f (x) x y f(x+dx) f(x) Df 関数 接線 x Dx x 1 x x y f f x (1) x x 0 f (x + x) f (x) f (2) f (x + x) f (x) + f = f (x) + f x (3) x f
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- ふじよし かんけ
- 4 years ago
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1 f fd f Df 関数 接線 D f f 0 f f f 2 f f f f f 3 f lim f f df 0 d 4 f df d 3 f d f df d 5 d
2 c f f t t f df d 6 d t dt 7 f df df d d df dt lim f 0 t df d d dt d t 8 dt 9.2 f,, f 0 f 0 lim 0 lim 0 f 0 f 2 0 3, 4 f d, d f, d d 5 f f, u v u, v u, v 5 f u u u u v 6 v u v 7 v u u v v 8 v
3 c u u u v v v F C F,, C 2 F,, 0 F F F F 0 22 F/ 0 F/ F/ F/ F/ F/ F F F d d d
4 c u 2 a 2 a > 0 2 u a b u 3/2 2 a 2 3/2 d 2 du u / a 2 2u d a b u a du 2u a 2 a b 29 a u 2 b 2u d a b au a du 2 u 3 a 3 3 bu 2 a 2b a b 3a u d 3/2 a b u 3 a du 2 au 2 a 3 a b u 2 b 2u d 3/2 a b au 3 a du 2 a 2 u b 2 a 2b u a 2 32 a b 3 a > 0 log 2 a 2 d log 2 a a 2 d 2 log 2 a 2 2 a 2 a 2 2 a 2 d log 2 a 2 a 2 2 d 2 2 a 2 d log 2 a tan a a > 0 a tan π/2 2 a 2 3/2 d π/2 2 cos, /2 dd cos 3 a a 3 cos 2 d π/2 a 2 sin 2 π/2 a π /2 dd /2 2π cos d a tan π/2 2 a 2 d cos 2 a π/2 a 2 cos 2 d π/2 a π/2 π a 36
5 c , 2, cos q ,,,, cos φ 3,, φ 3 φ 3,, 3,, φ cos φ 42 sin φ 43 cos 44
6 c q cosq sinqsinf f 0 0 sinq 0 sinq sinqcosf e, e 2 e, e e cos e sin e 45 e sin e cos e 46 e q q q q e 4 2 e e e cos e sin e 47 e sin e cos e 48,, 0 e e cos e sin e cos e sin e cos 2 e cos sin e sin 2 e cos e e cos sin 0 e sin cos 0 e e 50 e 0 0 e
7 c cos sin 0 e sin cos 0 e 0 0 cos sin 0 e sin cos 0 e e 5 e 0 0 e cos φ sin sin φ cos e cos cos φ cos sin φ sin e e 52 e φ sin φ cos φ 0 e e sin cos φ sin sin φ cos e e cos cos φ cos sin φ sin e 53 e sin φ cos φ 0 e φ sin cos φ cos cos φ sin φ e sin sin φ cos sin φ cos φ e 54 cos sin 0 e φ f, dd f, dd 55 S S f 2π 0 f dd f 2π d 56 2π d d f,, ddd f,, ddd V V f,, φ 2 sin dddφ 57 V 3 f 2π π 0 0 f 2 sin dddφ f 4π 2 d 58 4π 2 d d
8 c a b a b a b a b cos 59 a b b a 60 e e e e e e 6 e e e e e e a b a a e a e a e 63 b b e b e a e 64 2 a b a e a e a e b e b e a e a b a b a b 65 a 2 b 2 2 a b cos b a 2 66 e e e e e e 67 e e e e e e 0 68 e e e e e ϕ e φ 69 e e e e φ e e φ e e e e e 7 e e e e e 72 e e e e e 73 e e e e e e 0 74
