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
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- みりあ しげい
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1 009 IA 5 I, 3, 4, 5, 6, () 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 (x) cos x [nπ, (n+)π] h n (x) g n (x) gn (x) h n (x) h n (x) Arcsin x 3. () cos (Arcsin x) () tan (Arcsin x) ( < x < ) (3) sin (Arctan x) (4) sin ( Arctan x) x 4. f(x) = x 4 5 f D 0 3x + D D x f 0 f(x) D x f 0 f(x) D 5. log( + x) 0 x < x 6. x < x x x log ( + ) = x n=0 a n x n {a n} n=0 7. x y () x y 3 () sin(x y 3 ) (3) x y (4) x + y 8. f(x, y) g(x, y) (a, b) g(x, y) = ( x y )f(x, y) g g (a, b) = x y (a, b) = 0, a + b <
2 9. R x 0 {(x, 0) x 0} D D f(x, y)? () f y (x, y) = 0 f y x f(x, y) = g(x) x g () f x (x, y) = f y (x, y) = 0 f 0. f(x, y) { ( x + y sin f(x, y) = ) x +y (x, y) (0, 0) 0 (x, y) = (0, 0) f(x, y) (0, 0) (0, 0) x y. f(x, y) f(x, y) = xy x + y (x, y) (0, 0) 0 (x, y) = (0, 0) f(x, y) (0, 0) x y 0 u (0, 0) u (0, 0) () f(x, y) (0, 0) x y () f(x, y) 0 u (0, 0) u (3) f(x, y) (0, 0). f(x, y) f(x, y) = x y x 4 + y (x, y) (0, 0) 0 (x, y) = (0, 0) f(x, y) (0, 0) (0, 0) 3. f(x, y) f(x, y) = () f(x, y) (0, 0) (x y) 3 x + y (x, y) (0, 0) 0 (x, y) = (0, 0) () f(x, y) (0, 0) x y (3) f(x, y) (0, 0) 4. f f(x, y) = { ( x + y ) sin ( ) x +y (x, y) (0, 0) 0 (x, y) = (0, 0) f (0, 0) x f x y f y (0, 0)
3 009 IA 5 I, 3, 4, 5, 6, () π 6, () π, (3) π 6, (4) 3 4 π, (5) π 6, (6) π 3 () Arcsin sin θ = π θ π θ Arcsin = π 6 () () y 0 tan x = y 0 x tan x 0 = y 0 x 0 x = x 0 + mπ m Arctan y 0 tan x = y 0 x π/ < x < π/ tan x = y 0 x Arctan y 0 x = Arctan y 0 + mπ m f n (x) (nπ π/, nπ + π/) nπ π < f n (y 0 ) < nπ + π fn (y 0 ) = Arctan y 0 + nπ y 0 fn (x) = Arctan x + nπ
4 5 () y 0 y 0 sin x = y 0 x sin x 0 = y 0, π/ x 0 π/ x 0 x = mπ + ( ) m x 0 m cos x = sin (x + π/) ( cos x = y 0 sin x + π ) = y 0 cos x = y 0 x x 0 x = mπ + ( ) m x 0 π = m π + ( ) m x 0 m sin x = y 0 x π/ x π/ Arcsin y 0 x 0 Arcsin y 0 x x x = mπ + ( ) m Arcsin y 0, x = m Arcsin y 0 + ( ) m Arcsin y 0 m nπ π g n (y 0 ) nπ + π, nπ h n (y 0 ) (n + )π n gn (y 0 ) x m = n h n (y 0 ) x m = n + gn (y 0 ) = nπ + ( ) n Arcsin y 0, h n (y 0 ) = n + π + ( ) n+ Arcsin y 0 y 0 [, ] gn (x) = ( ) n Arcsin x + nπ, h n (x) = ( ) n+ Arcsin x + n + π h 0 (x) = Arccos x Arccos x = Arcsin x + π y = x
5 5 3 y f f π Arctan x = f 0 f O x f Arcsin x = g 0 g g g y O g x h h y O h 0 = Arccos x x h 3 : f n, gn, h n () sin(arcsin x) = x cos θ sin θ cos θ π/ θ π/ 0 cos θ = sin θ π θ π π/ Arcsin x π θ Arcsin x cos (Arcsin x) = sin (Arcsin x) = x () cos(arcsin x) (cos(arcsin x)) = cos (Arcsin x) Arcsin x = sin(arcsin x) cos(arcsin x) = x + C = x x x
