View
0
Download
0
Category
Preview:
Citation preview
!
!
tbrbs\wfmpt!ufrojlvsj!vojwfstjufuj!
!
r9tywjubsj<f8!h9rbstfmb<f8!j9tjhvb8!
f9fmfseb-wjmj8!n9ynjbeb-wjmj8!{9\fejb-wjmj!
!
!
nb\fnbujlb!flpopnjtufcjtb\wjt!
)bnpdbobUb!lsfcvmj*!
obxjmj!JJ!
!
! ! ! ! ! ! ebnuljdfcvmjb!tuv.t!
!!!!!!!!!!!!!tbtxbwmp.nfUpevsj!
!!!!!!!!!!!!!!!!!!!!!!tbcXpt!njfs!
Ucjmjtj!!
3116!
!
!
jl[`e
\�^��1ZY^ \�[`u!cYq![ b �1gYZ m cY[%s:gYcY[`d!^`e`^km��`u![��Yd!^`ekjl_a^ m�[`p`[`u��C^���q4ngYi`^km�[ b [�[`cYq!fh[`ZYgYi`^kmae`u!gYikjl_�mzu!q!cYgY_a^`f��1[`ZYekj s:p`ZY^`_a^`[�jlcY[��!_ag nm�^�m�[km%\�[`p`_agYi`_agYi`^km#gYe`q!ZYq!cY^`ekjlu!^ b [�m�[`i`[`ZYe`q4nom�[� ^`ZY[`Z m�q��`u!q� ^kn_a^km i`[`e`[`_a[`p`u!^`[`d!^km�m�[� g��`jlu!^km<m�d4j b gYZYd!gYi`^km�[%s:p`^km����^`ZY[`[`u4m�q4ni`u!^`p`[ b�b [�m�d!u4j��Yd4jlu4jl_a[ b ^��1^�m�u4jl_���g m�[`i`[`cY^km�q!i`[���^`[K[`c�� [`cY[ bcYq��YcYg b ��1[`ZY[%s:_agYi`^km m�[`cY^`ZY^km�d!u!q4m cY^`gYu b [`cYd!e`^`fhgYikjl_ m�[���gY_�nc����!p`[`ZYgY_aq4m�s:[`Z�� cY[%s:gYcY[`d!^`e`[ gYe`q!ZYq!cY^km�d!gYi`^km�[%s:p`^km��! s:i`^`_a^km�^"�1_aq!i`[`_�n#�`u!^`ZYd!^"%$�&'&'&(��
u!g b [��Yd!q!u!^") �`u!q� g m�q!u!^ b ��ZY[`d!u!q*��p`^`_a^
u!gYfhgYZ,+�gYZYd!gYi`^")- ^.+�^`e`[knrcY[%s:gYcY[`d!^`e`^km�cYgYfhZY^`gYu!gYi`[%s:[ b q��Yd!q!u!^"�`u!q� g m�q!u!^ b ��ZY[`d!u!q*��p`^`_a^
^.+�^`e`[knrcY[%s:gYcY[`d!^`e`^km�cYgYfhZY^`gYu!gYi`[%s:[ b q��Yd!q!u!^"�`u!q� g m�q!u!^["�/�1[��1ZY^��Yg
0© �1[`cYq!cYfhgYcY_aq!i`[�� d!g��YZY^`ekjlu!^#jlZY^`p`gYu4m�^`d!gYd!^��1 xzy1y�{X,24365
s:[`p`^��#XYX
������������ ��������������������������������������������� ������� !��#"�$�%����������&�� !����������������
�'���� !)(*���� �+�,� !��,&�
- � $'�3\�[`u!cYq!gYikjl_a^km �1[`ZYcY[`u!d!gYi`^km �1[`cYq/.YgYZYgYi`^%s�^��`q!p`gts ��gYc b g n�1^ jlZ��Yfh^`^kmU\�[`u!cYq!gYikjl_a^")
1) f(x) = x2;
2) f(x) = x3;
3) f(x) = 2x;
4) f(x) = ex;
5) f(x) = −4x2 + 1;
6) f(x) = (2x + 3)2.
- � x �%��g��Yfhg jl_a^ jlZ��Yfh^`^km �1[]\�[`u!cYq!gYi`^km \�g m�^km �1[`cYq/.YgYZYgYi`^%s^��`q!p`gts ��gYc b g��1^ jlZ��Yfh^`gYi`^kmU\�[`u!cYq!gYikjl_agYi`^")
1) y = arctg x;
2) y = lnx;
3) y = arcsin x.
- � 04�4uAs3jl_a^ jlZ��Yfh^`^km-�1[]\�[`u!cYq!gYi`^km<\�g m�^km��1[`cYq/.YgYZYgYi`^%s��1[kncYqAs:p`[`_agts jlZ��Yfh^`^kmU\�[`u!cYq!gYikjl_a^")
1) xx (x > 0);1
�
2) xsin x (x > 0);
3) xln x (x > 0).
- � �1�4^��`q!p`gtsIekj s ��g�!u!q!cYgY_�m�[`f �YcYZY^kmf(x) = x2 jlZ��Yfh^`^km
�1u![� ^`e`^kmM
(
12; 1
4
) \�gYu!d!^`_6+�g �1[`cY[`p`[`_a^<c���gYi`^-[`ikm�fh^knm�[%s:[ �!gYu��Y^km b [ b gYi`^%s cY^`cY[`uAs3jl_agYi`[km�s:[`Z��
- � { �4^��`q!p`gts ekj s ��g�Au!q!cYgY_�m�[`f �YcYZY^kmf(x) =
√3x3 jlZ�� n
fh^`^km �1u![� ^`e`^kmM
(
13;√
327
) \�gYu!d!^`_6+�g��1[`cY[`p`[`_a^�c���gYi`^[`ikm�fh^km�[%s:[ �!gYu��Y^km b [ b gYi`^%s cY^`cY[`uAs3jl_agYi`[km�s:[`Z��
- � �4�4^��`q!p`gtsy = x2 − 2x + 12
jlZ��Yfh^`^km �1u![� ^`e`^kmM(1; 2)
\�gYu!d!^`_6+�g �1[`cY[`p`[`_a^Uc���gYi`^kmwekj s �`jlu!^Ue`q!g ^`fh^`gYZYd!^"�- � - �*�1[`cYqAs:p`[`_agts [`u���jlcYgYZYd!^km
∆xZY[.+�u b ^ b [ jlZ��Yfh^`^km ∆y
ZY[.+�u b ^ y = x2−2x+3 jlZ��Yfh^`^km�[%s:p`^km�s3j
x^`fhp`_agYi`[")
[�(�$]n b [`Z���nrc b g��i�( { n b [`Z 0�nrc b g��� ( y n b [`Z $]nrc b g��
- � �4�4^��`q!p`gtsf(x)
jlZ��Yfh^`^km�ZY[.+�[`u b ^ x\�gYu!d!^`_ ��^" s3j [`u4n
��jlcYgYZYd!^km�ZY[.+�u b ^`[ ∆x)
[�(f(x) = x3 �
i�(f(x) =
√x + 1
�
� (f(x) =
2
x− 1�
- � &4�*�1[]\�[`u!cYq!gYi`^km�\�g m�gYi`^km�[ b [��Y^`u!^%s:[ b ^AgY_agYcYgYZYd![`u4jl_a^ j�nZ��Yfh^`gYi`^km \�[`u!cYq!gYikjl_agYi`^km f ��u!^`_a^km �1[`cYq/.YgYZYgYi`^%s ^��`q4np`gts \�[`u!cYq!gYikjl_agYi`^")
1) y = x3 − 2x2 + 4;
2) y = x5 − x3 + 3x2 − 2;
3) y =2
6x6 − 3
4x4 + 2x;
�
4) y =1
4x4 − x2
2+ 7;
5) y = x2 + 2x + 3√
x;
6) y = 2√
x− 3√
x2 − x;
7) y = 2x3
2 − 3x4
3 + 4x5
4 ;
8) y = 3x5
3 + 4x7
4 − 5x8
5 ;
9) y =1
x+
1
x2− 1
x3;
10) y =2
x3/2+
4
x5/4− 6
x7/6;
11) y = 2 sin x + 6 cosx + x;
12) y = 4 tg x + 5 ctg x + 1;
13) y = 4x + 3 · 2x;
14) y = 6 ln x + 7ex;
15) y = 3 arcsin x− 4 arccos x;
16) y = 6 arctg x− 7 arcctg x;
17) y = (2x2 + x) sin x;
18) y = (3x3 + 2x) cos x;
19) y = (x4 + 2x3) lnx;
20) y = ex sin x;
21) y = 5x cos x + ln 5;
22) y = 3x lnx + ln 3;
23) y = ln x sin x;
24) y = cos x tg x;
25) y = x arcsin x;
�
26) y =√
x log2 x;
27) y = x3 3x;
28) y = (sin x + cos x) 2x;
29) y =x + 1
x− 2;
30) y =x + 3
x2 + 1;
31) y =x
1− x2;
32) y =x2
3x;
33) y =2x
sin x;
34) y =3x2 + x− 1
x;
35) y =x3 − 2x + 1
ex;
36) y =sin x
ex;
37) y =sin x + cos x
sin x− cos x;
38) y =ex − 1
ex + 1.
- � $ y �4^��`q!p`gts \�[`u!cYq!gYikjl_agYi`^")$ (
y = 2 lnx + (x2 + 2x)ex +x + 3
x− 1+ 1
�
x (y = 3 · 5x + x4 sin x +
cos x
ex− 2
�
0(y = 4 sinx + x7 · 7x +
sin x
cos x+ 3
�
�
� (y = 5 cosx + (x3 + 2x) 3x +
√x
sin x− ln 2
�
{ (y = 7 arcsinx + sin x ln x− ex
cos x+ 4
�
�(y = 2 arccosx− x3 · cos x− sin x
x2+ 6
�
-'(y = 5 arctg x− sin x cos x− 4x
x− 8
�
�(y = 3 arcctg x− ln x ex − sin x
ex+ 9
�- � $'$'�4^��`q!p`gts
f ′(x0)ls3j
$ (f(x) = x3 + 2x2 + 3, x0 = 3
�
x (f(x) =
1
x− 3
x2+ 5, x0 = 1
�
0(f(x) = 4x− 2
√x, x0 = 4
�
� (f(x) =
6
x 3√
x− 1
3√
x, x0 = 8
�
{ (f(x) = x(x2 − 2), x0 = 5
�
�(f(x) =
1
x + 1− 2
x2 + 3, x0 = 0
�
-'(f(x) =
x2 − 4x + 3
x, x0 = −2
�
�(f(x) =
4x
3x + 1, x0 = 1
�
&(f(x) =
4ex
x2 + 1, x0 = 0
�
$ y (f(x) = ln
3x
x− 1, x0 = −1
�
$'$ (f(x) =
√
x
x + 1, x0 = 1
�
�
$ x (f(x) =
√
1 + ln2 x , x0 = 1�
$ 0(f(x) = ln 4
√
1 + tg x
1− tg x, x0 = 0
�
$ � (f(x) = ln
(√1+ex−1
)
−ln(√
1+ex+1)
, x0 =0�
- � $ x �4^��`q!p`gts ��gYc b g��1^ jlZ��Yfh^`^kmU\�[`u!cYq!gYikjl_a^")$ (
y = (x3 + 2x2 − 3)4 � x (y=(5x4− 3x3 + 2x)3 �
0(y = sin 2x
� � (y = 5 cos
x
5
�
{ (y = ln(3x− 4)
� �(y = ln(9x2 + 5)
�
-'(y =
√2x + 3
� �(y =
√x2 − 4x + 1
�
&(y = sin3 4x
� $ y (y = cos2 2x
�
$'$ (y = ln(sin x)
� $ x (y = cos(ln x)
�
$ 0(y = 3x3+1 � $ � (
y = 4√
x6−1 �
$ { (y = 4sinx � $ �(
y = 5cos x �
$ -'(y = sin(x3+1)+lnx4 � $ �( y = ln(x3− 4)− e3x �
$�&(y = 2arcsin 4x+3arccos 2x � xzy ( y = 4arctg 3x+5arcctg 2x �
x $ (y = ln
√
1 + sin x
1− sin x
� x1x (y = ln ctg
√x�
x 0(y = x3 · e3x � x � (
y = x2 · 2√
x �
x1{ (y = arcsin
√x� x �(
y = ln
√
x
x− 1�
x -'(y = ex2 · cos 4x
� x �(y = 4x3−5 · arcsin 3x
�
x &(y =
sin x√sin 2x
�
0 y (y = arctg
x
a+
1
2ln(x2 + a2), a 6= 0
�
�
- � $ 04�4^��`q!p`gts jlZ��Yfh^`^kmUcYgYq!u!gCu!^��1^kmU\�[`u!cYq!gYikjl_a^")$ (
y = x3 − 2x2 + 3� x (
y = x5 +3x4−2x2 +x�
0(y =
√1− x2 � � (
y =√
3 + x3 �
{ (y = (x3 + 1)3 � �(
y = (2x− 3)4 �
-'(y = x4 · ln x
� �(y = x5 · ex �
&(y = arcsin x
� $ y (y = arctg x
�
$'$ (y = e3x+4 � $ x (
y = 26x+7 �
$ 0(y = ln(x3 + 5x)
� $ � (y = sin x + 3e5x �
$ { (y = 4 + sin2 x
� $ �(y = ex · sin x
�- � $ �1�4^��`q!p`gts jlZ��Yfh^`^km
nnojlu!^u!^��1^km�\�[`u!cYq!gYikjl_a^")
$ (y = ax � x (
y = amx �
0(y = sin x
� � (y = cos x
�
{ (y = sin ax
� �(y = cos ax
�
-'(y = e2x � �(
y = 23x �- � $ { �4^��`q!p`gts jlZ��Yfh^`^km b ^� gYu!gYZYfh^`[`_a^")
$ (y = 2x2 + 3x− 14
� x (y = 5
√x +
13√
x+ x
�
0(y = sin 3x + ln 2x
� � (y = ln 3x + 2x+1 �
{ (y = (x2 − x + 1)2 � �(
y = sin3 x�
-'(y = x2 · e4x � �(
y = arccos 3x+arctg 6x�
&(y = ln
ex − 1
ex
� $ y (y = ln cos x
�
$'$ (y = x4 log3 x
� $ x (y =
5
x + x2
�
$ 0(y =
3√
2x2 + 3� $ � (
y = 3sinx + 2cos 3x �
$ { (y =
x + 1
x− 1� $ �(
y =x3 − 3
x2 + 2�
���
- � $ �4� b ^� gYu!gYZYfh^`[`_a^kmam�[��1jl[`_agYi`^%s ^��`q!p`gts cY^`[���_aq!gYi`^%s:^CcYZY^kn��p`ZYgY_aq!i`gYi`^")$ (
6√
67, 84� x (
5√
255, 15�
0(1, 0157 � � (
0, 9956 �
{ (3√
28, 62� �(
e1,02 �
-'(ln(e + 0, 544)
�- � $ - �4cts:_a^`[`ZY^U[`cYq!ZY[��1gYi`^km jlZ��Yfh^`[`[
(TR) = f(Q) = 300Q− 2Q2.
[�(�u![km!j b u!^km�cY[`u*� ^`ZY[`_�jlu!^A[`cYq!ZY[��1gYi`^km6 jlZ��Yfh^`^km�cYZY^kn��p`ZYgY_aq!i`["�u!q!fh[
Q = 25�
i�( �1[`cYqAs:p`[`_agts [`cYq!ZY[��1gYi`^km- jlZ��Yfh^`^km<fhp`_a^`_agYi`[-cYq4ns ��q!p`ZY^km
∆Q = 5gYuAs:g jl_a^%s �1[.+�u b ^km�[kmhs3j�cYq4n
fhgYc jl_ cYq!cYgYZYd*��^cYqAs ��q!p`ZY[`[Q = 25
gYuAs:g jl_a^"�- � $ �4�4cYqAs ��q!p`ZY^km jlZ��Yfh^`['cYq!fhgYc jl_a^`[Ud!q!_aq!i`^%s
P = 80−Q.
[�(�cYq��YgYi`ZYgts cts:_a^`[`ZY^ [`cYq!ZY[��1gYi`^km jlZ��Yfh^`^km�[ b [ cY^knm�^ ��g m�[`i`[`cY^km�^�cY[`u*� ^`ZY[`_�jlu!^�[`cYq!ZY[��1gYi`^km jlZ��Yfh^`^km�1[`cYq4m�[��`jl_agYi`gYi`^ �
i�( �1[`cYqAs:p`[`_agts cY[`u*� ^`ZY[`_�jlu!^U[`cYq!ZY[��1gYi`^"lu!q!fh[Q =
20�
� (�^��`q!p`gts cts:_a^`[`ZY^ [`cYq!ZY[��1gYi`^km fhp`_a^`_agYi`[":s3j cYqAs3n��q!p`ZY[G^.+�u b gYi`[GgYuAs:^�gYuAs:g jl_a^%s b [���gY[ b [`u!gts ^��1^cY[`u*� ^`ZY[`_�jlu!^�[`cYq!ZY[��1gYi`^km cYZY^���p`ZYgY_aq!i`[km
Q = 20gYuAs:g jl_6+�g��
- � $�&4�*�1[`cYqAs:p`[`_agts cY[`u*� ^`ZY[`_�jlu!^ [`cYq!ZY[��1gYi`^"Gs3j cYqAs ��q!p`ZY^km jlZ��Yfh^`['cYq!fhgYc jl_a^`[ ��gYc b g��1^'d!q!_aq!i`^%s6)
� �
[�(P = 6− 2Q
�
i�(P =
500√3 + Q
�
� (P = 3
√200− 4Q .- � xzy �4cts:_a^`[`ZY^U[`cYq!ZY[��1gYi`^km jlZ��Yfh^`['cYq!fhgYc jl_a^`[Ud!q!_aq!i`^%s6)
(TR) = ln3
√
(
1− 3Q
1 + 3Q
)2
.
^��`q!p`gts cY[`u*� ^`ZY[`_�jlu!^U[`cYq!ZY[��1gYi`^"lu!q!fh[Q = 1
�- � x $'�0m�[]\�[`u!cYq4m c j b cY^`p`^ b [`ZY[���[`u��C^`[ (FC) = 500 $
��q4n_aq fhp`_agYi`[ b ^ b [`ZY[���[`u��C^��`u!q b j��Yfh^`^km gYuAs:g jl_6+�g
�
(V C) = 3 $�
[�(�^��`q!p`gts cts:_a^`[`ZY^ b [cY[`u*� ^`ZY[`_�jlu!^ b [`ZY[���[`u��CgYi`^ �i�( �1[`cYqAs:p`[`_agts cts:_a^`[`ZY^ b [`ZY[���[`u��C^"lu!q!fh[ Q = 40
�
� ( �1[`cYqAs:p`[`_agts cts:_a^`[`ZY^ b [`ZY[���[`u��C^kmUfhp`_a^`_agYi`["�s3jcYqAs ��q!p`ZY[ �1[`^.+�u b gYi`[ � y gYuAs:g jl_a^ b [`Z � 0 gYuAs:g njl_a[`c b g��
- � x1x �4p%s��Yp`[%s61\�[`u!cYq!gYi`^kmKm�[��1jl[`_aq b [`ZY[���[`u��C^km jlZ��Yfh^`[CcYq4nfhgYc jl_a^`[d!q!_aq!i`^%s6)
(AC) = 3Q + 4 +15
Q.
[�(�^��`q!p`gts cts:_a^`[`ZY^ b [`ZY[���[`u��C^km�[ b [ cY[`u*� ^`ZY[`_�jlu!^b [`ZY[���[`u��C^km jlZ��Yfh^`gYi`^ �
i�(�cYq��YgYi`ZYgts cY[`u*� ^`ZY[`_�jlu!^ b [`ZY[���[`u��C^km�m�[��1jl[`_agYi`^%s�1[`cYqAs:p`_a^`_a^ m�u4jl_a^ b [`ZY[���[`u��C^km fhp`_a^`_agYi`["�s3j\�[`u!cYq!gYikjl_a^ �`u!q b j��Yfh^`^km�u![`q b gYZYq!i`[ cYfh^`u b gYi`[xzy n b [`Z $ ��gYuAs:g jl_a[`c b g��
���
� ( �1[`cYqAs:p`[`_agtsEcts:_a^`[`ZY^ b [`ZY[���[`u��C^km +Tj�m�d!^ fhp`_a^kn_agYi`["�u!q!fh[
Q = 20nom b [ �`u!q b j��Yfh^`^km�u![`q b gYZYq4n
i`[CcYfh^`u b gYi`[ x gYuAs:g jl_a^%s6��^��`q!p`gts m���p`[`q!i`[;cts:_a^kn[`ZY^ b [`ZY[���[`u��C^km�+Tj�m�d b [CcY^`[���_aq!gYi`^%s cYZY^���p`ZYgY_aq4ni`gYikm ��q!u!^km�
- � x 04�3\�[`u!cYq!gYi`^km ^�� m�^`u!gYikjl_a^ b [`ZY[���[`u��C^`[ - { b q!_a[`u!^"��q4n_aq fhp`[`_agYi`[ b ^ b [`ZY[���[`u��C^!�`u!q b j��Yfh^`^km gYuAs:g jl_a^km\�[`u!cYq!gYi`^km�[%s:p`^km��
4 +3
Q
�[�( �1[`cYqAs:p`[`_agts cts:_a^`[`ZY^ b [`ZY[���[`u��C^ b [wcY[`u*� ^`ZY[`_�j�nu!^ b [`ZY[���[`u��C^ �`u!q b j��Yfh^`^km Q
u![`q b gYZYq!i`^km�[%s:p`^km �i�( �1[`cYqAs:p`[`_agts cts:_a^`[`ZY^ b [`ZY[���[`u��C^kmGfhp`_a^`_agYi`^km +Tj�nm�d!^cYZY^���p`ZYgY_aq!i`gYi`^ �$ (UcYqAs ��q!p`ZY^km
∆Q = 3gYuAs:g jl_a^%s��1[.+�u b ^km�[km
x (UcYqAs ��q!p`ZY^km∆Q = 4
gYuAs:g jl_a^%s ��gYcYfh^`u!g ni`^km�[km
s3j�[��!gYikjl_ cYq!cYgYZYd*��^��`u!q b j��Yfh^`^km�u!gY[`_a^.+�[`fh^`^km b q!ZYgY[Q = 50
gYuAs:g jl_a^"�- � x �1�� ^`u!cY^km�cY^`gYu b [��1g��1cY^`_a^ b [`ZY[���[`u��C^�cYq!^`fhgYcY[� jlZ��Yfh^`^%s6)
K(Q) = (TC) =9
3Q3 − 9Q2 + 3.
\�[`u!cYq!gYi`^km u![-cYq!f+jl_aq!i`^km b u!q4m ^��YZYgYi`[ b [`ZY[���[`u��CgYi`^cY^`ZY^`cY[`_�jlu!^��
- � x1{ �3\�[`u!cYq!gYi`^kmwcts:_a^`[`ZY^ b [`ZY[���[`u��C^km jlZ��Yfh^`[`[
K(Q) = 0, 03Q2 − 2Q + 300.
