f
[a, b]
[a, b] n n " N
P = {x0, x1, . . . , xn},
a = x0 < x1 < . . . < xn = b
"xk = xk - xk-1 - k P 1 k n,
�(P) = max{"xk : 1 k n} - P,
�k " [xk-1, xk] - k P 1 k n.
�1 �2 �3 �4 . . . �n-1 �n
a = x0 x1 x2 x3 x4 . . . xn-2 xn-1 b = xn
�(P)
f [a, b] P
f P
�k 1 k n
n
�(f, P) = f(�k)"xk.
k=1
f
Ox x = a x = b "xk
f(�k) 1 k n
y
f
f (�1) f(�2) f (�3) f (�4) f (�5)
�1 �2 �3 �4 �5 b
a
x
"x1 "x2 "x3 "x4 "x5
[a, b] n = 5
 f [a, b]
f [a, b]
b
n
f(x) dx = lim �(f, P) = lim f(�k)"xk
�(P)0 �(P)0
k=1
a
P
[a, b] �k 1 k n
a a b
f(x) dx = 0 f(x) dx = - f(x) dx a < b.
a b a
b
f(x) dx f(x) dx
a
[a,b]
[a, b]
[a, b]
T f
Ox x = a x = b |T |
"Tk
�(P) - 0
b
n n
|T | = lim |"Tk| = lim f(�k)"xk = f(x) dx.
�(P)0 �(P)0
k=1 k=1
a
Ox T
y
f
"Tk
f(�k)
�k
"xn "xn-1 b
a "x1 "x2 "x3 . . . "xk . . . -2 x
"xn
 V
f Ox x = a x = b |V |
"Vk
�(P) - 0
b
n n
|V | = lim |"Vk| = lim Ąf2(�k)"xk = Ą f2(x) dx.
�(P)0 �(P)0
k=1 k=1
a
y
y
f
f
f (�k)
"xk
a
b x �k
a
b x
"Vk
z
z
S [a, b]
v(t) t " [a, b] |S|
"Sk "tk v(�k)
�(P) - 0
b
n n
|S| = lim "Sk = lim v(�k)"tk = v(t) dt.
�(P)0 �(P)0
k=1 k=1
a
S v
Ot t = a t = b
f [a, b]
0, x = 0,
f(x) =
1
, 0 < x 1,
x
1
[a, b] x = n 2
n
 [a, b]
b
n
b - a b - a
f(x) dx = lim f a + k .
n"
n n
k=1
a
n
b - a b - a
f a + k f
n n
k=1
�n(f) �n
f
1
3x dx.
0
f(x) = 3x
[0, 1]
1
n n
" k
1 k 1
n
n
3x dx = lim 3 = lim 3 .
n" n"
n n
k=1 k=1
0
n
" k
n
3 n a1 =
k=1
" "
n n
3 q = 3 n an
a1(1-qn)
q Sn =
1-q
" " n "
n n "
n
n n
3 1 - 3
" k
3(1 - 3) 2 3
n
3 = " = " " .
=
n n n
1 - 3 1 - 3 3 - 1
k=1
1
" "
n n
1 2 3 2 3 2
3x dx = lim " = lim = .
1
n
n" n" -1 ln 3
n
n 3
3 - 1
1
0
n
f [a, b] F
b
f(x) dx = F (b) - F (a).
a
b
b
F (b) - F (a) F (x) F (x)
a
a
1
1 Ą Ą
1
dx = arctg x = arctg 1 - arctg 0 = - 0 = .
0
x2 + 1 4 4
0
1
1
(3x2 + 6x + 1) dx = (x3 + 3x2 + x) = 5 - 1 = 4.
-1
-1
 f g [a, b] ą �
b b b
[ąf(x) + �g(x)] dx = ą f(x) dx + � g(x) dx.
a a a
f g C1 [a, b]
b b
b
f (x)g(x) dx = f(x)g(x) - f(x)g (x) dx.
a
a a
(fg) = f g + fg
b b
b
f(x)g(x) = f (x)g(x) dx + f(x)g (x) dx.
a
a a
b b
b
f (x)g(x) dx = f(x)g(x) - f(x)g (x) dx.
a
a a
e e 3
e
1 1 1 1
x2 ln x dx = x3 ln x dx = x3 ln x - x3 � dx =
3 3 3 x
1
1 1 1
e
e
1 1 1 1 1 1 1 1
= e3 ln e - ln 1 - x2 dx = e3 - � x3 = e3 - (e3 - 1) =
3 3 3 3 3 3 1 3 9
1
1 1 1 2 1 2e3 + 1
= e3 - e3 + = e3 + = .
3 9 9 9 9 9
f : [a, b] - [c, d] �" R
f g : [c, d] - R
f(b)
b
g(f(t))f (t) dt = g(x) dx.
a
f(a)
 g G g
f
(g ć% f) � f G ć% f
f(b)
b
g(f(t))f (t) dt = G(f(b)) - G(f(a)) = g(x) dx.
a
f(a)
Ą
2
"
cos t 1 + sin t dt.
0
x = 1 + sin t = f(t) dx = cos t dt
Ą
Ą
f
( )
2 2
2
"
" "
2" 2 4" 2
cos t 1 + sin t dt = x dx = x dx = x3 = 2 - .
3 3 3
1
0 1
f(0)
f [a, b]
g f g
[a, b]
b b
g(x) dx = f(x) dx.
a a
f [a, b] c " (a, b)
b c b
f(x) dx = f(x) dx + f(x) dx.
a a c
f g
[a, b]
f(x) g(x) x " [a, b]
b b
f(x) dx g(x) dx.
a a
 f [a, b]
f [a, b]
b
1
f = f(x) dx.
b - a
a
f [a, b]
b - a
f Ox x = a x = b
f [a, b] c " (a, b)
f = f(c)
b
"c " (a, b) : f(x) dx = (b - a)f(c)
a
b
1
"c " (a, b) : f(c) = f(x) dx
b - a
a
f
a > 0
a
f(x) dx = 0,
-a
a > 0
a a
f(x) dx = 2 f(x) dx,
-a 0
T a " R
a+T T
f(x) dx = f(x) dx.
a 0
Ox a b
f [a, b] D �" R2
D = (x, y) " R2 : a x b, �(x) y �(x) ,
� � [a, b]
y
�
O
P
D
N
�
M
a
b x
|D| D
b
|D| = [�(x) - �(x)] dx.
a
ć%
x " [a, b]
0 �(x) �(x),
b b b
|D| = P abO - MabN = �(x) dx - �(x) dx = [�(x) - �(x)] dx.
a a a
ć%
� � K
x " [a, b]
0 �(x) + K �(x) + K.
�(x) - �(x) = [�(x) + K] - [�(x) + K] .
y
� + K
D
� + K
�
a
b x
D
�
|D| D
r = f(�) f ą � � � - ą 2Ą � = ą
� = �
�
1
|D| = f2(�) d�.
2
ą
r = f(�)
f(�)
f(�)
f(ą)
�
ą
x
O
 D
x = x(t), y = y(t), t1 t t2,
x, y C1 Ox
t2
|D| = |y(t) x (t)| dt.
t1
y
(x(t1), y(t1))
(x(t2), y(t2))
D
t1 t2 t
f(x) = x3 - x2 - x
g(x) = x
f(x) = g(x)
x3 - x2 - x = x
x3 - x2 - 2x =
x(x2 - x - 2) = 0
x = 0 x = -1 x = 2.
A(-1, -1) B(0, 0) C(2, 2)
D1 D2
f(x) = x3 - x2 - x
y
D2
-1
x
D1
2
|D| = |D1| + |D2|.
0 0 0
|D1| = f(x) - g(x) dx = (x3 - x2 - x - x) dx = (x3 - x2 - 2x) dx =
-1 -1 -1
0
1 1 1 1 5
= x4 - x3 - x2 = 0 - + - 1 = .
4 3 4 3 12
-1
x
=
)
x
(
g
2 2 2
|D2| = g(x) - f(x) dx = x - (x3 - x2 - x) dx = - x3 + x2 + 2x) dx =
0 0 0
2
1 1 8 8
= - x4 + x3 + x2 = -4 + + 4 - 0 = .
4 3 3 3
0
5 8 37
|D| = + =
12 3 12
f(�) = a�
0 � 2Ą a > 0 � = 2Ą
0 x
D
2Ą 2Ą
2Ą
1 1 1 1 1 4
|D| = f2(�) d� = a2�2 d� = a2 � �3 = a2 � 8Ą3 = a2Ą3.
2 2 2 3 6 3
0
0 0
x(t) = tet y(t) = te-t Ox x = e
Ox
y(t) = 0
te-t = 0 �!�! t1 = 0.
A Ox A = (x(0), y(0)) = (0, 0)
x = e
x(t) = e �!�! tet = e �!�! t2 = 1.
1
B x = e B = (x(1), y(1)) = e,
e
y
B
D
A
x
y(t) 0 x (t) = et + tet 0
0 t 1
1 1 1
1
1 1 1 1 5
|D| = |y(t)x (t)| dt = te-t(et + tet) dt = t + t2 dt = t2 + t3 = + = .
2 3 0 2 3 6
0 0 0
 x(t) = a cos t y(t) = b sin t a, b > 0 t " [0, 2Ą]
x y
= cos t, = sin t,
a b
2
x 2 y
+ = cos2 t + sin2 t,
a b
x2 y2
+ = 1.
a2 b2
a b
y
b
D
a x
-a
-b
Ą 2Ą Ą 2Ą
|D| = |y(t)x (t)| dt+ |y(t)x (t)| dt= |b sin t � a(- sin t)| dt+ |b sin t � a(- sin t)| dt=
0 Ą 0 Ą
Ą 2Ą Ą 2Ą 2Ą
= -ab sin2 t dt + -ab sin2 t dt = ab sin2 tdt + ab sin2 tdt = ab sin2 tdt.
0 Ą 0 Ą 0
1 1
sin2 t = - cos 2t
2 2
2Ą
2Ą
1 1 1 1 1 1
|D| = ab - cos 2t dt = ab t - sin 2t = ab � 2Ą - sin 4Ą = abĄ.
2 2 2 4 2 4
0
0
f [a, b]
� = (x, f(x)) : x " [a, b]
b
|�| = 1 + [f (x)]2 dx.
a
y
f
a
x
b
 f [a, b]
[a, b]
[a, b] = [x0, x1] *" [x1, x2] *" [x2, x3] *" . . . [xn-1, xn], x0 = a, xn = b.
"xk = xk - xk-1 k = 1, . . . , n k
(x0, f(x0)) (x1, f(x1)) (x2, f(x2)) . . . (xn, f(xn))
n
y
f(xk-1)
dk
"kf
f(xk)
"xk
x
a = x0 x1 x2 xk-1 xk xn-1 b = xn
[xk-1, xk]
�k " (xk-1, xk)
f(xk) - f(xk-1)
f (�k) = ,
xk - xk-1
f(xk) - f(xk-1) = f (�k)(xk - xk-1).
(xk-1, f(xk-1)) (xk, f(xk))
dk = (xk - xk-1)2 + [f(xk) - f(xk-1)]2 = (xk - xk-1)2 + [f (�k)]2 (xk - xk-1)2 =
= 1 + [f (�k)]2 � (xk - xk-1).
d1 + . . . + dn = 1 + [f (�1)]2(x1 - x0) + . . . + 1 + [f (�n)]2(xn - xn-1) =
n
= 1 + [f (�k)]2(xk - xk-1).
k=1
�n 1 + [f (x)]2
b
|�| = lim �n = 1 + [f (x)]2 dx.
n"
a
 f [ą, �]
|�| r = f(�) ą � �
�
|�| = [f(�)]2 + [f (�)]2 d�.
ą
r = f(�)
�
ą
x
O
|�|
x = x(t) y = y(t) t1 t t2 x = x(t) y = y(t)
[t1, t2]
[x (t)]2 + [y (t)]2 = 0

t2
|�| = [x (t)]2 + [y (t)]2 dt.
t1
y
(x(t2), y(t2))
(x(t1), y(t1))
a
x
b
|�|
x = x(t) y = y(t) z = z(t) t1 t t2 x = x(t) y = y(t) z = z(t)
[t1, t2]
[x (t)]2 + [y (t)]2 + [z (t)]2 = 0