9 c a b b a a b a b e a b a b e a b a b e 75 a e e e b a a a b b b A B OACB OA B h a b a b a 2 a 2 77 OACB S S OA h a b a b a b 78 a b Bb, b C 0, 0 O 5 h q A a, a h OB sin 79 a b a b sin 80 a b sin a b a b sin 8
10 c e e e 82 e e e 83 e e e 84 e e e e e e e e e φ 86 e e φ e 87 e φ e e 88 e e e e e φ e φ S S S S n S S n 90 a b 2 S S 2 a b 9 S 面積 S S b 面積 S a 6 OABC ABC AB AC b a c a 2 2 b c 2 c a 2 a b 92 OAB OBC OCA S S S e S e S e 93
11 c 208 C B O A 7 S S S S e S e S S cos 94 n e S S cos e S cos e S cos e ,, s s e e 96 s e e 97 0 d s d lim e 0 e e d d e 98 d 96 d s d e d e 99 d s ds ds 2 d s 2 d 2 d 2 00 d d ds ds dd 0
12 c d dq d d d s ds 8 d s d e d e 02 ds 2 d 2 d 2 03 ds dd ,, d, d, d d s ds d s d e d e d e 05 ds 2 d 2 d 2 d 2 06 d e d e dd e 07 d S ds d S dd e dd e dd e 08 ds 2 dd 2 dd 2 dd 2 09 ds e e e dd ds 2 dd 2
13 c dv dv ddd d s d e d e e 3 ds 2 d 2 d 2 d 2 4 d S dd e dd e dd e 5 ds 2 dd 2 dd 2 dd 2 6 dv ddd d s d e d e dφ e φ 8 ds 2 d 2 d 2 dφ 2 9 d S 2 sin ddφ e ddφ e dd e φ 20 dv 2 sin dddφ A t A A t e A t e A t e 22 da dt da t e da t e da t e 23 dt dt dt A dt A t dt e A t dt e A t dt e 24
14 c e cos e sin e 25 e sin e cos e 26 2π 2π 2π e d cos d e sin d e π 0 2π 0 2π e d sin d e cos d e C s s, s, s f,, C f,, ds f s, s, s d 2 d 2 d 2 C C d 2 f s, s, s C ds A A d s A d A d A d A d C C C 2 d ds C 2 d ds 29 ds A d A d 30 C 7.3 f S S f,, ds f,, dd 2 dd 2 dd 2 S S 2 2 f,, dd 3 S A A ds A dd A dd A dd A dd A dd A dd 32 S S S S S 8 8.
15 c e e e e 0 sin cos 0 e cos sin 0 e e e 34 e e 0 e e 0 e e 0 35 e e gad e e e φ e 0 cos cos φ cos sin φ sin e sin cos φ sin sin φ cos e e e 37 e φ e 0 e e φ e φ sin sin φ sin cos φ 0 cos sin φ cos cos φ 0 e cos φ sin φ 0 e 3 f 0 f sin e φ cos e φ sin e cos e f f 39 0 f 40 f f f f f 4 f 42
16 c f,,,,,, φ f e e e e e e e e e φ φ div 8.3. A e e e A e A e A e A A A A e e e A e A e A e A e e A e e e A e e A e e A e A e e A e e e A e e A A e e e A A A A A A A 50
17 c A e e e φ A e A e A φ e φ φ A e e A e e e A Aφ e φ φ e e φ A φ A e φ A e e A e e A e Aφ e φ φ e φ A e φ A cos e φ A A A A φ φ A cos A 2 A 2 sin A A φ φ ot 8.4. A e e e A e A e A e A e e A e e A e e A e e A e e A e e A A A e A A e A e e e e A A A 52
