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1 Ver 8 ( c T. Miw, ) - 1 -

2 : , 2.0, 3.0, 2.0, (%) ( 2.0) ( 3.0) + ( 2.0) {21, 25, 20, 21, 28} 5 x 1, x 2, x 3, x 4, x 5 x x x 1 + x 2 + x 3 + x 4 + x x j - 2 -

3 1.2 {30, 19, 25, 27, 29} : n x x 1 + x x n n 1 n x j n x 1, x 2,..., x n (x j x) 0 n x 1, x 2,..., x n x 1/n : V[ x] σ2 n 1/n (1) (2) (3) ( ) ( ) (x j + y j ) x j + y j ( ) (c x j ) c x j c n c - 3 -

4 n x 1, x 2,..., x n x j x 1 + x x n (1.1) (1.1) x j j 1 n 1 n x 1, x 2,..., x n T x : T x 1 + x x n : x x 1 + x x n n x j (1.2) 1 n x j T n (1.3) 2 x j j (j 1, 2,..., n) x j j 2 (j 1, 2,..., n) 1 n 2 j n j n 2 n (n + 1) 2 n (n + 1)(2n + 1) x j j i j k x i x k i, j, k, l,... k1 s t x i x j x k x s x α x 1 + x x n k1 s1 α1 j j 2-4 -

5 n x j j j x j x j x 2 x 1,..., x n x x n c, j y 1,..., y n n ( ) ( ) (1) (x j + y j ) x j + y j (1.4) (2) (3) (4) ( ) (c x j ) c x j c n c ( ) ( ) (c x j + y j ) c x j + y j (1.5) (1.6) (1.7) (1), (2), (3) 2 (1) (2) (4) (1) (2) (4) 3 x 1,..., x n x j x x j x (j 1,..., n) (evition) n (x j x) 0 (1.8) 3 (1.8) (sum of squres) (x j x) 2 x 2 j n x 2 x 2 j T 2 n (1.9) 4 (1.9) 4 n x j x (j 1,..., n) 2 n (1.8) 1 n 1 x 1 x,..., x n 1 x x n x n n 1 n 1 (egrees of freeom) - 5 -

6 (1.9) n x2 j T 2 /n n (x j x) 2 5 c, y j c x j + (j 1,..., n) x j c y j ȳ c x + (1.10) (y j ȳ) 2 c 2 (x j x) 2 (1.11) y j 5 (1.10) (1.11) {p 1,..., p n } {x 1,..., x n } p j (j 1,..., n) p j 0, p j 1 2 µ σ p 1,..., p n ( p j 1) 1 2 j n p 1 p 2 p j p n x 1 x 2 x j x n y 1 y 2 y j y n y j c x j + x j p j µ (1.12) (x j µ) 2 p j σ (1.13) (x j µ) p j 0 (1.14) σ 2 x 2 j p j µ 2 (1.15) 6 6 (1.14) (1.15) c, y j c x j + (j 1,..., n) y j (1.12) (1.13) µ y y j p j σ 2 y (y j µ y ) 2 p j - 6 -

7 y y j µ, σ 2 µ y c µ + σ 2 y c 2 σ 2 (1.16) (1.17) σy (1.16) (1.17) m n i j x ij (i 1,..., m; j 1,..., n) m n 1.3x ij 1 i i 1 m 2 j j 1 n 1.3 m n x ij j i 1 2 j n 1 x 11 x 12 x 1j x 1n T 1. x 1. 2 x 21 x 22 x 2j x 2n T 2. x i x i1 x i2 x ij x in T i. x i m x m1 x m2 x mj x mn T m. x m. T. 1 T. 2 T. j T. n T.. x. 1 x. 2 x. j x. n x.. i T i. x i. (i 1,..., m) T i. x ij x i1 + x i2 + + x in x i. T i./n. T i. x ij i j j T. j x. j (j 1,..., n) T. j x ij x 1j + x 2j + + x mj x. j T. j /m T.. ( ) T.. T i. x ij T. j ( m ) x ij x ij x ij - 7 -

8 i j m n x.. x.. T.. m n m n x ij x.. x ij x.. (x ij x..) 2 (x ij x i.) 2 + n ( x i. x..) 2 (1.18) (1.18) 9 8 x ij i j i n i n 1, n 2,..., n m 1 n 1 x 1j (j 1,..., n 1 ) 2 n 2 x 2j (j 1,..., n 2 ) i n i x ij (j 1,..., n i ) x ij x ij n i 1 x 11 x 1n1 n 1 T 1. x 1. T 1./n 1 2 x 21 x 2n2 n 2 T 2. x 2. T 2./n 2.. i x i1 x ini n i T i. x i. T i./n i..... m x m1 x mnm n m T m. x m. T m./n m N T.. x.. T../N T i. x i. n i T i. x ij x i1 + x i2 + + x ini x i. T i. n i ( n i ) T.. T i. x ij 2 n i i j T i. x ij i T.. T i. ) x ij ( n i

