Rank Agora on Aug / 67
24 (Section 3.) { ax + by = x, cx + dy = y, (3.),... a x + a 2 x a N x N = a, a 2 x + a 22 x a 2N x N = a 2,. a N x + a N2 x a NN x N = a N, ( ) Random Walk & Google Page Rank Agora on Aug / 67
25 (Section 3.) (3.), ax + by = x, cx + dy = y,, xy-, (3.).,. x = by dx ad bc, y = ay cx ad bc 2 ax + by = x, cx + dy = y, ax + by = x (3.). 3 ax + by = x, cx + dy = y,, (3.). ( ) Random Walk & Google Page Rank Agora on Aug / 67
26 (Section 3.). ax + by = x, cx + dy = y ad bc = 0 ax + by = x, cx + dy = y a : b : x = c : d : y (3.) ad bc 0, (3.) 2 ad bc = 0 by dx = 0 ( ay cx = 0, ax + by = x (x, y) (3.). 3 ad bc = 0 by dx 0 ( ay cx 0), (3.). ( ) Random Walk & Google Page Rank Agora on Aug / 67
27 (Section 3.) ad bc 0 ( ) ( ) x ax + by y cx + dy xy- (x,y) (ax + by,cx + dy) (ax+by,cx+dy) = (x,y ) (x,y) (x,y) (3.) ad bc 0 ( ) Random Walk & Google Page Rank Agora on Aug / 67
28 (Section 3.2) N {x i } N i= x x =. x N x = (x i ) (Definition 3.2., P. 3) x i x i ( ) Random Walk & Google Page Rank Agora on Aug / 67
29 (Section 3.2) N x R N = x =. : x i R x N N (Definition 3.2.2, P. 3) R N, 0 0 (Definition 3.2.4, P. 3) ( ) Random Walk & Google Page Rank Agora on Aug / 67
30 (Section 3.2) R 2 xy- xy- (x, y) ( x x = y) R 3 xyz- N R N ( ) Random Walk & Google Page Rank Agora on Aug / 67
31 (Section 3.2) R N x, y R N x + y x y x + y x =., y =. = x + y =. x N y N x N + y N a R, x R N x a x ax x =. = ax =. ax N x N (Definition 3.2.6, P. 4) ( ) Random Walk & Google Page Rank Agora on Aug / 67
32 (Section 3.2) ( ) Random Walk & Google Page Rank Agora on Aug / 67
33 (Section 3.2) R N W W W., 0 W. 2 W x, y W x + w W. 3 W x W a R ax W. R 2 xy- R 3 xyz- ( ) Random Walk & Google Page Rank Agora on Aug / 67
34 (Section 3.2, Example 3.2.9) x R 2 W = {ax : a R} xy-,, x ( ) {( )} a x = W = y = 2x 2 2a x, y R 3 x y W = {ax + by : a,b R} xyz-,, x, y x =, y = 0 W = a a + b 0 b x y + z = 0 ( ) Random Walk & Google Page Rank Agora on Aug / 67
35 (Section 3.2) ( ) Random Walk & Google Page Rank Agora on Aug / 67
36 (Section 3.2) R N k {x,... x k } a x + + a k x k = 0 = a = = a k = 0 (Definition 3.2., P. 5) {x,y} {x,y} ( ) Random Walk & Google Page Rank Agora on Aug / 67
37 (Section 3.2, Example 3.2.3) {x,y,z} ax + by + cz = 0 a, b, c 0 a 0, x = b a y c a z, x y z x z y ( ) Random Walk & Google Page Rank Agora on Aug / 67
38 (Section 3.2, Example 3.2.4) ( ) ( ) x =, y = 2 ( ) ( ) 2 x =, y = (y + 2x = 0) x =, y = 0 0 x =, y =, z = x =, y =, z = 3 (2x+y z = 0) 0 ( ) Random Walk & Google Page Rank Agora on Aug / 67
39 (Section 3.2) V k (dim V = k) V k V k + (Definition 3.2.5, P. 7) dim R N = N, R N N {x,...,x N } ( ) Random Walk & Google Page Rank Agora on Aug / 67
40 (Section 3.2) dim V = k V k (Definition 3.2.9, 8) R N 0 0 e =.,, e N =. 0 RN 0 R N ( ) Random Walk & Google Page Rank Agora on Aug / 67
41 (Section 3.2) dim V = k, {x,...,x k } x V x = a x + + a k x k ( ) Random Walk & Google Page Rank Agora on Aug / 67
42 (Section 3.2) R 2 (Example 3.2.2, P. 8) ( ( 0 e =, e 0) 2 = ) ( ( x =, x ) 2 = ) ( ( y =, y 0) 2 = ) ( ) 2 x = 3 x = 2e + 3e 2 = 5 2 x 2 x 2 = y + 3y 2 (Example , P. 8) ( ) Random Walk & Google Page Rank Agora on Aug / 67
43 (Section 3.2) R N x x x R N, x 2 x R N, k R, kx = k x 3 x, y R N, x + y x + y 4 x = 0 x = 0 (Definition , P. 9),, x 2 + y 2 ( ) Random Walk & Google Page Rank Agora on Aug / 67
