09 II 09/12/ (3D ) f(, y) = 2 + y 2 3D- 1 f(0, 0) = 2 f(1, 0) = 3 f(0, 1) = 4 f(1, 1) = 5 f( 1, 2) = 6 f(0, 1) = z y (3D ) f(, y) = 2 + y
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1 09 II 09/12/ I 2D II 3D f() = f(, y) = y + y 2 6 4y f(1) = = 2 y = f() f(3, 2) = = 8 z = f(, y) y 2 1 z y 1 ( y ) 1 (0, 0) 2 (1, 0) 3 (0, 1) 4 (1, 1) 5 ( 1, 2) 6 (0, 1) y y
2 09 II 09/12/ (3D ) f(, y) = 2 + y 2 3D- 1 f(0, 0) = 2 f(1, 0) = 3 f(0, 1) = 4 f(1, 1) = 5 f( 1, 2) = 6 f(0, 1) = z y (3D ) f(, y) = 2 + y 2 3D hiver.html
3 09 II 09/12/ I II I II 1 f() = log f(, y) = y + y 2 6 4y 2 f () = 2 { f (, y) = I II 3 Fermat f () = 0 = a 3 Fermat { f (, y) = 0 0 { =? y =? 4 f () 4 = a 5 Hesse f (a) + 5 Hesse 1 ( 5.4 ) f() = log 3 3 = log 3 + log 3. ({}}{ ) ({}} {) ( {}}{ f () = log 3 + log 3 {}}{ ) 1 = 3 = 1 3 Fermat f () = = 0 = 1 3 f () = ( 1 ) ( ) 3 = 1 3 = 1 2 = 1 2 = 1 3 Hesse f ( 1 3 ) = 1 ( 1 = 9 < 0 = 1 ( 3 )2 3 f(1 3 ) = log 3 1 ) 3 1 = log 1 1 = 1 ( ) 3 3 //
4 09 II 09/12/ y = f() = y = f () y = dy d { y y = dy d = 2. y = u 3 + u dy du = 3. z = d = 4. z = y 3 + y dy = 5. z = d =
5 09 II 09/12/ z = 2y 3 + 4y dy = 7. z = d = 8. z = y y dy = 9. z = 3 a + 2 b + 2 d = 10. z = ay 3 + by dy = 11. z = 3 a b 2 + c 4 d = 12. z = a 3 y 3 + b 2 y 2 + c 4 dy =
6 09 II 09/12/ * z = 3 y y 2 + y 4 d = y 14. * z = 3 y y 2 + y 4 dy = y * ( ) = = ( {}}{{}}{ c f()) = c f (), ( ) {}}{ f() c = f {}}{ () c, ( ) {}}{{}}{ {}}{{}}{ a f() b = a f () b ( 3 ) = 0 ( c ) = 0 ( 3 2) = 3 2 = 6 ( a 3 ) = a ( 3 4 ) = 3a 4 ( 2 3 ) = 2 3 = 6 ( 3 a 2) = 3 4 a 2( = 3a 2 4) ( ) ( = = 24 a3 3 b 2) = a 3 ( 3 4 )b 2 = 3a 3 b 2 4 =
7 09 II 09/12/ d 2006 IBM z f (, y) z f z = f(, y) = 2 y z y f y (, y) }{{} z y f y }{{} y 1 1. z f z f z y f y z f z y f y 2. Chain Rule d d the partial derivative of z with respect to 3. f f y f f y f(, y, z) f f y f z
8 09 II 09/12/ f(, y) = 4 y {}}{ y 3 y {}}{ 4 3y 2 = 3 4 y 2 y 2. f(, y) = 3 2 y 4 y y 3. f(, y) = 4 3 y y y
9 09 II 09/12/ f(, y) = y + y y = 6 + 2y + 4 y y + 0 = 2 + 2y y 5. f(, y) = y + y 2 6 4y y y 6. f(, y) = 2 + 4y + 9y y + 2 y y
10 09 II 09/12/ f(, y) = 3 y y y y 8. * f(, y) = α y β ( α) = α α 1 α β y y 9. f(, y) = 1 2 y 1 2 y y II
11 09 II 09/12/ f(, y) = 1 3 y 2 3 y y 11. f(, y) = e + log y y y 12. * f(, y) = α log + (1 α) log y y y
12 09 II 09/12/ f(, y) = e y log y y 14. f(, y) = y + y y y 15. f(, y) = 2y + y y y ( ) f = f g f g g g 2
13 09 II 09/12/
14 09 II 09/12/
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平成18年度「商品先物取引に関する実態調査」報告書
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「産業上利用することができる発明」の審査の運用指針(案)
1 1.... 2 1.1... 2 2.... 4 2.1... 4 3.... 6 4.... 6 1 1 29 1 29 1 1 1. 2 1 1.1 (1) (2) (3) 1 (4) 2 4 1 2 2 3 4 31 12 5 7 2.2 (5) ( a ) ( b ) 1 3 2 ( c ) (6) 2. 2.1 2.1 (1) 4 ( i ) ( ii ) ( iii ) ( iv)
09 II 09/11/ y = e x y = log x = log e x 2. ( e x ) = e x 3. ( ) log x = 1 x 1 Warming Up 1 u = log a M a u = M log a 1 a 0 a 1 a r+s 0 a r
