18
.
( 2- )
.
:
1: , , , .
2: , , , , .
: , , : + = .
3: . , , .
1( ): 1 (1) 2 (2) 12 = 2 1.
4: .
.
2 ( ): 1(1;1) 2(2;2) :
d = (2 1)2 + (2 1)2
3 ( ): : (1;1), (2;2), (3;3). :
S = ½ ((x2 x1)(y3 y1) (x3 x1)(y2 y1)).
4 ( ): (;) 12 . :
1 + 2 1 + 2
= 1 + = 1 +
-1-
5: - . (;),
( ) , ( ) ( ).
:
p = x2 + y2 x = p cosa
tg a = y/x y = p sina.
:
: + + = 0;
: = + ;
, : 1 = ( 1);
, : 1 1
2 1 = 2 - 1
: / + / = 1.
: k2 k1
: tga = 1 + k1k2
k1 = k2;
k2 = -1/k1.
5( ): (0;0) + + = 0 d. 0 + 0 +
|
|
d =
2 + 2
6: , , , , .
: 2/2 + 2/2 = 1.
7: , , , , .
: 2/2 - 2 /2 = 1.
8: , , , , .
: 2 = 2 (2 = 2).
-2-
:
1. (-5), (4), (-2). , , .
2. // , :
) (3), (11) ) (-5), (-3)
) (-1), (3) ) (1), (-3)
3. (0;0), (3;-4), (-3;4).
) ; ) ; ) .
4. , 5 (1;3).
5. , :
) (2;-3), (3;2), (-2;5); ) (3;-4), N(-2;3), (4;5).
6. 3, (3;1) (1;-3). , , .
7. , , 15 .. (-2;1), (2;2), (4;).
8. N |MK|: |KN| = 2: 3. , (7;4), N(-3;9).
9. , (1;-3) (4;3) . .
10. , (2;-1), (-2;-3), (2;5).
11. 6: 3. , (2;-3) (-4;1).
12. (-3;-5), (3;2). , .
13. ,
(-2;1), (2;-1), (4;3).
14. , :
(2; /2), (3; /4), (3; 3/4), D(4;0), F(2; 3/2), (3;).
15. (0;5), (-3;0), (-1;1),
(2:-3). .
16. (8; /2), (4; -/4), (2; /6). .
17. . :
) 2 3 + 5 = 0; ) 3 + 5 1 = 0; ) 12 -5 -65 =0
18. , =3 =/6.
19. , :
) = ¾ + 2; ) = -3/7 5;
) = 5/2 - 4; ) = -2 + 3.
20. , (2;1) = /4.
21. , :
)(1;3), (4;1); ) (-1;5), D(3;-7); ) (-3;0), N(0;5).
|
|
-3-
22. , (7;0), (3;6), (-1;1).
23. (-2;0), ((2;4), (4;0). , , D.
24. : ) 2 + 5 +20 = 0; ) 6 + -3 = 0;
) 3 4 12 = 0; ) 8 + 4 = 0.
25. :
) 5 +4 = 0 10 2 + 1 = 0; ) 2 + 1 = 0 2 + 1 = 0;
) 3 + 2 + 3 = 0 3 2 1 = 0; ) 5 + 4 = 0 + 5 1 = 0;
) 5 3 + 1 = 0 15 + 9 7 = 0; ) 3 + 2 +17 = 0 2 3 + 8 = 0.
26. :
) = 2 3 = /2 + 1;
) 5 + 7 = 0 2 3 + 1 = 0;
) 2 + = 0 = 3 4.
27. , (-6;-3), (6;7), (2;-1).
28. , (4;2) (1;-7), (-1;3) (8;6).
29. , (6;2) 4 - 7 = 0.
30. , (-4;3) + 2 +3 = 0.
31. , 3 5 + 2 = 0.
32. , (2;3) 4 + 3 12 = 0.
33. , (7;3) (-3;2), .
34. (4;3), (2;1), (4;0), (0;0)
3 + 4 10 = 0.
35. , 2 3 6 = 0 4 - 6 25 = 0 .
36. ,
4 + 3 = 7 3 + 2 = 5 ,
2 + = 5.
37. (-1;3), (4;-2), (0:1), . ?