9 N n i n 1 + n n m x.. T.. N x ij x.. n i (x ij x..) 2 n i (x ij x i.) 2 + n i ( x i. x..) 2 (1.19) (1.19) i 1 m j 1 n i j x ij i p i j q j x ij p i q j x ij x ij ( m ) ( ) p i q j p i q j (1.20) 10 (1.20) x j j 1 n n 1 x j x 0 + x x n 1 j0 j p j p 0 + p p n + j0-9 -

10 1.3 f(x) b 1.1 f(x) 0 ( x b) K f(x) x K f(x) 0 ( x b) 1.2 K f(x) x K f(x) K 0 f(x) K b 0 b f(x) f(x) x K 1 K 2 + K 3 f(x) K 1 K 3 0 K 2 b 1.3 f(x), b f(x) x f(t) t f(z) z i k f(x) f(x) F (x) F (x) f(x) x f(x) b f(x) x F (b) F () [ F (x) ] b (1.21)

11 F (b) F () [ F (x) ] b c, x g(x) (1) (2) (3) (4) {f(x) + g(x)} x {c f(x)} x c f(x) x + g(x) x (1.22) f(x) x (1.23) c x c (b ) (1.24) {c f(x) + g(x)} x c f(x) x + g(x) x (1.25) (1.21) (1) - (4) 1.2 (1) - (4) (4) (1) (2) (1) (2) (4) 1.2 (1) - (4) 11 f(x) 0 ( x b), f(x) x 1 f(x) 2 µ σ 2 x f(x) x (1.26) (x µ) 2 f(x) x (1.27) σ 2 (x µ) f(x) x 0 (1.28) x 2 f(x) x µ 2 (1.29) (1.28) (1.29) c, y g(x) c x + y g(x) (1.26) (1.27) µ y σ 2 y y f(x) x (y µ y ) 2 f(x) x g(x) f(x) x (g(x) µ y ) 2 f(x) x

12 µ y c µ + σ 2 y c 2 σ 2 (1.30) (1.31) σy (1.30) (1.31) c x F (x) x c f(t) t (1.32) t x x F (x) x f(x) f(t) F (x + x) F (x) c x x + x F (x) f(x) (1.33) x F (x + x) F (x) F (x) lim x x 0 x F (x + x) F (x) 1.4 x f(x) x 0 f(x) (1.32) F (x) b f(x) x F (b) F () [ F (x) ] b 1.5 F (x) f(x) G(x) G(x) f(x) x {G(x) F (x)} f(x) f(x) 0 x G(x) F (x) C C G(x) F (x) + C [ ] b b G(x) G(b) G(b) F (b) F () f(x) x f(x) 1.5 c F (b) F () b

13 F (x) f(x) x f(x) x F (b) F () [ F (x) ] b f(x) x x b 0 f(x) x f(x) x 0 f(x) x lim b f(x) x lim b 0 f(x) x f(x) x f(t) F (x) 1 t 0 x < t < 0 f(t) 1 0 t < t < x F (x) x f(t) t f(t) t 0 t 1 f(t) x t x 1.7 x 0 < x 0 F (x) f(t) t x 0 x x <

14 1.4 y f(x) x x f(x) lim f(x + x) f(x) x 0 x (1.34) (x, f(x)) 1.8 y x, f (x), f(x) f (x) f(x) 1.8 x x x f(x) f(x) f(x) f(x) x (1.34) f(x) x f(x) x x > 0 x 0 x < 0 x 0 f(x) x 1.9 f(x) x x 0 f(0) 0 f(0 + x) f(0) lim x>0 x x 0 lim x<0 x 0 f(0 + x) f(0) x x lim x>0 x 0 lim x<0 x 0 x lim x>0 x 0 x x lim x<0 x 0 x x 1 x x f(x) x 0 x f(x) x x 0 c, x g(x) (1) {f(x) + g(x)} f (x) + g (x) (1.35) (2) {c f(x)} c f (x) (1.36) (3) x c 0 (1.37) (4) {c f(x) + g(x)} c f (x) + g (x) (1.38) (1) (2)