[�(�^��`q!p`gts m�[��1jl[`_aq b [`ZY[���[`u��C^km�[ b [ cY[`u*� ^`ZY[`_�jlu!^b [`ZY[���[`u��C^km jlZ��Yfh^`gYi`^ �
� 1i�(�^��`q!p`gts m�[��1jl[`_aq b [`ZY[���[`u��C^ b [�cY[`u*� ^`ZY[`_�jlu!^ b [knZY[���[`u��C^"lu!q!fh[") �$ (
Q = 50 x (
Q = 100�
� (U\�[`u!cYq!gYi`^km u![�cYq!f+jl_aq!i`^km�[%s:p`^km�^��YZYgYi`[<m�[��1jl[`_aqb [`ZY[���[`u��C^UcY^`ZY^`cY[`_�jlu!^��
- � x �4�3\�[`u!cYq!gYi`^km�c j b cY^`p`^ b [`ZY[���[`u��C^`[ $ {zy b q!_a[`u!^"*��q!_aqfhp`[`_agYi`[ b ^ b [`ZY[���[`u��C^�� x b q!_a[`u!^" cYqAs ��q!p`ZY^km- j�nZ��Yfh^`[`[
P + 2Q = 82.�1[`cYq4m�[���gts cYq��1gYi`^km
РjlZ��Yfh^`[
Qnomm�[��1jl[`_agYi`^%s6�
[�(�^��`q!p`gts m�[��Yq!ZY_a^kmwu!['u![`q b gYZYq!i`[^��Y_agYp`[ xzy1y b q4n_a[`u!^kmwd!q!_ cYq��1gYi`[km �
i�( �`u!q b j��Yfh^`^km u![Vu![`q b gYZYq!i`[�j/+�u4jlZYp`gY_ .Yq� m�cYq4n�1gYi`^km�cY[�� m�^`cY[`_�jlu m�^ b ^ b g m �
- � x - �4cYqAs ��q!p`ZY^km� jlZ��Yfh^`[`[P = 447 − 2Q
��q!_aq m�[��1jl[`_aq
b [`ZY[���[`u��C^km jlZ��Yfh^`[ � (AC) =7
Q+ 3
�T^��`q!p`gts �`u!q4nb j��Yfh^`^km^kmUu![`q b gYZYq!i`["Tu!q!cYgY_a^`fVj/+�u4jlZYp`gY_ .Yq� mUcY[�� nm�^`cY[`_�jlu cYq��1gYi`[km�
- � x �4�4cYq!fhgYc jl_a^`[acts:_a^`[`ZY^C[`cYq!ZY[��1gYi`^km�[ b [Gcts:_a^`[`ZY^ b [`ZY[���[`u4n�C^km jlZ��Yfh^`gYi`^")
(TR) = −3Q2 + 40Q,
(TC) = 4Q + 5.^��`q!p`gts cYq��1gYi`^km jlZ��Yfh^`^km�cY[�� m�^`c jlcY^"�
- � x &4�*�`u!q b j��Yfh^`^km \�[`u!cYq!gYi`^km ^�� m�^`u!gYikjl_a^ b [`ZY[���[`u��C^`[(FC) = 80 $
��q!_aq!�`u!q b j��Yfh^`^km gYuAs:g jl_6+�g<fhp`[`_ag ni`[ b ^ b [`ZY[���[`u��C^ � (V C) = 4 $
�TcYqAs ��q!p`ZY^km jlZ��Yfh^`[`[
P = 300−Q.
� �
[�(�^��`q!p`gts cY[`u*� ^`ZY[`_�jlu!^�cYq��1gYi`^km jlZ��Yfh^`[ �i�(�cY[`u*� ^`ZY[`_�jlu!^AcYq��1gYi`^kmAm�[��1jl[`_agYi`^%s�cY^`[���_aq!gYi`^%s �1[kncYqAs:p`[`_agts cYq��1gYi`^km�fhp`_a^`_agYi`["�u!q!cYgY_a^`f ��gYg m�[`i`[`cYg ni`[-\�[`u!cYq!gYikjl_a^��`u!q b j��Yfh^`^km u![`q b gYZYq!i`^km fhp`_a^kn_agYi`[km
Q1 = 40gYuAs:g jl_a^ b [`Z Q2 = 42
gYuAs:g jl_a[`c nb g��
� ( �1[`cYqAs:p`[`_agts cYq��1gYi`^km�+Tj�m�d!^ fhp`_a^`_agYi`[Q1 = 40
gYuAs:g jl_a^ b [`Z Q2 = 42gYuAs:g jl_a[`c b g b [ ��gY[ b [`u!gts
i�( n ��^'cY^��!gYikjl_ cY^`[���_aq!gYi`^%s cYZY^���p`ZYgY_aq!i`[km�- � 0 y �4cYqAs ��q!p`ZY^km jlZ��Yfh^`[`[
P = 40−Q,
��q!_aq m�u4jl_a^ b [`ZY[���[`u��C^km jlZ��Yfh^`[ �
(TC) =1
2Q2 + 4Q− 10.
[�(U\�[`u!cYq!gYi`^km'u![ b q!ZYg!j/+�u4jlZYp`gY_ .Yq� m'cts:_a^`[`ZY^;[`cYq4nZY[��1gYi`^km�cY[�� m�^`c jlc m �
i�(U\�[`u!cYq!gYi`^km�u![Q0
b q!ZYg-j/+�u4jlZYp`gY_ .Yq� m�cY[�� m�^`cY[kn_�jlu cYq��1gYi`[km �U^��`q!p`gts cYq��1gYi`^km�g m�cYZY^���p`ZYgY_aq!i`["�
- � 04$'�4cYq*��cY[`u!gYi`^km jlZ��Yfh^`[cYq!fhgYc jl_a^`[d!q!_aq!i`^%s
C(Y ) = 0, 01Y 2 + 0, 1Y + 25,m�[ b [`f Y
gYu!q!p`Z jl_a^ ��gYcYq4m�[`p`[`_a^`["��1[`cYqAs:p`[`_agts cY[`u*� ^`ZY[`_�jlu!^�cY^ b u!gYe`^`_agYi`gYi`^�cYq*��cY[`u!g n
i`^km�[(MPC) b [ b [.+�q��1p`^km�[ b cY^ (MPS
(�`u!q b g m�[`f Y =
40�1�1[`[`[`ZY[`_a^.+�gts cY^��!gYikjl_a^ ��g b g��1gYi`^"�
- � 0 x � b [`ZY[.+�q��1^km jlZ��Yfh^`['cYq!fhgYc jl_a^`[Ud!q!_aq!i`^%s
S(Y ) = 0, 7Y + 90,m�[ b [`f Y
gYu!q!p`Z jl_a^ ��gYcYq4m�[`p`[`_a^`["�
� �
^��`q!p`gts cY[`u*� ^`ZY[`_�jlu!^EcY^ b u!gYe`^`_agYi`gYi`^ b [.+�q��1p`^km�[(MPS) b [cYq*��cY[`u!gYi`^km�[ b cY^ (MPC)
�- � 0 04�4cYq!fhgYc jl_a^`[cYqAs ��q!p`ZY^km jlZ��Yfh^`[
P = 500− 4Q.
[�( �1[`cYqAs:p`[`_agts cYqAs ��q!p`ZY^km;m�[��1jl[`_aq gY_a[km�d!^`ekjlu!q!i`[ [km�^kmfhp`_a^`_agYi`^kmUcY^`cY[`uAs6Ts3j �`u!q b j��Yfh^`^kmUgYuAs:g njl_a^km [km�^�e`_agYikjl_aq!ikm $ xzy b q!_a[`u!^ b [`Z $ y1y b q4n_a[`u![`c b g��
i�( �1[`cYqAs:p`[`_agts cYqAs ��q!p`ZY^km +/�!p`u4jl_a^�( gY_a[km�d!^`ekj�nu!q!i`["Cu!q!fh[
P = 120� �1[`[`[`ZY[`_a^.+�gts cY^��!gYikjl_a^
��g b g��1gYi`^"�- � 0 �1�4cYqAs ��q!p`ZY^km jlZ��Yfh^`[`[
P = 75− 3Q.
�1[`cYqAs:p`[`_agts cYqAs ��q!p`ZY^km +/�!p`u4jl_a^�(�gY_a[km�d!^`ekjlu!q!i`["u!q b g m�[`f
[�(P = 6
� i�(P = 60
�- � 0 { �4cYq!fhgYc jl_a^`[cYqAs ��q!p`ZY^km jlZ��Yfh^`[
P = 90−Q.
[�( �1[`cYqAs:p`[`_agts cYqAs ��q!p`ZY^km m�[��1jl[`_aq gY_a[km�d!^`ekjlu!q4ni`[ [km�^km#fhp`_a^`_agYi`^km#cY^`cY[`uAs6�s3j��`u!q b j��Yfh^`^km#gYu4ns:g jl_a^km [km�^ �1[`^.+�[`u b [�0 y b q!_a[`u!^ b [`Z 0 { b q4n_a[`u![`c b g��
i�( �1[`cYqAs:p`[`_agts cYqAs ��q!p`ZY^km +/�!p`u4jl_a^�( gY_a[km�d!^`ekjlu!q4ni`["�u!q b g m�[`f P = 30
�- � 0 �4�4cYqAs ��q!p`ZY^km jlZ��Yfh^`[`[
P = −Q2 − 2Q + 57.
� �
[�( �1[`cYqAs:p`[`_agts cYqAs ��q!p`ZY^km +/�!p`u4jl_a^�( gY_a[km�d!^`ekjlu!q4ni`[ [km�^km�cY^`cY[`uAs6lu!q!fh[
P = 22�
i�(�u!q��1q!u!^`[�cYqAs ��q!p`ZY^km �`u!q!fhgYZYd4jl_a^�fhp`_a^`_agYi`["1s3j [km�^km �`u!q!fhgYZYd4jl_a^Ufhp`_a^`_agYi`[`[
3 %�
- � 0 - �4cYq!fhgYc jl_a^`[cYqAs ��q!p`ZY^km jlZ��Yfh^`[
P = −2Q2 − 7Q + 1000.
[�( �1[`cYqAs:p`[`_agts cYqAs ��q!p`ZY^km +/�!p`u4jl_a^�( gY_a[km�d!^`ekjlu!q4ni`[ [km�^km�cY^`cY[`uAs6 s3j�cYqAs ��q!p`ZY[`[
Q = 20�
i�( [km�^km�u!q��1q!u!^ �`u!q!fhgYZYd4jl_a^ fhp`_a^`_agYi`[ �1[`cYq!^]\Tnp`gYpkm�cYqAs ��q!p`ZY^km
2 %nr^`[`Zafhp`_a^`_agYi`[km �
- � 0 �4�*�1[`cYqAs:p`[`_agts cYqAs ��q!p`ZY^km +/�!p`u4jl_a^�( gY_a[km�d!^`ekjlu!q!i`[ [km�^kmwcY^`cY[`uAs6 s3j
P = 12 b [cYqAs ��q!p`ZY^km jlZ��Yfh^`[cYq4n^`fhgYcY[d!q!_aq!i`^%s
P =√
500− 3Q .
- � 0'&4�4cYq!fhgYc jl_a^`[cY^]\�q b gYi`^km jlZ��Yfh^`[
Q = 0, 5P 2 + 2P + 10.
[�( �1[`cYqAs:p`[`_agts cY^]\�q b gYi`^kmm�[��1jl[`_aq gY_a[km�d!^`ekjlu!q!i`[ [km�^km ZY[.+�u b ^km cY^`cY[`uAs6#s3j �`u!q b j��Yfh^`^km gYuAs:g njl_a^km [km�^ �1[`^.+�[`u b [ � b q!_a[`u!^ b [`Z & b q!_a[`u![`c nb g��
i�( �1[`cYqAs:p`[`_agts cY^]\�q b gYi`^km +/�!p`u4jl_a^�( gY_a[km�d!^`ekjlu!q4ni`[ [km�^km�cY^`cY[`uAs6lu!q!fh[
P = 6�
- � � y �4^��`q!p`gts cY^]\�q b gYi`^kmAm�[��1jl[`_aq�gY_a[km�d!^`ekjlu!q!i`[6 [km�^km�ZY[.+�u4nb ^km�cY^`cY[`uAs6 s3j�cY^]\�q b gYi`^km jlZ��Yfh^`[`[
Q = 0, 01P 2 + 0, 5P + 60
� �
b [ �`u!q b j��Yfh^`^km gYuAs:g jl_a^km [km�^�e`_agYikjl_aq!ikm {zy b q4n_a[`u!^ b [`Z � � b q!_a[`u![`c b g����1[`cYqAs:p`[`_agts +/�!p`u4jl_a^�gY_a[km`nd!^`ekjlu!q!i`["lu!q b g m�[`f� [km�^`[ {zy b q!_a[`u!^"�
- � �1$'�4cY^]\�q b gYi`^km jlZ��Yfh^`[`[
Q = 0, 007P 2 + 3P + 4.
[�( �1[`cYqAs:p`[`_agts cY^]\�q b gYi`^km +/�!p`u4jl_a^�( gY_a[km�d!^`ekjlu!q4ni`[ [km�^km�cY^`cY[`uAs6lu!q!fh[
P = 15�
i�(�u!q��1q!u!^`[acY^]\�q b gYi`^km �`u!q!fhgYZYd4jl_a^afhp`_a^`_agYi`["�s3j [km�^km �`u!q!fhgYZYd4jl_a^Ufhp`_a^`_agYi`[`[
10 %�
- � � x �0m�[`e`q!ZYfhgYu!d!q b [`u!i`[.+�^'^`d!gYpkm & y1y1y cY[ . jlu!gYi`gY_�m�ls3j i`^kn_agts:^km6 [km�^!^��YZYgYi`[ � b q!_a[`u!^"�cY[���^`Zh^ .Y^ b gYi`[*- {zy1y i`^`_ag ns:^"�4i`^`_agts:^km� [km�^km x b q!_a[`u!^%s �1[`^`[� gYi`[`c �1[ .Y^ b jl_a^i`^`_agts:gYi`^km u![`q b gYZYq!i`[��1[.+�[`u b [ $ {zy1y nr^%s6�Cu![� [km�[ bjlZ b [ �1[`^ .Y^ b q4m<i`^`_agts:^" u!q!c;cts:_a^`[`ZY^�[`cYq!ZY[��1gYi`^w^ .Yq4mcY[�� m�^`cY[`_�jlu!^"3s3j b [`cYq!e`^ b gYikjl_agYi`[ i`^`_agts:^km [km`m�[ b [�1[ .Y^ b jl_a^Ui`^`_agts:gYi`^km�u![`q b gYZYq!i`[km ��q!u!^kmU\�u� ^`p`^`[��
- � � 04�0m��`q!u!d4jl_a^�e`q4m�d4jlcY^km [km�^-�1[`^.+�[`u b [�$ x b q!_a[`u!^ b [`Z$ { b q!_a[`u![`c b g��;[`cY^km �1[`cYq ZY[`fhp`_a[ b � y1y1y nr^km�[ s:p`g���^^ .Y^ b gYi`[ {zy1y1y fh[`_a^e`q4m�d4jlcY^"�lfhZYq!i`^`_a^`[" u!q!c b [`cYq!e`^knb gYikjl_agYi`[ [km`m�[ b [ �1[ .Y^ b jl_a^m��`q!u!d4jl_a^Ue`q4m�d4jlcYgYi`^kmu![`q b gYZYq!i`[km ��q!u!^kmU\�u� ^`p`^`["�l^��`q!p`gts6)[�(�cYqAs ��q!p`ZY^km jlZ��Yfh^`[ �i�(�u!q��1q!u!^ jlZ b [ ^ .Yq4m m��`q!u!d4jl_a^ e`q4m�d4jlcY^km� [km�^"u!q!cacts:_a^`[`ZY^U[`cYq!ZY[��1gYi`^U^ .Yq4m�cY[�� m�^`cY[`_�jlu!^��
- � � �1�0m�[`p`[`v`u!q ^`u!cY[km�[��Ypkm���gYZYq!i`["�u!q!cYgY_ ��^`f �1[`ZY_a[��1gYikjl_a^`[$ {zy q� ^km�^"�Cs3j .Yq!p`gY_Cs:p`^kjlu!^ �1[ b [km�[���[ b ^ s:^%s:q!g jl_
� �
q� ^km�+�g'^��YZYgYi`[ 0 {zy b q!_a[`u!^"hcY[���^`Z#.Yp`gY_a[-q� ^km�^ �1[��Y^knu![`p b gYi`["�3cY[`u!e`gYd!^`Z���jl_acY[ [`ZY[`_a^.+�cY[ [ � p`gYZY["0u!q!c �1[ b [knm�[���[ b ^km � b q!_a[`u!^%s �1[��Yp`^`u!gYi`[�^]\�p`gYpkm �1[��Y^`u![`p`gYikjl_a^q� ^km�gYi`^km x nr^%s-��gYcYfh^`u!gYi`[km��u![��1[ b [km�[���[ b ^!jlZ b [ b [`[]\�g nm�q4m ^`u!cY[`c�Au!q!c<cY^`^��!q4mVcY[�� m�^`cY[`_�jlu!^Vcts:_a^`[`ZY^�[`cYq4nZY[��1gYi`^�� ^���jl_a^km���cYgYi`["+u!q!c �1[ b [km�[���[ b ^ �1[��Y^`u![`p`gYikjl_a^q� ^km�gYi`^kmwu![`q b gYZYq!i`^km�\�u� ^`p`^ jlZ��Yfh^`[`["� (
- � � { �0m�[`p`[`v`u!q� ^`u!cY[ .Yq!p`gY_ s:p`g���^ .Y^ b ^km {zy1y fh[`_ c m`jlikj��:[`pknd!q!cY[`Z��Y[`ZY[km:s:^%s:q!g jl_�m x1{zy1y b q!_a[`u![ b �4cY[`u!e`gYd!^`Z���j�n_a^K[`ZY[`_a^.+�^ �1p`^ � p`gYZYgYikm�u!q!c xzy1y b q!_a[`u!^%s� [km b [`e`_ag ni`[<^]\�p`gYpkm �1[ .Y^ b jl_a^<[`p`d!q!cYq!i`^`_agYi`^km u![`q b gYZYq!i`^km {zy n^%s +�u b [km�[�(�^��`q!p`gts cYqAs ��q!p`ZY^km jlZ��Yfh^`["ls3jV^kmU\�u� ^`p`^`[ �i�(�u![ [km�^-jlZ b [ b [`g b q4m�c m`jlikj��U[`p`d!q!cYq!i`^`_�m4u!q!ccts:_a^`[`ZY^U[`cYq!ZY[��1gYi`^^ .Yq4m�cY[�� m�^`cY[`_�jlu!^��
� ( ^`u!cY^km .Yq!p`gY_Cs:p`^kjlu!^ b [`ZY[���[`u��CgYi`^km jlZ��Yfh^`[��1[`cYq4n^km�[���gYi`[Ud!q!_aq!i`^%s6)
K(Q) = 45000 + 100Q.
u!q��1q!u!^wjlZ b [<^ .Yq4m�[`p`d!q!cYq!i`^`_a^km [km�^"0u!q!ccY^knp`^��!qAs cY[�� m�^`cY[`_�jlu!^�cYq��1gYi`[��
- � � �4�*�kjlu!^km �Y[`u*��[`ZY[#.Yq!p`gY_ b �!^kjlu![ b .Y^ b ^km�0 y1y1y fh[`_ �kjlu4ms:^%s:q!g jl_�m y { b q!_a[`u![ b ���kjlu!^km [km�^km��1[.+�u b [`c y xb q!_a[`u!^%s �1[`cYq!^]\�p`^`[ �1[ .Y^ b jl_a^ �kjlu!^km u![`q b gYZYq!i`^km$ y1y nr^%s���gYcYfh^`u!gYi`["�[�(�^��`q!p`gts cYqAs ��q!p`ZY^km jlZ��Yfh^`["�s3j ^km �1[ .Y^ b jl_a^ �kj�nu!^km�u![`q b gYZYq!i`[.+�gC\�u� ^`p`[ b [`[ b [`cYq!e`^ b gYikjl_a^ �
� �
i�(�u![6 [km�^AjlZ b [ b [`g b q4m �kjlu4mzu!q!c4cYq��1gYi`[�^ .Yq4macY[�� nm�^`cY[`_�jlu!^"4s3j cY^km b [km�[`c,+�[ b gYi`_a[ b ^���[`u��CgYi`[ y 0b q!_a[`u!^��
- � � - �4cY[`d![`u!gYi`gY_ ��^V[`u!^km $ { p`[��1q!ZY^"��s:^%s:q!g jl_ p`[��1q!Z���^ � y[ b �1^`_a^`["�hs3j cY[`d![`u!gYi`_a^km<i`^`_agts:^km- [km�^w[`u!^km $ y b q4n_a[`u!^"hcY[���^`Z#^ .Y^ b gYi`[ { � y i`^`_agts:^"� +�[� �`jl_a^km-�`gYu!^`q b n��^�i`^`_agts:^km �!^`u!gYikjl_agYi`[ �1[.+�[`u b g m $ b q!_a[`u!^%s6zu![`cY[`f�1[`cYq!^]\�p`^`[ �1[ .Y^ b jl_a^Ai`^`_agts:gYi`^kmGu![`q b gYZYq!i`^km x &�nr^%s���g ncYfh^`u!gYi`["�[�(�^��`q!p`gts cYqAs ��q!p`ZY^km jlZ��Yfh^`["�s3j b [`cYq!e`^ b gYikjl_agYi`[
�1[ .Y^ b jl_a^�i`^`_agts:gYi`^km u![`q b gYZYq!i`[km�[ b [ i`^`_agts:^km [km`m ��q!u!^kmU\�u� ^`p`^`[ �
i�(�u!q��1q!u!^�jlZ b [�^ .Yq4m<i`^`_agts:^km [km�^" u!q!c'cts:_a^`[`ZY^[`cYq!ZY[��1gYi`^ ^ .Yq4m cY[�� m�^`cY[`_�jlu!^�� ^��`q!p`gts g m cY[�� m�^kncY[`_�jlu!^[`cYq!ZY[��1gYi`^"�
- � � �4�4^��`q!p`gts cYq!ZYq!d!q!Z jlu!q!i`^km �1jl[`_ag b gYi`^")$ (
y = 2x3 + 3x2 + 4x� x (
y = −x3 + 2x2 − 5x�
0(y = x3 − 27x + 2
� � (y =
1
3x3−7
2x2+10x+3
�
{ (y = (x− 5)2 � �(
y =1
2x + 3�
-'(y = ln x− 8
3x3 � �(
y = ln(4− x2)�
- � �&4�4^��`q!p`gts ��gYc b g��1^ jlZ��Yfh^`gYi`^km�g�� m�d!u!gYc jlcYgYi`^")$ (
y = 2x2 + 8x− 1� x (
y = 4x− x2 �
0(y = 2x3+3x2−36x+5
� � (y = x4−4x3 +4x2 +1
�
{ (y =
x4
4− x3 � �(
y =x4
4− 2x2 �
-'(y =
4√
x5 + 1� �(
y = 3x− 2√
x�
� �
&(y =
1
1 + x2
� $ y (y =
x3
1 + x2
�
$'$ (y =
x2
x− 3� $ x (
y = ex2 �
$ 0(y =
ex
x
� $ � (y = x · e−x �
$ { (y = ln x +
2
x
� $ �(y = x ln2 x
�- � {zy �4^��`q!p`gts ��gYc b g��1^� jlZ��Yfh^`^km j b ^ b g m�^ b [<jlcYfh^`u!g m�^ cYZY^kn
��p`ZYgY_aq!i`gYi`^UcY^%s:^%s:gYikjl_ �1jl[`_ag b ��^")$ (
y = x2 − 2x, [0; 3]�
x (y = x4 − 8x2 + 3, [−3; 3]
�
0(y = x +
√x, [0; 4]
�
� (y = (x− 3)2e−x, [0; 6]
�- � { $'�4^��`q!p`gts [`cYq�+�ZYg��Y^`_aq!i`^km b [ � [.+�ZYg��Y^`_aq!i`^km �1jl[`_ag b gYi`^
b [ �1[ b [��4jlZYp`^km�\�gYu!d!^`_agYi`^")$ (
y = x3 − 6x2 + 3� x (
y = 2x− 5x3 �
0(y = 3x5−5x4+3x−2
� � (y = x4 +6x3−60x2 +3
�
{ (y = x ex � �(
y = 2x2 + ln x�
-'(y =
x
x + 1� �(
y = x4 + x2 + ex �
&(y =
√x− 2
� $ y (y =
1
x2 + x + 1
�- � {1x �4^��`q!p`gts ��gYc b g��1^ jlZ��Yfh^`gYi`^km �1u![� ^`e`gYi`^km�[km�^`c��`d!q!d!gYi`^")
$ (y =
3
x + 5� x (
y =4
(x− 3)(x− 7)�
0(y =
2
x2 − 9� � (
y =x
x2 − 16�
� �
{ (y =
9(x2 − 5)
3x2 + 7
� �(y =
x
(x + 3)2
�
-'(y =
x2 − 3x + 5
x2 + x− 6
� �(y =
x3 + x + 2
2x3 − 16
�
&(y =
2− 3x
4 + 5x� $ y (
y =1− x2
1 + x2
�
$'$ (y = x +
1
x
� $ x (y = 2x +
2
x + 1�
$ 0(y =
x2 − 5x + 6
x− 2� $ � (
y =x3
x2 + x− 2�
$ { (y =
5x5
x4 − 9� $ �(
y =2x7
4 + x6
�
$ -'(y = ln(1− x2)
� $ �(y =
2x3 ln x
x2 + 1
�- � { 04�*�1[`cYq!^`e`p`_a^`gts ��gYc b g��1^� jlZ��Yfh^`gYi`^ b [![`[��1gts cY[%s:^��1u![� ^kn
e`gYi`^")$ (
y = x2 + x� x (
y = x2 +1
x
�
0(y = x3 − 12x2 + 36x
� � (y = x3−3x2−24x+1
�
{ (y =
x + 2
3x− 1� �(
y =2x + 1
x− 1
�
-'(y =
x2
1− x
� �(y =
x2
x− 3
�
&(y = x +
1
x
� $ y (y = e−x2/2 �
$'$ (y = ln(1 + x2)
� $ x (y =
3√
x3 − 3x�
s:[`p`^��#XYXYX
��� ��,&�� !)��,� ��"��� ���������������
�4� $'�*�1[`cYq4m�[���gts cY[`uAs:ekj s ��g b ^km S [`uAs:q!i`^"hu!q��1q!u!f cY^km�^
a�1p`gYu b ^km�[ b [ b b ^`[��1q!ZY[`_a^km jlZ��Yfh^`["�
�4� x �*�1[`cYq4m�[���gts cts:_a^`[`ZY^<[`cYq!ZY[��1gYi`^(TR)
u!q��1q!u!f m�[��Yq!Z n_a^km
Qu![`q b gYZYq!i`^km�[ b [�[`cCm�[��Yq!ZY_a^km<gYuAs:^�gYuAs:g jl_a^km
P [km�^km jlZ��Yfh^`["�
�4� 04�*�1[`cYqAs:p`[`_agts jlZ��Yfh^`^kmVcYZY^���p`ZYgY_aq!i`[�cY^%s:^%s:gYikjl_ \�gYu4nd!^`_ ��^")$ (
z = x2y3 − 4xy − 8x, M(1;−2);x (
z = ln(x2 + y2), M(−1; 2);0(z = (x2 + 1) ln(x + y2), M(0; 2).
