1
. z = f (x; y) (x; y).
(x; y) x ( y) x ( y) D x (D y)
(1)
. (1/)
: z¢ x, f ¢ x, ,
z¢ , f ¢ , , .
:
1. , , .
2. , .
z = f (x; y)
d x z = z ¢ x d x = d x - z = f (x; y) x (2)
d z = z ¢ d = d - z = f (x; y) y (2/)
z = f (x; y)
d z = d x z + d z
d z = d x + d y (3)
:
1. u = f (x; y;...; t), ,
d u = d x + d y + + d t (3/)
2. ,
d(u v) = du dv
d(u×v) = vdu + udv
d()= .
D z = f (x 0 +D x; y 0 +D y) f (x; y) z = f (x; y), D x D y
D z d z (4)
, , (4) :
f (x o +D x; y 0 +D y) f (x 0; y 0) + f x/(x 0; y 0)×D x + f /(x 0; y 0)×D (5)
Dx=dx, D=dy
1. .
1) z = x2 + 2y2 3xy 4x + 2y + ln 5 2) z = (x3 +y3 xy2)3 3) z = ln (x2 +2y3)
4) z = (1 + x2)y 5) z = arcsin 6) u = + -
7) u = arctg 8) u = ln 9) u = (x×y) Z
10) z = x + 11) z = ln (x + ) 12) z = (x - )e-x y
13) z = arcsin 14) u = sin2(3x + 2y - z).
2.
1) z = x + y + M0 (3; 4)
2) z = cos (3x 5y) M0 (; 0)
3) z = ln (x2 y2) M0 (2; -1)
4) u = ln (1 + x + y2 + z3) M0 (1;1; 1).
3. ,
|
|
) z = x ln x + y = z
) z = xy
) u = x + = 1.
4.
) z = 3x2y5
) u = 2xyz
) z = arccos
) z = x3 + xy2 + x2y
) u = sin2(xy2z3)
e) z =
) z = y ln2 x
) u =
) z = sin2 y cos2 x
5. :
) z = arcctg x = 1; y = 3; dx = 0,01; dy = - 0,05
) z = x = 2; y = 1; dx = - ; dy =
) u = (x; y; z) 0 (10; - 10; 5) 1 (9; - 11; 6).
6. , 0, :
) 1,942 0,12 ) sin1,59 tg3,09 ) 2,68sin0,05
z = x2ey, M0 (2; 0) z = sin x tg y, M0 (; ) z = xsin y , M0 (; 0).
7. :
) (1,02)3 × (0,97)3 ) ) 1,083,96 )