11
a b , :
a b = |a| |b| cos α
a b , a b.
a = {ax; ay} b = {bx; by} :
a b = ax bx + ay by
a = {ax; ay; az} b = {bx; by; bz} :
a b = ax bx + ay by + az bz
n -
n- a = {a1; a2;...; an} b = {b1; b2;...; bn} :
a b = a1 b1 + a2 b2 +... + an bn
1. :
a a ≥ 0
2. , :
a a = 0 <=> a = 0
3. :
a a = |a|2
4. :
a b = b a
5. , :
a ≠ 0, b ≠ 0, a b = 0 <=> a ┴ b
6. (αa) b = α(a b)
7. :
(a + b) c = a c + b c
12
. , :
1) ( , , );
2) (. ., , );
3) , .
, , . . .
.
1. .
2. , .
3. .
4. , . ( .)
5. , ,
, . .
6. , :
. (13)
|
|
31
- , - .
, .. . - . - , . - , , , . , ...
f x, x+Δx, Δx . y = f(x+Δx) f(x). f = lim(f(x+Δx) f(x))/Δx, Δx → 0. . , , , , f^(n) n- , n ≥ 0.
27
, .
: {an}, {yn}, an, yn.
b {yn}, n yn b:
.
lim limes - .
: n : y1 = 1; y2 = 0,5; y3 = 0,33¼; y4 = 0,25; ¼; y100 = 0,01; ¼; y1000 = 0,001; ¼ , 0:
.
: n , .
.
b {yn}, ynb, N, e: |ynb| < e n ³ N (N e).
, , , , .
1. : .
2. () () : .
3. : .
4. : , .
, y(x).
13
39
y = kx | C - =kx, k ≠ 0 - . k =1, .. , . | |||
y = kx + b | : k b - . k = 0.5, b = -1. | |||
y = x 2 | - . | |||
y = ax 2 + bx + c | : a - (a R, a ≠ 0), b, c - . | |||
y = x 3 | . | |||
y = x 1/2 | y = √ x | (x 1/2 = √ x). | ||
y = k/x | (1/x = x -1) - - . k = 1. | |||
y = ex | e - 2,7182818284590... | |||
y = ax | a > 0 a ≠ 1. a. y = 2x (a = 2 > 1). | |||
y = ax | a > 0 a ≠ 1. a. y = 0,5x (a = 1/2 < 1). | |||
y = lnx | e ( ) . | |||
y = log ax | a >0 a ≠1. a. y = log2 x (a = 2 > 1). | |||
y = log ax | a > 0 a ≠ 1. a. y = log0,5 x (a = 1/2 < 1). | |||
y = sinx | . | |||
y = cos x | . | |||
y = tg x | . | |||
y = tg x | . |
|
|
. | ||||
y = arcsin x | y = sin x. [−1; 1]. −π/2 π/2. | |||
y = arccos x | y = cos x. [−1; 1]. 0 π. | |||
y = arctg x | y = tg x. . (−π/2; π/2). . | |||
. | y = arcctg x | y = ctg x. . (0 π). . |
20
19
29
. , .
,
.. .. ,
: .
. , .
. , :
|
|
:
. .
..
..
28
:
, , , , , ,
- .
:
: , , , , .., ..;
- ;
;
.
:
:
24