18 c A e e e A e A e A e A e e A e e e e A e e e A e e A A A A A e A e e e e A e e e A e A e e A A e e A e e A e e A e e A e e A A e e A e e A A A e e A A e A e A A A e A A e A e A A e 53
19 c A e e e φ A e A e A φ e φ φ e A φ e A φ e A φ e e A e e e e A e e φ A φ e φ A e e e A φ e e φ A A φ e φ A φ e A φ A e φ e φ e φ e φ e φ e φ φ e A φ e e A e e e A φ e φ A e e A e e A φ e 0 e φ A φ e e φ A A φ e A e φ sin e φ e φ cos e φ A φ e φ sin e cos e A φ cos A φ A φ e e φ A φ A φ A φ eφ e A A A e e sin Aφ A e φ A φ Aφ e A A e φ ϕ ϕ 55 2 A gad div A ot ot A 56
20 c ϕ 2 ϕ ϕ e e e ϕ e e ϕ e ϕ A 2 A A A A A A A e A A e 2 A 2 A 2 A e 2 A 2 2 A A e A A A A e A A A A e A A A 2 A 2 2 A 2 A 2 e 2 2 A 2 e A e A e A e 2 A 2 2 A e A 2 2 A 2 2 A 2 2 ϕ 2 2 ϕ 2 2 ϕ 2 57 A A e A 2 A e 2 A 2 A 2 2 A 2 A 2 e 2 2 A 2 2 A 2 2 e e e e e e ϕ 2 ϕ e e e ϕ e e ϕ e ϕ 2 ϕ 2 ϕ 2 ϕ 2 2 ϕ 2 2 ϕ 2 ϕ 2 ϕ ϕ 2 ϕ 2 ϕ ϕ 2 60
21 c A A A e e A A A e e e A 2 A 2 A 2 A 2 2 A A e 2 2 A 2 A 2 2 e A e e A 2 A 2 A 2 A 2 A A A A A 2 A 2 A A A A A A e 2 2 A 2 A A 2 2 A e e 2 A A e e e A A 2 A 2 A 2 A 2 A 2 A 2 A 2 A 2 A A 2 A 2 2 A 2 2 A A 2 A 2 2 A A 2 A 2 2 A 2 2 A 2 2 A 2 2 A 2 2 A A 2 2 A 2 A 2 A 2 2 A ϕ 2 ϕ e e e φ ϕ 2 ϕ 2 2 ϕ ϕ e φ 2 ϕ 2 2 ϕ 2 ϕ 2 2 cos 2 sin 2 2 ϕ 2 2 sin e ϕ e φ sin ϕ cos ϕ ϕ sin ϕ 2 sin 2 2 ϕ φ 2 2 sin 2 ϕ φ 2 ϕ φ 2 2 ϕ φ 2 64
22 c A A A 2 A 2 sin A 2 A A cos A A 2 2 A 2 A 2 A 2 cos 2 sin A cos A φ 2 sin φ 2 A φ φ 2 2 A 2 2 A 2 sin A A φ 2 sin φ 2 A φ φ A φ φ A φ φ A A 2 2 A sin A cos A 2 A 2 A 2 sin A 2 A A cos A A 2 A 2 2 A 2 sin 2 A cos A φ 2 sin 2 φ 2 A φ 2 sin φ 2 A 2 2 A 2 sin 2 A cos A φ 2 sin 2 φ 2 A φ 2 sin φ A φ φ A φ φ cos A 2 sin 2 A sin sin A A φ φ φ 2 A 2 sin A 2 A A cos A A A φ φ A φ φ 2 A 2 sin φ 2 A φ cos A 2 sin 2 φ 2 A 2 sin φ 2 A φ 2 sin 2 φ 2 67
23 A A sin A cos A sin c A φ sin A φ Aφ A cos A sin 2 A cos A sin 2 A 2 2 A φ 2 A φ φ 2 A φ φ cos 2 sin A A 2 cos A 2 A cos A 2 sin 2 A A 2 sin 2 φ 2 A φ 2 sin φ 2 A φ φ 2 sin sin A cos A 2 A 2 sin A φ 2 sin φ 2 A φ φ sin A 2 A 2 sin 2 φ 2 68 A φ sin Aφ A A A φ cos A φ sin 2 φ 2 A φ φ 2 A sin 2 φ 2 2A 2 A 2 2 A cos A φ 2 sin 2 φ 2 A φ 2 sin φ 2 sin 2 cos 2 sin 2 A φ A φ φ 2 A φ 2 sin φ 2 sin 2 2 A φ 2 2 A φ 2 2 A 2 A A 2 A 2 A A sin φ Aφ sin Aφ A φ 2 A sin φ 2A φ 2 A φ 2 sin 2 A φ cos A φ 2 A φ 2 2 A φ 2 A φ 2 A φ 2 2 A φ 2 2 sin 2 A φ 2 A φ 2 sin 2 A φ cos A φ 2 sin 2 sin 2 A φ sin sin A φ 2 A φ 2 A φ 2 A 2 sin φ 69 70