t2
|�| = [x (t)]2 + [y (t)]2 + [z (t)]2 dt.
t1
Ą
f(x) = ln cos x 0 x
3
sin x
f (x) = - = -tg x.
cos x
sin2 x
[f (x)]2 = ,
cos2 x
sin2 x cos2 x + sin2 x 1
1 + [f (x)]2 = 1 + = = ,
cos2 x cos2 x cos2 x
1 1
1 + [f (x)]2 = = ,
cos2 x | cos x|
Ą Ą Ą
3 3 3
1 1
|�| = 1 + [f (x)]2 dx = dx = dx.
| cos x| cos x
0 0 0
1 1 1
= = ,
Ą
x Ą x Ą
cos x sin(x + )
2 sin + cos +
2
2 4 2 4
Ą
cos2 x +
2 4
1 1 1
Ą Ą
2
cos2 x + cos2 x +
1 ( ) ( )
2 4 2 4
= = .
x Ą x Ą
cos x
2tg + tg +
2 4 2 4
Ą
1
3 Ą
Ą
3
cos2 x +
( ) x Ą
2 4
|�| = dx = ln tg + =
x Ą
2 4
2tg +
0
2 4
0
"
Ą Ą Ą 5
= ln tg + - ln tg = ln tg Ą - ln 1 = ln(2 + 3) H" 1, 32.
6 4 4 12
r
r f(�) = r
0 � 2Ą
2Ą 2Ą 2Ą
"
2Ą
|�| = [f(�)]2 + [f (�)]2 d� = r2 + 02 d� = r d� = r� 0 = r(2Ą - 0) = 2Ąr.
0 0 0
x(t) = (t2 - 2) sin t + 2t cos t,
0 t 2Ą.
y(t) = (2 - t2) cos t + 2t sin t,
x (t) = 2t sin t + (t2 - 2) cos t + 2 cos t - 2t sin t = t2 cos t,
y (t) = -2t cos t - (2 - t2) sin t + 2 sin t + 2t cos t = t2 sin t.
2Ą 2Ą 2Ą
|�| = [x (t)]2 + [y (t)]2 dt = t4 cos2 t + t4 sin2 t dt = t4(cos2 t + sin2 t) dt =
0 0 0
2Ą 2Ą
" 2Ą
1 8
= t4 dt = t2 dt = t3 = Ą3.
3 0 3
0 0
x(t) = r cos t
y(t) = r sin t z(t) = kt t " [0, 2Ą] r k
x (t) = -r sin t,
y (t) = r cos t,
z (t) = k.
[x (t)]2 + [y (t)]2 + [z (t)]2 = r2 sin2 t + r2 cos2 t + k2 = r2 + k2,
"
[x (t)]2 + [y (t)]2 + [z (t)]2 = r2 + k2.
2Ą 2Ą
" "
|�| = [x (t)]2 + [y (t)]2 + [z (t)]2 dt = r2 + k2 dt = 2Ą r2 + k2
0 0
f : [a, b] - R |V |
V = {(x, y, z) " R3 : a x b, z2 + y2 f2(x)},
x = a x = b
f Ox
b
|V | = Ą f2(x) dx.
a
f C1 [a, b] |S|
b
|S| = 2Ą |f(x)| 1 + [f (x)]2 dx.
a
y
y
f
T
a
b x
a
b x
z
z
V
y = f(x) a x b Ox
f [a, b]
a 0 T f
Ox x = a x = b V
T Oy
b
|V | = 2Ą xf(x) dx.
a
y
y
f
T
a
x
a b
x
b
z
z
� x = x(t) y = y(t)
t " [t1, t2] y(t) x(t) C1
[t1, t2]
t2
|V | = Ą y2(t)|x (t)| dt,
t1
t2
|S| = 2Ą |y(t)| [x (t)]2 + [y (t)]2 dt.
t1
y(t) C1
[t1, t2]
S(x) x " [a, b]
V Ox x S
[a, b] V
b
|V | = S(x) dx.
a
f(x) = sin x x " [0, Ą] Ox
Ą Ą
Ą
1 1 Ą 1 Ą2
|V | = Ą sin2 x dx = Ą - cos 2x dx = x - sin 2x = .
2 2 2 2 2
0
0 0
f (x) = cos x sin x 0 x " [0, Ą]
Ą Ą Ą
" " "
|S| = 2Ą | sin x| 1 + cos2 x dx = 2Ą | sin x| 1 + cos2 x dx = 2Ą sin x 1 + cos2 x dx.
0 0 0
cos x = t sin xdx = -dt cos 0 = 1
cos Ą = -1
-1
"
|S| = -2Ą 1 + t2 dt.
1
 k = 0

" " "
1 1
k + x2 dx = x k + x2 - k ln x + k + x2 + C.
2 2
-1
" " "
1 1
|S| = -2Ą x 1 + x2 - ln x + 1 + x2 = 2 2 Ą.
2 2
1
"
1
Ox x(t) = t2 y(t) = t - t3 0 t 3
3
"
x (t) = 2t 0, 3
" " "
3 3 3
2
1 2 1 2 1
|V | = Ą t - t3 |2t| dt = 2Ą t2 - t4 + t6 t dt = 2Ą t3 - t5 + t7 dt =
3 3 9 3 9
0 0 0
"
3
1 1 1 9 27 81 9 9
= 2Ą t4 - t6 + t8 = 2Ą - + = 2Ą - 3 + =
4 9 72 4 9 72 4 8
0
18 24 9 3
= 2Ą - + = Ą.
8 8 8 4
" "
1 1 1
y(t) = t - t3 = - t t2 - 3 = - t t - 3 t + 3
3 3 3
"
t " 0, 3
" "
t
0
- 3 3
1 1
t - t3 = t - t3 y (t) = 1 - t2
3 3
" "
3 3
"
1 1
|S| = 2Ą t - t3 4t2 + (1 - t2)2 dt = 2Ą t - t3 4t2 + 1 - 2t2 + t4 dt =
3 3
0 0
" "
3 3
"
1 1
= 2Ą t - t3 1 + 2t2 + t4 dt = 2Ą t - t3 (1 + t2)2 dt =
3 3
0 0
" "
3 3
1 2 1
= 2Ą t - t3 � 1 + t2 dt = 2Ą t + t3 - t5 dt = 3Ą.
3 3 3
0 0
f [a, b]
"
" f : [a, b] -
 f [a, +")
B
F (B) = f(x) dx B +"
a
f [a, +")
+"
f(x) dx
a
+" B
f(x) dx = lim F (B) = lim f(x) dx.
B+" B+"
a a
+"
f(x) dx
a
(-", b] (-", +")
b b
f(x) dx = lim F (A) = lim f(x) dx
A-" A-"
-" A
+" c +" c B
f(x) dx = f(x) dx + f(x) dx = lim f(x) dx + lim f(x) dx,
A-" B+"
-" -" c A c
c " (-", +")
f g [a, +")
[a, b] b > a
0 f(x) g(x) x a,
+" +"
g(x) dx f(x) dx
a a
+" +"
f(x) dx g(x) dx
a a
+"
2
xe-x dx.
0
+" B
B
2 2 1 2 1 2 1 1
xe-x dx = lim xe-x dx = lim - e-x = lim - e-B + = .
B+" B+" B+"
2 2 2 2
0
0 0
0
x
dx.
1 + x2
-"
0 0
0
x x 1
dx = lim dx = lim ln(1 + x2) =
A-" A-"
1 + x2 1 + x2 2
A
-" A
1 1 1
= lim ln 1 - ln 1 + A2 = lim - ln 1 + A2 = -"
A-" A-"
2 2 2
+"
x
dx
1 + x2
0
+"
dx
.
x2 + 4x + 5
-"
(-", +")
+" 0 +"
dx dx dx
= + .
x2 + 4x + 5 x2 + 4x + 5 x2 + 4x + 5
-" -" 0
0 B
dx dx
F (A) = F (B) = ,
x2 + 4x + 5 x2 + 4x + 5
A 0
0 0
0
dx dx
lim F (A) = lim = lim = lim arctg (x + 2) =
A-" A-" A-" A-"
x2 + 4x + 5 (x + 2)2 + 1 A
A A
Ą
= lim [arctg 2 - arctg (A + 2)] = arctg 2 + .
A-"
2
B B
B
dx dx
lim F (B) = lim = lim = lim arctg (x + 2) =
B+" B+" B+" B+"
x2 + 4x + 5 (x + 2)2 + 1 0
0 0
Ą
= lim [arctg (B + 2) - arctg 2] = - arctg 2.
B+"
2
 lim F (A) lim F (B)
A-" B+"
0 +"
dx dx
x2 + 4x + 5 x2 + 4x + 5
-" 0
+"
dx Ą Ą
= arctg 2 + + - arctg 2 = Ą.
x2 + 4x + 5 2 2
-"
1
f(x) = y = 0 x = 1
x2
y
1
1
y =
x2
x
1 2 3 4
" B
B
1 1 1 1
|D| = dx = lim dx = lim - = lim - + 1 = 1.
B" B" B"
x2 x2 x B
1
1 1
1
f(x) = y = 0 x = 1
x
y
1
1
y =
x
x
1 2 3 4
" B
B
1 1
|D| = dx = lim dx = lim (ln |x|) = lim (ln |B| - ln |1|) = +".
B" B" B"
x x 1
1 1
|V | |S|
1
Ox f(x) = y = 0 x = 1
x
" B
B
1 1 1 1
|V | = Ą dx = Ą lim dx = Ą lim - = Ą lim - + 1 = Ą.
B" B" B"
x2 x2 x B
1
1 1
"
2
1 1 1 1 1 x4 + 1 1 x4 + 1
|f(x)| 1 + [f (x)]2 = 1 + - = 1 + = = .
x x2 |x| x4 |x| x4 |x| x2
" " "
" "
1 x4 + 1 1 x4 + 1
|S| = 2Ą |f(x)| 1 + [f (x)]2 dx = 2Ą dx = 2Ą dx =
|x| x2 x x2
1 1 1
" " " " " B
"
x4 + 1 x4 x2 1 1
= 2Ą dx 2Ą dx = 2Ą dx = 2Ą dx = 2Ą lim dx =
B"
x3 x3 x3 x x
1 1 1 1 1
B
= 2Ą lim (ln |x|) = 2Ą lim (ln |B| - ln |1|) = +".
B" B"
1
|S| = +"
+"
Ą
f [a, b)
lim f(x) = -" lim f(x) = +"
xb- xb-
�
F (�) = f(x) dx � b-
a
f a b
b
f(x) dx
a
b �
f(x) dx = lim F (�) = lim f(x) dx.
�b- �b-
a a
b
f
b
f(x) dx
a
f
(a, b]
b b
f(x) dx = lim F (�) = lim f(x) dx.
�a+ �a+
a �
x0 f [a, b]
b x0 b � b
f(x) dx = f(x) dx + f(x) dx = lim f(x) dx + lim f(x) dx.
�x0- �x0+
a a x0 a �
1
x
" dx.
1 - x2
0
x
f(x) = " [0, 1)
1 - x2
1
[ ]
x
0+
lim f(x) = lim " = +".
x1- x1-
1 - x2
1 �
�
" "
x x
" dx = lim " dx = lim - 1 - x2 = lim - 1 - �2 + 1 = 1.
�1-
1 - x2 �1- 1 - x2 �1-
0
0 0
 n
Rn = {(x1, x2, . . . , xn) : x1, . . . , xn " R}
n
D �" Rn
f : D (x1, x2, . . . , xn) - f(x1, x2, . . . , xn) " R,
(x1, x2, . . . , xn) " D
n
f
f Df
x2 + y2 - 1
f(x, y) = ln .
4 - x2 - y2
ńł
x2 + y2 - 1
�ł
x2 + y2 - 1
> 0
�!�! > 0 �!�!
4 - x2 - y2
ół 4 - x2 - y2
4 - x2 - y2 = 0

x2 + y2 - 1 > 0 x2 + y2 - 1 < 0
�!�! �!�!
4 - x2 - y2 > 0 4 - x2 - y2 < 0
x2 + y2 > 1 x2 + y2 < 1
�!�! �!�! 1 < x2 + y2 < 4.
x2 + y2 < 4 x2 + y2 > 4
(0, 0) 1 2
y
2
1
x
-2 -1 1 2
-1
-2
f : Df - R n
x1, x2, . . . , xn
{(x1, x2, . . . , xn, f(x1, x2, . . . , xn)) " Rn+1 : (x1, x2, . . . , xn) " Df}.
"
z = f(x, y) = Ax + By + C
Ł n = [-A, -B, 1]
(0, 0, C)
z
n
�
C
Ł : -Ax - By + z - C = 0
y
x
"
z = ą R2 - x2 - y2
+ -
R Oz
z
z
R
z = R2 - x2 - y2
R
R
y
y
R
R z = - R2 - x2 - y2
x
x R
"
z = a(x2 + y2)
z = ax2, y = 0
Oz
z
z = a(x2 + y2)
y
x
"
z = a x2 + y2
z = ax, y = 0 x 0
Oz
z
z = a x2 + y2
y
x
"
z = g x2 + y2
z = g(x), y = 0 x 0
Oz
z
z = g( x2 + y2)
y
x
" z = g(x) z = h(y)
z = g(x), y = 0 Oy
z = h(y), x = 0 Ox
z
z = g(x)
y
x
 z = f(x - a, y - b) + c z = f(x, y)
v = [a, b, c] z = -f(x - a, y - b) + c
z = f(x, y) Oxy
z z
z = f(x, y)
z = f(x - a, y - b) + c
z = f(x, y)
y
y
x
x
z = -f(x, y)
Rn
P = (x1, x2, . . . , xn) P = (x 1, x 2, . . . , x n)
�(P, P ) P P
n
�(P, P ) = (x i - xi)2.
i=1
r > 0 P0
O(P0, r) = {P : �(P0, P ) < r}.
P0 r
O(P0)
r > 0 P0
S(P0, r) = {P : 0 < �(P0, P ) < r}.
S(P0, r) = O(P0, r) \ {P0}
P0 A
A P0
A
P0 A
A
P0 A
A
 A
Rn
Rn
k k Pk = (xk, xk, . . . , xk)
1 2 n
(Pk) {Pk}
(Pk) Rn P0 Pk
P0
lim �(Pk, P0) = 0.
k"
lim �(Pk, P0) = 0 �!�! lim xk = x0 , 1 m n, m " N.
m m
k" k"
k
2
k
(xk, yk) = 2k + k3, 1 +
k
k2 + 3k - 5
(xk, yk) = , (-1)k
k2 + 1
k2 + k + 2 2k - 5k
(xk, yk) = ,
k + 1 10 � 4k + 32k
"
cos k 1 k + 2k2 k
(x1, x2, x3, x4) = , k sin , k2 + 1 - k,
k k k k
k + 1 k k + k2
k
2
k
lim (xk, yk) = lim 2k + k3, 1 + = (2, e2)
k" k"
k
k3 k3
k k
k
lim 2k + k3 = lim 2k 1 + = lim 2 � 1 + = 2.
k" k" k"
2k 2k
�!
1
k
k �2
2
2 [1"] 2
lim 1 + = 1 + = e2
k"
k k
k2 + 3k - 5
(xk, yk) = , (-1)k
k2 + 1
1 k
(-1)k =
-1 k
2
k + 1 +
k2 + k + 2 2k - 5k 2k - 5k
k
lim (xk, yk) = lim , = lim , =
1
k" k" k"
k + 1 10 � 4k + 32k 1 + 10 � 4k + 9k
k
k k
2 2 5
k + 1 + -
" + 1 + 0 0 - 0
k 9 9
lim , = , = (", 0)
1 k
4
k"
1 + 1 + 0 0 + 1
10 � + 1
k
9
cos k [0� ] 1
" lim = 0 cos k -1, 1
k"
k + 1 k + 1
0
1
sin
1
k
" lim k sin = lim = 1
1
k" k"
k
k
" "
"
k2 + 1 - k k2 + 1 + k
["-"] k2 + 1 - k2
" lim k2 + 1 - k = lim " = lim " =
k" k" k"
k2 + 1 + k k2 + 1 + k
1 1
lim " = = 0
k"
"
k2 + 1 + k
k + 2k2 k
" lim = [2"] = "
k"
k + k2
lim (x1, x2, x3, x4) = (0, 1, 0, ")
k k k k
k"
f : D - R D �" Rn P0 D
g
f P0 lim f(P ) = g f(P ) - g
P P0 P P0
{Pk} P0 Pk " D Pk = P0 k " N

{f(Pk)} g
lim f(P ) = g �!�! "{Pk} : "k " N : Pk " D, Pk = P0 Pk - P0 =�! f(Pk) - g .

