Rw
 k
101
10123 - 8
1
n
n2
1 1
0.01 n > k = 10 < = 0.01.
n2 k2
0
(an)ne"1 , (bn)ne"1 , (xn)ne"1 an, n e" 1, bn, n e" 1, xn, n e" 1
n
"
n + n3 1
n
an = , bn = 2n + 5n, cn = 1 + ,
2n - 3 n
f0 = 0, f1 = 1, fn = fn-1 + fn-2, n > 1 ,
w0 = 0, wn = 2wn-1 + 1, n e" 1
j1 = 1, j2n+1 = 2jn + 1, j2n = 2jn - 1, n e" 1.
a
(a - , a + )
> 0 b 2
|a - b| < ;
+" "
(K, ") K
(-", K), K
0, 99 1, 01 1.
0.998 1
[100, ") " (100, ").
a
(an)ne"1
> 0
|an - a| <
limn" an = a n
an, n e" 1 a.
an = a, n e" 1 a
limn" an = a
a.
1
> 0
n2
"
k = -1
-1 n > k n2 > k2 > -1 e" -1 an <
|an| < an > 0
an, n e" 0 a
bn = an+l, n e" 1 l bn, n e" 1
an, n e" 1
n an
 an, n e" 0 > 0
n k |an - a| < .
n > k - l |bn - a| = |an+k - a| < .
a (0, 1) an = an, n e" 1
log
> 0 k .
log a
(an)ne"1 (|an|)ne"1
> 0
||an| - 0| < . |an|
(|an|)ne"1
|an| <
(an)ne"1 .
an = (-1)n, n e" 1
n, an = 1 an = -1.
-1. a
1 1
(a - , a + )
4 4
1 1
|a - 1| < |a + 1| < .
4 4
"
an = n, n e" 1
a (0, 2a)
an
K k max(0, K)
n > k an > max(0, K) e" K
(K, ")
(0, 2a)
a < 0 a = 0
an, n e" 1
limn" an = "
 limn" an = a
(an)ne"1
|b-a|
a b a = b. = a

4
an
b
an
a b
ą �
(an)ne"1 (bn)ne"1 a b
(ąan + �bn)ne"1 (anbn)ne"1
lim (ąan + �bn) = ąa + �b, lim anbn = ab.
n" n"
an
b = 0

bn
an a
lim = .
n"
bn b
b = 0

an
bn, n e" 0
bn
n.
2n+n3
an =
3n3+7
an =
2bn+1 1 1
bn = cn = . bn
3+7cn n2 n3
1
dn =
n
cn = bndn

 2 3
2
3
n a n+1 |q| < 1
an = aqk. qan = aqk = an+1 - a =
k=0 k=1
qk+1-1
aqk+1 + an - a. an = a . |qn| = |q|n
q-1
a
limn" an = .
1-q
> 0
an, n e" 1 a bn, n e" 1 b

n |ą||an - a| < ą = 0
2

:=
|ą|

n |ą||an - a| <
2
n |ąan + �bn - (ąa + �b)| d" |ą(a - an)| +

|�(bn - b)| d" |ą||an - a| + |�||bn - b| < + =
2 2
" + " = ",
a" = " a > 0
a" = -" a < 0
a
= 0 a " R.
ą"
"
" - " 0"
"
a" = " a > 1 a" = 0 |a| < 1 1"
 (an + bn)ne"1
(an)ne"1 (bn)ne"1 : an = (-1)n
bn = (-1)n+1 0
(an)ne"1 (bn)ne"1 a b n
an d" bn a d" b.
1 1
an = - , bn = an < bn
n n
0 = 0.
a-b
a > b. =
4
a-b
=
Ą
an
a bn
b a- > b+ n an > bn
(an)ne"1 ,
(bn)ne"1 (cn)ne"1
an d" bn d" cn.
(an)ne"1 (cn)ne"1 g
(bn)ne"1 g
> 0
(cn)ne"1 |cn -g| < cn < g +
(cn)ne"1
cn < g + .
(an)ne"1 g
(cn)ne"1 g - < bn
n, |bn - g| < .
(bn)ne"1
"
n
a > 0 an = a
a > 1; a < 1 b =
1
> 1
a
"
1
n
limn" a = limn" "b =
n
1
= 1; a = 1
1
(1 + x)n e" 1 + nx, x e" 0, n e" 1;

n

n
(a + b)n = akbn-k.
k
k=0
"
n
a > 1 bn := a - 1 e" 0
x := y x y
"
n
bn bn = a-1
x = bn
(1 + bn)n - 1 a - 1
0 d" bn d" = .
n n
bn
an = bn + 1
"
n
an = 2n + 5n
" 2n + 5n d" 2 � 5n
n
an d" 5 2. 2n + 5n e" 5n
"
n
an e" 5. 5 d" an d" 5 2.
5
limn" an = 5.
an, n e" 1
M > 0 |an| d" M
M an d" M an e" M
n
k
 g
max {|g + 1|, |g - 1|, a1, ...., ak} |g +
1| |g - 1| g
an = (-1)n;
n |an| = 1 M 1.
an = n
0
an = -n
an = (-1)nn
limn" |an| = ",
(0, ") (-", 0)
(an)ne"1 limn" |an| = ".
an = [(-1)n - 1]n
a2k+1 = -2(2k + 1).
an, n e" 0
n an d" an+1,
1
an = 1 -
n
M0 n
|an| d" M0
M M0 d" M. > 0 M0 -
< M0 M0 -
k ak > M0 - .
ak+1 e" ak > M0 - an > M0 - ,
n e" k an d" M0 n.
n |an - M0| < ,
M0
(an)ne"1
(-an)ne"1
n
an =
3n
an+1 < an
n+1
< 3.
n
0
n+1 1
an+1 = an. g
n 3
1 1
1 g g g = g
3 3
g = 0.
n 1
n 1 bn = k=0 k! =
1
2+ , n e" 1 cn+1-cn =
k=2 k! (n+1)!
an, n e" 1
k! e" 2k-1;
k-1
n 1 1
n
n 1 2. k=2 k! d" k=2 2k-1
cn := cn+1 - cn = 2-n.
k=2
2k-1
1
2
= 1. (bn)ne"1
1
1-
2
e
2, 718...
bn
 1 n an =
1 +
n
1"
9
e. a1 = 2, a2 = , a3 =
4
64 625
, a4 = .
27 256

n n

n 1 n! 1
an = =
k nk (n - k)!k! nk
k=0 k=0
n

1 (n - 1)(n - 2)...(n - k + 1)
= 2 + .
k! nk
k=2

n

1 1 2 k - 1
an = 2 + 1 - 1 - ... 1 - .
k! n n n
k=2
an+1

n+1

1 1 2 k - 1
an+1 = 2 + 1 - 1 - ... 1 - .
k! n + 1 n + 1 n + 1
k=2
n
2 d" k d" n
1 1 2 k-1 1 1 2 k-1
(1- )(1- )...(1- ) d" (1- )(1- )...(1- );
k! n n n k! n+1 n+1 n+1
an+1 e" an
n 1 n 1
1 an d" 2 + = .
k=2 k=0 k!
k!
e,
an, n e" 1.
an, n e" 1 an d" bn
g g d" e.
n e" m
n

1 (n - 1)(n - 2)...(n - k + 1)
an = 2 +
k! nk
k=2
m

1 (n - 1)(n - 2)...(n - k + 1)
e" 2 + .
k! nk
k=2
 g m + 1
n "
m
m 1
g e" 2 + . bm.
k=2 k!
g e" bm m " g e" e
g = e.
n
1
n
1 n 1 1 1 - e-1;
1
1 - = =
n
1
n ( )n 1
(1+ )n-1 (1+ )
n-1
n-1 n-1
(an)ne"1
-1
n
limn"(1 + an)a e.
1 a b -1 a < b
a+b 2ab
b1 = a1 = a-1 + b-1 =
2 2 a+b
a1 < b1. b2 a1 b1
a2 (an)ne"1
(bn)ne"1 a < an < an+1 < bn+1 < bn < b
ą �
1 1
bn+1 = (an + bn), � = (ą + �) ą = �.
2 2
a1b1 = ab ab = anbn
"
ab = ą� = ą2 ab
(an)ne"1 (bn)ne"1
(an)ne"1
(bn)ne"1
(an)ne"1 (bn)ne"1
M
(bn)ne"1
0 d" |anbn| d" M|an|.
 (an)ne"1 (|an|)ne"1
(|anbn|)ne"1 (anbn)ne"1
sin n
an =
n
1
bn = cn = sin n
n
1
n | sin n| d" 1 an = (-1)n sin
n
1
bn = (-1)n cn = sin
n
1 1
0 d" sin d"
n n
an, n e" 1
1
sn = (a1 + ... + an).
n
n
> 0 g
(an)ne"1

n k + 1, |an - g| < 3
1
bn,k = ai. n e" k + 1,
n-k i=k+1
n n n

1 1 1
|bn,k - g| = | ai - g| d" |ai - g| d" .
n - k n - k n - k 3
i=k+1 i=k+1 i=k+1
1 n-k
sn = (a1+...+ak)+ bn,k |sn-bn,k| d"
n n
1 k
| (a1+...+ak)|+| bn,k. k
n n

n (an)ne"1
3
M
k
bn,k, n e" k+1 M.
n

n .
3
n |sn -g| d" |sn -bn,k|+|g -bn,k| < ,
an = (-1)n+1
n
bn = ak 1
k=1
n 0 n (bn)ne"1
1
an = n sin 1
n
1
sn = bn
n
limn" an
(an)ne"1 an > 0
g
"
n
lim a1a2 � � � an = lim an.
n" n"
an, n e" 1
limn" an+1 = g limn"
an
"
an
n
an g bn = , n e" 1
an-1
"
"
n
n
a0 = 1 b1 � � � b = an
"n
n
limn" n+1 = 1, limn" n = 1.
n
limn" (n+1)! = limn"(n + 1) = "
n!
"
n
limn" n! = ".
"
n
1 n!
n! an =
n nn
an+1 n
1
= (n+1 )n = limn" an+1 = e-1.
n
1
an an
1+
( )
n

"
n
1 n n!
limn" n n! = limn" nn = e-1.
xn, n e" 1 yn, n e" 1
n, yn+1 > yn,
limn" xn+1-xn = g limn" xn
yn+1-yn yn
g
n
xn = ak
k=1
yn = n yn yn+1 - yn = 1 xn+1 - xn =
an+1.
limn" an+1 = limn" an = g
1
limn" n n ak.
k=1
 (an)ne"1
" n
1
limn" nan = g limn" "n k=1 ak
n
2g. xn ak yn
k=1
"
" "
n. yn+1 - yn = n + 1 - n > 0
an+1
limn" "n+1-"
=
n

" "
"
n
limn" an+1( n + 1 + n) = g + limn" an+1 n + 1 = g + g =
n+1
2g.
 (an
n)ne"1
sn = ai, n e" 1
i=1
"
(an)ne"1 an.
n=1
s
"
an
n=1
(an)ne"1
"
an.
n=k
(sn)ne"1
(an)ne"1 (sn)ne"1?
a1 = s1 a2 = s2 - s1, a3 = s3 - s2
an = sn -sn-1, n e" 2.
 "
sn an.
n=1
"
(-1)n
n=1
((-1)n)ne"1 -1 0
"
a a
n=1
(a)ne"1 na
"
a = ".
n=1
" 1
.
n=1 n(n+1)
n 1
1 1 1
an = = - . sn = -
n(n+1) k=1 n
n 1 n 1 n+1 1 n n+1
1
= = 1 - ,
k=1 n+1 k=1 k=2 n n+1
" n -
an = limn" sn = 1.
n=1
"
aqn, a = 0, q " R

n=1
1-qn+1
sn = a q = 1; q = 1, sn = na

1-q
a
|q| < 1 . q e" 1
1-q
+ą" a
q < -1
q = -1.
"
an
n=1
limn" an = 0
(sn)ne"1 limn" an = limn"(sn - sn-1) = limn" sn -
limn" sn-1 = s - s = 0.
" n+1 n+1 1
limn" 2n-9 =
n=1 2n-9 2
"
(-1)n
n=1
limn"(-1)n
 (an)ne"1
"
an
n=1
" 1
n=1 n
1
an =
n
1 2n
n s2n - sn = + ... +
n+1 .
n
1 1
s2n - sn e" n = .
2n 2
3
2
1
2
1 1
;
2 2
1
2
"
an
n=1
> 0 n0
|sn - sk| = |ak+1 + ... + an| <
n e" k e" n0.
"
n=1
an
"
|an|.
n=1
> 0 n0 n e" k e" n0
|ak+1| + ... + |an| < . |ak+1 + ... + an| d"
"
|ak+1| + ... + |an|. an
n=1

(-1)n+1
n
 n

(-1)n -2n
n2 3n+5

(-1)n (-1)nn
n n2+1
" " "
an bn an
n=1 n=1 n=1
n
bn d" an
" "
bn
n=1
" n=1 an
"
" n=1 bn n=1 bn
an. n
n=1
"
" n=1 bn
an
n=1
(bn)ne"1
n
bn
(bn)ne"1 (an)ne"1;
(an)ne"1 (bn)ne"1 .
" 1
0 < ą < 1
n=1 ną
1 1
e" .
ną n
ą e" 0
" 1
n2
" 1 n=1 " 1 1 1
= e" , n e" 2
n=1 n(n+1)
" 1n=2 n(n-1) n(n-1) n2
ą > 1
n=1 ną
ą e" 2
" 1 Ą2
= ,
n=1 n2 6
n
limn" k=1 ak = ".
"
limn" nan = limn" an
1
"
n
" 1
"
n=1
n
"
an
n=1
" "
an bn
n=1 n=1
an
" limn" bn = g e" 0.
"

" n=1 bn n=1 an. g = 0
bn an.
n=1
"
" " n=1 bn
an
n=1
> 0 n
an
< g + . n
bn
(an)ne"1 ((g + )bn)ne"1
 "
an.
n=1
" n2+3
" 1 n=1 en4-Ą
n=1 n2
n2+3
en4-Ą
limn" = limn" n4+3n2 e-1
1
en4-Ą
n2
s
(an)ne"1
+" -"
(-1)n
-1 1
nĄ
an = sin -1, 0
2
1.
nĄ
an = sin
4
nĄ
an = sin ?
6
n
an = 2 + 3(-1) +
Ą
2
2sin n ?
 (0, 1)
1
2
1 2
,
3 3
1 2 3
, , ,
4 4 4
1 2 3 4
, , , ,
5 5 5 5
... .
[0, 1]
(an)ne"1 nk, k e" 1
an , k e" 1
k
(an)ne"1. s inR (an)ne"1
(an)ne"1 s.
(an)ne"1
(an)ne"1 (-an)ne"1
lim supn" an
limn"an lim infn" an limn"an
(-1)n 1 -1;
nĄ
sin 1
2
0.
nĄ
sin .
6

 (an)ne"1
g > 0
(an)ne"1 g + ?
g + (an)ne"1
g (an)ne"1.
g (an)ne"1 |g-an| <
an > g -
(an)ne"1.
(an)ne"1 g
> 0 (an)ne"1
an > g -
an < g + .
"
an g =
n=1
"
n
lim sup an = g g < 1 g > 1
g = 1
g < 1
"
n
> 0 q := g + < 1. g an ne"1
"
n
an < g + ,
"
an < qn qn
n=1
g > 1
q := g - > 1.
"
"
n n
an ne"1 an > g - = q,
an > qn qn
an 0
(an)ne"1 (an)ne"1
" xn
x " R,
n=1 n!

|x|
n

limn" xn = limn" "n! = 0 < 1
n!
"
n
limn" n! = "
 "
an
n=1
an+1
g = lim sup g < 1
an
g > 1.
" 2nn! " 2nn!
n=1 nn n=1 nn
an+1 n
= 2(n+1 ) - 2e-1 < 1
an
n"
an+1 n
= 2(n+1 ) - 3e-1 > 1.
an
n"
(an)ne"1
an+1 " " an+1
n n
lim inf d" lim inf an d" lim sup an d" lim sup .
n" n"
an an
n" n"
"
n
limn" an+1 limn" an
an
lim supn" an+1 < 1
an
"
n
lim supn" an < 1.
"