09 II 09/11/16 1 5.6 1. y = e x y = log x = log e x 2. e x ) = e x 3. ) log x = 1 x 1 Warming Up 1 u = log a M a u = M log a 1 a 0 a 1 a r+s 0 a r a s 1 a 2 f g) = f g + f g 1. fx) = x e x f x) = 2. fx)
( ) 2.1. C. (1) x 4 dx = 1 5 x5 + C 1 (2) x dx = x 2 dx = x 1 + C = 1 2 x + C xdx (3) = x dx = 3 x C (4) (x + 1) 3 dx = (x 3 + 3x 2 + 3x +
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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.
r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B
1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n
sin cos No. sine, cosine : trigonometric function π : π = 3.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd cos = cos : even.
08 No. : No. : No.3 : No.4 : No.5 : No.6 : No.7 : No.8 : No.9 : No.0 : No. : sin cos No. sine, cosine : trigonometric function π : π = 3.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin
I No. sin cos sine, cosine : trigonometric function π : π =.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd cos = cos : even.
I 0 No. : No. : No. : No.4 : No.5 : No.6 : No.7 : No.8 : No.9 : No.0 : I No. sin cos sine, cosine : trigonometric function π : π =.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd
CALCULUS II (Hiroshi SUZUKI ) f(x, y) A(a, b) 1. P (x, y) A(a, b) A(a, b) f(x, y) c f(x, y) A(a, b) c f(x, y) c f(x, y) c (x a, y b)
CALCULUS II (Hiroshi SUZUKI ) 16 1 1 1.1 1.1 f(x, y) A(a, b) 1. P (x, y) A(a, b) A(a, b) f(x, y) c f(x, y) A(a, b) c f(x, y) c f(x, y) c (x a, y b) lim f(x, y) = lim f(x, y) = lim f(x, y) = c. x a, y b
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1 1 7 1.1.................................. 11 2 13 2.1............................ 13 2.2............................ 17 2.3.................................. 19 3 21 3.1.............................
1.2 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 3 y 1 (x) y 1 (x) e R P (x)dx y 2
1 1.1 R(x) = 0 y + P (x)y + Q(x)y = R(x)...(1) y + P (x)y + Q(x)y = 0...(2) 1 2 u(x) v(x) c 1 u(x)+ c 2 v(x) = 0 c 1 = c 2 = 0 c 1 = c 2 = 0 2 0 2 u(x) v(x) u(x) u (x) W (u, v)(x) = v(x) v (x) 0 1 1.2
( ) 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
m d2 x = kx αẋ α > 0 (3.5 dt2 ( de dt = d dt ( 1 2 mẋ kx2 = mẍẋ + kxẋ = (mẍ + kxẋ = αẋẋ = αẋ 2 < 0 (3.6 Joule Joule 1843 Joule ( A B (> A ( 3-2
3 3.1 ( 1 m d2 x(t dt 2 = kx(t k = (3.1 d 2 x dt 2 = ω2 x, ω = x(t = 0, ẋ(0 = v 0 k m (3.2 x = v 0 ω sin ωt (ẋ = v 0 cos ωt (3.3 E = 1 2 mẋ2 + 1 2 kx2 = 1 2 mv2 0 cos 2 ωt + 1 2 k v2 0 ω 2 sin2 ωt = 1
a 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
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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 =,
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9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x
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() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y
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