6 5 4 C x 0 C = 0 cos(arcsin x) = x () () tan(arcsin x) = sin(arcsin x) cos(arcsin x) = x x (3) tan (Arctan x) = x sin θ tan θ π/ θ π/ cos θ 0 cos θ = cos θ cos θ = sin θ + cos θ cos = tan θ + θ sin θ = tan θ cos θ = θ = Arctan x sin (Arctan x) = tan θ tan θ + tan (Arctan x) tan (Arctan x) + = x x + (4) sin ( Arctan x) = sin (Arctan x) cos (Arctan x) sin (Arctan x) (3) cos (Arctan x) (3) cos θ = θ = Arctan x + tan θ cos (Arctan x) = π < θ < π x + x sin ( Arctan x) = x + = x + x x +
7 5 5 () () () f(x) = x x x = + x + x + + x n + xn x x = + x + ( x ) + + ( x ) n + ( x ) n x f 0 n R n (x) = xn x ( x ) n x ( ) = x n x n ( x) x < 0 0 D x < x 5 5 log( + x) ( ) log( + x) = = ( + x) + x ( ) log( + x) = ( + x) ( log( + x) ) = ( + x) 3 ( log( + x) ) = 3 ( + x) 4. ( log( + x) ) (n) = ( ) n (n )!( + x) n
8 5 6 log( + x) 0 n p n (x) p n (x) = log( + 0) + ( + 0) x + = x x + x3 xn + ( )n 3 n ( + 0) x + a ( + 0) 3 x 3 3! lim p n(a) = log( + a) < a n + + ( )n (n )!( + 0) n x n n! x x a a > p n (a) n log( + a) a > a n lim n n = 4 0 n=0 a n n a n 0 p n (a) n log( + a) < a p n (a) log( + a) R n+ (x) = log( + x) p n (x) < a lim R n+(a) = 0 n R n+ (x) R n+(x) = + x + x x + ( ) n x n = ( + x) ( x + x + ( ) n x n ) + x = ( ) n xn + x R n+ (0) = 0 R n+ (a) = R n+ (a) R n+ (0) = R n+ (c)a = ( ) n + c a c 0 a < a < c < n c n 0 R n+ (a) 0 cn 0 e x sin x, cos x S S N = N n=0 a n a N = S N S N+ N S S = 0
9 5 7 0 e x e x x x x3 3! xn n! = ec x n+ (n + )! c 0 x e c x n+ (n + )! e x x n+ (n + )! n 0 x e x 0 x e x 4 log( + x) 0 x < x < f f = g f (k) = g (k ) f(x) 0 n f(x) = f(0) + f (0)x + f (0) x + f (0) x f (n) (0) + R n+ (x) 3! n! g(x) = f (0) + f (0)x + f (0) x + + f (n) (0) (n )! xn + R n+(x) = g(0) + g (0)x + g (0) x + + g(n ) (0) (n )! xn + R n+(x) f(x) n f n f n f n log( + x) /( + x) ( x) n = ( + x)( x + x + ( ) n x n ) + x = x + x + ( ) n x n + ( ) n xn + x () x /( + x) 0 n lim x 0 x n x n + x = lim x 0 x + x = 0 0
10 5 8 () 0 x x log( + x) = x x + x3 xn + ( )n 3 n + t n 0 + t dt log( + x) 0 n R n+ (x) = x 0 t n + t dt n 0 0 Arctan x 0 x Arctan x () x x + x = x + x 4 + ( ) n x n + ( ) n xn + x n n 0 0 x Arctan x = x x3 3 + x5 xn + ( )n 5 n + ( )n x 0 t n + t dt n = n ) t n + t tn x 0 t n + t dt x 0 t n dt = x n+ n + x n 0 x Arctan x 0 Arctan x Arctan x [, ] 0 x > 5 a > 6 < X log( + X) = X X + X3 3 X4 Xn + + ( )n 4 n +
