graph

x6. •®«-®¥ ¨áá«¥¤®¢ -¨¥ äã-ªæ¨¨ ¨ ¯®áâ஥-¨¥ ¥ñ £à 䨪

41

„«ï - 宦¤¥-¨ï ¨-â¥à¢ «®¢ §- ª®¯®áâ®ï-áâ¢

à¥è ¥¬ -¥à ¢¥-á⢮ y > 0

¨«¨

x

> 0, ®âªã¤

2 < x < 0 ¨ x > 2. ”ã-ªæ¨ï ¯®«®¦¨â¥«ì- -

¨-â¥à-

x2 4

¢ « å ( 2;0) ¨ (2;+1) ¨ ®âà¨æ ⥫ì-

(¢ ᨫã -¥çñâ-®áâ¨) - ¨-â¥à¢ « å

(1; 2) ¨ (0;2).

| x = 2 ¨ x = 2.

5. ”ã-ªæ¨ï ¨¬¥¥â ¤¢¥ â®çª¨ à §àë¢

•®¢¥¤¥-¨¥ äã-ªæ¨¨ -

£à -¨æ¥ ®¡« á⨠®¯à¥¤¥«¥-¨ï:

lim =

x

= 0;

lim

=

x

= 0;

x2 4

x! 1

x!+1

x2 4

x lim2

=

x

= 1;

x lim2+

=

x

= +1;

x2

4

x2

4

!

!

x

;

lim

x

:

xlim2

= x2

4 = 1

4 = +1

x

!

2+ = x2

!

Žвбо¤ б«¥¤г¥в, зв® ¢ в®зª е x = 2 ¨ x = 2 дг-ªж¨п ¨¬¥¥в а §ал¢л ¢в®а®£® த . •ап¬л¥ x = 2 ¨ x = 2 п¢«повбп ¢¥ав¨ª «м-л¬¨ б¨¬¯в®-

⬨, ¯àï¬ ï y = 0 | £®à¨§®-â «ì-®© ᨬ¯â®â®©.

6.• ©¤ñ¬ ¯à®¨§¢®¤-ãî: y0 = (xx22+44)2 . •à®¨§¢®¤- ï ®âà¨æ ⥫ì- - ¢á¥© ç¨á«®¢®© ®á¨, ªà®¬¥ â®ç¥ª x = 2 ¨ x = 2, £¤¥ ®- -¥ áãé¥áâ¢ã¥â (¢ íâ¨å

â®çª å ¨ á ¬ äã-ªæ¨ï - áãé¥áâ¢ã¥â). ”ã-ªæ¨ï ¬®-®â®--® ã¡ë¢ ¥â ¢áî¤ã, £¤¥ ®- ®¯à¥¤¥«¥- .

7. • 室¨¬ ¢â®àãî ¯à®¨§¢®¤-ãî: y00 = 2x(x2+12). •¥è ¥¬ -¥à ¢¥-á⢠y00 >

(x2 4)3

> 0 ¨ y00 < 0. ˆ¬¥¥¬: y00 > 0 ¨«¨ 2x(x2+12) > 0, ®âªã¤ 2 < x < 0, x > 2;

(x2 4)3

y00 < 0, ®âªã¤ x < 2, 0 < x < 2.

•à¨à ¢-¨¢ ï ¢â®àãî ¯à®¨§¢®¤-ãî -ã«î, - 室¨¬ ªà¨â¨ç¥áªãî â®çªã ¢â®à®£® த : x = 0. ˆ§ á奬ë á«¥¤ã¥â, çâ® ¢ â®çª¥ x = 0 äã-ªæ¨ï ¨¬¥¥â ¯¥- ॣ¨¡ (íâ® á«¥¤ã¥â â ª¦¥ ¨ ¨§ -¥çñâ-®á⨠äã-ªæ¨¨). • ¨-â¥à¢ « å ( 2;0) ¨ (2;+1) äã-ªæ¨ï ¢ë¯ãª« ¢-¨§, - ¨-â¥à¢ « å (1; 2) ¨ (0;2) | ¢ë- ¯ãª« ¢¢¥àå. • ©¤ñ¬ ®à¤¨- âã â®çª¨ ¯¥à¥£¨¡ : y¯¥à = 0.