(-1)nan
n=n0
an
an+1 < an
limn" an = 0. |s - sn| d"
an+1.
n0 = 0. xn = s2n yn =
s2n+1, n e" 0. xn, n e" 0
 s2n+2 - s2n = (-1)2n+2a2n+2 + (-1)2n+1a2n+1 = a2n+2 - a2n+1 d" 0.
yn, n e" 0
xn e" s1 yn d" s0.
sn, n e" n0
|xn - yn| = |s2n - s2n+1| = |a2n+1| 0
n"
s2n
s2n+1

1 1 1
(-1)n , (-1)n , (-1)n sin
n + 1 n2 + 1 n
1 1 1 1 1 1
- + - + - + . . .
2 3 4 9 8 27
(-1)n(n+2)
an
5n+1
2
limn" (n+2) = ,
5n+1 5
limn"(-1)n (n+2)
5n+1
" (-1)n
n=1 n2+1
0, 01 n
|s - sn| 0, 01 an+1
1 1
< n
n+12+1 100
9.
fn, n e" 1
"
D R. fn(x)
n=1
n
sn(x) = fk(x)
k=1

1
sin nx
n2+x2
D = R
x
x
 D x
"
xn x " (-1, 1).
n=0
n
xn+1-1 1
Sn(x) = xk = S(x) = . |Sn(x) - S(x)| =
k=0 x-1 1-x


xn+1 |xn+1| |x|n+1
= =
1-x 1-x 1-x
" "
" n=1 an n=1 bn
(an sin nx+bn cos nx) x "
n=1
R. |an sin nx+bn cos nx| d" |an|+|bn|
" 1
D = R,
n=1 x2+n2
" 1
1 1
d"
n2+x2 n2
" sinn=1 n2
nx2 sin nx2 1
" 1 n=1 n! | n! | d" n!
n=1 n!
"
x
2ab d" a2 + b2
n=1 1+n4x2
(a - b)2
a = 1 b = n2|x|
|x|
1 1
d" .
1+n4x2 2 n2
" x2
x = 0 x = 0

n=1 (1+x2)n
"

an(x - x0)n.
n=1
an x0
" 1
xn
n=1 n!
"
1
an = x0 = 0. 3n(x + 1)n
n! n=1
x0 = -1 an = 3n.
x = x0. x = x0


n n
|an||x - xn|n = |x - xn| |an|.

n
g = lim sup |an|.
1
g 0 |x - x0| < ,
g
g = 0
x " R, g = ", x = x0.

1
0 < g < " |x - x0| >
g
1
R =
n
lim sup |an|
g = 0 g = "
R = 0,
x = x0. R = ",
x " R, 0 < R < " x " (x0 -R, x0 +R)
x " [x0-R, x0+R]. x = x0+R x = x0-R
(an)ne"1
an
limn" an+1
1
"
= R.
limn" n an
" 1
xn "
n=1 n!
limn" (n+1)! = limn"(n + 1) = ".
n!
"
x " R. 3n(x + 1)n
n=1
"
1 4
n
3 3
" an = 3. x = x0 - R = -
"
1
3n(- )n = (-1)n
n=1
" 3 n=1
2
.
n=1 3
" 1 x = x0 + R = -
4 2
3n(x + 1)n (- , - ).
n=1 3 3

" (-1)n
n 1
xn R = 1
n=1 n n
1, n x = -1
" 1 " (-1)n
x = 1
n=1 n n=1 n
(-1, 1].
" xn
n=1 n2
" (-1)n
xn
n=1 n
"

xn " (-1)nx2n+1 " (-1)nx2n
, . .
n! (2n + 1)! (2n)!
n=0 n=0 n=0
x " R. x " R
a2n = 0, n e" 0,





(-1)n 1
2n+1 2n+1
2n+1

|a2n+1| = = - 0.

n"
(2n + 1)! (2n + 1)!

n
limn" |an| = 0
ex exp x sin x
cos x
sin cos x ex
 R Rw
Rw Rw, w = 1, 2, 3, ...
w (x1, ..., xw). R1 =
R R2
R3 R4
R127
Rw R1, R2, R3
R4. R127
127
Rw
x

 Rw
x = (x1, ..., Rw
xw)
w"
y = (y1, ..., yw) " Rw (xk - yk)2 < .
k=1


w


d(x, y) = (xk - yk)2
k=1
x y.
"
a2 = |a|, x y R |x - y|
Rw

Rw an ne"1 , an " Rw
p p
p
p.

n
n2 1
an = (xn, yn) = , 1 +
2n2-2 n
1 1
( , e) limn" xn = limn" yn = e.
2 2
Z �" Rw
Rw s = (s1, ..., sw)
Z
Z s Z.
1 Z =
{-1, 1} Z
1
an = (-1)n, -1 1
-1 1
0 d" x d" 1
(0, 1)
(0, 1).
1
0 x =
n
n 0
 (n, k) n k
R2.
(0, 1) (0, 1] [0, 1)
[a, b] a d" b
xn, n e" 1
= {(x, y) " R2; 0 d"
x d" y d" 1} (0, 0), (1, 1) (0, 1)
R2). an = (xn, yn)
a = (x, y) limn" xn = x limn" yn = y
0 d" xn d" yn d" 1
0 d" x d" y d" 1 a = (x, y) .
Rw
p
p
Z �" Rw
Z
(-", a) (b, ") a d" b
Rw.
Z
Z ? p
Z Z p
p
Z . p Z ,
Rw Rw
 Z .
Z
Z
{(x, y) " R2; x > y}
{(x, y) " R2; x d" y}
(0, 1]
f : D R D �" Rw
f p
(an)ne"1 D limn" an = p
an = p.


f(an) .
ne"1
(an)ne"1
p limxp f(x).
p
x2-3x+2
f(x) =
x-2
x = 2. (an)ne"1 2, f(an) =
a2 -3an+2
n
= an - 1 1. limx2 x2-3x+2
x-2 x-2
1.
(x2-4) sin y
f(x, y) =
(x-2)y
y x = 2. limn" xn = 2 limn" yn =
(xn+2) sin yn sin y0
y0 = 0. f(xn, yn) = 4

yn y0
sin y0
lim(x,y)(2,y ) f(x, y) = 4 y0 = 0. xn

0
y0
x0 yn 0 limn" f(xn, yn) = 2 + x0
lim(x,y)(x ,0) f(x, y) = 2 + x0.
0
"1
f(x, y) = D =
xy
{(x, y)|xy > 0}
x y
(xn, yn) " D,
xnyn
"1
(x0, y0) lim(x,y)(x ,y0) xy = ".
0
g p
� x " D
p f(x) f(p).
lim (ąf(x) + �g(x)) = ą lim f(x) + � lim g(x).
xp xp xp
p f
g
g
p
{-1, 1}
1
x = 0 f( ) = n
n
1
x =
n
p f
f
f p
f(p).
x2-3x+2
f(x) =
x-2
x = 2 f
f x = 2
f(2) = 1. f
f(x) = x - 1,
x = 2.
f
k, l, m " N
f(x, y, z) = xkylzm
limn" xn = x, limn" yn
l m
= y limn" zn = z limn" xk ynzn = xkylzm
n
f g ąf + �g ą �
f g p g(p) = 0

f
.
g
 g y
f x g ć% f
f(x) = y g ć% f x
f
[a, b] f(a) < f(b). c
f(a) < c < f(b). x " [a, b] f(x) =
c. f(b) < f(a).
y = c
x3 - 3x +
1 = 0 f(x) = x3 -x+1 f(0) = 1 f(1) = -1.
x " [0, 1] f(x) = 0.
1 3 3
f(1 ) = - + 1 = > 0
2 8 8 4
1
[ , 1]
2
3 1 3 53
x = . f( ) = -
4 4 4 64
1 3 5 1
[ , ] x = .
2 4 8 4
x
0, 0001.
g 0 d" g(x) d" 1
x " [0, 1] f(x) = g(x) - x. f(0) = g(0) e" 0
f(1) = g(1) - 1 d" 0. f(0) = 0 g(0) = 0
0. f(1) = 0 g(1) = 1 1.
 f(0) > 0 > f(1) x " [0, 1],
f(x) = 0 g(x) = x.
R2 Rw
n e" 2
Z �" Rw
w = 1
ą")
Z.
Z �" Rw
M
w
x = (x1, ..., xw) " Z x2 d" M.
k=1 k
d = 1 Z
n e" 1 an " Z |an| e"
n limn" |an| = ", |an|, n e" 1
Z Z
p
Z. p
R
 x
f
> 0 �, f(y)
f(x) y x.
� x

�. x
x x
"
" "
f(x) = x. | x- 1| < < 1
� = min(1-(1- )2, (1+ )2-1) = min(2 - 2, 2+2 ) =

"
1
2 - 2. | x - | < < 1
4
1 1 1 1
� = min( -( - )2, (1 + )2 - ) = - 2 - .
2 2 2 2 4
1
� x = 1 > 0 2 - 2, x =
4
1
- 2 -
4
� x � = �(x, ).
f
�
x y f
|f(x) - f(y)| < y
x. �
x > 0. � = �( ) f
�( )
d(x, y) < �( ) |f(x) - f(y)| <
f.
f(x) = x
�( ) = .
f(x) = x3
� > 0
� f
x |y3 - x3| <
y |x-y| �
� � � �
y = x+ . (x+ )3-x3 = 3x2 � +3x( )2+( )3 <
2 2 2 2 2
� �
3x2 � + 3x( )2 + (2 )3 e" x2�
2 2
x
f(x) = x3
 x3 - y3 = (x - y)(x2 + xy + y2) |x3 - y3| d"
M|x - y| M |x2 + xy + y2| d" M. M
x2 y2 max(a2, b2),

|xy| max(a2, b2, ab). �( ) =
M
f.
sin cos
x-y x+y
sin x - sin y = 2 sin cos .
2 2
x-y x+y x-y |x-y|
| sin x - sin y| d" |2 sin cos | d" |2 sin | d" 2 = |x - y|.
2 2 2 2
�( ) = .
f : R �" D R
ą > 0 M
|f(x) - f(y)| d" M|x - y|ą, x, y " D. ą = 1
f f
1

M ą
�( ) = .

sin x, x x3
R,
"
f(x) = x
"
"
|x-y|
"
[a, ") a > 0. | x- y| = "
x+ y
1
d" |x - y|. [0, 1].
2a
M
" "

x - y d" M|x - y| x, y " [0, 1]. xn, n e" 1
limn" xn = 0.
xn
x = xn y =
2


"
xn
xn -

2
d" M.
xn

2
1
"
"
xn
xn+
2
n "
1
ą = M = 1 [0, ").
2
" "
"
x - y d" x - y x > y
"
y d" xy
"
"
Z a f(a) d"
f(x) x " Z b f(b) e" f(x) x " Z.
f(a) f f(b)
f.
Ą
sin [0, ] 0
2
1.
f(x) = x2-2x+1
[0, 3] 0 x = 1 f(3) =
4.
f [a, b]
g f.
m M f
f.
f
[a, b] [m, M]
g. g
yn, n e"
1, yn " [m, M] y g(y) = limn" g(yn).

g(yn), n e" 1
g(yn ), k e" 1
k
c " [m, M], c = g(y). f

f(g(yn )) = yn f(c)
k k
f f(c) = f(g(y)) = y

yn y
 S
S d : S � S R+
S
d(x, y) = d(y, x), x y S
d(x, y) = 0 x = y
d(x, z) d" d(x, y) + d(y, z), x, y z S.
d(x, y) x y.
x y y x
x x
x z y
x z
S
 R d(x, y) = |x-y|.
Rw, w e" 1
a1, ..., aw b1, ..., bw
2 2 2
w w w

akbk d" a2 b2 .
k k
k=1 k=1 k=1
w

f(t) = (akt - bk)2, t e" 0.
k=1
w w w w

f(t) = [a2t2 - 2akbkt + b2] = t2 a2 - 2t akbk + b2,
k k k k
k=1 k=1 k=1 k=1
f
"
w w w

4( akbk)2 - 4 a2 b2 d" 0.
k k
k=1 k=1 k=1
4
x1, ..., xw, y1, ..., yw
z1, ..., zw
�ł �ł2

w w w



�ł łł
(xk - zk)2 d" (xk - yk)2 + (yk - zk)2 .
k=1 k=1 k=1
xk - zk = (xk - yk) + (yk - zk)
w w w

(xk - yk)2 + (yk - zk)2 + 2 (xk - yk)(yk - zk).
k=1 k=1 k=1


w w w w



(xk - yk)2 + (yk - zk)2 + 2 (xk - yk)2 (yk - zk)2.
k=1 k=1 k=1 k=1
S
x1x2, ...
N
0, 1, ..., N - 1. N = 2 0
1 S
x1x2x3... y1y2y3...
"

|xn - yn|
d(x1x2x3..., y1y2y3...) = .
(N + 1)n+1
n=1
d
Rw
R2
d((x1, x2), (y1, y2)) = |x1 - y1| + |x2 - y2|.
Rw?
d((x1, x2), (y1, y2)) = max{|x1 - y1|, |x2 - y2|}
R2. Rw, w e" 1.
 x x
X
x = 0 x
ąx = |ą| x x ą,
x + y d" x + y .
X
X d(x, y) = x - y .
S
p r
q
d(p, q) d" r. q
d(p, q) < r. p
p

Rw
 {(k, n); k, n " N} " R2
(k, n)
"
" 2
2 2
9
R2

(pn)ne"1 S
p
p
(pn)ne"1 p
an = d(pn, p), n e" 1 0.
p
p.
R2
(pn)ne"1 Rw p
p
(pn)ne"1 .
R

 a b
[a, b].
f g
(f + g)(x) = f(x) + g(x).
f + g x
f g f + g
ą
ąf (ąf)(x) = ąf(x).
f [a, b]. |f|
|f|(x) |f(x)|
[a, b]
x0 " [a, b], |f|(x0) |f|
f f ,
|f|(x0).
[a, b]
C[a, b].
x0 f.
[0, 1], f f(x) = x2
|f|(1) = f(1) = 1 g , g(x) = 1-x |g|(0) =
1 1 1 1 1
g(0) = 1, h h(x) = |x - | - |h|( ) = -h( ) = .
2 2 2 2 2
f(x) = x2-3x+2
C[a, b] a b
(fn(x))ne"1 [a, b].
"
cn
n=1
fn d" cn.
x " [a, b], |fn(x)| d" cn
 "
n=1 fn(x), x " [a, b] sn(x) =
n
fn(x), n e" 1 s(x)
k=1
sn s C[a, b]
limn" sn -s . x " [a, b] |sn(x)- sm(x)|
m e" n.
m m m

|sn(x) - sm(x)| = | fk(x)| d" |fk(x)| d" ck.
k=n+1 k=n+1 k=n+1
"
cn > 0
n=1
m
n0 m e" n e" n0 ck <
k=n+1
|sn(x) - sm(x)| < . n0
x
x limm" sm(x) = s(x). |sn(x)-s(x)| <
, x n e" n0, sn - s <
n e" 1.
"
an(x - x0)n
n=1
R [x0 -
r, x0 + r] r R x

|x - x0| r
n
lim sup |an(x - x0)n| = < .
R R
n"
n
r
|an(x - x0)n| d" n
R
n
r
fn(x) = an(x-x0)n cn =
R
n.
C[x0 - r, x0 + r].
r R
C[a, b] [a, b], a < b
(x0 - R, x0 + R).
[x0-r, x0+r] [x0-r, x0+r]
[a, b]
 C[a, b]
C[a, b]
C[a, b]
fn, n e" 1 [0, 1]
fn(x) = xn x " [0, 1]
limn" fn(x) = 0 x " [0, 1)
limn" fn(1) = 1. C[0, 1]
f(x) = 0 x " [0, 1) f(1) = 1? f
C[0, 1]
fn, n e" 1 f(x) = 0
C[0, r] 0 d" r < 1.
C[a, b]
fn, n e" 1 f
C[a, b]. fn, n e" 1 x
f x
fn(x), n e" 1 fn(x).
n0 |fn(x) - f(x)| < n e" n0.
n0 x
n0 = n0(x, ) fn f
C[a, b]?
fn - f < . n0
x " [a, b] |fn(x) - f(x)| < . n0 x
. C[a, b]
n0 fn(x), n e" 1
f(x). C[a, b]
fn,
n e" 1 D f
D > 0 n0 n e" n0
x " [a, b]
|fn(x) - f(x)| < .
n0 x .
fn, n e" 1 f D
f [a, b] D fn, n e" 1
f.
fn, n e" 1 D
" "
cn |fn(x)| d" cn fn(x)
n=1 n=1
"
fn(x)
n=1
fn, n e" 1
f 0 < < 1
n, x |xn - 0| e" x
"
n
x = .
fn f f
S
fn, n e" 1 f d
x " S > 0 fn f

n y " S, |fn(y) - f(y)| < .
3
fn �

y " S d(x, y) < � |f(x) - f(y)| <
3
y S
d(x, y) < |f(x) - f(y)| < .