P P0 k" k"
lim f(P ) = g �!�! " � > 0 " � > 0 " P " D : �(P, P0) < � =�! |f(P ) - g| < �.
P P0
lim f(P ) = " �!�! "{Pk} : "k " N : Pk " D, Pk = P0 Pk - P0 =�! f(Pk) - " .

P P0 k" k"
lim f(P ) = " �!�! " � > 0 " � > 0 " P " D : �(P, P0) < � =�! f(P ) - ".
P P0
(x, y) - f(x, y) lim lim f(x, y)
xx0 yy0
lim lim f(x, y)
yy0 xx0
lim f(x, y)
(x,y)(x0,y0)
 lim(lim f(x, y)) lim(lim f(x, y))
xa yb yb xa
x2 + y2
f(x, y) = a = " b = "
x2 + y4
y2
x2
+
x2 + y2 0 + 0
y4 y4
lim lim = lim lim = lim = 0,
y4 x"
x2
x" y" x" y"
x2 + y4 0 + 1
+
y4 y4
y2
x2
+
x2 + y2 1 + 0
x2 x2
lim lim = lim lim = lim = 1.
y4 y"
x2
y" x" y" x"
x2 + y4 1 + 0
+
x2 x2
x2 2xy
lim lim
(x,y)(0,0) x2 + y2 (x,y)(0,0) x2 + y2
x
x2y x + y
lim lim
(x,y)(0,0) x2 + y2 (x,y)(",5) x
2
sin(xy) ex +y2 - 1
lim lim
(x,y)(0,3) x (x,y)(0,0) x2 + y2
x2
lim
(x,y)(0,0) x2 + y2
(0, 0) Df = R2 \ {(0, 0)}
1 1 1 2
(xn, yn) = , (x n, yn) = , .
n n n n
1 1
1
n2 n2
lim f (xn, yn) = lim = lim =
1 1 2
n" n" n"
+ 2
n2 n2 n2
1 1
1
n2 n2
lim f (x n, yn) = lim = lim = .
1 4 5
n" n" n"
+ 5
n2 n2 n2
f (xn, yn) f (x n, yn)
x2
lim
(x,y)(0,0) x2 + y2
2xy
lim
(x,y)(0,0) x2 + y2
2
1 1 2
n2
lim f , = lim = = 1,
1 1
n" n"
n n + 2
n2 n2
2
1 2 2
n2
lim f , = lim = .
1 4
n" n"
n n + 5
n2 n2
x2y
lim
(x,y)(0,0) x2 + y2
f(xn, yn) (xn, yn) (0, 0)
Df = R2 \ {(0, 0)}
1 0
lim f 0, = lim = lim 0 = 0,
1
n" n" n"
n
n2
1 0
lim f , 0 = lim = lim 0 = 0,
1
n" n" n"
n
n2
1
1 1 1
n3
lim f , = lim = lim = 0,
2
n" n" n"
n n n
n2
12
2 3 12
n3
lim f , = lim = lim = 0,
13
n" n" n"
n n 13n
n2 1
1
1 1 n2
n7 n7
lim f , = lim = lim = lim = 0.
1 1
n5+1
n" n" n" n"
n2 n3 + n5 + 1
n4 n9
n9
(xn, yn) f(xn, yn) 0
2
(xn, yn) x2 + yn = 0

n
lim xn = lim yn = 0 an := max{|xn|, |yn|}
n" n"
2
lim an = 0 |xn| an |yn| an |x2 + yn| a2
n n
n"
1 1
2
|x2 + yn| a2
n n
x2yn |xn|2|yn| a2 � an
n n
0 d" |f(xn, yn)| = d" d" = an - 0.
2 2
x2 + yn |x2 + yn| a2
n n n
|f(xn, yn)| 0 f(xn, yn) 0
x2y
(xn, yn) lim = 0
(x,y)(0,0) x2 + y2
x x y
x
x + y [1"] y y
y
lim = lim 1 + = lim 1 + = e5
(x,y)(",5) x (x,y)(",5) x (x,y)(",5) x
�!
e
sin(xy) sin(xy)
lim = lim �y = 1 � 3 = 3
(x,y)(0,3) x (x,y)(0,3) xy
�!
1
2
ex +y2 - 1
lim x, y f
(x,y)(0,0) x2 + y2
x2 + y2 t
x2 + y2 = t
x 0 y 0 t 0
2
0
[ ]
ex +y2 - 1 et - 1
0
lim = lim = lim et = e0 = 1.
t0 t0
(x,y)(0,0) x2 + y2 t
f : D - R D �" Rn P0
D f P0
 lim f(P )
P P0
f(P0)
lim f(P ) = f(P0)
P P0
xy
"
(x, y) = (0, 0)

x2+y2
f(x, y) =
0 (x, y) = (0, 0)
xy
(x, y) = (0, 0) f f(x, y) =

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

f (0, 0) f(0, 0) = 0
f (0, 0)
xy
lim = 0.
(x,y)(0,0)
x2 + y2
0 (x - y)2 x, y " R
x2 + y2
xy
2
xy x2 + y2 x2 + y2 x2 + y2 (x,y)(0,0)
0 = = - 0.
2
x2 + y2 2 x2 + y2 2
xy
lim = 0
(x,y)(0,0)
x2 + y2
R2
f : D - R D �" Rn
P0 " D O(P0) P0
P " O(P0) )" D f
P1 P2
f(P1) f(P2) f(P1) < 0 f(P2) > 0 f
f : D - R D �" Rn n (x1, . . . , xn)
O(P0) P0 = (x0, . . . , x0)
1 n
xk
"f f(x0, . . . , x0 , x0 + "xk, x0 , . . . , x0) - f(x0, . . . , x0)
1 k-1 k k+1 n 1 n
(P0) = lim .
"xk0
"xk "xk
 f : R2 - R
"f f(x0 + "x, y0) - f(x0, y0)
(x0, y0) = lim ,
"x0
"x "x
"f f(x0, y0 + "y) - f(x0, y0)
(x0, y0) = lim .
"y0
"y "y
"f
(P0) = fx (P0)
k
"xk
f(x, y) = x2 + y2
"f f(x + "x, y) - f(x, y) (x + "x)2 + y2 - x2 + y2
(x, y) = lim = lim =
"x0 "x0
"x "x "x
(x + "x)2 + y2 - x2 + y2 � (x + "x)2 + y2 + x2 + y2
= lim =
"x0
"x � (x + "x)2 + y2 + x2 + y2
(x + "x)2 + y2 - x2 - y2
= lim =
"x0
"x � (x + "x)2 + y2 + x2 + y2
x2 + 2x"x + ("x)2 - x2
= lim =
"x0
"x � (x + "x)2 + y2 + x2 + y2
2x"x + ("x)2
= lim =
"x0
"x � (x + "x)2 + y2 + x2 + y2
"x(2x + "x)
= lim =
"x0
"x � (x + "x)2 + y2 + x2 + y2
2x + "x 2x + 0
= lim = =
"x0
(x + "x)2 + y2 + x2 + y2 (x + 0)2 + y2 + x2 + y2
2x 2x x
= = = .
x2 + y2 + x2 + y2 2 x2 + y2 x2 + y2
"f f(x, y + "y) - f(x, y) x2 + (y + "y)2 - x2 + y2
(x, y) = lim = lim =
"y0 "y0
"y "y "y
x2 + (y + "y)2 - x2 + y2 � x2 + (y + "y)2 + x2 + y2
= lim =
"y0
"y x2 + (y + "y)2 + x2 + y2
x2 + (y + "y)2 - x2 - y2
= lim =
"y0
"y x2 + (y + "y)2 - x2 + y2
(y + "y)2 - y2
= lim =
"y0
"y x2 + (y + "y)2 + x2 + y2
y2 + 2y"y + ("y)2 - y2
= lim =
"y0
"y x2 + (y + "y)2 + x2 + y2
2y"y + ("y)2
= lim =
"y0
"y x2 + (y + "y)2 + x2 + y2
"y(2y + "y)
= lim =
"y0
"y x2 + (y + "y)2 + x2 + y2
2y + "y 2y y
= lim = = .
"y0
x2 + (y + "y)2 + x2 + y2 2 x2 + y2 x2 + y2
y
f(x, y) = arctg
x
"f
(x, y) y
"x
"f
(x, y) x
"y
"f 1 y 1 y x2 y -y
(x, y) = � - = � - = � - = ,
2
y
"x x2 x2 + y2 x2 x2 + y2 x2 x2 + y2
1 +
x
x2
"f 1 1 1 1 x2 1 x
(x, y) = � = � = � = .
2
y
"y x x2 + y2 x x2 + y2 x x2 + y2
1 +
x
x2
pv = RT
"p "v "T
-1
"v "T "p
p
RT
p = .
v
p T v
"p RT
= - .
"v v2
RT "v R
v = , = ,
p "T p
pv "T v
T = , = .
R "p R
"p "v "T RT R v RT
d � � = - � � = = -1.
"v "T "p v2 p R vp
f (x0, y0) ą
f
y = y0 (x0, y0, f(x0, y0)) Oxy �
f x = x0
"f "f
(x0, y0) = tg ą, (x0, y0) = tg �.
"x "y
"f
(x0, y0) f
"x
x y
"f
(x0, y0) n
"x
f : D - R D �" Rn
xk D
"f "f
: D P - (P ) " R
"xk "xk
f xk D
f : D - R D �" Rn n (x1, . . . , xn)
"f
"xk
f
"2f " "f "2f " "f "2f " "f "2f " "f
= , = , = , = .
"x2 "x "x "x"y "x "y "y2 "y "y "y"x "y "x
f : D - R D �" Rn n (x1, . . . , xn)
"2f
i = j

"xi"xj
f n D �" Rn
"2f "2f
"xi"xj "xj"xi
"2f "2f
= .
"xi"xj "xj"xi
n
n
(n - 1) n
n n
n
f (x0, y0) k
x l y k + l = n
"nf
(x0, y0)
"xk"yl
 Ck(D)
D k
f(x, y) = xy x, y > 0
"f
(x, y) = yxy-1,
"x
"f
(x, y) = xy ln x,
"y
"2f
(x, y) = y(y - 1)xy-2,
"x2
"2f
(x, y) = xy ln2 x,
"y2
"2f " 1
(x, y) = (xy ln x) = yxy-1 ln x + xy � = yxy-1 ln x + xy-1 = xy-1 (y ln x + 1) ,
"x"y "x x
"2f "
(x, y) = (yxy-1) = xy-1 + yxy-1 ln x = xy-1 (1 + y ln x) .
"y"x "y
f : R2 �" D - R
O(P0) P0 = (x0, y0) P = (x0 + "x, y0 + "y)
"f = f(x0 + "x, y0 + "y) - f(x0, y0)
f P0 P
f P0
A1 A2 "f
"f = A1"x + A2"y + o(�),
� = ("x)2 + ("y)2 P0 P o(�)
o(�)
� � 0 o(�) 0 lim = 0
�o
�
"x "y "f f
P0 "x "y df(P0)("x, "y) dP f("x, "y)
0
df(P0) dP f
0
"f "f
df(P0)("x, "y) = A1"x + A2"y A1 = (P0), A2 = (P0).
"x "y
"x "y dx dy
"f "f
df(P0)(dx, dy) = (P0)dx + (P0)dy.
"x "y
f P0 = (x0, y0)
"f = f(x0 + "x, y0 + "y) - f(x0, y0) = A1"x + A2"y + o(�).
"x = 0 "y = 0

f(x0 + "x, y0 + "y) - f(x0, y0) o(|"x|)
= A1 + .
"x "x
"f
"x "x 0 (P0) = A1
"x
"f
(P0) = A2
"y
(x0, y0)
f (x0, y0)
(x0, y0, f(x0, y0))
"f "f
(x0, y0)(x - x0) + (x0, y0)(y - y0) - z + f(x0, y0) = 0.
"x "y