44 (Section 3.2) (Example , P. 9) x = x + + x N, x 2 = x x N 2, x = max i=,...,n x i x = x 2 = x = ( ) Random Walk & Google Page Rank Agora on Aug / 67
45 (Section 3.3, Definition 3.3.) N N N a a 2 a N a 2 a 22 a 2N A = = ( a ) a... a N =. a a N a n2 a N NN a ij A (i,j) k a k =. k a k a Nk l a l = ( ) a l a Nl l A = (a ij ) ( ) Random Walk & Google Page Rank Agora on Aug / 67
46 (Section 3.3, Definition 3.3.3) A = (a ij ) a ii 0 E = O = ( ) Random Walk & Google Page Rank Agora on Aug / 67
47 (Section 3.3, Definition 3.3.4) N N A = (a ij ), B = (b ij ), A + B A + B = (a ij + b ij ) k R, N N A = (a ij ), ka ka = (ka ij ) ( ) ( ) 2 3 A =, B = ( ) ( ) A + B =, 3A = ( ) Random Walk & Google Page Rank Agora on Aug / 67
48 (Section 3.3) x = (x i ) R N N N A = (a ij ) Ax ( N ) Ax = a ik x k k= R N (Definition 3.3.5, P. 2) ( ) ( ) ( ) A =, x = Ax = ( ) ( ) ( ) a b x ax + by A =, x = Ax = c d y cx + dy ( ) Random Walk & Google Page Rank Agora on Aug / 67
49 (Section 3.3) N N A = (a ij ), B = (b ij ) AB ( N ) AB = a ik b kj k= N N (Definition 3.3.6, P. 2) ( ) ( ) a b x y A =, B = c d w z ( ) ( ) ax + bw ay + bz ax + cy bx + dy AB =, BA = cx + dw cy + dz aw + cz bw + dz ( ) ( ) A =, B = ( ) ( ) AB = BA = ( ) Random Walk & Google Page Rank Agora on Aug / 67
50 (Section 3.3) N N A = (a ij ), B = (b ij ) AB B = ( b b N ) AB = ( Ab Ab N ) N (Remark 3.3.2, P. 22) N N A = (a ij ) A T A T = (a ji ) (Definition 3.3.7, P. 2) ( ) a b A = c d A T = ( ) a c b d ( ) Random Walk & Google Page Rank Agora on Aug / 67
51 (Section 3.3, Example 3.3.3) { ax + by = x, cx + dy = y, (3.) A = ( ) a b, x = c d Ax = b ( ) x, b = y ( x y ) ( ) Random Walk & Google Page Rank Agora on Aug / 67
52 (Section 3.4) N N A = (a ij ) R N x Ax R N (Definition 3.4., P. 23). A0 = 0. 2 x, y R N, λ, µ R,. (Proposition 3.4.2, P. 23) A(λx + µy) = λax + µay ( ) Random Walk & Google Page Rank Agora on Aug / 67
53 (Section 3.4, Example 3.4.4) ( ) 2 A = 3 4 ( )( ) ( ) 2 Ae = =, Ae = ( )( ) ( ) 2 2 8, Ax = = {x i }, ( )( ) 0 = ( ) 2 4 A, {Ax i } (Remark 3.4.5, P. 25) E Ex = x O Ox = 0 ( ) Random Walk & Google Page Rank Agora on Aug / 67
54 (Section 3.4, Example 3.4.7) ( ) 0 A = 0, x x, y y, ( ) 0 A = x 0 x Ax Ax x ( ) Random Walk & Google Page Rank Agora on Aug / 67
55 (Section 3.4, Example 3.4.8) θ ( ) cos θ sin θ A = sin θ cos θ Ax x θ ( ) Random Walk & Google Page Rank Agora on Aug / 67
56 (Section 3.4) N N A, AA = A A = E N N A A A (Definition 3.4.9, P. 26) A, ( ) Random Walk & Google Page Rank Agora on Aug / 67
57 (Section 3.4) ( ) a b A =, A c d = AA = ( ) x y z w ( ) ax + bz ay + bw = cx + dz cy + dw ( ) 0 0 ad bc 0 ( ) ( ) A x y a b = = z w ad bc c d ad bc = 0 (Example 3.4.0, P. 26) ( ) Random Walk & Google Page Rank Agora on Aug / 67
58 (Section 3.4) A N N A Ax = b x = A b (Theorem 3.4.4, P. 27) ( ) Random Walk & Google Page Rank Agora on Aug / 67
59 (Section 3.4, Example 3.4.5) { x + 2y = 3, 3x + 4y = 4 ( )( ) ( ) 2 x 3 = 3 4 y 4 ( ) ( ) ( )( )( ) x 2 2 x = 3 y y ( 2 = )( ) 3 = 4 ( ) ( ) Random Walk & Google Page Rank Agora on Aug / 67
60 (Section 3.4) r(v i ) = r(v j ) v j v j B vi (2.) Ax = x,, ( ) Random Walk & Google Page Rank Agora on Aug / 67
61 (Section 3.5) ( ) a b 2 2 A = c d A. A A. 2 ad bc 0 ( ( a b 3 A a =, a c) 2 = d) A = (a a 2 ), {a, a 2 }. 4 A a = ( a b ), a 2 = ( c d ) ( ) a A =, {a a, a 2 }. 2 (Theorem 3.5., P. 28) ( ) Random Walk & Google Page Rank Agora on Aug / 67