|f(x)-f(y)| d" |f(x)-fn(x)|+|fn(x)-fn(y)|+|fn(y)-f(y)| < + + .
3 3 3
R
(x0 -
R, x0 + R) x " (x0 - R, x0 + R)
[x0 - r, x0 + r], 0 < r < R x
r = |x - x0|.

a < b
f " C[a, b]
> 0 w f - w < �
C[a, b].
C[a, b]
C[a, b]
w
f - w <

 C[a, b] R
w
f = limn" fn

n

1 k(b - a) n
fn(x) = f a + (x - a)n-k(b - x)k;
(b - a)n n k
k=0
n
k
n
n!
= .
k k!(n-k)!
fn
a = 0 b = 1 :

n

k n
fn(x) = f xn-k(1 - x)k.
n k
k=0
g
[a, b] f(x) = g((1 - x)a + xb)
x " [0, 1] f
C[0, 1] g C[a, b]
g f
f(x) = x2 [0, 1].
gn [a, b]
f g gn((1 - x)a + xb) fn(x)
n
x 0 d" x d" 1
Y
n
k pk = xk(1 - x)n-k. f " C[0, 1]
k

Y
X = f
n
k
k X f(n )
X xk

pk E X = xkpk.
Y
X = f .
n
n
n
n
E Y = k xk(1 - x)n-k
k=0 k
Y
E Y = nx E = x.
n
Y
Y
f f
n n
x.
Y
x
n
x(1-x)
(Y - x)2
n n
x (1 - x)2x + (-x)2(1 - x) = x(1 - x);
n nx(1 - x).
n
n2
Y
f
n
x n
n

xk(1 -
n 2 n
n n k k
x)n-k = 1 k xk(1 - x)n-k = nx - x xk(1 -
k=0 k k=0 n k
x(1-x)
x)n-k = .
n
� > 0


Y x(1 - x)

P - x e" � d" .

n n�2
|Y -x| � pk
n
k
(n - x)2 �.
Y
�2.
n
Y
n
2
n

k n
- x xk(1 - x)n-k.
n k
k=0
k
| - x| e" �
n
2
k
- x
n
�2. �2
k
pk | - x| e" �
n
x(1-x)
Y
n n
�2
f

> 0 � |f(x) - f(y)|
2
x, y [0, 1] |x - y| < �.
n
n k=0 pk = 1 f(x) =
f(x)pk f fn
k=1


n


f(x) - f k
pk.

n
k=1
Ł1
k
| - x| e" � Ł2
n

� Ł2
2
1 Ł2

. Ł1
2
2 f ,
x(1-x)
Ł1 2 f
n�2
1
x x(1 - x) [0, 1] x =
2
f f
1
Ł2 n e"
4 2n�2 �2

fn - f + = ,
2 2
x
ex = limn"(1+ )n
n
f
x " R. f
f(x + h) - f(x)
h
h " R).
x
(x, f(x)) (x+h, f(x+h)) f(x)
[x, x + h]
limh0 f(x+h)-f(x)
h
f x.
f
x.
df(x)
f x f (x) .
dx
f x
f
f x f (x)
x f f.
f(x) = a
0 0
n e" 1 f(x) = xn
x " R f (x) = nxn-1.
x

n

f(x + h) - f(x) (x + h)n - xn 1 n
= = hkxn-k
h h h k
k=1

n

n
= nxn-1 + hk-1xn-k - nxn-1.
k h0
k=2
x
limh0 f(x+h)-f(x)
h
f(x) = |x| x = 0
h > 0 1 h < 0
-1,
f(x) =
x - [x]
f g x ą �
ąf + �g fg
(ąf + �g)(x) = ąf (x) + �g (x)
(fg) (x) = f (x)g(x) + f(x)g (x).
f
g(x) = 0

g

f f (x)g(x) - f(x)g (x)
(x) = .
g g2(x)
1
[f(x + h)g(x + h) - f(x)g(x)]
h
1 1
= [f(x + h) - f(x)]g(x + h) + f(x) [g(x + h) - g(x)]
h h
g x


1 f(x + h) f(x) 1 f(x + h)g(x) - f(x)g(x + h)
- =
h g(x + h) g(x) h g(x + h)g(x)
1 1
[f(x + h) - f(x)]g(x) - f(x)[g(x + h) - g(x)]
h h
=
g(x + h)g(x)
g(x + h) h
g(x) = 0 g x).

2x2+3x
f(x) = ?
x2-3x+2
x = 1, 2. x
x
4x+3
2x - 3
(4x + 3)(x2 - 3x + 2) - (2x - 3)(2x2 + 3x)
f (x) = .
(x2 - 3x + 2)2
f
[a, b] (a, b). f(a)
= f(b) c " (a, b) f (c) = 0.
f [a, b]
f(a) f(b),
x " [a, b].
-f c
 f(c) e" f(x) x " [a, b]. c " (a, b)
f(c+h)-f(c)
c : ,
h
h.
c
h.
h 0 0.
f
f(a) = f(b).
c
f(b) - f(a)
f (c) = .
b - a
" f
"
f a b
x
f
f(b) - f(a)
g(x) = f(x) - (x - a) - f(a).
b - a
 g(a) = 0
g(b) = 0. g (a, b)
f c " (a, b)
f(b)-f(a)
g (c) = 0. g (x) = f (x) - .
b-a
 f
[x, x + h] a = x b = x + h h > 0
� " (0, 1)
f (x + �h)h = f(x + h) - f(x).
(x, x + h)
x + �h � " (0, 1). h < 0
x x + h � " (0, 1)
f(x) - f(x + h) = -hf (x + h - � h). � = 1 - �
(0, 1)
h
f1 f2
[a, b] c1 c2

f1(b) - f1(a) = (b - a)f1(c1) f2(b) - f2(a) = (b - a)f2(c2),

f1(b)-f1(a) f1(c1)
= .

f2(b)-f2(a) f2(c2)
c

f1(b) - f1(a) f1(c)
= .

f2(b) - f2(a) f2(c)
f2(x) = x f1(x) = f(x)
f1(b) - f1(a)
g(x) = f1(x) - f1(a) - (f2(x) - f2(a))
f2(b) - f2(a)
[a, b]
g(a) = g(b) = 0. c " (a, b)
f1(b)-f1(a)

g (c) = 0 g (x) = f1(x) - f2(x)
f2(b)-f2(a)
f [a, b]
d " (a, b] f
 [a, d]
f(d)-f(a)
c " (a, d), f (c) = . f (c) 0,
d-a
f(d) = f(a). d
f
x0
x0.
limxx f (x). x0
0
h f
x0 x0 +h h
� = �(h) " (0, 1) f (x0 + �h) =
1
[f(x0 +h)-f(x0)] h 0, �h 0, 0 d" |�h| d" |h|
h
f x0 f(x0 + �h)
(fn)ne"1
x f
f

fn
g f g.
x h x x + h
�n " (0, 1)

fn(x + h) - fn(x) = hfn(x + �nh).
� �n, n e" 1 �n , k e" 1
k
�, n e" 1. �n �n
k
k ",
f(x + h) - f(x) = hg(x + �h)

fn g g
g |�h| d" |h|,
h g(x) h 0.
 fn

fn.
1 1
( sin(nx)) = sin(nx) fn(x) = sin(nx)
n n
0
"
an(x-x0)n
n=0
R.
"
nan(x - x0)n-1
n=0
n
sn = ak(x - x0)k
k=0
=
n
n (x0 - R, x0 + R). s
kak(x - x0)k-1
k=0
(x0 -
"
R, x0 + R) nan(x -
n=0
x0)n-1
"
sin x (2n +
n=0
"
1)(-1)n x2n = (-1)n x2n = cos x (cos x) = - sin x
(2n+1)! n=0 (2n)!
(ex) = ex.
" x2n+1
sinh
n=0
" x2n (2n+1)!
cosh .
n=0 (2n)!
(sinh x) = cosh x (cosh x) = sinh x
ex+e-x ex-e-x
cosh x = sinh x = .
2 2
sin2 x + cos2 x
= 1 x.
f(x) = sin2 x + cos2 x.
cosh2 x - sinh2 x =
1 x " R.
1
(tg x) = tg x
cos2 x
sin x
.
cos x
 "
1
xn , x " (-1, 1).
n=0 1-x
an = 1
1
=
(1-x)2
" d "
d 1
= xn = nxn-1.
dx 1-x n=0 dx
n=1 "
"
n 1
S S = nxn-1
n=1 3n 3 n=1
1 1 9 3
x = , S = = .
3 3 4 4
J0(x) J1(x)
" "
2n+1

(-1)n x 2n (-1)n x
J0(x) = , J1(x) =
(n!)2 2 n!(n + 1)! 2
n=0 n=0
(-1)n 1
a2n = , a2n+1 = 0, n e" 0.
(n!)2 22n

n
|an|


1 1 1 1
n
2n
= ,
(n!)2 22n 2 n!
"
n
0 limn" n! = ".
d
x " R. J0(x) =
dx
-J1(x)
"
2n-1

(-1)n x2n-1 " (-1)n x

J0(x) = 2n =
(n!)2 22n n!(n - 1)! 2
n=1 n=1
"
2n+1

(-1)n x
= - ,
n!(n + 1)! 2
n=0
 x, y " R
" " " i

xn yk xj yi-j
exey = = (i = n + k)
n! k! j! (i - j)!
k=0 n=0 i=0 j=0

" i "

1 i ti(x + y)i
= xjyi-j = = ex+y.
i! j i!
i=0 j=0 i=0
ex > 0 x > 0. exe-x = e0 = 1
ex > 0 x " R. (ex) = ex
R. x > 0
xn
n e" 0 ex e" limx" xn = ", n e" 1
n! n!
1
limx" ex = " limx-" ex = limx" e-x = limx" ex = 0.
x " R ex e"
1 + x. f(x) = ex 1 - x.
"-
xn
f (x) = ex - 1 = .
n=0 n!
x < 0 x > 0 x = 0
0. (0, x), x > 0 f
(x, 0), x < 0
f(x) e" f(0) = 0 x " R,
f g
" g ć% f f
g
" f x
" g y = f(x) g
g ć% f x
g (f(x))f (x) = g (y)f (x).
h f(x + h)
�
h = f(x + h) - f(x).
�
g f(x)+h
( )-g(f(x))
g(f(x+h))-g(f(x))
=
h h
� " (0, 1)
�
f(x+h)-f(x)
h
� � �
g (f(x) + �h) = g (f(x) + �h). h 0 �h
h h
� � �
|�h| < h limh0 h = limh0(f(x + h) - f(x)) = 0.
 g
f (x)g (f(x))
g
cos2 x -2 sin x cos x
cos(x2) -2x sin(x2).
f
[a, b]
g
f, [m, M] m M
f. g
(m, M).
y x = g(y)
g(y+h)-g(y)
�
h . h = g(y + h) - g(y).
h
� �
y + h = f(g(y) + h) f h h
� �
g(y + h) - g(y) h h
= = ,
� �
h
f(g(y) + h) - y f(x + h) - f(x)
1
f (x)
1
g (y) = ,
f (x)
x = g(y).
x ex
(ex) = ex.
(0, ")
ln .
1 1 1
(ln y) = x = ln y (ln y) = = .
(ex) ex y
ln(st) = ln s + ln t, s, t > 0.
ex = s ey = t
ln x
loga a > 0 a = 1 loga x = .

ln a
a = 0 ln 1 = 0

ax, a > 0 ax = ex ln a.
a < 1 ax
a > 1
a
alog x = x, x > 0, loga
ax a = 1
ax loga x.
a
x xa x > 0
1
axa-1. xaxb = xa+b x-a = .
xa
x xx? xx =
ex ln x, (xx) = (x ln x) ex ln x = (ln x + 1)xx.
cos
cos 0 = 1 cos(-x) = cos x. cos
x = 2
2 4 16
sin 2 < (-1)n 4n = 1 - + < 0.
n=0 (2n)! 2! 24
c " (0, 2) cos c = 0.
cos c.
Ą 2c. x
Ą
cos x .
2
Ą Ą
x sin x [- , ] (sin x) = cos x
2 2
cos x
Ą -Ą
cos = cos = 0
2 2
Ą -Ą
 sin = ą1 sin = ą1.
2 2
Ą Ą Ą -Ą
[- , ] sin sin = 1 sin =
2 2 2 2
-1.
sin arc sin
Ą Ą
[-1, 1] - .
2 2
1
arc sin 0 = 0. arc sin y = =
sin x
1 1 1
" "
= = , |y| < 1
cos x
1-sin2 x 1-y2
cos x > 0 y = -1
y = 1
arc cos arc tg
 cosh x [0, ")
1
(cosh x) = (ex - e-x) ex >
2
e-x, x > 0 arc cos h
[1, +") cosh 0 = 1.

(ex)2 - 2yex + 1 = 0 ex = y ą y2 - 1.

arc cosh y = ln(y + y2 - 1).

