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1

2 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:// 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.

3

4 i

5 ii

6 iii [note] 1 3

7 iv x 0 sin x x z = f(x, y) 1 y = f(x)

8 v

9 vi A1 1 A 13 A A α N ν B β Ξ ξ Γ γ O o Δ δ Π π E ɛ, ε P ρ Z ζ Σ σ H η T τ Θ θ Υ υ I ι Φ ϕ, φ K κ X χ Λ λ Ψ ψ M μ Ω ω

10 a, b a b y = f(x) x a b x, y Δx, Δy Δx = b a, Δy = f(b) f(a) Δy Δx Δy Δx = f(b) f(a) b a x = a x = b f(x) y = f(x) A(a, f(a)), B(b, f(b)) x = a x = a + ( 0) f(x) Δy Δx = f(a + ) f(a) [note] (difference) Δ 3.1 f(x) =x (1) x = 1 x = () x = a x = a + () Δy Δx (1) Δy Δx = f() f( 1) ( 1) = f(a + ) f(a) = ( 1) 3 = (a + ) a =1 = a + =a f(x) (1) f(x) = 4x, x = x =+ () f(x) =x 5x, x = a x = a + y = f(x) x = a x = a + Δy y = f(x) A(a, f(a)), B(a +, f(a + )) Δx l

11 6 Δx = 0 l A l l A(a, f(a)) y = f(x) A f(a + ) f(a) 0 A y = f(x) 3.1 y = x A(, 4) y = x x = x =+ Δx = 0 lim Δx 0 Δy Δx 0 ( + ) (4 + ) =4 4 f(x) x = a x = a + lim Δx 0 Δy Δx 0 f(a + ) f(a) y = f(x) x = a x = a f(x) f (a) 3.1 f (a) 0 f(a + ) f(a) y = f(x) x = a f (a) y = f(x)

12 3 7 (a, f(a)) 3. f(x) =x x 1 x =3 f (3) 0 f(3 + ) f(3) {(3 + ) (3 + ) 1} (3 3 1) 0 (4 + ) =4 0 y = x x 1 x = f(x) ( ) x (1) f(x) =x + x (x =) () f(x) =x 3 (x =1) (3) f(x) = 1 x (x = 1) (4) f(x) = x (x =4) t y = f(t) [m] Δy = f(3 + ) f(3) t =3 t =3+ Δt = [s] Δy Δt = f(3 + ) f(3) [m/s] 0 f (3) 3 [m/s] 3.3 t [m] = f(t) f (10) t =10[s] x a 0 lim {f(x) f(a)} x a x a f(x) x = a x a = f(x) f(a) x a 0 f(a + ) f(a) (x a) = f (a) 0=0

13 8 lim x a f(x) =f(a) f(x) x = a 3. f(x) x = a f(x) x = a 3.4 f(x) = x lim +0 f(0 + ) f(0) =1, lim 0 f(0 + ) f(0) 0 0 = 1 f(0 + ) f(0) lim 0 f(x) = x x =0 O y = x 3. y = f(x) I y = f(x) I y = f(x) I x = a I a a x = a f (a) f(x) f (x) 3.3 f (x) 0 f(x + ) f(x) f(x) f (x) y, dy dx, d dx f(x)

14 3 9 y = f(x) x [note] dy dx lim Δx 0 Δy dy Δx dx 3.5 y = x y (x + ) x 0 (x +x + ) x 0 (x ) =x, 3.3 dy dx =x, 0 (x + ) =x d dx (x )=x (1) y = x y =1 () y = x 3 y =3x (x) =1,(x ) =x, (x 3 ) =3x n (x n ) = nx n 1 n =1 3.3 (1) k ( ) x k = kx k 1 ( x k+1 ) (x + ) k+1 x k+1 0 (x + ) (x + ) k x x k 0 { x (x + ) k x k} + (x + ) k 0 { x (x + } )k x k +(x + ) k 0 = x lim 0 (x + ) k x k = x kx k 1 + x k =(k +1)x k + lim 0 (x + ) k = x(x k ) + x k n = k +1 (x n ) = nx n 1 n (x n ) = nx n 1