11 5 9 5 X = /x < /x x < x ( log + ) = x x x + 3x 3 4x ( )n nx n + ( x x log + ) = x x x x + x x x 3x 3 + x 4x 4 ( )n x nx n = 3x + 4x + ( )n (n + )x n + a n = ( )n n + R n U R m f : U R m n m f n n n = n n 3 n = m = 0 () : f : U R U R (a, b) U A B f(a + u, b + v) f(a, b) Au Bv lim = 0 (u,v) (0,0) u + v (A B) (Df) (a,b) f (a, b). T (v) ( u ) v u Au + Bv Au + Bv = (A B) T (v) v T (A B) (A B) (Df) (a,b) ( ) u () u = R v d f(a + ut, b + vt) f(a + ut, b + vt) f(a, b) dt = lim t 0 t=0 t f (a, b) u f (a, b) (a, b) u (Df) (a,b) ( u) = Au + Bv A, B ()
12 5 0 (3) (a, b) (a, b) (4) f(x, y) (a, b) x f f(a + t, b) f(a, b) (a, b) := lim x t 0 t f x (a, b) f (a, b) y f f(a, b + t) f(a, b) (a, b) := lim y t 0 t f y (a, b) U (a, b) () f (a, b) (Df) (a,b) f (a, b) (a, b) f Df. i j x (, ) y (, ) (i, j) ( ) e j i x 0 ( ) 0 y (5) (a, b) x y A = f (a, b), x B = f (a, b) y A, B (). f : U R m U R n f m n f i (x,..., x n ) f p = (a,..., a n ) U (Df) p () T (Df) p ( v) = f x (p) f x (p). f m x (p) f f x (p) x n (p) f f x (p) x n (p)..... f m x (p) f m x n (p) v v. v n v = ( ) f f (a, b) (a, b) x y () (A B) (6) f f C v v. v n
13 5 7 7 () x x y 3 = xy 3 y x3 y 3 = 3x y () x sin(x y 3 ) = xy 3 cos(x y 3 ) y sin(x y 3 ) = 3x y cos(x y 3 ) (3) y xy = yx y y xy = y ey log x = (log x)e y log x = x y log x (4) x + y x = x (x + y ) = (x + y ) x x = x + y x y y x + y = y x +y (4) x y y f(x, y) 7 8 f(x, y) g(x, y) 5 (8) g(x, y) x + y g(x, y) x + y = (x, y) g(x, y) = ( x y )f(x, y) = 0 f(x, y) = 0 x +y < g(x, y) > 0 (x, y) x + y < g(x, y) < 0 (x, y) x + y < g(x, y) 0 x + y = a + b < (a, b) g(x, y) x + b < φ(x) = g(x, b) x + b <
14 5 x = a φ(x) φ (a) = 0 g(x, y) x y f(x, y) φ(x) φ (x) = g x (x, b) (a, b) g x (a, b) = 0 y ψ(y) ψ(y) = g(a, y) a + y < ψ(y) y = b ψ (b) = 0 ψ (y) = g y (a, y) g y (a, b) = 0 g g (a, b) = x y (a, b) = 0 a + b < 0 f(x, y) (a, b) f f (a, b) = (a, b) = 0 x y 9 () f(x, y) = { x x > 0 y > 0 0 f y (x, y) = 0 f(, ) = f(, ) = 0 f y () f(x, y) (x 0, y 0 ), (x, y ) f(x 0, y 0 ) = f(x, y )
15 5 3 (x 0, y 0 ) (, y 0 ) D f(x, y 0 ) x f(x 0, y 0 ) f(, y 0 ) = f x (c, y 0) c x 0 f x 0 0 f(x 0, y 0 ) = f(, y 0 ) (, y 0 ) (, y ) D f(, y) y f(, y 0 ) f(, y ) = f y (, c ) c y 0 y f y 0 0 f(, y 0 ) = f(, y ) (, y ) (x, y ) D f(, y ) = f(x, y ) f(x 0, y 0 ) = f(, y 0 ) = f(, y ) = f(x, y ) 0 ()
16 5 4 C = = = 0 0 (0, 0) C (0, 0) 0 (x, y) ( ) sin x + y x + y < ε = f(x, y) < ε f(x, y) (0, 0) x y f(x, y) x y x ( ) f(x, 0) f(0, 0) x + 0 sin ( ) x lim = lim +0 x = lim x 0 x x 0 x x 0 x sin x (0, 0) x z = x sin x x = 0 z = 0 x 0 z () f(x, y) x y x f(x, 0) f(0, 0) lim = lim x 0 x x 0 x 0 x +0 0 = lim 0 = 0 x x 0
17 5 5 x y f f (0, 0) = (0, 0) = 0 x y ( ) u () (0, 0) u = φ(t) := f(ut, vt) t v φ(t) t = 0 f φ(t) t 0 φ(t) = (ut)(vt) (ut) + (vt) = uv u + v φ(0) = 0 t 0 uv/(u + v ) t = 0 0 u v 0 uv 0 φ(t) 0 t = 0 f (0, 0) u (3) (0, 0) lim f(x, y) = f(0, 0) (x,y) (0,0) (x, y) (0, 0) f(0, 0) (x, y) (0, 0) f(0, 0) y = x (x, y) (0, 0) xx lim f(x, x) = lim x 0 x 0 x + x = 0 = f(0, 0) f(x, y) = c c (0, 0) ( ) u u = f (0, 0) u v ()
