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(y)

(y

)

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f(x) =

(x 2)2x

3

x = 3. D ] , 3[]3, [.x = 3

x = 0 : = y = (2)2

3 = 4

3

y = 0 : = 0 = (x 2)2

x

3

= (x 2)2 = 0 = x = 2 (2; 0)

0; 43

y =

2(x

2)(x

3)

(x

2)2

(x 3)2 =(x

2)(x

4)

(x 3)2y = 0 = x1 = 2, x2 = 4y = = x3 = 3

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y =(x 2)(x 4)

(x 3)2> 0

=//x2

34

+ +y :

ymax = f(2) =

(2

2)2

2 3 = 0.ymin = f(4) =

(4 2)24 3 = 4.

y

=(2x 6)(x 3)2 2(x 3)(x2 6x + 8)

(x 3)4=

2

(x 3)3y

= 0

y = =

2

(x 3)3 > 0=

//x +3] ; 3[

]3; [

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x = 3

k = limx

f(x)

x= lim

x(x 2)2x(x 3) =

(x2)2x2

x(x3)x2

= 1

b = limx

(f(x) k x) = limx

(x 2)2

x 3 x

= limx

1 + 1

x 3

= 1y = k x b = x 1

//x

OOy

x = 3

y = x 1

O

____________

4

4

2

4

3

8/14/2019 L12 101

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f(x) = x ex

D ] , [x = 0 : = y = 0 e0 = 0y = 0 : = 0 = x ex (ex = 0) = x = 0

O(0, 0)

y = ex + x

ex

(

1) = ex(1

x).

y = 0 = x = 1y = = x R

y = ex(1

x) > 0

=//x+ y : 1

]

; 1[

]1;

[

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x = 1

ymax = f(1) = 1 e1 = 1e

y

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

y

: //x +2] ; 2[

]2; [

k = limx

f(x)

x= lim

xx ex

x= lim

xex =

0 x + x

b = limx

(f(x)

k

x) = lim

x(x

ex

0

x) = 0

y = kx + b

k = 0 y = b = 0k = x = 0

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//x

OOy

1

_ _ _ _ _1

e

2

x = 0

y = 0

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f(x) =

4 + x2

4

x2

D ] ; 2[] 2;2[]2; [x = 0 : = y = 1 , OY

(0;1)

y = 0 : x

R , OX

y =2x (4 x2) (2x) (4 + x2)

(4 x2)2 =2x(4 x2 + 4 + x2)

(2 x)2(2 + x)2

//x +y : 0

] ; 0[

]0; [

x = 0

ymin = f(0) =4

4= 1

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y

= 16x(4 x2)2

=

16(3x2 + 4)

(2 x)3(2 + x)3

y

: //x

+2 2

0

]

;

2[

]2;

[,

] 2;2[,

x = 2

x = 2

k = limx

4 + x2

x(4

x2)

= 0

b = limx(f(x) kx) = limx 4+x24x2 = 1

y = 1

x =

2

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//x

OOy

y =1

x =2 x = 21O

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f(x) = 3x ex

2

2

D : ] ; [

x = 0

y = 0

y = 0 x = 0

y = 3(x 1)(x + 1) ex22 .

y

: //x+ +

1 1

0

ymax = f(1) = 3 e12 = 3e

ymin = f(1) = 3 e12

= 3

e

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y

=

3

(x2

3)

e

x2

2

y

: //x + +3 0 3

] 3;0[]3; [,

] ; 3[]0; 3[,

x = 0, x = 3

y

y(3) = 33 e3

2

y(0) =

3

0

e0 = 0

y(

3) = 33 e32

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k = limx 3

x

e

x2

2

x = 0b = lim

x(3x ex22 0) = 0

y = kx + b = 0

//x

OOy

_____ 3e

_ _ _ _ _ 3e

1

1

3

3

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f(x) =x

ln x

D :

ln x = 0x > 0

=

x = 1x > 0

= x ]0;1[]1; [

ln x = 0 x = e0 = 1

k = limx

x

x ln x= 0.

b = limx

xln x

0 x = limx

xlnx

= limx

11x

= limx

x = .

y = 1 ln x 1x x(ln x)2

= ln x 1(ln x)2

x = e ]0; e[,

]e;

[,

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ymin(e) =e

ln e

=e

1

= e.

y

=1x (ln x)2 2 ln x 1

x (ln x 1)

(ln x)4=

2(ln x 2)x (ln x)3

y

= 0 = lnx

= 2 =x

=e2

y

= = x = 0ln x = 0

= x = 0x = e0 = 1

=x = e2

x = 0, x = 1

x = e2, y =

e2

2

y : //x + 0 1 e2

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//x

OO y

_ _ _ _ _ _ _ _ _ _

________________

x = 1

e e2

e

e22

O

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y = f(x)

M(x0, y0)

M

k

y y0 = k (x x0)

k = tg = f(x0)

y y0 = k (x x0)

M

k

= 1k

= 1f(x0)

y = f(x)

M(x0, y0)

y y0 = 1f(x0)

(x x0)

QM

QP M R

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P Q

QP = y0 ctg = y0tg

=y0

f(x0).

T,

ST,

SN

ST = |

QP| =

y0

f(x0) ,

//x

OO y yyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyyy

8888

8888

8888

8888

8888

8888

888

kk

M(x0, y0)

P ROQ

T =

y20 +

y20|f(x0)|2 =

y0

f(x0)

(f(x0))2 + 1,

P R = y0 tg = y0f(x0) SN = |y0y0|,N =

y20 + y

20 y02 = |y0|

1 + y0

2.

y = x2 4x

x = 1

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y = x2 4x. x = 1 y(1) = 3. y = 2x 4.y

|x=1 =

2.

y+3 = 2(x1) 2x+y+1 = 0

k

= 1k

=1

2.

y + 3 =

1

2 (x

1)

x

2y

7 = 0