Методичка для геологов-заочников

Hkgh\gu_ ijZ\beZ ^bnn_j_gpbjh\Zgby

Imklv C = const,

u = u(x),

v = v(x)

- ^bnn_j_gpbjm_fu_

nmgdpbb

Lh]^Z

2) (Cu)/ = Cu / ;

3) (u ± v)/ = u / ± v / ;

1) C / = 0;

/

/

/

æ u ö

/

u / v - uv /

4) (uv) = u

v

+ uv

;

5)

ç

÷

=

;

v2

è v ø

_keb y = f (u),

u = u(x), lh y / (x) = y / (u) × u / (x)

( 7 )

ijZ\beh ^bnn_j_gpbjh\Zgby keh`ghc nmgdpbb

Ijhba\h^gZy kl_i_ggh-ihdZaZl_evghc nmgdpbb

(u v )/ = v ×u v−1 ×u / + u v × ln u × v / ,

( 8 )

]^_ u = u(x), v = v(x)

- ^bnn_j_gpbjm_fu_ nmgdpbb

AZ^Zqb k j_r_gb_f

AZ^ZqZ GZclb m]he f_`^m ijyfufb

: 4x + 2 y - 5 = 0

b 6x + 3y +1 = 0

; 3x - y - 2 = 0 b

3x + y -1 = 0 .

J_r_gb_ Ijb\_^_f mjZ\g_gby d \b^m mjZ\g_gbc ijyfuo k

m]eh\uf dhwnnbpb_glhf i

: 4x + 2 y - 5 = 0 Þ y = -2x +

5

Þ k = -2

2

1

6x + 3y +1 = 0 Þ y = -2x -

1

Þ k

2 = -2 .

3

tgϕ = -

3 -

3

=

- 2 3 = 3;ϕ = 60o.

1 -

3 ×

3

- 2

z = 1(cos 0o + i sin 0o )

Wk =

6

z =

6

æ

0 + 2kπ

0

+ 2kπ ö

=

1çcos

6

+ i sin

÷

è

6 ø

= cos 2kπ + i sin 2kπ

= cos kπ + i sin kπ , k = 0,1,2,...5

6

6

3

3

AZ^ZqZ J_rblv kbkl_fm mjZ\g_gbc

3x + 2 y - z = 0

ì

ï

+ 2 y + 9z = 0.

íx

ï

î x + y + 2z = 0

J_r_gb_ Bf__f

3

2

-1

= 3

2

9

- 2

1

9

-1

1

2

= -15 +14 +1 = 0.

1 2 9

1

2

1

2

1

1

1

1

2

Ke_^h\Zl_evgh

kbkl_fZ

bf__l

j_r_gby hlebqgu_ hl

3x + 2 y - z = 0

ì

.

í

îx

+ 2 y + 9z = 0

- 13 -

x = t

2

-1

= 20t; y = -t

3

-1

= -28t; z = t

3

2

= 4t.

2

9

1

9

1

2

AZ^ZqZ GZclb lim

sin(x - 3)

.

x→3 x 2 - 4x + 3

J_r_gb_ Ijb

x ® 3

x - 3 ® 0 ke_^h\Zl_evgh,

lim

sin(x - 3)

= lim

x - 3

= lim

1

=

1

.

x→3 x 2 - 4x + 3

x→3 (x - 3)(x -1)

x→3 x -1 2

AZ^ZqZ GZclb lim

cos 4x - cos 2x

.

x→0

arcsin2 3x

Ijb x ® 0

2

2

sin 3x ~ 3x, sin x ~ x, arcsin 3x ~ 3x

lh _klv

(arcsin 3x)2 ~ (3x)2 Ihwlhfm

lim

cos 4x - cos 2x

= lim - 2 sin 3x sin x = lim

- 2 ×3x × x = -

2

.

x→0

arcsin2 3x

x→0 arcsin2 3x

x→0

(3x)2

3

AZ^ZqZ GZclb ijhba\h^gu_ nmgdpbc

1)

y = (2x3 + 5)4

2)

y = ln(x 2 + 5)

3)

y = x x2 .

J_r_gb_

H[hagZqbf

2x3 + 5 = u

lh]^Z

y = u 4

Ih

ijZ\bem

^bnn_j_gpbjh\Zgby keh`ghc nmgdpbb iZjZ]jZn bf__f

y / = (u 4 )/ × (2x3 + 5)/ = 4u 3 (6x 2 )= 24x 2 (2x3

+ 5)3 .

u

x

2)

(ln(x 2 + 5))/ =

2x

.

x2 + 5

Ih nhjfme_ ihemqZ_f

(x x2 )/

= x 2 × x x2 −1 ×1 + x x2 × ln x × 2x

(x x2 )/

= x x2 +1 (1 + 2 ln x)

ijhoh^ys_c q_j_a lhqdm A(1;2)

2.

J_rblv mjZ\g_gb_ x4 + 81 = 0

3. J_rblv kbkl_fm mjZ\g_gbc

ì 7x -

4 y - z = 3

ï

+ 4z

= -1

í2x + 3y

ï

+ 2z

= -2

4.

î- x + y

GZclb

(

n )

Z lim

n + 1 -

n→∞

[ lim

x -1

x -1

x→1

\ lim

x -1

x -1

x→1 3

] lim

sin x - sin a

x→a

x - a

æ

x

öx

^ limç

÷

x→∞è x + 1

ø

5. GZclb ijhba\h^gu_ nmgdpbc

1

Z y = x

ln x

[ y = arctg

3x - x3

1 - 3x2

æ

1 ö

1

\ y = lnç1 -

÷ +

x

è

x ø

] y = ln sin

xtg

x − x

^ y = 2xtg 2x + ln cos 2x − 2x2 .

ì- 5x + y + z = 0

ï

= 0

í x - 6 y + z

ï

= 0

î x + y - 7z

4. GZclb

2x + 3

Z lim

x→+∞ x + 3

x

3

x2 - 23 x + 1

[ lim

(x

-1)2

x→1

\ lim

x2 - 2x + 6 - x2 + 2x - 6

x2 - 4x + 3

x→3

] lim

cos mx - cos nx

x→0

x2

] y =

x

sin 3x + 9

2.

GZclb - 2 + 2i

ì x + 2 y + z = 2

3.

ï

J_rblv kbkl_fm mjZ\g_gbc í 3x + 2 y + z = 4

ï

î4x + 3y - 2z = 9

æ x + 8

öx

Z lim

ç

÷

x→∞è x - 2

ø

[ lim

sin x - cos x

x→π

π - 4x

4

4 + x + x2 - 2

\ lim

x + 1

x→−1

] lim

1

- cos5x

- cos3x

x→0 1

^ lim

sin 2x

æ

x + 1 ö5

] y = ç

÷

è x +

x ø