24 c A A A A 2 2 A 2 2 A 2 sin sin A cos A 2 2 sin sin A cos A φ 2 sin φ 2 A 2 A φ φ 2 sin A 2 A 2 sin sin A A sin A 2 A 2 sin 2 A 2 sin 2 φ 2 2A sin 2 A A A A 2 A 2 2 A 2 sin 2 A 2 sin A cos sin 2 sin 2 cos A φ 2 sin 2 φ 2 A φ 2 sin φ 2 A 2 sin 2 φ A 2 2 A 2 sin A sin 2 A φ 2 A 2 sin φ 2 A φ cos 2 sin 2 2 A φ 2 2 A φ 2 sin 2 A φ 2 2 A φ 2 sin A φ sin A 2 A 2 sin 2 φ 2 A φ φ sin A A φ φ 2 A 2 sin 2 A φ φ 2 2 sin 7 2 A 2 sin 2 φ 2 2 A 2 2 sin 2 A 2 cos A φ 2 sin 2 φ 72 A A φ A φ A A φ 2 A 2 sin φ 2 A φ 2 sin 2 φ 2 sin A φ 2 sin 2 A φ 2 sin 2 φ 2 2 A 2 sin φ φ 2 sin 2 A φ cos A 2 sin 2 2 A φ φ A φ φ φ 2 sin 2 A φ ot gad ϕ 0 74 div ot A 0 75 ϕ ot gad ϕ ot e ϕ e ϕ e ϕ ϕ ϕ e ϕ ϕ e ϕ e 0 76
25 div ot A A div A A A c A e A A A A e A e A A 0 77 ϕ ot gad ϕ ot e e ϕ e ϕ φ φ ϕ sin ϕ e φ φ ϕ φ ϕ e φ ϕ ϕ e φ 0 78 div ot A sin Aφ div A e φ 2 2 sin Aφ A φ sin 2 sin 2 sin 2 sin sin cos A φ sin A φ A φ A φ sin Aφ A φ sin A φ A φ cos A φ sin A φ 2 A 2 sin 2 sin φ cos A φ sin A φ A φ 2 A φ 2 A φ A φ 2 sin Aφ A φ cos A φ 2 A φ cos A φ e A A A φ A A φ A 2 A φ 2 A φ cos 2 sin A φ A φ 2 A 2 sin φ cos A φ 2 A φ 2 A φ 2 A 2 sin φ cos 2 sin A φ A φ 2 cos A φ 2 A φ A 2 sin φ 2 A φ 2 A 2 sin φ 0 A e φ 79
c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l
![c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l](/thumbs/85/91372642.jpg "c y /2 ddy = = 2π sin θ /2 dθd /2 [ ] 2π cos θ d = log 2 + a 2 d = log 2 + a 2 = log 2 + a a 2 d d + 2 = l")
c 28. 2, y 2, θ = cos θ y = sin θ 2 3, y, 3, θ, ϕ = sin θ cos ϕ 3 y = sin θ sin ϕ 4 = cos θ 5.2 2 e, e y 2 e, e θ e = cos θ e sin θ e θ 6 e y = sin θ e + cos θ e θ 7.3 sgn sgn = = { = + > 2 < 8.4 a b 2
4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx
4 4 5 4 I II III A B C, 5 7 I II A B,, 8, 9 I II A B O A,, Bb, b, Cc, c, c b c b b c c c OA BC P BC OP BC P AP BC n f n x xn e x! e n! n f n x f n x f n x f k x k 4 e > f n x dx k k! fx sin x cos x tan
A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π
4 4.1 4.1.1 A = f() = f() = a f (a) = f() (a, f(a)) = f() (a, f(a)) f(a) = f 0 (a)( a) 4.1 (4, ) = f() = f () = 1 = f (4) = 1 4 4 (4, ) = 1 ( 4) 4 = 1 4 + 1 17 18 4 4.1 A (1) = 4 A( 1, 4) 1 A 4 () = tan
