. , , ,

,,,

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y sinx

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́, ́ - , , , , [2] . , - , . , , , .

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, , , , () : . . - , . . , , : .

, , . , , : . . , , , .

, , . , , , . , 12= (-1) 2, 1=1. , .

; ; .

. .

́́ ́ - , . x + iy, x y - , i - - , n n , . . , , - , , , , .

a + bi = c + di , a = c b = d ( , ).

(a + bi) + (c + di) = (a + c) + (b + d) i.

(a + bi) − (c + di) = (ac) + (bd) i.

.

- , - , .

: . : f (x) = x!

.

:

x 0 1 2 3 4 5 6 7 8 9
y 1 1 2 3 5 8 13 21 34 55

1) . - x ( x), y = f (x) .

- y, . .2) - , .3) - , .4) . ( ) - , . ( ) - , .5) () . - , f (-x) = f (x). . - , f (-x) = - f (x). .6) . , M, |f (x) | ≤ M x. , - .7) . f (x) - , T, x : f (x+T) = f (x). . . ( ).

.

- , c x, - y.

- y = ax + b. y a > 0 a < 0. b = 0 .0 (y = ax - )

- = 2 + b + . . > 0, , < 0 - . ( ) - ax2 + bx + =0

- . > I III , < 0 - II IV. - . - = ( > 0) - ( < 0).

y = logax (a > 0)

. . y = sin x (. 19). .


y = cos x . 20; , y = sin x /2.

. , , , .

. - x ( x), y = f (x) .

- y, .

.2) - , .3) - , .4) .

( ) - , .

( ) - , .5) () . - , f (-x) = f (x). . - , f (-x) = - f (x). .6) . , M, |f (x) | ≤ M x. , - .7) . f (x) - , T, x : f (x+T) = f (x). . . ( ).

. .

́ ́ ― , - , (). . : 2 ( ) T1 T2 (T1,T2). 2 ( ) . , , .

.

. : y = axn, a, n - . n = 1 : y = ax; n = 2 - ; n = 1 - . , - . , , , 1, c, n = 0 : y = a, . e. - , , (, , ?). ( a = 1) .13 (n 0) .14 (n < 0). x , :

.

́ ́ - , , . , :

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n- .

n- a , n- a.

: n- a , n- a.

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f (z) = zn . . , f (z) = z. . , , .

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- .

- , .

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- uv, 1695 .

, e. ( ).

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, =y. , , .. , x> x<. , a<ay a>ay. . , x=, .

, 0<a<1. . a=ay x=. 1=1y x=. a=ay, a>0 a≠1.

( ) () >0 : 0 < () < 1 f (x) g (x) , () > 1 - . () < 0. : , f (x) g (x) , , . , () = 0 () = 1 (, ), .

b a ( . λόγος - "", "" ἀριθμός - "" [1] ) , a, b. : . , . : , .

:

, .

f (x) = logax,

:

:

(1; 0)

:

, , . loga = b ( > 0, 1). x = ab.

, , loga = b ( > 0, 1) = b.

. , , , :

loga f () = loga g (), f () = g (), f () >0, g () >0, > 0, 1.

.

.

.

.

, , : loga f () > loga g ().

, . loga f () > loga g () f (x) > g (x) > 0 a > 1 0 < f (x) < g (x) 0 < < 1.

. , , , .

. ( ) - 1/360 . , 360. 60 ( ); - 60 ( ).

. (. " " " . "), l, r : = l / r.

. , l = r, = 1, ,  1 , : = 1 . , :

, (AmB = AO, .1). , , , .

 

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:

 

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:

 

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(; .

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- . (sin x), (cos x), (tg x), (ctg x), , , .


y sinx

:

1. D (y) =R.

2. () = [-1; 1].

3. = sinx - , sin (-x) = - y/R = - sinx, R - , - ().

4. = 2 - . ,

sin (x+p) = sinx.