8. •®«¥§-® ¯à®¢¥á⨠í᪨§¨à®¢ -¨¥ í⮣® £à 䨪 . ’ ª ª ª y =	x	,
	(x 2)(x+2)

â® ¯à¨ x ! 0 ¨¬¥¥¬ y 4x; ¯à¨ x ! 2 ¯®«ã稬 y 12 x 12; ¯à¨ x ! 2

-室¨¬ y 12 x+21 ; ¯à¨ x ! 1 ¨¬¥¥¬ y x1 ! 0.

’ª¨¬ ®¡à §®¬, áâ -®¢¨âáï ¯®-ïâ-ë¬ ¯®¢¥¤¥-¨¥ äã-ªæ¨¨ ¯à¨ x = 0, ¢

®ªà¥áâ-®á⨠â®ç¥ª à §àë¢ ¨ - ¡¥áª®-¥ç-®áâ¨. ƒà 䨪 äã-ªæ¨¨ ¨§®¡à ¦ñ- - à¨áã-ª¥.

42	x6. •®«-®¥ ¨áá«¥¤®¢ -¨¥ äã-ªæ¨¨ ¨ ¯®áâ஥-¨¥ ¥ñ £à 䨪

•à¨¬¥à 5. •à®¢¥á⨠¯®«-®¥ ¨áá«¥¤®¢ -¨¥ ¨ ¯®áâநâì £à 䨪 äã-ªæ¨¨

y = 2(xx+1)3 2 .

•¥è¥-¨¥. 1. Ž¡« áâì ®¯à¥¤¥«¥-¨ï | ¢áï ç¨á«®¢ ï ®áì, ªà®¬¥ â®çª¨ x = 1.

2.”ã-ªæ¨ï -¥ ï¥âáï ¯¥à¨®¤¨ç¥áª®©.

3.”ã-ªæ¨ï -¥ ï¥âáï -¨ çñâ-®©, -¨ -¥çñâ-®©.

4.”ã-ªæ¨ï ¨¬¥¥â ®¤-ã â®çªã ¯¥à¥á¥ç¥-¨ï á ®áﬨ ª®®à¤¨- â | â®çªã (0;0).

”ã-ªæ¨ï ¯®«®¦¨â¥«ì-

¯à¨ x > 0 ¨ ®âà¨æ ⥫ì- ¯à¨ x < 0.

5. ”ã-ªæ¨ï ¨¬¥¥â à §àë¢ ¢ â®çª¥ x = 1.

•®¢¥¤¥-¨¥ äã-ªæ¨¨ -

£à -¨æ¥ ®¡« á⨠®¯à¥¤¥«¥-¨ï:

x3

;

lim

x3

;

x lim

= 2(x + 1)2 = 1

x

=

2(x + 1)2 = +1

+

1

! 1

x3

!

x3

x lim1

=

= 1;

x lim1+

=

= 1:

2(x + 1)2

2(x + 1)2

!

!

Žâáî¤ á«¥¤ã¥â, çâ® ¢ â®çª¥ x = 1 äã-ªæ¨ï ¨¬¥¥â à §àë¢ ¢â®à®£® த . •àï¬ë¥ x = 1 ï¥âáï ¢¥à⨪ «ì-®© ᨬ¯â®â®©.