62 (Section 3.5) 2 2 A = ( ) a b c d, det A det A = ad bc (Definition 3.5.2, P. 28) N N 3 3 deta = a a 22 a 33 + a 2 a 23 a 3 + a 3 a 2 a 32 a 3 a 22 a 3 a 2 a 2 a 33 a a 23 a 32 N N, aij N N! (Remark 3.5.3, P. 28) ( ) Random Walk & Google Page Rank Agora on Aug / 67
63 (Section 3.5) N N A = (a ij ), A. A A. 2 deta 0 3 A N a i =.. a i a Ni A = (a...a N ), {a,..., a N } 4 A N a j = ( ) a j a jn a A =.., {a,...,a N } a N (Theorem 3.5.4, P. 29) ( ) Random Walk & Google Page Rank Agora on Aug / 67
64 (Section 3.5). dete = 2 (deta)(det B) = (det AB) 3 deta T = deta (Theorem 3.5.6, P. 29) ( ) Random Walk & Google Page Rank Agora on Aug / 67
65 (Section 3.5) N N A N {a k } N k= A = ( ) a a N, {a k } N k= A (Definition 3.5.7, P. 29) N N A, A. A A. 2 ranka = N. (Theorem 3.5.8, P. 29) ( ) Random Walk & Google Page Rank Agora on Aug / 67
66 (Section 3.5) N N A f A : R N R N, x Ax ranka = N (det A 0 ) f A (R N ) = R N ranka < N (det A = 0 ) f A (R N ) R N deta = 0 Ax = b b f A (R N ) or b f A (R N ) ( ) Random Walk & Google Page Rank Agora on Aug / 67
67 (Section 3.5) N N A fa f A (R N ) f A Im f A Im A {x R N : f A (x) = 0} = {x R N : A(x) = 0} f A, kerf A kera (Definition 3.5.9, P. 30) N N A,. dimim A = ranka, 2 N dimkera = dimima. 3 deta 0 dim Im A = N, dimkera = 0. (Theorem 3.5., P. 30) ( ) Random Walk & Google Page Rank Agora on Aug / 67
68 (Section 3.5) Ax = b A, b R N, x = A b. 2 A. 2. b ImA,. 2.2 b ImA,. x 0, y ker A, x = x 0 + y, kera. (Theorem 3.5.3, P. 3) ( ) Random Walk & Google Page Rank Agora on Aug / 67
69 2 2 (Section 3.5) ( )( ) a b x c d y ( x = y ) ad bc 0, b R 2, x = A b. 2 ad bc = 0 ««a x 2. c y ««a x 2.2 c y.. x 0 «, x b x = x 0 + k. a (Corollary 3.5.4, P. 3) ( ) ( ) a b Im A, ker A c a ( ) Random Walk & Google Page Rank Agora on Aug / 67
70 2 2 (Section 3.5) (x,y ) ker A Im A ker A Im A (x,y ) ker A ker A ImA Im A a + d a + d = (cf. Example 3.5.5) (cf. Example 3.5.6) ( ) Random Walk & Google Page Rank Agora on Aug / 67
71 2 2 (Section 3.5, Example 3.5.5) ( )( ) ( ) 2 x a = 2 y b deta = 0 {( )} {( kera = span, Im A = span 2 )} ( ) a b = Im A a = b b ( ) ( ) a x = + t a 2 kera 2x + y = 0, (a, a) a = b = 0 Ax = 0 0 = dim ker A ( ) Random Walk & Google Page Rank Agora on Aug / 67
72 ( ) Random Walk & Google Page Rank Agora on Aug / 67
73 (Section 3.6) (2.) Ax = x N N A, λ R x R N (x 0), λ A, Ax = λx (3.2) x, λ (Definition 3.6., P. 35) x A ( ) Random Walk & Google Page Rank Agora on Aug / 67
74 (Section 3.6) x λ = A(kx) = kax = k(λx) = λ(kx) for all k R kx (Remark 3.6.2, P. 35) ( ) Random Walk & Google Page Rank Agora on Aug / 67
75 (Section 3.6) Ax = tx t, x Ax tex = 0 2 (A te)x = 0 3 det(a te) 0 x = (A te) 0 = 0 det(a te) = 0 λ N N A, λ t N det(a te) = 0 (Theorem 3.6.4, P. 36) ( ) Random Walk & Google Page Rank Agora on Aug / 67
76 (Section 3.6) N t N + a n t n + + a t + a 0 = 0, N (Theorem 3.6.5, P. 36) N N A,, N. (Remark 3.6.6, P. 36), t N det(a te) = 0, N ( ) Random Walk & Google Page Rank Agora on Aug / 67
77 (Section 3.6, Example 3.6.7) ( ) 2 A =, 2 ( ) t 2 det(a te) = det = t 2 2t 3 = (t+)(t 3) 2 t, A, 3. ( ) 2 A =, 2 ( ) t 2 det(a te) = det = t 2 2t t, A + 2i, 2i. ( ) Random Walk & Google Page Rank Agora on Aug / 67
78 (Section 3.6, Example 3.6.7) 3 A = 3, 3 det(a te) = t 3 + 9t 2 24t + 6 = (t )(t 4) 2, A 4, 4,. ( ) Random Walk & Google Page Rank Agora on Aug / 67