38. , (2;4) d = 2.
39. , , :
) 32+ 162 = 192; ) 16 2 25 2 = 400; ) 162 92= 144;
) 32 42 =12; ) 22+ 2 = 32; ) 92 + 252 = 225.
40. , , 16 (-10;0), (10;0).
41. , 2 = 6, | F1F2| 8.
-4-
.
:
1: , , m n .
2: , .
3: , .
4: .
5: , aij : i j .
:
1) + = + 5) =
2) ( + ) + = + ( + ) 6) () = ()
3) + 0 +
4) + (- ) + 0 7) (+) = +
6: 11 12
( ) 21 22
, :
|
|
D = 1122 1221.
7: 11 12 13
: = 21 22 23
31 32 33
( ) :
Det A = a11a22a33 + a12a23a31 + a21a32a13 a13a22a31 a23a32a11 a12a21a33.
:
1) , (.. ).
2) ( ) .
3) ( ) .
4) ( ) .
-5-
5) - ( ) ( ), , .
8: ij , , .
9: ij , (-1) (i + j), i j .
i+j
ij = (-1) ij.
:
( ) D .
D = ai1Ai1 + ai2Ai2 + + ainAin
:
, , . : , , .
-1
2- 3- ():
1. .
2. .
3. .
4. 1/D.
= :
-1
=
1. , .
2. .
3. , .
, :
1. , (1;1), (2;2):
1
1 1 1 = 0
2 2 1
-6-
2. (1; 1), (2;), (3;3).
X1 1 1
SABC = 2 2 1
3 3 1
3. (1;1), (2;2), (3;3) :
X1 1 1
2 2 1 = 0
3 3 1
4. 1 + 1 + 1 = 0, 2 + 2 + 2 = 0,
3 + 3 + 3 = 0 :
1 1 1
2 2 2 = 0
3 3 3
5. , (1;1;Z1), (2;2;Z2),
(3;3;Z3):
X - X1 Y - 1 Z - Z1
2 X1 2 - Y1 Z2 Z1 = 0
3 X1 3 Y1 Z3 Z1
:
1. , :
) 2 4 -1 3 ) 2 -1 4 1
= - 1 3 = 1 -4 = 3 0 = -3 -1
5 8 2 3
) 3 1 0 4 2 -3
= 2 -7 4 = 5 7 0 ) = 1 0 3 2 -1
|
|
6 5 2 0 0 1 2 4 8 = 3 5
0 -8
2. 1 3; -2; - 1; 5.
3. :
) 3 2 ) 2 ) 2 + 3
2 -4 0 4 -1 -2 1 -1 2
1) = -1 5 1 = 0 -3 5 = 3 -4 2
0 3 7 2 0 -4 -2 1 5
6 -4 0 -1 2 5 -1
2) = 3 -2 = -2 5 = 4 -2 8
-1 5 4 0
-7-
4. , :
-1 2
) = 3 -1 = 1 1 ) = 3 2 1 = 2 0
1 1 3 1 0 1 2 -3 1
0 -1 2 3 1
) = 2 1 1 = 2 1
3 0 1 1 0
3 7 1
5. = 2 + 2, = 2 -1 = -7 4
0 35 -3
-1 2
6. 3* 2, = 2 0 = 2 -1 0
-3 1 3 2 1
7. :
) -1 4 ) 1 2 3 ) 2 3 -4
5 2 4 5 6 5 6 7
7 8 9 8 0 3
) 3 -1
4 -5 ) 3 2 1 ) 5 0 0
2 5 3 3 2 0
) 2 0 3 4 3 0 7 -1
1 -3
8. :
) 3 2 = 5, ) 5 + 3 = 7,
6 - 4 = 11; 10 6 = 2;
) 5 + 3 = 12, ) 2 3 + z = -7,
2 - = 7; x + 4y + 2z = -1,
x 4y = -5;
) 2 + 3 = 7,
4 5 = 2; ) 2x 7y + z = -4,
3x + y z = 17,
) 2 + 5 = 3, x y + 3z = 3.
4 + 10 = 6;
9. , :
) (2;-3), (4;1) ) (-5;-1), (2;3) ) (8;-2), (-4; 1) ) (0; -2), (3;5).