�4� �1�4^��`q!p`gts jlZ��Yfh^`^km �1[`Z m�[.+/�!p`u!^km�[`u!g�)
1) z = x2 + y;
2) z =1
x2 + y2;
3) z =xy
x− y;
4) z = ln(16− x2 − y2);
5) z =√
25− x2 − y2;
6) z =1
√
x2 + y2 − 9;
� �
� 1
7) z =2
√
x2 + y2 − 5;
8) z =5
x2 − 4y2;
9) z = ln x(y2 + 1);
10) z = ln x2(y − 2)2 ;
11) z = ln(x− 3)y2.
�4� { �*�1[`cYqAs:p`[`_agts +/�!p`[`u!^")$ (
limx→1y→2
(3x2 − 2xy + y2)�
x (lim
x→−1y→0
(5x2 + 7xy − 8y2)�
0(lim
x→−1y→2
√
8x2 + 4y2 �
� (limx→1y→0
ln(x + ey)√
4x2 + y2
�
{ (limx→1y→2
3xy
x2 + y2 + 5
�
�(limx→1y→2
3−√xy + 4
x2 + y2 + 1�
�4� �4�%��gY^km%\�[`p`_agts
f(x, y) =
xy
x2 + y2,u!q!fh[
x2 + y2 6= 0
0,u!q!fh[
x2 + y2 = 0
jlZ��Yfh^`^kmj�\.Yp`gYd!q!i`^kmUm�[`e`^%s ��^"�
� �
�4� - �%��gY[`cYq:\�cYgts6Tj�\.Yp`gYd!^`[�s3j�[`u![ jlZ��Yfh^`[
f(x, y) =
x2y
x4 + y2,u!q!fh[
x4 + y2 6= 0,
0,u!q!fh[
x4 + y2 = 0.�4� �4�4^��`q!p`gts� jlZ��Yfh^`^km �`^`u!p`gY_a^�u!^��1^km-e`gYu��Yq \�[`u!cYq!gYikjl_ag n
i`^")$ (
z = x2y3 − 4xy2 + 7x3 + 10�
x (x = 4x3y2 − 2xy − 3y2 + 5
�
0(z = 5x2 + 7xy4 − 14x + 15y + 11
�
� (z = x2(y3 − x3)
�
{ (z = (y − 3) ln2 x
� �( x2
3x2 − y2
�
-'(z = y3e4x2 � �(
z =x2 + 5y
4x2 + 7y2
�
&(z =
4x2 + 5y3
4x− 8� $ y (
z =4y2
5x2 − 18�
$'$ (z =
x2 + y3
4xy
� $ x (z =
3xy
4x2 + 3y3
�
$ 0(z = (5x2 + y3)4 � $ � (
z = (4x2− 7y2 +5)3 �
$ { (z=
√
4x2+7y3+11� $ �(
z = 4x2e5y �
$ -'(z = −3y3e5x2
+ 4xy� $ �(
z = ex
y�
$�&(z = e5x2+7y3+8xy+15 � xzy (
z = 3xy2+5xy+7 �
x $ (z = ln
5x
y2 + 5� x1x (
z = ln4x2
7x2 + 2y�
x 0(z = y2 log5 x
� x � (z = 7x3 log2 y
�
x1{ (z = xy � x �(
z = logx y�
� �
�4� &4�*�1[`cYqAs:p`[`_agts jlZ��Yfh^`^km'm�u4jl_a^ b ^� gYu!gYZYfh^`[`_a^cY^%s:^%s:g nikjl_ \�gYu!d!^`_ ��^")$ (
z = 7x2y2, M(1;−1)�
x (z = x3 − 2xy + y2, M(−2; 1)
�
0(z =
1
2ln(x2 + y2), M(1; 2)
�
� (z =
x
y+ 5x2, M(2; 1)
�
{ (z = 4e5x2+y3
, M(3; 2)�
�(z = x3y, M(2; 1)
�
-'(z = 5x3e8y + 11, M(1; 2)
�
�(z = 3 log2 x + 4y, M(4; 2)
��4� $ y �4^��`q!p`gts jlZ��Yfh^`^kmUcYgYq!u!gCu!^��1^km�e`gYu��Yq \�[`u!cYq!gYikjl_agYi`^")
$ (z = 3x2y4 + 4xy − 9x + 17y
�
x (z = 7xy3 − 11x2 + 9
�
0(z = 4x ln y
�
� (z = ln(x2 + y3)
�
{ (z =
y3
x2 − 9�
�(z = 4ex ln y
�
-'(z = 5 ln(x2 + 9xy)
�
�(z = 2e8xy �
&(z = (x2 − 9y3)4 �
$ y (z =
x3
2 + y3+ e4y �
� �
�4� $'$'�4^��`q!p`gts uAs3jl_a^ jlZ��Yfh^`^kmU\�[`u!cYq!gYikjl_a^")
$ (z =
5x2
7y3, x = 2et, y = t2
�
x (z = ln(x3 + y) , x = t2, y =
√t�
0(z = ln(5x2 + 7xy) , x = et, y = ln t
�
� (z = 5x2y3 + 11xy − 9x2 , x = 4t, y = t3
�
{ (z = 7xy − 15x + 12y − 11 , x = 5t, y = t2
�
�(z = (x + y)4 , x = et, y =
4√
t�
-'(z =
√
x + y
x− y, x = t4, y = t2
�
�(z = 4y ln x , x = 4et, y = t
��4� $ x �4^��`q!p`gts [`u![`f ��[ b ^ jlZ��Yfh^`^km\�[`u!cYq!gYikjl_a^")
$ (x2 − 2xy + 7y2 = 10x3 − 11x + 12y − 5
�
x (4x3 − 2xy − 9 = 11x2 + 12y − 18xy2 �
0(y3 + xy = ln(x2 + 2y)
�
� (4y3 − 15x2 + 17xy2 = 0
�
{ (ey + xy = 0
�
�(x ln y + 7 · 2x + 15 = 0
�
-'(4ex + 9x2y = xy − 15
�
�(7e3x + 4y3 = ln(3x− 4y)
�
&(8ex2+y3
= ln(x + y)�
$ y (3 log2 y = x · 5y �
� �
�4� $ 04�4[ � p`gYZYgts6�u!q!cz = f(x, y)
[`u!^kmmu!^��1^kmGgYuAs��1p`[`u!q!p`[`ZY^
jlZ��Yfh^`[")$ (
z =5x2 + 8xy + 4y2
10x + y, m = 1
�
x ( √
5x2 + 7xy − 10y2 , m = 1�
0(z = 5x + 10y, m = 1
�
� (z =
ax + by
cx + dy, m = 0
�
{ (z = x2 + 2xy + 3y2, m = 2
�
�(z =
x3 + 4xy2
x + y, m = 2
�
-'(z = 7x4 + 7x3y + 8y4, m = 4
�
�(z =
√
2x4y2 + 4x3y3 , m = 3�
�4� $ �1�4[ � p`gYZYgts6Au!q!c<p`^�� m�gY_�
e`q���
b [��1_a[km�^km� jlZ��Yfh^`[ \�[`u!cYq4n[ b �1gYZ m m
u!^��1^km�gYuAs��1[`u!q!p`[`Z jlZ��Yfh^`[km)$ (
Q = CK1
4 L1
2 , m =3
4�
x (Q = CK
1
3 L2
5 , m =11
15�
�4� $ { �4cYqAs ��q!p`ZY^km jlZ��Yfh^`[km�[��Ypkmm�[���g
Q = 270− 3P − 4PA + 0, 2Y.
^��`q!p`gts cYqAs ��q!p`ZY^km[�(�e`gYu��Yq gY_a[km�d!^`ekjlu!q!i`[ [km�^km�cY^`cY[`uAs �i�( �Cp`[`u!g b ^`ZY^UgY_a[km�d!^`ekjlu!q!i`[ �� (�e`gYu��Yq gY_a[km�d!^`ekjlu!q!i`[ ��gYcYq4m�[`p`_a^kmwcY^`cY[`uAs6ls3j
P = 15 $, PA = 20 $, Y = 500 $.
� �
�1[`cYqAs:p`[`_agtsIcYqAs ��q!p`ZY^km �`u!q!fhgYZYd4jl_a^ fhp`_a^`_agYi`["�s3jm�[��Yq!ZY_a^km [km�^ �1[`^.+�u b gYi`[ { gYuAs:g jl_a^%s6�
�4� $ �4�4cYq!fhgYc jl_a^`[cYqAs ��q!p`ZY^km jlZ��Yfh^`[
Q = 400− 5P − 3PA + 0, 01Y 2.
^��`q!p`gts cYqAs ��q!p`ZY^km[�(�e`gYu��Yq gY_a[km�d!^`ekjlu!q!i`[ [km�^km�cY^`cY[`uAs �i�( �Cp`[`u!g b ^`ZY^UgY_a[km�d!^`ekjlu!q!i`[ �� (�e`gYu��Yq gY_a[km�d!^`ekjlu!q!i`[ ��gYcYq4m�[`p`_a^kmwcY^`cY[`uAs6ls3j
P = 10 $, PA = 20 $, Y = 100 $.
�1[`cYqAs:p`[`_agtsIcYqAs ��q!p`ZY^km �`u!q!fhgYZYd4jl_a^ fhp`_a^`_agYi`["�s3j[`_ad!gYu!ZY[`d!^kjl_a^ �`u!q b j��Yfh^`^km [km�^���gYcYfh^`u b [ $ y�� nr^%s6�
�4� $ - �4cYq!fhgYc jl_a^`[cYqAs ��q!p`ZY^km jlZ��Yfh^`[
Q = 700− 6P − 8PA + 0, 04Y.
^��`q!p`gts cYqAs ��q!p`ZY^km[�(�e`gYu��Yq gY_a[km�d!^`ekjlu!q!i`[ [km�^km�cY^`cY[`uAs �i�( �Cp`[`u!g b ^`ZY^UgY_a[km�d!^`ekjlu!q!i`[ �� (�e`gYu��Yq gY_a[km�d!^`ekjlu!q!i`[ ��gYcYq4m�[`p`_a^kmwcY^`cY[`uAs6ls3j
P = 20 $, PA = 10 $, Y = 1000 $.
�1[`cYqAs:p`[`_agtsIcYqAs ��q!p`ZY^km �`u!q!fhgYZYd4jl_a^ fhp`_a^`_agYi`["�s3j��gYcYq4m�[`p`[`_a^ ��gYcYfh^`u b [ xzy�� nr^%s6�
�4� $ �4�0m�[]\�[`u!cYq!q jlZ��Yfh^`[km�[��Ypkmm�[���g
Q = 3K2
5 L3
4 .
^��`q!p`gts6)[�(�e`[��`^`d![`_a^km +/�!p`u4jl_a^ �`u!q b j��Yd!^ (MPK)
�
i�( ��u!q!cY^km +/�!p`u4jl_a^ �`u!q b j��Yd!^ (MPL)�
� �
� ( +/�!p`u4jl_a^ ��gYZY[`fhp`_agYi`^kmwZYq!u!cY[(MRTS)
�s3j
K = 32, L = 16.
�4� $�&4�0m�[]\�[`u!cYq!q jlZ��Yfh^`[km�[��Ypkmm�[���g
Q = 3LK + L1
3 .
^��`q!p`gts6)[�(�e`[��`^`d![`_a^km +/�!p`u4jl_a^ �`u!q b j��Yd!^ (MPK)
�
i�( ��u!q!cY^km +/�!p`u4jl_a^ �`u!q b j��Yd!^ (MPL)�
� ( +/�!p`u4jl_a^ ��gYZY[`fhp`_agYi`^kmwZYq!u!cY[(MRTS)
�s3j
K = 2, L = 27.
b []\�gYu!gts ^.+�q!e`p`[`ZYd!gYi`^km ��g m�[`i`[`cY^km�^ �1[`ZYd!q!_agYi`["� �1[kncYqAs:p`[`_agts6�u![4m�^ b ^ b ^%s jlZ b [��1[`^.+�[`u b q4m K
e`[��`^`d![`_a^"s3j ��u!q!cY^km [��Yd!q!u!^���gYcYfh^`u b gYi`[ $�&-gYuAs:g jl_a^%s ^���j�n_a^km���cYgYi`["lu!q!ca[`cafhp`_a^`_agYi`[kmw[`u jlZ b ['cYq�� .Yp`g mU\�[`u!cYq4ngYikjl_a^ �`u!q b j��Yfh^`^km b q!ZY^km#fhp`_a^`_agYi`["�g�� ^"� (L, K)
\�gYu4nd!^`_a^U^`fhp`_agYi`[Uc���q!_aq b ^�� m�^`u!gYikjl_ ^.+�q!e`p`[`ZYd�+�g (��
�4� xzy �4[`[��1gts ��g m�[`i`[`cY^km�^�^.+�q!e`p`[`ZYd!^")[�(
Q = LK1
2 , Q0 = 3�
i�(Q = L
1
2 + 2K1
2 , Q0 = 2�
� (Q = 5LK + L2, Q0 = 5
��4� x $'�� ^`u!cY[�gYuAs3m�[ b [�^`cY[`p`g#m�[��Yq!ZYgY_�m�m�[���^`ZY[`q b [�m�[��1[`u!gYq
i`[.+�[`u�+�g .Y^ b ^kmm���p`[ b [km���p`[ [km�[ b �s:^%s:q!g jl_a^ i`[.+�u!^km cYqAs ��q!p`ZY[
(Qi) b [� [km�^(Pi)gYuAs:cY[`ZYgts!s:[`Z b [`e`[`p���^`u!gYikjl_a^`[6��gYc b g��1^�cYqAs ��q!p`ZY^km j�n
Z��Yfh^`gYi`^%s6)[�(
P1 = −4Q1 + 30
P2 = −8Q2 + 89�
1 �i�(
P1 = −2Q1 + 36
P2 = −3Q2 + 68�
^`u!cY^km�cts:_a^`[`ZY^ b [`ZY[���[`u��CgYi`^-�1[`cYq!^%s:p`_agYi`[� q!u4nc jl_a^%s6)[�(
(TC) = Q21 + 10Q1 + 4Q2
2 + 41Q2�
i�((TC) = Q2
1 + Q22 + 2Q1Q2
��1[`Z m�[.+/�!p`u!gts ^`u!cY^km�^km�gts:^4m�[� [km�q �`q!_a^`d!^`e`["�u!q4n
cYgY_a^`f j/+�u4jlZYp`gY_ .Yq� m ^`u!cY^km�cY[�� m�^`cY[`_�jlu cYq��1gYi`[km b [^��`q!p`gts [`cacYq��1gYi`^kmUm�^ b ^ b g��
�4� x1x �� ^`u!cY[kmVe`[��`^`d![`_a^km $ gYuAs:g jl_6+�g<g���[`u��CgYi`[ x$*��q!_aq
��u!q!cY^km $ gYuAs:g jl_6+�g � {$�� ^`u!cY^km m�[]\�[`u!cYq!q! jlZ�� n
fh^`[`[
Q = 6LK + 5L2.
\�[`u!cYq!gYi`^kmK b [ L
[��Yd!q!u!gYi.+�g ��[`u��C^km�[%s:p`^km �1[`cYq4n.Yq� ^`_a^�[��Ypkm ^�� m�^`u!gYikjl_a^ s:[`Z���[ xzy1y
$�3^��`q!p`gts \�[`u4n
cYq!gYi`^km�q��`d!^`cY[`_�jlu!^�u!g�� ^`cY^"��4� x 04�� ^`u!cY[kmVe`[��`^`d![`_a^km $ gYuAs:g jl_6+�g<g���[`u��CgYi`[�$
$*��q!_aq
��u!q!cY^km $ gYuAs:g jl_6+�g � x$�� ^`u!cY^km m�[]\�[`u!cYq!q! jlZ�� n
fh^`[`[
Q = 8K1
2 + L1
2 .
\�[`u!cYq!gYi`^kmK b [ L
[��Yd!q!u!gYi.+�g ^`u!cY[km �1[`cYq/.Yq� ^`_a^[��Ypkm� ^�� m�^`u!gYikjl_a^ s:[`Z���[ � $ y1y
$� ^��`q!p`gts \�[`u!cYq!gYi`^km
q��`d!^`cY[`_�jlu!^�u!g�� ^`cY^"��4� x �1�� ^`u!cY^kmU\�[`u!cYq!gYi`^km jlZ��Yfh^`[`[
Q = 2√
K +√
L .
1 �
Ke`[��`^`d![`_a^km $;gYuAs:g jl_a^ �!^`u4m �
$'��q!_aq
L��u!q!cY^km $
gYuAs:g jl_a^�� x$� ^��`q!p`gts \�[`u!cYq!gYi`^km�q��`d!^`cY[`_�jlu!^ u!g n
� ^`cY^"zs3j ^`u!cY[kmG\�[`u!cYq!gYi`^kmK b [ L
[��Yd!q!u!gYi.+�g*�1[`cYq4n.Yq� ^`_a^[��Ypkm ^�� m�^`u!gYikjl_a^�s:[`Z���[ � � � y
$�
�4� x1{ �� ^`u!cY[Gm�[��Yq!ZYgY_�m .Y^ b ^km#q!u�i`[.+�[`u�+�g!m���p`[ b [km���p`[� [km�[ b �s:^%s:q!g jl_a^ i`[.+�u!^km cYqAs ��q!p`ZY^km jlZ��Yfh^`[`[
P1 = 400 −2Q1
P2 = 650 − 3Q2
�� ^`u!cY^km�cts:_a^`[`ZY^ b [`ZY[���[`u��C^km jlZ��Yfh^`[`[
(TC) = 50+20(Q1+Q2)��^��`q!p`gts \�[`u!cYq!gYi`^km
u!g�� ^`cY^"�u!q!cYgY_a^`f�j/+�u4jlZYp`gY_ .Yq� m'cY[�� m�^`cY[`_�jlu cYq��1gYi`[km��4� x �4�� ^`u!cY[�.Y^ b ^kmVq!u!^ m�[���^kmVm�[��Yq!ZYgY_�m��s:^%s:q!g jl_6+�g cYqAs3n
��q!p`ZY^km jlZ��Yfh^`[`[P1 = 320− Q1
b [ P2 = 240− 2Q2�
^`u!cY^kmGcY^`gYu �1[]\�g jl_a^�cts:_a^`[`ZY^ b [`ZY[���[`u��C^ �1[`cYq!^km�[���gYi`[ q!u!c jl_a^%s
(TC) = 60 + 20(Q1 + Q2).
�1[`Z m�[.+/�!p`u!gts ^`u!cY^km<^km�gts:^�m�[� [km�q �`q!_a^`d!^`e`["0u!q!cYg n_a^`f�j/+�u4jlZYp`gY_ .Yq� mKcY[�� m�^`cY[`_�jlu�cYq��1gYi`[km b [G^��`q!p`gts [`ccY[�� m�^`cY[`_�jlu!^�cYq��1gYi`^kmm�^ b ^ b g��
�4� x - �� ^`u!cY[ .Y^ b ^kmCq!u!^�m�[���^kmam�[��Yq!ZYgY_�m��s:^%s:q!g jl_6+�g!cYqAs ��q4np`ZY^km jlZ��Yfh^`[`[
P1 = 900 − 4Q1b [ P2 = 700 − 2Q2
� ^`u!cY^kmGcY^`gYu �1[]\�g jl_a^�cts:_a^`[`ZY^ b [`ZY[���[`u��C^ �1[`cYq!^km�[���gYi`[ q!u!c jl_a^%s
(TC) = 90 + 20(Q1 + Q2).
