Математика для юристов - Д.А. Ловцова

A B

D AoB

F ºB

G ºA

H FoG

S (G H)

0

0

1

1

1

1

1

0

1

1

0

1

1

1

1

0

0

1

0

0

1

1

1

1

0

0

1

1

Ȼ. ɉɪɨɰɟɧɬɵ

Ɂ ɚ ɞ ɚ ɱ ɚ . ɐɟɧɚ ɧɚ ɧɟɮɬɶ ɜ

Ɍɚɛɥɢɰɚ ɤ ɬɟɫɬɭ 3,ɛ

ɩɟɪɜɵɣ ɞɟɧɶ ɬɨɪɝɨɜ ɢɡɦɟɧɢɥɚɫɶ ɧɚ

W

p1%, p2%

W

p1%, p2%

p1 ɩɪɨɰɟɧɬɨɜ, ɚ ɜɨ ɜɬɨɪɨɣ – ɧɚ p2

0

40, 25

10

15, 40

ɩɪɨɰɟɧɬɨɜ (ɫɦ. ɬɚɛɥ. ɤ ɬɟɫɬɭ 3,ɛ).

ɇɚ ɫɤɨɥɶɤɨ 'p ɩɪɨɰɟɧɬɨɜ ɢɡɦɟɧɢ-

1

15, 60

11

40, 30

ɥɚɫɶ ɟɟ ɰɟɧɚ ɡɚ ɷɬɢ ɞɜɚ ɞɧɹ?

2

20, 50

12

35, 40

ɉɪɢɦɟɱɚɧɢɹ. Ɂɞɟɫɶ: ɡɧɚɱɟ-

3

25, 16

13

20, 10

ɧɢɟ pk 0 ɨɡɧɚɱɚɟɬ, ɱɬɨ ɰɟɧɚ ɭɩɚ-

4

30, 20

14

40, 15

ɥɚ, ɚ pk!0 – ɰɟɧɚ ɜɨɡɪɨɫɥɚ ɧɚ

5

25, 24

15

50, 20

pk ɩɪɨɰɟɧɬɨɜ (k {1,2}).

6

15, 40

16

20, 10

Ɉɛɪɚɬɢɬɶ ɜɧɢɦɚɧɢɟ ɧɚ ɬɨ, ɱɬɨ

ɜ ɷɬɨɣ ɡɚɞɚɱɟ ɛɚɡɚ ɩɪɢ ɜɵɱɢɫɥɟ-

7

40, 30

17

40, 25

ɧɢɢ ɢɡɦɟɧɟɧɢɣ ɜ ɩɪɨɰɟɧɬɚɯ ɰɟɧɵ ɜ

8

35, 60

18

15, 20

ɩɟɪɜɵɣ ɢ ɜɨ ɜɬɨɪɨɣ ɞɧɢ ɦɟɧɹɟɬɫɹ.

9

24, 50

19

20, 10

W

f(x)

a,

x1,

W

f(x)

a,

x1,

b

x2

b

x2

4

1.0,

1.7,

2

8

0.4,

0.6,

0

4 x

3.0

3.5

10

8ux

8u x

2.0

2.4

x

2

x2

1

x

2

16

16

0.5,

2.0,

11

8

x2

8

6.5,

6.0,

x

4.5

5.5

2

x 8 2

2.5

1.5

2

cos2(x)usin(x)

0.3,

6.0,

12

e x e

x

0.0,

1.0,

0.9

1.0

3

6.0

7.5

0.5,

0.8,

§ x

·

3.0,

1.0,

3

xue2-x

13

x ucos¨

¸

2

2.5

3.0

©

¹

9.0

7.0

2

0.4,

1.0,

64ux

8.0,

10,

4

sin

(x)ucos(x)

2.0

2.2

14

x2 16 2

0.0

1.0

5

xue

x2

2.0,

0.8,

15

4ux

5.0,

1.0,

x2 5

0.0

0.3

7.0

10.0

6

2ux

2

108

59

5.0,

6.0,

16

2ux

2

4ux

2

1.0,

6.0,

x2

1.0

2.5

x

0.2

8.0

x3

8

9.0,

10.0,

3

§ x ·

4.0,

1.0,

7

8ux

1.0

2.0

17

sin

¨

¸ucos(x)

7.2

2.8

8

x2

© 4 ¹

2 x

4

0.8,

0.2,

x3

7

1.0,

1.5,

8

x 2 2

0.8

1.0

18

7ux

5.0

6.0

7

x2

15ux

3

2.0,

1.0,

§ x

·

12,

11,

9

19

x usin¨

¸

18

16

x

2

0.4

0.2

2

1

©

¹

Ɂ ɚ ɞ ɚ ɱ ɚ .