10. , :
) (1;1), (6;4), (8;2) ) (2;-1), (-5;0), (-1;2).
11. , :
) (2;1), (-1;0), (5;2) ) (1;2), (0;0), (-2;-1) ) (2;-1), (1;2), (3;2).
-8-
12. , :
) 2 5 1 = 0, = 0, + 1 = 0;
) 2 4 = 0, + 1 = 0, + 1 = 0.
13. :
) (2;), (3;1), (-2;4); ) (-1;1), (3;7), (;0)?
14. :
) 2 3 -1 = 0, 2 3 -2 = 0, 2 = 0;
) 5 4 = 0, - + 5 = 0, + 1 = 0;
) + 2 3 = 0, 2 + 2 + = 0, = 4.
) + + = 2, = 1, = -2?
15. , :
) (1;1;2), (4;-1;3), (1;-1; -2); ) (-1;-1;-1;), (2;3;1), (-1;2;2)
16. , :
) 2 -1 ) 3 -4 ) 1 2 3
1 3 1 2 0 -1 2
3 0 7
17. :
) 1 2 7 ) -1 1 3
3 4 * = 17 2 0 * = -2
18. :
) 31 52 = 13, ) 31 42 = -6,
21 + 72 = 81; 31 + 42 = 18.
-9-
.
:
1: 1,2,3,, , n, 1, 2 n, .
2: yn, N( ), : | yn a |< .
Lim (Yn) = a
3: ( ) , m M , m M. , ( M), ( m).
4: ( ) , ( ) c N , n > N : | | > .
Lim (Xn) = 00
5: ( ) , ( ) N , n > N : |n| < .
Lim (Yn) = 0
:
1) lim (X + Y + Z) = limX + limy = limZ
2) lim (XYZ) = limX * limY * limZ
3) lim (X/Y) = (lim X)/(limy)
4) lim (cX) = c limX ( )
n n
5) lim (X) = (limX)
6: , , , .
|
|
7: b f(x) , > 0 > 0, , | x a| < |f(x) b| < .
Lim f(x) = b
x a
(.. b f(x) , ,
-10-
, f(x) b.
8: f(x) , , >0, > 0, |x| > M | A f(x)| < .
Lim f(x) = A
X 00
(.. y = f(x)
( , ), f(x) .)
(0) () :
0 + 0 = 0 + = :
+ 0 = + = 0×
0 × 0 = 0 + 0 = 0/
× 0 = 0 × = /0
0/ = 0 × = /
/0 = / = 0/0
/ = 0
:
1) lim (sin x)/ = 1 2) lim (1 + 1/ x)x =
0 x oo
, P(x) Q(x)
3) lim P(x)/Q(x) = 0, P(x) Q(x)
c1/2, P(x) Q(x)
P(x) Q(x) , 1, 2 .
9: f(x) , , ..
Lim f(x) = f(x)
x a
:
, , .. lim f(x) = lim f(x).
0 + 0
10: 0 1- , f(x) . 0 2- .
-11-
:
1. :
) lim (x2 7x +4); ) lim (x2 + x + 2) / (x2 + x + 1); ) lim x + 5,
x 0 x 1 x -5
) lim 5 / ( -1); ) lim (x2 + 3) / 7; ) lim 12 / (5 )
1
) lim (x2 + x) / 23; ) lim (x3 + 1) / ( 2); ) lim (x + 6) /
x oo 2 -6
2. :
) lim (x2 6x + 9) / (x2 3x); ) lim (x 1 - 2) / (x 5)
x 3 x 5
) lim (x2 5x + 6) / ( 2); )lim (x + 2 - 1) / (x + 1)
2 x -1
) lim (x2 + 2x) / (2 4); ) lim x + 1 - x
-2 x oo
3. :
) lim (x4 3) / (x2 + 5x); ) lim (5x3 + x 1) / (2x3 + 5x2);
x oo x oo
) lim (2x3 x + 5) / (3x3 + 7x + 1); ) lim (x2 + 4) / x;
x oo x oo
) lim (x3 + 3x) / (6 + x x7); ) lim (x2 2x4) / x2;
x oo x oo
) lim (x4 2x7) / (x7 3x5); ) lim (5x + cos x) / x;
x oo x oo
) lim (x2 + 8x 1) / (x5 + 7x3 + 11); ) lim (cos 2x 6x) / (2x + 5);
x oo x oo
e) lim (4x3 + x2 2) / (3x2 + 5x 2); ) lim (x2 cos 2x) / (4 x2).
