wjt! - BPA · 2) xsinx (x > 0); 3) xlnx (x > 0): 1 4^ `q!p`gtsIekj s g!u!q!cYgY_ m [`f YcYZY^km f(x) = x2 jlZ Yfh^`^km 1u![ ^`e`^km M 1 2; 4 \gYu!d!^`_ 6+ g 1[`cY[`p`[`_a^

$Page 1: wjt! - BPA · 2) xsinx (x > 0); 3) xlnx (x > 0): 1 4^ `q!p`gtsIekj s g!u!q!cYgY_ m [`f YcYZY^km f(x) = x2 jlZ Yfh^`^km 1u![ ^è`^km M 1 2; 4 \gYu!d!^`_ 6+ g 1[`cY[`p`[`_a^$
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\�^��1ZY^ \�[ù!cYq![ b �1gYZ m cY[%s:gYcY[`d!^è`^km��ù![��Yd!^èkjl_a^ m�[`p`[ù��C^��q4ngYi`^km�[ b [�[`cYq!fh[`ZYgYi`^kmaeù!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^èkjlu!^ b [�m�[ì`[`ZYe`q4nom�[� ^`ZY[`Z m�q��ù!q� ^kn_a^km i`[è`[`_a[`pù!^`[`d!^km�m�[� g��`jlu!^km<m�d4j b gYZYd!gYi`^km�[%s:p`^km��^`ZY[`[ù4m�q4niù!^`p`[ b�b [�m�d!u4j��Yd4jlu4jl_a[ b ^��1^�m�u4jl_��g m�[ì`[`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!^è`[ gYe`q!ZYq!cY^km�d!gYi`^km�[%s:p`^km��! s:i`^`_a^km�^"�1_aq!i`[`_�n#�ù!^`ZYd!^"%$�&'&'&(��

u!g b [��Yd!q!u!^") �ù!q� g m�q!u!^ b ��ZY[`d!u!q*��p`^`_a^

u!gYfhgYZ,+�gYZYd!gYi`^")- ^.+�^è`[knrcY[%s:gYcY[`d!^è`^km�cYgYfhZY^`gYu!gYi`[%s:[ b q��Yd!q!u!^"�ù!q� g m�q!u!^ b ��ZY[`d!u!q*��p`^`_a^

^.+�^è`[knrcY[%s:gYcY[`d!^è`^km�cYgYfhZY^`gYu!gYi`[%s:[ b q��Yd!q!u!^"�ù!q� g m�q!u!^["�/�1[��1ZY^��Yg

0© �1[`cYq!cYfhgYcY_aq!i`[�� d!g��YZY^èkjlu!^#jlZY^`p`gYu4m�^`d!gYd!^��1 xzy1y�{X,24365

s:[`p`^��#XYX

�� !��#"�$�%��&�� !��

�'�� !)(*�� +�,� !��,&�

- � $'�3\�[ù!cYq!gYikjl_a^km �1[`ZYcY[ù!d!gYi`^km �1[`cYq/.YgYZYgYi`^%s�^��`q!p`gts ��gYc b g n�1^ jlZ��Yfh^`^kmU\�[ù!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[]\�[ù!cYq!gYi`^km \�g m�^km �1[`cYq/.YgYZYgYi`^%s^��`q!p`gts ��gYc b g��1^ jlZ��Yfh^`gYi`^kmU\�[ù!cYq!gYikjl_agYi`^")

1) y = arctg x;

2) y = lnx;

3) y = arcsin x.

- � 04�4uAs3jl_a^ jlZ��Yfh^`^km-�1[]\�[ù!cYq!gYi`^km<\�g m�^km��1[`cYq/.YgYZYgYi`^%s��1[kncYqAs:p`[`_agts jlZ��Yfh^`^kmU\�[ù!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![� ^è`^kmM

(

12; 1

4

) \�gYu!d!^`_6+�g �1[`cY[`p`[`_a^<c��gYi`^-[ìkm�fh^knm�[%s:[ �!gYu��Y^km b [ b gYi`^%s cY^`cY[ùAs3jl_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![� ^è`^kmM

(

13;√

327

) \�gYu!d!^`_6+�g��1[`cY[`p`[`_a^�c��gYi`^[ìkm�fh^km�[%s:[ �!gYu��Y^km b [ b gYi`^%s cY^`cY[ùAs3jl_agYi`[km�s:[`Z��

- � �4�4^��`q!p`gtsy = x2 − 2x + 12

jlZ��Yfh^`^km �1u![� ^è`^kmM(1; 2)

\�gYu!d!^`_6+�g �1[`cY[`p`[`_aÛc��gYi`^kmwekj s �`jlu!Ûe`q!g ^`fh^`gYZYd!^"�- � - �*�1[`cYqAs:p`[`_agts [ù��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[.+�[ù b ^ x\�gYu!d!^`_ ��^" s3j [ù4n

��jlcYgYZYd!^km�ZY[.+�u b ^`[ ∆x)

[�(f(x) = x3 �

i�(f(x) =

√x + 1

�

� (f(x) =

2

x− 1�

- � &4�*�1[]\�[ù!cYq!gYi`^km�\�g m�gYi`^km�[ b [��Y^ù!^%s:[ b ÂgY_agYcYgYZYd![ù4jl_a^ j�nZ��Yfh^`gYi`^km \�[ù!cYq!gYikjl_agYi`^km f ��u!^`_a^km �1[`cYq/.YgYZYgYi`^%s ^��`q4np`gts \�[ù!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 \�[ù!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\�[ù!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\�[ù!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!û!^��1^km�\�[ù!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^`[`ZYÛ[`cYq!ZY[��1gYi`^km jlZ��Yfh^`[`[

(TR) = f(Q) = 300Q− 2Q2.

[�(�u![km!j b u!^km�cY[ù*� ^`ZY[`_�jlu!Â[`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�[ì`[`cY^km�^�cY[ù*� ^`ZY[`_�jlu!^�[`cYq!ZY[��1gYi`^km jlZ��Yfh^`^km�1[`cYq4m�[��`jl_agYi`gYi`^ �

i�( �1[`cYqAs:p`[`_agts cY[ù*� ^`ZY[`_�jlu!Û[`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 [ù!gts ^��1^cY[ù*� ^`ZY[`_�jlu!^�[`cYq!ZY[��1gYi`^km cYZY^��p`ZYgY_aq!i`[km

Q = 20gYuAs:g jl_6+�g��

- � $�&4�*�1[`cYqAs:p`[`_agts cY[ù*� ^`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^`[`ZYÛ[`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[ù*� ^`ZY[`_�jlu!Û[`cYq!ZY[��1gYi`^"lu!q!fh[Q = 1

