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 + y 1 + z a b c z f(x, b),z f(a, y) 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 x r cos θ, y r sin θ xy sin θ x + y 1
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 x r cos θ, y r sin θ xy sin θ x + y f(x, y) A (x, y) (a, b) f(x, y) A lim (x,y) (a,b) (x, y) (a, b) (x a) +(y b) < (x a) +(y b) r f(x, y) A M(r) lim M(r) r + (ε-δ r δ, M(r) ε ) f(x, y) xy xy ab (x a)(y b)+(x a)b+a(y b) x a y b + x a b + y b a r +( a + b )r f(x, y) f(a, b) f(x, y) (a, b) lim (x,y) (a,b) f(x, y) U f(x, y) U U f(x, y) ( 7.3) y x y b 1 x f(a + h, b) f(a, b) f(x, b) f(x, b) x lim h h f(x, y) (x, y) (a, b) x f(x, y) (x, y) (a, b) x f (a, b) f(x, y) (x, y) (a, b) y x f(a, b + k) f(a, b) lim f (a, b) k k y f x (a, b),f y (a, b) f(x, y) f x xy (x, y) x y (x, y) f (x, y) y
(1/1) f(x, y) { y x sin x, f(x, y) x x f(x, y) a f f (a, b) (a, b) x y f y (,b) f (, ) x b f (,b) x a, b g(t) f(at, bt) g () f(x, y) (x, y) (, ) 3
1/19 1 f(x, y) (x, y) (a, b) 1. 1 f (a, b)(x a)+ f (a, b)(y c)+f(a, b) x y f(x, y) lim (x,y) (a,b) ( f x ) (a, b)(x a)+ f (a, b)(y c)+f(a, b) y (x a) +(y b) z f (a, b)(x a)+ f (a, b)(y c)+f(a, b) x y ( 4.4 p.18) f(x, y) x, y g(t),h(t): t f(g(t),h(t)) f(x, y) g(t),h(t) f(g(t),h(t)) (a, b) (g(c),h(c)) g(t),h(t) t c f(x, y) (x, y) (a, b) f(g(t),h(t)) t c d dt f(g(t),h(t)) tc x f(a, b)g (c)+ y f(a, b)h (c). [ ] (x a) +(y b) r f(x, y) f(a, b) x f(a, b)(x a) f(a, b)(y b) y k(r) (x a) +(y b) k(r) F (t) f(g(t),h(t)) F (t) F (c) f(a, b)(g(t) g(c)) f(a, b)(h(t) h(c)) x y k(r) (g(t) g(c)) +(h(t) h(c)) 4
t c t c lim F (t) F (c) t c t c x f(a, b)g (c) y f(a, b)h (c) k(r) (g (c) + h (c). [ ] f(x, y) (f(a, b)+f x (a, b)(x a)+f y (a, b)(y b)) (f(x, b) f(a, b) f x (a, b)(x a)) + (f(x, y) f(x, b) f y (a, b)(y b)) f(x, b) f(a, b) f x (a, b)(x a) (x a) +(y b) f(x, b) f(a, b) f x(a, b)(x a) x a f(x, y) f(x, b) f y (a, b)(y b) (f y (x, t) f y (a, b))(y b) t y b f y (x, y) (x, y) (a, b) (x, y) (a, b) (x, t) (a, b) (f y (x, t) f y (a, b))(y b) (x a) +(y b) f y (x, t) f y (a, b). 5
1/6 F (t) f(a+t(x a),b+t(y b)) F (1) F () F (t) f(x, y) f(a, b) x f(a + t(x a),b+ t(y b))(x a)+ f(a + t(x a),b+ t(y b))(y b). y f x (x, y) f xx (x, y),f xy (x, y) f yx (x, y),f yy (x, y) f(x, y), f(x, y) x x y ([ ] 6 p.118) f(x, y) f xy (a, b) f yx (ab) [ ] f(x, y) f(x, b) g(x) f(x, y) f(a, y) f(x, b)+f(a, b) g(x) g(a) g (s)(x a) ( ( ) f(s, y) f(s, b))(x a) f(s, t) (x a)(y a) x x x y ( ) f(x, y) y x f(x, y) f(a, y) f(x, b)+f(a, b) lim (x,y) (a,b) (x a)(y a) Taylor ([ ] 1 p.13) ( ) f(a, b) x y f(a + h, b + k) f(a, b)+f x (a, b)h + f y (a, b)k + 1 f xx(a, b)h + f xy (a, b)hk + 1 f yy(a, b)k + n 1 n (n m)!m! x n m y f(a + ht, b + m kt)hn m k m m <t<1 1 t F (t) f(a + ht, b + kt) Taylor 6