15 : (x 1,..., x n ) f(x)... :... (1) ( 1 + 2) ( 1) + ( 2) (2) ( ) () (1) (2) (liner) (1) (2) (4) (4) ( ) 1 ( 1) + 2 ( 2) (1) (2) (4) 3 () {f(x) g(x)} f (x) g(x) + f(x) g (x) (1.39) { 1 } g (x) (b) g(x) g(x) 2 (1.40) { f(x) } f (x) g(x) f(x) g (x) (c) g(x) g(x) 2 (1.41) () y f(z), z g(x) y f(g(x)) y x y z z (1.42) x (e) x f 1 1 (x) f (f 1 (1.43) (x)) 13 (1.39) - (1.43) 13 (1.39) (1.21) f (x) g(x) x ( ) {f(x) g(x)} f(x) g (x) x {f(x) g(x)} x f(x) g (x) x [ f(x) g(x) ] b f(x) g (x) x (1.44) 14 (1.42) f(z) z c 13 f(g(x)) g (x) x (1.45) z g(x), g(c), b g() F (z) f(z) F (z) z f(z) z F (b) F ()

16 z g(x), g(c), b g() F F (g(x)) x z z x f(z) g (x) f(g(x)) g (x) c f(g(x)) g (x) x F (g()) F (g(c)) F (b) F () f(z) z 14 x xn n x n 1 n x x x 1 x ex e x, x x (ln ) x x ln x 1 x sin x cos x, x cos x sin x, x 1 x sin 1 (x) 1 x 2 ( 1 < x < 1) ( 1 ) x x 2 x, 1/2 x tn x 1 cos 2 x f(x) f (x) x f (x) < 0 f (x) 0 f (x) > 0 f(x) f(x) f(x) f(x) x b 1) f(x) x 3 x 0 2) 1.10 f(x) 1.10 b 15 n x 1,..., x n f() (x j ) 2 (1.46)

17 x 1,..., x n f() f() f() (x j ) 2 (1) (x j ) 2 y z 2, z (x j ) 2 z 2 (x j ) (x j ) z x j ( 2) (x j ) 2 ( n x j ) 2 (n x n ) 2 n ( x) x f () 0 f() < x x > x f () 0 + f() (x j x) 2 n (x j x) 0 f() f() (x j ) 2 (x j x) 2 + (x j x) 2 + (x j x + x ) 2 ( x ) ( x ) ( x ) 2 (x j x) (x j x) 2 + n ( x ) x f() f() 2 f() x f() n x 1,..., x n x x j 2 f() (x j ) exp(x) e x e x > 1 + x (x > 0) (1.47) e x > 1 + x + x2 2 (x > 0) (1.48)

18 e 0 1, e x > 1 (x > 0), f 1 (x) e x (1 + x) f 1 (0) e f 1(x) e x 1 > 0 (x > 0) x ex e x 1.6 x > 0 f 1 (x) > 0 f 1(x) > 0 (x > 0) (1.47) f 2 (x) e x (1 + x + x2 2 ) f 2 (0) e f 2(x) e x (1 + x) f 1 (x) > 0 (x > 0) f 2 (x) > 0 (x > 0) (1.48) x e x x e x < x 0 (x ) (1.49) 1 + x + x2 2 n e x > 1 + x + x xn n! x n e x 0 (x ) (x > 0) e x x x n ln z z 1 ln z z 0 1 lim 0 ln z z 1 x ln z (z e x ) e c 0 (c ) z x ex 1 ln z z 0 c x z x x 0 c x e x x f (x) e x, g(x) x [ x e x] 0 0 c e x x [ x e x] 0 c [ e x] 0 c c c e c 1 + e c 1 (c )

19 1.5 2 x, y z f(x, y) y y b f(x, b) x x f(x + x, b) f(x, b) f(x, b) lim x x 0 x (1.50) x x y b 1.11 x b b y f(x + x, y) f(x, y) f(x, y) lim x x 0 x (1.51) f(x, y) f(x, b) b 1.11 x y b (1.50) (1.51) x y 18 f(x, y) x 2 + 3x y + 2y 2 y f(x, y) 2x + 3y x y x f(x, y) 3x + 4y y y 18 (x, y) D f(x, y) f(x, y) 1.11 x, y f(x, y) 0, x f(x, y) 0 y D n (x 1, y 1 ), (x 2, y 2 ),..., (x n, y n ) b f(, b) (y j b x j ) 2 (1.52) x j, y j (j 1,..., n) b 14 ˆb f(, b) b b (x j x) (y j ȳ), â ȳ ˆb x (x j x)