�- � x $'�0m�[]\�[ù!cYq4m c j b cY^`p`^ b [`ZY[��[ù��C^`[ (FC) = 500 $

��q4n_aq fhp`_agYi`[ b ^ b [`ZY[��[ù��C^��ù!q b j��Yfh^`^km gYuAs:g jl_6+�g

�

(V C) = 3 $�

[�(�^��`q!p`gts cts:_a^`[`ZY^ b [cY[ù*� ^`ZY[`_�jlu!^ b [`ZY[��[ù��CgYi`^ �i�( �1[`cYqAs:p`[`_agts cts:_a^`[`ZY^ b [`ZY[��[ù��C^"lu!q!fh[ Q = 40

�

� ( �1[`cYqAs:p`[`_agts cts:_a^`[`ZY^ b [`ZY[��[ù��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\�[ù!cYq!gYi`^kmKm�[��1jl[`_aq b [`ZY[��[ù��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[��[ù��C^km�[ b [ cY[ù*� ^`ZY[`_�jlu!^b [`ZY[��[ù��C^km jlZ��Yfh^`gYi`^ �

i�(�cYq��YgYi`ZYgts cY[ù*� ^`ZY[`_�jlu!^ b [`ZY[��[ù��C^km�m�[��1jl[`_agYi`^%s�1[`cYqAs:p`_a^`_a^ m�u4jl_a^ b [`ZY[��[ù��C^km fhp`_a^`_agYi`["�s3j\�[ù!cYq!gYikjl_a^ �ù!q b j��Yfh^`^km�u![`q b gYZYq!i`[ cYfh^ù b gYi`[xzy n b [`Z $ ��gYuAs:g jl_a[`c b g��

��

� ( �1[`cYqAs:p`[`_agtsEcts:_a^`[`ZY^ b [`ZY[��[ù��C^km +Tj�m�d!^ fhp`_a^kn_agYi`["�u!q!fh[

Q = 20nom b [ �ù!q b j��Yfh^`^km�u![`q b gYZYq4n

i`[CcYfh^ù b gYi`[ x gYuAs:g jl_a^%s6��^��`q!p`gts m��p`[`q!i`[;cts:_a^kn[`ZY^ b [`ZY[��[ù��C^km�+Tj�m�d b [CcY^`[��_aq!gYi`^%s cYZY^��p`ZYgY_aq4ni`gYikm ��q!u!^km�

- � x 04�3\�[ù!cYq!gYi`^km ^�� m�^ù!gYikjl_a^ b [`ZY[��[ù��C^`[ - { b q!_a[ù!^"��q4n_aq fhp`[`_agYi`[ b ^ b [`ZY[��[ù��C^!�ù!q b j��Yfh^`^km gYuAs:g jl_a^km\�[ù!cYq!gYi`^km�[%s:p`^km��

4 +3

Q

�[�( �1[`cYqAs:p`[`_agts cts:_a^`[`ZY^ b [`ZY[��[ù��C^ b [wcY[ù*� ^`ZY[`_�j�nu!^ b [`ZY[��[ù��C^ �ù!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[��[ù��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^ù!g ni`^km�[km

s3j�[��!gYikjl_ cYq!cYgYZYd*��^��ù!q b j��Yfh^`^km�u!gY[`_a^.+�[`fh^`^km b q!ZYgY[Q = 50

gYuAs:g jl_a^"�- � x �1�� ^ù!cY^km�cY^`gYu b [��1g��1cY^`_a^ b [`ZY[��[ù��C^�cYq!^`fhgYcY[� jlZ��Yfh^`^%s6)

K(Q) = (TC) =9

3Q3 − 9Q2 + 3.

\�[ù!cYq!gYi`^km u![-cYq!f+jl_aq!i`^km b u!q4m ^��YZYgYi`[ b [`ZY[��[ù��CgYi`^cY^`ZY^`cY[`_�jlu!^��

- � x1{ �3\�[ù!cYq!gYi`^kmwcts:_a^`[`ZY^ b [`ZY[��[ù��C^km jlZ��Yfh^`[`[

K(Q) = 0, 03Q2 − 2Q + 300.

[�(�^��`q!p`gts m�[��1jl[`_aq b [`ZY[��[ù��C^km�[ b [ cY[ù*� ^`ZY[`_�jlu!^b [`ZY[��[ù��C^km jlZ��Yfh^`gYi`^ �

� 1i�(�^��`q!p`gts m�[��1jl[`_aq b [`ZY[��[ù��C^ b [�cY[ù*� ^`ZY[`_�jlu!^ b [knZY[��[ù��C^"lu!q!fh[") �$ (

Q = 50 x (

Q = 100�

� (U\�[ù!cYq!gYi`^km u![�cYq!f+jl_aq!i`^km�[%s:p`^km�^��YZYgYi`[<m�[��1jl[`_aqb [`ZY[��[ù��CÛcY^`ZY^`cY[`_�jlu!^��

- � x �4�3\�[ù!cYq!gYi`^km�c j b cY^`p`^ b [`ZY[��[ù��C^`[ $ {zy b q!_a[ù!^"*��q!_aqfhp`[`_agYi`[ b ^ b [`ZY[��[ù��C^�� x b q!_a[ù!^" cYqAs ��q!p`ZY^km- j�nZ��Yfh^`[`[

P + 2Q = 82.�1[`cYq4m�[��gts cYq��1gYi`^km

Π jlZ��Yfh^`[

Qnomm�[��1jl[`_agYi`^%s6�

[�(�^��`q!p`gts m�[��Yq!ZY_a^kmwu!['u![`q b gYZYq!i`[^��Y_agYp`[ xzy1y b q4n_a[ù!^kmwd!q!_ cYq��1gYi`[km �

i�( �ù!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[��[ù��C^km jlZ��Yfh^`[ � (AC) =7

Q+ 3

�T^��`q!p`gts �ù!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[��[ù4n�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�*�ù!q b j��Yfh^`^km \�[ù!cYq!gYi`^km ^�� m�^ù!gYikjl_a^ b [`ZY[��[ù��C^`[(FC) = 80 $

��q!_aq!�ù!q b j��Yfh^`^km gYuAs:g jl_6+�g<fhp`[`_ag ni`[ b ^ b [`ZY[��[ù��C^ � (V C) = 4 $

�TcYqAs ��q!p`ZY^km jlZ��Yfh^`[`[

P = 300−Q.

� �

[�(�^��`q!p`gts cY[ù*� ^`ZY[`_�jlu!^�cYq��1gYi`^km jlZ��Yfh^`[ �i�(�cY[ù*� ^`ZY[`_�jlu!ÂcYq��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�[ì`[`cYg ni`[-\�[ù!cYq!gYikjl_a^��ù!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 [ù!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[��[ù��C^km jlZ��Yfh^`[ �

(TC) =1

2Q2 + 4Q− 10.