• ©¤ñ¬ ¯ à ¬¥âàë - ª«®--®©	ᨬ¯â®âë:
			x3			1
k = lim yx = lim					=				;
			2x(x + 1)2
x!1	x!1					2
			x3			1
b = xlim (y kx) = xlim							x = 1:
		2(x + 1)2				2
!1	!1

x6. •®«-®¥ ¨áá«¥¤®¢ -¨¥ äã-ªæ¨¨ ¨ ¯®áâ஥-¨¥ ¥ñ £à 䨪					43
“à ¢-¥-¨¥ - ª«®--®© ᨬ¯â®âë:2 y = 21x 1.
6. • ©¤ñ¬ ¯à®¨§¢®¤-ãî: y0 =			x (x+3)	. •¥è ¥¬ -¥à ¢¥-á⢠: y0 > 0 ¨ y0	< 0.
			3
			2(x+1)
ˆ¬¥¥¬: y0 > 0 ¨«¨	x2(x+3)	> 0, ®âªã¤ x < 3, 1 < x < 0, x > 0; y0			< 0,
	2(x+1)3
®âªã¤ 3 < x < 1. •		¨-â¥à¢ « å (1; 3), ( 1;0) ¨ (0;+1) äã-ªæ¨ï
¬®-®â®--® ¢®§à áâ ¥â, -		¨-â¥à¢ «¥ ( 3; 1) | ¬®-®â®--® ã¡ë¢ ¥â.

•à¨à ¢-¨¢ ï¯à®¨§¢®¤-ãî -ã«î, - 室¨¬ ªà¨â¨ç¥áª¨¥ â®çª¨ ¯¥à¢®£® à®- ¤ : x = 0, x = 3. ˆ§ á奬ë (à¨á. )) á«¥¤ã¥â, çâ® ¢ â®çª¥ x = 3 äã-ªæ¨ï ¨¬¥¥â ¬ ªá¨¬ã¬, ¢ â®çª¥ x = 0 íªáâ६㬠-¥â. • ©¤ñ¬ ®à¤¨- âã â®çª¨

¬ªá¨¬ã¬ : ymax = 338.

7.• 室¨¬ ¢â®àãî ¯à®¨§¢®¤-ãî: y00 = (x+1)3x 4 . ‚â®à ï ¯à®¨§¢®¤- ï ¯®«®- ¦¨â¥«ì- - ¨-â¥à¢ «¥ (0;+1) ¨ ®âà¨æ ⥫ì- - ¨-â¥à¢ « å (1; 1) ¨

( 1;0). Šà¨â¨ç¥áª ï â®çª ¢â®à®£® த | x = 0. ˆ§ á奬ë (à¨á. ¡)) á«¥- ¤ã¥â, çâ® ¢ â®çª¥ x = 0 äã-ªæ¨ï ¨¬¥¥â ¯¥à¥£¨¡. • ¨-â¥à¢ « å (1; 1)

¨ ( 1;0) äã-ªæ¨ï ¢ë¯ãª«	¢¢¥àå, - ¨-â¥à¢ «¥ (0;+1) | ¢ë¯ãª« ¢-¨§.
Žà¤¨- â â®çª¨ ¯¥à¥£¨¡	y¯¥à = 0.

8. ƒà 䨪 äã-ªæ¨¨ ¨§®¡à ¦ñ- - à¨á. ¢).

•à¨¬¥à 6. •à®¢¥á⨠¯®«-®¥ ¨áá«¥¤®¢ -¨¥ ¨ ¯®áâநâì £à 䨪 äã-ªæ¨¨

1

y = e x 1 .

•¥è¥-¨¥. 1. Ž¡« áâì ®¯à¥¤¥«¥-¨ï | ¢áï ç¨á«®¢ ï ®áì, ªà®¬¥ â®çª¨ x = 1.

2.”ã-ªæ¨ï -¥ ï¥âáï ¯¥à¨®¤¨ç¥áª®©.

3.”ã-ªæ¨ï -¥ ï¥âáï -¨ çñâ-®©, -¨ -¥çñâ-®©.

4.”ã-ªæ¨ï -¥ ¨¬¥¥â -ã«¥©. Ž- ¯®«®¦¨â¥«ì- - ¢á¥© ç¨á«®¢®© ®á¨, ªà®¬¥ â®çª¨ x = 1. ”ã-ªæ¨ï ¯¥à¥á¥ª ¥âáï á ®áìî Oy ¢ â®çª¥ 0; 1e .