79 (Section 3.6) det(a te) A A λ k, λ k, λ,...,λ j A, det(a te) = (t λ ) k (t λ j ) k j t λ i λ i (Definition 3.6.8, P. 37) ( ) Random Walk & Google Page Rank Agora on Aug / 67
80 (Section 3.6) λ R det(a λe) = 0, dim ker(a λe) > 0, ker(a λe) x dim ker(a λe), λ k dim ker(a λe) k = ( ) Random Walk & Google Page Rank Agora on Aug / 67
81 (Section 3.6, Example 3.6.0) ( ) 2 A =. {, 3} 2 ( ) ( 2 2 x A + E = x =, 2 2 y) 2x + 2y = 0. λ = ( ) x = ( ) λ = 3 x = ( ) Random Walk & Google Page Rank Agora on Aug / 67
82 (Section 3.6, Example 3.6.0) ( ) 2 A =. { + 2i, 2i} 2 ( ) 2i 2 A ( + 2i)E = λ = + 2i ( 2 2i i x = ) ( ) i λ = 2i x = ( ) Random Walk & Google Page Rank Agora on Aug / 67
83 (Section 3.6) N N A,. A A T. 2 T, A T AT. (Theorem 3.6.2, P. 39) ( ) Random Walk & Google Page Rank Agora on Aug / 67
84 (Section 3.6) Ax = x (2.),. A, A 2 A x 3 (Problem 3.6.4, P. 40) ( ) Random Walk & Google Page Rank Agora on Aug / 67
85 n (Section 3.7, Example 3.7.) ( ) ( ) 3 A =, T =, B = T AT, A n. ( ) ( ) 0 0 B =, B 0 2 n = 0 2 n A n = (TBT ) n = TBT TB BT = TB n T B, 2 A T A T 2 A 2 T T AT, ( ) Random Walk & Google Page Rank Agora on Aug / 67
86 (Section 3.7) N N A,. λ i x i.,, T AT =. (Corollary 3.7.3, P. 4) T = ( x x N ) λ... λ N ( ) Random Walk & Google Page Rank Agora on Aug / 67
87 (Section 3.7) ( ) λ 0 L = 0 λ 2 λ λ 2 Ae = λ e, Ae 2 = λ 2 e 2, 2 2 A λ, λ 2 λ i a i Aa = λ a, Aa 2 = λ 2 a 2, (3.5) ( ) Random Walk & Google Page Rank Agora on Aug / 67
88 (Section 3.7) T = ( a b ) (3.5) AT = TL, T AT = L {a,a 2 }, A 3a a a2 2a2 ( ) Random Walk & Google Page Rank Agora on Aug / 67
89 (Section 3.7) ( ) cos θ sin θ A = cos θ ± isin θ = e sin θ cos θ ±iθ 2 2 e ±iθ : θ ( ) Random Walk & Google Page Rank Agora on Aug / 67
90 (Section 3.8) A.. A. 0.. A = ( ) Random Walk & Google Page Rank Agora on Aug / 67
91 (Section 3.8, Theorem 3.8.) A {λ i } N i= λ > λ 2 λ N. x 0 y k+ = Ax k, x k+ = y k+ y k+ x 0, x N λ. ( ) Random Walk & Google Page Rank Agora on Aug / 67
92 (Section 3.8) (Theorem 3.8.3, P. 44) N N A = (a ij ), B i = {z C : z a ii N j=,j i a ij }., A λ, N B i. ( ) Random Walk & Google Page Rank Agora on Aug / 67
93 (Section 3.8) N N A = (a ij ) (a ij 0), A λ C λ max i=,...,n. (Theorem 3.8.4, P. 45) N j= a ij ( ) Random Walk & Google Page Rank Agora on Aug / 67
94 ( ) Random Walk & Google Page Rank Agora on Aug / 67
95 (Section 4.) V, E : E e E, V o(e), t(e) V G = (V,E) V E o(e) e t(e) e (Definition 4.., P. 47) ( ) Random Walk & Google Page Rank Agora on Aug / 67
96 (Section 4.) ( ) Random Walk & Google Page Rank Agora on Aug / 67
97 (Section 4.) G = (V,E), A G G. V,,...,N.., N V. 2 N N A G, e E, o(e) = i, t(e) = j a ji =, a ji = 0 (Definition 4..4, P. 48) ( ) Random Walk & Google Page Rank Agora on Aug / 67
98 (Section 4.) 2 (Example 4..5, P. 48) A G = ( ) Random Walk & Google Page Rank Agora on Aug / 67
99 (Section 4., Example 4..6) A G = , A G 2 = ( ) Random Walk & Google Page Rank Agora on Aug / 67
100 (Section 4., Example 4..6) D,2 = D,2 A G2 D,2 = A G D,2 D,2 = E D,2 A G 2 D,2 = A G T A G2 T = A G ( ) Random Walk & Google Page Rank Agora on Aug / 67
101 (Section 4., Example 4..6) 0 N N A A G N aji =, j i G A G A = A G ( ) Random Walk & Google Page Rank Agora on Aug / 67
102 (Section 4.2, Definition 4.2.) G = (V,E) x, y x y G = (V,E) x, y x y ( ) Random Walk & Google Page Rank Agora on Aug / 67