�1[`Z m�[.+/�!p`u!gts ^`u!cY^km<^km�gts:^�m�[� [km�q �`q!_a^`d!^`e`["0u!q!cYg n_a^`f�j/+�u4jlZYp`gY_ .Yq� mKcY[�� m�^`cY[`_�jlu�cYq��1gYi`[km b [G^��`q!p`gts [`ccY[�� m�^`cY[`_�jlu!^�cYq��1gYi`^kmm�^ b ^ b g��
�4� x �4�� ^`u!cY[kmVe`[��`^`d![`_a^km $ gYuAs:g jl_6+�g<g���[`u��CgYi`[ x$*��q!_aq
��u!q!cY^km $�gYuAs:g jl_6+�g � 0$��m�[]\�[`u!cYq!q jlZ��Yfh^`[`[
Q =
1 �
4KL + 2L2 � \�[`u!cYq!gYi`^km [��Yd!q!u!gYi.+�g b [`ZY[���[`u��C^� ^�� nm�^`u!gYikjl_a^`[ b [ 0 xzy $
nr^km�d!q!_a^`["�K b [ L
m�^ b ^ b gYgYi`^kmu![wcYZY^���p`ZYgY_aq!i`gYi`^km�[%s:p`^km�cY^`^��:\�gYp`[�cY[�� m�^`cY[`_�jlu!^-cYq��1g ni`["ls3j��1[ .Y^ b jl_a^ �`u!q b j��Yfh^`^kmwgYuAs:g jl_a^km [km�^`[ { $
)[�(
K = 4 $, L = 3 $, Q = 3KL + K2,
(TC) = 450 $�
i�(K = 2 $, L = 3 $, Q = 3KL + 2L2,
(TC) = 400 $�
� (K = 1 $, L = 3 $, Q = KL + L2,
(TC) = 150 $�
s:[`p`^ X��
�������( � �� ����� ����������',������ !������������������ ����'���������� !�� " ����� �( ��#� �������� ����� !������ �����( ������ ������������ ������,������ ���� �� ��,� !��
����+&��"���������"��������������������� !���&�/�#���,�� ��������������+&��"�)" � +&������ �� !���" ��( �� �,����&�
&4� $'�*�Y^`u!^%s:[ b ^;^`ZYd!g��1u!gYi`^km'f ��u!^`_a^km �1[`cYq/.YgYZYgYi`^%s��1[`cYqAs:p`[kn_agts ��gYc b g��1^'^`ZYd!g��1u![`_agYi`^")$ ( ∫ √
x dx�
x ( ∫ x + 13√
xdx
�
0( ∫ (
x2 − 2 3√
x +4
x
)
dx�
� ( ∫ (
2x + 5√
x +3
x− 4√
x+ 8
)
dx�
{ ( ∫ √x + 1
xdx
�
�( ∫ (1 + x√
x)2
x2dx
�
-'( ∫(1− 2
√x)2dx
�
�( ∫(1− x2)3dx
�
&( ∫x2(√
x− 1)2dx�
1 1
1 �
$ y ( ∫(1 + x)(1 + 2x)(1 + 3x) dx
�
$'$ ( ∫ ( 1
x+
2
x2+
3
x3
)
dx�
$ x ( ∫ (1− x
x
)2
dx�
$ 0( ∫ 2√
x− 3 3√
x + 44√
xdx
�
$ � ( ∫ √
2√
x dx�
$ { ( ∫ √
x 3√
x dx�
$ �( ∫ (
1− 1
x
)
√
x√
x dx�
$ -'( ∫ √x4 + x−4 + 2
x3dx
�
$ �( ∫ x2
1 + x2dx
�
$�&( ∫ x2
1− x2dx
�
xzy ( ∫(3x + 4x)2dx
�
x $ ( ∫ 2x+2 − 5x−2
10xdx
�
x1x ( ∫ e3x + 1
ex + 1dx�
&4� x � b [`[`cYd!e`^`fhgts6lu!q!cCs3j∫
f(x) dx = F (x) + C,
cY[���^`Z∫
f(ax + b) dx =1
aF (ax + b) + C (a 6= 0).
1 �
&4� 04�*�1[`cYqAs:p`[`_agts ^`ZYd!g��1u![`_agYi`^� � [km�cY^km ��gYu*��^%s�(�)$ ( ∫ √
3x− 2 dx� x ( ∫ dx
(2x− 5)3
�
0((4x− 3)10dx
� � ( ∫ 5√
1− 5x dx�
{ ( ∫ 13√
1− 2xdx
� �( ∫e5xdx
�
-'( ∫34x+5dx
� �( ∫(e−x + e−2x) dx
�
&( ∫(ex − e−x)2dx
� $ y ( ∫ ln3 x dx
x
�
$'$ ( ∫ e1
x
x2dx
� $ x ( ∫xe−x2
dx�
$ 0( ∫ dx
x ln4 x
� $ � ( ∫ exdx
2 + ex
�
$ { ( ∫ dx
ex + 1� $ �( ∫ ex − x−x
ex + e−xdx
�
$ -'( ∫ex3
x2dx� $ �( ∫
x(3x2 − 1)5dx�
$�&( ∫2−x4
x3dx� xzy ( ∫
√1 + ln x
xdx
�
x $ ( ∫ x dx
3− 2x2
� x1x ( ∫ x dx
(1 + x2)2
�
x 0( ∫x2 3√
1 + x3 dx� x � ( ∫ x dx√
1− x2
�
x1{ ( ∫ e√
xdx√x
� x �( ∫ 4x + 3
(x− 2)3dx
�
x -'( ∫√
1 +√
x√x
dx� x �( ∫ ln 2x
ln 4x
1
xdx
�
1 �
x &( ∫ dx
(x + 2)(x− 1)� 0 y ( ∫ 1 + x
1− xdx
�
04$ ( ∫ x2
1 + xdx
� 0 x ( ∫ (1 + x)2
1 + x2dx
�
0 0( ∫x(1− x)10dx
� 0 � ( ∫x√
2− 5x dx�
&4� �1�4ZY[]\�^`_aq!i`^%s:^^`ZYd!g��1u!gYi`^km���gYu*��^%s �1[`cYqAs:p`[`_agts���gYc b g��1^^`ZYd!g��1u![`_agYi`^")$ ( ∫
xexdx� x ( ∫
x3xdx�
0( ∫ln x dx
� � ( ∫xn ln x dx (n 6= −1)
�
{ ( ∫ ( ln x
x
)2
dx� �( ∫
ln(x2 + 1)dx�
-'( ∫ √x ln2 x dx
� �( ∫xe−xdx
�
&( ∫x2e−2xdx
� $ y ( ∫ln2 x dx
�
$'$ ( ∫ ln x√x
dx� $ x ( ∫
e√
xdx�
$ 0( ∫ ln(ln x) dx
x
�&4� { �0m�[]\�[`u!cYq4mUcY[`u*� ^`ZY[`_�jlu!^ b [`ZY[���[`u��C^ Q
gYuAs:g jl_a^km\�[`u4ncYq!gYi`^km b u!q4m�[`u!^km
0, 03Q2 + 4Q + 150 b q!_a[`u!^�( .
�1[`^��1gts b [`ZY[���[`u��C^��`^`u!p`gY_a^ xzy gYuAs:g jl_a^km m�[]\�[`u!cYq4ngYi`_a[ b ;s3j �`u!q b j��Yfh^`^km �`^`u!p`gY_a^ gYuAs:g jl_a^km \�[`u!cYq4ngYi`[.+�gC^���[`u��CgYi`[ x1{1x y $ b q!_a[`u!^"�
&4� �4�4cY[`u*� ^`ZY[`_�jlu!^ b [`ZY[���[`u��C^km jlZ��Yfh^`[kmU[��YpkmUm�[���g
(MC) = K ′(Q) = 6Q2 − 10Q + 260,
1 �
m�[ b [`f K(Q)[`u!^kmwcts:_a^`[`ZY^ b [`ZY[���[`u��C^km jlZ��Yfh^`["4��q4n
_aqQ
� \�[`u!cYq!gYikjl_a^ �`u!q b j��Yfh^`^km u![`q b gYZYq!i`["� ^kn� m�^`u!gYikjl_a^ b [`ZY[���[`u��C^`[�$ {zy1y1y b q!_a[`u!^"� �1[`cYqAs:p`[`_agtsb [`ZY[���[`u��C^"lu!q!cYgY_a^`f ��gYg m�[`i`[`cYgYi`[ �`u!q b j��Yfh^`^km �`^`u!p`g n_a^ $ y gYuAs:g jl_a^km�\�[`u!cYq!gYi`[km�
&4� - �4cY[`u*� ^`ZY[`_�jlu!^U[`cYq!ZY[��1gYi`^km jlZ��Yfh^`[km�[��YpkmUm�[���g
(MR) = f(Q) = 80− 0, 4Q.
u!q��1q!u!^w^��YZYgYi`[wcts:_a^`[`ZY^�[`cYq!ZY[��1gYi`^ �`^`u!p`gY_a^ {zy gYuAs:g njl_a^km �1[ .Y^ b p`^km ��gYc b g�� �
&4� �4�4cY[`u*� ^`ZY[`_�jlu!^ b [`ZY[���[`u��C^km jlZ��Yfh^`[`[
(MC) = K ′(Q) = 200− 0, 6Q + 0, 09Q2.
^��`q!p`gts b [`ZY[���[`u��C^km jlZ��Yfh^`^km�ZY[.+�u b ^��`u!q b j��Yfh^`^km�cYq4nf+jl_aq!i`^km xzy gYuAs:g jl_a^ b [`Z {zy gYuAs:g jl_a[`c b g*�1[.+�u b ^km ��g ncts ��p`gYp`[���^"�
&4� &4�� ^`u!cY[`c b [`[ b �1^`ZY["1u!q!c4cY[`u*� ^`ZY[`_�jlu!^ b [`ZY[���[`u��C^ QgYu4n
s:g jl_a^km m�[]\�[`u!cYq!gYi`_a[ b jlZ b [ ^ .Yq4m (MC) = 3, 5 −0, 04Q
�4�1[`^��1gts6Tu![`c b gYZY^ b [`^���[`u��CgYi`[ �`u!q b j��Yfh^`^km �`^knu!p`gY_a^ $ y1y gYuAs:g jl_a^km-m�[]\�[`u!cYq!gYi`_a[ b 4s3j��`u!q b j��Yfh^kn^km �`^`u!p`gY_a^ x gYuAs:g jl_a^km'm�[]\�[`u!cYq!gYi`_a[ b ^���[`u��CgYi`[ $ � yb q!_a[`u!^��
&4� $ y �4^��`q!p`gts cYq*��cY[`u!gYi`^km jlZ��Yfh^`[H(Y )
zs3j-fhZYq!i`^`_a^`["zu!q!ccY[`u*� ^`ZY[`_�jlu!^GcY^ b u!gYe`^`_agYi`[GcYq*��cY[`u!gYi`^km�[ b cY^6�1[`cYq!^km�[���g ni`[d!q!_aq!i`^%s
(MPC) = H ′(Y ) = 0, 6 +0, 2√
Y,
1 �
m�[ b [`f YgYu!q!p`Z jl_a^���gYcYq4m�[`p`[`_a^`["� [`cY[km�s:[`Z��fhZYq!i`^`_a^`["
u!q!c�s3j gYu!q!p`Z jl_a^ ��gYcYq4m�[`p`[`_a^ $ y1y gYuAs:g jl_a^kmd!q!_a^`["cY[���^`ZCcYq*��cY[`u!gYi`['jld!q!_ b gYi`[ � { gYuAs:g jl_�m�
&4� $'$'�4^��`q!p`gts cYq*��cY[`u!gYi`^km6 jlZ��Yfh^`["�s3jwfhZYq!i`^`_a^`["�u!q!chcY[`u*� ^knZY[`_�jlu!^AcY^ b u!gYe`^`_agYi`[ b [.+�q��1p`^km�[ b cY^ �1[`cYq!^%s:p`_agYi`[Ad!q4n_aq!i`^%s
(MPS) = 0, 7− 0, 023√
Y ,
m�[ b [`f Y[`u!^km gYu!q!p`Z jl_a^���gYcYq4m�[`p`[`_a^"�![`cY[km�s:[`Z�!fhZYq4n
i`^`_a^`["�u!q!cGu!q b g m�[`f gYu!q!p`Z jl_a^ ��gYcYq4m�[`p`[`_a^ � ��gYuAs:g njl_a^`["�cY[���^`Z!cYq*��cY[`u!gYi`^km� jlZ��Yfh^`[ � y gYuAs:g jl_a^km;d!q!_a^`["�
&4� $ x �4^��`q!p`gts(TC)
cts:_a^`[`ZY^ b [`ZY[���[`u��C^"as3j cY[`u*� ^`ZY[`_�jlu!^b [`ZY[���[`u��C^`[
(MC) = 6e0,3Q,
��q!_aq ^�� m�^`u!gYikjl_a^ b [`ZY[���[`u��C^ 0 y gYuAs:g jl_a^km�d!q!_a^`["�&4� $ 04�4cYq!ZYq��`q!_a^km�d4jlu!^ ^`u!cY^km cY[`u*� ^`ZY[`_�jlu!^ [`cYq!ZY[��1gYi`^ cYq4n
^`fhgYcY[d!q!_aq!i`^%s6)[�(
(MR) = 100− 4Q�
i�((MR) = 80− 6Q
�
� ((MR) =
12√Q + 9
�
b ( (MR) =3√
Q + 4
�[��
Q[`u!^kmKu!gY[`_a^.+�gYikjl_a^ �`u!q b j��Yfh^`^kmKu![`q b gYZYq!i`[ [`Z j
cYqAs ��q!p`ZY[�(��z^��`q!p`gts(TR)
cts:_a^`[`ZY^G[`cYq!ZY[��1gYi`^km jlZ��Yfh^`[b [ b [`[ b �1^`ZYgts ��g m�[`i`[`cY^km�^�cYqAs ��q!p`ZY^km jlZ��Yfh^`^kmm�[���g��
1 �
&4� $ �1�*�1[`cYqAs:p`[`_agts �1[`Z m�[.+/�!p`u4jl_a^�^`ZYd!g��1u![`_a^")
$ ( 4∫
1
(x4 +√
x) dx� x (
1∫
0
√1 + x dx
�
0( 4∫
0
(x2 − 2√
x + 8)dx� � (
7∫
4
dx
(x− 3)2
�
{ ( 6∫
1
dx√x + 3
� �(4∫
0
ex
4 dx�
-'( 1∫
0
(1 + 3x)3dx� �(
4∫
1
1 +√
x
xdx
�
&( 9∫
4
x− 1√x + 1
dx� $ y (
10∫
0
dx√16 + 2x
�
$'$ ( 9∫
0
dx√x + 16−√x
� $ x (2∫
1
e1
x
x2dx
�
$ 0( e5∫
e
dx
x ln x
� $ � (16∫
4
dx√x− 1
�
$ { ( 9∫
1
dx
(1 +√
x)2
� $ �(5∫
1
x dx√5 + 4x
�
$ -'( 1∫
0
xexdx� $ �(
e2∫
1
ln x dx�
$�&( 1∫
0
x2exdx� xzy (
e∫
1
ln2 x dx�
� �
&4� $ { �*�1[`cYqAs:p`[`_agts ^`c ^���jlu!^km [`uAs:q!i`^"'u!q!cYgY_a^`f ��gYcYq4nm�[.+/�!p`u4jl_a^`[ ��gYc b g��1^#\�^`u!gYi`^%s6)
y = 3x2, y = 0, x = 1, x = 2.
&4� $ �4�*�1[`cYqAs:p`[`_agts [`uAs:q!i`^"%u!q!cYgY_a^`f ��gYcYq4m�[.+/�!p`u4jl_a^`[y =
9− x2 �`[`u![`i`q!_a^%s:[ b [ y = 0\�u� ^%s6�
&4� $ - �*�1[`cYqAs:p`[`_agts ^`c ^���jlu!^km [`uAs:q!i`^"'u!q!cYgY_a^`f ��gYcYq4nm�[.+/�!p`u4jl_a^`[ ��gYc b g��1^#\�^`u!gYi`^%s6)[�(
y =1
x, y = 0, x = 2, x = 10
�
i�(y = 3x, y = 0, x = 3, x = 4
�
� (y =
2
x2, y = 0, x = 3, x = 9
�
b ( y = 5x, y = 0, x = 1, x = 3�
g (y = x2, y = 2− x2 �
p�(y = ln x, y = 0, x = e2 �
&4� $ �4�*�1[`cYqAs:p`[`_agts [`uAs:q!i`^"%u!q!cYgY_a^`f ��gYcYq4m�[.+/�!p`u4jl_a^`[y =
ex y = e−x \�^`u!gYi`^%s:[ b [ x = 1\�u� ^%s6�
&4� $�&4�*�1[`cYqAs:p`[`_agts [`uAs:q!i`^"%u!q!cYgY_a^`f ��gYcYq4m�[.+/�!p`u4jl_a^`[y =
x2 + 4x�`[`u![`i`q!_a^%s:[ b [ y = x + 4
\�u� ^%s6�&4� xzy �*�1[`cYqAs:p`[`_agts [`uAs:q!i`^"%u!q!cYgY_a^`f ��gYcYq4m�[.+/�!p`u4jl_a^`[
y =
x2 − 2x�`[`u![`i`q!_a^%s:[ b [ 0x
�!gYu��Y^%s6�&4� x $'�4cYqAs ��q!p`ZY^km jlZ��Yfh^`[km�[��Ypkm ��gYc b g��1^#m�[���g
P = fD(Q) = 7− Q
5.
p`^��`q!p`qAs(CS)
cYq!c���cY[`u!gYi`_a^km b [`ZY[.+�q��1^"�u!q b g m�[`f<p`[`vknu!q!i`^km b q!ZYg Q0 = 20
nr^km d!q!_a^`["�;[`[��1gts cYqAs ��q!p`ZY^kmcYu4j b ^ b [6�1[`cYq4m�[���gts�cYq!c���cY[`u!gYi`_a^km b [`ZY[.+�q��1^ [`uAs:q4ni`^kmUm�[���^%s6�
� �
&4� x1x �4cYqAs ��q!p`ZY^km jlZ��Yfh^`['cYq!fhgYc jl_a^`[Ud!q!_aq!i`^%s
P = fD(Q) = 1000− 0, 4Q− 0, 003Q2.
^��`q!p`gts cYq!c���cY[`u!gYi`_a^km(CS) b [`ZY[.+�q��1^"As3j p`[`v`u!q!i`^km
b q!ZYg cYqAs ��q!p`ZY^km b q!ZYg ( xzy1y nr^kmUd!q!_a^`["�&4� x 04�4cY^]\�q b gYi`^km jlZ��Yfh^`[cYq!fhgYc jl_a^`[
P = fS(Q) = 8 +1
2
√
Q
q!u!c jl_a^%s6�%^��`q!p`gts(PS)
c \�[`u!cYq!gYi`_a^kmG[`cYq!ZY[��1gYi`^km�ZY[`cYg nd!^"ls3j��`u!q b j��Yfh^`^kmwgYuAs:g jl_a^km [km�^`[ $ y b q!_a[`u!^"�
&4� x �1�4^��`q!p`gts c \�[`u!cYq!gYi`_a^km<[`cYq!ZY[��1gYi`^km�ZY[`cYgYd!^(PS)
s3j cY^kn\�q b gYi`^km jlZ��Yfh^`[`[
P = fS(Q) = 40 + 8Q
b [ �`u!q b j��Yfh^`^kmwgYuAs:g jl_a^km [km�^`[ xzy1y b q!_a[`u!^"�&4� x1{ �� ^`u!cY[ e`p`^`u![���^ .Y^ b ^km xzy1y1y cY[`fh^`p`[`u4m � s:^%s:q!g jl_�m
x1{zy b q!_a[`u![ b ��cY[`u!e`gYd!^`Z��1^�[ � p`gYZYgYikm�u!q!c $ y b q!_a[`u!^%s [km�^km ��gYcYfh^`u!gYi`[K^]\�p`gYpkm �1[ .Y^ b jl_a^;cY[`fh^`p`u!gYi`^km'u![`q b g nZYq!i`^km�+�u b [km e`p`^`u![���^ xzy1y gYuAs:g jl_a^%s6�C^��`q!p`gts cYqAs3n��q!p`ZY^km jlZ��Yfh^`[ b [��1[`cYqAs:p`[`_agts (CS)
cYq!c���cY[`u!gYi`_a^kmb [`ZY[.+�q��1^"+s3jVp`[`v`u!q!i`^km b q!ZYgY[ Q0 = 220
��^���jl_a^km���cYg ni`["�u!q!cCcYqAs ��q!p`ZY^km jlZ��Yfh^`[#\�u� ^`p`^`["�
&4� x �4�4cY^]\�q b gYi`^km jlZ��Yfh^`[`[
P = fS(Q) = 2Q2 + 4Q + 1.
^��`q!p`gts(PS)
c \�[`u!cYq!gYi`_a^km [`cYq!ZY[��1gYi`^km ZY[`cYgYd!^"�s3jQ0 = 8
�
� �
&4� x - �4cYqAs ��q!p`ZY^km jlZ��Yfh^`[`[
P = fD(Q) =200
Q + 3.