[�(U\�[ù!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\�[ù!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[ù!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[ù*� ^`ZY[`_�jlu!^�cY^ b u!gYe`^`_agYi`gYi`^�cYq*��cY[ù!g n

i`^km�[(MPC) b [ b [.+�q��1p`^km�[ b cY^ (MPS

(�ù!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[ù*� ^`ZY[`_�jlu!ÊcY^ b u!gYe`^`_agYi`gYi`^ b [.+�q��1p`^km�[(MPS) b [cYq*��cY[ù!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!^èkjlu!q!i`[ [km�^kmfhp`_a^`_agYi`^kmUcY^`cY[ùAs6Ts3j �ù!q b j��Yfh^`^kmUgYuAs:g njl_a^km [km�^�e`_agYikjl_aq!ikm $ xzy b q!_a[ù!^ b [`Z $ y1y b q4n_a[ù![`c b g��

i�( �1[`cYqAs:p`[`_agts cYqAs ��q!p`ZY^km +/�!pù4jl_a^�( gY_a[km�d!^èkj�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 +/�!pù4jl_a^�(�gY_a[km�d!^èkjlu!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!^èkjlu!q4ni`[ [km�^km#fhp`_a^`_agYi`^km#cY^`cY[ùAs6�s3j��ù!q b j��Yfh^`^km#gYu4ns:g jl_a^km [km�^ �1[`^.+�[ù b [�0 y b q!_a[ù!^ b [`Z 0 { b q4n_a[ù![`c b g��

i�( �1[`cYqAs:p`[`_agts cYqAs ��q!p`ZY^km +/�!pù4jl_a^�( gY_a[km�d!^èkjlu!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 +/�!pù4jl_a^�( gY_a[km�d!^èkjlu!q4ni`[ [km�^km�cY^`cY[ùAs6lu!q!fh[

P = 22�

i�(�u!q��1q!u!^`[�cYqAs ��q!p`ZY^km �ù!q!fhgYZYd4jl_a^�fhp`_a^`_agYi`["1s3j [km�^km �ù!q!fhgYZYd4jl_aÛfhp`_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 +/�!pù4jl_a^�( gY_a[km�d!^èkjlu!q4ni`[ [km�^km�cY^`cY[ùAs6 s3j�cYqAs ��q!p`ZY[`[

Q = 20�

i�( [km�^km�u!q��1q!u!^ �ù!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 +/�!pù4jl_a^�( gY_a[km�d!^èkjlu!q!i`[ [km�^kmwcY^`cY[ùAs6 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!^èkjlu!q!i`[ [km�^km ZY[.+�u b ^km cY^`cY[ùAs6#s3j �ù!q b j��Yfh^`^km gYuAs:g njl_a^km [km�^ �1[`^.+�[ù b [ � b q!_a[ù!^ b [`Z & b q!_a[ù![`c nb g��

i�( �1[`cYqAs:p`[`_agts cY^]\�q b gYi`^km +/�!pù4jl_a^�( gY_a[km�d!^èkjlu!q4ni`[ [km�^km�cY^`cY[ùAs6lu!q!fh[

P = 6�

- � � y �4^��`q!p`gts cY^]\�q b gYi`^kmAm�[��1jl[`_aq�gY_a[km�d!^èkjlu!q!i`[6 [km�^km�ZY[.+�u4nb ^km�cY^`cY[ùAs6 s3j�cY^]\�q b gYi`^km jlZ��Yfh^`[`[

Q = 0, 01P 2 + 0, 5P + 60

� �

b [ �ù!q b j��Yfh^`^km gYuAs:g jl_a^km [km�^�e`_agYikjl_aq!ikm {zy b q4n_a[ù!^ b [`Z � � b q!_a[ù![`c b g��1[`cYqAs:p`[`_agts +/�!pù4jl_a^�gY_a[km`nd!^èkjlu!q!i`["lu!q b g m�[`f� [km�^`[ {zy b q!_a[ù!^"�

- � �1$'�4cY^]\�q b gYi`^km jlZ��Yfh^`[`[

Q = 0, 007P 2 + 3P + 4.

[�( �1[`cYqAs:p`[`_agts cY^]\�q b gYi`^km +/�!pù4jl_a^�( gY_a[km�d!^èkjlu!q4ni`[ [km�^km�cY^`cY[ùAs6lu!q!fh[

P = 15�

i�(�u!q��1q!u!^`[acY^]\�q b gYi`^km �ù!q!fhgYZYd4jl_aâfhp`_a^`_agYi`["�s3j [km�^km �ù!q!fhgYZYd4jl_aÛfhp`_a^`_agYi`[`[

10 %�

- � � x �0m�[è`q!ZYfhgYu!d!q b [ù!i`[.+�^'^`d!gYpkm & y1y1y cY[ . jlu!gYi`gY_�m�ls3j i`^kn_agts:^km6 [km�^!^��YZYgYi`[ � b q!_a[ù!^"�cY[��^`Zh^ .Y^ b gYi`[*- {zy1y i`^`_ag ns:^"�4i`^`_agts:^km� [km�^km x b q!_a[ù!^%s �1[`^`[� gYi`[`c �1[ .Y^ b jl_aî`^`_agts:gYi`^km u![`q b gYZYq!i`[��1[.+�[ù 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_aÛi`^`_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[`^.+�[ù b [�$ x b q!_a[ù!^ b [`Z$ { b q!_a[ù![`c b g��;[`cY^km �1[`cYq ZY[`fhp`_a[ b � y1y1y nr^km�[ s:p`g��^^ .Y^ b gYi`[ {zy1y1y fh[`_aê`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_aÛe`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^`[`ZYÛ[`cYq!ZY[��1gYi`Û^ .Yq4m�cY[�� m�^`cY[`_�jlu!^��

- � � �1�0m�[`p`[`vù!q ^ù!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[ù!^"hcY[��^`Z#.Yp`gY_a[-q� ^km�^ �1[��Y^knu![`p b gYi`["�3cY[ù!e`gYd!^`Z��jl_acY[ [`ZY[`_a^.+�cY[ [ � p`gYZY["0u!q!c �1[ b [knm�[��[ b ^km � b q!_a[ù!^%s �1[��Yp`^ù!gYi`[�^]\�p`gYpkm �1[��Y^ù![`p`gYikjl_a^q� ^km�gYi`^km x nr^%s-��gYcYfh^ù!gYi`[km��u![��1[ b [km�[��[ b ^!jlZ b [ b [`[]\�g nm�q4m ^ù!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^ù![`p`gYikjl_a^q� ^km�gYi`^kmwu![`q b gYZYq!i`^km�\�u� ^`p`^ jlZ��Yfh^`[`["� (