1 1 1 1
" "
(arc cosh y) = = = = .
(cosh x) sinh x
cosh2 x-1 y2-1
1 y 1
(arc cosh y) = � (1 + ) = .
y + y2 - 1 y2 - 1 y2 - 1
arc tgh arc sinh
f g
[a, b] f(a) = g(a) = 0
limxa+ f (x) = g limxa+ f(x)
g (x) g(x)
g
limxa+ f(x) f
(a, a + h), h > 0
g > 0 � > 0
|f(x) - g| < a < x < a + �
f(x) f(x)-f(a) f (a+�(x-a))
x � " (0, 1), = = .
g(x) g(x)-g(a) g (a+�(x-a))
x a �(x - a) 0
 g
"
"
limxa+ f (x)
g (x)
limxa+
f(x)
f g
g(x)
H
0
limx0 sin x = = limx0 cos x = cos 0 = 1.
x 0 1
1
H
"
x
limx" ln x = = limx" 1 = 0.
x "
x
limn"(1 + )n
n
x
ln(1+ )
n
x x x limn" x
x
n"
n n
n
lim (1 + )n = lim eln(1+ )n = elim ln(1+ )n = e
n" n"
n
1
ln(1+y)
1+y
H
y 1
= ex limy0 = ex limy0 = ex.
exp
l
x l(1 + x).
x x2
l(1 + )2 = l(1 + 2x + ) e" l(1 + x).
2 4
n
x x
l(1 + )n x > 0 (1 + )n, n e" 1
n n
x?
l lex.
 f
f . f f
f
n
dn
f f(n)
dxn
f(n) = (f(n-1)) , n e" 2 f
f = f(0). n
n
(cos x)(1) = - sin x, (cos x)(2) = - cos x,
(cos x)(3) = sin x, (cos x)(4) = cos x.
(cos x)(n+4) = (cos x)(n), n e" 0
sin .
dn
ex = ex, n e" 0.
dxn
dn k!
xk = xn-k, n = 0, 1, ..., k
dxn (n-k)!
dn
xk = 0 n e" k + 1.
dxn
J0

xJ0 (x) + J0(x) + xJ0(x) = 0
"

(-1)n x2n

J0 (x) = -J1(x) = - (2n + 1) .
n!(n + 1)! 22n+1
n=0
2n + 1 = 2(n + 1) - 1
"

(-1)n x2n " (-1)n x2n

J0 (x) = - 2(n + 1) +
n!(n + 1)! 22n+1 n!(n + 1)! 22n+1
n=0 n=0
1
= -J0(x) + J1(x), x = 0.

x
x = 0.

x = 0 J0
J0
n
n
(ąf + �g)(n) = ąf(n) + �g(n),

n

n
(fg)(n) = f(k)g(n-k),
k
k=0
(xex)(n) = ex(1 + x)
(x3ex)(n).
 n
n, w(x) = akxk an = 0.

k=0
w(k)(0) = k!ak.
n

f(k)(0)
w(x) = xk,
k!
k=0
w x = 0
n f n - 1
[a, b]. f(n-1)
c " (a, b)
n-1

f(k)(a) f(n)(c)
f(b) = (b - a)k + Rn, Rn = (b - a)n.
k! n!
k=0
Rn
n = 1
n-1

f(k)(x)
g(x) = (b - x)k - f(b),
k!
k=0
n-1

f(k)(x) g(a)
h(x) = (b - x)k - (b - x)n.
k! (b - a)n
k=0
x = a
x = b
[a, b] x = a
 h
h(a) = h(b) = f(b) c " (a, b), h (c) = 0.
x " (a, b),
n-1 n-1

f(k+1)(x) f(k)(x)
g (x) = (b - x)k - (b - x)k-1
k! (k - 1)!
k=0 k=1
n-1 n-2

f(k+1)(x) f(k+1)(x)
= (b - x)k - (b - x)k
k! k!
k=0 k=0
f(n)(x)
= (b - x)n-1
(n - 1)!
f(n)(x) g(a)
h (x) = (b - x)n-1 + n (b - x)n-1.
(n - 1)! (b - a)n
f(n)(c) g(a)
c (b - c)n-1 + n (b - c)n-1 = 0,
(n-1)! (b-a)n
f(n)(c)
g(a) = - (b - a)n. x = a
n!
n-1

f(k)(x) g(a)
hm(x) = (x - a)k - (b - x)m,
k! (b - a)m
k=0
m 1, ..., n
f(n)(c)
Rn = (b - c)n-m(b - a)m.
(n - 1)!m
c m n m = n
m = 1
f(n)(c)
Rn = (b - c)n-1(b - a).
(n - 1)!
a = 0
 a b g(x) =
f(-x) [-b, -a]
d " (-b, -a)
n-1
g(n)(d)
g(-a) = g(k)(-b)(-a + b)k + (-a + b)n. g(k)(x) =
k=1 n!
n-1
f(n)(c)
(-1)kf(k)(-x) f(a) = f(k)(b)(a - b)k + (a -
k=1 n!
b)n, c = -d " (a, b).
f
[a, b].
"

f(n)(a)
(b - a)n.
n!
n=0
f(b)

1
e- x
, x > 0,
f(x) =
0, x d" 0.
x =

1
0. x = 0 limx0+ e- x
=
limy-" ey = 0 limx0- f(x) = 0

1
1
e- x
, x > 0,
x2
f (x) =
0, x < 0.
1 1 y2 H 2y 2
H
x
lim e- = lim = lim = lim = 0
x0+ y" y" y"
x2 ey ey ey
f (0) 0.

f(b)
 f(n)(x), x > 0
f(n)(x), x > 0
1
1
f(n)(x) = wn(x )e- x
, wn
limx0+ f(n)(x) = 0, f
x = 0 0.
0;
f x > 0.
lim Rn = 0.
n"
ex
[a, b].
n

ea ec
eb = (b - a)k + (b - a)n.
k! n!
k=0
n c " (a, b) n
ec
exp Rn = (b - a)n
n!
eb eb
(b - a)n. xn = (b - a)n, n e" 1 n > (b - a)
n! n!
0; xn+1 =
b-a
xn limn" xn = limn" xn+1 = 0 limn" xn = 0.
n+1
" ea
Rn eb = (b - a)k.
k=0 k!
a = 0 exp
"
ea 1 (b - a)k =
k=0 k!
eaeb-a a = x b = y + x ex+y = exey,
f [a, b]
M |f(n)(x)| d" M x " [a, b]
n e" 1. f.
M(b-a)n
|Rn| d"
n!
 sin cos .
f(x) = (1 + x)a
n n
n n x " R (1 + x)n =
xk. = 0 k > n
k=0 k k

"

n
(1 + x)n = xk.
k
k=0
n
n(n-1)...(n-k+1)
n!
= = .
k k!(n-k)! k!

a
n =
k
a(a-1)...(a-k+1)
, a " R, k " N. 
k!
n
=
k
0 k > 0 k
n
0.
k
(1 +
a
"
x)a = xk. a x
k=0 k
a
x " R a
a
an =
n
limn" an+1 = limn" a-n = -1
an n

n
limn" |an+1| = 1 limn" |an| = 1 = R-1, R = 1.
|an|
|x| d" 1;
x = -1 a

"

a
(1 + x)a = xk, a " R, |x| < 1.
k
k=0
limn" Rn =
0. x > 0
a a
f(n)(x) = n! (1 + x)a-n, (1 + cn)a-nxn =
n n
 n
a
(1 + cn)a x c
n 1+cn
n cn cn (0, x).
x (1 + x)a
1] a > 0 a < 0. Rn
[0,

a
C xn C = 2a a > 0 C = 1 a < 0
n
a

"
cn > 1 1 + cn > 1. xn
n=0 n
a
|x| < 1 limn" n xn = 0.
limn" Rn = 0 x > 0.
x < 0

f(n)(cn) a
Rn = (x - cn)n-1x = n (1 + cn)a-n(x - cn)n-1x.
(n - 1)! n
cn
x < cn < 0, �n = 0 < �n < 1
x
cn = �nx Rn
n-1
a a 1 - �n
n (1 + cn)a-n(1 - �n)n-1xn = n xn(1 + cn)a-1 .
n n 1 + x�n
(1 + cn)a
a
C |Rn| Cn xn(1 +
n
n-1
a
cn)a-1 1-�n n xn
1+x�n n
n-1
1-�n
-1 <
1+x�n
x < 0 -x�n < �n 0 < 1 - �n < 1 + x�n
1
1
a = -
2

1 3 1-2n
(- )(- )...( )
a (2n - 1)!! (2n - 1)!!
2 2 2
= = (-1)n = (-1)n ,
n n! 2nn! (2n)!
(2n - 1)!! 1
2n - 1 (2n)! 2 2n
"

1 (2n - 1)!!
" = 1 + (-1)n xn, |x| < 1.
(2n)!!
1 + x
n=1
-x2 x
"

1 (2n - 1)!!
" = 1 + x2n, |x| < 1.
(2n)!!
1 - x2
n=1
 f
(x0 -r, x0 +r)
"

f(x) = an(x - x0)n.
n=0
"
nan(x - x)n-1.
n=0
f
"

f(k)(x) = n(n - 1)...(n - k + 1)!an(x - x0)n-k, k e" 0.
n=0
k
(x - x0)0
n!an.
f(n)(x0)
an = .
n!
f(x) =
2 2 2
ex -4x x = 2? f(x) = ex -4x = ex -4x+4e-4 =
2
e(x-2) e-4. x " R
"

(x - 2)2n
f(x) = e-4 .
n!
n=0
(x - 2)
f 2
0.
122
(x - 2)122 n = 61
e-4 1 f(122)(2) = e-4 122! .
61! 61!
f.
100 x12ex
3
x = 0 x10e(x )?
"

(2n - 1)!! x2n+1
x + , |x| < 1.
(2n)!! 2n + 1
n=1
1
"
. arc sin
1-x2
(-1, 1)
arc sin arc sin 0 = 0
x = 0,
arc sin x x " (-1, 1).
10012 10013 arc sin x
x = 0?
f
x0

x0
x0.
limn" Rn = 0 Rn
f
x x f
> 0 |h| < , f(x+h)
f(x + h) d" f(x) h = 0

f.
 sin x =
Ą
+kĄ k x = kĄ.
2
f(x) = |x| x = 0.
f(x) =
|(x - 1)(x + 3)|
x
f f x f (x) = 0.
f (x) = 0 x
f(x) = x3 x = 0
f
(x - �, x]
[x, x+�) x
x
f (y) e" 0 y e" x x f (y) d" 0
y d" x x x f
x
x
 f x
f
x f (x) = 0
|h|
f (x + �h)
f(x + h) = f(x) + h2,
2!
� " (0, 1)
f (x + �h). f (x) > 0 h
f h = 0

f (x) > 0 x f (x) < 0 x
f(x) = x3, x = 0
f (x) = 0 f(x) = x4, f(x) =
-x4 x = 0 f (x) = 0
f
x
f (x + �h)
f(x + h) = f(x) + h3,
3!
� " (0, 1)
h.
f (x) > 0.
h
h, h3
x f (x) < 0
f(n)(x + �h)
f(x + h) = f(x) + hn.
n!
 n
n
f x f(n)(x) < 0
f(n)(x) > 0. n
h x
x f (x) = 0
f, f
x
x
x
f(x) = x3
x = 0 f (0) = 6.
f.
x
x = 0.
(a, b)
a = -" b = "
f
u, v " (a, b) 0 d" ą d" 1,
f(ąu + (1 - ą)v) d" ąf(u) + (1 - ą)f(v).
u < v 1 e" ą > 0 x = ąu + (1 - ą)v
[u, v]
1-ą
u v ą = 0 x = v
ą
y = ąf(u) + (1 - ą)f(v)
[f(u), f(v)].
f
 f -f
f(x) = x2 f(x) = |x|. x3
(0, ") (-", 0).
f(x) = sin x (-", ")
(0, Ą) (Ą, 2Ą)
Ą 3
( , Ą).
2 2
f (a, b)
a < u1 d" u2 d" u3 < b,
u2 - u1 u3 - u2
f(u2) d" f(u3) + f(u2).
u3 - u1 u3 - u1
f (a, b)
�
f(x) = f(a + b - x). a = -", b = ",
�
a+b = 0. f f
t
f f(u) = f(2t - x). f
f
(a, b) u " (a, b).
f(u) - f(x)
F (x) = Ff,u(s) = , x " (a, u),
u - x
f(y) - f(u)
G(y) = Gf,u(y) = , y " (u, b).
y - u
F G F (x) d" G(y)
x t F G
a < x < x < u
u1 = x1, u2 = x2 u3 = u,
F. G
Gf (x) = -Ff,a+b-u(a + b - x).
�
� �
f(a + b - u) - f(a + b - x) f(u) - f(x)
Ff,a+b-u(a + b - x) = = .
�
a + b - u - (a + b - x) x - u
u1 = x, u2 = u, u3 = y.
 u " (a, b)
f,
f(x) = |x| x = 0.
f
f
u " (a, b) Ff,u
x < u x <

� < u -f (�)(u - x) + f(u) - f(x) = 0. Ff,u(x) =
-f (x)(u-x)+f(u)-f(x)
e" 0, Ff,u
(u-x)2
(a, u).
f (a, b)
f(x) = x2,
2. (-", ").
x+y
x, y " (0, Ą) sin
2
1 1
sin x + sin y? x, y " (Ą, 2Ą)?
2 2
x, y " R,
3 1
x+ y 3 1
4 4
e d" ex + ey.
4 4
x
f (x - �, x) f
(x, x + �)
x = 0
f(x) = x3.
 x
f.
f
x0
x = x0
/ f
+" a " R
b := lim (f(x) - ax).
x"
y = ax + b
f. a = 0
ax+b limx"(f(x)-
ax - b)

f(x) - ax - b f(x) b
lim = lim - a - .
x" x"
x x x
b
limx" x = 0
f(x)
a = lim ;
x"
x
f a = limx" f (x)
a
y = ax + b
f.
 a f.
f
y = ax + b


x2-2x+3
f(x) = - x
x-1
n
x0, x1, ..., xn
a0, ..., an.
Pi = (xi, ai), i = 0, 1, ..., n.
n a0, ..., an
n
Pi = (xi, ai), i = 0, 1, ..., n,
xi ai, i = 1, ..., n.
n
x1, ..., xn
w0(x) = (x - x1)(x - x2)...(x - xn).
 w0(x0)
1
p0(x) = w0(x) 0
w0(x0)
x0;
1.
wk(x) = (x - x0)(x - x1)...(x - xk-1)(x - xk)...(x - xn), k = 1, ..., n
xk
1
pk(x) = wk(x)
wk(xk)
xk pk 1.
n

w(x) = akpk(x),
k=0
ak xk, k = 0, ..., n.
n pk n
w n
pk
ak, k = 0, ..., n
f ak =
f(xk). w
f?
w
f n + 1 [a, b]
(n + 1) (6.19)
[a, b]. x " [a, b] c " (a, b)
(x, b)
f(n+1)(c)
f(x) = w(x) + W (x),
(n + 1)!
W (x) = (x - x0)(x - x1)...(x - xn).
f(n+1)(c)
W (x)
(n+1)!
W (x) f
x c
f(x) = w(x) W (x) = 0.
x
g(y) = f(y) - w(y) - CW (y), y " [a, b]
f(x)-w(x)
C = C(x) = W (x) x
W (x)
g n + 2
n + 1 y = x,
n + 1
g g
n+1
n (n +
1) g (a, b)
c w n
w(n+1) = 0 W n + 1
(n + 1) xn+1
1, (n+1)!. (n+1) g
f(x)-w(x)
f(n+1) -(n+1)!C. c f(n)(c) = ,
W (x)
(n + 1) f [a, b]
M
M
|f(x) - w(x)| d" w(x).
(n + 1)!
f(x) = 27x5 - 3x3 + 4
1 2 M
0, , , 1
3 3 (n+1)!
n = 3, a = 0, b = 1.
n
f
 f
f(n+1)(c)
c
(n+1)!
xn+1
W (x).
f x a c
x a c
(a, b).
f
f
n + 1
 F (x, y) = 0.
Rw
(x1, x2, x3) (x2 + x2, x3 - x2)
1 2

Ąx2
1
(x1, x2) cos x1, sin(x1 + x2), tg .
4(x2 + x2)
1 2
1
Rw
2
Rw w1 w2
w1 = 3 w2 = 2 w1 = 2 w2 = 3.
1 2 1
f : Rw �" D(F ) Rw f Rw
2
Rw
Rw
(x1, ..., xw) + (y1, ..., yw) = (x1 + y1, ..., xw + yw)
ą(x1, ..., xw) = (ąx1, ..., ąxw).
 Rw
Rw Rw
Rw
-

x x x " Rw
1 2
A : Rw Rw
- -

1
x y Rw ą �
- - - -
A(ą + �) = ąA() + �A().
x y x y
1
Rw
2
Rw
- - - -
A(ą + �) = ąA + �A.
x y x y
A w1
w2
-
-
x A
x
1 2
Rw Rw .
1 2
Rw Rw A w1
w2
A
w1 = w2 = 1
R R x ax, a
x ax + b
b = 0.
(x1, x2, x3, x4) (x1 + x2, x1 -
x4, 2x2, -x2 - x3).
 AB
�ł �ł
1 1

�ł-1 -1�ł
3 -1 2 1
�ł �ł
A = , B = .
�ł łł
-1 0 5 -1 0 0
-2 -2
BA
a
f(a + h) - f(a)
f (a) = lim .
h0 h
a, h " R
-

-

1 2
a , h " Rw R Rw

-

h
-
------
- --
-
- -
f( + h ) - f()
a a
h h .
h = |h| h
w1 = w2 = 2 w1 = w2 = 3
f(a + h) - f(a) h f(a + h) - f(a)
lim = lim .
h0 |h| h0 |h| h
h h
limh0 |h| limh0+ |h| = 1
h
limh0- |h| = -1.
f(a + h) - f(a)
lim = f (a)
h0+ |h|
f(a + h) - f(a)
lim = -f (a).
h0- |h|
h
limh0ą |h| f (a) = ąf (a)
f(a + h) - f(a) - f (a)h
lim = 0.
h0 |h|
f (a) f a
f
a.
1 2
f : D(f) �" Rw Rw
a " D(f) D(f). A
f a
f(a + h) - f(a) - Ah
lim
h0 h
a h
- -
------ -
- --
- -
- -
f( + h ) - f() - A h
a a
lim .
h0 h
A f a
h Ah
(x, y) (x2y, x + y)