18 5 6 u φ(t) := f(ut, vt) t φ(t) t = 0 t 0 φ(t) = (ut) (vt) (ut) 4 + (vt) = u vt u 4 t + v φ(0) = 0 φ (0) v 0 φ(t) φ(0) u v lim = lim t 0 t t 0 u 4 t + v = u v v = 0 φ(t) 0 φ (0) = 0 (0, 0) (3) (x, y) (0, 0) f(x, y) f(0, 0) y = x x 0 f(x, x ) = x x x 4 + (x ) = x 0 f(0, 0) = 0 f(x, y) f(x, y) = c 3 () f(0, 0) = 0 ε x + y < δ = f(x, y) < ε δ f(x, y) = (x y) 3 ( x + y )3 x + y x + y x x + y y x + y x y x + y ( x + y ) 3 f(x, y) x + y = 8 x + y
19 5 7 δ = ε 8 x + y < δ f(x, y) 8 x + y < 8δ = ε (0, 0) () f x (0, 0), f y (0, 0) f(x, y) f f(x, 0) f(0, 0) (0, 0) = lim = lim x x 0 x x 0 x x = lim = x 0 x 3 f f(0, y) f(0, 0) ( y) 3 (0, 0) = lim = lim y y 0 y y 0 y y = lim( ) = y 0 (3) f(x, y) (0, 0) P (x, y) P (x, y) = (x 0) + ( )(y 0) + f(0, 0) = x y (x y)3 f(x, y) P (x, y) = x + y (x y) = (x y)((x y) (x + y )) x + y xy(x y) = x + y y = x, x > 0 (x, y) (0, 0) f(x, y) P (x, y) lim = lim (x,y) (0,0) x + y x +0 y= x x>0 4x 3 x x = lim x +0 4x 3 x 3 = 0 f(x, y) (0, 0)? No g(x, y) = y(3x y ) x + y (x, y) (0, 0) 0 (x, y) = (0, 0) (0, 0)
20 5 8 4 (0, 0) (0, 0) (0, 0) x y f(x, y) (0, 0) ( ) f f(x, 0) f(0, 0) (0, 0) = lim = lim x sin x x 0 x x 0 x θ sin θ ( ) 0 x sin x x ( ) lim x sin x 0 x = 0 f (0, 0) = 0 x f(x, y) x y f (0, 0) = 0 y f(0, 0) = 0 0 f(x, y) (0, 0) lim (x,y) (0,0) f(x, y) x + y = 0 f(x, y) lim x + y sin (x,y) (0,0) ( ) x + y 0 f(x, y) (0, 0) f x (x, y) (0, 0) (x, y) (0, 0) f x (x, y) ( ) f (x, y) = x sin x x + y x ( ) x + y cos x + y (0, 0) ε δ a + b < δ f x (a, b) f x (0, 0) ε (a, b)
21 5 9 ( f x (a, b) f x (0, 0) = a sin a + b ) a ( ) a + b cos a + b (a, b) a = 0 0 b = 0 b = 0 ( ) f x (a, 0) f x (0, 0) = a sin a ( ) a cos a a a n = nπ a = a nπ a > 0 f x ( ), 0 nπ f x (0, 0) = nπ n a δ n /(δ π) a = / nπ a + 0 < δ f x (a, 0) f x (0, 0) = nπ f x (x, y) (0, 0) f(x, y) x y f y (x, y) (0, 0) C 0 z = x sin x x = 0 z = 0 x 0 z C
22 5 0 C :
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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
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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II 1 3 2 5 3 7 4 8 5 11 6 13 7 16 8 18 2 1 1. x 2 + xy x y (1 lim (x,y (1,1 x 1 x 3 + y 3 (2 lim (x,y (, x 2 + y 2 x 2 (3 lim (x,y (, x 2 + y 2 xy (4 lim (x,y (, x 2 + y 2 x y (5 lim (x,y (, x + y x 3y
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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.....................................