- � � { �0m�[`p`[`vù!q� ^ù!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[ù![ b �4cY[ù!e`gYd!^`Z��j�n_a^K[`ZY[`_a^.+�^ �1p`^ � p`gYZYgYikm�u!q!c xzy1y b q!_a[ù!^%s� [km b [è`_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^`[`ZYÛ[`cYq!ZY[��1gYi`^^ .Yq4m�cY[�� m�^`cY[`_�jlu!^��

� ( ^ù!cY^km .Yq!p`gY_Cs:p`^kjlu!^ b [`ZY[��[ù��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[ù*��[`ZY[#.Yq!p`gY_ b �!^kjlu![ b .Y^ b ^km�0 y1y1y fh[`_ �kjlu4ms:^%s:q!g jl_�m y { b q!_a[ù![ b ��kjlu!^km [km�^km��1[.+�u b [`c y xb q!_a[ù!^%s �1[`cYq!^]\�p`^`[ �1[ .Y^ b jl_a^ �kjlu!^km u![`q b gYZYq!i`^km$ y1y nr^%s��gYcYfh^ù!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�ÂjlZ b [ b [`g b q4m �kjlu4mzu!q!c4cYq��1gYi`[�^ .Yq4macY[�� nm�^`cY[`_�jlu!^"4s3j cY^km b [km�[`c,+�[ b gYi`_a[ b ^��[ù��CgYi`[ y 0b q!_a[ù!^��

- � � - �4cY[`d![ù!gYi`gY_ ��^V[ù!^km $ { p`[��1q!ZY^"��s:^%s:q!g jl_ p`[��1q!Z��^ � y[ b �1^`_a^`["�hs3j cY[`d![ù!gYi`_a^km<i`^`_agts:^km- [km�^w[ù!^km $ y b q4n_a[ù!^"hcY[��^`Z#^ .Y^ b gYi`[ { � y i`^`_agts:^"� +�[� �`jl_a^km-�`gYu!^`q b n��^�i`^`_agts:^km �!^ù!gYikjl_agYi`[ �1[.+�[ù b g m $ b q!_a[ù!^%s6zu![`cY[`f�1[`cYq!^]\�p`^`[ �1[ .Y^ b jl_aÂi`^`_agts:gYi`^kmGu![`q b gYZYq!i`^km x &�nr^%s��g ncYfh^ù!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^ù!g m�^ cYZY^kn

��p`ZYgY_aq!i`gYi`ÛcY^%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![� ^è`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!^è`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[ùAs:ekj s ��g b ^km S [ùAs: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�[.+/�!pù!^km�[ù!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`[ù!^")$ (

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�[è`^%s ��^"�

� �

�4� - �%��gY[`cYq:\�cYgts6Tj�\.Yp`gYd!^`[�s3j�[ù![ 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 �`^ù!p`gY_a^�u!^��1^km-e`gYu��Yq \�[ù!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 \�[ù!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\�[ù!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 [ù![`f ��[ b ^ jlZ��Yfh^`^km\�[ù!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)

[ù!^kmmu!^��1^kmGgYuAs��1p`[ù!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^`[ \�[ù!cYq4n[ b �1gYZ m m

u!^��1^km�gYuAs��1[ù!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.

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P = 15 $, PA = 20 $, Y = 500 $.

� �

�1[`cYqAs:p`[`_agtsIcYqAs ��q!p`ZY^km �ù!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.


P = 10 $, PA = 20 $, Y = 100 $.

�1[`cYqAs:p`[`_agtsIcYqAs ��q!p`ZY^km �ù!q!fhgYZYd4jl_a^ fhp`_a^`_agYi`["�s3j[`_ad!gYu!ZY[`d!^kjl_a^ �ù!q b j��Yfh^`^km [km�^��gYcYfh^ù b [ $ y�� nr^%s6�

�4� $ - �4cYq!fhgYc jl_a^`[cYqAs ��q!p`ZY^km jlZ��Yfh^`[

Q = 700− 6P − 8PA + 0, 04Y.


P = 20 $, PA = 10 $, Y = 1000 $.

�1[`cYqAs:p`[`_agtsIcYqAs ��q!p`ZY^km �ù!q!fhgYZYd4jl_a^ fhp`_a^`_agYi`["�s3j��gYcYq4m�[`p`[`_a^ ��gYcYfh^ù b [ xzy�� nr^%s6�

�4� $ �4�0m�[]\�[ù!cYq!q jlZ��Yfh^`[km�[��Ypkmm�[��g

Q = 3K2

5 L3

4 .

^��`q!p`gts6)[�(�e`[��`^`d![`_a^km +/�!pù4jl_a^ �ù!q b j��Yd!^ (MPK)

�

i�( ��u!q!cY^km +/�!pù4jl_a^ �ù!q b j��Yd!^ (MPL)�

� �

� ( +/�!pù4jl_a^ ��gYZY[`fhp`_agYi`^kmwZYq!u!cY[(MRTS)

�s3j

K = 32, L = 16.

�4� $�&4�0m�[]\�[ù!cYq!q jlZ��Yfh^`[km�[��Ypkmm�[��g

Q = 3LK + L1

3 .

^��`q!p`gts6)[�(�e`[��`^`d![`_a^km +/�!pù4jl_a^ �ù!q b j��Yd!^ (MPK)

�

i�( ��u!q!cY^km +/�!pù4jl_a^ �ù!q b j��Yd!^ (MPL)�

� ( +/�!pù4jl_a^ ��gYZY[`fhp`_agYi`^kmwZYq!u!cY[(MRTS)

�s3j

K = 2, L = 27.

b []\�gYu!gts ^.+�q!e`p`[`ZYd!gYi`^km ��g m�[ì`[`cY^km�^ �1[`ZYd!q!_agYi`["� �1[kncYqAs:p`[`_agts6�u![4m�^ b ^ b ^%s jlZ b [��1[`^.+�[ù b q4m K

e`[��`^`d![`_a^"s3j ��u!q!cY^km [��Yd!q!u!^��gYcYfh^ù b gYi`[ $�&-gYuAs:g jl_a^%s ^��j�n_a^km��cYgYi`["lu!q!ca[`cafhp`_a^`_agYi`[kmw[ù jlZ b ['cYq�� .Yp`g mU\�[ù!cYq4ngYikjl_a^ �ù!q b j��Yfh^`^km b q!ZY^km#fhp`_a^`_agYi`["�g�� ^"� (L, K)

\�gYu4nd!^`_aÛ^`fhp`_agYi`[Uc��q!_aq b ^�� m�^ù!gYikjl_ ^.+�q!e`p`[`ZYd�+�g (��

�4� xzy �4[`[��1gts ��g m�[ì`[`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 $'�� ^ù!cY[�gYuAs3m�[ b [�^`cY[`p`g#m�[��Yq!ZYgY_�m�m�[��^`ZY[`q b [�m�[��1[ù!gYq

i`[.+�[ù�+�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 [è`[`p��^ù!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�

^ù!cY^km�cts:_a^`[`ZY^ b [`ZY[��[ù��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�[.+/�!pù!gts ^ù!cY^km�^km�gts:^4m�[� [km�q �`q!_a^`d!^è`["�u!q4n

cYgY_a^`f j/+�u4jlZYp`gY_ .Yq� m ^ù!cY^km�cY[�� m�^`cY[`_�jlu cYq��1gYi`[km b [^��`q!p`gts [`cacYq��1gYi`^kmUm�^ b ^ b g��

�4� x1x �� ^ù!cY[kmVe`[��`^`d![`_a^km $ gYuAs:g jl_6+�g<g��[ù��CgYi`[ x$*��q!_aq

��u!q!cY^km $ gYuAs:g jl_6+�g � {$�� ^ù!cY^km m�[]\�[ù!cYq!q! jlZ�� n

fh^`[`[

Q = 6LK + 5L2.

\�[ù!cYq!gYi`^kmK b [ L

[��Yd!q!u!gYi.+�g ��[ù��C^km�[%s:p`^km �1[`cYq4n.Yq� ^`_a^�[��Ypkm ^�� m�^ù!gYikjl_a^ s:[`Z��[ xzy1y

$�3^��`q!p`gts \�[ù4n

cYq!gYi`^km�q��`d!^`cY[`_�jlu!^�u!g�� ^`cY^"��4� x 04�� ^ù!cY[kmVe`[��`^`d![`_a^km $ gYuAs:g jl_6+�g<g��[ù��CgYi`[�$

$*��q!_aq

��u!q!cY^km $ gYuAs:g jl_6+�g � x$�� ^ù!cY^km m�[]\�[ù!cYq!q! jlZ�� n

fh^`[`[

Q = 8K1

2 + L1

2 .

\�[ù!cYq!gYi`^kmK b [ L

[��Yd!q!u!gYi.+�g ^ù!cY[km �1[`cYq/.Yq� ^`_a^[��Ypkm� ^�� m�^ù!gYikjl_a^ s:[`Z��[ � $ y1y

$� ^��`q!p`gts \�[ù!cYq!gYi`^km

q��`d!^`cY[`_�jlu!^�u!g�� ^`cY^"��4� x �1�� ^ù!cY^kmU\�[ù!cYq!gYi`^km jlZ��Yfh^`[`[

Q = 2√

K +√

L .

1 �

Ke`[��`^`d![`_a^km $;gYuAs:g jl_a^ �!^ù4m �

$'��q!_aq

L��u!q!cY^km $

gYuAs:g jl_a^�� x$� ^��`q!p`gts \�[ù!cYq!gYi`^km�q��`d!^`cY[`_�jlu!^ u!g n

� ^`cY^"zs3j ^ù!cY[kmG\�[ù!cYq!gYi`^kmK b [ L

[��Yd!q!u!gYi.+�g*�1[`cYq4n.Yq� ^`_a^[��Ypkm ^�� m�^ù!gYikjl_a^�s:[`Z��[ � � � y

$�

�4� x1{ �� ^ù!cY[Gm�[��Yq!ZYgY_�m .Y^ b ^km#q!u�i`[.+�[ù�+�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

�� ^ù!cY^km�cts:_a^`[`ZY^ b [`ZY[��[ù��C^km jlZ��Yfh^`[`[

(TC) = 50+20(Q1+Q2)��^��`q!p`gts \�[ù!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�� ^ù!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�

^ù!cY^kmGcY^`gYu �1[]\�g jl_a^�cts:_a^`[`ZY^ b [`ZY[��[ù��C^ �1[`cYq!^km�[��gYi`[ q!u!c jl_a^%s

(TC) = 60 + 20(Q1 + Q2).

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P1 = 900 − 4Q1b [ P2 = 700 − 2Q2

� ^ù!cY^kmGcY^`gYu �1[]\�g jl_a^�cts:_a^`[`ZY^ b [`ZY[��[ù��C^ �1[`cYq!^km�[��gYi`[ q!u!c jl_a^%s

(TC) = 90 + 20(Q1 + Q2).

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��u!q!cY^km $�gYuAs:g jl_6+�g � 0$��m�[]\�[ù!cYq!q jlZ��Yfh^`[`[

Q =

1 �

4KL + 2L2 � \�[ù!cYq!gYi`^km [��Yd!q!u!gYi.+�g b [`ZY[��[ù��C^� ^�� nm�^ù!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^ �ù!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��

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x

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)

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)

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xzy ( ∫(3x + 4x)2dx

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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 �

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3x− 2 dx� x ( ∫ dx

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{ ( ∫ 13√

1− 2xdx

� �( ∫e5xdx

�

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� �( ∫(e−x + e−2x) dx

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� $ y ( ∫ ln3 x dx

x

�

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x

x2dx

� $ x ( ∫xe−x2

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$ 0( ∫ dx

x ln4 x

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2 + ex

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ex + e−xdx

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dx� x �( ∫ ln 2x

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x &( ∫ dx

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0 0( ∫x(1− x)10dx

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xexdx� x ( ∫

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0( ∫ln x dx

� � ( ∫xn ln x dx (n 6= −1)

�

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x

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dx� �( ∫

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x

�&4� { �0m�[]\�[ù!cYq4mUcY[ù*� ^`ZY[`_�jlu!^ b [`ZY[��[ù��C^ Q

gYuAs:g jl_a^km\�[ù4ncYq!gYi`^km b u!q4m�[ù!^km

0, 03Q2 + 4Q + 150 b q!_a[ù!^�( .

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&4� �4�4cY[ù*� ^`ZY[`_�jlu!^ b [`ZY[��[ù��C^km jlZ��Yfh^`[kmU[��YpkmUm�[��g

(MC) = K ′(Q) = 6Q2 − 10Q + 260,

1 �

m�[ b [`f K(Q)[ù!^kmwcts:_a^`[`ZY^ b [`ZY[��[ù��C^km jlZ��Yfh^`["4��q4n

_aqQ

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&4� - �4cY[ù*� ^`ZY[`_�jlu!Û[`cYq!ZY[��1gYi`^km jlZ��Yfh^`[km�[��YpkmUm�[��g

(MR) = f(Q) = 80− 0, 4Q.

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&4� �4�4cY[ù*� ^`ZY[`_�jlu!^ b [`ZY[��[ù��C^km jlZ��Yfh^`[`[

(MC) = K ′(Q) = 200− 0, 6Q + 0, 09Q2.