2 1
a = (1, 1) A = .
1 1
h = (h1, h2). f(a + h) = f(1 + h1, 1 + h2) = ((1 + h1)2(1 + h2), 2 +

h1 + h2) f(a) = (1, 2), Ah = (2h1 + h2, h1 + h2) h = h2 + h2.
1 2

f(a + h) - f(a) - Ah h2 + 2h1h2 + h2h2
1 1
= , 0 .
h
h2 + h2
1 2
2|h1h2| d" h2 + h2 |h2h2| d" h2
1 2 1 1
|h2| d" 1. h2

2 h2 + h2
1 2
h
a = (2, 2)

8 4
A = .
2 2
A w1 w2
- -
2 1
f : Rw Rw f() = A.
x x
A.
Ah h, f(a + h) - f(a).
h
R.
2
Rw w2
(ą, r) (r cos ą, r sin ą), ą " R, r > 0,
(ą, r) r sin ą (ą, r) r cos ą
 f : Rw R.
k = 1, ..., w
f(a1, .., ak-1, ak + h, ak+1..., aw) - f(a1, .., ak-1, ak, ak+1..., aw)
lim ;
h0 h
h
f a = (a1, ..., aw)
ak; w w
"f(a)
"ak
fa .
k
"f(r,ą)
f(r, ą) = r cos ą = -r sin ą
"ą
"f(r,ą)
= cos ą.
"r
x2y
f(x, y) = ,
x4+y2
x2 + y2 = 0 f(x, y) = 0 x y

f(0, y) = 0 f(x, 0) = 0 x y
"f(0,0) "f(0,0)
= = 0.
"x "y
xn, n e" 1 xn = 0 yn = x2

n
1
(xn, yn), n e" 1 (0, 0) f(xn, yn) = .
2
1
xn = 0 yn = (xn, yn), n e" 1 (0, 0) f(xn, yn) = 0
n
f
(0, 0)
(0, 0)
A
1 2
f : Rw �" D(f) Rw w2
1
fk : Rw R, k = 1, ..., w2.
 a " D(f). f
�ł �ł
"f1(a) "f1(a) "f1(a)
...
"a1 "a2 "aw1
�ł �ł
"f2(a) "f2(a) "f2(a)
�ł �ł
...
"a1 "a2 "aw1
�ł �ł
�ł �ł
�ł �ł
�ł łł
"fw2 (a) "fw2 (a) "fw2 (a)
...
"a1 "a2 "aw1

-r sin ą cos ą
.
r cos ą sin ą

2xy x2
.
y x
w
-

a v = (v1, ..., vw)
-
f(a + h) - f(a)
v
lim
h0 h
h
-


f v f- . v

v
k
1
-

k v = (0, ..., 1, ..., 0)
"f
-

k f- = . v = 0 v

v "xk
0.
 ą
-

- =

v fą ąf- ą =
v v
0
f
-
- h3ą2�
a = (0, 0) v = (ą, �) = (0, 0) f(a + h) = ,
v
h4ą4+h2�2
1 - 1 - ą2� 1
(f(a+h)-f(a)) = f(a+h) = . limh0 h (f(a+
v v
h h h2ą4+�2
-
h)-f(a)) 0 � = 0 � = 0
v
ą2
.
�
A f : Rw �" D(f)
-

R a v
- -

v f- = A.
v

v
-

v = 0 t 0


- - - -

f(a + t) - f(a) - tA f(a + t) - f(a) - tA
v v v v

=

t |t|

- -

f(a + t) - f(a) - tA
v v
-

=
v
-

t
v

- -
-

f(a+ h )-f(a)-A h
-
h = t t 0.
v
h
(ą, r)
(v1, v2)
(-rv1 sin ą + v2 cos ą, rv1 cos ą + v2 sin ą).
(r, �, Ć) (r cos � cos Ć, r cos � sin Ć, r sin �).
(1, -1, -2)
Ą Ą
(1, , )?
4 4
1 2
f : Rw �" D(f) Rw
a " D(f) f
2 3
g : Rw �" D(g) Rw g
f(a). a
g ć% f B A :
(g ć% f) (a) = BA B g f(a) B
f a.
f(t) = (sin t, cos t) g(x, y) = x2 +y2.
g (2x, 2y), f
(cos t, - sin t)
g(f(t)) = sint + cos2 t 2 sin t cos t - 2 sin t cos t = 0
f w
x1, ..., xn x1, ..., xn t.
f
df(t) "f dx1 "f dx2 "f dxw
= + + ... + .
dt "x1 dt "x2 dt "xw dt
x sin z
f(x, y, z) = x(t) = t, y(t) = t2 z(t) =
y2+1
arc tg t
df(t) "f dx "f y "f dz
= + +
dt "x dt "y dt "z dt
sin z(t) 2y(t)x(t) sin z(t) x(t) cos z(t) 1
= - 2t +
y2(t) + 1 (y2 + 1)2 y2(t) + 1 1 + t2
sin arc tg t 2t3 sin arc tg t t cos arc tg t 1
= - 2t + .
t4 + 1 (t4 + 1)2 t4 + 1 1 + t2
x, y z f,
x2 + y2 - 5 = 0.
(x, y)
 x
y.
" (x0, y0)
5
x
" " y
"
(0, 5), (0, - 5), ( 5, 0)
"
(- 5, 0) x
y y x.
"
x = 5 - y2 y = 5 - x2
x, y (x0, y0).
" "
x = 5 - y2 y = - 5 - x2 (0, 5)
"
(x, y) y = 5 - x2
x y

5 - y2
" "
(0, - 5). ( 5, 0)

x y x = 5 - y2
y x.
"
(- 5, 0).
x
y
x y
y x
x2e2y - y2ex = 0?
f
"f(x0,y0)
(x0, y0) = 0

"y
y x y(x)
"f(x,y)
fx
"x
y (x) = - = - .
"f(x,y)
fy
"y
 - y = y(x)
x
x y
"
5 f(x, y) =
"f(x0,y0)
x2 + y2 - 5 = 2y0
"y
" "
y0 = 0, ( 5, 0) (- 5, 0)

y x
x
y (x) = -
y(x)
"f(x0,y0)
y. = 2x0
"x
" "
(0, 5) (0, - 5) x y
dx y
= - .
dy x(y)
x + y = ex-y.
y x? f(x, y) =
ex-y -x-y y -ex-y -1.
x = y y

1-ex-y
x y (x) .
1+ex-y
y = x
1
x = y = .
2
g(x) = f(x, y(x)) = 0.
"f "f
g (x) = (x, g(x)) + (x, g(x))g (x) = 0,
"x "x
y g.
"2f "2f "2f "2f "2f "2f
:= , , , := .
"x2 "x"x "y"x "x"y "y2 "y"y
"2f "2f "2f "f
+ 2 y (x) + [y (x)]2 + y (x) = 0
"x"x "y"x "y2 "y
"2f "2f "2f
+ 2 y (x) + [y (x)]2
"x"x "y"x "y2
y (x) = - .
"f
"y
"f
fx =
"x
fxx + 2fxyy + fyy(y )2
y = - ,
fy
2 2
fxxfy - 2fxyfxfy + fyyfx
y = - .
3
fy
fxx
y = 0 y = - .
fy
"
(x, y) = (0, 5)
y(x) x2 + y2 = 5
"
fxx ex-y e- 5
"
- = = > 0.
3
fy ex-y+1
(e- 5+1)3
y
f w + 1 x1, ..., xw
y
y
f(x1, ..., xw, y) y
y(x1, ..., xn)
"f(x1,...,xn,y)
"y(x1, ..., xn)
"xk
= - .
"y(x1,...,xn)
"xk
"y
 x, y z
Ć(cx - az, cy - bz) = 0, Ć = Ć(x1, x2)
a, b c
z Ć? z
"z "z
a + b = c,
"x "y
f(x, y, z) = Ć(cx - az, cy - bz).
"f(x, y, z) "Ć(x1, x2) "Ć(x1, x2)
= -a - b
"y "x1 "x2
x1 = cx - az x2 = cy - bz.
0
"Ć(x1,x2)
1
ac"Ć(x ,x2) bc
"z "z
"x1 "x1
a + b = + = c,
"Ć(x1,x2) "Ć(x1,x2) "Ć(x1,x2) "Ć(x1,x2)
"x "y
a + b a + b
"x1 "x2 "x1 "x2
f(x, y) = 0.
x y
f(x, y)
f(x, y) = 0
(x0, y0)
f(x0, y0) = 0, fx(x0, y0) = 0, fy(x0, y0) = 0.
y
x
f(x, y) = 0
" = fxxfyy - fxy.
" " > 0 (x0, y0)
" " < 0
" " = 0
f(x, y) = (x2 + y2)(x - y + 1)
(0, 0) y = x + 1.
"
" < 0 fyy = 0

fxx + 2fxyt + fyyt2 t1 t2.
fyy = 0
f(x, y) = 0
f(x, y) = 0
f(x, y) = x3 + y3 - 3xy. fy = 3y2 - 3x
x = y2 y
-
" " x = y2, y6 2y3 = 0 (0, 0)
3 3
( 4, 2). fx = 3x2 - 3y
(0, 0). " < 0 fxx = fyy = 0 fxy = -3
3t = 0,
y = 0 x = 0
(x, y)
x3 + y3 - 3xy
x = 0 y = 0
(0, 0)
" "
3 3
( 4, 2)
x y
y = x.
fy = 0
y-x2
y (x) .
y2-x
y < 0
x x
f(x, y) = 0 y x2+y2-3x = 0 < 0
y y
y2 - 3x > 0
"
y = x
" "
3 3
( 2, 4).
" "
3 3
" " " y2-x = 0 (0, 0) ( 4, 2)
"
3 3 3 3
(0, 0) ( 2, 4) ( 4, 2)
"
y = x
"
3
x x = 2
fxx 6x
y(x) = - = - .
fy y2-x
y = -x - 1
 R3
 f (x1, ..., xw)
a = (a1, a2, ..., aw) � > 0, h < �
f(a + h) d" f(a) a
f h = 0
a
f(x, y) =
|x - 2| + |y + 1| (2, -1)
f(x, y) = |x - y|.
0
 w
f x1, ..., xw.
f
g x1 g(x1) = f(x1, a2, a3, ..., aw); a
f a1 g
f a g a1
0 g
f a
f 0;
f.
f(x, y) = x2 +xy+y2 -3x-6y
fx(x, y) =
2x + y - 3, fy(x, y) = x + 2y - 6.

2x + y = 3,
x + 2y = 6
(0, 3)
R2.
f
a
"f "f
(a) = ... = (a) = 0.
"x1 "xw
6. f
1
f(a + h) - f(a) = dfah + d2fa+�h(h).
2
� (0, 1), dfah
w "f
(a)hi d2fa+�h(h)
i=1 "xi
 w w "2f
(a + �h)hihj.
i=1 j=1 "xi"xj
a
d2fa+�h(h)
h a
a
f
d2fa+�h(h)
fx ,x1(a + �h)h2 + 2fx ,x1(a + �h)h1h2 + fx ,x2(a + �h)h2.
1 1 1 2 2
h
fx ,x1(a)h2 + 2fx ,x1(a)h1h2 + fx ,x2(a)h2;
1 1 1 2 2

fx ,xj (a) h
i
Ah2 +2Bh1h2 +Ch2, A, B C
1 2
h
h2 = 0 h1 = 0

Ah2 + 2Bh1h2 + Ch2 = h2(At2 + 2Bt + C)
1 2 2
h1
t = .
h2
A > 0 " = 4B2 - 4AC
A < 0 " < 0
" > 0
" -4W W

fx ,x1(a) fx ,x2(a)
1 1
.
fx ,x1(a) fx ,x2(a)
2 2
a

 fx ,x1(a) > 0 W > 0 a
1
fx ,x1(a) < 0 W > 0 a
1
W < 0 a
W = 0
f

2 1
. W = 3
1 2
(0, 3) f
f(x, y) = x4 + y4 fx = 4x3, fy = 4y3
(0, 0)
W = 0
(0, 0) f
f(x, y) = x2 - y2 fx = 2x, fy = -2y
(0, 0)

2 0
, W = -4
0 -2
f
(0, 0) (x, y)
0zx
z = x2.
z = x2 - y2 x = staa
0zx
z = f(x, y)
2
w w

ąi,jhihj,
i=1 j=1
�ł �ł
ą1,1 ą1,2 ... ą1,w
�łą2,1 ą2,2 ... ą2,w �ł
�ł �ł
�ł �ł ,
�ł łł
ąw,1 ąw,2 ... ąw,w
"2f
f a ąi,j = (a).
"xi"xj
h
ą1,1

ą1,1 ą1,2 ... ą1,w


ą2,1 ą2,2 ... ą2,w
ą1,1 ą1,2


ą1,1 > 0, > 0, ..., > 0.
ą2,1 ą2,2


ąw,1 ąw,2 ... ąw,w
a
h = 0,

k�k -1
(-1)k

ą1,1 ą1,2 ... ą1,w


ą2,1 ą2,2 ... ą2,w
ą1,1 ą1,2

ą1,1 < 0, > 0, ..., (-1)w > 0,

ą2,1 ą2,2


ąw,1 ąw,2 ... ąw,w
f(x, y, z) = x2 + y2 + z2 - 4x + 6y - 2z.
fx = 2x-4, fy = 2y+6, fz = 2z-2
(2, -3, 1).
�ł �ł
2 0 0
�ł0 2 0 łł
0 0 2
2, 4 8.
x2 + y2 + z2 - xy + x - 2z
ńł
�ł2x - y + 1 = 0,
�ł
�ł
2y - x = 0,
�ł
�ł
ół2z - 2 = 0,
2 1
(- , - , 1).
3 3
�ł �ł
2 -1 0
�ł-1 2 0łł .
0 0 2
1
2, 3 6 (-2 , - , 1)
3 3
f.
x2+y
f(x, y) =
x2+y2+1
|x| d" 1, |y| d" 1?
2x(y2-y+1) (1-y)(1+y+x2)
fx = , fy = ,
(x2+y2+1)2 (x2+y2+1)2
fx x = 0 fy
y = 1
x2+1 1
f(x, 1) = = 1- .
x2+2 x2+2
1
x = 0 .
2
2
x = ą1 .
3
x2-1 3
f(x, -1) = = 1 - .
x2+2 x2+2
1
- 0.
2
1+y
f(-1, y) = ;
2+y2
y2+2y-2
y -
(2+y2)2
"
y = -1ą 3
[-1, 1]
+ - y = -1 y = 1
2
0 .
3
"
" "3 1+"3 f(1, 3-
"
1) = f(-1, 3-1) = = . f(0, -1) =
4
6-2 3
1
- .
2
f
f
f(p, V, T ) p V T
p, V T
pV = nRT.
2P
x, y z
xyz
x, y z
2xy + 2yz + 2zx = 2P.
x
y xy + yz + xz = P x = 2y
 w
f(x1, ..., xw) m Ći, i = 1, ..., m
S x = (x1, ..., xw)
Ći(x) = 0, i = 1, ..., m
a " S f
� > 0 |h| < � a + h " S
f(a + h) d" f(a);
f Ći, i = 1, ..., m
L(x1, ...., xw, 1, ..., m) w + m
m