11/ ([ ] 4.5 p.138) f(x, y) (x, y) (a, b) (a, b) (x, y) f(a, b) f(x, y) f(x, y). f(x, y) (x, y) (a, b) f x (a, b) f y (a, b). 1 f(x, b),f(a, y) f(x, y). f x (a, b) f y (a, b) A f xx (a, b),b f xy (a, b),c f yy (a, b) 1. AC B > A> f(x, y) (x, y) (a, b). AC B > A< f(x, y) (x, y) (a, b) 3. AC B < f(x, y) (x, y) (a, b) A, B, C f(x, y) 1 (Ax +Bxy + Cy ) A f xx (a, b),b f xy (a, b),c f yy (a, b) 1. AC B > A> f(x, y) (x, y) (, ). AC B > A< f(x, y) (x, y) (, ) 3. AC B < f(x, y) (x, y) (, ) [ ] I. Ah +Bhk + Ck 1 A ((Ah + Bk) +(AC B )k ) II. ( ) ( ) A B A t P P B C C 7
[ ] 1. Taylor f(a + h, b + k) f(a, b) 1 f xx(a + th, b + tk)h + f xy (a + th, b + tk)hk + 1 f yy(a + th, b + tk)k <t<1 f xx (x, y) > f xx (x, y)f yy (x, y) f xy (x, y) > A f xx (a + th, b + tk),b f xy (a + th, b + tk),c f yy (a + th, b + tk) A h +B hk + C k (h, k) (, ) 3 AC B < Ah +Bhk + Ck > (h, k) Ah + Bh k + Ck > (h,k ) F (t) f(a + ht, b + kt) F () > F (t) t G(t) f(a + h t, b + k t) G(t) t f(x, y) (x, y) (a, b) 8
(1/6) x, y f(x, y) (x + y ) 4xy (1) f (x, y) x () f(x, y) (x, y) (1, ) (3) f(x, y) (4) 1 x 1, 1 y 1 f(x, y) 9
(1/6) 1 (1) f x (x, y) 4x(x + y ) 4y () f(1, ) 5 817,f x (1, ) 81,f y (1, ) 4 436 z 1(x 1) + 36(y ) + 17 1x +36y 67 (3) f x (x, y) 4x(x + y ) 4y f y (x, y) 4y(x + y ) 4x x y x + y 1 x y (x, y) (, ), ( 1 1, ), ( 1, 1 ) 3 ( ) ( ) fxx (, ) f xy (, ) 4 16 < (, ) f xy (, ) f yy (, ( ) 4 fxx ( 1, ) 1 ) f xy ( 1 1 ( ), ) 8 f xy ( 1 1, ) f yy ( 1 1, 64 > ) 8 ( 1 1, ) 1 1 ( 1, 1 ) 1 (4) (3) ( 1 1, ) ( 1, 1 ) x ±1 y ±1 f( 1 1, )f( 1, 1 ) 1 y 1 f(x, 1) (x +1) 4x x +1 x x ±1 f(x, 1) (x +1) 4x 4x 4x 1 x 1/ x f(x, 1) > 1 f( 1 1, )f( 1, 1 ) 1 1 <x<1, 1 <y<1 y 1 f(x, 1) (x +1) 4x 1 x 1 f xx (x, 1) 1x +4> f(x, 1) f(1, 1) 4 f( 1, 1) 8 8 1 x 1, 1 y 1 f( 1, 1) f(1, 1) 8 f( 1 1, )f( 1, 1 ) 1 1
11/9 ([ ] 5.1 p.146) D {(x, y) a x b, c y d} f(x, y) D a a a 1 a n b, c c c 1 c m d D a 1 a,...,a n a n 1,c 1 c,...,c m c m 1 D ij {(x, y) a i 1 x a i,c j 1 y c j } (x ij,y ij ) n i1 m f(x ij,y ij )(a i a i 1 )(c j c j 1 ) j1 D f(x, y) f(x, y)dxdy D f(x, y)dxdy D [ ] b d f(x, y)dxdy a c x y D ij f(x, y) m ij, M ij n m m ij (a i a i 1 )(c j c j 1 ) i1 j1 n m f(x ij,y ij )(a i a i 1 )(c j c j 1 ) i1 n j1 i1 j1 m M ij (a i a i 1 )(c j c j 1 ). (M ij m ij )(b a)(d c) (M ij m ij ) D 11