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1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1
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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,
I, II 1, 2 ɛ-δ 100 A = A 4 : 6 = max{ A, } A A 10
1 2007.4.13. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 0. 1. 1. 2. 3. 2. ɛ-δ 1. ɛ-n
I y = f(x) a I a x I x = a + x 1 f(x) f(a) x a = f(a + x) f(a) x (11.1) x a x 0 f(x) f(a) f(a + x) f(a) lim = lim x a x a x 0 x (11.2) f(x) x
11 11.1 I y = a I a x I x = a + 1 f(a) x a = f(a +) f(a) (11.1) x a 0 f(a) f(a +) f(a) = x a x a 0 (11.) x = a a f (a) d df f(a) (a) I dx dx I I I f (x) d df dx dx (x) [a, b] x a ( 0) x a (a, b) () [a,
() 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 ()
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 ),
, 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,
No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y
No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ
(2000 )
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f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a
3 3.1 3.1.1 A f(a + h) f(a) f(x) lim f(x) x = a h 0 h f(x) x = a f 0 (a) f 0 (a) = lim h!0 f(a + h) f(a) h = lim x!a f(x) f(a) x a a + h = x h = x a h 0 x a 3.1 f(x) = x x = 3 f 0 (3) f (3) = lim h 0 (
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
( ) 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
1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x
![, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x](/thumbs/93/113428934.jpg ", 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x")
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9.. x + y + 0. x,y, x,y, x r cos θ y r sin θ xy x y x,y 0,0 4. x, y 0, 0, r 0. xy x + y r 0 r cos θ sin θ r cos θ sin θ θ 4 y mx x, y 0, 0 x 0. x,y 0,0 x x + y x 0 x x + mx + m m x r cos θ 5 x, y 0, 0,
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A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
Chap11.dvi
. () x 3 + dx () (x )(x ) dx + sin x sin x( + cos x) dx () x 3 3 x + + 3 x + 3 x x + x 3 + dx 3 x + dx 6 x x x + dx + 3 log x + 6 log x x + + 3 rctn ( ) dx x + 3 4 ( x 3 ) + C x () t x t tn x dx x. t x
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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 +
(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
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 =
5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 0 g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)
5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h 0 f(a + h, b) f(a, b) h............................................................... ( ) f(x, y) (a, b) x A (a, b) x
29
9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n
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. 00 3 9 [] sinh x = ex e x, cosh x = ex + e x ) sinh cosh 4 hyperbolic) hyperbola) = 3 cosh x cosh x) = e x + e x = cosh x ) . sinh x sinh x) = e x e x = ex e x = sinh x 3) y = cosh x, y = sinh x y =
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))
no35.dvi
p.16 1 sin x, cos x, tan x a x a, a>0, a 1 log a x a III 2 II 2 III III [3, p.36] [6] 2 [3, p.16] sin x sin x lim =1 ( ) [3, p.42] x 0 x ( ) sin x e [3, p.42] III [3, p.42] 3 3.1 5 8 *1 [5, pp.48 49] sin
2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p
2012 IA 8 I 1 10 10 29 1. [0, 1] n x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 2. 1 x = 1 (n = 1, 2, 3,...) 2 f(x) = n 0 [0, 1] 1 0 f(x)dx 3. < b < c [, c] b [, c] 4. [, b] f(x) 1 f(x) 1 f(x) [, b] 5.