x cos y
f(x, y) = (x0, y0) = (1, Ą)
R2
"f
(x, y) = ex cos y cos y
"x
"f
(x, y) = ex cos y(-x sin y)
"y
"f "f
 : (x0, y0)(x - x0) + (x0, y0)(y - y0) - z + f(x0, y0) = 0,
"x "y
 : ecos Ą cos Ą(x - 1) + ecos Ą(- sin Ą)(y - Ą) - z + ecos Ą = 0,
 : -e-1(x - 1) + e-1 � 0 � (y - Ą) - z + e-1 = 0,
 : -e-1x - z + 2e-1 = 0
 : x + ez - 2 = 0.
n
n
 n f : Rn �" D - R
O(P0) P0 = (x0, . . . , x0) P
1 n
(x0 + "x1, . . . , x0 + "xn) f P0
1 n
A1, A2, . . . , An � =
n
("xi)2
i=1
n
"f = Ai"xi + o(�).
i=1
f
df(P0)("x1, . . . , "xn) dP f("x1, . . . , "xn)
0
"f = df + o(�),
"f - df = o(�).
"f - df df - "f � 0
�
df - "f
lim = 0.
�0
�
"f H" df
df - "f � � 0
"f
"x "y
� � 0
f(x0 + "x, y0 + "y) H" f(x0, y0) + d(x ,y0)f("x, "y)
0
"x "y
� = ("x)2 + ("y)2
(1, 02)3 + (1, 97)3.
f(x, y) = x3 + y3 (x0, y0) = (1, 2) f
(x0, y0) f(1, 2) = 3
"f 3x2 "f 1
(x, y) = , (1, 2) = ,
"x 2
2 x3 + y3 "x
"f 3y2 "f
(x, y) = , (1, 2) = 2.
"y
2 x3 + y3 "y
 f (1, 2) "x = 0, 02 "y =
-0, 03
"f "f 1
d(1,2)f(0, 02, -0, 03) = (1, 2)"x + (1, 2)"y = � 0, 02 + 2 � (-0, 03) = -0, 05.
"x "y 2
(1, 02)3 + (1, 97)3 H" f(1, 2) + d(1,2)f(0, 02, -0, 03) = 3 - 0, 05 = 2, 95.
(1, 02)3 + (1, 97)3 H" 2, 9507.
"x x x0
"x = x - x0.
|"x| = |x - x0|.
|"x|
�x = .
|x0|
|"x|
�x = � 100%.
|x0|
U
R = ,
I
U = 220 dU = ą2
I = 30 dI = ą0, 5
dU dI "R "R
dU dI
"R dR
"R H" dR.
1 U
dR = dU - dI,
I I2
1 U
|dR| |dU| + |dI|
I I2
1 U
|"R| |dU| + |dI|.
I I2
1 220
|"R| � 2 + � 0, 5 = 0, 19&!.
30 302
&!
|"R| |dU| |dI|
+ .
R U I
|"R| 2 0.5
�x = + = 0, 026.
R 220 30
U = 220 I = 30
220
R = ą 0, 19 H" 7, 33 ą 0, 19&!.
30
f : Rn �" D - R
D df
"xi i = 1, 2, . . . , n D
d2f(P0)("x1, . . . , "xn)
d2 f("x1, . . . , "xn)
P0
d2f = d(df).
n
dnf = d(dn-1f), n = 2, 3, 4, . . . .
"f "f " "f "f " "f "f
d2f = d(df) = d dx + dy = dx + dy dx + dx + dy dy =
"x dy "x "x "y "y "x "y
"2f "2f "2f "2f
= dx + dy dx + dx + dy dy.
"x2 "x"y "y"x "y2
"2f "2f "2f
d2f = (dx)2 + 2 dxdy + (dy)2.
"x2 "x"y "y2
"3f "3f "3f "3f
d3f = (dx)3 + 3 (dx)2dy + 3 dx(dy)2 + (dy)3.
"x3 "x2"y "x"y2 "y3
n
n "nf
dnf = (dx)n-k(dy)k.
k "xn-k"yk
k=0
 z = f(x, y) C1
O(P0) P0 = (x0, y0) P0s
P0 s P
P0 O(P0)
f(P ) - f(P0)
lim
P P0
�(P0P )
f P0s
df
(P0)
ds
df f(P ) - f(P0)
(P0) = lim .
P P0
ds �(P0P )
|v| v = [v1, v2, . . . , vn] �" Rn
2 2
2
|v| = v1 + v2 + � � � + vn.
s [sx, sy]
f P0 = (x0, y0) s
df f(x0 + tsx, y0 + tsy) - f(x0, y0)
(x0, y0) = lim
ds t0+ t
� Oxy
f x = x0
y = y0 s
df
(x0, y0) = tg �.
ds
s
f(x1, x2, . . . , xn) C1
P0 " Rn
"f "f
(P0), . . . , (P0)
"x1 "xn
f P0 f(P0)
n = 2
"f "f
f(x0, y0) = (x0, y0)� + (x0, y0)5
,
"x "y
 � = [1, 0] 5
= [0, 1] Ox Oy Oxy n = 3
"f "f "f
Ć
f(x0, y0, z0) = (x0, y0, z0)� + (x0, y0, z0)5
+ (x0, y0, z0)k,
"x "y "z
Ć
� = [1, 0, 0] 5
= [0, 1, 0] k = [0, 0, 1] Ox Oy Oz
Oxyz
z = 2 - x2 - y2
P = (1, 1, 0)
(1, 1)
"f "f
(x, y) = -2x, (x, y) = -2y.
"x "y
("f)(1, 1) = [-2, -2] .
"f "f
(x0, y0)
"x "y
s = [sx, sy]
df
(x0, y0) = f(x0, y0, z0) ć% s,
ds
df "f "f
(x0, y0) = (x0, y0), (x0, y0) ć% [sx, sy].
ds "x "y
ć%
"f "f "f "f
(x0, y0), (x0, y0) ć% [sx, sy] = (x0, y0) � sx + (x0, y0) � sy.
"x "y "x "y
f : Rn �" D - R Cn
O(P0) P0 = (x0, . . . , x0) P = (x0 + h1, . . . , x0 + hn) " O(P0)
1 n 1 n
� " (0, 1)
1 1 1 1
f(P ) = f(P0) + dP f(h) + d2 f(h) + . . . + dn-1f(h) + dn f(h),
0 P0 P0 P0+�h
1! 2! (n - 1)! n!
h = (h1, . . . , hn)
f : D - R D �" R2 Cn
O(P0) P0 = (x0, y0) P = (x0 + h, y0 + k) " O(P0)
� " (0, 1)
1 1 1
f(x0+h, y0+k)=f(x0, y0)+ d(x ,y0)f(h, k)+ d2 f(h, k)+. . .+ dn-1 f(h, k)+Rn,
0 (x0,y0)
(x0,y0)
1! 2! (n - 1)!
1
Rn = dn )f(h, k)
(x0+�h,y0+�k
n!
n
 P0 = (0, . . . , 0)
f(x, y) = ex+y
(1, -1) n = 4 R4
P0 = (x0, y0) = (1, -1) P = (x0 + h, y0 + k) = (x, y) h = x - x0 =
x - 1 k = y - y0 = y + 1
"f "2f "3f
0
(x0, y0) = (x0, y0) = (x0, y0) = ex +y0 = e0 = 1,
"x "x2 "x3
"f "2f "3f
0
(x0, y0) = (x0, y0) = (x0, y0) = ex +y0 = e0 = 1,
"y "y2 "y3
"2f "3f "3f
0
(x0, y0) = (x0, y0) = (x0, y0) = ex +y0 = e0 = 1.
"x"y "x2"y "x"y2
(1, -1) f(x0, y0) = f(1, -1) = 1
"f "f
d(x ,y0)f(h, k) = (x0, y0)h + (x0, y0)k = x - 1 + y + 1 = x + y,
0
"x "y
"2f "2f "2f
d2 f(h, k) = (x0, y0)h2 + 2 (x0, y0)hk + (x0, y0)k2 =
(x0,y0)
"x2 "x"y "y2
= (x - 1)2 + 2(x - 1)(y + 1) + (y + 1)2 = (x + y)2,
"3f "3f "3f "3f
d3 f(h, k) = (x0, y0)h3 + 3 (x0, y0)h2k + 3 (x0, y0)hk2 + (x0, y0)k3 =
(x0,y0)
"x3 "x2"y "x"y2 "y3
= (y + 1)3 + 3(x - 1)2(y + 1) + 3(x - 1)(y + 1)2 + (y + 1)3 = (x + y)3.
1 1
ex+y = f(x0, y0) + d(x ,y0)f(h, k) + d2 f(h, k) + d3 f(h, k) + R4 =
0 (x0,y0) (x0,y0)
2 6
1 1
= 1 + x + y + (x + y)2 + (x + y)3 + R4 =
2 6
1 1 1 1 1 1
= 1 + x + y + x2 + y2 + xy + x3 + y3 + x2y + xy2 + R4,
2 2 6 6 2 2
1
R4 = e�(x+y) (x - 1)4+4(x - 1)3(y + 1)+6(x - 1)2(y + 1)2+4(x - 1)(y + 1)3+(y + 1)4
24
f : D - R D �" R2 x, y
(x0, y0) f (x0, y0)
S(x0, y0) (x0, y0) (x, y)
f(x, y) f(x0, y0).
f (x0, y0)
S(x0, y0) (x0, y0) (x, y)
f(x, y) f(x0, y0).
 f : D - R D �" R2 D
"(x, y) " D f(x, y) f(x0, y0) f (x0, y0)
"(x, y) " D f(x, y) f(x0, y0)
f (x0, y0)
f : D - R D �" R2 (x0, y0)
"f "f
(x0, y0) = 0 (x0, y0) = 0.
"x "y
P0 = (x0, y0)
f
f
ńł
"f
�ł
�ł
(x0, y0) = 0
�ł
"x
.
"f
�ł
�ł
ół (x0, y0) = 0
"y
f f(x, y) = x2 + y2 (0, 0)
f (0, 0)
z
f(x, y) = x2 + y2
y
x
f (0, 0)
"f f(0 + "x, 0) - f(0, 0) f("x, 0) - f(0, 0)
(0, 0) = lim = lim =
"x0 "x0
"x "x "x
"
("x)2 + 02 - 02 + 02 ("x)2 |"x|
= lim = lim = lim =
"x0 "x0 "x0
"x "x "x
1, "x > 0
= .
-1, "x < 0
x y
 f : D - R
D �" R2 (x, y) " D W
fxx(x, y) fxy(x, y) 2
W (x, y) = = fxx(x, y) � fyy(x, y) - fxy(x, y)
fyx(x, y) fyy(x, y)
f
f : D - R D �" R2 (x0, y0) C2
fx(x0, y0) = 0 fy(x0, y0) = 0
fxx(x0, y0) fxy(x0, y0)
W (x0, y0) = > 0
fyx(x0, y0) fyy(x0, y0)
f (x0, y0)
" fxx(x0, y0)
" fxx(x0, y0)
W (x0, y0) < 0 f (x0, y0)
f W (x0, y0) = 0
f
f
f(x, y) = x2 + y2 - 8 ln(xy)
1ć%
xy > 0 �! (x > 0, y > 0) (" (x < 0, y < 0).
f (x, y) x = 0 y = 0
2ć%
"f 1 8
= 2x - 8 � � y = 2x - ,
"x xy x
"f 1 8
= 2y - 8 � � x = 2y - .
"y xy y
3ć%
ńł
ńł
"f 8
�ł
�ł �ł
2x
�ł = 0 �ł - = 0
x2 - 4 = 0 (x + 2)(x - 2) = 0
"x x
�! �! �! �!
8
"f
�ł �ł
�ł ół y2 - 4 = 0 (y + 2)(y - 2) = 0
2y - = 0
= 0
ół
y
"y
x = -2 (" x = 2
�! .
y = -2 (" y = 2
P1 = (2, 2) P2 = (-2, -2) P3 = (2, -2) P4 = (-2, 2)
P1 P2 f
P1 P2
4ć%
"2f 8
= 2 + ,
"x2 x2
"2f 8
= 2 + ,
"y2 y2
"2f
= 0,
"x"y
"2f
= 0.
"y"x
5ć% P1 P2
P1
�ł łł �ł łł
8 8
2 + 0 2 + 0
4 0
�ł śł �ł śł
x2 x2
det (P1) = det (2, 2) = det = 16 > 0.
�ł �ł �ł �ł
8 8
0 4
0 2 + 0 2 +
y2 y2
"2f
f P1 (P1) = 4 > 0
"x2
f(P1) = f(2, 2) = 8 - 8 ln 4
P2
�ł łł �ł łł
8 8
2 + 0 2 + 0
4 0
�ł śł �ł śł
x2 x2
det (P2) = det (-2, = 16 > 0.
�ł �ł �ł �ł -2) = det
8 8
0 4
0 2 + 0 2 +
y2 y2
"2f
f P2 (P2) = 4 > 0
"x2
f(P2) = f(-2, -2) = 8 - 8 ln 4
n
n f : D - R
D �" Rn n n 2 a " D f
a S(a)
"x " S(a) f(x) f(a) "x " S(a) f(x) f(a) .
n
a f : D - R
D �" Rn a " D
a f a daf = 0
f
f a
n
f : D - R D �" Rn a " D
daf = 0
d2f < 0 d2f > 0
a a
 f a
f
(x0, y0, z0)
"f "f "f
(x0, y0, z0) = 0 (x0, y0, z0) = 0 (x0, y0, z0) = 0
"x "y "z
"2f
A = (x0, y0, z0),
"x2
�ł łł
"2f "2f
�ł śł
"x2 "x"y
�ł śł
B = det (x0, y0, z0),
�ł �ł
"2f "2f
"y"x "y2
�ł łł
"2f "2f "2f
�ł śł
"x2 "x"y "x"z
�ł śł
�ł śł
"2f "2f "2f
�ł śł
C = det (x0, y0, z0).
�ł śł
"y"x "y2 "y"z
�ł śł
�ł �ł
"2f "2f "2f
"z"x "z"y "z2
A > 0, B > 0, C > 0 f
A < 0, B > 0, C < 0 f
A, B, C ABC = 0 f