5. :

: sinx = 0; = pn, nÎZ;

Oy: = 0, = 0,6. :

sinx > 0, xÎ (2pn; p + 2pn), nÎZ;

sinx < 0, Î (p + 2pn; 2p+pn), nÎZ.

> 0 .

< 0 .

7. :

y = sinx [-p/2 + 2pn; p/2 + 2pn],

nÎz [p/2 + 2pn; 3p/2 + 2pn], nÎz.

8. :

 

xmax = p/2 + 2pn, nÎz; ymax = 1;

ymax = - p/2 + 2pn, nÎz; ymin = - 1.

= cosx :

:

1. D (y) = R.

2. () = [-1; 1].

3. = cosx - , cos (-a) = x/R = cosa ()

4. = 2p - . ,

cos (x+2pn) = cosx.

5. :

: cosx = 0;

= p/2 + pn, nÎZ;

: = 0, = 1.

6. :

 

cosx > 0, Î (-p/2+2pn; p/2 + 2pn), nÎZ;

cosx < 0, Î (p/2 + 2pn; 3p/2 + 2pn), nÎZ.

(). :

x > 0 a .

x < 0 a .

7. :

y = cosx [-p + 2pn; 2pn],

nÎz [2pn; p + 2pn], nÎz.

= tgx : -

1. D (y) = (xÎR, x ¹ p/2 + pn, nÎZ).

2. E (y) =R.

3. y = tgx -

4. = p - .

5. :

 

tgx > 0 Î (pn; p/2 + pn;), nÎZ;

tgx < 0 xÎ (-p/2 + pn; pn), nÎZ.


.

6. :

y = tgx

(-p/2 + pn; p/2 + pn),

nÎz.

7. :

.

8. x = p/2 + pn, nÎz -

= ctgx :

:

1. D (y) = (xÎR, x ¹ pn, nÎZ). 2. E (y) =R.

3. y = ctgx - .

4. = p - .

5. :

 

ctgx > 0 Î (pn; p/2 + pn;), nÎZ;

ctgx < 0 Î (-p/2 + pn; pn), nÎZ.


.

6. = ctgx (pn; p + pn), nÎZ.

7. = ctgx .

8. = ctgx , y= tgx p/2 (-1) ()

,

́ ́ ́ ( , ) - , . : ́, ́, ́, . "-" ( . arc - ). , ( , ), . sin−1 ..; , −1.


y=arcsinX, .

m x, y = sinx . y = arcsinx . ( ).

y=arccosX, .

m x,

y = cosx . y = arccosx . cos (arccosx) = x arccos (cosy) = y D (arccosx) = [− 1; 1], ( ), E (arccosx) = [0; π]. ( ). arccos ( -

y=arctgX, .

m α, . .

           

arctg

,

.

y=arcctg, .

m x,

.

. 0 < y < π arcctg ( - x.

.

.

. sin x = a; cos x = a; tg x = a; ctg x = a, x - , aR, .


 

. sin x = a; cos x = a; tg x = a; ctg x = a, x - , aR, .

 


: , . , , , .

1. , , , . 2. ,

B

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. , , .


= .

.6

 

3. , , .

= a .

.7

 

1. , . 2. , .

,

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2.3 , . a α, a || α. 2.4 . - , . b α, a || b a α ( 2.2.1). . a α, a α A. A b, a || b. a b . . 2.5 β a, α, b, b || a. , a b , β. , , a || α. 2.4 b β α.

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: , , , . , , , . , , , .

(1): , , , .

(2): , , .

(3): , .

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1- .

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2- .

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, , , .

AB - α, AC - c - α, C BC. CK AB. CK α ( AB), , , , CK c. AB CK β ( , ). c , β, BC CK , , , , , AC.

, , , .

- a, - - a, . , . a ( , ), , , . b ( , ). b, , , , . , a.

.

, , , , . , , .

, , , , . , , . , , , .

1. , , . , , .

2. , , , , , . AB - α.

AC - , CB - .

- , B - .

.

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.

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. . ,

 

 

 

! , , , .
. , :