ȼɵɱɢɫɥɢɬɶ lim f(x) ɞɥɹ f(x) ɧɨɦɟɪ W ɩɪɢ ɬɪɟɯ ɡɧɚ-

ɱɟɧɢɹɯ a.

xoa

Ɍɚɛɥɢɰɚ ɤ ɬɟɫɬɭ 4,ɛ

W

f(x)

a

W

f(x)

a

0

x3 3ux2 13ux 15

f,

10

2ux3 5ux2 x 6

f,

3,

5,

4ux

2 9ux 9

2ux3 x2 5ux 2

3

1

1

x3

4ux2 17ux 60

f,

11

4ux3 12ux2 5ux 6

f,

2,

2,

2ux

2 7ux 3

4ux3 9ux2 30ux 8

3

2

2

x3 3ux2

13x 15

f,

12

5ux2 17ux 6

3,

3,

f,

4ux

2 7ux 3

4ux3 9ux2 58ux 15

1

3

3

5ux3 x2

14ux 8

f,

13

2ux2 11ux 15

2,

2,

f,

4ux

2 5ux 6

4ux3 11ux2 6ux 9

2

3

4

2ux2

11ux 15

f,

14

2ux2 3ux 5

f,

1,

1,

5ux

3 39ux

2 67ux 15

5ux3 8ux2 19ux 6

3

1

5

4ux2 9ux 9

1,

15

2ux2 x 6

f,

f,

3,

5ux

3 34ux

2 53ux 12

3ux3 5ux2 16ux 12

3

2

6

4ux2

x 3

1,

16

3ux3 2ux2 53x 60

f,

f,

1,

5x

3 6ux2 23ux 12

5ux2 14ux 3

1

3

7

4ux2 5ux 6

1,

17

x3 x2 17ux 15

3,

f,

f,

5x

3 11ux2 38ux 8

5ux3 16ux2 x 12

2

1

8

4ux3 7ux2 62ux 15

1,

18

2ux3 9ux2 13ux 6

1,

f,

f,

4ux

3 31ux

2 52ux 15

5ux3 18ux2 7ux 6

3

1

9

4ux3 9ux2 58ux 15

1,

19

2ux3 5ux2 x 2

f,

f,

3,

3ux3 16ux2 x 60

4ux2 x 3

3

1

ɡɧɚɱɟɧɢɹɯ a.

f(x)

4ux3 25ux2 38ux 8

,

a

{ f, 2, 2}.

4ux

2 11ux 6

Ɋ ɟ ɲ ɟ ɧ ɢ ɟ .

§

4ux3 25ux2 38ux 8

·

a

f,

lim ¨

¸

¢n 3, m

2, n m 1,

2

¨

4ux

11ux 6

¸

xo f©

¹

ɧɟɱɟɬɧɨɟ.² f.

§

4ux3 25ux2 38ux 8

·

a

2,

¨

¸

¢ɩɪɹɦɚɹ ɩɨɞɫɬɚɧɨɜ-

lim ¨

4ux

2

11ux 6

¸

xo 2©

¹

ɤɚ.²

4u 2 3 25u 2 2 38u 2 8

54

.

4u 2 2 11u 2 6

11

§ 4ux3 25ux2 38ux

8 ·

a

2,

¨

¸

¢ɩɪɹɦɚɹ ɩɨɞɫɬɚɧɨɜ-

lim ¨

4ux

2

11ux 6

¸

xo2©

¹

ɤɚ.²

4u 2 3 25u 2 2 38u 2 8

0

!

(ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ

0

)

4u 2 2 11u 2 6

0

0

ɬɟɥɶ, ɢ ɡɧɚɦɟɧɚɬɟɥɶ f(x) ɞɟɥɹɬɫɹ ɧɚ (x 2):

4ux3 25ux2 38ux 8

x 2

4ux2 11ux 6

x 2

4ux2 17ux 4

4ux 3

4ux3 8ux2

4ux2 8ux

17ux2 38ux

3ux 6

17ux2 34ux

3ux 6

4ux 8

0

4ux 8

0

¢ɩɪɹɦɚɹ ɩɨɞɫɬɚɧɨɜɤɚ.²

4u 2 2 17u 2 4

14

.

4u 2 3

5

Ɍɚɛɥɢɰɚ ɤ ɬɟɫɬɭ 5,ɚ

W

f(x)

W

f(x)

2

§

1

·

0

sin(x

)u

¨

¸

10

tg(x2)u(x 3)2

©

2ux ¹

e

3 x2

§

1 ·

x2

2

1

u¨

¸

11

x2

cos(x2 )

©

¹

2

2 x2

12

4 x2

4

sin(x2 )

2 ln(x)

3

ln(x3 3)

13

3

2

x2 7

x 1

x2 1

4

x2 3

14

e

x

cos2(x)

5ux2 4ux

5

sin2(x)

15

2 x

2

x2 9

3

u x 14

6

e

x

2

u

x

2

3

16

3 x2

2

2x2 2

7

cos2(x)

17

cos(x)

x2 9

x2 2

8

2

4

18

3 x 3 2

ln (x)u(x 3)

x2 2

cos(x2)u

x

x 2 2

9

2

19

tg x2

ɉ ɪ ɢ ɦ ɟ ɪ .

ȼɵɱɢɫɥɢɬɶ

ɩɪɨɢɡɜɨɞɧɭɸ

ɨɬ f(x) ɧɨɦɟɪ W 20:

§

cos

x ·c

ɫɜɨɣɫɬɜɚ,

cos

x c u3 x cos x u 3 x c

¨

3

¸

ɬɚɛɥɢɰɚ

2

¨

x

¸

ɩɪɢɡɜɨɞɧɵɯ

©

¹

3 x

sin x u

1

u

x

u3 x cos

x u

1

u

3 x

x

2

x

3

.

3 x 2

Ɍɚɛɥɢɰɚ ɤ ɬɟɫɬɭ 5,ɛ

W

f(x)

W

f(x)

0

5ux3 18ux2 7ux 6

10

3ux3 16ux2 x 40

1

4ux3 9ux2 58ux 15

11

2ux3 6ux2 48ux 5

2

3ux3 2ux2 53ux 60

12

5ux3 19.5ux2 18ux 40

3

2ux3 5ux2 x 2

13

4ux3 7ux2 62ux 15

4

5ux3 34ux2 53ux 2

14

3ux3 5ux2 16ux 12

5

4ux3 11ux2 6ux 9

15

2ux3 27ux2 108ux 120

6

3ux3 16ux2 x 60

16

5ux3 8ux2 19ux 6

7

2ux3 3ux2 36ux 3

17

4ux3 6.5ux2 4ux 3

8

5ux3 34ux2 53ux 12

18

2ux3 15ux2 36ux 27.5

9

4ux3 31ux2 52ux 15

19

x3 3.5ux2 6ux 11

f(x) x3 0.5ux2 4ux 1

Ɋ ɟ ɲ ɟ ɧ ɢ ɟ .

f(x) –

ɤɭɛɢɱɟɫɤɚɹ ɩɚɪɚɛɨɥɚ, ɨɫɨɛɟɧɧɨɫɬɟɣ

ɧɟ ɢɦɟɟɬ.

ɇɚɯɨɞɢɦ ɧɭɥɢ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ f(x).

f’(x)

3ux2 x 4.

3ux2 x 4 0.

3ux2 x 4 0.

x0

1 1 48

4

,

x1

1 1 48 1,

6

3

6

Ɉɬɫɬɭɩɚɟɦ

ɨɬ ɤɨɪɧɹ

x0 ɜ ɬɨɱɤɭ

xL

6

2. Ɂɞɟɫɶ f’( 2) 5 0

3

f’( 2) 0

f’(x0)

0

f’(0)!0

f’(2) 0

x0

x1

3ux

6 5 4 3 2 1 0

1

2

3

4

5

6

Ɋɢɫ. ɗ