f(x, y, z) = 2x2 + y2 + z2 + 2xy - 4y + z.
R3
"f "f "f
(x, y, z) = 4x + 2y, (x, y, z) = 2y + 2x - 4, (x, y, z) = 2z + 1.
"x "y "z
ńł
"f
�ł
�ł
(x, y, z) = 0
�ł
�ł
"x
�ł
�ł
"f
.
(x, y, z) = 0
�ł
"y
�ł
�ł
�ł
"f
�ł
ół
(x, y, z) = 0
"z
(x0, y0, z0) = -2, 4, -1
2
(x0, y0, z0)
"2f "2f "2f
(x0, y0, z0) = 4, (x0, y0, z0) = 2, (x0, y0, z0) = 2,
"x2 "y2 "z2
"2f "2f
(x0, y0, z0) = (x0, y0, z0) = 2,
"x"y "y"x
"2f "2f
(x0, y0, z0) = (x0, y0, z0) = 0,
"x"z "z"x
"2f "2f
(x0, y0, z0) = (x0, y0, z0) = 0.
"y"z "z"y
"2f 1
A = -2, 4, - = 4 > 0,
"x2 2
�ł łł
"2f "2f
�ł śł
1
4 2
"x2 "x"y
�ł śł
B = det -2, 4, - = det = 4 > 0.
�ł �ł
"2f "2f
2 2
2
"y"x "y2
�ł łł
"2f "2f "2f
�ł śł �ł łł
"x2 "x"y "x"z
�ł śł
4 2 0
�ł śł
1
"2f "2f "2f
�ł śł �ł �ł
C = det -2, 4, - = det 2 2 0 = 8 > 0.
�ł śł
2
"y"x "y2 "y"z
�ł śł
0 0 2
�ł �ł
"2f "2f "2f
"z"x "z"y "z2
A B C f -2, 4, -1
2
33
f -2, 4, -1 =
2
4
"2f 1 "2f 1 "2f 1
d2f -2, 4, -1 = -2, 4, - (dx)2 + -2, 4, - (dy)2 + -2, 4, - (dz)2 +
2
"x2 2 "y2 2 "z2 2
"2f 1 "2f 1
+2 -2, 4, - dxdy + 2 -2, 4, - dxdz +
"x"y 2 "x"z 2
"2f 1
+2 -2, 4, - dydz = 4 (dx)2 + 4dxdy + 2 (dy)2 + (dz)2 =
"y"z 2
= 4 (dx)2 + 4dxdy + (dy)2 + (dy)2 + (dz)2 =
= (2dx + dy)2 + (dy)2 + (dz)2 > 0.
-2, 4, -1
2
f
33
f -2, 4, -1 =
2
4
f : D - R D �" Rn
D f D
D
D f
D
D
f(x, y) = x2 - y2
x2 + y2 d" 4
1ć%
"f "f
= 2x, = -2y.
"x "y
ńł
"f
�ł
�ł
�ł = 0
"x
(0, 0) f(0, 0) = 0
"f
�ł
�ł
= 0
ół
"y
2ć%
"
x " (-2, 2) y = 4 - x2
"
g(x) = f(x, 4 - x2) = x2 - 4 + x2 = 2x2 - 4
x = 0 (0, 2) f
f(0, 2) = -4.
"
x " (-2, 2) y = - 4 - x2
"
h(x) = f(x, - 4 - x2) = x2 - 4 + x2 = 2x2 - 4
x = 0 (0, -2) f
f(0, -2) = -4.
(2, 0) (-2, 0)
f(2, 0) = f(-2, 0) = 4.
max{-4, 0, 4} = 4, min{-4, 0, 4} = -4
f
f -4
F : R2 �" D R2
y = f(x)
F (x, y) = 0
f
F (x, y) = 0
3x-y -4 = 0
y = 3x - 4 x - y2 - = 0
" "1
[1, ") y = x - 1 y = - x - 1
x4 + y2 + 2 = 0
 F
C1 (x0, y0)
F (x0, y0) = 0
Fy(x0, y0) = 0

F (x, y) = 0 x0
y = f(x) x0 y0 = f(x0)
x0
Fx(x, f(x))
f (x) = - .
Fy(x, f(x))
F C2 f
Fxx(Fy)2 - 2FxyFxFy + Fyy(Fx)2
f (x) = - ,
(Fy)3
(x, f(x)))
F C2
F (x0, y0) = 0 Fx(x0, y0) = 0 Fy(x0, y0) = 0

Fxx(x0, y0)
I(x0, y0) = - = 0

Fy(x0, y0)
f F (x, y) = 0 x0
y0
" I(x0, y0) < 0
" I(x0, y0) > 0
y = f(x) F (x, y) = 0
F
F (x, y) = 0
.
Fx(x, y) = 0
(x1, y1) (x2, y2) (xn, yn)
F x
Fxx(x, y)
I(x, y) = - .
Fy(x, y)
 (x1, y1) (x2, y2) (xn, yn)
I(x, y)
f
x2 - 2xy - 3y2 + 4 = 0.
F (x, y) = x2 - 2xy - 3y2 + 4.
Fx(x, y) = 2x - 2y, Fy(x, y) = -2x - 6y.
F (x, y) a" x2 - 2xy - 3y2 + 4 = 0
.
Fx(x, y) a" 2x - 2y = 0
P1 = (1, 1) P2 = (-1, -1)
F x
Fxx(x, y) = 2,
Fxx(x, y) 2 1
I(x, y) = - = - = .
Fy(x, y) -2x - 6y x + 3y
1
I(x, y) P1 P2 I(P1) = > 0 f
4
x1 = 1 P1 y1 = 1
P1 I(P2) = -1 < 0 f x2 = -1
4
P2 y2 = -1 P2
lim(lim f(x, y)) lim(lim f(x, y))
xa yb yb xa
xy
f(x, y) = a = " b = 0+
1 + xy
xy 1 1
lim lim = lim = ,
x" x"
y0+ 1 + xy 1 + 1 2
xy 1 1
lim lim = lim lim = lim = 1.
y0+ x" 1 + xy y0+ x" 1 + 1 y0+ 0 + 1
xy
x3 + y3 1 - cos(x2 + y2)
lim lim
(x,y)(0,0) x2 + y2 (x,y)(0,0) (x2 + y2)2
9 + x2 + y2 - 3
lim
(x,y)(0,0) x2 + y2
x3 + y3
lim = 0 xn yn
(x,y)(0,0) x2 + y2
2
x2 + yn > 0 lim xn = lim yn = 0
n
n" n"
an := max{|xn|, |yn|} lim an = 0
n"
3
x3 + yn |xn|3 + |yn|3 a3 + a3
n n n
0 d" |f(xn, yn)| = d" d" = 2an.
2 2
x2 + yn |x2 + yn| a2
n n n
|f(xn, yn)| 0 f(xn, yn) 0
x3 + y3
{xn} {yn} lim = 0
(x,y)(0,0) x2 + y2
1 - cos(x2 + y2)
lim
(x,y)(0,0) (x2 + y2)2
x2 + y2 = t
0 0
1 - cos(x2 + y2) 1 - cos t [ ] sin t [ ] cos t 1
0 0
lim = lim = lim = lim =
t0 t0 t0
(x,y)(0,0) (x2 + y2)2 x2+y2=t t2 2t 2 2
1 - cos(x2 + y2) [1 - cos(x2 + y2)] [1 + cos(x2 + y2)]
lim = lim =
(x,y)(0,0) (x2 + y2)2 (x,y)(0,0) (x2 + y2)2 [1 + cos(x2 + y2)]
1 - cos2(x2 + y2)
= lim =
(x,y)(0,0) (x2 + y2)2 [1 + cos(x2 + y2)]
sin2(x2 + y2)
= lim =
(x,y)(0,0) (x2 + y2)2 [1 + cos(x2 + y2)]
sin2(x2 + y2) 1 1 1
= lim � = 1 � = .
(x,y)(0,0) (x2 + y2)2 2 2
1 + cos(x2 + y2)
�!
�!
1
1
9 + x2 + y2 - 3 9 + x2 + y2 - 3 9 + x2 + y2 + 3
lim = lim � =
(x,y)(0,0) x2 + y2 (x,y)(0,0) x2 + y2
9 + x2 + y2 + 3
9 + x2 + y2 - 9
= lim =
(x,y)(0,0)
(x2 + y2)( 9 + x2 + y2 + 3)
x2 + y2
= lim =
(x,y)(0,0)
(x2 + y2)( 9 + x2 + y2 + 3)
1 1
= lim = .
(x,y)(0,0) 6
9 + x2 + y2 + 3
3
2
f(x, y) = (x2 + y2)
R2
1 1
"f 3
2 2
(x, y) = x2 + y2 2x = 3x x2 + y2
"x 2
1 1
"f 3
2 2
(x, y) = x2 + y2 2y = 3y x2 + y2
"y 2
("f)(x, y) f(x, y)
1 1
"f "f
2 2
("f)(x, y) = (x, y), (x, y) = 3x x2 + y2 , 3y x2 + y2 .
"x "y
 A = {(x, y) : |("f)(x, y)| = 2} A
|("f)(x, y)| = 2,
2 2
1 1
2 2
3x (x2 + y2) + 3y (x2 + y2) = 2,
2 2
1 1
2 2
3x x2 + y2 + 3y x2 + y2 = 4,
9x2(x2 + y2) + 9y2 x2 + y2 = 4,
(9x2 + 9y2) x2 + y2 = 4,
2
9 x2 + y2 = 4,
4
2
x2 + y2 = ,
9
2
x2 + y2 = .
3
2
(0, 0)
3
"
y
f(x, y) = (x2 + y)
y
1ć% > 0 y f = R2
f
2ć%
"
"f
y
= 2x ,
"x
" " "
"f 1 1 1" y
y
y y y
"
= + (x2 + y) = + (x2 + y) = (2 + x2 + y).
y
"y 2 2
2
3ć%
ńł
" "
ńł
"f
�ł
y y
�ł
2x = 0 / :
�ł = 0 �ł "
x = 0 x = 0
y
=0