^��`q!p`gts(CS)
cYq!c���cY[`u!gYi`_a^km b [`ZY[.+�q��1^"�s3j-�`u!q b j��Yfh^kn^km�gYuAs:^gYuAs:g jl_a^U^ .Y^ b gYi`[ $ y b q!_a[`u![ b �
&4� x �4�*�1[`cYqAs:p`[`_agts(PS)
c \�[`u!cYq!gYi`_a^km�[`cYq!ZY[��1gYi`^km�ZY[`cYgYd!^"�s3jcY^]\�q b gYi`^km jlZ��Yfh^`[`[
P = fS(Q) = 2 + 0, 03Q2,
��q!_aq p`[`v`u!q!i`[ � b gYi`[ Q0 = 10 b q!ZYg,+�g��a[`[`[��1gts ��g nm�[`i`[`cY^km�^ ZY[���[.+�^ b [-�1[`cYq4m�[���gtsEc \�[`u!cYq!gYi`_a^km [`cYq!ZY[��1g ni`^kmwZY[`cYgYd!^ [`uAs:q!i`^kmm�[���^%s6�
&4� x &4�0m�[`d!u![km�d!q q!Z b ^ ��\�_a^km �1[`ZYcY[`p`_aq!i`[���^^�� b ^kmws:[`Z���[km0 y1y1y b q!_a[`u!^km�^`ZYd!gYZ m�^`p`q!i`^%s6�%m�[`u��1gYi`_a^km�\�_a^kjlu!^AuAs3j�n_a^ �1[`ZY[`e`p`gts:^`[ $ y � j�\.Yp`gYd!^ b [`u!^`f ��p`^km<\�g m�^%s�(�� �1[knb [�� b [ ^]\.YgYi`[ gYuAs:^ \�_a^km ��gYc b g��1�:^��`q!p`gts m�[`d!u![km�d!q q!Z b ^km b ^km�e`q!ZYd!^`u!gYikjl_a^ �!^`u!gYikjl_agYi`["�
&4� 0 y �0m�[`d!u![km�d!q q!Z b cY[ajlZ b [K^`cYq��YcYg b q4m { \�_a^km �1[`ZYcY[`p`_aq4ni`[���^ b [���gYc b �1q!cY^ { \�_a^km �1[`ZYcY[`p`_aq!i`[���^'jlZ b [U^���[ b q4ms:[`Z���[�$ y1y1y1y b q!_a[`u!^kmK^`ZYd!gYZ m�^`p`q!i`^%s6��m�[`u��1gYi`_a^kmK\�_a^knjlu!^#uAs3jl_a^��1[`ZY[`e`p`gts:^`[ $ � � j�\.Yp`gYd!^ b [`u!^`f ��p`^km'\�g nm�^%s�(��h^��`q!p`gts m�[`d!u![km�d!q q!Z b ^km�s:[`Z���[ �`^`u!p`gY_a^�m�[`cY^\�_a^km ��gYc b g��1�
&4� 04$'�0m��`q!u!d4m�cYgYZY^<[� q!u!cYgYikm<m�[���gY_ [km�q e`q!ZYd!u![��Yd4m�3[`ce`q4ncYd!u![��Yd!^km �`^`u!q!i`gYi`^kmas:[`ZY[���cY[ b ^��1^!cY^`^��!gYikmGs:[`Z���[km�u!q4ncYgY_a^`f<^.+�u b gYi`[�s:[`ZY[`i`u![ b `j�\.Yp`gYd![ b�b [:\�u� ^`p`[ b \�_a^knjlu!^�$ y1y1y1y1y b q!_a[`u!^ b [`Z b [ [��:\�gYpkm x1x1{zy1y1y b q!_a[`u4m
� 1{ \�_a^km ��gYc b g��1��[`cY^`d!q!cCcY^km�^ ��gY_ [km�^U[`u!^km
F (t) = 100000(
1 +1
4t)
b q!_a[`u!^"�
^��`q!p`gts e`q!ZYd!u![��Yd!^km m�[]\.Y^km�^ �!^`u!gYikjl_agYi`["u!q!cYgY_a^`f��gYg m�[`i`[`cYgYi`[!m�[`u��1gYi`_a^km�\�_a^kjlu!^�uAs3jl_a^*- � nr^`[`ZY^��1[`ZY[`eknp`gts:^%s j . \�p`gYd b [`u!^`f ��p`[km� cY^%s:^%s:gYi`[") ^km�[`u��1gYi`_agts^`cY^%s6Au!q!c ��gY_ [km�^km�+�u b ^km�m�^ � �Y[`u!gY[ ^`ZYd!gYZ m�^`p`q!i`[`[�(F ′(t) = 25000
� (&4� 0 x �4^��`q!p`gts�jlp`[ b q c j b cY^`p`^ � y1y1y b q!_a[`u!^km'ZY[`e`[ b ^km#m�[]\.Y^kn
m�^��!^`u!gYikjl_agYi`["�s3j�m�[`u��1gYi`_a^kmUuAs3jl_a^;\�_a^kjlu!^ �1[`ZY[`eknp`gts:^`[ � � j�\.Yp`gYd!^ b [`u!d!^`f ��p`^%s6�
&4� 0 04�4^`ZYp`g m�d!^`fh^`^km ZY[`e`[ b ^ cYq!fhgYc jl_a^`[ d!q!_aq!i`^%sI(t) =
6000t1
2
�1�1[`cYqAs:p`[`_agts6)[�(m�[`[��Yfh^`q e`[��`^`d![`_a^km b [��1u!q!p`gYi`[��`^`u!p`gY_a^!\�_a^km�i`q4n_aq b [`ZCcYgYf ��u!gG\�_a^kmwi`q!_aq!c b g��
i�(�u![`c b gYZY^ \�gY_a^`[ m�[`v`^`u!q ^`cY^km�s:p`^km'u!q!c�m�[`[��Yfh^`qe`[��`^`d![`_acY[ �1[ b [`[`v`[`u!i`q4m $ xzy1y1y b q!_a[`u4m �
&4� 0 �1�4^`ZYp`g m�d!^`fh^`^km�ZY[`e`[ b ^cYq!fhgYc jl_a^`[Ud!q!_aq!i`^%s
I(t) = 12000t1
3 .
[�(�^��`q!p`gts e`[��`^`d![`_a^km b [��1u!q!p`gYi`[�cYgYq!u!g \�_a^kmGi`q!_aq4nb [`ZacYg���p`^ b ga\�_a^kmwi`q!_aq!c b g��
i�(�u![`c b gYZY^:\�_a^km���gYc b g�� �1[ b [`[`v`[`u!i`gYikm�m�[`^`ZYp`g m�d!^`fh^`qe`[��`^`d![`_a^ & y1y1y1y b q!_a[`u4m �
&4� 0 { �4^��`q!p`gts e`[��`^`d![`_a^km b [��1u!q!p`gYi`[ t = T1cYq!cYgYZYd!^ b [`Z t =
T2cYq!cYgYZYd![`c b g��s3j�^`ZYp`g m�d!^`fh^`^kmGZY[`e`[ b ^�cYq!fhgYc jl_a^`[!d!q4n
_aq!i`gYi`^%s6)[�(
I(t) = Atα�
� �
i�(I(t) = Aeαt
A b [ α b [ b gYi`^%s:^Uc j b cY^`p`gYi`^`[�(��
&4� 0 �4�0m�[`d!u![km�d!q q!Z b ^CcYq��YcYg b gYikm b �!gY^ b [`Z $ y \�_a^km �1[`ZYcY[`pkn_aq!i`[���^ b [ ^�� b ^km s:[`Z���[km {zy1y1y b q!_a[`u!^kmV^`ZYd!gYZ m�^`p`q4ni`^%s���gYc b �1q!cY^#-U\�_a^km �1[`ZYcY[`p`_aq!i`[���^"�Tm�[`u��1gYi`_a^km�uAs3j�n_a^ \�_a^kjlu!^ �1[`ZY[`e`p`gts:^`[ $ x � j�\.Yp`gYd!^ b [`u!^`f ��p`^%s�(��^��`q!p`gts6)[�(m�[`d!u![km�d!q q!Z b ^kmm�[]\.Y^km�^ �!^`u!gYikjl_agYi`[ �i�(m�[`d!u![km�d!q q!Z b ^km-m�^ b ^ b g �`^`u!p`gY_a^ { \�_a^km ��gYc nb g�� �
� (m�[`d!u![km�d!q q!Z b ^km#m�[]\.Y^km�^ �!^`u!gYikjl_agYi`["ls3j�ZY[`f+np`_a[ b - \�_a^km�[ ^��1^ ^`cYq��YcYg b gYikm c j b cY^`p`[ b ��gYcYq4nj�m�[.+/�!p`u!gY_a^ b u!q!^km �1[`ZYcY[`p`_aq!i`[���^�(��
&4� 0 - �*�1[`u!e`p`g jl_a^ �`u!q b j��Yfh^`^kmwcYqAs ��q!p`ZY^km jlZ��Yfh^`[`[
P = fD(Q) = 1000− 0, 2Q− 0, 06Q2.
^��`q!p`gts cYq!c���cY[`u!gYi`_a^km b [`ZY[.+�q��1^"hs3j p`[`v`u!q!i`^km b q!ZYgY[Q0 = 100
�&4� 0 �4�4^��`q!p`gtsEcYq!c���cY[`u!gYi`_a^km
(CS) b [`ZY[.+�q��1^":u!q b g m�[`f gYu4ns:g jl_a^km [km�^`[
P0 = 6 ��q!_aq cYqAs ��q!p`ZY^km jlZ��Yfh^`[")
[�(P = 38− 4Q
�
i�(P =
18√Q + 2
�&4� 0'&4�4^��`q!p`gts cYq!c���cY[`u!gYi`_a^km
(CS) b [`ZY[.+�q��1^"�s3j�cYqAs ��q!p`ZY^kmfD(Q)
jlZ��Yfh^`[ b [acYqAs ��q!p`ZY^km Q0b q!ZYg!cYq!fhgYc jl_a^`[6��g n
c b g��1^#m�[���^%s6)[�(
P = fD(Q) = 60− 5Q, Q0 = 8�
i�(P = fD(Q) = 50−Q2, Q0 = 6
�
� �
&4� � y �:s:[`p`^km`j [`_a^#e`q!ZYekjlu!gYZYfh^`^km �`^`u!q!i`gYi���^#i`[.+�u!^km'cYqAs ��q4np`ZY^km jlZ��Yfh^`[`[
P = fP (Q) = 30− 2Q,
��q!_aq cY^]\�q b gYi`^km jlZ��Yfh^`[ �
P = fS(Q) = 10 + 2Q.
�1[`cYqAs:p`[`_agts Q0
b q!ZYg,+�gap`[`v`u!q!i`^km�[km (�)[�(�cYq!c���cY[`u!gYi`_a^km
(CS) b [`ZY[.+�q��1^ �i�(�c \�[`u!cYq!gYi`_a^km
(PS)[`cYq!ZY[��1gYi`^kmwZY[`cYgYd!^"�
&4� �1$'�4cYq!fhgYc jl_a^`[cYqAs ��q!p`ZY^km jlZ��Yfh^`[
P = fD(Q) = −Q2 − 5Q + 70
b [cY^]\�q b gYi`^km jlZ��Yfh^`[
P = fS(Q) = Q2 + 3Q + 6.
fhZYq!i`^`_a^`["lu!q!c �1p`[��Ypkmm�u4jl_ .Yq� ^`_a^�e`q!ZYekjlu!gYZYfh^`["�^��`q!p`gts
Q0b q!ZYg,+�gCp`[`v`u!q!i`^km�[km (�)
[�(�cYq!c���cY[`u!gYi`_a^km(CS) b [`ZY[.+�q��1^ �
i�(�c \�[`u!cYq!gYi`_a^km(PS)
[`cYq!ZY[��1gYi`^kmwZY[`cYgYd!^"�&4� � x �*�1[`cYqAs:p`[`_agts [`u![km�[`ekj s:u!^`p`^�^`ZYd!g��1u![`_agYi`^")
$ ( +∞∫
2
dx
x
� x (+∞∫
3
dx
x3
�
0( +∞∫
0
3−xdx� � (
+∞∫
2
dx
x
�
{ ( +∞∫
1
dx√
(1 + x)3
� �(+∞∫
−∞
2x
(1 + x2)4dx
�
� �
-'( 0∫
−∞
e5xdx� �(
1∫
−∞
dx
(3x− 5)2
�
&( +∞∫
2
dx
x ln x
� $ y (+∞∫
5
dx
x ln2 x
�
$'$ ( +∞∫
1
ln x dx
x2
� $ x (16∫
0
14√
xdx
�
$ 0( 36∫
0
dx√x
� $ � (3∫
0
x dx√9− x2
�
$ { ( 2∫
0
dx√4− x2
� $ �(1∫
0
x dx√1− x2
�
$ -'( 0∫
−1
e−1
x
x2dx
� $ �(1∫
0
ln x dx�
$�&( 3∫
1
x dx√x− 1
� xzy (e2
∫
1
dx
x√
ln x
�
x $ ( 2∫
1
dx
x ln x
� x1x (1∫
−1
dx
x2
�
x 0( 1∫
−1
dx3√
x2
� x � (3∫
0
dx3
√
(x− 2)2
�
s:[`p`^ �
"�$�%����������&�� ���� (*������ � !���&���
��� ������
�� !��������������
$ y � $'�4[`e`cY[ .Yq� ^`_agYikm s3j [`u![-cYq!fhgYc jl_ b ^� gYu!gYZYfh^`[`_�jlu �1[`Z nd!q!_agYi`[km
y = f(x) jlZ��Yfh^`[��
$ (xy′ = 2y, y = 5x2 �
x (y′′ = x2 + y2, y =
1
x
�
0(y′′ − 2y′ + y = 0, y = xex �
� (y′′ − 4y = 0, y = e2x �
{ ((x− y + 1)y′ = 1, y = x + 6ex �
�(x + y + x
dy
dx= 0, y = e3x + 4
�
-'(y′′ − 2y′ + y = 0, y = (1 + 2x)ex �
�(y′′ + 2y = 0, y = xex �
$ y � x �4^��`q!p`gts b ^� gYu!gYZYfh^`[`_�jlu!^��1[`ZYd!q!_agYi`^km +�q��1[ b ^A[`cYq!ZY[��`nm�gYZY^")$ (
y′ = 2x + 5�
x (y′ = x + e2x �
0(y′ = 6
�
� (y′ = 4x + 3
�
{ (y′′ = 3− 2x
�
� �
� �
�(y′′ = 6x + e2x �
$ y � 04�4^��`q!p`gts b ^� gYu!gYZYfh^`[`_�jlu!^-�1[`ZYd!q!_agYi`^km e`gYu��Yq [`cYq!ZY[��`nm�gYZY^")$ (
y′ = 4x3 + 3x2 �s3j y(0) = 1�
x (y′ = xex �s3j y(0) = 2
�
0(y′ =
2x
1 + x2
�s3jy(1) = 0
�
� (y′ = ln x
ls3jy(1) = 1
�
{ (y′′ = 3x2 + 4
ls3jy(0) = 2
y′(0) = 1
�
�(y′′ = 2 + ex ls3j y(0) = 3
y′(0) = 1
�$ y � �1�4[`cYq*�`m�gYZY^%s �1[`ZYfh[`_agYi`[ b fhp`_a[ b gYi`^`[`ZY^ b ^� gYu!gYZYfh^`[`_�jlu!^
�1[`ZYd!q!_agYi`[")$ (
xyy′ = 1− x2 �
x (xy′ = y2 ls3j y(1) = 1
�
0(yy′ =
1− 2x
y
�s3jy(0) = 3
�
� ((xy2 + x)dx + (y − x2y)dy
�
{ (yy′ + x = 0
�
�((1 + y2)dx + xy dx = 0
�
-'(y′ = ex+y �
�((1 + ex)yy′ = ex �
$ y � { �4[`cYq*�`m�gYZY^%s��`^`u!p`gY_a^�u!^��1^km<gYuAs��1p`[`u!q!p`[`ZY^ b ^� gYu!gYZYfh^kn[`_�jlu!^ �1[`ZYd!q!_agYi`gYi`^")$ (
x dy − y dx = y dy�s3j
y(2) = 1�
x (y′ =
y
x− 2
�
� �
0(y′ =
x + y
x− y
�
� ((y − 2x)dy + 2y dx = 0
ls3jy(0) = 1
�
{ ((y − x)y dx + x2dx = 0
�
�(y dx + (2
√xy − x)dy = 0
�
-'(y2dx + (x2 − xy)dy = 0
�
�((x− y)dx + (x + y)dy = 0
�s3jy(1) = 0
�$ y � �4�4^��`q!p`gts �`^`u!p`gY_a^#u!^��1^km#\�u� ^`p`^ b ^� gYu!gYZYfh^`[`_�jlu!^ �1[`Z n
d!q!_agYi`^kmw[`cYq!ZY[��`m�gYZY^")$ (
y′ − y
x= x
�
x (y′ +
y
x=
ex
x
�
0(y′ +
2
xy = x3 �
� (y′ +
2x
1− x2y = x + 1
�
{ (y′ + y = e−x �s3j y(0) = 2
�
�(y′ + 2xy = 3x2e−x2 ls3j
y(0) = 1�
-'(xy′ + y = x + 1
�s3jy(2) = 3
�
�(xy′ + 2y = 3x
ls3jy(−2) = 0
�$ y � - �4[`cYq*�`m�gYZY^%s ��gYc b g��1^ b ^� gYu!gYZYfh^`[`_�jlu!^ �1[`ZYd!q!_agYi`gYi`^")
[�(y′′ = 3x2 + 2x
�
i�(y′′ = x + ex �
� (y′′ = 6 + 4e2x �
b ( y′′ =1
x3− 4.
$ y � �4�4[`cYq*�`m�gYZY^%s cYgYq!u!gu!^��1^km�c j b cY^`p`e`q!g ^`fh^`gYZYd!gYi`^`[`ZY^<\�u4n ^`p`^gYuAs��1p`[`u!q!p`[`ZY^ b ^� gYu!gYZYfh^`[`_�jlu!^ �1[`ZYd!q!_agYi`gYi`^")
� �
$ (y′′ − 9y′ = 0
ls3jy(0) = 1
y′(0) = 2
�
x (y′′ + 4y′ = 0
�s3jy(0) = 1
y′(0) = 4
�
0(y′′ − 4y′ + 3y = 0
�
� (y′′ + 5y′ + 6y = 0
�
{ (4y′′ + 4y′ + y = 0
�
�(y′′ − 6y′ + 9y = 0
ls3jy(0) = 1
y′(0) = 3
�
-'(4y′′ − 12y′ + 9y = 0
�
�(y′′ + 2y′ + y = 0
�s3jy(0) = 2
y′(0) = 1
�
&(y′′ + 4y′ = 0
�s3jy(0) = 1
y′(0) = 2
�
$ y (y′′ + 2y′ + 5y = 0
�
$'$ (y′ + y = 0
�s3jy(0) = 0
y′(0) = 1
�
$ x (y′′ + 4y′ + 8y = 0
�$ y � &4�4^��`q!p`gts cYgYq!u!gAu!^��1^km'c j b cY^`p`e`q!g ^`fh^`gYZYd!gYi`^`[`ZY^;\�u� ^`p`^
[`u![`gYuAs��1p`[`u!q!p`[`ZY^ b ^� gYu!gYZYfh^`[`_�jlu!^ �1[`ZYd!q!_agYi`^km +�q4n�1[ b ^[`cYq!ZY[��`m�gYZY^")$ (
y′′ − 4y′ + 4y = x2 �
x (y′′ − y = x2 − x + 1
�
0(y′′ + y′ = 3
�
� (y′′ − 3y′ = 2− 6x
�
{ (y′′ + 2y′ + y = e2x �
�(y′′ + y = 4ex �
-'(y′′ − 2y′ − 3y = 6xex �
�(y′′ − 2y′ + y = ex �
$ y � $ y �0m�[���gY_ac \�^� q4m-i`u4jlZYp`[���^�[��YpkmwgYu!q!p`Z jl_a^�p`[`_�jld!^km �<cY^kn_a^`[`u b ^�gYuAs:g jl_a^"�Ccts:[`p`u!q!i`[`c �1[ b []\.Yp`^`d![ i`u4jlZYp`[���^
� �
��gYcYq!^��!q4m� jl_a^km;[���[`_a^KZY^���ZYgYi`^"� .Yq!p`gY_ b �!^kjlu![ b i`[`ZYekn��^ ��g b ^km { cY^`_a^`q!ZY^'gYuAs:g jl_a^km ��g m�[`i`[`cY^km�^ �Yp`gY_a^'ekj��`^knjlu!["�cY^kmwZY[`fhp`_a[ b i`u4jlZYp`[���^ �1[ b ^km ��g m�[`i`[`cY^km�^�u![`q b g nZYq!i`^kmw[���[`_a^ jl_a^km�ZY^���ZYgYi`^"�$ ( ��gY[ b �1^`ZYgts b ^� gYu!gYZYfh^`[`_�jlu!^��1[`ZYd!q!_agYi`[" u!q!cYg n_a^`f [��:\�gYu4m�[���[`_a^Up`[`_�jld!^km�i`u4jlZYp`[���^ ��g m�p`_a[km �
x (�u![ b u!q b [km�v`^`u b gYi`[ �Yp`gY_a^'p`[`_�jld!^km � y � nr^%s���g nfhp`_a[km �
$ y � $'$'�0m�[���gY_ac \�^� q4mwi`u4jlZYp`[���^[��Ypkm �wcY^`_a^`[`u b ^UgYuAs:g jl_a^p`[kn_�jld!["�Gm�[`v`^`u!q![ ��gYcYq!^��!gYikjl_ ^��YZY[km jl_a^km [���[`_a^ ZY^kn��ZYgYi`^"� .Yq!p`gY_ b �!^kjlu![ b i`[`ZYe���^ ��g b ^km $ y cY^`_a^`q!ZY^VgYu4ns:g jl_a^km ��g m�[`i`[`cY^km�^ �Yp`gY_a^Uekj��`^kjlu!["+cY^kmwZY[`fhp`_a[ b i`u4j�nZYp`[���^ �1[ b ^km ��g m�[`i`[`cY^km�^wu![`q b gYZYq!i`^km-[���[`_a^ jl_a^km�ZY^kn��ZYgYi`^"�$ (�u![`c b gYZ ��[`Z���^ b [`u � gYi`[�i`u4jlZYp`[���^ �Yp`gY_a^�p`[`_�jld!^km
14
�
x (�u![ b u!q b [km�v`^`u b gYi`[ �Yp`gY_a^ p`[`_�jld!^km cts:_a^`[`ZY[ b��gYfhp`_a[km �
$ y � $ x �0m�[���gY_ac \�^� q4mwi`u4jlZYp`[���^[��Ypkm �wcY^`_a^`[`u b ^UgYuAs:g jl_a^p`[kn_�jld!["�0m�[`v`^`u!q![-��gYcYq!^��!gYikjl_ ^��YZY[km jl_a^km [���[`_a^�ZY^��1nZYgYi`^"��.Yq!p`gY_ b �!^kjlu![ b i`[`ZYe���^ ��g b ^km � cY^`_a^`q!ZY^�gYuAs:g njl_a^km ��g m�[`i`[`cY^km�^ �Yp`gY_a^ ekj��`^kjlu!["'cY^km ZY[`fhp`_a[ b i`u4j�nZYp`[���^ �1[ b ^km ��g m�[`i`[`cY^km�^wu![`q b gYZYq!i`^km-[���[`_a^ jl_a^km�ZY^kn��ZYgYi`^"�4u![`c b gYZY^ b �!^km���gYc b g�� ^��YZYgYi`[�i`u4jlZYp`[���^ q!u��CgYucYgYd!^[���[`_a^Uekj��`^kjlu![ �Yp`gY_Cs:[`Z6��g b [`u!gYi`^%s6�
$ y � $ 04� b [`pkj1��p`[%s t b u!q!^km�[%s:p`^km<m�[���gY_ac \�^� q*��^-cYq4m�[���_agYq!i`^kmu![`q b gYZYq!i`[`[ P = P (t)
lfhZYq!i`^`_a^`[" u!q!c ��q!i`[ b q!i`[ b [
� �
m�^`e`p b ^`_a^`[`ZYq!i`[ b u!q!^kmKgYuAs:g jl_ ��^6�`u!q��`q!u!fh^kjl_a^`[KcYq4nm�[���_agYq!i`^km�u![`q b gYZYq!i`^km��`u!q��`q!u!fh^kjl_aq!i`^km�e`q!g ^`fh^kngYZYd!^`["���g m�[`i`[`cY^km�[ b 1
16b [ 1
48
%�1[`u b [ [`cY^km�["0m�[���gY_ac n\�^� q*��^w[`u!^km-gYcY^��1u![`fh^`[" u!q!cY_a^kmwm�^ � �Y[`u!gY[ $ {zy1y e`[`fh^\�gY_a^]\�[ b ��^"���1[`ZYpkm�[.+/�!p`u!qAs cYq4m�[���_agYq!i`^km u![`q b gYZYq!i`[b u!q!^kmwZYgYi`^km�cY^`gYu!^UcYq!cYgYZYd!^km�[%s:p`^km+s3jVfhZYq!i`^`_a^`["lu!q!ct = 0
cYq!cYgYZYd!^km�[%s:p`^kmUcYq4m�[���_agYq!i`^km�u![`q b gYZYq!i`['d!q!_a^`[35 · 106 �
$ y � $ �1�4p%s��Yp`[%s6Am�[`f ��q!p`u!gYi`gY_a^ �1[`u!gYcYq4m�[ b [ m�[`e`p`gYi`^ u!g m`jlu4nm�gYi`^km ��g,+/�4j b jl_aq!i`^km �1[`cYq cYq4m�[���_agYq!i`^km cY[�� m�^`cY[`_�j�nu!^ u![`q b gYZYq!i`["Au!q!cYgY_a^`f ^`[`u4m�gYi`gYikm b [ �1[`cYq!^`e`p`gYi`gYi`[[`u4m�gYikjl_a^Uu!g,+�gYu!p`gYi`^%s6�d!q!_a^`[ { cY^`_a^`q!ZY^km�lcYq!fhgYc jl_a^t b u!q!^km�[%s:p`^km<cYq4m�[���_agYq!i`^km<u![`q b gYZYq!i`[`[ P (t)
1��q!_aqcYq4m�[���_agYq!i`^km +�u b ^kmAm�[ � �Y[`u!g��`u!q��`q!u!fh^kjl_a^`[ 1
250P (t)·
[5 · 106 − P (t)]m�^ b ^ b ^km�/�1[`ZYpkm�[.+/�!p`u!qAs cYq4m�[���_agYq!i`^km
u![`q b gYZYq!i`[ tcYq!cYgYZYd!^km�[%s:p`^kmls3j�m�[]\.Y^km
t = 0cYq!cYgYZYd4n
��^cYq4m�[���_agYq!i`^km�u![`q b gYZYq!i`[`[ 4, 5 · 106 �
s:[`p`^ �#X
� ���%�,�)" ������ �( � ������������� !��������������
$'$'� $'� b [���d!u!^���gts jld!q!_aq!i`^%s �1[`Z m�[.+/�!p`u4jl_a^�ZY[���gYp`[`u4m�^`i`u!d4n.Yg�)$ (
x ≥ 5� x (
y < −3�
0( −2x ≥ 7� � ( −y
5≤ 1, 2
�
{ (2x− 3y ≥ 3
� �(4x− 5y ≤ 1
�
-'(y ≥ 2x
� �(y ≤ x + 1
�
&( −x + 40y ≤ 160� $ y (
0, 1x + 0, 2y ≥ 2, 75�
$'$ (0, 3x + 2, 1y − 3 ≤ 0
� $ x (2x + 4y ≥ 5x− 1
�$'$'� x � b [���d!u!^���gts b [km�[���p`gYiVcYZY^���p`ZYgY_aq!i`[%s:[�[`u!g b [ ^��`q!p`gts
cY^km�^'\�p`gYu!q!gYi`^")
$ ({
x + y ≤ 5,
x ≥ 0;
x ({
2x + y ≤ −3,
4x− 2y ≥ 12;
0({
7x + y ≤ 11,
2x− 3y ≥ −10;
� (
x + 3y ≤ 9,
4x− y ≥ 10,
y ≥ 0;
{ (
4x + 2y ≤ 5,
−x + y ≤ 0,
y ≥ 0;
�(
2x + 3y ≤ −5,
3x− 2y ≥ 12,
x ≥ 0,
y ≥ 0;
� 1
� �
-'(
3x + 2y ≥ 6,
3x + y ≤ 9,
x ≥ 0,
y ≥ 0;
�(
x + y ≤ 4,
x + y ≥ 1,
−x + y ≥ −1;
&(
x + y ≥ 1,
x + y ≤ 4,
−x + y ≤ 1,
y ≥ 0;
$ y (
2x + y ≤ 10,
x + y ≤ 7,
x + 2y ≤ 12,
x ≥ 0,
y ≥ 0;
$'$ (
3x + y ≤ 21,
x + y ≤ 9.
x + 3y ≤ 21,
x ≥ 0,
y ≥ 0;
$ x (
x + y ≥ −3,
x− y ≤ 5,
y ≤ 3,
x ≥ 0;
$ 0(
2x + y ≥ 8,
2x− 3y ≤ 0,
y ≤ 10,
x ≤ 12,
x ≥ 0;
$ � (
2x + y ≤ 12,
4x− 3y ≤ 0,
y ≤ 10,
x ≤ 3,
x ≥ 0.
