Chap10.dvi
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- くうしょう てっちがわら
- 4 years ago
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1 =0. f = 2 +3 { f = 1 =0 { sin 0 3 f = 1 =0 2 sin f = 0 =0 { f = 0 =0 f 3 2 lim = lim =0 =0. f 0 = 0. 2 = f 1 lim 0 0 = lim 0 sin 2 cos 1 = lim 0 2 sin = lim =0 0 2 =0. f 0 = 0. f 0 lim = lim sin 0 sin1/ 0. =0. f 0 = 0. 5 =0. 1
2 =0. sin 1 0 f = 0 =0 1 + sin f = 0 =0 3 f = f 0 lim = lim sin 0, =0. 2 =0. 3 f 0 lim = lim , 0. =0, f 0 = logtan 1 3 arcsin a a 2 arcsin a y = /2 1+2 = y tan sec 2 = = tan tan = 2 sin 2 2
3 3 y 1 1 = = y =, log y = log. y /y = log +1. y = ylog +1= log +1 5 y = 1 2 a 2 2 a a2 = a 1 /a 2 a 2 log log 3 cos + sin 2 4 e arctansec + tan = y = log log y = log log. y /y = log log + 1/. y = log log log + 1 log log 3 y = 2cos + sin sin + cos 4 y = e y 1 = 1 + sec + tan sec tan + 2 sec2 = 1 2 n, =0. e +1 2 cos log1 + 3
4 y = e +1, y n = ee y = e + e + e2 2 + e3 6 + e y = cos 2 = 1 2 cos , yn =2 n 1 cos 2 + nπ 2 y = y =1+ 3 log1 +, y = log , y = 61 + log , y = 6 log , y n = 6 1n n 4! 1 + n 3 n 4 y = =0. arctan 2 sin 3 3 e sin y = arctan y n+2 +2n +1y n+1 + nn +1y n =0. y0 = 0, y 0 = 1, y 0 = y 4 0 = 0, y 0 = 2. 2 y = sin 3 = 3 4 sin 1 sin 3. 4 y n = 3 4 sin + nπ 2 y = 3 3 3n 4 sin 3 + nπ 2 y = 3 + 4
5 3 y = e sin. y n =2 n/2 e sin + nπ 4 y = R, φ = dist, Z T = n=0 φ10 n 10 n R,. R, 0 φ10 n 1 2 n=0 φ10 n 10 n R. φ φ10 n R., T R. φ φ10 n + 1 = φ10 n, T. 0 <1 T. [0, 1 =0.a 1 a 2 a n 0 a n 9 0.a 1 a 2 a n 99 0.a 1 a 2 a n φ10 n = { 0.an+1 a n+2 a n a n+1 a n+2 a n+1 5 { 10 m a m =4, 9 h m = 10 m a m 4, 9 + h m =0.a 1 a 2 a ma m+1, a m = 5 { am 1 a m =4, 9 a m +1 a m 4, 9
6 a m a m =10 m h m φ10 n + h m φ10 n 0 n m = 10 n m a m a m =10 n h m n < m, a n n m a m a m= 10 n h m n < m, a n+1 5 m 1 T + h m T = ± 10n m 1 h m = ±1 h m 10 n h m n=0 m. m h m 0, m 1 ±1 T 0 <1. n=0 n=0 R f = b k cos a k π, a, b R, b < 1, ab < 1 k=0 R C 1. n b < 1 f n = b k cos a k π R f k=0. f R. n ab < 1 f n = πab k sin a k π R k=0 g = πab k sin a k π k=0. f =g f R C 1. f = i vi. 6
7 i, f f. ii f =0, f. iii f =0, f. iv f. v f. vi f. i R \{ 1, 1}. f f f = , f = ii f =0 = 3, 0, 3. f 3 < 0 = 3 f 3 = f 0 = 0, =0 f0 = 0 f 3 > 0 = 3 f 3 = iii f iv < 1 f < 0 1 <<0 f > 0 0 <<1 f < 0 lim 1 0 lim < f > 0 =0 f 0 = =, lim =, f = + 2 1, lim ± lim 1+0 y = f. 3 = =1, = = 1, 2 1 =0 y =, 2 1 7
8 v f y y y vi f.., f, f = 2 e, R 2 f = log, 0, f 2 = e 2 f = 1 log,, >0 < arctan <. 1+2 f = arctan, f, > f 2 2 = > 0. f0 = 0, f , >0 f > 0. g = arctan, g, >0 g = 2 > 0. g0 = 0, g, >0 1+2 g > 0. 0 <<π/2 2 π < sin. 8
9 f = sin 2 π, f, f cos sin = 2. g = cos sin g, 0<<π/2 g = sin <0. g0 = 0, g 0 <<π/2 g < 0. 0 <<π/2 fπ/2 = 0, f, f > 0. y 2 f = 0, 0. y 2 +y 2, y 0, 0 0, y =0, 0 2 fh, 0 f0, f 0, 0 = lim = lim =0 h 0 h h 0 h f0,k f0, f y 0, 0 = lim = lim =0 k 0 k k 0 k fh, 0 f0, 0 h f 0, 0 = lim = lim h 0 h h 0 h =0 f0,k f0, 0 f y 0, 0 = lim k 0 k k = lim k 0 k =0. sin y y sin 1, y 0, 0 2 f = 2 +y 2 0, y =0, 0 9