15 30 y = c (c) 0 c c =0 3.4 x n n c (x n ) = nx n 1, (c) =0 3.6 ( x 4 ) =4x 3, ( x 5 ) =5x f(x), g(x) cf(x), f(x) ± g(x) c (1) {cf(x)} = cf (x) () {f(x) ± g(x)} = f (x) ± g (x) (1) F (x) =cf(x) F (x + ) =cf(x + ) {cf(x)} F (x + ) F (x) 0 cf(x + ) cf(x) 0 {cf(x)} = cf (x) f(x + ) f(x) = c lim 0 = cf (x) () F (x) =f(x)+g(x) F (x + ) =f(x + )+g(x + ) {f(x)+g(x)} F (x + ) F (x) 0 {f(x + )+g(x + )} {f(x)+g(x)} 0 0 f(x + ) f(x) + lim 0 g(x + ) g(x) = f (x)+g (x) {f(x)+g(x)} = f (x)+g (x) {f(x) g(x)} = f (x) g (x)

16

17 f(x) ( ) (1) f(x) =x 3 +6x +3x (x = 1) () f(x) =x 4 (x =1) 10.4 f(x) c 0 x f(x) =f(0) + f (0) 1! x + f (0)! x 0 f (n+1) (c) (n + 1)! f(x) f(0) + f (0) 1! x + + f (n) (0) n! x + f (0)! x n + f (n+1) (c) (n + 1)! x n+1 0 x + + f (n) (0) n! x n x n+1 f(x) n f(x) 10.9 cos x cos x =1 x! + x4 4! +( 1) n xn (n)! + cos x cos x 1 x! x =0. cos 0. cos ! = I f(x) = 1+x

18 f(x) = 1+x f (x) = 1 (1 + x) 1 f (x) = 1 4 (1 + x) 3 f(0) = 1 f (0) = 1 f (0) = 1 4 f(x) = 1+x 1+x 1+ 1 x 1 8 x x = f(x) ( ) 3 (1) f(x) =e x ( 4 e ) () f(x) =log(1+x) (log1.1) (3) f(x) = 1 ( ) 1 1+x II (1) f(t) = 1+t () (1) L, a, x a x L ( L + x + a ) ( L + x a ) ax L (1) f (t) = t 1+t, f (t) = 1 ( 1+t ) 3 f(t) f(0) + f (0)t + 1 f (0)t =1+ 1 t () 1 L ( x ± a ) 0 (1)

19 156 4 L + ( x ± a ) { ( = L 1+ 1 x ± a )} L { L 1+ 1 { ( 1 x ± a )} } L = L + 1 ( x ± a ) L ( L + x + a ) ( L + x a ) { ( 1 x + a ) ( x a ) } L = ax L [note] (1) f(t) = 1 t () (1) x, R x R R R x x R f(x) = 1 1 x f(0.) = 1 = n =1 3 f n (x) f(x) = 1 1 x x = f 1 (x) =1+x f (x) =1+x + x f 3 (x) =1+x + x + x sin x sin x f(x) =sinx sin x = x x3 3! + x5 5! x7 7! + f(x) =sinx f 1 (x) =x, f 3 (x) =x x3 3!, f 5(x) =x x3 3! + x5 5!

20 y = f 1 (x), y = f 3 (x), y = f 5 (x) n y = f n (x) y =sinx n f(x) f n (x) = R n+1 (x) = f (n+1) (c) (n + 1)! x n+1 R n+1 (x) sin x 5 sin 1 4 sin x x x3 3! + x5 5! sin ! + 1 5! (sin x) (6) = sin x sin x 1 R 6 (1) = sin c 6! 1 6 < 1 6! = cos x 6 cos 1 5

21

22 z = f(x, y) xy xy f(x, y) f(x, y) 0 z = f(x, y) xy xy V n k k ΔS k (k =1,,...,n) (x k,y k ) f(x k,y k ) ΔS k f(x k,y k ) ΔS k n f(x k,y k ) ΔS k k=1 V