20 ˆb 4 x 1,..., x n ˆb, (x 1, y 1 ),..., (x n, y n ) 20 19, b f(, b) (y j b x j ) 2 (y j â ˆb x j ) 2 + (y j â ˆb x j + â + ˆb x j b x j ) 2 19 { } 2 (â ) + (ˆb b) xj (y j â ˆb x j ) 2 f(â, ˆb) â, b ˆb f(, b) y j â ˆb x j y j ȳ ˆb (x j x) {y j ȳ ˆb (x j x)} 0 {y j ȳ ˆb (x j x)} x j {y j ȳ ˆb (x j x)} (x j x) (1.52) f(, b) b b y j b x j z j f(, b) f(, b) (y j b x j ) 2 (z j ) 2 15 b â z ȳ b x f(, b) g(b) (z j z) 2 {y j ȳ b (x j x)} 2 b b b 2 ˆb (x j x)(y j ȳ) (x j x) 2 g(b)

21 1.6 ( > 0, 1) f(x) f(x) x > 0 ( < x < ) (1.53) 1 x 1 x 1 x ( < x < ) f(x) 0.5 x e e x 2 2 x (1) f( x) x 1 x (2) f(0) x (3) > 1 f(x) x lim x x 0 lim x x (4) 0 < < 1 f(x) x lim x x lim x x 0 (5) x, y f(x + y) x+y x y f(x) f(y) e f(x) e x exp(x) ( < x < ) (1.54) e x exp(x) e x exp(x) x ex e x (1.55) x (1.55) e φ(x) 1 e x2 /2 1 exp ( x2 ) 2π 2π φ(x) 1/ 2π (1.56) x

22 exp( x 2 /2) x 0 lim x exp( x2 /2) 0 lim x exp( x2 /2) 0 (1.55) (1.42) exp( x 2 /2) x exp ( x2 2 ) exp ( x2 2 ) ( x 2 x 2 ) x exp ( x2 2 ) 22 ( > 0, 1) x y x y y f(x) log x (1.57) x > 0 f(x) log x y ( < y < ) f(x) log 2 x ln x log 0.5 x (y log x, x y ) (1) f(1/x) log (1/x) log x (2) f(1) log 1 0 (3) > 1 f(x) log x lim log x x 0 lim log x x (4) 0 < < 1 f(x) log x lim log x x 0 lim log x x (5) x > 0, y > 0 f(x y) log (x y) log x + log y f(x) + f(y) 10 log 10 x log 10 x log x e log e x ln x log e x ln x log x ln x log 10 x x

23 f(x) ln x x ln x 1, (x 0) (1.58) x x ln x 1 x x ex e x 1.7 n r n P r n P r np r n (n 1) (n 2) (n r + 1) }{{} r (1.59) A 3 n A 1 A 2 A n n A 1, A 2,..., A n r r 1 2 r n n 1 n r A 3 A 1 A 4 A 4 A 3 r r 3 r A n 1 2 n 1 r r 1 n (r 1) n r np n n (n 1) (n 2) n! (1.60) n! 1 n n n P r np r n (n 1) (n 2) (n r + 1) n! (n r)! (1.61) 0! 1 15 (1.61)

24 n r r n C r n C r ( ) n nc r n P r n! r r! r! (n r)! (1.62) n P r 1 r r! n C r n P r /r! 1.15 r 3 A 1, A 3, A 4 3! 6 {A 1, A 3, A 4 }, {A 1, A 4, A 3 }, {A 3, A 1, A 4 }, {A 3, A 4, A 1 }, {A 4, A 1, A 3 }, {A 4, A 3, A 1 } ( ) n n C r ( ) r n r ( ( n r) n r) r 0 nc 0 n! 0! (n 0)! n! 0! n! 1 (1.62) ( ) ( ) n n nc r n C n r, r n r r n 10 C 8 10C 8 10 C 2 10 P 2 2! (1.63) 2 (p + q) n n {}}{ (p + q) (p + q) (p + q) nc r p r q n r (1.64) r0 (p + q) n n + 1 p r q n r (r 0, 1,..., n) p r q n r n (p + q) r p n C r p r q n r n C r

25 1.8 [] / [] α A lph β B bet / γ Γ gmm δ elt ε, ɛ E epsilon -lòn, ] / ζ Z zet / η H et / θ Θ thet / ι I iot κ K kpp λ Λ lmb µ M mu [mju:] / [mju:, mu:] ν N nu [nju:] / [nu:, nju:] ξ Ξ xi [si, ksi, gzi, zi] / [zi, si] o O omicron Ómik-] / [ÓmikrÒn, óumikròn] π Π pi [pi] ρ P rho [rou] σ Σ sigm τ T tu [tu, to:] υ Υ upsilon -lòn, 2ps-] / [ φ, ϕ Φ phi [fi] χ X chi [ki] ψ Ψ psi [psi] / [si, psi] ω Ω omeg / α β γ δ ε ζ η θ ι κ λ µ ν ξ o π ρ σ τ υ φ χ ψ ω γ χ r x γ r χ x X λ 1 RAMUDA ?

