1. :
-- n=(,,) 0(0,0,z0)
(-0)+ (-0)+ (z z0)
-- /+/b+z/c=1
-- ++z+D=0.
2. 1 + 1 + 1z + D1 = 0 (1) 2+ 2 + 2z + D2 = 0 (2).
φ, , :
12+12+12
s φ = ―――――――――――.
√12+12+12 √22+22+22
:
1 1 1
― = ― = ―
2 2 2
: 12+12+12 = 0.
4. :
-- :
11+1+1z+D1 = 0,
22+2+2z+D2 = 0;
-- (1,1,z1) s = (m,n,p):
1 1 z z1
―― = ―― = ――
m n p
5. s1 = (m1,n1,p1) s2 = (m2,n2,p2).
φ :
m1m2+n1n2+p1p2
s φ = ―――――――――――
√m12+n12+p12 √m22+n22+p22
:
m1 n1 p1
― = ― = ―
m2 n2 p2
:
m1m2 + n1n2 + p1p2 = 0.
6. 1 1 z z1
―― = ―― = ―― + +z + D = 0.
m n p
φ :
│m + Bn + Cp│
sin φ = ――――――――――
√2+2+2 √m2+n2+p2
:
m + n + Cp = 0.
:
― = ― = ―.
m n
5. . .
5. . .
1. 3
2. : 1) ; 2)
3) ; 4) ; 5)
6) ; 7)
1. () X () Y, , X = f(); X , Y y.
2. y =f() , f(-) =f(), , /(-) = = -f(x). f() .
3. =f() () X, () f(). .
4. f() X, > , |f()| < , X. .
|
|
5. = f(u) ( U Y), , , = φ() ( X U), X =f[φ()] .
6. :
) = n;
) = , > 0, ≠ 1
( = (-∞;+∞); Y = (0;+∞));
) = log ax, > 0, ≠ 1
( = (0;+∞); Y = (-∞;+∞));
) = sin , = cos , y= tg , = ctg x ) = arcsin x, y = arccos x, y = arctg x, y = arcctg x.
7. , , .
S. = f(x) ≠ 0,
e f(x+ ) =f(x) X.
9. :
a) y=f(x+a)- =f(x) a , ( > 0 , < 0 );
) f(x) + b - = fix) b (b>0 , b < 0 );
) = cf(x) ( ≠ 0) - ( > 1) (0 < < 1) =f(x) ; < 0 ;
) =f(kx) (k≠0) k (k > 1) (0 < k < 1) =f(x) ; k < 0 . 10. () :
x, ≥0 |x|= , <0.
6.
1. n n , (n).
2. (n)
ξ> 0 N, ξ, n > N
│an-A│<ξ(lim an=A
3. y=f(x) > ∞, ξ > 0 S > , ξ, , || > S, │f(x)-A│<ξ (lim f(x)=A)
4. () > 0 ( > ∞), lim () = 0
6. F) > 0, > 0 δ > 0, o , # | - | < δ
│f(x)│>M(lim F(x)=∞.
.
Lim sin x/x =1
u→0
Lim (1+1/x)x=e lim (1+y)1/y=e.
x→∞ y→0
. , , ;
|
|
.
6. .
6. .
. .
1. 4
2. , : 1) y=C, C=const; 2) ; 3) y=sinx
x=2 x=0:
1) ; 2) ; 3)
: 1) y=sin5x; 2) y=cos5x; 3) y=ln(x2+1); 4) y=78x-3; 5) y=(1-2x)50
.
1. y = f(x) ( , ):
Δy f (x+Δx) f (x)
y' = f'(x) = lim ― = lim ――――――.
Δx→0 Δx Δx→0 Δx
x0 ( ) , ( ).
2. y = f ' (x) 0 ( ), ( ). , .