x oo x oo
4. :
a) lim (sin 2x) / x; ) lim (sin 5x) / 3x;
x 0 x 0
) lim (sin x) / (3x); ) lim (sin 3x) / (sin 7x);
x 0 x 0
) lim (sin 17x) / 8x ) lim (tg 2x) / x.
x 0 x 0
-12-
5. :
x 2x
) lim (x / (x + 1)); ) lim ((x + 2) / x);
x→ ∞ x → ∞
x 3x
) lim (1 + 5/x); ) lim (1 + 2/(3x));
x→ ∞ x → ∞
3x 5x²
) lim (1 + 2/x); ) lim ((x³ - 2x)/ x³).
x → ∞ x → ∞
6. :
) y = (x + 2) / (x 3); ) y = x + 1, ≥ 2
1, x < 2.
) y = lnx / (x - 5);
) y = 2 x, < 1
) y = x / (x + 1); lg x, x ≥ 1.
x
) y = √ x 2 / (x + 7); ) y = (1/2), ≤ -1
2, x > -1
) y = 5 - x², ≥ 1
+ 3, < 1
) y = 1/x, x ≥ -1
x ², x< -1
) y = 2/x + 1, x ≥ 2
x³ - 6, x < 2
) y = x / (x² - 1), x < 2
lg (x 1), x ≥ 2
-13-
.
:
1.( ).
y = f(x) Δ Δ , 0, ..
΄ = f ΄(x) = lim Δy /Δx.
Δx → 0
:
y = f(x) , , ..
k = f ΄(x) = tg φ, φ .
:
.
2: , .
: y y0 = y' (x0)(x x0).
: y y0 = -1/ y'(x0) * (x x0).
:
1) ' = 0 4) (Cu)' = C u', -
2) ()' = 1 5) (uv)' = u'v + v 'u
3) (u + v - w)' = u' + v ' w ' 6) (u/v)' = (u'v v 'u) / v²
7) y(g(x)) = y'(g)×g'(x)
:
n n 1
1) (x) = n (x), 7) (sin x)' = cos x,
2) (√ x)' = 1/2√x, 8) (cos x)' = - sin x,
x x
3) (a)' = a × ln a, 9) (tg x)' = 1/ cos² x,
x x
4) (e)' = e, 10) (ctg x)'= - 1/ sin² x.
5) (ln x)' = 1/x,
6) (log a x)' = 1 / (x lna),
:
1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
6 1 6 15 20 15 6 1
-14-
1 ( ): f(x) g(x) , , g' (a) ≠ 0 f(a) = g(a) = 0 ( = ∞), f(x) g(x) , ..
lim f(x) / g(x) = lim f '(x) / g'(x).
x → a x → a
3: () f(x),
f(x) < f(xo) (f(x) > f(xo)).
2( ): y = f(x) , : f '(xo) = 0/
3 ( ( )): f(x) n , (n-1) , n- . n ( ), . n , , : , , , .
4: y = f(x) , (.. ). , , . , .
4 ( ): y = f(x) , , , .
5: , , .
:
=
y = b
y = kx + b, k = lim f(x)/x, b = lim (f(x) kx) - .
→∞ →∞
-15-
:
1. 1- :
) = (3 )/x²; ) y = x²sinx;
) = 4/x² - 1/x³ + 4x 6; ) y = x²(5 - 4x + x³);
) = (9 - ²)³; ) y = ln²sin4x;
) y = 2sin 5x; ) y = (cos3x 2)³;
5x
) y = sin²3x; ) y = 4 e - 3/x³ + 1;
) y = log 5 x; 3x
) y = ln x³; o) y = 2 + 5.
2. = ² - 1 45?
3. , = 2³ - 4 (1; -2).
4. , y = f(x) , :
) y = √x +5x, xo = 4; ) y = sin 3x 2x, xo = 0.
.