^��`q!p`gts b [`ZY[��[ù��C^km jlZ��Yfh^`^km�ZY[.+�u b ^��ù!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�� ^ù!cY[`c b [`[ b �1^`ZY["1u!q!c4cY[ù*� ^`ZY[`_�jlu!^ b [`ZY[��[ù��C^ QgYu4n

s:g jl_a^km m�[]\�[ù!cYq!gYi`_a[ b jlZ b [ ^ .Yq4m (MC) = 3, 5 −0, 04Q

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&4� $ y �4^��`q!p`gts cYq*��cY[ù!gYi`^km jlZ��Yfh^`[H(Y )

zs3j-fhZYq!i`^`_a^`["zu!q!ccY[ù*� ^`ZY[`_�jlu!^GcY^ b u!gYe`^`_agYi`[GcYq*��cY[ù!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^`["

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&4� $'$'�4^��`q!p`gts cYq*��cY[ù!gYi`^km6 jlZ��Yfh^`["�s3jwfhZYq!i`^`_a^`["�u!q!chcY[ù*� ^knZY[`_�jlu!ÂcY^ 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[ù!^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[ù!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[��[ù��C^"as3j cY[ù*� ^`ZY[`_�jlu!^b [`ZY[��[ù��C^`[

(MC) = 6e0,3Q,

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^`fhgYcY[d!q!_aq!i`^%s6)[�(

(MR) = 100− 4Q�

i�((MR) = 80− 6Q

�

� ((MR) =

12√Q + 9

�

b ( (MR) =3√

Q + 4

�[��

Q[ù!^kmKu!gY[`_a^.+�gYikjl_a^ �ù!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�[ì`[`cY^km�^�cYqAs ��q!p`ZY^km jlZ��Yfh^`^kmm�[��g��

1 �

&4� $ �1�*�1[`cYqAs:p`[`_agts �1[`Z m�[.+/�!pù4jl_a^�^`ZYd!g��1u![`_a^")

$ ( 4∫

1

(x4 +√

x) dx� x (

1∫

0

√1 + x dx

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0

(x2 − 2√

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4

dx

(x− 3)2

�

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1

dx√x + 3

� �(4∫

0

ex

4 dx�

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0

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1 +√

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xdx

�

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4

x− 1√x + 1

dx� $ y (

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0

dx√16 + 2x

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dx√x + 16−√x

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x

x2dx

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dx

x ln x

� $ � (16∫

4

dx√x− 1

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1

dx

(1 +√

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� $ �(5∫

1

x dx√5 + 4x

�

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0

xexdx� $ �(

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1

ln x dx�

$�&( 1∫

0

x2exdx� xzy (

e∫

1

ln2 x dx�

� �

&4� $ { �*�1[`cYqAs:p`[`_agts ^`c ^��jlu!^km [ùAs:q!i`^"'u!q!cYgY_a^`f ��gYcYq4nm�[.+/�!pù4jl_a^`[ ��gYc b g��1^#\�^ù!gYi`^%s6)

y = 3x2, y = 0, x = 1, x = 2.

&4� $ �4�*�1[`cYqAs:p`[`_agts [ùAs:q!i`^"%u!q!cYgY_a^`f ��gYcYq4m�[.+/�!pù4jl_a^`[y =

9− x2 �`[ù![ì`q!_a^%s:[ b [ y = 0\�u� ^%s6�

&4� $ - �*�1[`cYqAs:p`[`_agts ^`c ^��jlu!^km [ùAs:q!i`^"'u!q!cYgY_a^`f ��gYcYq4nm�[.+/�!pù4jl_a^`[ ��gYc b g��1^#\�^ù!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 [ùAs:q!i`^"%u!q!cYgY_a^`f ��gYcYq4m�[.+/�!pù4jl_a^`[y =

ex y = e−x \�^ù!gYi`^%s:[ b [ x = 1\�u� ^%s6�

&4� $�&4�*�1[`cYqAs:p`[`_agts [ùAs:q!i`^"%u!q!cYgY_a^`f ��gYcYq4m�[.+/�!pù4jl_a^`[y =

x2 + 4x�`[ù![ì`q!_a^%s:[ b [ y = x + 4

\�u� ^%s6�&4� xzy �*�1[`cYqAs:p`[`_agts [ùAs:q!i`^"%u!q!cYgY_a^`f ��gYcYq4m�[.+/�!pù4jl_a^`[

y =

x2 − 2x�`[ù![ì`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[ù!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[ù!gYi`_a^km b [`ZY[.+�q��1^ [ùAs: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[ù!gYi`_a^km(CS) b [`ZY[.+�q��1^"As3j p`[`vù!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 \�[ù!cYq!gYi`_a^kmG[`cYq!ZY[��1gYi`^km�ZY[`cYg nd!^"ls3j��ù!q b j��Yfh^`^kmwgYuAs:g jl_a^km [km�^`[ $ y b q!_a[ù!^"�

&4� x �1�4^��`q!p`gts c \�[ù!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 [ �ù!q b j��Yfh^`^kmwgYuAs:g jl_a^km [km�^`[ xzy1y b q!_a[ù!^"�&4� x1{ �� ^ù!cY[ e`p`^ù![��^ .Y^ b ^km xzy1y1y cY[`fh^`p`[ù4m � s:^%s:q!g jl_�m

x1{zy b q!_a[ù![ b ��cY[ù!e`gYd!^`Z��1^�[ � p`gYZYgYikm�u!q!c $ y b q!_a[ù!^%s [km�^km ��gYcYfh^ù!gYi`[K^]\�p`gYpkm �1[ .Y^ b jl_a^;cY[`fh^`pù!gYi`^km'u![`q b g nZYq!i`^km�+�u b [km e`p`^ù![��^ 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[ù!gYi`_a^kmb [`ZY[.+�q��1^"+s3jVp`[`vù!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 \�[ù!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[ù!gYi`_a^km b [`ZY[.+�q��1^"�s3j-�ù!q b j��Yfh^kn^km�gYuAs:^gYuAs:g jl_aÛ^ .Y^ b gYi`[ $ y b q!_a[ù![ b �

&4� x �4�*�1[`cYqAs:p`[`_agts(PS)

c \�[ù!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`[`vù!q!i`[ � b gYi`[ Q0 = 10 b q!ZYg,+�g��a[`[`[��1gts ��g nm�[ì`[`cY^km�^ ZY[��[.+�^ b [-�1[`cYq4m�[��gtsEc \�[ù!cYq!gYi`_a^km [`cYq!ZY[��1g ni`^kmwZY[`cYgYd!^ [ùAs: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[ù!^km�^`ZYd!gYZ m�^`p`q!i`^%s6�%m�[ù��1gYi`_a^km�\�_a^kjlu!ÂuAs3j�n_a^ �1[`ZY[è`p`gts:^`[ $ y � j�\.Yp`gYd!^ b [ù!^`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!^ù!gYikjl_a^ �!^ù!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[ù!^kmK^`ZYd!gYZ m�^`p`q!i`^%s6��m�[ù��1gYi`_a^kmK\�_a^knjlu!^#uAs3jl_a^��1[`ZY[è`p`gts:^`[ $ � � j�\.Yp`gYd!^ b [ù!^`f ��p`^km'\�g nm�^%s�(��h^��`q!p`gts m�[`d!u![km�d!q q!Z b ^km�s:[`Z��[ �`^ù!p`gY_a^�m�[`cY^\�_a^km ��gYc b g��1�

&4� 04$'�0m��`q!u!d4m�cYgYZY^<[� q!u!cYgYikm<m�[��gY_ [km�q e`q!ZYd!u![��Yd4m�3[`ce`q4ncYd!u![��Yd!^km �`^ù!q!i`gYi`^kmas:[`ZY[��cY[ b ^��1^!cY^`^��!gYikmGs:[`Z��[km�u!q4ncYgY_a^`f<^.+�u b gYi`[�s:[`ZY[ìù![ b `j�\.Yp`gYd![ b�b [:\�u� ^`p`[ b \�_a^knjlu!^�$ y1y1y1y1y b q!_a[ù!^ b [`Z b [ [��:\�gYpkm x1x1{zy1y1y b q!_a[ù4m

� 1{ \�_a^km ��gYc b g��1��[`cY^`d!q!cCcY^km�^ ��gY_ [km�Û[ù!^km

F (t) = 100000(

1 +1

4t)

b q!_a[ù!^"�

^��`q!p`gts e`q!ZYd!u![��Yd!^km m�[]\.Y^km�^ �!^ù!gYikjl_agYi`["u!q!cYgY_a^`f��gYg m�[ì`[`cYgYi`[!m�[ù��1gYi`_a^km�\�_a^kjlu!^�uAs3jl_a^*- � nr^`[`ZY^��1[`ZY[èknp`gts:^%s j . \�p`gYd b [ù!^`f ��p`[km� cY^%s:^%s:gYi`[") ^km�[ù��1gYi`_agts^`cY^%s6Au!q!c ��gY_ [km�^km�+�u b ^km�m�^ � �Y[ù!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[ù!^km'ZY[è`[ b ^km#m�[]\.Y^kn

m�^��!^ù!gYikjl_agYi`["�s3j�m�[ù��1gYi`_a^kmUuAs3jl_a^;\�_a^kjlu!^ �1[`ZY[èknp`gts:^`[ � � j�\.Yp`gYd!^ b [ù!d!^`f ��p`^%s6�

&4� 0 04�4^`ZYp`g m�d!^`fh^`^km ZY[è`[ 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`[��`^ù!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`^ù!q ^`cY^km�s:p`^km'u!q!c�m�[`[��Yfh^`qe`[��`^`d![`_acY[ �1[ b [`[`v`[ù!i`q4m $ xzy1y1y b q!_a[ù4m �

&4� 0 �1�4^`ZYp`g m�d!^`fh^`^km�ZY[è`[ 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`[ù!i`gYikm�m�[`^`ZYp`g m�d!^`fh^`qe`[��`^`d![`_a^ & y1y1y1y b q!_a[ù4m �

&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[è`[ 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:Ûc 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[ù!^kmV^`ZYd!gYZ m�^`p`q4ni`^%s��gYc b �1q!cY^#-U\�_a^km �1[`ZYcY[`p`_aq!i`[��^"�Tm�[ù��1gYi`_a^km�uAs3j�n_a^ \�_a^kjlu!^ �1[`ZY[è`p`gts:^`[ $ x � j�\.Yp`gYd!^ b [ù!^`f ��p`^%s�(��^��`q!p`gts6)[�(m�[`d!u![km�d!q q!Z b ^kmm�[]\.Y^km�^ �!^ù!gYikjl_agYi`[ �i�(m�[`d!u![km�d!q q!Z b ^km-m�^ b ^ b g �`^ù!p`gY_a^ { \�_a^km ��gYc nb g��

� (m�[`d!u![km�d!q q!Z b ^km#m�[]\.Y^km�^ �!^ù!gYikjl_agYi`["ls3j�ZY[`f+np`_a[ b - \�_a^km�[ ^��1^ ^`cYq��YcYg b gYikm c j b cY^`p`[ b ��gYcYq4nj�m�[.+/�!pù!gY_a^ b u!q!^km �1[`ZYcY[`p`_aq!i`[��^�(��

&4� 0 - �*�1[ù!e`p`g jl_a^ �ù!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[ù!gYi`_a^km b [`ZY[.+�q��1^"hs3j p`[`vù!q!i`^km b q!ZYgY[Q0 = 100

�&4� 0 �4�4^��`q!p`gtsEcYq!c��cY[ù!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[ù!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 �`^ù!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`[`vù!q!i`^km�[km (�)[�(�cYq!c��cY[ù!gYi`_a^km

(CS) b [`ZY[.+�q��1^ �i�(�c \�[ù!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`[`vù!q!i`^km�[km (�)

[�(�cYq!c��cY[ù!gYi`_a^km(CS) b [`ZY[.+�q��1^ �

i�(�c \�[ù!cYq!gYi`_a^km(PS)

[`cYq!ZY[��1gYi`^kmwZY[`cYgYd!^"�&4� � x �*�1[`cYqAs:p`[`_agts [ù![km�[èkj s:u!^`p`^�^`ZYd!g��1u![`_agYi`^")

$ ( +∞∫

2

dx

x

� x (+∞∫

3

dx

x3

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0( +∞∫

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3−xdx� � (

+∞∫

2

dx

x

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1

dx√

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2x

(1 + x2)4dx

�

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e5xdx� �(

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dx

(3x− 5)2

�

&( +∞∫

2

dx

x ln x

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dx

x ln2 x

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1

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x2

� $ x (16∫

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14√

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$ 0( 36∫

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� $ � (3∫

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x dx√9− x2

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dx√4− x2

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x dx√1− x2

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x2dx

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� xzy (e2

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� x1x (1∫

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x2

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�

s:[`p`^ �

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��

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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 �

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y′ = 2x + 5�

x (y′ = x + e2x �

0(y′ = 6

�

� (y′ = 4x + 3

�

{ (y′′ = 3− 2x

�

� �

� �

�(y′′ = 6x + e2x �

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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`[")$ (

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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

