"
1 1 n
n
an = (n, (-1)n) bn = ( n, , ln )
n n+1
R2
{(x, y) : x2 + y2 < 2}
{(x, y) : x2 + y2 2}
{(x, y) : x2 + y2 > 2}
{(x, y) : 1 x2 + y2 < 2}
{(x, y) : x + y = 1}

" " "
"
f(x, y) = x sin y f(x, y) = arcsin y - x f(x, y) = ln( x + y)
xy (xy)2
x
lim(x,y)(0,0) x+y lim(x,y)(0,0) x2+y2 lim(x,y)(0,0) x2+y2
sin(xy)
, x = 0

x
lim(x,y)(0,0) f(x, y) f(x, y) =
0 , x = 0
f : R2 - R

x2 + y2 , x 0
f(x, y) =
2 , x < 0
x
a) f(x, y) = arccos b) f(x, y, z) = xy - zx c) f(x, y) = x sin(x + 2y)
y
"
3
d) f(x, y) = y2 cos(2x - y) e) f(x, y) = f) f(x, y) = 3xy
2xy

"
g) f(x, y) = y2x h) f(x, y) = x + y + x

3
f(x, y) = x3 - y3 (x0, y0) = (0, 0)

x3+y
, (x, y, z) = (0, 0, 0)

x2+y2+z2
f(x, y, z) = (x0, y0, z0) = (0, 0, 0)
0 , (x, y, z) = (0, 0, 0)

xy(x2-y2)
, (x, y) = (0, 0)

x2+y2
f(x, y) = .
0 , (x, y) = (0, 0)
"2f "2f
(0, 0) = (0, 0)
"x"y "y"x
f(x, y) = x2 - y2 (x0, y0) = (1, -2)

xy
"
, (x, y) = (0, 0)

x2+y2
f(x, y) = (x0, y0) = (0, 0)
0 , (x, y) = (0, 0)

x y z z = f(u, v) = euv u = ln x2 + y2
x
v = arc tg
y

f(x, y) = x2 + y2
"
3
(x0, y0) = (0, 0) = (1, - )
v
2 2
f(x, y) = sin x cos y
"
3
(0, Ą) = (-1, )
v
2 2
a) f(x, y) = 3x3 + 3x2y - y3 - 15x; b) f(x, y) = 3(x - 1)2 + 4(y + 2)2;
c) f(x, y) = x3 + 3xy2 - 51x - 24y; d) f(x, y) = x3 + y3 - 3xy.

f(x, y) = 2 - 3x2 + 4y2 f(x, y) = x8 - y4
f(x, y) = x2 + y2 - xy + x + y
x = 0 y = 0 y = -x - 3
f(x, y) = 2xy
D = {(x, y) : x2 + y2 d" 1}
f y = f(x) y3 - 4xy + x2 = 0
y = f(x) y4 - 8xy - 4y + 8x2 = 0
z = f(x, y) 5x2 + 5y2 + 5z2 - 2xy - 2xz - 2yz - 72 = 0
f(x, y) = xy
x2 + y2 = 2