103 (Section 4.2, Definition 4.2.3) G = (V,E) G x, y V G = (V,E) G x, y V G = (V,E) (Proposition 4.2.4, P. 50) ( ) Random Walk & Google Page Rank Agora on Aug / 67
104 (Section 4.2, Example 4.2.6) A G = , A2 G = A 3 G = A G + A 2 G + A 3 G = ( ) Random Walk & Google Page Rank Agora on Aug / 67
105 (Section 4.2, Example 4.2.6) A G2 = , A2 G 2 = A 3 G 2 = A G2 + A 2 G 2 + A 3 G 2 = ( ) Random Walk & Google Page Rank Agora on Aug / 67
106 (Section 4.2, Example 4.2.6) A G3 = , A2 G 3 = A 3 G 3 = A G3 + A 2 G 3 + A 3 G 3 = ( ) Random Walk & Google Page Rank Agora on Aug / 67
107 (Section 4.2) G A G A G + A 2 G + + AN G G N G (Theorem 4.2.7, P. 52) ( ) A O N N X B C X X (Definition 4.2.8, P. 52) ( ) Random Walk & Google Page Rank Agora on Aug / 67
108 (Section 4.2) G G, G A G (Theorem 4.2.0, P. 53) 0 N N A A ( ) Random Walk & Google Page Rank Agora on Aug / 67
109 (Section 4.2, Example 4.2.9) A G = , A G 2 = A G3 = G G 2 G 3 ( ) Random Walk & Google Page Rank Agora on Aug / 67
110 ( ) Random Walk & Google Page Rank Agora on Aug / 67
111 (Section 4.3) G G = (V,E) (Example 4.3., P. 53) v v3 v2 t = n G t = n +, G i,, ( ) Random Walk & Google Page Rank Agora on Aug. 20 / 67
112 (Section 4.3) t = n v i p(n,i) p(n +,) 0 /2 /2 p(n, ) p(n +,2) = /2 0 /2 p(n, 2) (4.) p(n +,3) /2 /2 0 p(n, 3) π(n + ) = P G π(n) (4.2) ( ) Random Walk & Google Page Rank Agora on Aug / 67
113 (Section 4.3) π = (p i ) R N p + + p N =, 0 p i π (Definition 4.3.2, P. 54) N N P = (p ij ) P 0 pij N i= p ij = (Definition 4.3.4, P. 54) P G π, P P π (Proposition 4.3.5, P. 55) ( ) Random Walk & Google Page Rank Agora on Aug / 67
114 (Section 4.3) π(n + ) = P G π(n) π(0), n π(n), π(n), (Definition 4.3.3, P. 54) ( ) Random Walk & Google Page Rank Agora on Aug / 67
115 (Section 4.3, Definition 4.3.7) G = (V,E) P G i j p ji = /K K i i j p ji = 0 P G G P G π(n + ) = P G π(n) (4.4) G ( ) Random Walk & Google Page Rank Agora on Aug / 67
116 (Section 4.3) v v2 G P G A G v3 0 /2 /2 /2 0 /2 /2 / v v2 v3 0 /2 /2 0 0 /2 / v v2 v3 0 /2 /3 /2 0 /3 /2 /2 /3 0 0 ( ) Random Walk & Google Page Rank Agora on Aug / 67
117 (Section 4.3) N N P π(n + ) = P π(n) (Definition 4.3.3, P. 57) P 0 A 2 A G G G AG A G = A ( ) Random Walk & Google Page Rank Agora on Aug / 67
118 (Section 4.3) 0 3/4 /5 P = /3 0 4/5 2/3 /4 0 0 A = 0 0 v /3 3/4 v 2 2/3 /4 /5 v 3 4/5 ( ) Random Walk & Google Page Rank Agora on Aug / 67
119 (Section 4.3, Definition 4.3.5) P π(n + ) = P π(n) t = 0 π(0) t = n π(n) = P n π(0) π = P π π ( ) Random Walk & Google Page Rank Agora on Aug / 67
120 (Section 4.3) G π(n + ) = P G π(n), π = lim n π(n) π = PG π π P π = P π π P ( ) Random Walk & Google Page Rank Agora on Aug / 67
121 (Section 4.3) π(0), π = lim t π(t) (Problem 4.3.7, P. 58) ( ) Random Walk & Google Page Rank Agora on Aug / 67
122 (Section 4.3) P, π(0), (Problem 4.3.8, P. 58) (cf. Theorem 3.8., P. 44) π = lim n P n π(0) ( ) Random Walk & Google Page Rank Agora on Aug / 67
123 (Section 4.4) N N A a ij 0 A (Definition 4.4., P. 59) P A λ(a) λ(a), > 0. 2 A µ, µ λ(a). 3 λ(a). 4 λ(a) x,. 5 λ(a),. ( Theorem 4.4.2, P. 59) ( ) Random Walk & Google Page Rank Agora on Aug / 67
124 (Section 4.4) P,. P,. 2 P µ µ. 3 P. (Theorem 4.4.3, P. 59), λ(p ) = ( ) Random Walk & Google Page Rank Agora on Aug / 67
125 (Section 4.4), λ(p ) λ(p ) > 0. 2 λ(p ) x,, λ(p ). 3 P, e = (,...,) T P T e = e. P T ( ) Random Walk & Google Page Rank Agora on Aug / 67