.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
(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {
![(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {](/thumbs/94/118399854.jpg "(1) D = [0, 1] [1, 2], (2x y)dxdy = D = = (2) D = [1, 2] [2, 3], (x 2 y + y 2 )dxdy = D = = (3) D = [0, 1] [ 1, 2], 1 {")
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1 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 =
1 12 ( )150 ( ( ) ) x M x 0 1 M 2 5x 2 + 4x + 3 x 2 1 M x M 2 1 M x (x + 1) 2 (1) x 2 + x + 1 M (2) 1 3 M (3) x 4 +
( )5 ( ( ) ) 4 6 7 9 M M 5 + 4 + M + M M + ( + ) () + + M () M () 4 + + M a b y = a + b a > () a b () y V a () V a b V n f() = n k= k k () < f() = log( ) t dt log () n+ (i) dt t (n + ) (ii) < t dt n+ n
( ) 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 +
(.. C. ( d 5 5 + C ( d d + C + C d ( d + C ( ( + d ( + + + d + + + + C (5 9 + d + d tan + C cos (sin (6 sin d d log sin + C sin + (7 + + d ( + + + + d log( + + + C ( (8 d 7 6 d + 6 + C ( (9 ( d 6 + 8 d
0.6 A = ( 0 ),. () A. () x n+ = x n+ + x n (n ) {x n }, x, x., (x, x ) = (0, ) e, (x, x ) = (, 0) e, {x n }, T, e, e T A. (3) A n {x n }, (x, x ) = (,
[ ], IC 0. A, B, C (, 0, 0), (0,, 0), (,, ) () CA CB ACBD D () ACB θ cos θ (3) ABC (4) ABC ( 9) ( s090304) 0. 3, O(0, 0, 0), A(,, 3), B( 3,, ),. () AOB () AOB ( 8) ( s8066) 0.3 O xyz, P x Q, OP = P Q =
() 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 ()
さくらの個別指導 ( さくら教育研究所 ) A a 1 a 2 a 3 a n {a n } a 1 a n n n 1 n n 0 a n = 1 n 1 n n O n {a n } n a n α {a n } α {a
... A a a a 3 a n {a n } a a n n 3 n n n 0 a n = n n n O 3 4 5 6 n {a n } n a n α {a n } α {a n } α α {a n } a n n a n α a n = α n n 0 n = 0 3 4. ()..0.00 + (0.) n () 0. 0.0 0.00 ( 0.) n 0 0 c c c c c
1. z dr er r sinθ dϕ eϕ r dθ eθ dr θ dr dθ r x 0 ϕ r sinθ dϕ r sinθ dϕ y dr dr er r dθ eθ r sinθ dϕ eϕ 2. (r, θ, φ) 2 dr 1 h r dr 1 e r h θ dθ 1 e θ h
IB IIA 1 1 r, θ, φ 1 (r, θ, φ)., r, θ, φ 0 r
1 B64653 1 1 3.1....................................... 3.......................... 3..1.............................. 4................................ 4..3.............................. 5..4..............................
