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II ̳ ...

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5. ii i i z = x2 + y 2 2y.

ii i z = C i Oxy, i x2 + y 2 2y = C x2 + (y 1)2 = C + 1. i i (0; 1) i i C + 1 (. 2), C 1.

(0; 1) ii i, ii ii i z, i (0; 1). ii i

   

i i i . i i i i i.

i i.

i i (x; y; z), R3, X i i.

i i , i ii (x; y; z) ii u X.

x, y i z i (), u i (i), i, X i, X = {u R : u = f (x, y, z), (x; y; z) }.

i i , i i i i: u = f (x, y, z), w = w(x, y, z), (x; y; z).

6. i u = ln(1x2 y 2 z 2 ).

, i, i i i i, 1 x y 2 z 2 0, x2 + y 2 + z 2 1. , 2 i ii i (i ) i .

i i n i.

i n i , (x1 ; x2 ;... ; xn ) i .

i n i ii i i i: u = f (x1, x2,..., xn ), u = u(x1, x2,..., xn ).

ii, i n i u = f (x1, x2,..., xn ) n 3 i x = (x1 ; x2 ;... ; xn ) n-i Rn i u = f (x), x.

2. i i.

i i.

   

i i i ii, , i i.

i i i i i, i i ii i i i i i i. i i i i.

i z = f (x, y) = {(x; y) : |x x0 | a1, |y y0 | a2 }, i, iii x = x0, y = y0. i i i y i f (x, y) i i i x.

i i y, , 0 |y y0 | a2, i i f (x, y) x x0, , i , i y,

   

i i i i -i i Oxy, i i i i , i .

M0 i f, i i i i i.

   

x2 + y 2 = 0, f i i, i . i i i i (0; 0). i 2 , lim f (x, y) i, , f i (0; 0).

(x;y)(0;0) i f i, i i i .

i f i M0

   

i i f i M0 : i f i M0 (x0 ; y0 ), i i i i i,

   

i i i i i i, i i i i.

2. i z = f (x, y) i i i, i i:

1) , |f (x, y)| A, (x; y), A ;

2) m i i M ;

3) i i i - , i i m i M, i , i i i .

3. i ii i i i i i z = f (x, y), (x; y).

i i, , y, y = y0. i i f (x, y0 ) i i i x. x x,

   

, i i i i i i i i , i, i . , i z i i i i = fx (x, y), y = const, x z = fy (x, y), x = const. i y i i i i i i i, i .

   

2z 2z ii i i xy yx i .

6. i ii i z = x2 y 2.

i i, :

   

x + y 2 0.

i z = f (x, y) i i (x0 ; y0 ), i z = f (x0 + x, y0 + y) f (x0, y0 ) i i i

   

, i z = f (x, y) i i (x0 ; y0 ).

1. i z = f (x, y) i i (x0 ; y0 ), i i i i (i) i (x0 ; y0 ; z0 ), z0 = f (x0, y0 ), i

   

2. i i i i.

i i i f i i, i i.

, i i i:

   

1, 023 + 1, 973 3 + 0, 02 2 0, 03 = 2, 95.

i i i i ii i i i, . i, z = f (x, y), (x; y), i i. i, i ii, i (x0 ; y0 ) ii

   

i ii (dz) i x = dx, y = dy i .

(6) d2 z:

   

5.1. i i. i i f (x, y), (x; y), i i i x = (t), y = (t), t T R. , t T ((t); (t)). i i z = f ((t), (t)), t T. i f i, ii i.

, i f i , i ii T. , i i z = f ((t), (t)), t T, i i T, i ii f f z (t) = ((t), (t)) (t) + ((t), (t)) (t), t T. (1) x y i, ii t T t , t + t T, i i

   

i z i u i v , , (u; v) D , ((u, v); (u, v)). i , , ,

   




140 .


5.2. I ii i. ii 1, 1, .1.1.10, i, , i i i i, i F (x, y) = 0.

(x0 ; y0 ) i F (x, y) = 0. i i = {(x0 ; y0 ) : |x x0 |, |y y0 | 1 } i i F (x, y) = 0 y i i x, x (x0 ; x0 + ) i y (y0 1 ; y0 + 1 ) , (x; y) i. , i x2 y 2 = 0. i y = x, y = x, y = |x|, y = |x|, x R. , i (1; 1) i y = x, - i (0; 0) i . ǒ i i, . F (x, y) = x2 y 2. i Fy (x, y) = 2y, Fy (1, 1) = 0, Fy (0, 0) = 0. , i i i F (x, y) = 0 i y i (x0 ; y0 ) ii Fy (x0, y0 ) = 0.

(i i ii i). : 1) i F (x, y) i i U (x0 ; y0 ); 2) i ii Fx i Fy i i (x0 ; y0 ); 3) F (x0, y0 ) = 0, Fy (x0, y0 ) = 0. i i = {(x; y) : |x x0 |, |y y0 | 1 } U, i F (x, y) = 0 i y = f (x), f (x0 ) = y0, i f (x) i ii (x0 ; x0 + ) i i

   

6.2. . ii i i i , i ii i i , i . .

4. i z = x2 + y 2, x + y 1 = 0.

x + y 1 = 0 y = 1 x. i i z, i z = x2 +(1x)2 z = 2x2 2x+1.

i z = 4(x 1/2), z = 4. i , i x =

   

i f (ii) i x0 = (x0 ; x0 ;... ; x0 ), i i x0, i n 12 x , i (2), ii f (x0 ) f (x) (f (x0 ) f (x)).

i i

   

k, k {1,..., m}, , i . , i f, k, k {1,..., m}, i i i .

ii n + m i

   

= 2 1 0 = 20 0, i ii i (1; 2). i (1; 2) i 2 = = 2 1 0 = 20 0, (1; 2) .

6. i i S1 i S2 i ii x i y.

i C = 10x + xy + 10y, i i ii p1 = 50x+y i p2 = 30 + 2x y. i , i x + y = 15.

, i ii i S1 i S2.

i, (x, y) = P1 + P2 C = p1 x + p2 y C = (50x+y)x+(30+2xy)y10xxy10y = 40xx2 +2xy+20yy 2.

, i i (x, y) = 40x x2 + 2xy + 20y y 2 , x + y = 15.

x + y = 15 y x i i i : y = 15x, = 40xx2 +2x(15x)+20(15x)(15x)2 = 80x 4x2 + 75.

i i () .

(x) = 80 8x. i i , i, x0 = 10. i (x) = 8 0, i i x0 i , max = (10) = 80104102 +75 = 475.

i , i i (10; 5) i max = 475.

6.3. i. ii i i , i , i i i, , i i i ..

   

i i i (2) i i i i f (x, y) i D i f (x, y)dxdy.

D , ᒺ i Q i i,

   

, i i Q i ii i i. i i i i i.

i i i i ( i䒺 ).

D , z = f (x, y)

i, i i.

iᒺ D i n Dk, i i ii , S(Dk ),

   

i D i i.

i i i i, , - i D i i i i.

i i i i ii ii i. i .

1) A i , f (x, y) i i D, i i Af (x, y) i D i

   

i i i.

6) i f (x, y) i i D, |f (x, y)| i D i ii

   

x2 i, i i i ii y i x, ii x2 . ,

   



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