5.”ã-ªæ¨ï ¨¬¥¥â à §àë¢ ¢ â®çª¥ x = 1.

44	x6. •®«-®¥ ¨áá«¥¤®¢ -¨¥ äã-ªæ¨¨ ¨ ¯®áâ஥-¨¥ ¥ñ £à 䨪

•®¢¥¤¥-¨¥ äã-ªæ¨¨ - £à -¨æ¥ ®¡« á⨠®¯à¥¤¥«¥-¨ï:

1

1

lim = e

= 1;

lim

= e

= 1;

x 1

x 1

x! 1

x!+1

1

1

lim = e

= 0

;

lim = e

= +

1

:

x 1

x 1

x

1

x

!

1+

!

Žâáî¤

á«¥¤ã¥â, çâ® ¢ â®çª¥ x = 1 äã-ªæ¨ï ¨¬¥¥â à §àë¢ ¢â®à®£® த .

•àï¬ ï x = 1 ï¥âáï ¢¥à⨪ «ì-®©

ᨬ¯â®â®©, ¯àï¬ ï y = 1 | £®à¨§®--

â «ì-®©

ᨬ¯â®â®©.

1

6. • ©¤ñ¬ ¯à®¨§¢®¤-ãî: y0

1

=

e

x 1

. •à®¨§¢®¤- ï ®âà¨æ ⥫ì- -

(x 1)2

¢á¥© ç¨á«®¢®© ®á¨, ªà®¬¥ â®çª¨ x = 1. ‘«¥¤®¢ ⥫ì-®, äã-ªæ¨ï ¬®-®â®--® ã¡ë¢ ¥â ¢áî¤ã, £¤¥ ®- ®¯à¥¤¥«¥- .

1

7. • 室¨¬ ¢â®àãî ¯à®¨§¢®¤-ãî: y00 =

(2xx 1)14

e

. •¥è ¥¬ -¥à ¢¥-áâ¢

x 1

1

y00

> 0 ¨ y00 < 0. ˆ¬¥¥¬: y00 > 0 ¨«¨

2x 14

e

> 0, ®âªã¤

x >

1

, x = 1;

x 1

2

y00

< 0, ®âªã¤ x < 1.

(x 1)

6

2

•à¨à ¢-¨¢ ï¢â®àãî ¯à®¨§¢®¤-ãî -ã«î, - ©¤ñ¬ ªà¨â¨ç¥áªãî â®çªã ¢â®- ண® த : x = 12. ˆ§ á奬ë á«¥¤ã¥â, çâ® - ¨-â¥à¢ « å 12;1 ¨ (1;+1)

x6. •®«-®¥ ¨áá«¥¤®¢	-¨¥ äã-ªæ¨¨ ¨ ¯®áâ஥-¨¥ ¥ñ £à 䨪			45
®¡à §®¬, ¢ â®çª¥ x =	21	äã-ªæ¨ï ¨¬¥¥â ¯¥à¥£¨¡.	Žà¤¨- â	â®çª¨ ¯¥à¥£¨¡
äã-ªæ¨ï ¢ë¯ãª« ¢-¨§,		- ¨-â¥à¢ «¥ 1; 21	| ¢ë¯ãª« ¢¢¥àå. ’ ª¨¬

y¯¥à = e 2 0; 135.

8. ƒà 䨪 äã-ªæ¨¨ ¨§®¡à ¦ñ- - à¨áã-ª¥.