126 (Section 4.4) 4 µ µ n p ji = j= P T 5 P T P, P λ(p ) = 6, P x,, π =, π ( ) Random Walk & Google Page Rank Agora on Aug / 67
127 (Section 4.4) P, ( ) Random Walk & Google Page Rank Agora on Aug / 67
128 (Section 4.4) ( ) 0 P G = 0 ( ) π(0) = 0 = {, } π(2n) = ( ) π(2n + ) = 0 ( ) 0, (Example 4.4.4, P. 60) 2 {,e 2πi/3,e 2πi/3 } (Example 4.4.5, P. 6) ( ) Random Walk & Google Page Rank Agora on Aug / 67
129 (Section 4.4) µ = µ Theorem 3.8. (P. 44) λ > λ 2 λ N, π = lim n P n π(0) µ µ < ( ) Random Walk & Google Page Rank Agora on Aug / 67
130 (Section 4.5),. P k i = gcd{k N : p (k) ii > 0} P k = p (k) ij, k i i i, (Definition 4.5., P. 62) P k i = k j (Proposition 4.5.2, P. 62) ( ) Random Walk & Google Page Rank Agora on Aug / 67
131 (Section 4.5) k i 2 P, k i = (Definition 4.5.4, P. 63) = 3 4 (Example 4.5.8, P. 63) (Example 4.5.0, P. 64) ( ) Random Walk & Google Page Rank Agora on Aug / 67
132 (Section 4.5) P, P {p ii },. (Theorem 4.5.5, P. 63) P, P λ λ = λ =., π(0), t, π. (Theorem 4.5.6, P. 63) ( ) Random Walk & Google Page Rank Agora on Aug / 67
133 (Section 4.5) P k, e 2πi/k. e 2πi/k = cos(2π/k) + isin(2π/k) k = 5 (Example 3.7.4, P. 43) ( ) Random Walk & Google Page Rank Agora on Aug / 67
134 (Section 4.6) P, π 2 π(0) lim P n π(0) n. (Theorem 4.6., P. 65) π = lim n P n π(0) ( ) Random Walk & Google Page Rank Agora on Aug / 67
135 (Section 4.6), Theorem 4.6., π. (Theorem 4.6.2, P. 66) ( ) Random Walk & Google Page Rank Agora on Aug / 67
136 (Section 4.6) ( ) Random Walk & Google Page Rank Agora on Aug / 67
137 Google Google ( ) Random Walk & Google Page Rank Agora on Aug / 67
138 Google Goolge (Section 5.) ( ) Random Walk & Google Page Rank Agora on Aug / 67
139 Google Goolge (Section 5.), G G H, (Definition 5.., P. 67) H (Remark 5..2, P. 67), (Example 5..3, P. 67) ( ) Random Walk & Google Page Rank Agora on Aug / 67
140 Google Goolge (Section 5.) G = (V,E) v, v v. (Example 5..3, P. 67) / H = / , π = ( ) Random Walk & Google Page Rank Agora on Aug / 67
141 Google Goolge (Section 5.) S d = (di ) R N d i = { i, 0 i, S = N det, e =.. R N S = ( s s N ) s i = N e i, 0 i, ( ) Random Walk & Google Page Rank Agora on Aug / 67
142 Google Goolge (Section 5.) P P = H + S P Example 5..4 P / / / /2 0 /6 0 S = / /6 0,P = / /2 0 / / / / /6 0 ( ) Random Walk & Google Page Rank Agora on Aug / 67
143 Google Goolge (Section 5.), P H S P = H + S P P ( ) Random Walk & Google Page Rank Agora on Aug / 67
144 Google Goolge (Section 5.) T T = /N /N N eet =..... /N /N (Definition 5..5, P. 70) T ( ) Random Walk & Google Page Rank Agora on Aug / 67
145 Google Goolge Google (Section 5.) Google G(α) G(α) = αp + ( α)t 0 < α < (Definition 5..6, P. 70) α : ( ) Random Walk & Google Page Rank Agora on Aug / 67
146 Google Goolge Google (Section 5.) Example 5..4 (α = 9/0) / /2 0 /6 0 P = / /2 0 /6 0, T = / /6 0 G = 9 0 P + 0 T = ( ) Random Walk & Google Page Rank Agora on Aug / 67
147 Google Goolge Google (Section 5.) 0 < α < G(α) G(α) G(α) G(α) π π = lim n G n π(0) (Theorem 5..8, P. 70) Google G(α) (Definition 5..0, P. 7) ( ) Random Walk & Google Page Rank Agora on Aug / 67
148 Google Goolge (Section 5.) G( 9 0 ) = , π = ( ) Random Walk & Google Page Rank Agora on Aug / 67
149 Google Goolge α (Section 5.) G(α) = αp + ( α)t α α d dα π (α), α 0 d dα π (α) (Remark 5.., P. 7) Google α = 0.85 ( ) Random Walk & Google Page Rank Agora on Aug / 67
