Αθ.Κεχαγιας - users.auth.gr · Αθ.Κεχαγιας ΠΕΡΙΕΧΟΜΕΝΑ ii 7...
204
Αθ.Κεχαγιας Σημειωσεις : Λογισμός Συναρτήσεων Μιας Μεταβλητής με παράρτημα Αναλυτικής Γεωμετρίας v . 0.86 Θ. Κεχαγιας Απριλης 2010
Transcript of Αθ.Κεχαγιας - users.auth.gr · Αθ.Κεχαγιας ΠΕΡΙΕΧΟΜΕΝΑ ii 7...
-
.
:
v. 0.86
.
2010
-
. iii
v
1 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 132.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 243.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 334.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 395.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6 526.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
i
-
.
ii
7 677.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
8 758.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
9 839.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10 9110.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9110.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9310.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
11 10111.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10111.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10411.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
12 Taylor 11112.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11112.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11312.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
123.1 . . . . . . . . . . . . . 123
126.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
145.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
160.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
-
. , 1 . ,
. (
) .
1. .
2. .
3. .
, . , , . -, , . . , ( . ).
( ) ... .
: , ... . , . , . .
- .
1.
iii
-
.
iv
, .
. . - ( ) . v.0.85 ., - , beta. v.1.00. () .
, 2009
-
. ( Calculus) , 2 .
. , ( ) : f (x) f = f (x+ x)f (x) x x x , . .
- ( ) 3. 17 . , . - (.. , ) ( ) .
. f (x) : f : R R. - . .
, - . .
1. 1 i:1 = i, i2 = 1.
2. R C.
3. :N
n=1 an = a1 + a2 + ...+ aN .
2 .3.. .
v
-
.
vi
4. .
-
. 1
ex . - lnx .
( sinx), ( cosx), (tanx = sinx
cosx), (cotx = cosx
sinx),
(secx = 1cosx
) (cscx = 1sinx
). . ( ) .
.
1.1
1.1.1. ex .
1.1.2. x 1
fn (x) =(
1 +x
n
)(
1 +x
n
) ...
(1 +
x
n
) .
n
x R f1 (x), f2 (x) , ...
0 f1 (x) f2 (x) ... fn (x) fn+1 (x) ... ex
, n , fn (x) ex, :(1 +
x
n
)nn
ex.
1.1.3. lnx :
y = lnx x = ey.
.1 11.
1
-
.
1. 2
1.1.4.
eix = cosx+ i sinx (1.1)
cos (x) = cos x, sin (x) = sinx. (1.2)
(1.1), (1.2) . 12.
1.1.5. (1.1), (1.2)
cosx =eix + eix
2, sinx =
eix eix
2i. (1.3)
1.1.6.
sinh (ix) = i sinx,
sin (ix) = i sinhx,
cosh (ix) = cos x,
cos (ix) = cosh x.
1.1.7. .
arcsin(x) = y x = sin(y)arccos(x) = y x = cos(y)arccos(x) = y x = tan(y)arccot(x) = y x = cot(y)arcsec(x) = y x = sec(y)arccsc(x) = y x = csc(y).
1.1.8. , , .
: sinh(x) =ex ex
2
: cosh(x) =ex + ex
2
: tanh(x) =ex ex
ex + ex
: coth(x) =ex + ex
ex ex
: sech(x) =2
ex + ex
: csch(x) =2
ex ex.
-
.
1. 3
1.1.9. .
arc sinh (x) = y x = sinh(y)arc cosh (x) = y x = cosh(y)arc tanh(x) = y x = tanh(y)arc coth(x) = y x = coth(y)arc sech(x) = y x = sech(y)arc csch(x) = y x = csch(y).
1.1.10.
arc sinh (x) = ln(x+x2 + 1
), < x 1 x < 1 (1.7)
arc sech(x) = ln
(1
x+
1 x2x
), 0 < x 1 (1.8)
arc csch(x) = ln
(1
x+
1 + x2
|x|
), x 6= 0. (1.9)
1.2
1.2.1. (1.3) (1.1) x x
eix = cos (x) + i sin (x) = cos x i sinx (1.10)
( (1.2) ). (1.1) (1.10)
cosx =eix + eix
2.
sinx =eix eix
2i.
1.2.2. cos2 x+ sin2 x = 1.
cos2 x+ sin2 x =
(eix + eix
2
)2+
(eix + eix
2i
)2=ei2x + ei2x + 2
4 e
i2x + ei2x 24
=4
4= 1.
-
.
1. 4
1.2.3. sin(a+ b) = sin a cos b+ cos a sin b
sin a cos b+ cos a sin b =eia eia
2i e
ib + eib
2+eia + eia
2 e
ib eib
2i
=ei(a+b) ei(a+b) + ei(ab) ei(a+b)
4i
+ei(a+b) + ei(a+b) ei(ab) ei(a+b)
4i
=2ei(a+b) 2ei(a+b)
4i= sin (a+ b) .
1.2.4. sin a = tan a1+tan2 a
tan a1 + tan2 a
=sin a/ cos a
1 + sin2 a/ cos2 a=
sin a/ cos a(cos2 a+ sin2 a
)/ cos a
= sin a.
1.2.5. sin(2a) = 2 sin a cos a
2 sin a cos a = 2 eia eia
2i e
ia + eia
2
=eia+ia eiaeia + eiaia eaia
2i
=ei2a ei2a
2i= sin (2a) .