1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ
1 (1) ( i ) 60 (ii) 75 (iii) 15 () ( i ) (ii) 4 (iii) 7 1 ( () r, AOB = θ 0 < θ < ) OAB A OB P ( AB ) < ( AP ) (4) 0 < θ < sin θ < θ < tan θ 0 x, 0 y (1) sin x = sin y (x, y) () cos x cos y (x, y) 1 c
S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt
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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
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(,,
φ s i = m j=1 f x j ξ j s i (1)? φ i = φ s i f j = f x j x ji = ξ j s i (1) φ 1 φ 2. φ n = m j=1 f jx j1 m j=1 f jx j2. m
2009 10 6 23 7.5 7.5.1 7.2.5 φ s i m j1 x j ξ j s i (1)? φ i φ s i f j x j x ji ξ j s i (1) φ 1 φ 2. φ n m j1 f jx j1 m j1 f jx j2. m j1 f jx jn x 11 x 21 x m1 x 12 x 22 x m2...... m j1 x j1f j m j1 x
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all.dvi
38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t
t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ
4 5 ( 5 3 9 4 0 5 ( 4 6 7 7 ( 0 8 3 9 ( 8 t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ S θ > 0 θ < 0 ( P S(, 0 θ > 0 ( 60 θ
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S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d
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1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy
f f x, y, u, v, r, θ r > = x + iy, f = u + iv C γ D f f D f f, Rm,. = x + iy = re iθ = r cos θ + i sin θ = x iy = re iθ = r cos θ i sin θ x = + = Re, y = = Im i r = = = x + y θ = arg = arctan y x e i =
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 +
. p.1/15
. 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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[] x < T f(x), x < T f(x), < x < f(x) f(x) f(x) f(x + nt ) = f(x) x < T, n =, 1,, 1, (1.3) f(x) T x 2 f(x) T 2T x 3 f(x), f() = f(t ), f(x), f() f(t )
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1 1.1 [] f(x) f(x + T ) = f(x) (1.1), f(x), T f(x) x T 1 ) f(x) = sin x, T = 2 sin (x + 2) = sin x, sin x 2 [] n f(x + nt ) = f(x) (1.2) T [] 2 f(x) g(x) T, h 1 (x) = af(x)+ bg(x) 2 h 2 (x) = f(x)g(x)
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED)
rational number p, p, (q ) q ratio 3.14 = 3 + 1 10 + 4 100 ( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED) ( a) ( b) a > b > 0 a < nb n A A B B A A, B B A =
II (10 4 ) 1. p (x, y) (a, b) ε(x, y; a, b) 0 f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a 2 x a 0 A = f x (
II (1 4 ) 1. p.13 1 (x, y) (a, b) ε(x, y; a, b) f (x, y) f (a, b) A, B (6.5) y = b f (x, b) f (a, b) x a = A + ε(x, b; a, b) x a x a A = f x (a, b) y x 3 3y 3 (x, y) (, ) f (x, y) = x + y (x, y) = (, )
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di-problem.dvi
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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.)
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arctan arctan arctan arctan 2 2000 π = 3 + 8 = 3.25 ( ) 2 8 650 π = 4 = 3.6049 9 550 π = 3 3 30 π = 3.622 264 π = 3.459 3 + 0 7 = 3.4085 < π < 3 + 7 = 3.4286 380 π = 3 + 77 250 = 3.46 5 3.45926 < π < 3.45927
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![(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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Chap10.dvi
=0. f = 2 +3 { 2 +3 0 2 f = 1 =0 { sin 0 3 f = 1 =0 2 sin 1 0 4 f = 0 =0 { 1 0 5 f = 0 =0 f 3 2 lim = lim 0 0 0 =0 =0. f 0 = 0. 2 =0. 3 4 f 1 lim 0 0 = lim 0 sin 2 cos 1 = lim 0 2 sin = lim =0 0 2 =0.