"x
�! �!�! �! .
"
1" y
"f
�ł ół
y
�ł 2 + x2 + y = 0 y = -2
(2 + x2 + y) = 0 / :
= 0
ół
2
"y
P (0, -2)
4ć%
"
"2f
= 2 ey,
"x2
"2f 1 1 1" y 1" y 1" y 1" y
y
= " (2 + x2 + y) + = (2 + x2 + y) + = (2 + x2 + y + 2),
y
"y2 2 2 4 2 4
2
"
"2f 1
y
y
"
= 2x = x ,
y
"x"y
2
"
"2f 1" y
y
= � 2x = x .
"y"x 2
5ć% P
"2f 2
(0, -2) = ,
"x2
"2f 1
(0, -2) = ,
"y2 2
"2f
(0, -2) = 0,
"x"y
"2f
(0, -2) = 0.
"y"x
�ł łł
�ł łł
"2f "2f
2
0
�ł śł
1
�ł śł
"x2 "x"y
�ł śł
(P ) = det (0, -2) = det = > 0.
�ł �ł
�ł �ł 1 2
"2f "2f
0
2
"y"x "y2
"2f 2
f P (0, -2) = > 0
"x2
2
f(P ) = f(0, -2) = -
-x2-xy-y2
f(x, y) =
1ć% = R2
f
2ć%
"f
-x2-xy-y2
= (-2x - y),
"x
"f
-x2-xy-y2
= (-x - 2y).
"y
3ć%
ńł
"f
�ł
�ł
-x2-xy-y2
�ł = 0
(-2x - y) = 0 -2x - y = 0 x = 0
-x2-xy-y2
>0
"x
�! �!�! �! .
"f
�ł -x2-xy-y2
-x - 2y = 0 y = 0
�ł (-x - 2y) = 0
= 0
ół
"y
P (0, 0)
4ć%
"2f
-x2-xy-y2 -x2-xy-y2 -x2-xy-y2
= (-2x - y)(-2x - y) + (-2) = (4x2 + 4xy + y2 - 2),
"x2
"2f
-x2-xy-y2 -x2-xy-y2 -x2-xy-y2
= (-x - 2y)(-x - 2y) + (-2) = (x2 + 4xy + 4y2 - 2),
"y2
"2f
-x2-xy-y2 -x2-xy-y2 -x2-xy-y2
= (-2x-y)(-x-2y)+ (-1) = (2x2+5xy+2y2-1),
"x"y
"2f
-x2-xy-y2 -x2-xy-y2 -x2-xy-y2
= (-x-2y)(-2x-y)+ (-1) = (2x2+5xy+2y2-1).
"y"x
5ć% P
"2f
(0, 0) = -2,
"x2
"2f
(0, 0) = -2,
"y2
"2f
(0, 0) = -1,
"x"y
"2f
(0, 0) = -1.
"y"x
�ł łł
"2f "2f
�ł śł
-2 -1
"x2 "x"y
�ł śł
(P ) = det (0, 0) = det = 3 > 0.
�ł �ł
"2f "2f
-1 -2
"y"x "y2
"2f
f P (0, 0) = -2 < 0
"x2
f(P ) = f(0, 0) = 1
f(x, y) = 1 - x2 + y2
R2 x2+y2 e" 0
x y
"f -x
= ,
"x
x2 + y2
"f -y
= .
"y
x2 + y2
ńł R2 \ {(0, 0)}
"f
�ł
�ł
�ł = 0
"x
(0, 0)
"f
�ł
�ł
= 0
ół
"y
f (0, 0)
f(0, 0) = 1 (0, 0) f
f(0, 0) = 1 f (0, 0)
x y z x > 0 y > 0
z > 0 4x + 4y + 4z = 48 z = 12 - x - y
V (x, y) = xy(12 - x - y)
x y
"V
= 12y - 2xy - y2,
"x
"V
= 12x - x2 - 2xy.
"y
ńł
"V
�ł
�ł
�ł = 0
"x
P1 = (0, 0) P2 = (0, 12) P3 = (12, 0) P4 = (4, 4)
"V
�ł
�ł
= 0
ół
"y
P1, P2, P3 V
"2V
= -2y,
"x2
"2V
= -2x,
"y2
"2V
= 12 - 2x,
"x"y
"2V
= 12 - 2x.
"y"x
P4
�ł łł
"2V "2V
�ł śł
-8 4
"x2 "x"y
�ł śł
W (P4) = det (4, 4) = det > 0.
�ł �ł
"2V "2V
4 -8
"y"x "y2
P4 = (4, 4) V z = 12 - 4 - 4 = 4
"
"
f(x, y) = x - y
"
f(x, y) = e2x - ey
f(x, y) = ln(x2 - y2) + 4 ln2 y
x+2
f(x, y) = arccos
y+2
f(x, y) = arcsin xy
"
f(x, y) = Ąy - 4yarctg x
ln y
f(x, y) =
ln(y-x)
ln(x+y)
f(x, y) =
arccos(y-x)
f(x, y) = 3x2y - y2
"
f(x, y) = x3 + xy - 2y
f(x, y) = yex+xy
f(x, y) = x2ex-3y
arccos x
f(x, y) =
x+y
f(x, y) = x arcsin(y - Ąx)
f(x, y) = (y + x) ln2(1 - x - y)
x ln y
f(x, y) =
4+xy ln x
f(x, y) = e3xarctg (xy)
"
2
f(x, y) = (xy2 + 1)arctg (y x)
f(x, y) = x4y2
f(x, y) = 2x2 + 3xy - 4y2
f(x, y) = y ln x + x2 ln y
f(x, y) = x2 sin2 y
y
f(x, y) = arctg
x
f(x, y) = 2xy-3
f(x, y) = ln(x2 + xy + y2)
"f "f
x (x, y) + y (x, y) = 2.
"x "y
f(x, y) = xyyx
"f "f
x (x, y) + y (x, y) = (x + y + ln x)f(x, y).
"x "y
g C2
z = y � g(x2 - y2)
1 "z 1 "z z
+ = .
x "x y "y y2
x
y2
z = e
"z "z
2x + y = 0.
"x "y
x - y
u = x +
y - z
"u "u "u
+ + = 1.
"x "y "z
x
f(x, y) = 2 cos2 y -
2
"2f "2f
2 (x, y) + (x, y) = 0.
"x2 "x"y
"2u "2u
(x, y) + (x, y) u
"x2 "y2
"u
"2u "2u
"u = + .
"x2 "y2
"u = 0
n n > 2
f(x, y) = ln(x2 + y2)
"
f(x, y) = xy
21 805
x = y =
10 100
(1, 02)3,01
 f(x, y) = x2 + 2xy + 3y2
f(x, y) = 3x2 + 2xy + 2y2 - 3x - 2y
f(x, y) = x3 + y3 + 3xy
f(x, y) = x3 + 3xy - y3
f(x, y) = 2xy - x2 - 2y3 - 4y2
f(x, y) = x4 + y3 + 32x - 9y
f(x, y) = 2x2 + 8xy + ln y
1 8
f(x, y) = xy + +
x y
f(x, y) = yx2 + y3 - 4y2 y > 0
f(x, y) = xy2 - 3y - 2 ln y - ln x
2
f(x, y) = e4x-x -y2
f(x, y) = ex sin y
f(x, y) = 8xy - 4y4
f(x, y) = 2x3y + 2x3 + 3x2 + y2 + 2y
f(x, y) = y3 + 12x2 + 3y2 - 12xy - 12x + 3y
Ą Ą
f(x, y) = cos x + cos y + cos(x + y) 0 < x < 0 < y <
2 2
f(x, y) = e2x-y(2x2 - y2)
2x 4
f(x, y) = xy - y + +
x-1 y
2
f(x, y) = (x2 + y2)ex -y2
"
f(x, y) = 2x y - x2 - 2y + 3x
f(x, y) = 3x + 5 ln(x2 + y2 + 1)
1
f(x, y) = x + + y2
x
1 2
f(x, y) = 4x + - 8y -
x y
f(x, y) = (y2 + 4x)e2x
f(x, y) = 3 ln y + 2 ln x + ln(6 - 3y - x)
f(x, y) = arctg (xy2) - arctg x
D
f(x, y) = x2 + 2xy + 3y2 D = {(x, y) : -2 x 4, -1 y 3}
f(x, y) = x2 + 2y2 D = {(x, y) : x2 + y2 25, y 3}
f(x, y) = xy - 2x D y = 0, x = 1, y - x = 3
f(x, y) = 4x3 - 2x2y + y2 D y = x2, y = 16
 f(x, y) = x2 + 4y2 - x + 2y D = {(x, y) : x2 + 4y2 1}
3
2
y = f(x) xy = yx
x = 1
y = y(x)
x3 + y3 - 6xy = 0.
"f "f
= 6xy = 3x2 - 2y
"x "y
"f y "f
1 1 x
= 3x2 + =
"x 2 x "y 2 y
"f "f
= y(1 + y)ex+xy = (1 + xy)ex+xy
"x "y
"f "f
= (x2 + 2x)ex-3y = -3x2ex-3y
"x "y
"f
1 arccos x
"
= -arccos x "f = -
"x (x+y)2 "y
(x+y) 1-x2 (x+y)2
"f "f
Ąx x
" "
= arcsin(y - Ąx) - =
"x
1-(y-Ąx)2 "y 1-(y-Ąx)2
"f 2(y+x) ln(1-x-y) "f 2(y+x) ln(1-x-y)
= ln2(1 - x - y) - = ln2(1 - x - y) -
"x 1-x-y "y 1-x-y
"f ln y x ln y(y+y ln x) "f x2 ln x ln y
x
= - = -
"x 4+xy ln x (4+xy ln x)2 "y y(4+xy ln x) (4+xy ln x)2
"f y "f
= e3x 3arctg xy + = e3x x
"x 1+x2y2 "y 1+x2y2
" " " " "
"f y "f
2 2 2 2
"
= y2arctg (y x) + arctg (y x) = 2xyarctg (y x) + 2 xarctg (y x)
"x x "y
"2f "2f "2f "2f
= 12x2y2 = 2x4 = = 8x3y
"x2 "y2 "x"y "y"x
"2f "2f "2f "2f
= 4 = -8 = = 3
"x2 "y2 "x"y "y"x
2
"2f y "2f "2f "2f
1 2x
= -x + 2 ln x = -x = = +
"x2 2 "y2 y "x"y "y"x x y
"2f "2f "2f "2f
= 2 sin2 y = 2x2 cos 2y = = 2x sin 2x
"x2 "y2 "x"y "y"x
"2f 2xy "2f -2xy "2f "2f y2-x2
= = = =
"x2 (x2+y2)2 "y2 (x2+y2)2 "x"y "y"x (x2+y2)2
"2f "2f "2f "2f
= y-32x ln2 2 = 12 � 2xy-5 = = -3 ln 2 � 2xy-4
"x2 "y2 "x"y "y"x
21 805
f , = 4, 1125
10 100
(1, 02)3,01 = 1, 06
(0, 0)
2 3
,
5 10
(-1, -1)
(1, -1)
(0, 0)
"
(-2, 3)
1
, 4
2
(2, 0)
(0, -1)
(1, 1)
(-2, -2)
(2, 2)
(0, 0)
9
3,
4
-1, 0
3
(1, 0)
1 1
, -1 -1,
2 2 2 2
-1, 0
2
(2, 1)
 (0, 0)
67 (4, 3) 0 (0, 0)
50 (0, 5) 18 (0, 3)
6 (-3, 0) -2 (1, 0)
256 (0, 16) -512 (-4, 16)
" "
"
2 2
1 + 2 (- , ) -1
2 4 2
(1, -1)
2 4
10 20
yx ln y-yxy-1
f (x) = f(1) = 1
xy ln x-xyx-1
" "
3 3
x = 2 2 2 4
A � B = {(a, b) : a " A, b " B}
A B
K
K � K K
(K, +, �) K + �
K
" x, y, z " K (x + y) + z = x + (y + z)
" x, y " K x + y = y + x
" e " K " x " K x + e = x
" x " K " y " K x + y = e
" x, y, z " K (x � y) � z = x � (y � z)
" x, y, z " K x � (y + z) = x � y + x � z
" x, y " K x � y = y � x
" f " K \ {e} " x " K f � x = x
" x " K \ {e} " y " K x � y = f
e f K
y -x e
0
y x-1 f
R
(C, +, �) C = R � R + �
C
" (x1, y1), (x2, y2) " C : (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)
" (x1, y1), (x2, y2) " C : (x1, y1) � (x2, y2) = (x1x2 - y1y2, x1y2 + x2y1)
(C, +, �) C
z w
 x (x, 0)
(0, 1) i
i2 = (0, 1) � (0, 1) = -1. z = x + iy z = (x, 0) +
(0, 1) � (y, 0) z = (x, y)
z = (x, y)
z = (x, y) = x + iy.
z1 = x1 + iy1 z2 = x2 + iy2
z1 ą z2 = (x1 ą x2) + i(y1 ą y2),
z1 � z2 = (x1x2 - y1y2) + i(x1y2 + x2y1).
z = x + iy x z
y z
Re z = x, Im z = y.
x - iy z = x + iy
z
z = x + iy = x - iy = (x, -y).
z1 = x1 + iy1 z2 = x2 + iy2
z1
z2
z1 z1 � z2 (x1 + iy1)(x2 - iy2) x1x2 - ix1y2 + ix2y1 - i2y1y2
= = = =
2
z2 z2 � z2 (x2 + iy2)(x2 - iy2) x2 - i2y2
2
x1x2 + y1y2 + i(-x1y2 + x2y1) x1x2 + y1y2 x2y1 - x1y2
= = + i .
2 2 2
x2 + y2 x2 + y2 x2 + y2
2 2 2
z1 = 2 - 3i z2 = 3 + 4i z3 = -1 + 2i
z1 + z2 = (2 - 3i) + (3 + 4i) = 2 - 3i + 3 + 4i = 5 + i = (5, 1),
z2 - 3 � z3 = (3 + 4i) - 3 � (-1 + 2i) = (3 + 4i) - 3 � (-1 - 2i) = 3 + 4i + 3 + 6i =
= 6 + 10i = (6, 10),
z2 � z3 = (3 + 4i) � (-1 + 2i) = -3 + 6i - 4i + 8i2 = -3 + 2i + 8i2 = -3 + 2i - 8 =
= -11 + 2i = (-11, 2),
z3 -1 + 2i -1 + 2i 2 - 3i (-1 + 2i) � (2 - 3i) (-1 + 2i) � (2 + 3i)
= = � = = =
z1 2 - 3i 2 - 3i - 3i (2 - 3i) � 2 - 3i
(2 - 3i) � (2 + 3i)
2
-2 - 3i + 4i + 6i2 -2 + i - 6 -8 + i 8 1 8 1
= = = = - + i = - , .
4 - 9i2 4 + 9 13 13 13 13 13
 z = x+iy x2 + y2
|z|
z � z = (x + iy)(x - iy) = x2 - i2y2 = x2 + y2 = |z|2
z = x + iy = 0