1 1 3 ABCD ABD AC BD E E BD 1 : 2 (1) AB = AD =, AB AD = (2) AE = AB + (3) A F AD AE 2 = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD 1 1
ABCD ABD AC BD E E BD : () AB = AD =, AB AD = () AE = AB + () A F AD AE = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD AB + AD AB + 7 9 AD AB + AD AB + 9 7 4 9 AD () AB sin π = AB = ABD AD
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63 6 6.1 6.1.1 v = v 0 =v 0x,v 0y, 0) t =0 x 0,y 0, 0) t x x 0 + v 0x t v x v 0x = y = y 0 + v 0y t, v = v y = v 0y 6.1) z 0 0 v z yv z zv y zv x xv z xv y yv x = 0 0 x 0 v 0y y 0 v 0x 6.) 6.) 6.1) 6.)
Quiz x y i, j, k 3 A A i A j A k x y z A x A y A z x y z A A A A A A x y z P (x, y,z) r x i y j zk P r r r r r r x y z P ( x 1, y 1, z 1 )
Quiz x y i, j, k 3 A A i A j A k x y z A x A y A z x y z A A A A A A x y z P (x, y,z) x i y j zk P x y z P ( x 1, y 1, z 1 ) Q ( x, y, z ) 1 OP x1i y1 j z1k OQ x i y j z k 1 P Q PQ 1 PQ x x y y z z 1 1
grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )
2 9 2 5 2.2.3 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t))
, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,,
6,,3,4,, 3 4 8 6 6................................. 6.................................. , 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p,
29
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Gmech08.dvi
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九州大学学術情報リポジトリ Kyushu University Institutional Repository 物理工科のための数学入門 : 数学の深い理解をめざして 御手洗, 修九州大学応用力学研究所 QUEST : 推進委員 藤本, 邦昭東海大学基盤工学部電気電子情報工学科 : 教授 http://hdl.handle.net/34/500390 出版情報 : バージョン :accepted
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145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (
1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +
0 = m 2p 1 p = 1/2 p y = 1 m = 1 2 d ( + 1)2 d ( + 1) 2 = d d ( + 1)2 = = 2( + 1) 2 g() 2 f() f() = [g()] 2 = g()g() f f () = [g()g()]
![0 = m 2p 1 p = 1/2 p y = 1 m = 1 2 d ( + 1)2 d ( + 1) 2 = d d ( + 1)2 = = 2( + 1) 2 g() 2 f() f() = [g()] 2 = g()g() f f () = [g()g()]](/thumbs/92/110082312.jpg "0 = m 2p 1 p = 1/2 p y = 1 m = 1 2 d ( + 1)2 d ( + 1) 2 = d d ( + 1)2 = = 2( + 1) 2 g() 2 f() f() = [g()] 2 = g()g() f f () = [g()g()]")
8. 2 1 2 1 2 ma,y u(, y) s.t. p + p y y = m u y y p p y y m u(, y) = y p + p y y = m y ( ) 1 y = (m p ) p y = m p y p p y 2 0 m/p U U() = m p y p p y 2 2 du() d = m p y 2p p y 1 0 = m 2p 1 p = 1/2 p y
1 1. x 1 (1) x 2 + 2x + 5 dx d dx (x2 + 2x + 5) = 2(x + 1) x 1 x 2 + 2x + 5 = x + 1 x 2 + 2x x 2 + 2x + 5 y = x 2 + 2x + 5 dy = 2(x + 1)dx x + 1
. ( + + 5 d ( + + 5 ( + + + 5 + + + 5 + + 5 y + + 5 dy ( + + dy + + 5 y log y + C log( + + 5 + C. ++5 (+ +4 y (+/ + + 5 (y + 4 4(y + dy + + 5 dy Arctany+C Arctan + y ( + +C. + + 5 ( + log( + + 5 Arctan
K E N Z U 01 7 16 HP M. 1 1 4 1.1 3.......................... 4 1.................................... 4 1..1..................................... 4 1...................................... 5................................