26 1.9 1 (1) (x j + y j ) (x 1 + y 1 ) + + (x n + y n ) ( ) ( ) (x x n ) + (y y n ) x j + y j (2) (3) ( ) (c x j ) c x c x n c (x x n ) c x j c c } + {{ + } c n c n 2 ( ) ( ) (4) (c x j + y j ) c x j + y j ( ) ( ) c x j + y j (1) (2) (4) c 1 (1) 0 (2) 3 (x j x) x j x T n x 0 1 (4) 2 (3) 3 4 x j x j n x (x j x) 2 (x 2 j 2 x x j + x 2 ) x 2 j 2 x x j + x 2 x 2 j 2 x (n x) + n x 2 x 2 j n x 2 x 2 j T 2 /n 5 ȳ 1 y j 1 (c xj + ) c xj + 1 c x + n n n n (y j ȳ) 2 {(c x j + ) (c x + )} 2 {c (x j x)} 2 c 2 (x j x)

27 6 p j 1 (x j µ) p j x j p j µ p j µ µ p j µ µ 0 σ 2 (x j µ) 2 p j (x 2 j 2µ x j + µ 2 ) p j x 2 j p j 2µ x j p j + µ 2 p j x 2 j p j 2µ µ + µ 2 x 2 j p j µ 2 7 µ y σy 2 y j p j (c x j + ) p j c x j p j + p j c µ + (y j µ y ) 2 p j {(c x j + ) (c µ + )} 2 p j {c (x j µ)} 2 p j c 2 (x j µ) 2 p j c 2 σ 2 8 (x ij x..) 2 (x ij x i. + x i. x..) 2 (x ij x i.) (x ij x i.) 2 + n ( x i. x..) 2 (x ij x i.)( x i. x..) + ( x i. x..) 2 x i. x.. j (x ij x i.)( x i. x..) ( x i. x..) (x ij x i.) 0 ( x i. x..) 2 n ( x i. x..) 2 9 n i (x ij x..) 2 n i (x ij x i. + x i. x..) 2 n i n i (x ij x i.) (x ij x i.)( x i. x..) + n i (x ij x i.) 2 + n i ( x i. x..) 2 n i ( x i. x..)

28 8 n i n i (x ij x i.)( x i. x..) 0 ( x i. x..) 2 n i ( x i. x..) 2 10 ( ) x ij ( ) p i q j p i j ) (p i q j q j i ( m ) ( ) p i q j (1.39) (1.42) { } f(x + x) g(x + x) f(x) g(x) f(x) g(x)) lim x x 0 x f(x + x) g(x + x) f(x) g(x + x) + f(x) g(x + x) f(x) g(x) lim x 0 x {f(x + x) f(x)} g(x + x) f(x){g(x + x) g(x)} lim + lim x 0 x x 0 x f (x) g(x) + f(x) g (x) (1.40) (1.41) (1.39) (1.42) y f(z), z g(x) g(x + x) z + z f(g(x + x)) f(g(x)) x f(g(x + x)) f(g(x)) g(x + x) g(x) g(x + x) g(x) x f(z + z) f(z) g(x + x) g(x) y z x z z x f(f 1 (x)) x x 14 f(, b) n (y j b x j ) 2 b f(, b) 2 (y j b x j )

29 b b f(, b) 2 x j (y j b x j ), b â, ˆb (1) (2) (yj â ˆb x j ) n ȳ n â n ˆb x 0 xj (y j â ˆb x j ) x j y j n â x ˆb x 2 j 0 (1) (3) â ȳ ˆb x (3) (2) (4) xj y j n x ȳ ˆb ( x 2 j n x 2 ) x 2 j n x 2 (x j x) 2 (xj x)(y j ȳ) x j y j x y j ȳ x j + n x ȳ x j y j n x ȳ (4) (5) ˆb (x j x)(y j ȳ) (x j x) 2 15 np r n (n 1) (n 2) (n r + 1) n (n 1) (n 2) (n r + 1) (n r) 2 1 (n r) 2 1 n! (n r)!

基礎数学I

基礎数学I I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............

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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.   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)

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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 = 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

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