L(x1, ...., xw, 1, ..., m) = f(x1, ...., xw) + iĆi(x1, ...., xw).
i=1
f Ći, i = 1, ..., m a
i, i = 1, ..., m
Lx (a) = 0, i = 1, ..., w, Ći(a) = 0, i = 1, ..., m.
i
a
d2La(h) h = (h1, ..., hw) = 0

w

"Ći
(a)hj = 0, i = 1, ..., m;
"xj
j=1
L x1, ..., xw i
h a
a
a
 f(x, y, z) =
2x + y - 2z x2 + y2 + z2 = 36.
L(x, y, z, ) = 2x + y - 2z + (x2 + y2 + z2 - 36)
Lx = 2 + 2x, Ly = 1 + 2y, Lz = -2 + 2z.
ńł
�ł
�ł2 + 2x = 0,
�ł
�ł
�ł1 + 2y = 0,
�ł-2 + 2z = 0,
�ł
�ł
�ł
ółx2 + y2 + z2 36.
=
(x + z) = 0.
 = 0
2 = 0 x = -z.
2
1
y = - z
2
9
z2 = 36 z = ą4.
4
1
(-4, -2, 4) (4, 2, -4);  =
4
 = -1 .
4
2
(-4, -2, 4) (4, 2, -4)
-2 2.
f(x, y, z) = xyz, Ć1(x, y, z) = xy+xz+yz-P, Ć2(x, y, z) = x-2y.
L(x, y, z, , �) = xyz + (xy + xz + yz - P ) + �(x - 2y).
L
 �
ńł ńł
�łyz + (y + z) + � = 0, �łyz - 2y2 + 3� = 0,
�ł �ł
�ł �ł
�ł
�łxz + (x + z) - 2� = 0, �ł 4y2 6� = 0,
�ł4yz - -
�ł �ł
�ł �ł
�ł �ł
xy + (x + y) = 0, 2y2 + yz + 2yz = P,
�ł �ł
�ł �ł
�łx = 2y, �ł = - 2
�ł �ł
y,
�ł �ł
3
�ł �ł
�ł �ł
ółxy + yz + xz = P. ółx = 2y.
 x = 2y y
2
y > 0  = - y. x = 2y,  = -2 y
3 3
4
6yz - 8y2 = 0 z = y.
3

P
y =
6
2
� = y2.
9
4
f (2p, p, p)
3

P 2 2
p =  = - p � = p2.
6 3 9
L
�ł �ł
0 z +  y + 
�łz +  0 x + łł ,
y +  x +  0
4 1 2
x +  = p, y +  = p z +  = p.
3 3 3
2
p(2h1h2+h1h3+
3
4h1h3). h = (h1, h2, h3)
7 10 9
h1 = 2h2 ph1 + ph2 + ph3 = 0 Ć1(x, y, z) = x - 2y
3 3 3
"Ć1 "Ć1 "Ć2 "Ć2
Ć2(x, y, z) = xy+yz+xz-P = 1, = -2, = y+z, =
"x "y "x "y
"Ć2 7
x + z = x + y y + z = p, x + z =
"z 3
10 8
p x + y = 3p. h1 = 2h2 h3 = - h2.
3 3
2 16 32
ph2(4 - - ) < 0 h2 = 0 h2

2
3 3 3

4 P
h (2p, p, p) p =
3 6
f
f(x, y, z) = xyz
x + y + z = 4 xy + yz + zx = 5.
f(x, y, z) = xy2z3
x + 2y + 3z = 12 x, y, z > 0
f(x, y, z) =
xyz x2 + y2 + z2 = 1 x + y + z = 0.
 x + 2y + z = 4 3x + y + 2z =
3 l
x2 + y2 = 1 x + y + z = 1.
" x1 + x2 + � � � + xn
n
x1x2 � � � xn d" , n " N, x1, x2, ..., xn > 0.
n
c > 0
x1 + x2 + � � � + xn
Ć(x1, ..., xn) = - c, f(x1, ..., xn) = x1x2 � � � xn.
n
f Ć = 0 cn.
L(x1, ..., xn, ) = f(x1, ..., xn) + Ć(x1, ..., xn).
x1���xn 
Lx = + , f
i
xi n
ńł
�łx1x2 � � � xn = -  x1,
�ł
n
�ł
�ł
�łx x2 � � � xn = -  x2,
�ł
�ł
1
�ł n
. . .
�ł
�ł
�łx1x2 � � � xn = - 
�ł
xn,
�ł
n
�ł
�ł
ół x1+���+xn
= c.
n
n
x1 = x2 = � � � = xn
x1 = x2 = � � � = xn = c  = -ncn-1. (c, ..., c)
f(c, ..., c) = cn
x1���xn
Lx ,xi = 0, i = 1, ..., n Lx ,xj = i = j.

i i
xixj
L (c, ..., c)
cn-2

cn-2 hihj.
i =j,i,j=1,...,n
h = (h1, ..., hn) = 0 h


(h1 + h2 + � � � + hn) = h1 + h2 + � � � + hn = 0
n

Ćx = . h1 = -(h2 + h3 + ... + hn).
i
n
cn-2
h1(h2 + � � � + hn) + h2(h1 + h3 + ... + hn) + ... + hn(h1 + ... + hn-1).
h1
-(h2 + � � � + hn)2 - h2 - � � � - h2 .
2 n
h
(c, ..., c)
x, y z
3
x3 + y3 + z3 x + y + z
e" .
3 3
 A " R
m M a " A
m d" a d" M.
m M
(0, 1]
1
0.
A
A
0 1.
 A
" A
" A
" A
" A
" A
" A
A
K > 0 a " A
|a| d" K.
A B
mA, mB MA MB
A *" B A )" B.
f
[a, b] a < b. P
xi, i = 0, ..., n a = x0 < x1 < x2 <
... < xn = b. [xi-1, xi] Oi
�i = xi - xi-1, i = 1, ..., n
n n

SP = Mi�i, sP = mi�i,
i=1 i=1
mi Mi
f Oi.
f.
SP e" sP.
�i " Oi xi-1 d"
�i d" xi
n

ŁP = f(�i)�i
i=1
SP e" ŁP e" sP.
ŁP �i
 mi m
f b - a
m(b - a).
M(b - a) M f.
m(b - a) d" sP d" ŁP d" SP d" M(b - a)
f
[a, b]
x x = a
x = b
P [a, b]
Oi
SP.
P �i SP
P
Oi
Oi
sP.
ŁP
Mi
mi.
a = 0 b = 1 f(x) = x
i
n e" 1 xi = , i = 0, ..., n. f
n
[xi, xi+1];
1
�i
n
n-1 n-1 n-1

1 i 1
sP = xi�i+1 = = i
n n n2
i=0 i=0 i=0
n n n

1 i 1
SP = xi�i = = i.
n n n2
i=1 i=1 i=1
1
�i = (xi + xi-1)
2
Oi.
n

1 1
ŁP = (xi + xi-1)�i = (sP + SP).
2 2
i=1
n(n-1) n(n+1)
n-1 n+1
sP = SP = . n
2n2 2n 2n2 2n
sP SP ŁP
1 1
2 2
0x x = 1
(0, 0), (1, 0)
(1, 1)
[a, b]
[a, b]
Oi
Oi,j, j = 1, ..., k f
Mi Oi
j = 1 k Oi,j
Oi,j Mi
�i.
�i
 [a, b].

b
f(x) dx = lim SP.
max{�1,...,�n}0
a

a b
x
dx x
f

b
f(x) dx = lim sP.
max{�1,...,�n}0
a
a = 0 b = 1
f(x) = 1 x " [0, 1] f(x) = 0 x
[a, b]
 1 1.
1 0,
f [a, b]

b
f(x) dx.
a

f


b
c dx f(x) = c
a
c(b - a). c�i
c
�i b - a, c(b - a)
c(b-a)

b
c dx = c(b - a).
a
b f(x) = x [a, b]
1
x dx = [b2 - a2].
a 2
f
[a, b]
> 0
.
f(b) > f(a)

(b-a)(f(b)-f(a))
n n > .

b-a
P xi = a + i , i = 0, ..., n
n
SP - sP e" 0 f
f Oi f(xi); f(xi-1).
SP - sP
n n n

b - a b - a b - a
f(xi) - f(xi-1) = [f(xi) - f(xi-1)]
n n n
i=1 i=1 i=1
b - a
= [f(x1) - f(x0) + f(x2) - f(x1) + ... + f(xn) - f(xn-1)]
n
b - a b - a
= [f(xn) - f(x0)] = [f(b) - f(a)] < .
n n
sP SP,
SP - sP <
f
[a, b]
f
�

|x - y| < �, x, y " [a, b] |f(x) - f(y)| d" .
b-a
1
n e"
�

Mi mi .
n b-a
n
SP sP �i = �i = .
i=1 b-a b-a i=1
k
d
d M m
6
f [a, b].
yi i (yi - , yi + )
[a, b]
[a, b]
b-a
n
n

SP - sP = (Mi - mi)�i
i=1
n
(Mi - mi)�i
i=1
 n
2
k(2 + ) Mi mi
n
M - m.
1
2k( + )(M - m).
n
n 4k (M - m)
SP sP (1 + 4k(M - m)) .
(1 + 4k(M - m)) . (1 + 4k(M - m))
[0, 1] f(x) = x.
i
xi = , i = 0, ..., n, n " N.
n
n
1 1
1
. x dx = .
2 0 2
[0, 1] f(x) = xe-x.
i
xi = , i = 0, ..., n, n " N.
n-1 n-1 -n n-1
i
1 1
n
f(xi)�i+1 =
i=0 n2 i=0 n2
ie = i=0 iqi
1
n-1
q = e- n
. iqi
i=0
q
q2, q2
q3, q3, q3
qn-1, qn-1, � � � qn-1,
 i i qi.
1-qn-1
q + q2 + ... + qn-1 = q ,
1-q
q2 + ... + qn-1 = q2 1-qn-2
n-1 qi-qn 1 n-1 1-q
(n-1)qn q-qn (n-1)qn
= qi - = - .
i=1 1-q 1-q i=1 1-q (1-q)2 1-q

1
1 e- n - e-1 (n - 1)e-1
lim - .
1 1
n"
n2 (1 - e- n
)2 1 - e- n
1
1
H
n
limn" n(1 - e- n
) = limn" 1-e- = limh0 1-e-h =
1
h
n
limh0 e-h = 1, 1 - e-1 - e-1 = 1 - 2e-1,
1
xe-x dx.
0
a a > 0 b " R,
ebx dx
0
"
b
f(x) dx = F (f)
a
" [a, b] f g
ą � ąf + �g

b b b
(ąf(x) + �g(x)) dx = ą f(x) dx + � g(x) dx.
a a a
" f [a, b]

b
x " [a, b] f(x) e" 0 f(x) dx
a
"
[a, b]
" [a, b]
c " [a, b]
" f [a, b] f(a)
1
+ f(b).
2
" f [a, b] f(a)
-f(b).
f(x) = x - a
[a, b]
0 - (b - a) = a - b < 0.
� ąf +�g
ą�1+��2 �1
f �2 g.
b
b
�1 f(x) dx �2 f(x) dx
a a

b
� ą f(x) dx +
a

b
� g(x) dx ąf + �g
a

b b
ą f(x) dx + � g(x) dx
a a
f
[a, c] b a < b < c f
[a, b] [b, c]
 [a, c]

c b c
f(x) dx = f(x) dx + f(x) dx.
a a b
f [a, c]
b
b
b b.
f(x) dx
a

c
f(x) dx
b
a < b f [a, b]

a b
f(x) dx = - f(x) dx.
b a
x " [a, b]
f(x) e" g(x) f g

b b
f(x) dx e" g(x) dx.
a a
g(x) = 0, x " [a, b]
b f - g b
b
[f(x) - g(x)] dx = f(x) dx - g(x) dx
a a a

b b
f(x) dx - g(x) dx e" 0
a a

b
g(x) dx
a
f
[a, b]
 M " R x " [a, b]
f(x) d" M.

b
f(x) dx d" M(b - a).
a
f m " R
x " [a, b] f(x) e" m

b
f(x) dx e" m(b - a).
a
M e" f(x)
M(b - a)
f


b b

f(x) dx d" |f(x)| dx.


a a
f
d" f(x) d" |f(x)|.
b -|f(x)|
b b
- |f(x)| dx d" f(x) dx d" |f(x)| dx
a a a
f (a, b) x0 " (a, b)

x
g(x) = f(y) dy.
x0
g g (x) = f(x).
1
[g(x + h) - g(x)].
h

x+h
1
f(y) dy. > 0
h x
� > 0 |y - x| < � y " (a, b) |f(x) - f(y)| < .
x+h
f(x) dy = f(x)h y f(x)
x
|h| �


x+h x+h x+h
1 1 1

dx - f(x) = dx - f(x) dy

h h h
x x x


x+h
1

= (f(y) - f(x)) dy

h
x

x+h
1

d" |f(y) - f(x)| dy

|h|
x

x+h
1 1

d" dy = |h | = .

|h| |h|
x

f(x)
f
(a, b)
g f.
g
� = b2 - 4ac
ax2+bx+c =
0. f
g = f.
g = f

x
g(x) = f(x) dx
x0
 g = f f
(a, b). g
f
(a, b)
g g (x) = f(x) x " (a, b)
c < d c, d " (a, b)

d
f(y) dy = g(d) - g(c).
c

x
c h(x) = f(y) dy - g(x) + g(c).
c
f(x) - g (x) = 0.
h(c) = 0

d
x = d f(y) dy - g(d) + g(c) = 0,
c
x
R
"
-
"
f(x) = R2 x2.
R
2 R2 - x2 dx.
-R
"
R2 - x2 dx
 x = R sin t. x (t) = R cos t,


R2
R2 - x2 dx = R2 cos t dt = (1 + cos 2t) dt
2
R2 1
= (t + sin 2t) + C
2 2

1 x
= (R2 arc sin + x R2 - x2) + C.
2 R

x
(R2 arc sin + x R2 - x2)x=R = R2(arc sin 1 - arc sin(-1)) = ĄR2.
x=-R
R
 A
"n+1
an =
n2+1
2n + 3 1 5n + 3
an = , bn = + 3, cn = 2n+1, dn = .
3n + 4 n + 2 3 - 2n
{an} {bn}
1 1 1 1
an = 1 + + + ... + , bn = an + .
1! 2! n! n!
{an} {bn}

p! e" 2p-1 {an}
p"N
2n + 3 2 1 + 3n 3
lim = , lim = - ,
n" n" - 5n 5
3n + 4 3 1
"
3n2 - 7 n2 + 1 + 1
lim = 3, lim " = 1,
n" n"
n2 + n + 2
n2 + 1 - 1
) lim ln(n4 + 3) = ".
n"
Ą
an = (-1)3n+1, an = cos n.
2
an2-1
an =
(a-1)n2+n
a lim an = 2.
n"
a
5an-10
lim , lim an = 2.
a2 -4
n
n" n"
3an+6
lim an -2
4-a2
n
n"
{an} {bn}
lim anbn = 0
n"
1
lim an = " lim = 0.
an
n" n"
"
n2 + 6 - n 3 � 4n-1 - 3n
an = " , an = ,
2n+1 + 22n
n2 + 3 - n

1 1
1 + 3 + 5 + ... + (2n - 1) 1 + + ... +
2 2n
an = , an = " " ,
n + 1
n + 2 - n + 1
"
(n3 - 1)10
n
an = , an = 3n + 4n + 5n,
(n5 - 1)6

n sin n!
n
an = 3n2 + sin5 n, an = ,
n2 + 1
3n+2
n + 3 2n + 1
an = sin(3n - 4), an = ,
n2 - 4 2n + 3
2
n +n n
5n2 + 7n + 13 5
an = , an = 1 - ,
5n2 + n + 3 n2
n+7
n2 + 5n - 2
an = n[ln(n + 3) - ln(n + 2)], an = ,
n2 - 2n


3 3
an = (n + 1)2 - (n - 1)2, an = n2 + n + cos n - n,
7n + cos 2n + arctg(5n2 + 4) log2(2n + 1)
an = " , an = ,
3
8 � 7n + sin(ln( n + 3)) log2(4n + 1)