�

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�((1 + ex)yy′ = ex �

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x dy − y dx = y dy�s3j

y(2) = 1�

x (y′ =

y

x− 2

�

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0(y′ =

x + y

x− y

�

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ls3jy(0) = 1

�

{ ((y − x)y dx + x2dx = 0

�

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√xy − x)dy = 0

�

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�

�((x− y)dx + (x + y)dy = 0

�s3jy(1) = 0

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d!q!_agYi`^kmw[`cYq!ZY[��`m�gYZY^")$ (

y′ − y

x= x

�

x (y′ +

y

x=

ex

x

�

0(y′ +

2

xy = x3 �

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2x

1− x2y = x + 1

�

{ (y′ + y = e−x �s3j y(0) = 2

�

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x = −2[ù!^kmwcY^`ZY^`c jlcY^km�\�gYu!d!^`_a^"

y(−2) = −9�

x (x = 2

[ù!^km�cY[�� m�^`c jlcY^km�\�gYu!d!^`_a^"y(2) = 4

�

0(x = −3

[ù!^km�cY[�� m�^`c jlcY^km�\�gYu!d!^`_a^"y(−3) = 86

x = 2

[ù!^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

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2�

{ (x = 3

[ù!^km�cY^`ZY^`c jlcY^km�\�gYu!d!^`_a^"y(3) = −6

3

4

�

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cY^`ZY^`c jlcY^km�\�gYu!d!^`_agYi`^`["y(−2) =

y(2) = −4

x = 0[ù!^km cY[�� m�^`c jlcY^km \�gYu!d!^`_a^"

y(0) = 0�

-'(x = 0

[ù!^km�cY^`ZY^`c jlcY^km�\�gYu!d!^`_a^"y(0) = 1

�

�(x =

1

9[ù!^km�cY^`ZY^`c jlcY^km�\�gYu!d!^`_a^"

y(1

9

)

= −1

3�

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�

$ y (U[ù![[��Ypkm�g�� m�d!u!gYc jlcY^km�\�gYu!d!^`_agYi`^ �$'$ (

x = 6[ù!^kmUcY^`ZY^`c jlcY^km\�gYu!d!^`_a^"

y(6) = 12x = 0

[ù!^km�cY[�� m�^`c jlcY^kmw\�gYu!d!^`_a^"y(0) = 0

�

$ x (x = 0


�

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[ù!^km�cY^`ZY^`c jlcY^km�\�gYu!d!^`_a^"y(1) = e

�

$ � (x = −2

[ù!^kmwcY[�� m�^`c jlcY^km�\�gYu!d!^`_a^"y(1) =

1

e

�

� �

$ { (x = 2

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�

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x = e−2

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4

e2

�- � {zy � $ (

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x (ymin = y(−2) = y(2) = −13,

ymax = y(−3) = y(3) = 12�

0(ymin(0) = 0, ymax(4) = 6

�

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(−∞; 2)�1jl[`_ag b ��^ b [ � [.+�ZYg n

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(0; 0)�

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(1;−1)�

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2

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)

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$ y (\�^ù!^w[`cYq�+�ZYg��Y^`_a^`[(−1; 0)

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�- � {1x � $ (

x = −5, y = 0� x (

x = 3, x = 7, y = 0�

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�

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1

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5, y = −3

5

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x (

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$ y (

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$ x (

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� x (ln 5

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5� 0(

2√

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z′x = 2xy3 − 4y2 + 21x2 z′y = 3x2y2 − 8xy�

x (z′x = 12x2y2 − 2y

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�

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z′y = 28xy3 + 15

�

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� �

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2

x(y − 3) ln x

z′y = ln x2 �

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−2xy2

(3x2 − y2)2

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2x2y

(3x2 − y2)2

�

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z′y = 3y2e4x2 �

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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�

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(5x2 − 18)2

z′y =

8y

5x2 − 18�

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x2 − y3

4x2y

z′y =

2y3 − x2

4xy2

�

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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�

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� �

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�

x $ (z′x =

1

x

z′y = − 2y

y2 + 5

�

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4y

x(7x2 + 2y)

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7x2 + 2y�

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y2

x ln 5

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7x3

y ln 2�

x1{ (z′x = yxy−1 z′y = xy ln x

�

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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�

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�

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�

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�

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4 ln 2dx + 16 ln 4 dy

��4� $ y � $ (

z′′xx = 6y4 z′′xy = 24xy3 + 4

z′′yy = 36x2y2 �

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�

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z′′xy =

4

y

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y2

�

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2(y3 − x2)

(x2 + y3)2

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(x2 + y3)2

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(x2 + y3)2

�

� �

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6y3(x2 + 3)

(x2 − 9)3

z′′xy = − 6xy2

(x2 − 9)2

z′′yy =6y

x2 − 9�

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4ex

y

z′′yy = −4ex

y2

�

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2x2 + 18xy + 81y2

(x2 + 9xy)2

z′′xy = − 45x2

(x2 + 9xy)2

z′′yy = − 405x2

(x2 + 9xy)2

�

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$ y (z′′xx =

6x

2 + y3

z′′xy = − 9x2y2

(2 + y3)2

z′′yy =12x3y(y3 − 1)

(2 + y3)2

�

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7t6

(

1− 3

t

)

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x ( 1

t6 +√

t

(

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(

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t

)

�

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�

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)

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√

t2 − 1

t2 + 1

t

(t2 − 1)2

�

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�

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2(x− 7y + 6)�

x ( 6x2 + 9y2 − 11x− y

x− 18xy + 6�

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2y(6y − 7x)

�

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ey + x

�

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x(ln y + 7 · 2x ln 2)

�

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x(1− 9x)�

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(x + y)

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�

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�

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� i�( ≈ 0, 17�

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nr^%s6�

� �

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� i�(4, 5

� � (3, 75

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�

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34K+

L1

3 = 165�Ke`[��`^`d![`_a^!jlZ b [��1[`^.+�[ù b q4m ≈ 4, 8 %

n^%s6�

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i�(

� (

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�

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L = 30, K = 25�

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L = 160, K = 40�

�4� x1{ �Q1 = 95, P1 = 210

�

Q2 = 105, P2 = 305�

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�

P1 = 170, P2 = 130�

Πmax = 28490�

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�

P1 = 460, P2 = 360�

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�

i�(L = 50, K = 75

�

� (L = 120, K = 20

�

b ( L = 37, 5, K = 37, 5�

�� '(1*�&

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x + C�

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x− 3

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∣

x + 1

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3

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44(4x− 3)11 + C

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6(1− 5x)

6

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4 ln 3· 34x+5 + C

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x + C�

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3ex3

+ C�

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36(3x2 − 1)6 + C

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3(1− ln x)

3

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4(1 + x3)

4

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x + C�

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x− 2− 11

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3(1 +

√x)

3

2 + C�

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∣

∣

∣

∣

x− 1

x + 2

∣

∣

∣

∣

+ C�

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04$ ( (x + 1)2

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�

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11+

(1− x)12

12+ C

�

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375(2− 5x)

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