� " [0, 2Ą)
x y
cos � = , sin � = .
|z| |z|
z z
z = { z + 2kĄ, k " N}
z
z = |z| (cos � + i sin �)
z
"
z = 2 - 2i 3
" 2 "
"
|z| = 22 + -2 3 = 4 + 12 = 16 = 4,
ńł ńł ńł
�łcos � = 2 �łcos � = 1 �łcos � = 1
�ł �ł �ł
5
4 2 2
" "
�!�! �!�! �!�! � = Ą .
�łsin � = -2 3 �łsin � = - 3 �ł� " 3Ą, 2Ą 3
ół ół ół
2
4 2
5 5
z = 4 cos Ą + i sin Ą, .
3 3
z = 1 + i
"
1 3
z = i -
2 2
z = 2010 i
z = -2
z = 1 + i
" "
|z| = 12 + 12 = 2,
ńł "
1 2
�ł
�ł
sin "
�ł ą = =
Ą
2
2
"
�!�! ą = .
�ł 1 2 4
�ł
ół ą = =
cos "
2
2
"
Ą Ą
z = 2 cos + i sin .
4 4
"
1 3
z = i -
2 2
" 2
2
3 1
|z| = - + = 1,
2 2
ńł
1
�ł
1
�ł 2
�ł
sin ą = =
5
1 2
"
"
�!�! ą = Ą .
3
�ł - 6
3
�ł
2
ół
cos ą = = -
1 2
5 5 5 5
z = 1 cos Ą + i sin Ą = cos Ą + i sin Ą.
6 6 6 6
z = 2010 i
"
|z| = 02 + 20102 = 2010,
ńł
2010
�ł
�ł
sin ą = = 1
Ą
2010
�!�! ą = .
0
�ł 2
ół
cos ą = = 0
2010
Ą Ą
z = 2010 cos + i sin .
2 2
z = -2
|z| = (-2)2 + 02 = 2,
ńł
0
�ł
�ł
sin ą = = 0
2
�!�! {ą = Ą} .
-2
�ł
ół
cos ą = = -1
2
z = 2 (cos Ą + i sin Ą) .
 z = |z| (cos � + i sin �) n " N
zn = |z|n (cos n� + i sin n�) .
"
5 5
z = 2 - 2i 3 z = 4 cos Ą + i sin Ą
3 3
2011 � 5 2011 � 5 10055Ą 10055Ą
z2011 = 42011 cos Ą + i sin Ą = 24022 cos + i sin =
3 3 3 3
1675 � 3 � 2Ą + 5Ą 1675 � 3 � 2Ą + 5Ą
= 24022 cos + i sin =
3 3
5Ą 5Ą
= 24022 cos 1675 � 2Ą + + i sin 1675 � 2Ą + =
3 3
"
"
5Ą 5Ą 1 3
= 24022 cos + i sin = 24022 - i = 24021 - 24021 3i.
3 3 2 2
z = 1+i
"
Ą Ą
z = 2 cos + sin
4 4
" 2010
2010Ą 2010Ą 2010Ą 2010Ą
z2010 = 2 cos + i sin = 21005 cos + i sin =
4 4 4 4
1005Ą 1005Ą 1004Ą Ą 1004Ą Ą
= 21005 cos + i sin =21005 cos + + i sin + =
2 2 2 2 2 2
Ą Ą
= 21005 cos 502Ą + + i sin 502Ą + =
2 2
Ą Ą Ą Ą
= 21005 cos 251 � 2Ą + + i sin 251 � 2Ą + = 21005 cos + i sin =
2 2 2 2
= 21005 (0 + i) = 21005i
n z n " N
"
n
w wn = z z w wn = z
n z
z = |z| (cos � + i sin �) n " N
"
n
z = {w0, w1, . . . , wn-1},
� + 2kĄ � + 2kĄ
n
wk = |z| cos + i sin , k = 0, 1, . . . , n - 1.
n n
{w0, w1, . . . , wn-1} n
n
|z|
"
5 5
z = 2 - 2i 3 z = 4 cos Ą + i sin Ą
3 3
"
5
z = {w0, w1, w2, w3, w4},
5Ą 5Ą
"
+ 2kĄ + 2kĄ
5
3 3
wk = 4 cos + i sin , k = 0, 1, 2, 3, 4,
5 5
5Ą 5Ą 5Ą 5Ą
" "
+ 2 � 0 � Ą + 2 � 0 � Ą
5 5
3 3 3 3
w0 = 4 cos + i sin = 4 cos + i sin =
5 5 5 5
" " " "
5 5
" "
Ą Ą 1 3 4 4 � 3
5 5
= 4 cos + i sin = 4 + i = + i ,
3 3 2 2 2 2
5Ą 5Ą 5Ą 6Ą 5Ą 6Ą
" "
+ 2 � 1 � Ą + 2 � 1 � Ą + +
5 5
3 3 3 3 3 3
w1 = 4 cos + i sin = 4 cos + i sin =
5 5 5 5
"
11Ą 11Ą
5
= 4 cos + i sin ,
15 15
5Ą 5Ą 5Ą 12Ą 5Ą 12Ą
" "
+ 2 � 2 � Ą + 2 � 2 � Ą + +
5 5
3 3 3 3 3 3
w2 = 4 cos + i sin = 4 cos + i sin =
5 5 5 5
"
17Ą 17Ą
5
= 4 cos + i sin ,
15 15
5Ą 5Ą 5Ą 18Ą 5Ą 18Ą
" "
+ 2 � 3 � Ą + 2 � 3 � Ą + +
5 5
3 3 3 3 3 3
w3 = 4 cos + i sin = 4 cos + i sin =
5 5 5 5
"
23Ą 23Ą
5
= 4 cos + i sin ,
15 15
5Ą 5Ą 5Ą 24Ą 5Ą 24Ą
" "
+ 2 � 4 � Ą + 2 � 4 � Ą + +
5 5
3 3 3 3 3 3
w4 = 4 cos + i sin = 4 cos + i sin =
5 5 5 5
"
29Ą 29Ą
5
= 4 cos + i sin .
15 15
{w0, w1, w2, w3, w4}
"
5
(0, 0) 4
z1 = |z1| (cos �1 + i sin �1) z1 = |z1| (cos �1 + i sin �1)
z1 � z2 = |z1||z2| [cos(�1 + �2) + i sin(�1 + �2)]
z2 = 0

z1 |z1|
= [cos(�1 - �2) + i sin(�1 - �2)] .
z2 |z2|
"
z1 = 2 - 2i 3 z2 = 1 + i
5 5
z1 = 4 cos Ą + i sin Ą z2
3 3
"
(1, 1) 2
Ą
y = x
4
"
1 1
0x z2 = 2 cos Ą + i sin Ą
4 4
z1 z2
5 5
4 cos Ą + i sin Ą
z1 3 3 4 5 1 5 1
"
= = cos Ą - Ą + i sin Ą - Ą =
z2 " 1 1 3 4 3 4
2
2 cos Ą + i sin Ą
4 4
"
17 17
= 2 2 cos Ą + i sin Ą .
12 12
z1
z2
z1 z2
" " " "
z1 2 - 2i 3 (2 - 2i 3)(1 - i) 2 - 2i - 2i 3 + 2i2 3)
= = = =
z2 1 + i (1 + i)(1 - i) 1 - i2
" "
" "
2 - 2 3 - i(2 + 2 3)
= 1 - 3 - i(1 + 3)
2
" " 2 " " " "
2
1 - 3 + 1 + 3 = 1 - 2 3 + 3 + 1 + 2 3 + 3 = 8 = 2 2
ńł " " "
�łcos � = 1 - 3 = 2 - 6
�ł
"
�ł
4
2 2" " " .
�ł
�łsin � = -1 + 3 = - 2 + 6
ół
"
4
2 2
ez = ex+iy = ex � eiy = ex(cos y + i sin y).
"
z5 - 2 + 2i 3 = 0
"
z5 = 2 - 2i 3,
"
5
z = 2 - 2i 3.
"
2 - 2i 3
" " "
"
5 5
5
4 4� 3 11Ą 11Ą
4 cos + i sin
15 15
" 17Ą 17Ą " z = 2 + i 2 z = "
5 5 5
23Ą 23Ą 29Ą 29Ą
z = 4 cos + i sin z = 4 cos + i sin z = 4 cos + i sin
15 15 15 15 15 15
w3 - 1 = 0
w3 - 1 = (w - 1)(w2 + w + 1) = 0 �!�! w = 1 w2 + w + 1 = 0 .
 w2
"
" " -1+"3i+ w + 1 = 0 3 -1-"3i 1 "3 " = -3 = 3i2
1
" = ą 3i w = = -"+ w = = -
2 2 2 2
" "-2 2
"
" = 3i " = - 3i
"
3
z = 1 z = -1 +
2 2
"
3
z = -1 -
2 2
"
3
w = 1.
z = 1 = 1 + 0i
OX
1 = cos 0 + i sin 0,
"
0 + 2kĄ 0 + 2kĄ 2kĄ 2kĄ
3
1 = 1 � cos + i sin = cos + i sin , k " {0, 1, 2}.
3 3 3 3
k = 0 w = cos 0 + i sin 0 = 1
"
2Ą 2Ą 3
k = 1 w = cos + i sin = -1 +
3 3 2 2
"
4Ą 4Ą 3
k = 2 w = cos + i sin = -1 -
3 3 2 2
z2 - (1 + i)z + 6 + 3i = 0.
" = [-(1 + i)]2 - 4(6 + 3i) = 1 + 2i + i2 - 24 - 12i = -24 - 10i.
"
"
"
"
" = -24 - 10i = a + ib,
-24 - 10i = a2 + 2iab + i2b2,
-24 - 10i = a2 - b2 + 2iab.
a2 - b2 = -24
.
2ab = -10
a = 1 a = -1
b = -5 b = 5
" "
" = 1 - 5i " = -1 + 5i = -(1 - 5i)
1 + i - (1 - 5i) 6i 1 + i + (1 - 5i) 2 - 4i
z = = = 3i z = = = 1 - 2i.
2 2 2 2
i(z + z) + i(z - z) = 2i - 3.
Ż Ż
z = x + iy
i(x + iy + x - iy) + i(x + iy - x + iy) = 2i - 3,
2xi + 2yi2 = 2i - 3,
2xi - 2y = 2i - 3.
-2y = -3
,
2x = 2
3 3
x = 1 y = z = 1 + i
2 2
"
ńł ńł
1
2
�ł �ł
�ł �ł
sin x =
sin x = -
sin x = 0
2
"
2
"
3
cos x = -1
�ł �ł 2
ół ół
cos x =
cos x =
2"
2
"
ńł ńł ńł
1
2
�ł 3 �ł �ł
�ł �ł �ł
sin x = -
sin x = -
sin x = -
2
"
2
"
2
1 3
�ł �ł 2 �ł
ół ół ół
cos x = - cos x =
cos x = -
2 2
2
z1 = 2+3i z2 = 1-2i z3 = 5+12i z4 = (-2, 1)
z1 + 3z2 2z4 - z1 z3 � z1
z1 "
z12 + z23 z2
z2
x y
(2 + 3i)x + (5 - 2i)y = -8 + 7i 2x2 + y2 - 2yi = 12 - 4i
x y 1 + yi
+ = 1 = 3i - 1
2 - 3i 3 + 2i x - 2i
2
3 + i 4 - i
(2 + yi) � (x - 3i) = 7 - i x + y = 1 + i
3 - i 1 - 3i
x = 1 y = -2 x = 2 y = 2 x = -2 y = 2 x = 2 y = 3 x = 5 y = 17
x = 2 y = 3
" " "
1 3 2 2
z = 1 + i z = i - z = - - i
2 2 2 2
z = 2010 - 2010 i z = -16 z = -2i
"
Ą Ą
z = -1 - i 3 z = 1 + itg ą, ą " [0, ] z = tg ą + i, ą " [0, ]
2 2
" Ą Ą "
5Ą 5Ą 5Ą 5Ą 7Ą 7Ą
2(cos + i sin ) cos + i sin cos + i sin 2010 2(cos + i sin )
4 4 6 6 4 4 4 4
3Ą 3Ą 4Ą 4Ą 1
16(cos Ą + i sin Ą) 2(cos + i sin ) 2(cos + i sin ) (cos ą + i sin ą)
2 2 3 3 cos ą
1
(cos(Ą - ą) + i sin(Ą - ą))
sin ą 2 2
" 11 "
1 3 (1 + 3 i)14
(1 + i)2010 - i
2 2 (-1 - i)20
"
10
"
3 + 3i 3 3 - i
" -2 + 2 3 i
-2 + 2i
- 3 + i
" "
"
3 4
2i -4 1
" "
z3 + i 6 + 2 = 0 �3 + 1 = 0
x2 + 2x + 5 = 0 z4 + z2 + 1 = 0
"
(i - 1)3z3 = (1 + 3 i)6 zz + (z - z) = 3 + 2i
2 + i 1 - i
z2 + (1 + 4i)z - (5 + i) = 0 =
z - 1 + 4i 2z + i
"
z4 - 4i 3z2 - 16 = 0 (z2 - 6z + 11)(z3 + 1) = 0
" " " "
7 1
z = 1 - i z = -2 - 3i - i 1 + i 3 -1 - i 3 3 + i - 3 - i
6 6
" "
" "
1 3 1 3
3 + 2i 3 - 2i -1 + i - i
2 2 2 2
A = {z " C : zz + (Im z)2 e" 1
Ą
A = z " C : |z - i| > 1 '" d" z < Ą
4
4 + 3i
A = z " C : |z - 2| d" Re
2 + i
A = {z " C : z2 - 2i Re (z - i) Im (z + 4) e" 1}
sin 3x = 3 sin x - 4 sin3 x
cos 4x = cos4 x - 6 sin2 cos2 x + sin4 x
sin 6x = 6 cos5 x sin x - 20 cos3 x sin3 x + 6 cos x sin5 x
ctg4x - 6ctg2x + 1
ctg4x =
4ctg3x - 4ctgx
 K = K(t) t
K = rK,
r t
K0
t
K0 (1 + r)t .
n t
nt
r
K0 1 + .
n
t
n "
n rt
nt
r r
r
K(t) = lim K0 1 + = lim K0 1 + = K0ert.
n" n"
n n
t
K (t) = K0ert � r = K(t) � r
K = rK.
n n " N
F (x, y, y , y , . . . , y(n)) = 0
x y = y(x) y = y (x)
y = y (x) y(n) = y(n)(x) F : Rn+2 - R
y = y(x) x
F
y = y(x)
x
" y + x2y = sin x
" y + 3y - 2y = ln x
"
" y + 3xy - 2 y = 0
F (x, y, y , y , . . . , y(n)) = 0
[a, b] y = y(x)
" x " [a, b] F x, y(x), y (x), y (x), . . . , y(n)(x) = 0.
y = �(x, C1, C2, . . . Cn) C1, C2, . . . Cn
F (x, y, y , y , . . . , y(n)) = 0
y(x0) = y0, y (x0) = y1, y (x0) = y2, . . . , y(n-1)(x0) = yn-1,
x0 " [a, b] y0, y1, y2, . . . yn-1
F (x, y, y ) = 0
x y = y(x) y = y (x) F : R3 - R
y = f(x, y),
f : R2 - R
y = f(x, y),
y(x0) = y0
"f
f (x, y)
"y
D �" R2 (x0, y0) " D
(x0, y0)
D
x0
"
" x y
"
"
"
y = g(x)h(y)
h(y) = 0

y = g(x)h(y),
dy
= g(x)h(y),
dx
1
dy = g(x)dx.
h(y)
g f h(y) = 0 y

1
dy = g(x)dx + C,
h(y)
C
 g f (a, b) (c, d)
h(y) = 0 y " (c, d) x0 " (a, b) y0 " (c, d)

y = g(x)h(y),
y(x0) = y0
y + y ctgx = 0,
Ą
y = -1
3
y + y ctgx = 0,
dy
y + ctgx = 0,
dx
ydx + ctgxdy = 0,
ctgxdy = -ydx,
tg x
ctgxdy = -ydx � , y = 0,

y
1
dy = -tg xdx.
y
y = 0
y
1
dy = -tg xdx,
y
ln |y| = ln | cos x| + C1, C1 " R,
ln |y| = ln | cos x| + ln C2, C1 = ln C2, C2 " R+,
ln |y| = ln C2| cos x|,
|y| = C2| cos x|,
y = ąC2 cos x,
y = C3 cos x, C3 = ąC2 = 0.