5. , y=f(x) x:
2x
) y = -x³ + 4, x = 1; ) y = lnx -2x, x = ½; ) y = e +x, x =0.
6. y = x²/2 3 2 2 = 0?
7. 3- :
2x x -x
) y = e; ) y = ln x; ) y = e × cos x; ) y = 2 / x;
) = sin²x; ) y = cos 5x; ) y = x lnx; ) y = x² lnx.
8. 4- :
) y = x³/ (2-x); ) y = sin x × ln x; ) y = x² cos x.
9. :
) lim (sin 7x / tg x); ) lim 4ⁿ / (2n); ) lim (2ⁿ -4) / (n -2);
x → 0 n → ∞ n → 2
) lim (ln x / x); ) lim (ln n + n²) / e²ⁿ; ) lim n /(eⁿ - 1).
x → ∞ n → ∞ n → 0
10. :
4 2 2 4 -
) = - 8; ) = 2 - ; ) = ²;
) = 2³ + 6² - 18 + 120; ) = (4)/(1 + ²); ) = lnx/;
4 4 3
) = (3 + 1)/³; ) = 3 - 4; ) = .
11. :
) = ³ - 3² + 1; ) = ³ - 3² - 9 + 11; ) = ( 1)³; ) = (² +4)/.
4 2 8
) = - ³ + 3²; ) = - ; ) = (2 );
12. :
) = ³ -12 + 4; ) = 1/3³ - 2²; ) = /(9 + ²);
) = 2³ - 6; ) = 1/(1+²); ) = ln(² +1).
-16-
5. :
:
1: F(x) f(x) , x F΄(x) = f(x).
2: F(x) + C f(x) : ∫ f(x) dx, f(x) , f(x) dx ,
x .
:
1. , .. ∫ f(x) dx)΄ = f(x).
2. , .. ∫ mf(x) dx = m ∫ f(x) dx.
3. , .. ∫(f(x) φ(x)) dx = ∫f(x) dx ∫φ(x) dx.
4. ∫ f (kx + b) dx = 1/k F(kx +d) +C
3: F(x) + C f(x), F(b) F(a) x x = a x = b
b
∫ f(x) dx, ..
b a
∫ f(x) dx = F(b) F(a)
a
a , b .
B b
∫ f(x) dx = F(x)│= F(b) F(a) .
A a
:
n n+1
1. ∫ dx = (x) / (n +1) + C n ≠ -1
x
2. ∫ dx = e + C
x x
3. ∫ a dx = a / ln a + C
4. ∫ (dx) / = ln |x| + C
5. ∫ cos x dx = sin x + C
6. ∫ sin x dx = - cos x + C
4: , .
5: f(x)dx + φ(x) = 0 .
-17-
6: f(x)F(x) + φ(x)() = 0 .
:
1. dx dy.
2. .
3. .
4. .
5. , .
7: ' + = q, q , .
:
1. ' + = q.
2. = uv, ' = u'v + v'u .
3. , v u . , .
4. .
5. , v u = uv.
6. , .
:
1. , :
) ∫(3 + 1) / dx ) ∫(x2 + x + 5) / 2x dx
) ∫ (x5 + 3ex)dx ) ∫ (x3 + 2x) dx
) ∫3 dx ) ∫ (2/ + 8 +5 -3/5)dx
) ∫sin3x dx ) ∫cos (5 2x)dx
2. ( )
) ∫ sin x cos x dx ) ∫ sin2x cos x dx
) ∫ cos 3x dx ) ∫ tg x dx
) ∫ (2 +3)4dx ) ∫ (9 23)4 2 dx
3. :
) ∫ cos x dx ) ∫ 2 dx
) ∫ sin x dx
4. , :
) = -2 + 4, = 0;
) = 1/, = 0, = 1, = 3;
-18-
) = 6, + 7 = 0
5. :
) d + d = 0; ) 2d = 32d;
) d/ = d /( 1); ) d = d.
6. ( ):
) ' = 2;
) 2d + 2d = 0
) (1 + 2)d -2d = 0;
) 1 + ' + + ' = 0
) d + 2d = 0;
) ' 1 = 0.
7. :
) ' (3/) = ;
) ' + tg = cos2;
) ' + 2/ = 2, ≠0.
-19-