I. ([ ] 6 p.15) b ( d ) f(x, y)dxdy f(x, y)dy dx D a c d c ( b a ) f(x, y)dx dy. (1) [ ] F (x) d f(x, y)dy c b F a (x)dx D {(x, y) a x b, c y d} a a a 1 a n b, c c c 1 c m d a i 1 x i a i F (x i ) d f(x c i,y)dy m cj j1 c j 1 f(x i,y)dy. c j c j 1 f(x i,y)dy f(x i,y ij )(c j c j 1 ) c j 1 y ij c j F (x i ) m j1 f(x i,y ij )(c j c j 1 ). b F (x)dx a n i1 F (x i)(a i a i 1 ) n m i1 j1 f(x i,y ij )(a i a i 1 )(c j c j 1 ) f(x, y)dxdy D (1) b dx d f(x, y)dy, d dy b f(x, y)dx. a c c a 1
11/16 D a x b, c(x) y d(x) b d(x) f(x, y)dxdy dx f(x, y)dy. a c(x) a x b,c(x) y d(x) : x +y 1 1 1 1 x y dxdy 1 1 π(1 x )dx π[x x3 3 ]1 1 4π 3. 1 x dx 1 x y dy 1 x : [ ] p.155 7 () y x π π π π π π y cos(x y)dxdy dy y cos(x y)dx y[sin(x y)] π y dy y cos ydy π. π x y cos(y x)dy ([y sin(y x)] x sin(y x)dy)dx π π sin ydy)dx [ cos y] x dx 1 cos xdx π. y sin ydy [ y cos y] π π dx π x ( x II 1 f(x, y)dxdy D E f(r cos θ, r sin θ)rdrdθ. 13
11/3 11/3 f(x, y)dxdy D x +y 1 1 π dr 1 x y dxdy 4π[ 1 3 (1 r ) 3 ] 1 4π 3. E f(r cos θ, r sin θ)rdrdθ. r 1, θ π 1 1 r rdθ 4π 1 r rdr 1 x y dxdy 4 (x 1 ) +y 1 4 π cos θ π π dθ 3 π 4 3 3 π 8 9. π 1 r rdr 4 [ 1 3 (1 r ) 3 ] cos θ dθ 4 3 π π dθ r cos θ, π θ π cos θ (1 sin 3 θ)dθ 1 r rdr 1 r rdrdθ 1 r rdrdθ (sin θ cos θ sin θ)dθ 3 π 4 3 [ cos θ + 1 3 cos3 θ] π 14
1 III (1/7) I 1 y x I xdxdy x 1 y x (x 1) + y 1 y x ydxdy y x x r cos θ, y r sin θ r x r cos θ, y r sin θ θ I x + y 1 y x +1 x + y 1 x,y ydxdy ydxdy y x +1 x,y ydxdy 1 x r cos θ, y r sin θ 15
(1/7) 1 I 1 1 x dx xdy 1 x 3 dx 1 4. I 1 I 1 I 1 1 1 (x 1) dx [x 1 x 1 1 1 y dy xdx y dy 1 4. (x 1)3 3 y dy ydx 1 1 y 1 1 ydy (1 (x 1) x )dx ] 1 x3 1 3 (1 1 3 1 3 )1 6. 1 1 3 1 1 + y 1 y dy. y(y 1+ 1 y )dy y sin θ I 1 π/ 6 + sin θ cos θdθ 1 [ ] π/ 6 + cos3 θ 1 3 6 + 1 3 1 6. x r cos θ, y r sin θ (x 1) +y 1 r r cos θ r cos θ y x θ π/4 I r cos θ π/4 θ π/ π/ π/4 r sin θrdrdθ 8 3 cos3 θ sin θdθ π/ π/4 [ ] π/ 3 cos4 θ 1 π/4 6 cos θ dθ r sin θdr 16
I r cos θ π/4 θ π/ r sin θrdrdθ r [ cos θ] Arccos r π/4 dr 1 y x +1 x,y x + y 1 x,y r dr π π/ ydxdy Arccos r dr r sin θdθ π/4 r ( 1 r 3 4 )dr 3 8 r 1 π/ θ π sin θdθ 1 3 [ cos θ]π π/ 1 3. r sin θ rdrdθ 1 6. ydxdy 1 6 I 1 3 1 6 1 6. 17
1/14 1/1 x g(s, t),y h(s, t). (s, t) (x, y) st E xy D 1 1 f(x, y)dxdy f(g(s, t),h(s, t)) J(s, t) dsdt. 1.. D J(s, t) det ( ) x A y (x, y) (s, t) (x, y) (s, t) E g(s, t) s h(s, t) s g(s, t) s g(s, t) t h(s, t). t h(s, t) t ( (r cos θ, r sin θ) cos θ r sin θ (r, θ) sin θ rcos θ ( ) ( ) s a b, A t c d (x, y) (s, t) A. g(s, t) t h(s, t). s ), J(r, θ) r. ( ) ( ) ( a c, a c b d det b d) 1 x y x, x + y 3 π cos x y dxdy 3 x s + t, y s +t π π t cos s 3dsdt 3 dt cos sds 3 s,t,s+t π π [sin s] π t dt 3 π 18 cos tdt 3.