‡ ¤ ç¨ ¤«ï á ¬®áâ®ï⥫ì-®£® à¥è¥-¨ï

•à®¢¥á⨠¯®«-®¥ ¨áá«¥¤®¢ -¨¥ ¨ ¯®áâநâì £à 䨪 äã-ªæ¨¨:

69.y = x3 3x;

70.y = x33 x2 3x;

71.y = x3 + 6x2 + 9x;

72.y = x33 + x2;

73.y = 1 + 2x2 x44 ;

74.y = x44 + x3;

75.y = x44 x33 ;

76.y = x44 2x2;

77.y = 3x5 5x3;

78.y = x55 x4 + x3;

79.y = (x2 1)3;

80.y = 32x2(x2 1)3; p

81.

y = x + 2 x;

82.

xp

x

;

y = 6p

1

x

83.

y =

;

x+2

p

p

84.

2

+ 1

2

y = p

x

+ p

x

1;

85.

2

2

y = 3

x

1

x

+ 1;

86.

y = px2

1;

p

87.y = 1 3 (x 4)2; p

88.y = 2x 3 3 x2;

46 x6. •®«-®¥ ¨áá«¥¤®¢ -¨¥ äã-ªæ¨¨ ¨ ¯®áâ஥-¨¥ ¥ñ £à 䨪

p

89. y = 1 + 3 (x 1)2;

22

90.y = (x 2)3 (x + 2)3 ;

22

91.y = (x 2)3 + (x + 2)3 ;

2

92. y = x 3 (1 x);

2

93. y = x(x 1)3 ;

94. y = 1 xx2 ;

95. y = x2x 4;

96. y = x2x+1;

97. y = (2xx 1)12 ; 98. y = (3x 22)x2 ; 99. y = (x x2)(x1 5);

100. y = x ;

(x 1)(4 x)

101.y = x2x2 1;

102.y = (xx2+1)+2x2 ;

103.y = (xx2+11)2 ;

104.y = x(2x 43)x+52 ;

105.y = 3xx2 xx2+13;

106.y = xe x2 ;

107.y = (x + 1)e x;

108.y = x2e x;

109.y = (x + 4)2e x2 ;

110.y = xe x22 ;

3 x2

111. y = xe 2 ;

112. y = (1 x)ex; 113. y = (x 2)2ex;

114. y = x3ex;

115. y = x3e x;

116. y = exx ;

117. y = xex2; 118. y = 4(1ex x);

119. y = (1exx)2 ; 120. y = xe2 x3;

121. y = ex+e x ;

ex e x

122. y = ex e x ;

ex+e x

123. y = ex(x2+3);

x6. •®«-®¥ ¨áá«¥¤®¢ -¨¥ äã-ªæ¨¨ ¨ ¯®áâ஥-¨¥ ¥ñ £à 䨪

47

124. y = x2e x2; 125. y = x lnx;

126. y = x lnx;

127. y = x ln2 x;

128. y = x2 ln2 x;

129. y = lnxx ;

130. y = x2 lnx;

131. y = 1+lnx x ;

132. y = lnxx ;

133. y = lnxjxj;

134. y = lnx2x ;

135. y = ln((xx 1)1)2 ;

136. y = lnx2 x ;

137. y = lnx ;

px

138.y = x 32 e x;

139.y = x2 + x2;

140.y = 2x + x12 ;

141.y = 1x3x2 ;

142.y = 1+x3x2 ;

143.y = (xx31)2 ;

144.y = 2((xx 2)1)2 ;

145.y = x + arctgx;

146.y = ((xx+1)1)32 ;

147.y = (1+x4x)3 ;

148.y = x2 + arcctgx;

149.y = x arctg2x;

150.y = x 2arctgx;

1

151. y = (x + 2)e x ;

2

152. y = 1 + xe x ;

1

153. y = e x x;

p

154. y = 3 x2(1 x);

p

155. y = 3 x(1+ x2);

p

156.y = 3 x(3 x)2;

157.y = x 2tgx;

158.y = x + sin2x;

p

159. y = 3 x(2 x2).

Žâ¢¥âë

1. ”ã-ªæ¨ï çñâ- ï. 2. ”ã-ªæ¨ï -¥ ï¥âáï -¨ çñâ-®©, -¨ -¥çñâ-®©.