$'$'� 04�4^��`q!p`gtsf(x, y) = −x + 2y
cY^.+�ZY^km jlZ��Yfh^`^km-j b ^ b g m�^b [;jlcYfh^`u!g m�^'cYZY^���p`ZYgY_aq!i`gYi`^"ls3j x b [ y
[`e`cY[ .Yq� ^`_agYikm��gYc b g�� �`^`u!q!i`gYikm)
$ (
2x + 3y ≤ 18,
y ≥ 0,
x ≥ 0;
x (
x− y ≤ 1,
x + y ≤ 6,
x ≥ 1;
� �
0(
2x + 5y ≤ 0,
2x− 5y ≤ 20,
x ≥ 0.$'$'� �1�4^��`q!p`gts ��gYc b g��1^'[`cYq!fh[`ZY^km�[`cYq!ZY[��`m�gYZY^")$ (
max(x + 2y)x (
max(10x + 15y)
x− y ≤ 0,
x ≥ 0,
y ≥ 0;
2x + y ≤ 100,
0, 1y ≤ 4,
0, 1x ≥ 1,
y ≥ 20;0(
max(2x + 3y)� (
min(8x + 12y)
x + y ≤ 7,
x + 2y ≤ 12,
2x + y ≤ 10,
x ≥ 0,
y ≥ 0;
x + y ≤ 10,
x ≥ 1,
x ≤ 3,
y ≥ 0;
{ (min(5x + 15y)
�(min(x + y)
x + y ≤ 0,
−2x + y ≥ 1,
x ≥ 0,
y ≥ 0;
x + 2y ≥ 11,
3x + 2y ≥ 17,
3x + y ≥ 10,
x ≥ 0,
y ≥ 0;-'(
max(5x + 50y)�(
min(100x + 125y)
−x + y ≥ −4,
−x + y ≤ 0,
y ≥ 1,
y ≤ 4;
x + y ≥ 5,
−2x + 2y ≤ 2,
−2x + y ≤ 2,
y ≥ 0;
� �
&(max(x + 2y)
$ y (max(5x + 10y)
x + y ≤ 6,
x− 2y ≥ 0,
y ≤ 4,
x ≥ 0,
y ≥ 0;
x + 3y ≤ 36,
x + y ≤ 16,
2x + y ≤ 24,
x ≥ 0,
y ≥ 0.
$'$'� { �� ^`u!cY[ []\�[`u!cYq!gYikmVq!u!^A b [ B
d!^��`^km��`u!q b j��Yd4m� Ad!^��`^km �`u!q b j��Yd!^km b [km�[`c,+�[ b gYi`_a[ b m�[`v`^`u!q![Am�[]\�[`u!cYq!q�`u!q!fhg m�^km $;cY[`Z��Y[`ZY[km�[`[%s:^ b [ ��e`[`f+m�[`[%s:^"��q!_aq B
d!^kn�`^km �`u!q b j��Yd!^km b [km�[`c,+�[ b gYi`_a[ b � x cY[`Z��Y[`ZY[km�[`[%s:^ b [0 e`[`f+m�[`[%s:^"� ^`u!cY^km .Yq!p`gY_ae`p`^`u!g jl_a^ m�[]\�[`u!cYq!q m�^`c n�Y_a[`p`u!gY[ � y cY[`Z��Y[`ZY[km�[`[%s:^ b [ x � y e`[`f+m�[`[%s:^"�TjlZ b [[��!^knZY^���ZYq4m�u!q!c
Ad!^��`^km �`u!q b j��Yd�+�gac j b cY^`p`[ b [`u4m�gYi`q!ikm
cYqAs ��q!p`ZY[ xzy gYuAs:g jl_a^km q b gYZYq!i`^%s64u!q!cYgY_a^`f ^`u!cY[`cjlZ b [ b [`[`e`cY[ .Yq� ^`_aq4m� A d!^��`^km �`u!q b j��Yd!^kmwm�[`u!gY[`_a^kn+�[`fh^`q [km�^`[ � y b q!_a[`u!^"1��q!_aq B
d!^��`^km�[ � � y b q4n_a[`u!^"��u!q��1q!u jlZ b [ b [��1g��1cYq4m ^`u!cY[`c�.Yq!p`gY_ae`p`^`u!g jl_a^\�[`u!cYq!gYi`["lu!q!ca[`cYq!ZY[��1gYi`^U^ .Yq4m�cY[�� m�^`cY[`_�jlu!^"�
$'$'� �4�0m�[]\�[`u!cYqE[`c,+�[ b gYikm �`^.�C[`e`gYikm�[ b [ ��[`u!p`_agYikm�4gYuAs:^��`^kn�C[`e`^km ��g m�[`e`gYu![ b m�[`v`^`u!q![ { cYgYd!u!^ � m�q!p`^`_a^ b [ {zy�1u![`cY^ �Y[� ^"agYuAs:^���[`u!p`_a^km ��g m�[`e`gYu![ b e`^ � x cYgYd4nu!^ � m�q!p`^`_a^ b [�$ y �1u![`cY^ �Y[� ^"��m�[]\�[`u!cYq4m �1[`[ � ZY^`[ � y1ycYgYd!u!^ � m�q!p`^`_a^ b [ { e`^`_aq��1u![`cY^��Y[� ^"��fhZYq!i`^`_a^`["Tu!q!c��[`u!p`_agYi`^kmKu![`q b gYZYq!i`[�[`u�jlZ b [�[��!gYcY[`d!gYi`q b ^km 0 {zy gYu4ns:g jl_�m��gYuAs:^��`^.�C[`e`^kmGu!gY[`_a^.+�[`fh^`^km ��g b g��1[ b cY^��!gYikjl_a^��gYcYq4m�[`p`[`_a^`[ $ y1y b q!_a[`u!^"'��q!_aq gYuAs:^6��[`u!p`_a^km;e`^ �
� �
� y b q!_a[`u!^"�4u![`c b gYZY^ �`^.�C[`e`^ b [ ��[`u!p`[`_a^-jlZ b [ ��gYe`g nu!q4m<m�[]\�[`u!cYq!cUcY[�� m�^`cY[`_�jlu!^ ��gYcYq4m�[`p`_a^km cY^��!gYi`^km�cY^.+TnZY^%s �U^��`q!p`gts cY[�� m�^`cY[`_�jlu!^ ��gYcYq4m�[`p`[`_a^"�
$'$'� - �0m�[`[`u � gYp`ZYq e`[`c��`[`ZY^`[���^ cYq!ZY[]\�^`_ag �`[`u!d!^`gYikm m�[���gY_ac \�^kn q m�s:[`p`[.+�q!ikmj [km�q m�[`u!gYe`_a[`cYq gts:gYu4md!gY_agYp`^.+�^`^%s:[b [ u![ b ^`qAs:^"�4s:^%s:q!g jl_a^��`[`u!d!^`^km�[%s:p`^km \�^`ZY[<m�[`[`u � g np`ZYq �`gYu!^`q b ��^!m�[`u!gYe`_a[`cYq b u!q�[`u jlZ b [�[��!gYcY[`d!gYi`q b g m�C[`c���^ { m�[`[%s3m�[`cY[km�s:[`Z�`m�[`d!gY_agYp`^.+�^`qVu!gYe`_a[`cY^km b [km�[��1np`gYi`^<cY[�� m�^`cY[`_�jlu!^�q b gYZYq!i`[`[ xzy1y \Tj s:^" ��q!_aq u![ b ^`qu!gYe`_a[`cY^km � x1{zy \Tj s:^"�hd!gY_agYu!gYe`_a[`cY^km b u!q [`u jlZ b [[��!gYcY[`d!gYi`q b g mCu![ b ^`q!u!gYe`_a[`cY^km {zy�� nom��fhZYq!i`^`_a^`["zu!q!cd!gY_agYu!gYe`_a[`cY[ { n �CgYu j u!q g g��Yd4jlu!^`[":p`^ b u!g�u![ b ^knq!u!gYe`_a[`cY["�zu![���[`Z��1u��Y_a^`p`q!i`^kmKd!gY_ag b [�u![ b ^`q!u!gYe`_a[`cY[j/+�u4jlZYp`gY_ .Yq� m \�^`ZY[km�[`[`u � gYp`ZYqLe`[`c��`[`ZY^`^km cY[�� m�^`cY[`_�jlum�[`u��1gYi`gY_�m �
$'$'� �4�4[`p`d!q��Y[`u*��[`ZY[ j1��p`gYikm q!u!^A b [ B
cYq b gY_a^km [`p`d!q!cYq4ni`^`_�m�zm�[]\�[`u!cYq!q��`u!q!fhg m�^ ��jl_a^km���cYq!ikm#Xhm�[`[`c��Yu!q*��^ �Y[knu!^km[]\.Yq!i`[km�XYX:m�[`[`c��Yu!q*��^ ��g��!gYi`p`[km b [#XYXYX4m�[`[`c��Yu!q*��^[`p`d!q!cYq!i`^`_a^km�d!g��YZY^`ekjlu �1[`cY[`uAs:p`[km�
AcYq b gY_a^km'[`p`d!q!cYq!i`^`_a^km �1[`cYq4m�[���p`gYi`[ b X3m�[`[��Yu!q4m
g m�[`v`^`u!q!gYi`[ x m�s6�XYX3m�[`[`c��Yu!q4m � �4m�s6�XYXYX0m�[`[`c��Yu!q4m �x m�s6�
BcYq b gY_a^km#[`p`d!q!cYq!i`^`_a^km �1[`cYq4m�[���p`gYi`[ b X0m�[`[��Yu!q4m
g m�[`v`^`u!q!gYi`[ �4m�s6�XYX3m�[`[`c��Yu!q4m � x m�s6�XYXYX0m�[`[`c��Yu!q4m �x m�s6�
� �
m�[`[`c��Yu!q!gYi`^km'c j1��[`q!i`[ b u!q*��^ ��g,+/�4j b jl_a^`[")�e`p`^`u!^km�1[`ZYcY[`p`_aq!i`[���^KXhm�[`[`c��Yu!q c j1��[`q!ikm;[`u![kjlcYgYd!g m � y m�[`[%s:^knm�["zXYX+m�[`[`c��Yu!q � � �Cm�[`[%s:^km�["�XYXYX+m�[`[`c��Yu!q � - $�m�[`[%s:^knm�["�
AcYq b gY_a^kmw[`p`d!q!cYq!i`^`_a^km �1[ .Y^ b p`^%s cY^��!gYikjl_a^�cYq4n
�1gYi`[`[ � y1y1y b q!_a[`u!^" ��q!_aq BcYq b gY_a^km �1[ .Y^ b p`^%s �
� y1y1y b q!_a[`u!^"�cYgYZYg,�CcYgYZYd4mwm`jlu4m b [`[ b �1^`ZYq4mhs3j�u![`c b gYZY^wfh[`_a^ A
b [ BcYq b gY_a^km [`p`d!q!cYq!i`^`_a^ jlZ b [ �1[`cYq4j1��p`[km [`p`d!q4n
�Y[`u*��[`ZY[`cVe`p`^`u!^km �1[`ZYcY[`p`_aq!i`[���^"'u![%s:[ cY^`^��!q4m cY[�� m�^kncY[`_�jlu!^UcYq��1gYi`["�
$'$'� &4�0m�[`^`ZYp`g m�d!^`fh^`q e`q!c��`[`ZY^`[km xzy cY^`_a^`q!ZY^ b q!_a[`u!^km ^`ZYp`g m`nd!^`u!gYi`^km�[%s:p`^km�[��Ypkm ��gYc b g��1^'q!u!^[`_ad!gYu!ZY[`d!^`p`[")
[`_ad!gYu!ZY[`d!^`p`[ cYq4m�[`_aq b ZYgY_a^cYq��1gYi`[ � (
b [`i`[`Z b gYi`^kmcY[�� m�^`cY[`_�jlu!^q b gYZYq!i`[ cY_aZ�� (
gYZYgYu��1gYd!^`e`[ $ y $ x {e`[`p���^`u��1[`i`c jl_aq!i`[ � $ {
[`cY[km�s:[`Z�%m���p`[`q!i`[AgYZYgYu��1gYd!^`e`[km�[ b [!e`[`p���^`u��1[`i`c jl_aq4ni`^kmVm gYu!q!gYi���^ b [`i`[`Z b gYikjl_ s:[`Z���gYikm���q!u!^km [`uIjlZ b [[��!gYcY[`d!gYi`q b g m { cY^`_a^`q!Z b q!_a[`u4m�Te`q!c��`[`ZY^`[km#m`jlu4m b [kn[ b �1^`ZYq4mAs3j u!q��1q!uIcYq![�� b ^`ZYq4m jl_a^km�^`ZYp`g m�d!^`u!gYi`[cY[�� m�^`cY[`_�jlu!^;cYq��1gYi`^km;cY^.+�ZY^%s6��u!^km�^Cd!q!_a^C^��YZYgYi`[KcY[�� m�^kncY[`_�jlu!^UcYq��1gYi`[��
$'$'� $ y �� gYu!cY[km-[��Ypkm {zy �Yu!q*��^km�[ b [ xzy1y f ��p`u!^km�m�[ b �1q!cY^"� g nu!cY^km m�[��Yq!p`u!gYi`^km [`uAs:q!i`^`[!- x �`["�K\�_a^km �1[`ZYcY[`p`_aq4ni`[���^�gYuAs:^��Yu!q*��^km �1[`cYq4m�[`e`p`gYi`[ b m�[`v`^`u!q![ $ �`[<m�[��Yq4np`[`u!^" ��q!_aq gYuAs:^ f ��p`u!^km �1[`cYq4m�[`e`p`gYi`[ b � y x �`["�
� �
gYu!cY[km ��g j��Y_a^`[ [`[`ZY[.+/�![kjlu!q4m $ y1y1y1y m�[`c j1��[`q m�[`[%s:^\�_a^km �1[`ZYcY[`p`_aq!i`[���^"�3gYuAs:^ �Yu!q*��^km cYq4m�[`p`_agY_a[ b m�[`v`^knu!q![ $ {zy m�[`c j1��[`q m�[`[%s:^"/��q!_aq gYuAs:^Uf ��p`u!^km�[%s:p`^km-e`^� x1{ m�[`c j1��[`q m�[`[%s:^"�fhZYq!i`^`_a^`["u!q!c gYuAs:^ �Yu!q*��[^��Y_agYp`[ x1{zy b q!_a[`u cYq��1gYi`[km ��q!_aqIf ��p`[`u!^ e`^ � � {b q!_a[`u4m .Yq!p`gY_G\�_a^kjlu![ b (���u![`c b gYZY^ �Yu!q*��[ b [Cf ��p`[`u!^jlZ b [ � .Y[`p b g m� gYu!cY[kmzu![%s:[acYq��1gYi`[G^ .Yq4mKcY[�� m�^`cY[`_�jlu!^��
$'$'� $'$'�0m�[]\�[`u!cYq [`c,+�[ b gYikm'cY[��1^ b gYikm�[ b [Ce`[`u![ b gYikm�TgYuAs:^CcY[��1^knb ^km b [km�[`c,+�[ b gYi`_a[ b m�[`v`^`u!q![ x c 3 ��^km�cY[km�[`_a[ b [ y �e��w\�gYi`q TgYuAs:^;e`[`u![ b ^km b [km�[`c,+�[ b gYi`_a[ b e`^ � �!c 3 ��^kmcY[km�[`_a[ b [�$' { e�� \�gYi`q ��m�[]\�[`u!cYq4m ��g j��Y_a^`[ �1[`cYq!^ .Yg nZYq4m � xzy c 3 ��^km'cY[km�[`_a[ b [ $ 0 { e��U\�gYi`q ��fhZYq!i`^`_a^`["�u!q!ccY[��1^ b gYi`^km-u![`q b gYZYq!i`[�cYgYd!^U[`Z;d!q!_a^jlZ b [�^ .Yq4m�m�[`gYu4ns:q u![`q b gYZYq!i`^km 2
3
nom� gYuAs:^�cY[��1^ b ^km u!gY[`_a^.+�[`fh^`^km ��g nb g��1[ b cY^��!gYikjl_a^ cYq��1gYi`[`[ $ {zy b q!_a[`u!^" ��q!_aq gYuAs:^e`[`u![ b ^km u!gY[`_a^.+�[`fh^`^km ��g b g��1[ b cY^��!gYikjl_a^ cYq��1gYi`[ e`^� {zy1y b q!_a[`u!^"��u![`c b gYZY^#cY[��1^ b [ b [#e`[`u![ b [;jlZ b [ b [kn[`c,+�[ b q4m m�[]\�[`u!cYq!c cY[�� m�^`cY[`_�jlu!^ cYq��1gYi`^km cY^km�[��!gYi`[ b �u!^km�^Ud!q!_a^`[cY[�� m�^`cY[`_�jlu!^�cYq��1gYi`[��
$'$'� $ x �� ^`u!cY[ [`c,+�[ b gYikm �Y[`_a^km�[ b [�cY[`cY[`e`[`fh^km�m�p`^`d!gYu!gYikm�!s:^kns:q!g jl_a^:m�p`^`d!gYu!^km�[%s:p`^kma^���[`u��CgYi`[ y -Ge�� �Y[� ^"%[`cY[km�s:[`Z�$Ufh[`_a^ �Y[`_a^km�m�p`^`d!gYu!^km-cYq4m�[�� m�q!p`[ b m�[`v`^`u!q![ y x cY[`Z n�Y[`ZY[km�[`[%s:^" ��q!_aq $�fh[`_a^-e`[`fh^km�m�p`^`d!gYu!^km�[%s:p`^km e`^ �y { cY[`Z��Y[`ZY[km�[`[%s:^"�4 ^`u!cY^km .Yq!p`gY_ b �!^kjlu!^KcY[`u![��1^`[ 0 { e���Y[� ^ b [ $ � cY[`Z��Y[`ZY[km�[`[%s:^"�3m�[`i`[.+�u!qEcYqAs ��q!p`ZY^`_agYi`^ b [`Z�1[`cYq!c b ^`ZY[`u!g ^`u!cY[`c b [`[ b �1^`ZY["Au!q!c �Y[`_a^km m�p`^`d!gYu!^kmu![`q b gYZYq!i`[A[`u jlZ b [A[��!gYcY[`d!gYi`q b g m�cts:gY_a^��`u!q b ^��Yfh^`^km
� �
� y � nom� �Y[`_a^kmwm�p`^`d!gYu!^kmwm�[`u!gY[`_a^.+�[`fh^`q [km�^`[ 0 y b q4n_a[`u!^"/��q!_aq cY[`cY[`e`[`fh^km�e`^ � {zy b q!_a[`u!^"�lu![Uu![`q b g nZYq!i`^km m�p`^`d!gYu!gYi`^ jlZ b [ []\�[`u!cYq!q4m ^`u!cY[`c%.Yq!p`gY_ b �!^knjlu![ b �u!q!c6��gYcYq4m�[`p`[`_a^U^ .Yq4m�cY[�� m�^`cY[`_�jlu!^"�
��� ��� �'����
����� '(1*�&
- � $'� $ (2x� x (
3x2 � 0(2x ln 2
�
� (ex � { ( −8x
� �(2(2x + 3)
- � x � $ ( 1
1 + x2
� x ( 1
x
� 0( 1√1− x2
�- � 04� $ (
xx(ln x + 1);x (
xsinx(
cos x ln x +1
xsin x
)
;0(
2xlnx−1 ln x.- � �1�45◦
�- � { �
30◦�
- � �4�0�
- � - � $ (3�9� x ( −2
� −12� 0(
1� −1
�- � �4� [�(
3x2∆x + 3x(∆x)2 + (∆x)3 �
i�( √x + ∆x−√x
�!� ( −2∆x
(x− 1)(x + ∆x− 1)
�- � &4� $ (
3x2 − 4x� x (
5x4 − 3x2 + 6x�
0(2x5 − 3x3 + 2
� � (x3 − x
�
{ (2x + 2x ln 2 +
3
2√
x
� �( 1√x− 2
3 3√
x− 1
�
-'(3√
x− 4 3√
x + 5 4√
x� �(
53√
x2 + 74√
x3 − 85√
x3 �
� �
� �
&( − 1
x2− 2
x3+
3
x4
� $ y ( − 3√x5− 5
4√
x9+
76√
x13
�
$'$ (2 cosx− 6 sin x + 1
� $ x ( 4
cos2 x− 5
sin2 x
�
$ 0(4x ln 4 + 3 · 2x ln 2
� $ � ( 6
x+ 7 ex �
$ { ( 7√1− x2
� $ �( 13
1 + x2
�
$ -'((4x + 1) sin x + cos x(2x2 + x)
�
$ �((9x2 + 2) cos x− (3x3 + 2x) sin x
�
$�&((4x3 + 6x2) lnx + x3 + 2x2 �
xzy (ex(sin x + cos x)
� x $ (5x ln 5 cosx− 5x sin x
�
x1x (3x ln 3 lnx +
3x
x
�
x 0( 1
xsin x + cos x ln x
� x � (cos x
�
x1{ (arcsin x+
x√1− x2
� x �( 1
2√
xlog2 x+
1√x ln 2
�
x -'(3x(3x2 + x3 ln 3)
�
x �((cos x− sin x) 2x + 2x ln 2(sin x + cos x)
�
x &( − 3
(x− 2)2
� 0 y ( −x2 − 6x2 + 1
(x2 + 1)2
�
04$ ( x2 + 1
(1− x2)2
� 0 x ( 2x− x2 ln 3
3x
�
0 0( 2x(ln 2 sin x− cos x)
sin2 x
� 0 � ( 3x2 + 1
x2
�
0 { ( −x3 + 3x2 + 2x− 3
ex
� 0 �( cos x− sin x
ex
�
� 1
0 -'( − 2
(sin x− cos x)2
� 0 �( 2ex
(ex + 1)2
�
- � $ y � $ ( 2
x+ ex(x2 + 4x + 2)− 4
(x− 1)2
�
x (3 ln 5 · 5x + 4x3 sin x + x4 cos x− sin x + cos x
ex
�
0(4 cosx + 7x · x6(7 + ln 7 · x) +
1
cos2 x
�
� ( −5 sin x + 3x(x3 ln 3 + 3x2 + 2x ln 3 + 2)+sin x− 2x cos x
2√
x sin2 x
�
{ ( 7√1−x2
+ cos x ln x +1
xsin x−ex(sin x+cos x)
cos2 x
�
�( − 2√1−x2
−3x2 cos x+x3 sin x−x cos x−2 sin x
x3
�
-'( 5
1 + x2− cos2 x + sin2 x− 4x(ln 4 · x− 1)
x2
�
�( − 3
1 + x2− ex
(1
x+ ln x
)
− cos x− sin x
ex
�
- � $'$'� $ (39� x (
5� 0(
31
2
� � ( 1
48
� { (73�
�( −1� -'( 1
4� �( 1
4� &(
4� $ y ( −1
2�
$'$ ( 1
4√
2
� $ x (0� $ 0( 1
2
� $ � ( 1√2
�
- � $ x � $ (4(x3 + 2x2 + 3)3(3x2 + 4x)
�
x (3(5x4 − 3x3 + 2x)2(20x3 − 9x2 + 2)
�
0(2 cos 2x
� � ( − sinx
5
� { ( 3
3x− 4
�
� �
�( 18x
9x2 + 5� -'( 1√
2x + 3
�
�( x− 2√x2 − 4x + 1
� &(6 sin 8x sin 4x
�
$ y ( −2 sin 4x� $'$ (
ctg x� $ x ( −1
xsin(ln x)
�
$ 0(3 ln 3 · x2 · 3x2+1 � $ � (
3 ln 4x√
x6 − 1· 4
√x6−1 �
$ { (ln 4 · cos x · 4sinx � $ �( − ln 5 · sin x · 5cos x �
$ -'(3x2 cos(x3 + 1) +
4
x
� $ �( 3x2
x3 − 4− 3e3x �
$�&(4 ln 2
2arcsin 4x
√1− 16x2
− 2 ln 33arccos 2x
√1− 4x2
�
xzy ( 3 ln 4 · 4arctg 3x − 2 ln 5 · 5arcctg 2x
1 + 9x2
�
x $ ( 1
cos x
� x1x ( − 1√x sin 2
√x
�
x 0(3x2e3x(1 + x)
� x � (x · 2
√x(
2 +1
2ln 2
√x)
�
x1{ ( 1
2√
x− x2
� x �( − 1
2x(x− 1)�
x -'(2ex2
(x cos 4x− 2 sin 4x)�
x �(4x3−5
(
3 ln 4 · x2 · arcsin 3x− 3√1− 9x2
)
�
x &( sin x√sin3 2x
� 0 y x + a
x2 + a2
�
- � $ 04� $ (6x− 4
� x (20x3 + 36x2 − 4
�
0( − 1√
(1− x2)3
� � ( 3x4 + 36x
4√
(3 + x3)3
�
{ (72x7 + 90x4 + 18x
� �(48(2x− 3)2 �
-'(12x2 ln x + 7x2 � �(
ex · x3(x2 + 10x + 20)�
� �
&( x
(1− x2)3
2
� $ y ( − 2x
(1 + x2)2
� $'$ (9e3x+4 �
$ x (36 ln2 2 · 26x+7 � $ 0( −3x4 − 25
(x3 + 5x)2
�
$ � ( − sin x + 75e5x � $ { (2 cos 2x
� $ �(2ex cos x
�- � $ �1� $ (
ax(ln a)n � x ((m ln a)namx �
0(sin
(
x +πn
2
)
� � (cos
(
x +πn
2
)
�
{ (an sin
(
ax +πn
2
)
� �(an cos
(
ax +πn
2
)
�
-'(2ne2x � �(
(3 ln 2)n23x �- � $ { � $ (
(4x + 3)dx� x ( ( 5
2√
x− 1
3x 3√
x+ 1
)
dx�
0( (
3 cos 3x +1
x
)
dx� � ( (1
x+ 2x+1 ln 2
)
dx�
{ ((4x3 − 6x2 + 6x− 2)dx
�
�(3 sin2 x cos xdx
� -'(2xe4x(1 + 2x)dx
�
�((
− 3√1− 9x2
+6
1 + 36x2
)
dx� &( 1
ex − 1dx
�
$ y ( − tg x dx� $'$ (
x3(
4 log3 x +1
ln 3
)
dx�
$ x ( −5(1 + 2x)
(x + x2)2dx
� $ 0( 4x
3 3
√
(2x2 + 3)2dx
�
$ � ( (
cos x · 3sinx ln 3− 3 ln 2 sin 3x · 2cos 3x)
dx�
$ { ( − 2
(x− 1)2dx
� $ �( x4 + 6x2 + 6x
(x2 + 2)2dx�
- � $ �4� $ ( ≈ 2, 02� x ( ≈ 3, 03
� 0( ≈ 1, 11� � ( ≈ 0, 97
�
{ ( ≈ 3, 06� �( ≈ 2, 77
� -'( ≈ 1, 2�
- � $ - � [�((MR) = 200
� i�(∆(TR) = 950
�
� �
- � $ �4� [�((TR) = 80Q−Q2; (MR) = 80− 2Q
�
i�((MR) = 40
� � (∆(TR) = 39
�- � $�&4� [�( −4Q + 6
� i�( 250(6 + Q)√
(3 + Q)3
�
� ( 600− 16Q
3 3
√
(200− 4Q)2
�
- � xzy �(MR) = 0, 5
�- � x $'� [�(
(TC) = 500 + 3Q; (MC) = 3�
i�((MC) = 3
� � (∆(TC) = 9
�- � x1x � [�(
(TC) = 3Q2 + 4Q + 15; (MC) = 6Q + 4;i�( ≈ −248
�
� ( +Tj�m�d!^fhp`_a^`_agYi`[`[ −236��m���p`[`q!i`[`[
12�
- � x 04� [�((TC) = 78 + 4Q; (MC) = 4
�
i�( �$ (12 x ( −16
�- � x �1�
Q = 2�
- � x1{ � [�(K(Q) = 0, 03Q−2+
300
Q
�K ′(Q) = 0, 06Q−2
�
i�( �$ (K(50) = 5, 5; K ′(50) = 1
�
x (K(100) = 4; K ′(100) = 4
�� (
Q = 100�
- � x �4�Π = −2Q2 + 40Q− 52
�[�(
Q1 = 5; Q2 = 35� i�(
Q = 10�
- � x - �Q = 111
�- � x �4�
Πmax = 103�
- � x &4� [�(Π′(Q) = 296− 2Q
� i�( ≈ 432�
� ( +Tj�m�d!^fhp`_a^`_agYi`[`[428
�- � 0 y � [�(
Q = 20�Ii�(
Q0 = 12, Π(6) = 226�
- � 04$'�(MPC) = 0, 9; (MPS) = 0, 1
�- � 0 x �
(MPS) = 0, 7; (MPC) = 0, 3�
� �
- � 0 04� [�(0, 3158
� i�(0, 3158
�
- � 0 �1� [�( 2
23� i�(
4�
- � 0 { � [�( 1
2
� i�( 1
2
�
- � 0 �4� [�( 11
30� i�( 33
30%�
- � 0 - � [�( 1
29� i�(
58 %�
- � 0 �4� 72
89
�- � 0'&4� [�(
1, 425� i�(
1, 2�
- � � y � ≈ 0, 682�
- � �1$'� [�( ≈ 0, 952� i�(
9, 52 %�
- � � x �P = 9
�- � � 04� [�(
P = 30− 3
1000Q� i�(
P = 15�
- � � �1�P = 475
�- � � { � [�(
P = 4500− 4Q� i�(
P = 2250� � (
P = 2300�
- � � �4� [�(P = 6, 5− 0, 002Q
�Ii�(3, 4
�
- � � - � [�(P = 30− Q
29
�Ii�(P = 15; (TR) = 6525
�
- � � �4� $ ( +�u b [ b ^`[ (−∞; +∞)�1jl[`_ag b ��^ �
x (�e`_agYi`[ b ^`[ (−∞; +∞)�1jl[`_ag b ��^ �
0( +�u b [ b ^`[ (−∞;−3) b [ (3; +∞)�1jl[`_ag b gYi���^"�e`_ag n
i`[ b ^`[ (−3; 3)�1jl[`_ag b ��^ �
� ( +�u b [ b ^`[ (−∞; 2) b [ (5; +∞)�1jl[`_ag b gYi���^"�e`_ag n
i`[ b ^`[ (2; 5)�1jl[`_ag b ��^ �
{ ( +�u b [ b ^`[ (5; +∞)�1jl[`_ag b ��^" e`_agYi`[ b ^`[ (−∞; 5)�1jl[`_ag b ��^ �
� �
�(�e`_agYi`[ b ^`[(
−∞;−3
2
)
b [(
− 3
2; +∞
) �1jl[`_ag b g ni���^ �
-'( +�u b [ b ^`[ (
0;1
2
) �1jl[`_ag b ��^"+e`_agYi`[ b ^`[(1
2; +∞
)
�1jl[`_ag b ��^ ��( +�u b [ b ^`[ (−2; 0)
�1jl[`_ag b ��^" e`_agYi`[ b ^`[ (0; 2)�1j�n
[`_ag b ��^"�- � �&4� $ (
x = −2[`u!^kmwcY^`ZY^`c jlcY^km�\�gYu!d!^`_a^"
y(−2) = −9�
x (x = 2
[`u!^km�cY[�� m�^`c jlcY^km�\�gYu!d!^`_a^"y(2) = 4
�
0(x = −3
[`u!^km�cY[�� m�^`c jlcY^km�\�gYu!d!^`_a^"y(−3) = 86
x = 2
[`u!^km�cY^`ZY^`c jlcY^km�\�gYu!d!^`_a^"y(2) = −39
�
� (x = 0 b [ x = 2
cY^`ZY^`c jlcY^km \�gYu!d!^`_agYi`^`["y(0) =
y(2) = 1x = 1
[`u!^kmGcY[�� m�^`c jlcY^kmG\�gYu!d!^`_a^"y(1) =
2�
{ (x = 3
[`u!^km�cY^`ZY^`c jlcY^km�\�gYu!d!^`_a^"y(3) = −6
3
4
�
�(x = −2 b [ x = 2
cY^`ZY^`c jlcY^km�\�gYu!d!^`_agYi`^`["y(−2) =
y(2) = −4
x = 0[`u!^km cY[�� m�^`c jlcY^km \�gYu!d!^`_a^"
y(0) = 0�
-'(x = 0
[`u!^km�cY^`ZY^`c jlcY^km�\�gYu!d!^`_a^"y(0) = 1
�
�(x =
1
9[`u!^km�cY^`ZY^`c jlcY^km�\�gYu!d!^`_a^"
y(1
9
)
= −1
3�
&(x = 0
[`u!^km�cY[�� m�^`c jlcY^km�\�gYu!d!^`_a^"y(0) = 1
�
$ y (U[`u![[��Ypkm�g�� m�d!u!gYc jlcY^km�\�gYu!d!^`_agYi`^ �$'$ (
x = 6[`u!^kmUcY^`ZY^`c jlcY^km\�gYu!d!^`_a^"
y(6) = 12x = 0
[`u!^km�cY[�� m�^`c jlcY^kmw\�gYu!d!^`_a^"y(0) = 0
�
$ x (x = 0
[`u!^km�cY^`ZY^`c jlcY^km�\�gYu!d!^`_a^"y(0) = 1
�
$ 0(x = 1
[`u!^km�cY^`ZY^`c jlcY^km�\�gYu!d!^`_a^"y(1) = e
�
$ � (x = −2
[`u!^kmwcY[�� m�^`c jlcY^km�\�gYu!d!^`_a^"y(1) =
1
e
�
� �
$ { (x = 2
[`u!^km�cY^`ZY^`c jlcY^km�\�gYu!d!^`_a^"y(2) = 1 + ln 2
�
$ �(x = 1
[`u!^km�cY^`ZY^`c jlcY^km�\�gYu!d!^`_a^"y(1) = 0
x = e−2
[`u!^km�cY[�� m�^`c jlcY^kmw\�gYu!d!^`_a^"y(e−2) =
4
e2
�- � {zy � $ (
ymin(1) = −1, ymax(3) = 3�
x (ymin = y(−2) = y(2) = −13,
ymax = y(−3) = y(3) = 12�
0(ymin(0) = 0, ymax(4) = 6
�
� (ymin(3) = 0, ymax(0) = 9
�- � { $'� $ (U\�^`u!^ [`cYq�+�ZYg��Y^`_a^`[
(−∞; 2)�1jl[`_ag b ��^ b [ � [.+�ZYg n
�Y^`_a^`[(2; +∞)
�1jl[`_ag b ��^"'�1[ b [��4jlZYp`^kmC\�gYu!d!^`_a^`[(2;−13)
�
x (U\�^`u!^ [`cYq�+�ZYg��Y^`_a^`[(0; +∞)
�1jl[`_ag b ��^ b [ � [.+�ZYg n�Y^`_a^`[
(−∞; 0)�1jl[`_ag b ��^""�1[ b [��4jlZYp`^kmC\�gYu!d!^`_a^`[
(0; 0)�
0(U\�^`u!^ [`cYq�+�ZYg��Y^`_a^`[(−∞; 1)
�1jl[`_ag b ��^ b [ � [.+�ZYg n�Y^`_a^`[
(1; +∞)�1jl[`_ag b ��^"'�1[ b [��4jlZYp`^kmC\�gYu!d!^`_a^`[
(1;−1)�
� (U\�^`u!^w[`cYq�+�ZYg��Y^`_a^`[(−5; 2)
�1jl[`_ag b ��^ b [ � [.+�ZYg��Y^kn_a^`[
(−∞; 5) b [ (2; +∞)�1jl[`_ag b gYi���^" �1[ b [��4jlZ n
p`^kmU\�gYu!d!^`_agYi`^`[(−5;−1622) b [ (2;−173)
�
{ (U\�^`u!^G[`cYq�+�ZYg��Y^`_a^`[(−∞;−2)
�1jl[`_ag b ��^ b [ � [.+�ZYg n�Y^`_a^`[
(−2; +∞)�1jl[`_ag b ��^"%�1[ b [��4jlZYp`^km \�gYu!d!^kn
_a^`[(−2;−2e−2)
�
�(U\�^`u!^-[`cYq�+�ZYg��Y^`_a^`[ (
0;1
2
) �1jl[`_ag b ��^ b [ � [.+�ZYg��Y^kn
_a^`[ (1
2; +∞
) �1jl[`_ag b ��^"4�1[ b [��4jlZYp`^km\�gYu!d!^`_a^`[(1
2;1
2− ln 2
)
�
� �
-'(U\�^`u!^G[`cYq�+�ZYg��Y^`_a^`[(−1; +∞)
�1jl[`_ag b ��^ b [ � [.+�ZYg n�Y^`_a^`[
(−∞;−1)�1jl[`_ag b ��^"��1[ b [��4jlZYp`^km \�gYu!d!^kn
_a^[`u [��Ypkm ��(U\�^`u!^[`cYq�+�ZYg��Y^`_a^`[
(−∞; +∞)�1jl[`_ag b ��^ �
&(U\�^`u!^w[`cYq�+�ZYg��Y^`_a^`[(2; +∞)
�1jl[`_ag b ��^" �1[ b [��4jlZ np`^kmU\�gYu!d!^`_a^U[`u [��Ypkm �
$ y (\�^`u!^w[`cYq�+�ZYg��Y^`_a^`[(−1; 0)
�1jl[`_ag b ��^ b [ � [.+�ZYg��Y^kn_a^`[
(−∞;−1) b [ (0; +∞)�1jl[`_ag b gYi���^"�1[ b [��4j�n
ZYp`^kmU\�gYu!d!^`_agYi`^`[(−1; 1) b [ (0; 1)
�- � {1x � $ (
x = −5, y = 0� x (
x = 3, x = 7, y = 0�
0(x = 3, x = −3, y = 0
�
� (x = 4, x = −4, y = 0
�
{ (y = 3
� �(x = −3, y = 0
�
-'(x = 2, x = −3, y = 1
� �(x = 2, y =
1
2�
&(x = −4
5, y = −3
5
� $ y (y = −1
�
$'$ (x = 0, y = x
� $ x (x = −1, y = 2x
�
$ 0(x = 2, y = x− 3
�
$ � (x = −2, x = 1, y = x− 1
�
$ { (x =
√3, x = −
√3, y = 5x
� $ �(y = 2x
�
$ -'(x = 1, x = −1
�
$ �(U[km�^`c��`d!q!d!gYi`^U[`u [��Ypkm�
� �
- � { 04� $ (
x (
0(
� �
� (
{ (
� 1�(
-'(
� �
�(
&(
� �
$ y (
$'$ (
$ x (
� �
������� '(1*�&
�4� $'�S = a
√b2 − a2 �
�4� x �(TR) = P Q
��4� 04� $ ( −8
� x (ln 5
� 0(ln 4
��4� �1� $ (�cts:gY_a^#m�^`i`u!d/.Yg��
x (�cts:gY_a^#m�^`i`u!d/.YgCe`q!q!u b ^`ZY[`dAs:['m�[%s:[`p`^km �1[`u b [ �0(�cts:gY_a^#m�^`i`u!d/.Yg���1[`u b [ y = x
\�u� ^km�[ �� (U\�u!g�Gu!q!cY_a^km�u![ b ^kj�m�^`[ � b [ fhgYZYd!u!^ e`q!q!u b ^knZY[`dAs:[#m�[%s:[`p`g���^� m�[.+/�!p`u!^km �1[`u!g���g ( �
{ (�^`c0\�u!^km�[ b [�\�u!g \�^`u!^kmG\�gYu!d!^`_agYi`^"zu!q!cY_a^kmafhgYZYd4nu!^�e`q!q!u b ^`ZY[`dAs:[Um�[%s:[`p`gY[ b [�u![ b ^kj�m�^ { nr^kmwd!q4n_a^`[ �
�(�cts:gY_a^ m�^`i`u!d/.Yg� �1[`u b [ ^`c \�u!^km�[ b [ \�u!g \�^`u!^knm�["lu!q!cY_a^kmwfhgYZYd!u!^e`q!q!u b ^`ZY[`dAs:[m�[%s:[`p`g���^`[ b [u![ b ^kj�m�^�0�nr^kmUd!q!_a^`[ �
-'(�cts:gY_a^:m�^`i`u!d/.Yg� �1[`u b [A^`c0\�u!g \�^`u!^km�["�u!q!cY_a^kmGfhg nZYd!u!^e`q!q!u b ^`ZY[`dAs:['m�[%s:[`p`gY[ b [u![ b ^kj�m�^`[ { �
�(�cts:gY_a^#m�[`i`u!d/.Yg���1[`u b [ y = ±1
2x\�u� gYgYi`^km�["�
�4� { � $ (3� x (
5� 0(
2√
6�
� (0� { (
0, 6� �( 3−
√6
6�
�4� �4�4[`u [`u!^kmj�\.Yp`gYd!^"��4� - �4[`u [`u!^kmj�\.Yp`gYd!^"��4� �4� $ (
z′x = 2xy3 − 4y2 + 21x2 z′y = 3x2y2 − 8xy�
x (z′x = 12x2y2 − 2y
z′y = 8x3y2− 2x− 6y
�
0(z′x = 10x + 7y4 − 14
z′y = 28xy3 + 15
�
� (z′x = 2xy3 − 5x4 z′y = 3x2y2 �
� �
{ (z′x =
2
x(y − 3) ln x
z′y = ln x2 �
�(z′x =
−2xy2
(3x2 − y2)2
z′y =
2x2y
(3x2 − y2)2
�
-'(z′x = 8xy3e4x2
z′y = 3y2e4x2 �
�(z′x =
2xy(7y − 20)
(4x2 + 7y2)2
z′y =20x2 − 14x2y − 35y2
(4x2 + 7y2)2
�
&(z′x =
16x2 − 64x− 20y3
(4x− 8)2
z′y =
15y2
4x− 8�
$ y (z′x = − 40xy2
(5x2 − 18)2
z′y =
8y
5x2 − 18�
$'$ (z′x =
x2 − y3
4x2y
z′y =
2y3 − x2
4xy2
�
$ x (z′x =
3y(3y3 − 4x2)
(4x2 + 3y3)2
z′y =
6x(2x2 − 3y3)
(4x2 + 3y3)2
�
$ 0(z′x = 40x(5x2 + y3)3 z′y = 12y2(5x2 + y3)3 �
$ � (z′x = 24x(4x2 − 7y2 + 5)2 z′y = −42y(4x2 − 7y2 + 5)2 �
$ { (z′x =
4x√
4x2 + 7y2 + 11
z′y =21y2
2√
4x2 + 7y2 + 11
�
$ �(z′x = 8xe5y z′y = 20x2e5y �
$ -'(z′x = −30xy3e5x2
+ 4y
z′y = −9y2e5x2
+ 4x�
$ �(z′x =
ex
y
y
z′y = − x
y2e
x
y�
$�&(z′x = (10x + 8y)e5x2+7y3−8xy+15 z′y = (21y2 + 8x)e5x2+7y3−8xy+15 �
� �
xzy (z′x = (y2 + 5y)3xy2+5xy+7 ln 3
z′y = (2xy + 5x)3xy2+5xy+7 ln 3
�
x $ (z′x =
1
x
z′y = − 2y
y2 + 5
�
x1x (z′x =
4y
x(7x2 + 2y)
z′y =−2
7x2 + 2y�
x 0(z′x =
y2
x ln 5
z′y = 2y log5 x�
x � (z′x = 21x2 log2 4
z′y =
7x3
y ln 2�
x1{ (z′x = yxy−1 z′y = xy ln x
�
x �(z′x = − ln y
x ln2 x
z′y =
1
y ln x
��4� &4� $ (
21 dx− 14 dy�
x (10 dx + 6 dy
�
0( 1
5dx +
2
5dy�
� (21 dx− 2 dy
�
{ (120e53 dx + 48e53 dy
�
�(12 dx + 24 ln 2 dy
�
-'(15e16 dx + 40e16 dy
�
�( 3
4 ln 2dx + 16 ln 4 dy
��4� $ y � $ (
z′′xx = 6y4 z′′xy = 24xy3 + 4
z′′yy = 36x2y2 �
x (z′′xx = −22
z′′xy = 21y2 z′′yy = 42xy
�
0(z′′xx = 0
z′′xy =
4
y
z′′yy = −4x
y2
�
� (z′′xx =
2(y3 − x2)
(x2 + y3)2
z′′xy = − 6xy2
(x2 + y3)2
z′′yy =3y(2x2 − y3)
(x2 + y3)2
�
� �
{ (z′′xx =
6y3(x2 + 3)
(x2 − 9)3
z′′xy = − 6xy2
(x2 − 9)2
z′′yy =6y
x2 − 9�
�(z′′xx = 4ex ln y
z′′xy =
4ex
y
z′′yy = −4ex
y2
�
-'(z′′xx = −5
2x2 + 18xy + 81y2
(x2 + 9xy)2
z′′xy = − 45x2
(x2 + 9xy)2
z′′yy = − 405x2
(x2 + 9xy)2
�
�(z′′xx = 128y2e8xy z′′xy = 16e8xy + 128y2e8xy z′′yy = 128x2e8xy �
&(z′′xx = 8(x2 − 9y3)3 − 48x2(x2 − 9y3)2 z′′xy = −648y2(x2 − 9y3)2 z′′yy = −216y(x2 − 9y3)3 + 8748y4(x2 − 9y3)2 �
$ y (z′′xx =
6x
2 + y3
z′′xy = − 9x2y2
(2 + y3)2
z′′yy =12x3y(y3 − 1)
(2 + y3)2
�
�4� $'$'� $ ( 40e2t
7t6
(
1− 3
t
)
�
x ( 1
t6 +√
t
(
6t5 +1
2√
t
)
�
0( et
5t4 + 7et ln t
(
10t + 7 ln t +7
t
)
�
� (t(880t9 + 176t2 − 288)
�
{ (105t2 + 24t− 75
�
�(4(et +
4√
t)3(
et +1
44√
t3
)
�
� �
-'( −2
√
t2 − 1
t2 + 1
t
(t2 − 1)2
�
�(8t + 4 ln 4
�
�4� $ x � $ ( 2x− 2y − 30x2 + 11
2(x− 7y + 6)�
x ( 6x2 + 9y2 − 11x− y
x− 18xy + 6�
0( 2x− x2y − 2y2
3x2y2 + 6y3 + x3 − 2xy − 2�
� ( 7y2 − 30x
2y(6y − 7x)
�
{ ( −y
ey + x
�
�( −y
x(ln y + 7 · 2x ln 2)
�
-'( 4ex + 18xy − y
x(1− 9x)�
�( 3− 21e3x(3x− 4y)
12y2(3x− 4y) + 4y ln 4�
&( 1− 16xex2+y2
(x + y)
24y2ex2+y2(x + y)− 1
�
$ y ( y 5y ln 2
3− xy5y ln 2 ln 5�
�4� $ { � [�( ≈ 0, 184� i�( ≈ 0, 327
�
� ( ≈ 0, 4� b ( ��gYcYfh^`u b gYi`[ ≈ 6, 13 %
nr^%s6��4� $ �4� [�( ≈ 0, 128
� i�( ≈ 0, 15�
� ( ≈ 0, 5� b (6�1[`^.+�u b gYi`[ ≈ 1, 5 %
nr^%s6��4� $ - � [�( ≈ 0, 26
� i�( ≈ 0, 17�
� ( ≈ 0, 09� b ( ��gYcYfh^`u b gYi`[ ≈ 1, 8 %
nr^%s6�
� �
�4� $ �4� [�(1, 2
� i�(4, 5
� � (3, 75
��4� $�&4� [�(
81� i�( ≈ 6, 04
�
� ( ≈ 0, 075� ��g m�[`i`[`cY^km�^�^.+�q!e`p`[`ZYd!^km6�1[`ZYd!q!_agYi`[`[
34K+
L1
3 = 165�Ke`[��`^`d![`_a^!jlZ b [��1[`^.+�[`u b q4m ≈ 4, 8 %
n^%s6�
�4� xzy � [�(
i�(
� (
�4� x $'� [�(P1 = 22 $, P2 = 73 $
�
i�(P1 = 28 $, P2 = 38 $
��4� x1x �
L = 30, K = 25�
� �
�4� x 04�L = 25, K = 50
��4� x �1�
L = 160, K = 40�
�4� x1{ �Q1 = 95, P1 = 210
�
Q2 = 105, P2 = 305�
�4� x �4�Q1 = 150, Q2 = 55
�
P1 = 170, P2 = 130�
Πmax = 28490�
�4� x - �Q1 = 110, Q2 = 170
�
P1 = 460, P2 = 360�
�4� x �4� [�(K = 40, L = 80, 128000
�
i�(L = 50, K = 75
�
� (L = 120, K = 20
�
b ( L = 37, 5, K = 37, 5�
��� '(1*�&
&4� $'� $ ( 2
3x√
x + C�
x ( 3
5x
5
3 +3
2x
2
3 + C�
0(x3 − 3
2x
4
3 + 4 ln |x|+ C�
� (x2 +
5
6x
6
5 + 3 ln |x| − 8√
x + 8x + C�
{ (2√
x + ln |x|+ C�
�( − 1
x2+ 4
√x +
x2
2+ C
�
-'(x− 8
3x
3
2 − 2x2 + C�
� 1
�(x− x3 +
3x5
5+
x7
7+ C
�
&( x4
4− 4
7x
7
2 +x3
3+ C
�
$ y (x + 3x2 +
11x3
3+
3x4
2+ C
�
$'$ (ln |x| − 2
x− 3
2x2+ C
�
$ x ( −1
x− 2 ln |x|+ x + C
�
$ 0( 8
5x
5
4 − 36
13x
13
12 +16
3x
3
4 + C�
$ � ( 4√
2
5x
5
4 + C�
$ { ( 3
5x
5
3 + C�
$ �( 4
7x
7
4 − 4
3x
3
4 + C�
$ -'(ln |x| − 1
4x5+ C
�
$ �(x− arctg x + C
�
$�&( −x− 1
2ln
∣
∣
∣
∣
x + 1
x− 1
∣
∣
∣
∣
+ C�
xzy ( 9x
ln 9+ 2
12x
ln 12+
16x
ln 16+ C
�
x $ ( − 4
5x ln 5− 1
25 · 2x ln 2+ C
�
x1x ( 1
2e2x − ex + x + C
&4� 04� $ ( 2
9(3x− 2)
3
2 + C�
� �
x ( − 1
4(2x− 5)2+ C
�
0( 1
44(4x− 3)11 + C
�
� ( −1
6(1− 5x)
6
5 + C�
{ ( −3
4(1− 2x)
2
3 + C�
�( 1
5e5x + C
�
-'( 1
4 ln 3· 34x+5 + C
�
�( −e−x − 1
2e−2x + C
�
&( 1
2e2x − 2x− 1
2e−2x + C
�
$ y ( 1
4ln4 x + C
�
$'$ ( −e1
x + C�
$ x ( −1
2e−x2
+ C�
$ 0( − 1
3 ln3 x+ C
�
$ � (ln(2 + ex) + C
�
$ { (x− ln(1 + ex) + C
�
$ �( 3
2ln(1 + e2x)− x + C
�
$ -'( 1
3ex3
+ C�
$ �( 1
36(3x2 − 1)6 + C
�
� �
$�&( − 2−x4
4 ln 2+ C
�
xzy ( 2
3(1− ln x)
3
2 + C�
x $ ( −1
4ln |3− 2x2|+ C
�
x1x ( − 1
2(1 + x2)+ C
�
x 0( 1
4(1 + x3)
4
3 + C�
x � ( −√
1− x2 + C�
x1{ (2e√
x + C�
x �( − 4
x− 2− 11
2(x− 2)2+ C
�
x -'( 4
3(1 +
√x)
3
2 + C�
x �(ln |x| − ln 2 + C
�
x &( 1
3ln
∣
∣
∣
∣
x− 1
x + 2
∣
∣
∣
∣
+ C�
0 y ( −x− 2 ln |x− 1|+ C�
04$ ( (x + 1)2
2− 2(x + 1) + ln |x + 1|+ C
�
0 x (1− x− 2 ln |1− x|+ C
�
0 0( −(1− x)11
11+
(1− x)12
12+ C
�
0 � ( −8 + 30x
375(2− 5x)
3
2 + C�
&4� �1� $ (ex(x− 1) + C
�
x ( x · 3x
ln 3− 3x
ln2 3+ C
�
� �
0(x(ln x− 1) + C
�
� ( xn+1
n + 1ln x− xn+1
(n + 1)2+ C
�
{ ( −1
x(ln x + 1)2 + C
�
�(x ln(x2 + 1)− 2x + 2 arctg x + C
�
-'( 2
3x
3
2
(
ln2 x− 4
3ln x +
8
9
)
+ C�
�( −x(e−x + 1) + C�
&( −e−2x
2
(
x2 + x +1
2
)
+ C�
$ y (x(ln2 |x| − 2 ln |x|+ 2) + C
�
$'$ (2√
x(ln |x| − 2) + C�
$ x (2e√
x(√
x− 1) + C�
$ 0(ln x(ln(ln x)− 1) + C
�&4� { �
3980 $�
&4� �4�19100 $
�&4� - �
3500 $�
&4� �4�8880 $
�&4� &4�
323, 08 $�
&4� $ y �0, 6Y + 0, 4
√Y + 1
�&4� $'$'�
0, 3Y +3
20Y
3√
Y + 16, 96�
&4� $ x �20e0,3Q + 10
�&4� $ 04� [�(
(TR) = 100Q− 2Q2, P = fD(Q) = 100− 2Q�
i�((TR) = 80Q− 3Q2, P = fD(Q) = 80− 3Q
�
� ((TR) = 24(
√
Q + 9− 3), P =24√
Q + 9 + 3
�
� �
b ( (TR) = 6(√
Q + 4− 2), P =6√
Q + 4 + 2
�
&4� $ �1� $ (55
2
3� x ( 2
3(2√
2− 1)�
0(42
2
3� � ( 3
4�
{ (2(√
3−√
2)� �(
4(e− 1)�
-'(21, 25
� �(ln 4 + 2
�
&(18� $ y (
2�
$'$ ( 11
3� $ x (
e−√
e�
$ 0(ln 5
� $ � (2(ln 3 + 2)
�
$ { (2 ln 2 +
1
2� $ �(
2, 8(3)�
$ -'(1� $ �(
e2 + 1�
$�&(e− 2
� xzy (e− 2
�&4� $ { �
7�
&4� $ �4�36�
&4� $ - � [�(ln 5
� i�( 54
ln 3� � ( 2
9� b ( 120
ln 5�
&4� $ �4�e +
1
e− 2
�&4� $�&4�
15, 5�
&4� xzy � 4
3�
&4� x $'�40 $
�&4� x1x �
24000 $�
&4� x 04� ≈ 1, 17 $�
&4� x �1�1600 $
�&4� x1{ �
1210 $�
&4� x �4� ≈ 811 $�
� �
&4� x - �200 ln
20
3− 170 $
�&4� x �4�
20 $�
&4� x &4�30000(e0,1 − e−0,7) $
�&4� 0 y � 5
7(e0,7 − e1,4) · e0,42 $
�
&4� 04$'� 25
7(1− e0,35) · 105 $
�&4� 0 x �
100000 $�
&4� 0 04� [�(104000 $
� i�( ≈ 4, 5\�gY_a^"�
&4� 0 �1� [�( ≈ 98280 $�Ii�( ≈ 5, 63
\�gY_a^"�&4� 0 { � [�( A
1 + α(T α+1
2 − T α+11 )
�
i�( A
α(eαT2 − eαT1)
�
&4� 0 �4� [�( 5
12(e−1,2 − e−2,04) · 105 $
�
i�( 5
12(e−0,6 − e−1,44) · 105 $
�
� ( 5
12e−1,2 · 105 $
�
&4� 0 - �41000 $
�&4� 0 �4� [�(
128 $� i�( ≈ 15 $
�&4� 0'&4� [�(
240 $� i�(
144 $�
&4� � y � [�(100 $
� i�(100 $
�&4� �1$'� [�( ≈ 82, 7 $
� i�( ≈ 66, 7 $�
&4� � x � $ ( 1
2� x ( 1
18� 0( 1
ln 3�
� ( �1[`Z���_a[ b ^`[ � { ( 1√2
� �(0�
-'( 1
5� �( 1
6� &(6�1[`Z���_a[ b ^`[ �
� �
$ y ( 1
ln 5� $'$ (
1� $ x ( 32
3�
$ 0(12� $ � (
3� $ { ( π
2
�
$ �(1� $ -'(
e� $ �( −1
�
$�&( 16− 2√
2
3� xzy ( 2
√2� x $ ( �1[`Z���_a[ b ^`[ �
x1x ( �1[`Z���_a[ b ^`[ � x 0( 6� x � (
3(1 + 3√
2)�
� '(1*�&
$ y � $'� $ ( b ^`[�� � x (�[`u![ � 0( b ^`[�� � � ( b ^`[�� �{ (�[`u![ � �(�[`u![ � -'( b ^`[�� � �(�[`u!["�
$ y � x � $ (y = x2 + 5x + C
�
x (y =
1
2(x2 + e2x) + C
�
0(y = 6x + C
�
� (y =
4x
ln 4+ 3x + C
�
{ (y =
3
2x2 − 1
3x3 + C1x + C2
�
�(y =
1
4e2x + x2 + C1x + C2
�$ y � 04� $ (
y = x4 + x3 + 1�
x (y = xex − ex + 3
�
0(y = ln
1 + x2
2�
� (y = x ln x− x + 2
�
{ (y =
1
4x4 + 2x2 + x + 2
�
�(y = x2 + ex + 2x
�
� �
$ y � �1� $ ( 1
2(x2 + y2) + ln x + C
�
x (ln x +
1
y= 1
�
0( y3
3= x− x2 + 9
�
� ( 1 + y2
1− x2= C
�
{ ( x2
2+
y2
2= C
�
�(x√
1 + y2 = C�
-'(ex + e−y = C
�
�( y2
2= ln(1 + ex) + C
�
$ y � { � $ ( x
y+ ln |y| = 2
�
x (y = Cx− 2x ln |x| �
0(arctg
y
x− ln
√
x2 + y2 = C�
� (ln y +
2x
y= 0
�
{ (ln |x| − x
y= C
�
�(√
x
y+ ln y = C
�
-'(y = x ln
y
C
�
�( √
x2 + y2 = e− arctgy
x�
$ y � �4� $ (y = Cx + x2 �
� �
x (y =
C + ex
x
�
0(y =
C
x2+
x4
6�
� (y = (1− x2)[C − ln |1− x|] �
{ (y = e−x(2 + x)
�
�(y = e−x2
(1 + x3)�
-'(y =
2
x+
x
2+ 1
�
�(y =
8
x2+ x
�
$ y � - � [�(y =
1
4x4 +
1
3x3 + C1x + C2
�
i�(y =
1
6x3 + ex + C1x + C2
�
� (y = 3x2 + e2x + C1x + C2
�
b ( y =1
2x− 2x2 + C1x + C2
�
$ y � �4� $ (y =
5e3x + e−3x
6
�
x (y = 2− e−4x �
0(y = C1e
x + C2e3x �
� (y = C1e
−3x + C2e−2x �
{ (y = (C1 + C2x)e−
x
2�
�(y = e3x �
-'(y = (C1 + C2x)e1,5x �
�(y = (2 + 3x)e−x �
&(y = cos 2x + sin 2x
�
� �
$ y (y = (C1 cos 2x + C2 sin 2x)e−x �
$'$ (y = sin x
�
$ x (y = (C1 cos 2x + C2 sin 2x)e−2x �
$ y � &4� $ (y = (C1 + C2x)e2x +
1
4x2 +
1
2x +
3
8�
x (y = C1e
x + C2e−x − x2 + x− 3
�
0(y = C1 + C2e
−x + 3x�
� (y = C1 + C2e
3x + x2 �
{ (y = (C1 + C2x)e−x +
e2x
9�
�(y = C1 cos x + C2 sin x + 2ex �
-'(y = C1e
3x + C2e−x − 3
2xex �
�(y =
(
C1 + C2x +x2
2
)
ex �
$ y � $ y � [�( dx
dt= 5 · 106 � i�(
600 b �!g�� � (800 b �!g��
$ y � $'$'� [�(200 b �!g�� i�(
800 b �!g��$ y � $ x �1000 b �!g��
$ y � $ 04�P (t) = e
t
24 (3464 · 104 + 36 · 103e−t
24 )�
$ y � $ �1�P (t) =
45 · 105
e−2·104t + 9
�
� 1��� '(1*�&
$'$'� $'� $ (
x (
0(
� �
� (
{ (
�(
-'(
� �
�(
&(
$ y (
$'$ (
� �
$ x (
$'$'� x � $ (
x (
� �
0(
� (
{ (
�(∅
� �
-'(
�(
&(
� �
$ y (
$'$ (
��� �
$ x (
$ 0(
$ � (
��� �
$'$'� 04� $ (j b ^ b g m�^ cYZY^���p`ZYgY_aq!i`[`[ f(0; 6) = 12ajlcYfh^`u!g m�^
cYZY^���p`ZYgY_aq!i`[`[f(9; 0) = −9
�
x (j b ^ b g m�^KcYZY^���p`ZYgY_aq!i`[`[ f(1; 5) = 9zjlcYfh^`u!g m�^CcYZY^kn
��p`ZYgY_aq!i`[`[f(1; 0) = −1
�
0(j b ^ b g m�^KcYZY^���p`ZYgY_aq!i`[`[ f(0; 0) = 0zjlcYfh^`u!g m�^CcYZY^kn
��p`ZYgY_aq!i`[`[f(0;−4) = −8
�$'$'� �1� $ (�[`cYq!ZY[��`m�gYZY^U[`u �1[`[ � ZY^`[ �
x (f(30; 40) = 900
�
0(f(2; 5) = 19
�
� (f(1; 0) = 8
�
{ ( b [km�[���p`gYiUcYZY^���p`ZYgY_aq!i`[%s:['m�^`cYu![`p`_agKfh[`u!^`gY_a^`[ ��(
f(3; 4) = 7�
-'(f(8; 4) = 240
�
�(f(5; 0) = 500
�
&(f(4; 2) = 8
�
$ y (f(6; 10) = 130
�$'$'� { �� ^`u!cY[`c�jlZ b [�[]\�[`u!cYq!q4m A
d!^��`^km � � b [ Bd!^��`^km $ �
gYuAs:g jl_a^ �`u!q b j��Yd!^"Gu!q!c cY^`^��!q4m�cY[�� m�^`cY[`_�jlu!^ [`cYq4nZY[��1gYi`^ x � � y b q!_a[`u!^kmwq b gYZYq!i`^%s6�
$'$'� �4� xzy �`^.�C[`e`^ b [ 0 {zy ��[`u!p`[`_a^"�cY[�� m�^`c jlcY^���gYcYq4m�[`p`[`_a^`[x 0 y1y1y b q!_a[`u!^"�
$'$'� - � $ y1y \Tj s:^d!gY_agYu!gYe`_a[`cY[" xzy1y \Tj s:^Uu![ b ^`q!u!gYe`_a[`cY["�$'$'� �4�
AcYq b gY_a^km�$ y [`p`d!q!cYq!i`^`_a^" B
cYq b gY_a^km $ { [`p`d!q!cYq4ni`^`_a^"�lcY[�� m�^`cY[`_�jlu!^�cYq��1gYi`[`[ $ � y1y1y1y b q!_a[`u!^"�
$'$'� &4�4gYZYgYu��1gYd!^`e`[���^ $ x { cY^`_a^`q!ZY^ b q!_a[`u!^"���q!_aq e`[`p���^`u��1[`iknc jl_aq!i`[���^#- { cY^`_a^`q!ZY^ b q!_a[`u!^"��cY[�� m�^`cY[`_�jlu!^;cYq��1gYi`[`[$' � { cY^`_a^`q!ZY^ b q!_a[`u!^"�
$'$'� $ y � � y �Yu!q*��["�$ � y f ��p`[`u!^"�lcYq��1gYi`[`[ $ - xzy1y b q!_a[`u!^"�
��� �
$'$'� $'$'� $ y1y cY[��1^ b [" {zy e`[`u![ b ["lcYq��1gYi`[`[ � y1y1y1y b q!_a[`u!^"�$'$'� $ x � 0 y �Y[`_a^kmVm�p`^`d!gYu!^" xzy cY[`cY[`e`[`fh^kmVm�p`^`d!gYu!^"� $�& y1y b q4n
_a[`u!^"�
��� 1)+&���(�(15;6�&
s:[`p`^ �#XYX � cY[`u*� ^`ZY[`_�jlu!^ jlZ��Yfh^`gYi`^"�� jlZ��Yfh^`^km\�[`u!cYq!gYikjl_a^"� b ^� gYu!gYZYfh^`[`_a^"� jlZ��Yfh^`^kmm�u4jl_a^ �1[`cYq!e`p`_agYp`[ ���������.�������������������������.��������������� 0
s:[`p`^ �#XYXYX � cYu![`p`[`_a^Ufhp`_a[ b ^km jlZ��Yfh^`gYi`^ �������������������������.��� x1xs:[`p`^UX�� � ^`ZYd!g��1u![`_�jlu!^U[��!u!^`f ��p`^kmwgY_agYcYgYZYd!gYi`^"�
cYq!c���cY[`u!gYi`_a^km b [`ZY[.+�q��1^"�lc \�[`u!cYq!gYi`_a^km[`cYq!ZY[��1gYi`^kmwZY[`cYgYd!^"�Tj�\.Yp`gYd!^���gYcYq4m�[`p`_a^kmZY[`e`[ b ^kmwcY^`c b ^`ZY[`u!g �!^`u!gYikjl_agYi`["�^`ZYp`g m�d!^`fh^`^kmwZY[`e`[ b ^ b [e`[��`^`d![`_a^kmb [��1u!q!p`gYi`[ �������������.�����������������������.�������������������������.� 0 0
s:[`p`^�� � b ^� gYu!gYZYfh^`[`_�jlu �1[`ZYd!q!_agYi`[%s:[�s:gYq!u!^`^kmgY_agYcYgYZYd!gYi`^ �������������������.�������������������������.������������������� � -
s:[`p`^��#X � \�u� ^`p`^ b [��`u!q��1u![`cYgYi`^km�gY_agYcYgYZYd!gYi`^ �������.��������� { 0
�`[km`j1��gYi`^ �����������������.�������������������������.�����������������������.��������������������� �4$�#XYXGs:[`p`^������������������.�������������������������.�������������������������.������� �4$�#XYXYXGs:[`p`^ �.�������������������������.�������������������������.����������������������- �X��Is:[`p`^ �������.�������������������������.�����������������������.��������������������� � x�Is:[`p`^ �����.�������������������������.�������������������������.����������������������� �'&�#XGs:[`p`^ �������.�������������������������.�����������������������.��������������������� & 0
Recommended