" "
an = n + 6 n + 1 - n ,


an = ( n2 + 11 - n2 + 6) sin( n2 + 11 + n2 + 6) ,
1 1 1
an = " + " + ... + " ,
n2 + 1 n2 + 2 n2 + n
1 1 1
an = + + ... + ,
n + log(n + 1) n + log(n + 2) n + log(n + n)
" " "
n + n + 2n - 1 n + n + 3 n + n + 1
an = " + ... + " + " .
n6 + n2 n6 + n2 n6 + n2
1 1 1
an = + + ... + ,
e + Ą e2 + Ą en + Ą
1 1 1
an = + + ... + ,
5 + Ą 52 + Ą 5n + Ą
1 1 1
ln(1 + ) ln(1 + ) ln(1 + )
n n n
an = + + ... + ,
1 1 1
1 + 2 + n +
n2 n2 n2
| sin 1| | sin 2| | sin n|
an = + + ... + ,
2 22 2n
2n!
an = .
nn
{an}

3
a1 = , an = 3an-1 - 2, n e" 2,
2
"
"
a1 = 2, an+1 = 2an, n " N.
{an}
2(2an + 1)
a1 = 1, an+1 = , n " N.
an + 3

1 + 2 + ... + n 1 1 1
" " "
lim , lim 1 + + ... + ,
n" n"
n2 n n
2
1k + 2k + ... + nk
lim .
n"
nk+1
"
" " "

n - n2 - 1 1 3n - 1
, , ,
(3n - 2)(3n + 1) 5n
n(n + 1)
n=1 n=1 n=1

" " "

2n + 1 2Ą 4Ą 1
, sin cos , ln 1 + .
n2(n + 1)2 3n 3n n
n=1 n=1 n=1
" " "

n - 2 1 1
, sin , n sin ,
2n3 + 1 n n2
n=1 n=1 n=1
" " "

1 1 1
"
tg2 , n sin3 , (1 - cos ).
n n n
n=1 n=1 n=1
" " "


n2 + 1 1
, cos(sin ), n n2 + 1 - n2 - 1 .
2n2 + 5 n
n=1 n=1 n=1
" "

10n " (n!)2 Ą
, , n tg ,
n! (2n)! 2n
n=1 n=1 n=1
" "

n! (n!)(3n)!
, .
nn [(2n)!]2
n=1 n=1
" "

n
n2
arc tg(n2 + 1) , ,
1
(2 + )n
n
n=1 n=1
2
n
" "

n5 n - 1
, Ąn .
2n + 3n n
n=1 n=2
 n
" " "

3 nn n + 4
, , ,
n! 2n - 1
n(n3 + 1)
n=1 n=1 n=1
n2
" " "

n + 1 n! - n3 (n - 5)n
2n, , " ,
n n2n + n!
nn
n=1 n=1 n=1
"

1
" .
n
n
n=1
2
n
" "

(-1)n (-1)n n
, ,
1 + n2 3n n - 1
n=1 n=2
"
" "

n + 1 (-3)n
(-1)n , ,
n + 2 (2n)!
n=0 n=1

" "

n2 1
(-1)n , (-1)n+1 sin ,
n2 + 1 n
n=1 n=1
n " n
"

2n n + 7
(-1)n , (-1)n+1 .
n + 5 3n + 5
n=1 n=1
" " "

(x - 1)n 3n n2xn
, (x + 2)n , ,
2n + 1 n! (n + 1)22n
n=0 n=0 n=1
"
" "

n(2n + 1) 2nxn
x2n, .
6n (2n)25n
n=1 n=1
n
(-1)n
an = 1 + , an = (-1)n[(-1)n + 1],
n
n
nĄ (1 - (-1)n)2n + 1
an = cos , an = .
3 2n + 3
1 1
lim log2 |x| = -", lim cos x = 1, lim = ,
x0 x0 x1
x + 2 3
Ą 1
lim arctg x = - , ) lim = +".
x-" x1 - 1)2
2 (x
1 1
lim , lim cos x2, lim sin , lim ex sin 2x.
x0 x" x"
x3 x0+ x

sin 2x
, x " (-Ą , 0),
2x 2
f(x) =
2x Ą
, x " (0, ).
|x| 2
x0 = 0
" "
4

1 + x + 2 x4 + 1
lim " , lim , lim ( x2 + 1 + x),
x" x-"
1 + x2 x-" x

cos x sin 5x
lim ) lim , ) lim ( x2 + x - 6 - x)
Ą
x0 x-"
x - 2x sin 3x
Ą
2
tg 3x 2x + 3x 1
5x
lim , lim , lim (1 + x) ,
x" x0
x0- x3 3x + 1
x+1
3x + 5 1 - cos x x2 - x + 4
lim ) lim , ) lim "
x" x0 x"
3x + 7 x2
x3 + 2
"
Ą
tg(x - ) 1
1 - 1 - x
3
sin x
lim , lim lim (cos x) ,
Ą
x0 x0
x - 2 cos x sin 4x
1
3
"
cos x - 1 |x - 2| Ąx
lim , ) lim , ) lim |1 - x| tg .
x0 x2 - 4 2
x1
sin x x2
2 ln(2x + 1) ln 2
lim ex+sin x = 0, lim = .
x-" x"
ln(3x + 1) ln 3
X = R \ {0}
"
3
tg 3x x + 1 - 1
f(x) = f(x) =
5x x
x0 = 0
ńł
�łx - 1, x d" 0,
�ł
Ą �ł
sin x, |x| d" ,
2
) f(x) = 2 ) f(x) =
0, 0 < x < 2,
x Ą
�ł
4 , |x| < ,
�ł
Ą 2
ół2x - 4, x e" 2,
ńł
�łx, x d" 0,
�ł
x �ł
, x = 0,

|x|
x
) f(x) = ) f(x) =
, 0 < x < 1,
x-1
�ł
0, x = 0.
�ł
ółx2
- 4, x e" 1

1
x sin , x " R\{0},
x
f(x) =
0, x = 0

x, x " Q,
f(x) =
-x, x " R\Q
x = 0
a, b, c p

x3-27
, x = 3,

x-3
) f(x) =
p, x = 3,

sin 3x Ą Ą
, x = 0, x " (- , )

sin 5x 5 5
) f(x) =
p, x = 0,
"
1+x-1
, x = 0, |x|, x d" 1,

x
) f(x) = ) f(x) =
a, x = 0. x2 - a, x > 1,
ńł
sin ax
�ł
, x < 0,
�ł
x
�ł
�ł
�ł x4-1
, 0 d" x < 1,
x2-4x+3
) f(x) =
�łc,
x = 1,
�ł
�ł
�ł
ół2 - 2bx, x > 1.
a b
ńł
"
x2+5-3
�ł
�ł , x " R \ {-2, 2},
�ł x2-4
1 1
f(x) =
ln2 a - ln a, x = 2,
2 3
�ł
�ł
ół 1
sin b x = -2.
3
n, m " N am, bn = 0

ńł
�ł
�ł+", m > n, am � bn > 0,
�ł
�ł
amxm + am-1xm-1 + ... + a0 �ł-", m > n, am � bn < 0,
lim =
x" �ł
bnxn + bn-1xn-1 + ... + b0 am n = m,
�ł
bn
�ł
�ł
ół0 m < n,
1
f(x) = sin x + cos x [0, Ą]
3
1
f(x) = 2x + x2 [1, 3] .
10
f(x) = ln x + x2 - 1 [1, e] Ą.
 x3 + ax2 + bx + c = 0
1
x4 = 4x, (-", 0) ex = , (-1, 0)
x
3x + x = 3, (0, 1) 4x = x2, [-1, 0).
(0, 1)
1 1 1
x
f(x) = ex, f(x) = sin , f(x) = x sin f(x) = e .
x x
0, ")
"
f(x) = x, f(x) = x sin x, f(x) = x2.
f g
[a, b] f + g f � g
f � g (a, ").
 d : R2 R d(x, y) = |x2-y2|
R
X d :
X � X R

1, x = y,

d(x, y) =
0, x = y.
d X
r " (0, 1),
r = 1 r > 1
(X, d)
: X � X R
d(x, y)
(x, y) =
1 + d(x, y)
X
(R, d)

3x 3y
-

d(x, y) =
3x
1 + - 3y

1
0
3
d : R2 < 0, ")

(x
d(x, y) = - 1)3 - (y - 1)3 .

(R+, d)
K(1, 2) \ K(0, 1)
(N+, d),
|n1 - n2|
d(n1, n2) =
n1 � n2
1
3
(R2 , d)
+

|y1 - y2|, x1 = x2,
d((x1, y1), (x2, y2)) =
|y1| + |x1 - x2| + |y2|, x1 = x2,

R2
1
2
(R2 , d)
+


x1 y1

d((x1, y1), (x2, y2)) = log2 + log2

x2 y2
R2
2

nx3+x
(-", +")
n

1
x2 + x +
n
[0, ")
" "

cos nx 1
a) , x " R, b) x " R,
n2 + x2 n2 + x2
n=1 n=1
" "

sin nx x
c) x " R, d) x " R+ *" {0},
n! 1 + n4x2
n=1 n=1
" "

(-1)n x2 x
e) , x " (-2, "), f) sin , x " (0, 2],
x + 2n n n
n=1 n=1
" "

(-1)n+1n! 1
g) cos 2nx, x " R, h) " , x " [0, "),
n2n
2n-1 1 + nx
n=1 n=1
" "

nx ln(1 + nx)
i) , x " [a, "), a > 0, j) , x " [a, "), a > 1.
1 + n5x2 nxn
n=1 n=1
 x0

f(x) = cos 2x, f(x) = x3 - x,
"
3
f(x) = ln x, f(x) = x,
1 1
"
) f(x) = , ) f(x) = .
x sin x

x2 + x + 1, x e" 1,
f(x) = x0 = 1,
3x3, x < 1,

1
x arctg , x = 0,

x
f(x) = , x0 = 0,
0, x = 0,

1
x2 sin , x = 0,

x
f(x) = , x0 = 0,
0, x = 0,
"
x - 1, x e" 1,
f(x) = , x0 = 1,
x(x-1)
, x < 1,
2

-2x2 + 3x + 1, x e" 1,
f(x) = , x0 = 1,
x2 - 3x + 4, x < 1,
f(x) = |x|3, x0 = 0.
a b c d
R
ńł
�ł4x,
�ł x d" 0,
�ł
f(x) =
ax2 + bx + c, 0 < x < 1,
�ł
�ł
ół3 - 2x,
x e" 1,
ńł
�łax + 1, x < -2,
�ł
�ł
f(x) = - x, -2 d" x < 3,
3
�ł
�ł
ółx2
+ x + b, x e" 3,
ńł
�łax + b, x d" 0,
�ł
�ł
f(x) =
cx2 + dx, x " (0, 1],
�ł
�ł
ół1 - 1
, x > 1.
x

f(x) = ln(x + x2 + 1) + x x2 + 1, f(x) = logx(arctg x),
"
4
f(x) = xarc sin x, f(x) = arc sin 1 - 5x,

1 2x + 1
) f(x) = (arctg x) arctg , ) f(x) = sin7 ,
x 3x + 1
arc sin x
f(x) = 3sin x, f(x) = ,
ex
f(x) = (sin x)cos x + (cos x)sin x, f(x) = logx(sin x),

) f(x) = cos x 1 + sin2 x, ) f(x) = etg x logx arctg x,
2

"
1 - x2
) f(x) = 5ln x, ) f(x) = arc cos ,
1 + x2
f(x) = ln(ln(ln x)), f(x) = (arctg x)x,
"
sin x
5
f(x) = , f(x) = log4 ctgh x7,
2 + ln x
) f(x) = ln x � arc cosh x, ) f(x) = arc sin tg(x2 + 5).
(arc sin x) , (arc tg x) , (ln x) , (arc tgh x) , (tgh x) .
"
ln x
3
f(x) = (x + 1) 3 - x (-1, f(-1)), f(x) = (e, f(e)).
x
y = ln x
y = 2x
f(x) = x3 - 3x2 -
9x + 2 Ox
b c y =
x3 + bx + c y = x (1, 1).

f(x) = |x| - 1
[-1, 1] c
f(x) = 1+xm(x-1)n, m n
f (x) = 0 (0, 1)
3
f(x) = sin3 x + cos2 x
4
[0, Ą] c
f(x) = arc sin x
[-1, 1]
c
f(x) = x3 -4x2 +x+6
[0, 3]
c
x
) < ln(1 + x) < x, x > 0
x + 1
x
) x d" arc sin x d" " , 0 d" x < 1
1 - x2
) ex > ex, x > 1,
a - b a a - b
) < ln < , a < x < b.
a b b
2x
f(x) = 2 arctg x + arc sin
1 + x2
(1, +")
1 - x Ą
arctg x + arctg = x " (-1, +").
1 + x 4
) f(x) = x2e-x, ) f(x) = x2 ln x, ) f(x) = x + sin x,

1 x3
) f(x) = , ) f(x) = , ) f(x) = x 2x - x2.
x ln x 3 - x2
x ctg x - 1 2x - 22-x
lim , lim ,
x0
x2 x1- (x - 1)2
"
3
tg x - 1 1 1
lim , lim - ,
Ą
x x0
x sin x x2
2 sin2 x - 1
4

1 1
) lim - , ) lim (tg x)tg 2x
Ą
x0 x
x ex - 1
4
x x
1 2
lim ln , lim arctg ,
x"
x0+ x Ą
1

x
1
tg x
"
x
lim , lim (x + 1) ,
x0 x"
x

ln cos2 x 1
x
) lim , ) lim (x + b)e - x
x0 x0
sin2 x


Ąx cos x ex
lim ln x tg , lim - ,
x1- 2 x0+ x sin x
2x tg x
1 1
lim 1 + , lim ,
x"
x x0+ tg x
1

1

1 x
x
x
arctg x (1 + x)
lim , lim .
x0 x0
x e
1
x2 sin
x + sin x
x
) lim , ) lim .
x0 x" - sin x
sin x x
x
f(x) = , x0 = 2, n = 3,
x - 1
"
f(x) = x, x0 = 1, n = 3,
1
f(x) = , x0 = 2 n = 3,
x
Ą
f(x) = ecos x, x0 = , n = 2.
2
f(x) = xex, Rn,
f(x) = etg x, R2,
x
f(x) = , Rn,
ex
f(x) = sinh x, R2.
W (x)
P (x)
W (x) = 3x5 + 2x + 3, P (x) = x + 2,
W (x) = 3x4 - 8x3 - 6x2 + 24x - 4, P (x) = x - 1,
W (x) = x4 - 5x3 + x2 - 3x + 4, P (x) = x - 4,
W (x) = 2x4 + x3 - 2x2 - x + 2 P (x) = x - 1.
x2 x4 Ą
cos x = 1 - + , |x| d" ,
2 24 6
x2 1
e-x = 1 - x - , 0 < x d" ,
2 10
x2 1
cosh x = 1 - , 0 d" x .
2 2
 10-4
"
cos 5ć%, e, 1.01.
n-1

xk
) ex > , x > 0, n " N
k!
k=0
"
1 x2 1 x2 x3
) 1 + x - < x + 1 < 1 + x - + , x > 0.
2 8 2 8 16
f(x) = 2 + |x - 1| f(x) = |x2 - 1|,

1 - x2, x = 0, -x, x d" 1,

f(x) = f(x) =
0, x = 0, 2 - x, x > 1,
f(x) = |x2 - 1|ex, f(x) = |x|(x - 1)2,
1
x
f(x) = 1 + | arc tg(x - 1)|, f(x) = x .
x " R
x2
) 2x arctg x e" ln(1 + x2), ) cos x e" 1 - .
2
V

3Ą
f(x) = 2 sin x + sin 2x 0, ,
2
f(x) = x2 ln x, [1, e],
x
f(x) = arctg x - , [0, 2],
2
f(x) = x2|x2 - 1|, [-2, 3],
"
f(x) = x - 2 2 [0, 5],
f(x) = (x - 3)2e|x| [-1, 4].
1
x
f(x) = x2 ln x, f(x) = e- ,
1
x-1
f(x) = (x - 1)e , f(x) = 2 - |x5 - 1|.
xąy� d" ąx + �y, 0 d" ą d" 1, ą + � = 1, x, y e" 0.
2x x - 1
f(x) = ln , f(x) = arc cos ,
x - 2 2x - 1
1
x
f(x) = xe , f(x) = |ex - 1|.

a) f(x, y) = x - y2 + ln(x + 3y), b) f(x, y) = x sin y,
x2y y - 1
c) f(x, y) = , d) f(x, y) = arc sin ,
x
x2 + y2 - 25

x x
e) f(x, y) = - 1, f) f(x, y) = arc cos .
x2 + y2 + 2x 2x + y

a) f(x, y) = 9 - x2 - y2, b) f(x, y) = x2,

c) f(x, y) = 3 - x - y, d) f(x, y) = x2 + y2,
x2 y2
e) f(x, y) = + , f) f(x, y) = x2 - y2,
9 4
g) f(x, y) = x2 + y2.
a) f : R2 R2, f(x, y) = (x, y2),
b) f : R2 R2, f(x, y) = (x2 - y2, 2xy),
c) f : R3 R3, f(r, �, z) = (r cos �, r sin �, z),
d) f : R3 R2, f(x, y, z) = (x2 + y3 + z4, 2x2 + 3yz2),
e) f : R2 R3, f(x, y) = (xy2, x - xy, x2 + 2y2).