10 2 f 0, 0 = lim h 0 fh, 0 f0, 0 h f y 0, 0 = lim k 0 f0,k f0, 0 k = lim h 0 sin h h =1 = lim k 0 sin k k = 1 f 0, 0 = lim h 0 fh, 0 f0, 0 h f y 0, 0 = 0. h 0 sin 1 = lim 0 h h 0 h =0. sin5 + e y 2 e 2 +y 2 logy y 3 arctan 4 y y f = 5 cos5 + e y, f y = e y cos5 + e y 2 f = 2 logy+ 1 e 2 +y 2 f y = 3 f = y 2 + y 2, f y = 2y logy+ 1 y 2 + y 2 e 2 +y 2 4 f = y y y + log y, f y = y y y + log. a + by c + dy 2 log y 3 e a sin by + cos by 10
11 4 arcsin log y ad bc f = c + dy, f ad bcy 2 = c + dy 2 2 f = 1 log y, f y = log ylog y 2 3 f = ae a sin by + cos by, f y = be cos by sin by log y 4 f = 1 log y, f 2 y = y 1 log y 2. e log1 + y y 2 f0, 0 = 0, f 0, 0 = 0, f y 0, 0 = 1, f 0, 0 = 0, f y 0, 0 = 1, f yy 0, 0 = 1 e log1 + y =y y y2 + 2 f0, 0 = 1, f 0, 0 = 0, f y 0, 0 = 0, f 0, 0 = 1, f y 0, 0 = 0, f yy 0, 0 = y 2 = y 2 +. e a cos by sin y f0, 0 = 1, f 0, 0 = a, f y 0, 0 = 0, f 0, 0 = a 2, f y 0, 0 = 11
12 0, f yy 0, 0 = b 2 e a cos by = a2 2 b 2 y 2 2 f0, 0 = 0, f 0, 0 = 0, f y 0, 0 = 1, f 0, 0 = 0, f y 0, 0 = 1/2, f yy 0, 0 = sin y = y y R 2 f, y = 4 + y y 2y 2. f f, y =43 + y, f y f y, y =4y3 y +, f = f y =0, y = 2, 2,, y = 2, 2,, y =0, 0. f 2 f 2, y = 432 1, f yy 2 f y 2, y = 43y2 1 f y 2 f, y =4, y f f, y Hf, y f f y f y f yy = y 2 f, y =4 y y , 2, 2, 2, 0, 0 f 12
13 2, 2, 2, 2 det Hf 2, 2 = 384 > 0, f 2, 2 = 20 > 0 det Hf 2, 2 = 384 > 0, f 2, 2 = 20 > 0 det Hf0, 0 = 0. f, y ± 2, 2, f± 2, 2 = 8. f0, 0 = 0 = y 0 f, =2 4 > 0 =0, 0 y < 2 f0, y=y 2 y 2 2 < 0. 0, 0 f, y f0, 0 = 0 f0, 0 = 0. R 2 f, y = 3 + y y y 2. f = y, f y =3y 2 2y +, f = f y =0, y =0, 0,, y =. f, y Hf, y = 1 3, y 2 1 0, 0 3, 1 f 3 det Hf0, 0 = 3 > 0, f 0, 0 = 2 < 0 13
14 f0, 0 = 0. f det Hf 1 3, 1 = 1 < , 1 = y +3y 2 =1 2 0,y y 2,. F, y, λ = 2 + y 2 λ2 2 +2y +3y 2 1 F =2 λ4+2y, F y =2y λ2+6y, F λ = y +3y 2 1, F = F y = F λ =0 0,y 0. F + yf y, F λ = y2 0 = λ F = F y =0 λ y y 0 = y2 0 y 0 =0. 0,y 0 0, y y y =0 14
15 y0 2 = 10 4l,., y, z +y +z = l, S =2y+yz+z. F, y, z, λ=2y + yz + z λ + y + z l, F = F y = F z = F λ =0 λ = 4 l. 3 = y = z = l
( ) 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
, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f
,,,,.,,,. R f : R R R a R, f(a + ) f(a) lim 0 (), df dx (a) f (a), f(x) x a, f (a), f(x) x a ( ). y f(a + ) y f(x) f(a+) f(a) f(a + ) f(a) f(a) x a 0 a a + x 0 a a + x y y f(x) 0 : 0, f(a+) f(a)., f(x)
.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
[ ] 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 + 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 ()
1 2 1 No p. 111 p , 4, 2, f (x, y) = x2 y x 4 + y. 2 (1) y = mx (x, y) (0, 0) f (x, y). m. (2) y = ax 2 (x, y) (0, 0) f (x,
No... p. p. 3, 4,, 5.... f (, y) y 4 + y. () y m (, y) (, ) f (, y). m. () y a (, y) (, ) f (, y). a. (3) lim f (, y). (,y) (,)... (, y) (, ). () f (, y) a + by, a, b. + y () f (, y) 4 + y + y 3 + y..3.