y = C3 cos x, C3 = 0 y = 0

y = C3 cos x, C3 = 0,

�!�! {y = C cos x, C " R},
y = 0,
y + y ctgx = 0
RORR : y = C cos x, C " R.
Ą
y = -1
3
Ą
-1 = C cos ,
3
1
-1 = C � ,
2
C = -2.
y = -2 cos x.
y
y = g
x
x y
y = ux, u = u(x)
xu = g(u) - u.
g (a, b)
g(u) = u

y
y = g , y(x0) = y0,
x
y0
a < < b
x0
y y
y = + tg .
x x
y
Ą Ą
= + kĄ, k " Z y = x + kĄx, k " Z

x 2 2
x y
dy y
= f .
dx x
f f(u) = u + tg u
x y
y
u =
x
y = xu,
u x y
x
y = u + xu .
u + xu = u + tg u,
xu = tg u,
du
x = tg u,
dx
1
ctgudu = dx, u = kĄ, k " Z,

x
cos u 1
du = dx.
sin u x
 u = kĄ, k " Z
cos u 1
du = dx,
sin u x
ln | sin u| = ln |x| + C1, C1 " R,
| sin u| = eln |x|+C1,
1
| sin u| = eln |x| � eC ,
1
| sin u| = eC |x|,
1
sin u = ąeC x,
1
sin u = Cx, C = ą eC = 0,

u = arcsin Cx.
u = arcsin Cx, C = 0,

u = kĄ, k " Z.
y
y = x arcsin Cx, C = 0,

y = kĄx, k " Z.
y + p(x)y = g(x)
g = 0

g = 0
y = g(x)f(y) g(x) = -p(x), h(y) = y
y a" 0
dy
+ p(x)y = 0
dx
y = 0

dy
= -p(x)y,
dx
dy = -p(x)ydx,
1
dy = -p(x)dx,
y
1
dy = -p(x)dx,
y
ln |y| = -p(x)dx + C1, C1 " R,
ln |y| = -p(x)dx + ln C2, C1 = ln C2, C2 > 0,
ln |y| - ln C2 = -p(x)dx,
|y|
ln = -p(x)dx,
C2
|y|
-p(x)dx
= e ,
C2
-p(x)dx
|y| = C2e ,
-p(x)dx
y = ąC2e ,
-p(x)dx
y = C3e , ąC2 = C3 = 0.

y = 0
y = Ce- p(x)dx, C " R.
p q (a, b) x0 " (a, b) y0 " R
y + p(x)y = g(x), y(x0) = y0,
(a, b)
.
"
"
y = Ce- p(x)dx,
 C
C C C(x)
y = C(x)e- p(x)dx,
C(x)
C(x)
dy
= C (x)e- p(x)dx - C(x)e- p(x)dxp(x)
dx
C (x)e- p(x)dx - C(x)e- p(x)dxp(x) +p(x) C(x)e- p(x)dx = g(x),
y y
p(x)dx
C (x) = g(x)e .
p(x)dx
C(x) = g(x)e dx.
C(x)
p(x)dx
y = g(x)e dx � e- p(x)dx,
p(x)dx
y = Ce- p(x)dx + g(x)e dx � e- p(x)dx.
2
y - xy - xex = 0,
y(0) = 0.
2
y - xy = xex
: y - xy = 0.
dy
- xy = 0,
dx
dy 1
= xy, � , y = 0,

dx y
1
dy = xdx.
y
 y = 0
y
1
dy = xdx,
y
1
ln |y| = x2 + C1, C1 " R,
2
1
x2+C1
2
|y| = e , C1 " R,
1
x2
1
2
|y| = eC e ,
1
x2
1
2
y = ą eC e ,
1
x2
1
2
y = C2e , C2 = ą eC = 0.

1
x2
2
y = C2e , C2 = 0 y = 0

1
x2
2 1
y = C2e , C2 = 0,

x2
2
�!�! {y = Ce , C " R},
y = 0,
y - xy = 0
1
x2
2
: y = Ce , C " R.
C C C(x)
1
x2
2
C(x) y = C(x)e
1
x2
2
y = C(x)e
1 1
x2 x2
2 2
y = C (x)e + C(x) � xe .
y y
1 1 1
x2 x2 x2 2
2 2 2
C (x)e + C(x)xe - xC(x)e = xex .
C (x)
1
x2 2 1
2 2
C (x)e = xex , �e- x2 ,
1
x2
2
C (x) = xe .
C(x)
1 1
x2 x2
2 2
C(x) = xe dx = e .
C(x)
1 1
x2 x2 2
2 2
: y = e � e = ex .
1 1
x2 x2
2 2
: y = Ce + e , C " R.
 y (0) = 0
0 = Ce0 + e0,
0 = C + 1,
C = -1.
2 1
x2
2
y = ex - e .
y - y = ex.
y - y = ex
y - y = 0
y - y = 0,
dy
- y = 0,
dx
dy
= y, : y � dx, y = 0 ( y = 0 ),

dx
dy
= dx,
y
dy
= dx,
y
ln |y| = x + C1, C1 " R,
1
|y| = ex+C ,
1
|y| = eC � ex,
1
y = ąeC � ex,
1
y = C2 � ex, C2 = ąeC = 0.

x
y = C � , C " R.
C = C(x)
y = C(x) � ex,
y = C (x) � ex + C(x) � ex,
C (x) � ex + C(x) � ex - C(x) � ex = ex,
C (x) � ex = ex,
C (x) = 1,
C(x) = x.
y = x � ex
y = Cex + ex = (C + 1)ex, C " R.
 p = p(x)
y
y + py = g(x)
g(x)
g(x) n Pn(x)
g(x) = Pn(x) = anxn + an-1xn-1 + . . . + a1x + a0
ys(x) = Qn(x),
Qn(x) = bnxn +bn-1xn-1 +. . .+b1x+b0 n
g(x) = eąxPn(x) Pn(x) n
eąxQn(x), p = -ą,

ys(x) =
xeąxQn(x), p = -ą,
Qn(x n
g(x) = keąx
meąx, p = -ą,

ys(x) =
mxeąx, p = -ą,
m
g(x) = k cos bx + l sin bx
ys(x) = m cos bx + n sin bx
m n
g(x) = eąx (k cos bx + l sin bx)
ys(x) = eąx (m cos bx + n sin bx)
m n
g(x) = Pn(x) cos bx+Qn(x) sin bx Pn(x) Qn(x)
n n
ys(x) = Rn(x) cos bx + Sn(x) sin bx
Rn(x) Sn(x) Pn(x) Qn(x)
y + 2y = 3ex.
y + 2y = 0,
dy dx
= -2y, � , y = 0, ( y = 0 )

dx y
dy
= -2dx,
y
dy
= -2dx,
y
ln |y| = -2x + C1, C1 " R,
1
|y| = e-2x+C ,
1
y = ą eC � e-2x,
1
y = C2 � e-2x, C2 = ą eC = 0.

yj = C � e-2x, C " R.
.
ys = Aex,
A A ys
Aex +2 Aex = 3ex,
ys ys
3Aex = 3ex,
3A = 3,
A = 1.
ys = ex,
y = yj + ys = C � e-2x + ex, C " R.
y + y = cos 2x.
 y + y = 0,
dy dx
= -y, � , y = 0, ( y = 0 )

dx y
dy
= -dx,
y
dy
= -1dx,
y
ln |y| = -x + C1, C1 " R,
1
|y| = e-x+C ,
1
y = ą eC � e-x,
1
y = C2 � e-x, C2 = ą eC = 0.

yj = C � e-x, C " R.
ys = A sin 2x + B cos 2x,
A B A, B
ys
2A cos 2x - 2B sin 2x + A sin 2x + B cos 2x = cos 2x,
ys ys
(2A + B) cos 2x + (A - 2B) sin 2x = cos 2x.
2
A = ,
2A + B = 1,
5
�!�!
1
A - 2B = 0,
B = ,
5
2 1
ys = sin 2x + cos 2x.
5 5
2 1
y = yj + ys = C � e-x + sin 2x + cos 2x, C " R.
5 5
ys1(x) ys2(x)
dy dy
+ p(x)y = g1(x) + p(x)y = g2(x),
dx dx
y = ys1(x) + ys2(x)
dy
+ p(x)y = g1(x) + g2(x),
dx
y + p(x)y = g(x)yr,
r " R \ {0, 1} r
z = y1-r,
z(x)
dz
+ (1 - r)p(x)z = (1 - r)g(x).
dx
"
2 y
2y
y + - = 0.
x cos2 x
1
r =
2
1 "
2
z = y1- = y,
z x x
y
z = ,
"
2 y
y
"
2 y
2y 1
y + = , � , y = 0, (y = 0 )

"
x cos2 x 2 y
y 1 " 1
+ � y = ,
"
2 y x cos2 x
z
z
1 1
z + � z = .
x cos2 x
1
z + � z = 0,
x
dz 1 1
= - � z, � , z = 0,

dx x z
1 1
dz = - dx.
z x
z = 0
 z
1 1
dz = - dx,
z x
ln |z| = - ln |x| + C1, C1 " R,
1
ln |z| = ln + ln C2, C1 = ln C2, C2 " R+,
|x|
C2
ln |z| = ln , C2 " R+,
|x|
C2
|z| = , C2 " R+,
|x|
1
z = ą C2 , C2 " R+,
x
1
z = C3 , C3 = ą C2 = 0.

x
C3
z = , C3 = 0 z = 0

x
C3
C
z = , C3 = 0,

�!�! z = , C " R ,
x
x
z = 0,
1
z + � z = 0
x
C
: z = , C " R.
x
C
C(x)
z =
x
C (x) � x - C(x)
z = .
x2
z z
C (x) � x - C(x) 1 C(x) 1
+ � = ,
x2 x x cos2 x
C (x)x C(x) C(x) 1
- + = ,
x2 x2 x2 cos2 x
C (x) 1
= ,
x cos2 x
x
C (x) = .
cos2 x
C(x)
x
C(x) = dx = xtg x + ln | cos x|.
cos2 x
C(x)
xtg x + ln | cos x| + C
z = , C " R.
x
 y
" xtg x + ln | cos x| + C
y = .
x
2
xtg x + ln | cos x| + C
y = , C " R y = 0.
x
P (x, y)dx + Q(x, y)dy = 0
F (x, y)
F (x, y) = 0,
F
ńł
"F
�ł
�ł
(x, y) = P (x, y),
�ł
"x
�ł "F
�ł
ół (x, y) = Q(x, y).
"y
"Q
"P
P Q
"y "x
D �" R2
(x, y) " D
"P "Q
(x, y) = (x, y).
"y "x
y y
- (2y - x )y = 0.
y
P (x, y) =
y y
Q(x, y) = -(2y - x ) = x - 2y
"P
y
= ,
"y
"Q
y
= ,
"x
"P "Q
=
"y "x
F (x, y)
ńł
"F
�ł y
�ł
= , (1)
�ł
"x
�ł "F
y
�ł
ół = x - 2y. (2)
"y
 (1)
y y
F (x, y) = dx = x + �(y),
� y F
� (2)
"F
y y
= x + � (y) = x - 2y.
"y
� (y) = -2y,
�(y) = -y2 + C, C " R.
y
F F (x, y) = x -y2 +C C " R
y
y = y(x) F (x, y) = 0 x - y2 + C = 0
(2xy - 3x2)dx + (x2 + 3y2 + 1)dy = 0.
P (x, y) =
2xy - 3x2 Q(x, y) = x2 + 3y2 + 1
"P
= 2x,
"y
"Q
= 2x,
"x
"P "Q
=
"y "x
F (x, y)
ńł
"F
�ł
�ł
= 2xy - 3x2, (1)
�ł
"x
�ł "F
�ł
ół = x2 + 3y2 + 1. (2)
"y
(1)
F (x, y) = (2xy - 3x2)dx = x2y - x3 + �(y),
� y F y
(2)
"F
= x2 + � (y) = x2 + 3y2 + 1.
"y
� (y) = 3y2 + 1,
�(y) = y3 + y + C, C " R.
F F (x, y) = x2y - x3 + y3 + y + C C " R
y = y(x) F (x, y) = 0
x2y - x3 + y3 + y + C = 0
 p1 p2 q
y + p1(x)y + p2(x)y = q(x),
q(x) = 0 q(x) = 0

" y - 2x2y + 3y = 0
" y - 2x2y = sin x
p1 p2 q
(a, b)
ńł
y + p1(x)y + p2(x)y = q(x),
�ł
y(x0) = y0,
ół
y (x0) = y1,