([ ] p.17) S f x (x, y) + f y (x, y) +1dxdy. D ( ) ( S ) ( 1 x y ) 1 x + y +1dxdy x +y 1 x y x 1 x y + y 1 x y +1dxdy 4π x +y 1 r 1, θ π 1 1 1 r rdrdθ r 1 r dr 4π[ 1 r ] 1 4π h, k f x (x, y)h f y (x, y)k. 1 a a, b 1 S. c b a b 1 c a, b S (c ) (a, b, c 6 ) det(a bc). S (1+a + b )/ 1+a + b 1+a + b. 19
1/11 cos x e x n x n n!. ( 1) n x n, sin x (n)! n x < 1 n 1 1+x ( 1) n x n log(1 + x) n ( 1) n1 n 1 xn a,a 1,a,... 1 a n x n n ( 1) n x n+1 (n + 1)! n a n x n n a n x n n r x <r a n x n x >r a n x n n x a n x n r n n a n x n n a n x n n n x n n! x n 1 n
a n+1 lim n a n l ( lim n a n 1 n r 1, (l ) r, (l ) l a n x n n r l) a n x n n 1 na n x n 1 a n, n +1 xn+1 r n1 n r <x<r f(x) f(x) a n x n f(x) r <x<r n f (x) na n x n 1, n1 x f(x)dx n a n n +1 xn+1 arctanx x x3 3 + x5 5 x7 7 +. 1
r x < r f(x) III (1/11) n n 1 n(n 1) xn f(x) (1 x) log(1 x)+x 1 n(n 1) xn f (x) x
1/18 n a n s m m n a n nm a n n a n n a n a n,a n > n ( 1)n a n n1 ( 1)n 1 1 log n a n b n n b n n a n [] x, y x < y a n y n a n x n n a n y n M n a n y n M a n x n M x n y a n n x n n a n+1 x n+1 [] x < 1/l lim n a n x n < 1. m <s<1 n m a n+1 x n+1 a n x n s n m a n x n a m x m s n m a n+1 x n+1 x >1/l lim n a n x n > 1 a nx n (n ), [] 1 r na n x n 1 n1 (1) x <r na n x n 1 (n ) () x <r a n x n (n ) n 3
() x <r a n x n 1 (n ) a n x n (n ) (1) x <t<r n N a n t n < 1 N na n x n 1 n( x t )n 1 t (n ) f n (x) n k a kx k x, a s<t<r n N a n t n 1 N n N f(x) f n (x) a k x k a k s k kn+1 kn+1 kn+1 kn+1 a k t k ( s t ( s ) k ( s ) n+1 1 t t 1 s ( ε n ) t ) k x a f(x) f(a) f(x) f n (x) + f n (x) f n (a) + f(a) f n (a) f n (x) f n (a) +ε n lim f(x) f(a) ε n (n ). x a a n F (x) n +1 xn+1 x t<r x x n f(x)dx (f(x) f n (x))dx x x f n (x)dx + x (f(x) f n (x))dx f(x) f n (x) dx ε n x na n x n 1 n1 4 (n ).
([1] p.44 (5), p.194 3, [] p.73, [3] p.39 5.11.) ( ) a (1 + x) a x n n n a(a 1) 1+ax + x a(a 1)(a ) + x 3 a(a 1)(a )(a 3) + x 4 +. 3! 4! 1 ( ) 1 ( x ) n 1 3 (n 1) x n, 1 x n n n! n n 1+ 1 x + 3 4 x4 + 1 3 5 4 6 x6 +. ([1] p.195 11(3), [] p.75, [3] p.39 5.14.) Arcsinx x n 1 x dx 1 x ( n 1 3 (n 1) 4 n (n +1) xn+1 ) 1 3 (n 1) x n dx 4 n x + x3 3 + 1 3 4 5 x5 + 1 3 5 4 6 7 x7 +. ([1] p.197, [] p.134.) f(x) f (x) f(x) f (x) f(x). f(x) Ce x (C ( ) f(x) ex f (x) e x f(x) e x e x f (x) f(x) e x f(x) e x f(x) f(a),f (a) f (x) f(x) f() 1 f(x) e x e x x n n!. f(x) x n n! f (x) f(x) f() 1 n n (1 + x) a 1 <x<1 f() 1 (1 + x)f (x) af(x) 1 5
( f(x) (1 + x) a n ) (1 + x)a f (x) a(1 + x) a 1 f(x) (1 + x) a (1 + x)f (x) af(x) (1 + x) a+1 f(x) (1 + x) a ( ) a n x n ( a n+1) / ( a n 1 <x<1 f(x) n ) a n 1 1 n+1 ( ) a x n n f() 1 ( ) a f (x) nx n 1. n n n ( ) ( a n a a 1 ) ( n 1, a 1 ) ( n 1 + a 1 ) ( n a ) n ( ) ( ) a 1 a 1 (1 + x)f (x) (a + a )x n a n 1 n n n ( ) a x n af(x). n (1 + x) a n ( ) a x n n 6