23 194 6 V n lim n k=1 n f(x k,y k ) ΔS k 1 V 1 f(x, y) 0 f(x, y) (x k,y k ) 1 f(x, y) 1 f(x, y) f(x, y) ds f(x, y) f(x, y) 13.1 k (k =1,, 3,...,n) k (x k,y k ) k ΔS k f(x, y) f(x, y) ds n k=1 n f(x k,y k ) ΔS k n

24 13. I k (1) kf(x, y) ds = k f(x, y) ds () {f(x, y)+g(x, y)} ds = f(x, y) ds + g(x, y) ds II 1, f(x, y) ds = f(x, y) ds + 1 f(x, y) ds f(x, y) 0 f(x k,y k )ΔS k 0(k =1,,...,n) f(x, y)ds , f(x, y) 0 f(x, y)ds f(x, y)ds 13.1 a x b, ϕ 1 (x) y ϕ (x) f(x, y) 0 f(x, y) ds z = f(x, y) z a x b (x, 0, 0) x A(x)

25 196 6 f(x, y) ds = b a A(x) dx A(x) z = f(x, y) x y y = ϕ 1 (x) y = ϕ (x) A(x) = f(x, y) ds = b ϕ(x) ϕ 1(x) A(x) dx = f(x, y) dy b a a ϕ 1(x) { } ϕ(x) f(x, y) dy dx f(x, y) c y d, ψ 1 (y) x ψ (y) { d } ψ(y) f(x, y) ds = f(x, y) dx dy c ψ 1(y) f(x, y) 0 x, y f(x, y) dxdy

26 f(x, y) (1) = {(x, y) a x b, ϕ 1 (x) y ϕ (x)} { b } ϕ(x) f(x, y) dxdy = f(x, y) dy dx a ϕ 1(x) () = {(x, y) c y d, ψ 1 (y) x ψ (y)} { d } ψ(y) f(x, y) dxdy = f(x, y) dx dy c ψ 1(y) ={(x, y) a x b, c y d} { b } d { d } b f(x, y) dxdy = f(x, y) dy dx = f(x, y) dx dy a c c a 13.1 (x y +x) dxdy, ={(x, y) 0 x, 1 y } y y =1 y = x x =0 x = { } (x y +x) dxdy = (x y +x) dy dx 0 1 { [ = x y ] [ ] } +x y dx 0 ( = 3 +x) x dx 0 = 3 [ x 3 ] [ + x ] 0 1 =4+4=8

27 (1) () (3) (x + y ) dxdy, = {(x, y) 0 x 1, 1 y } e x+y dxdy, = {(x, y) 0 x, 0 y 1} sin(x + y) dxdy, = { (x, y) 0 x π, 0 y π } 1 ={(x, y) a x b, ϕ 1 (x) y ϕ (x)} y ={(x, y) c y d, ψ 1 (y) x ψ (y)} x 13. (1) (y x) dxdy, = {(x, y) 1 x, x y x} () { ydxdy, = (x, y) 0 y π }, 0 x cos y (1) 1 x y = x y =x y } (y x) dxdy = = = { x x [ (y x) 3 3 (y x) dy ] x x dx 1 3 x3 dx = 1 [ x dx ] 1= 5 4

28 () 0 y π x =0 y x =cosy x } ydxdy= = π { cos y 0 π 0 [ y x [ = y sin y 0 ] cos y ] π 0 0 = π [ cos y ydx dy = π 0 ] π 0 dy π 0 sin ydy = π 1 y cos ydy 13. (1) () (3) (4) xy dxdy, = {(x, y) 0 x 1, 1 y x } xdxdy, = {(x, y) 0 y π, 0 x sin y} x dxdy, = {(x, y) 1 x, 0 y x} x + y x + ydxdy, = {(x, y) 0 y 1, 0 x y} f(x, y) = {(x, y) 0 x, 0 y 1 } x +1 f(x, y) dxdy = 0 { 1 0 x+1 } f(x, y) dy dx 1 y = 1 x +1 x x = y + ={(x, y) 0 y 1, 0 x y +} 1 { y+ } f(x, y) dxdy = f(x, y) dx dy 0 0

29

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医系の統計入門第 2 版 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, 第 2 版 1 刷発行時のものです.

医系の統計入門第 2 版 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, 第 2 版 1 刷発行時のものです. 医系の統計入門第 2 版 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. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987

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