150 Google (Section 5.2) G(α) /N 2 π(0) =.. /N R N 3 G n π(0) π n+ = G(α)π n (Theorem 3.8., P. 44) ( ) Random Walk & Google Page Rank Agora on Aug / 67
151 Google (Section 5.2) Theorem 3.8. π k π k π 0 λ 2 k. (Theorem 5.2.6, P. 74) λ G(α) λ = π k π O ( λ 2 k) ( ) Random Walk & Google Page Rank Agora on Aug / 67
152 Google (Section 5.2) ǫ > 0 ǫ N N n > N n a n a < ǫ {a n } n= a (Definition 5.2.2, P. 72) r < r n 0 ǫ > 0 n > log r ǫ = r n < ǫ (Example 5.2.3, P. 73) ( ) Random Walk & Google Page Rank Agora on Aug / 67
153 Google (Section 5.2, Example 5.2.4) ǫ = /000 = 0 3 n log r ǫ = 3log 0/log r r n r = 0.7 r = 0.8 r = 0.9 e-50 e-00 e-50 e-200 e ( ) Random Walk & Google Page Rank Agora on Aug / 67
154 Google (Section 5.2) G(α) P {,λ 2,λ 3,...,λ N } G(α) = αp + ( α)t {,αλ 2,αλ 3,...,αλ N } G(α) α (Theorem 5.2.8, P. 75) λ 2 = 0.999, (0.999) k k k = 9899 α = 0.85 (0.85) k k k = 6 ( ) Random Walk & Google Page Rank Agora on Aug / 67
155 Google (Section 5.2) N N G π R N Gπ, O(N 2 ) (Proposition 5.2.9, P. 75) Intel Core i7 50Gflops ( 500 ) N = 55.5 (Remark 5.2.0, P. 75) (Remark 5.2.6, P. 77) N = : 800 (Proposition 5.2., P. 76) ( ) Random Walk & Google Page Rank Agora on Aug / 67
156 Google (Section 5.2) N N 0 O(N),, 0 (Definition 5.2.2, P. 76) H Google G G π ( α Gπ = αhπ + (e T π) N d + α ) N e O(N) (Proposition 5.2.4, P. 76), (Proposition 5.2.5, P. 76) ( ) Random Walk & Google Page Rank Agora on Aug / 67
157 Google (Section 5.2) G(α) π(n + ) = Gπ(n), α π(k) π = O(α k ) π(k + ) π(k) ( α)ǫ π(k) π ǫ (Remark 5.2.8, P. 78) ( ) Random Walk & Google Page Rank Agora on Aug / 67
158 Google (Section 5.3, Example 5.3.) / /2 0 / H = / /3 0 0 /2, d = /2 0 / / ( ) Random Walk & Google Page Rank Agora on Aug / 67
159 Google (Section 5.3, Example 5.3.) 0 /6 / P = H + /2 /6 / dt e = /2 / /6 /3 0 0 /2 0 /6 0 /2 0 /2 0 /6 0 /2 0 α = π = ( ) Random Walk & Google Page Rank Agora on Aug / 67
160 Google (Section 5.3, Example 5.3.) Page rank v v2 v3 v4 v5 v ( ) Random Walk & Google Page Rank Agora on Aug / 67
161 Google (Section 5.3, Example 5.3.) Iteration count - Error e-06 e-08 e-0 e-2 e ( ) Random Walk & Google Page Rank Agora on Aug / 67
162 Google (Section 5.3, Example 5.3.2) N = 5757, = 2445, = 4.25 ( ) Random Walk & Google Page Rank Agora on Aug / 67
163 Google (Section 5.3, Example 5.3.2) Iteration count - Error e-05 e-06 e-07 e-08 e-09 e ( ) Random Walk & Google Page Rank Agora on Aug / 67
164 Google (Section 5.3, Example 5.3.2) 000 Elasped time ( ) Random Walk & Google Page Rank Agora on Aug / 67
165 Goolge Google 0 0 ( ) Random Walk & Google Page Rank Agora on Aug / 67
166 Google, Google (?) ( ) Random Walk & Google Page Rank Agora on Aug / 67
167 ( ) Random Walk & Google Page Rank Agora on Aug / 67
i I II I II II IC IIC I II ii 5 8 5 3 7 8 iii I 3........................... 5......................... 7........................... 4........................ 8.3......................... 33.4...................
() 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 ()
II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K
II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F
ad 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
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(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y