3. ”ã-ªæ¨ï çñâ- ï.

4. ”ã-ªæ¨ï -¥çñâ- ï.

5. ”ã-ªæ¨ï -¥çñâ- ï.

6. ”ã-ªæ¨ï çñâ- ï.

7. ”ã-ªæ¨ï -¥ ï¥âáï -¨ çñâ-®©, -¨ -¥çñâ-®©.

8. ”ã-ªæ¨ï -¥çñâ- ï.

9. ”ã-ªæ¨ï -¥çñâ- ï.

10. ”ã-ªæ¨ï -¥çñâ- ï.

11.

”ã-ªæ¨ï çñâ- ï.

12. ”ã-ªæ¨ï -¥ ï¥âáï -¨ çñâ-®©, -¨ -¥çñâ-®©.

13.

”ã-ªæ¨ï -¥ ï¥âáï -¨ çñâ-®©, -¨ -¥çñâ-®©.

14.

.

15. 2 .

.

2 .

2

16.

2 .

17.

18. 2 .

19. .

20.

21.

.

22. 3 .

3

3

2

23.

2 .

24. 0 | à §àë¢ ¢â®à®£® த .

25. 0 | à §àë¢ ¯¥à¢®£® த

(᪠箪).

26. 0 | à §àë¢ ¯¥à¢®£® த (᪠箪).

27. 1 | à §àë¢ ¯¥à¢®£®

த

(ãáâà -¨¬ë© à §àë¢).

28.

2n2 1

(n | 楫®¥) | à §àë¢ë ¢â®à®£®

த .

29. 2, 2 | à §àë¢ë ¢â®à®£® த .

30. 2 | à §àë¢ ¯¥à¢®£®

த

(᪠箪).

31. 2 | à §àë¢ ¢â®à®£® த .

32. 0 | à §àë¢ ¢â®à®£®

த .

33. 0 | à §àë¢ ¯¥à¢®£® த (ãáâà -¨¬ë© à §àë¢).

34. 2, 2 |

à §àë¢ë ¯¥à¢®£® த

(ãáâà -¨¬ë¥ à §àë¢ë); 0 | à §àë¢ ¢â®à®£® த .

35.

x = 0, y = 1.

36. y = 1.

37. x = 0, y = 1.

38. x = 21,

y = 2.

439. x = 0,5

y = x.

40. x = 1, y = x 1.

41. x = 0, y = x 1.

42.

x =

7, y = 7.

43. y = x.

44.

y = x.

45. €á¨¬¯â®â

-¥â.

46. y = x + , y = x .

47. y = 4; ¯àï¬ ï x = 5 -¥

ᨬ¯â®â .

48.

y = 0.

49. y = 2x, y = 2x.

50. x = 0, y = x.

51. x = 0,1

y = 1.

52. x = 0, y = x.

53. €á¨¬¯â®â -¥â.

54. x = 2, y = 2.

55.

x = 1, y = x+12 .

56. x = 2, x = 2, y = 1.

57. x = 1, x = 1,

y =

x.

58. y

min

= y(0) =

1, y

max

= y(1) = 1.

59. ymin = y(3) =

1,

2

56

2

88

ymax = y(1) = 3.

60. ymin = y

3

=

9 , ymax = y

3

= 9 .

61. ymin =

=

y

(0) = 1,

y

max =

y

(1) = 8.

62. y

min =

y

(

1) =

0, y

max

= y( 2) = 17.

p

2

63.

ymin = y( ) = 2 , ymax = y( ) = 2 .

(3+ 2)

min

=

4 =

,

2

64. y

y

ymax

= y

= 1.

65. ymin = y( ) =

, ymax = y(0) = 0.

2

3

66.

y

= y(1) = 2, y = y(0; 1) = 10; 1.

67. y

= y(1) =

1,

ymax

= y( 1) =

31.

68. ymin = y(1) = 1, ymax = y(e) = 0.

min

max

min

48