ac b 0 r = r a 0 b 0 y 0 cy 0 ac b 0 f(, y) = a + by + cy ac b = 0 1 ac b = 0 z = f(, y) f(, y) 1 a, b, c 0 a 0 f(, y) = a ( ( + b ) ) a y ac b + a y
01 4 17 1.. y f(, y) = a + by + cy + p + qy + r a, b, c 0 y b b 1 z = f(, y) z = a + by + cy z = p + qy + r (, y) z = p + qy + r 1 y = + + 1 y = y = + 1 6 + + 1 ( = + 1 ) + 7 4 16 y y y + = O O O y = y
66 σ σ (8.1) σ = 0 0 σd = 0 (8.2) (8.2) (8.1) E ρ d = 0... d = 0 (8.3) d 1 NN K K 8.1 d σd σd M = σd = E 2 d (8.4) ρ 2 d = I M = EI ρ 1 ρ = M EI ρ EI
65 8. K 8 8 7 8 K 6 7 8 K 6 M Q σ (6.4) M O ρ dθ D N d N 1 P Q B C (1 + ε)d M N N h 2 h 1 ( ) B (+) M 8.1: σ = E ρ (E, 1/ρ ) (8.1) 66 σ σ (8.1) σ = 0 0 σd = 0 (8.2) (8.2) (8.1) E ρ d = 0... d = 0 (8.3)
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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
高等学校学習指導要領解説 数学編
5 10 15 20 25 30 35 5 1 1 10 1 1 2 4 16 15 18 18 18 19 19 20 19 19 20 1 20 2 22 25 3 23 4 24 5 26 28 28 30 28 28 1 28 2 30 3 31 35 4 33 5 34 36 36 36 40 36 1 36 2 39 3 41 4 42 45 45 45 46 5 1 46 2 48 3
春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim n an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16,
春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16, 32, n a n {a n } {a n } 2. a n = 10n + 1 {a n } lim an
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( ) x y f(x, y) = ax
013 4 16 5 54 (03-5465-7040) nkiyono@mail.ecc.u-okyo.ac.jp hp://lecure.ecc.u-okyo.ac.jp/~nkiyono/inde.hml 1.. y f(, y) = a + by + cy + p + qy + r a, b, c 0 y b b 1 z = f(, y) z = a + by + cy z = p + qy
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18 10 01 ( ) 1 2018 4 1.1 2018............................... 4 1.2 2018......................... 5 2 2017 7 2.1 2017............................... 7 2.2 2017......................... 8 3 2016 9 3.1 2016...............................
D 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
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1 1.. ( ) ( ) ( ) (A) E icb φ Et = + cdiva ct (H3) (B) ( ) ct ct ' ctct ' + ' = ' ct ' + ct ' i( ') (H3,H18) 3 (i) cosh Ψ = cosh ΘcoshΩ sinhψ sinhθ sinhω cosh Ψ cosh Θ cosh Ω = sinhψ sinhθ sinhω tanhψ
ORIGINAL TEXT I II A B 1 4 13 21 27 44 54 64 84 98 113 126 138 146 165 175 181 188 198 213 225 234 244 261 268 273 2 281 I II A B 292 3 I II A B c 1 1 (1) x 2 + 4xy + 4y 2 x 2y 2 (2) 8x 2 + 16xy + 6y 2
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
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4 ( ) ( ) ( ) ( ) 4 5 5 II III A B (0 ) 4, 6, 7 II III A B (0 ) ( ),, 6, 8, 9 II III A B (0 ) ( [ ] ) 5, 0, II A B (90 ) log x x () (a) y x + x (b) y sin (x + ) () (a) (b) (c) (d) 0 e π 0 x x x + dx e