![(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y](/thumbs/101/151465237.jpg "(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y")
(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = a c b d (a c, b d) P = (a, b) O P a p = b P = (a, b) p = a b R 2 { } R 2 x = x, y R y 2 a p =, c q = b d p + a + c q = b + d q p P q a p = c R c b
数学Ⅱ演習（足助・09夏）
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.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,
[ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b
(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y
![(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y](/thumbs/98/136512264.jpg "(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y")
(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = a c b d (a c, b d) P = (a, b) O P a p = b P = (a, b) p = a b R 2 { } R 2 x = x, y R y 2 a p =, c q = b d p + a + c q = b + d q p P q a p = c R c b
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1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim
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2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i
[ ] (2016 3Q N) a 11 a 1n m n A A = a m1 a mn A a 1 A A = a n (1) A (a i a j, i j ) (2) A (a i ca i, c 0, i ) (3) A (a i a i + ca j, j i, i ) A 1 A 11 0 A 12 0 0 A 1k 0 1 A 22 0 0 A 2k 0 1 0 A 3k 1 A rk
( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) (
6 20 ( ) sin, cos, tan sin, cos, tan, arcsin, arccos, arctan. π 2 sin π 2, 0 cos π, π 2 < tan < π 2 () ( 2 2 lim 2 ( 2 ) ) 2 = 3 sin (2) lim 5 0 = 2 2 0 0 2 2 3 3 4 5 5 2 5 6 3 5 7 4 5 8 4 9 3 4 a 3 b
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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
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Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
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06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,
y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
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),
1 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
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
Microsoft Word - 触ってみよう、Maximaに2.doc
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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
5 36 5................................................... 36 5................................................... 36 5.3..............................
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1/1 lim f(x, y) (x,y) (a,b) ( ) ( ) lim limf(x, y) lim lim f(x, y) x a y b y b x a ( ) ( ) xy x lim lim lim lim x y x y x + y y x x + y x x lim x x 1
1/5 ( ) Taylor ( 7.1) (x, y) f(x, y) f(x, y) x + y, xy, e x y,... 1 R {(x, y) x, y R} f(x, y) x y,xy e y log x,... R {(x, y, z) (x, y),z f(x, y)} R 3 z 1 (x + y ) z ax + by + c x 1 z ax + by + c y x +
[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s
![[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s](/thumbs/94/118579915.jpg "[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s")
[ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =
x () 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 ),
linearal1.dvi
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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
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
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