18 ( ) 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(,,
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 ),
Chap9.dvi
.,. f(),, f(),,.,. () lim 2 +3 2 9 (2) lim 3 3 2 9 (4) lim ( ) 2 3 +3 (5) lim 2 9 (6) lim + (7) lim (8) lim (9) lim (0) lim 2 3 + 3 9 2 2 +3 () lim sin 2 sin 2 (2) lim +3 () lim 2 2 9 = 5 5 = 3 (2) lim
(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)
2017 12 9 4 1 30 4 10 3 1 30 3 30 2 1 30 2 50 1 1 30 2 10 (1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) (1) i 23 c 23 0 1 2 3 4 5 6 7 8 9 a b d e f g h i (2) 23 23 (3) 23 ( 23 ) 23 x 1 x 2 23 x
2009 I 2 II III 14, 15, α β α β l 0 l l l l γ (1) γ = αβ (2) α β n n cos 2k n n π sin 2k n π k=1 k=1 3. a 0, a 1,..., a n α a
009 I II III 4, 5, 6 4 30. 0 α β α β l 0 l l l l γ ) γ αβ ) α β. n n cos k n n π sin k n π k k 3. a 0, a,..., a n α a 0 + a x + a x + + a n x n 0 ᾱ 4. [a, b] f y fx) y x 5. ) Arcsin 4) Arccos ) ) Arcsin
i
14 i ii iii iv v vi 14 13 86 13 12 28 14 16 14 15 31 (1) 13 12 28 20 (2) (3) 2 (4) (5) 14 14 50 48 3 11 11 22 14 15 10 14 20 21 20 (1) 14 (2) 14 4 (3) (4) (5) 12 12 (6) 14 15 5 6 7 8 9 10 7
di-problem.dvi
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( ) 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
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
2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h)
1 16 10 5 1 2 2.1 a a a 1 1 1 2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h) 4 2 3 4 2 5 2.4 x y (x,y) l a x = l cot h cos a, (3) y = l cot h sin a (4) h a
= M + M + M + M M + =.,. f = < ρ, > ρ ρ. ρ f. = ρ = = ± = log 4 = = = ± f = k k ρ. k
7 b f n f} d = b f n f d,. 5,. [ ] ɛ >, n ɛ + + n < ɛ. m. n m log + < n m. n lim sin kπ sin kπ } k π sin = n n n. k= 4 f, y = r + s, y = rs f rs = f + r + sf y + rsf yy + f y. f = f =, f = sin. 5 f f =.
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
211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
「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)
高等学校学習指導要領解説 数学編
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III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2 lim. (x,y) (1,0) x 2 + y 2 lim (x,y) (0,0) lim (x,y) (0,0) lim (x,y) (0,0) 5x 2 y x 2 + y 2. xy x2 + y
III No (i) (ii) (iii) (iv) (v) (vi) x 2 3xy + 2. (x,y) (1,0) x 2 + y 2 5x 2 y x 2 + y 2. xy x2 + y 2. 2x + y 3 x 2 + y 2 + 5. sin(x 2 + y 2 ). x 2 + y 2 sin(x 2 y + xy 2 ). xy (i) (ii) (iii) 2xy x 2 +
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9 8 3............................................. 3.......................................... 4.3............................................ 4 5 3 6 3..................................................
.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g(
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,
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. p./5 [ ] x y y x x y fx) y fx) x y. p.2/5 [ ] x y y x x y fx) y fx) x y y x y x y fx) f y) x f y) f f f f. p.2/5 [ ] x y y x x y fx) y fx) x y y x y x y fx) f y) x f y) f f f f [ ] a > y a x R ). p.2/5
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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
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
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))
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
2009 IA 5 I 22, 23, 24, 25, 26, (1) Arcsin 1 ( 2 (4) Arccos 1 ) 2 3 (2) Arcsin( 1) (3) Arccos 2 (5) Arctan 1 (6) Arctan ( 3 ) 3 2. n (1) ta
009 IA 5 I, 3, 4, 5, 6, 7 6 3. () Arcsin ( (4) Arccos ) 3 () Arcsin( ) (3) Arccos (5) Arctan (6) Arctan ( 3 ) 3. n () tan x (nπ π/, nπ + π/) f n (x) f n (x) fn (x) Arctan x () sin x [nπ π/, nπ +π/] g n
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
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