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x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s
... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z
5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................
5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)
II 2 II
II 2 II 2005 yugami@cc.utsunomiya-u.ac.jp 2005 4 1 1 2 5 2.1.................................... 5 2.2................................. 6 2.3............................. 6 2.4.................................
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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 =
M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 -
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f : R R f(x, y) = x + y axy f = 0, x + y axy = 0 y 直線 x+y+a=0 に漸近し 原点で交叉する美しい形をしている x +y axy=0 X+Y+a=0 o x t x = at 1 + t, y = at (a > 0) 1 + t f(x, y
017 8 10 f : R R f(x) = x n + x n 1 + 1, f(x) = sin 1, log x x n m :f : R n R m z = f(x, y) R R R R, R R R n R m R n R m R n R m f : R R f (x) = lim h 0 f(x + h) f(x) h f : R n R m m n M Jacobi( ) m n
f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f
22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )
Acrobat Distiller, Job 128
(2 ) 2 < > ( ) f x (x, y) 2x 3+y f y (x, y) x 2y +2 f(3, 2) f x (3, 2) 5 f y (3, 2) L y 2 z 5x 5 ` x 3 z y 2 2 2 < > (2 ) f(, 2) 7 f x (x, y) 2x y f x (, 2),f y (x, y) x +4y,f y (, 2) 7 z (x ) + 7(y 2)
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
IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + (
IA 2013 : :10722 : 2 : :2 :761 :1 23-27) : : 1 1.1 / ) 1 /, ) / e.g. Taylar ) e x = 1 + x + x2 2 +... + xn n! +... sin x = x x3 6 + x5 x2n+1 + 1)n 5! 2n + 1)! 2 2.1 = 1 e.g. 0 = 0.00..., π = 3.14..., 1
x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n
![x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n](/thumbs/92/108343288.jpg "x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n")
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II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R
![II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R](/thumbs/95/125540821.jpg "II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R")
II Karel Švadlenka 2018 5 26 * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* 5 23 1 u = au + bv v = cu + dv v u a, b, c, d R 1.3 14 14 60% 1.4 5 23 a, b R a 2 4b < 0 λ 2 + aλ + b = 0 λ =
< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3)
< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3) 6 y = g(x) x = 1 g( 1) = 2 ( 1) 3 = 2 ; g 0 ( 1) =
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
B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 (
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. 28 4 14 [.1 ] x > x 6= 1 f(x) µ 1 1 xn 1 + sin + 2 + sin x 1 x 1 f(x) := lim. 1 + x n (1) lim inf f(x) (2) lim sup f(x) x 1 x 1 (3) lim inf x 1+ f(x) (4) lim sup f(x) x 1+ [.2 ] [, 1] Ω æ x (1) (2) nx(1
高校生の就職への数学II
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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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2010 IA ε-n I 1, 2, 3, 4, 5, 6, 7, 8, ε-n 1 ε-n ε-n? {a n } n=1 1 {a n } n=1 a a {a n } n=1 ε ε N N n a n a < ε
00 IA ε-n I,, 3, 4, 5, 6, 7, 8, 9 4 6 ε-n ε-n ε-n? {a } = {a } = a a {a } = ε ε N N a a < ε ε-n ε ε N a a < ε N ε ε N ε N N ε N [ > N = a a < ε] ε > 0 N N N ε N N ε N N ε a = lim a = 0 ε-n 3 ε N 0 < ε
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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
B line of mgnetic induction AB MN ds df (7.1) (7.3) (8.1) df = µ 0 ds, df = ds B = B ds 2π A B P P O s s Q PQ R QP AB θ 0 <θ<π
8 Biot-Svt Ampèe Biot-Svt 8.1 Biot-Svt 8.1.1 Ampèe B B B = µ 0 2π. (8.1) B N df B ds A M 8.1: Ampèe 107 108 8 0 B line of mgnetic induction 8.1 8.1 AB MN ds df (7.1) (7.3) (8.1) df = µ 0 ds, df = ds B