-

f(x, y) = |x2 - y2| v

f- (0, 0)
v
-

f(x, y, z) = |x + y + z| v

f- (x0, y0, z0) x0 + y0 + z0 + 0
v
x
a) f(x, y) = arc sin , (x, y) = (0, 0),

x2 + y2
y
z
b) f(x, y) = x , x e" 0, z = 0

z
c) f(x, y) = xy , x e" 0, y e" 0,

d) f(x, y) = F ( x2 + y2), R,
e) f(x, y) = G(x + y, x - y), R2.

a) f(x, y) = x3 + 3xy - y2, b) f(x, y) = 2 sin y - cos x,

x - y
c) f(x, y) = , d) f(x, y) = ln(x + x2 + y2),
x + y
x + y
e) f(x, y, z) = sin(x tg(y cos z)), f) f(x, y, z) = arc tg ,
1 - xyz
g) f(x, y, z) = e3x+4y cos zx.

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

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

x3y
, (x, y) = (0, 0),

x2+y2
f(x, y) =
0, (x, y) = (0, 0).

a) f(x, y) = cos x cos y, b) f(x, y) = x2 + y2,
2(x2 - y2) x
c) f(x, y) = , d) f(x, y) = arc tg .
x2 + y2 y
arc tg 0, 9 sin 2, 3 arc tg 0, 7
a) " , b) ,
4, 02 e2,3
c) (1, 02)4,05, d) cos 29o tg 46o.

"z
w = x + 1 - y2 "z
"x "y
u = ln x, v = arc sin y
�(u) �(v)
y y "2z
z = x�( )+�( ) x2 "2z +2xy +y2 "2z = 0
x x "x2 "x"y "y2
y
z(x, y) = xnf( )
x2
"z "z
x + 2y = nz
"x "y
R

x3y
, (x, y) = (0, 0),

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

F (0) F (1) F (t) = f(t2, 2t2)

2(x-1)y
, (x, y) = (1, 0),

(x-1)2+y2
f(x, y) =
0, (x, y) = (1, 0).

F (0) F (t) = f(t + 1, t)
F (x, y) = f(u(x, y), v(x, y))
a) f(u, v) = u + v, u(x, y) = 2x, v(x, y) = 3y
b) f(u, v) = cos(u + v), u(x, y) = x, v(x, y) = x - y
v
c) f(u, v) = arc tg , u(x, y) = x + y, v(x, y) = 3y.
u

"z "z y
a) (x + y) = (x - y) , u(x, y) = ln x2 + y2, v(x, y) = arc tg
"x "y x
"2z "2z x y
b) + = 0, u(x, y) = , v(x, y) =
"x2 "y2 x2 + y2 x2 + y2
"2z "2z
c) + y2 = 0, u(x, y) = 2x, v(x, y) = -y2.
"x2 "y2
D = {(x, y) : (x-1)2+(y-1)2 < 1 '" y e" 1}

x 1
ln - x + y d" (y - 1)2.
y 2
(x,y)"D

1
|e1-x cos(y - 1) + x - 1| d" (x + y - 2)2.
2
xe"1,ye"1
1
y + sin y - x = 0
2
y = f(x) x " R
f (x), f (x), f (Ą) f (Ą).
 f (0), f (0) y = f(x)
x2 - xy + 2y2 + x - y = 1 f(0) = 1
z2 - xyz + y2 = 16
z = f(x, y) (1, 4, 2)
"z "2z
(x, y), (x, y).
"x "x2
a) f(x, y) = ex-y(x2 - 2y2),
b) f(x, y) = xy ln (x2 + y2),
1 1
c) f(x, y) = x5 + xy4 + 3x + 2,
5 4
d) f(x, y) = xy2(1 - x - y)3
e) f(x, y, z) = x2 + y2 + z2 + 4,
f) f(x, y, z) = x2 + 2x + y2 - 4y - z2 + 9z + 7,
g) f(x, y, z) = xyz(4 - x - y - z).
f(x, y) = xy -x(x+1)-y(y +1)
x = 0, y = 0, x + y = -4
f(x, y) = x2 + (y - 1)2
A(-1, 0), B(2, 0), C(0, 2)
f(x, y) = x3 + y2 - 3x - 2y - 1 D = {(x, y) : x e" 0 '" y e"
0 '" x + y d" 3}
f(x, y) = x2-xy+y2-2x+y D = {(x, y) : x d" 0'"|x|+|y| d" 1}
f(x, y) = x2 - y2 D = {(x, y) : x2 + y2 d" 4}
a) f(x, y) = xy2, x + y = 1,
b) f(x, y) = x2 + xy2 + y2, x + y = 1,
c) f(x, y) = x + y, ex+y - xy = 1,
d) f(x, y) = xy, x2 + y2 = 2,
e) f(x, y, z) = x + y + 2z, x2 + y2 + z2 = 1,
f) f(x, y, z) = xyz, x2 + y2 + z2 = 3,
g) f(x, y, z) = xyz, x + y + z = 5 xy + xz + yz = 8,
h) f(x, y, z) = xyz, x + y + z = 0 x2 + y2 + z2 = 1


1 arc sin x
x3
3
x2e dx, dx,
1 - x2

e-x cos xdx, cos2 xdx,

) tg xdx, ) arctg 2xdx,

ln xdx
x arctg xdx, " ,
x 1 + 2 ln x

arctg xearctg x
dx, arc cos2 xdx,
1 + x2

ln(arctg x)
) dx, ) sin ln xdx,
x2 + 1

1 + ln x x
dx, dx,
x ln x
sin2 x

sin 2xecos xdx, x3 ln(x2 + 1)dx.

x3 - 2x2 + 4 x3 + 5x + 3
dx, dx,
x3(x - 2)2 x4 + 6x2 + 9

dx dx
, ,
(x + 1)2(x2 + 1) (x - 1)2(x2 + x + 1)

x4 - 3x2 - 3x - 2 x4
) dx, ) dx.
x3 - x2 - 2x (x2 - 1)(x + 2)


x - 1 1 + x dx
x dx, ,
x + 1 x x
"

x2 + 1 + x dx
" dx, " " ,
3 3
1 + x x + 1 + x - 1

dx dx
) " , ) " ,
x2 + 5x + 7 4x2 - 8x + 7

4x - 5 x - 3
" dx, " dx,
-x2 - 6x - 8 x2 + 2x + 5


x2 - 6x - 7dx, 4x - x2dx,


"
x - 1 dx
3 3
) , ) x x + 1dx.
x + 1 x - 1

dx dx
, ,
(1 + cos x) sin x 3 cos x + sin x + 1

dx
, sin3 xdx,
sin2 x cos2 x

) sin2 x cos4 xdx, ) sin3 x cos2 xdx,

sin 3x cos 5xdx, cos 3x cos 6xdx,

dx cos xdx
, ,
5 + 4 cos x
1 + sin2 x

dx dx
) , ) ,
sin x cos x

"
dx
cos x sin3 xdx, ,
4 + sin2 x

cos3 xdx 1 - cos x
, dx,
1 + cos x
sin5 x

cos xdx dx
" , .
3
sin3 x cos x
sin2 x
 (an)ne"1
"
3
n2 sin(n2)
an = , n e" 1
n-1
2n+1
an = , n e" 1,
4n
"
n n
an = 2n + 2(-1) +n, n e" 1
n
1
an = 1 + , n e" 1.
2n+1
5x =
x2 - 1
0.25
" (x-1)n
x ?
n=0 n22n
2
(n+1) "
" "

1 (-1)n+1 n + 1 (n - 5)n
" , , " .
n
n 3n n
nn
n=0 n=0 n=0
f

x
, x = 0,

|x|
f(x) =
0, x = 0.
f
ńł
�łx - 1, x d" 0,
�ł
�ł
f(x) =
0, 0 < x < 2,
�ł
�ł
ół2x - 4, x e" 2.
5x = x3 +
2 0.25
" (x+1)n
"
x ?
n=0
n2n 2n+1
(an)ne"1
"
7
Ą
n3 cos(n )
8
an = , n e" 1
2n2-3n+1
n2
an = , n e" 1,
7n
"
n n
an = 3n + 3(-1) +n, n e" 1,
2
n
1
an = 1 - , n e" 1.
3n-4
1 2 n
a) lim " + " + ... + " ,
n"
n4 + 1 n4 + 2 n4 + n
2n
3
b) lim 1 + sin .
n"
n

1
1 1
1- 1-
1-
2 22 2n
lim + + ... + .
1+2 1+22 1+2n
n"
"
n n
an = 1 + 2n(-1) .
" " "
n

1 4n Ą
a) 2n sin , b) , c) n!.
3n 3n + 5n n
n=1 n=1 n=1
"

" "
(-1)n
" ( n + 3 - n + 1).
n + 1
n=1
sin2 x-tg2 x
� " N lim
x�
x0
1
Ą
x-2
a) lim (x - 1) , b) lim (x - ) tg x.
Ą 2
x2+ x
2
"

anxn
n=1
"

3
2
n anxn.
n=1
2 22 2n
a) lim " + " + ... + " ,
n"
4n + 2 4n + 22 4n + 2n

5
b) lim n2 cos - 1 .
n"
n
3
{an} a1 = , 3an+1 = a2 + 2, n " N.
n
2

" "

1 1 n5
3
a) sin tg , b) ,
n2
1
n n
1 +
n=1 n=1
2n

"

Ąn " n(n + 1)
n
c) , d) .
n! Ąe
n=1 n=1
"

(4x-8)n
4n(n+1)
n=1
" " "
"
2 - 1 + cos x 1 + 4x2
x
a) lim , b) lim , c) lim 1 - 2x.
x0 x-" x0
2x
sin2 x
p

ln(1+tg2(3x))
, x = 0,

x2
f(x) =
p2 - 3, x = 0.

n
a) lim (ln(n2 + 3) - ln(n2 + 2)),
n"
2
3n 3n
b) lim + ... +
Ą Ą
n"
n4 + n cos(n ) + 1 n4 + n cos(n ) + n
4 4
"
3
{an} a1 = 3, an+1 = an + 6, n " N.

" "

n3 n
n
a) n n , b) ,
1 1
100
+
n=1 2 3 n=1

" "
4

3n cos n n(n + 3)(n + 2)
c) , d) " .
5
n! n n
n=1 n=1
n
"

7n x + 3
" .
4
2n + 4
n=1
"
4
1
1 + 4x4 ln(cos x)
1-x3
a) lim 2 , b) lim , c) lim .
x1 x-" x0
2x x2
p
ńł
"
"
3
�ł sin x-
2 Ą
, x = ,

Ą
x- 3
f(x) =
3
ółp,
Ą
x = .
3
a) lim (2n2 + 1)(ln(n2 + 3) - ln(n2 + 2)),
n"
5n 5n
b) lim + ... +
n"
n4 + n sin(n!) + n n4 + n sin(n!) + 1
"
3
{an} a1 = 4, an+1 = an + 24, n " N.

" "
3

5n sin n (n + 3)(n + 2)
a) , b) " ,
4
n!
n7
n=1 n=1

" "

2n7 n
n
c) n n , d) .
1 1
222
+
n=1 5 4 n=1
"
n

n
"5 x-2
n+2 3
n=1
"
4
2
9x4 - 5 ln(cos x)
8-x3
a) lim 3 , b) lim , c) lim .
x2 x-" x0
3x x2
p
ńł
"
"
1
�ł - sin x
2 Ą
, x = ,

Ą
x- 6
6
f(x) =
ółp, Ą
x = .
6
e 0, 0001
0 < e < 3.
f(x) = 1 - |2x - 1|.
f
1 2
x0 = 0, x1 = , x2 = , x3 = 1.
3 3
f.
 f(x) = ex.
n-1 f(k)(0)
f(x) = xk + Rn
k=0 k!
f(n)(�x)
1
Rn = =
n! n!
n-1 1 e�x � " (0, 1). x = 1
e = + Rn Rn = e�. � " (0, 1),
k=0 k!
3
0 < Rn <
n!
5! = 120, 6! = 720 7! = 5040
3 1
8! > 8 � 5000 = 40000, R8 < < .
40000 10000
1 1 1
e H" 1 + + + +
2 3 4
1 1 1
+ + .
5 6 7
f(0) = f(1) = 0

1 2
w(x) = f p1(x) + f p2(x)
3 3
1 2
pk = wk(x), w1(x) = x(x - )(x - 1), w2(x) = x(x -
wk(xk) 3
1 1 2 2 2
)(x 1). w1(x1) = w1( ) = , w2(x2) = w2( ) = - ,
3 3 27 3 27
- 2 2
1
f = f =
3 3 3
2 27 2 2 27 1
w(x) = x(x - )(x - 1) - x(x - )(x - 1)
3 2 3 3 2 3
2 1
= 9x(x - 1)[(x - ) - (x - )]
3 3
= -3x(x - 1) = 3x(1 - x).
3
3 k
2 2
f3(x) = f x3-k(1-x)k = 3x2(1-x)+ 3x(1-
k=0 3 k 3 3
x)2 = 2x(1 - x)[x + 1 - x] = 2x(1 - x).
f3 f
1
x =
2
1
|f3(x)-f(x)| = |2x(1-x)-2x| = |-2x2| = 2x2 [0, ].
2
1 1
x =
2 2
w(x)-f(x) = 3x(1-x)-2x = x-3x2 = x(1-3x)
1
[0, ].
2
1
0
3
1 1
x =
6 12
3x
a) arc tg x > " , x > 0.
1 + 2 1 + x2

a - b a - b
< ctg b - ctg a < .
sin2 b sin2 a
Ą
02
1
f(x) = (2x + ) arc ctg 7x.
x-3
f(x) = x| ln x - 3|.
Ą Ą Ą
f(x) = 2(x - )4 sin(x - ) cos(x - )
4 4 4
Ą
x = .
4
"

nx
sin ,
1+xn
n=1
x " [b, "), b > 1.
12x
a) " < 2 arc tg x, x > 0.
2 + 2 4 + 4x2

ą - � ą - �
< tg ą - tg � < .
cos2 � cos2 ą
Ą
0<�<ą<
2
1 x
f(x) = (x + ) arc ctg .
x+2 3
f(x) = 3x| ln 2x - 6|.
Ą Ą Ą
f(x) = (x - )3(cos2(x - ) - sin2(x - ))
3 3 3
Ą
x = .
3
"

x
sin ,
1+nxn
n=1
x " [2, ").
x < 0
1 - x2
arc cos > 2 arc tg x.
1 + x2

x + y
x ln x + y ln y > (x + y) ln , x, y > 0.
2
x+ln x
lim
2x+sin x
x"
f(x) = (x - 4)e|5-x|.
d : R2 � R2 0; ")
+ +
| log2 x1 | | log2 x2 |
y1 y2
d ((x1, x2), (y1, y2)) = + .
a b
(R2 , d)
+
x+1
p(x) = 2, 3, 4.
x-1