1.2.6. sin a cos b = sin(ab)+sin(a+b)2
sin(a b) + sin(a+ b)2
=sin a cos b cos a sin b+ (sin a cos b+ cos a sin b)
2= sin a cos b.
1.2.7. sin a+ sin b = 2 sin(a+b
2
)cos(ab
2
) 1.2.6
sinA cosB =sin(AB) + sin(A+B)
2.
A = a+b2
, B = ab2
, AB = b, A+B = a,
sin
(a+ b
2
)cos
(a b
2
)=
sin(a) + sin(b)
2
.
-
.
1. 5
1.1: coshx.
1.2.8. coshx. coshx = e
x+ex
2. 1.1.
ex, ex. coshx, . limx coshx = . coshx cosh 0 = 1. coshx ( !) , , :
coshx 1 = ex + ex
2 1 = e
x + ex 22
=
(ex/2 ex/2
)22
0
coshx 1 = 0 ( coshx = 1)
ex/2 ex/2 = 0 ex/2 = ex/2 ex = 1 x = 0.
1.2.9. sinhx. sinhx = e
xex2
. 1.2. ex, ex. sinhx, . limx sinhx = limx sinhx =.
0 = sinhx =ex ex
2 ex = ex x = 0
. sinhx = 0 x = 0.
1.2.10. tanhx.
-
.
1. 6
1.2: sinhx.
tanhx =sinhx
coshx=ex ex
ex + ex.
limx
tanhx = limx
tanhx = limx
ex ex
ex + ex= lim
x
1 e2x
1 + e2x=
1 01 + 0
= 1.
limx tanhx = 1 tanhx = 0 x = 0. 1.3.
1.2.11.
sinh (x) = sinhx, cosh (x) = coshx. (1.11)
sinh (x) = ex e(x)
2= e
x ex
2= sinhx.
cosh (x) = cosh x .
1.2.12. cosh2 x sinh2 x = 1.
cosh2 x sinh2 x =(ex + ex
2
)2(ex ex
2
)2=e2x + e2x + 2
4 e
2x + e2x 24
=4
4= 1.
-
.
1. 7
1.3: tanhx.
1.2.13. sinh(x+ y) = sinh x cosh y + coshx sinh y
sinhx cosh y + coshx sinh y =ex ex
2 e
y + ey
2+ex + ex
2 e
y ey
2
=ex+y ex+y + exy e(x+y)
4
+ex+y + ex+y exy e(x+y)
4
=2ex+y 2e(x+y)
4= sinh (x+ y) .
1.2.14. sinh(2x) = 2 sinhx coshx 1.2.13 y = x
sinh (2x) = sinh(x+ x) = sinh x coshx+ coshx sinhx = 2 sinh x coshx.
1.2.15. sinhx+ sinh y = 2 sinh(x+y
2
)cosh
(xy
2
)
2 sinh
(x+ y
2
)cosh
(x y
2
)= 2 e
x+y2 ex+y2
2 e
xy2 + e
xy2
2
=e
x+y+xy2 exy+xy2 + ex+yx+y2 exyx+y2
2
=ex ex
2+ey ey
2= sinhx+ sinh y.
-
.
1. 8
1.2.16. arc sinh(x) = ln
(x+x2 + 1
) z = arc sinh(x).
x = sinh z =ez ez
2 ez 2x ez = 0.
a = ez, a 2x a1 = 0 a,
a2 2xa 1 = 0.
a
ez = a = xx2 + 1.
a = xx2 + 1 ( a = ez > 0).
a = ez = x+x2 + 1 z = ln
(x+x2 + 1
).
1.2.17. arc sinh (x) R. arc sinh (x) = ln
(x+x2 + 1
).
x2 + 1 > |x| , x +
x2 + 1 > 0
ln(x+x2 + 1
) x R (
). 1.4 arc sinh (x), sinhx x = y ( ;). x R arc sinhx.
-
.
1. 9
1.4: arc sinhx.
1.2.18. arc cosh (x) [1,). arc cosh (x) = ln
(x+x2 1
). , x
x2 1 0 x+x2 1 > 0.
, |x| 1. |x| >x2 1,
x > 0. x > 0 |x| 1 x 1, . , arc coshx, 1.5.
1.5: arc coshx.
1.2.19. arc tanh(x) = 12
ln(
1+x1x
), 1 < x < 1.
arc tanhx, 1.6 ( tanhx x = y).
-
.
1. 10
1.6: arc tanhx.
x 1+x1x > 0 ( !).
1.3
1.3.1. .
1. sin(a b) = sin a cos b cos a sin b
2. cos(a b) = cos a cos b sin a sin b
3. tan(a b) = tan atan b1tan a tan b
1.3.2. cos a = 11+tan2 a
.
1.3.3. .
1. cos(2a) = cos2 a sin2 a = 2 cos2 a 1
2. tan(2a) = 2 tan a1tan2 a .
1.3.4. .
1. sin(3a) = 3 sin a 4 sin3 a
2. cos(3a) = 4 cos3 a 3 cos a
3. tan(3a) = 3 tan atan3 a
13 tan2 a .
1.3.5. .
1. sin a sin b = cos(ab)cos(a+b)2
-
.
1. 11
2. cos a cos b = cos(ab)+cos(a+b)2
1.3.6. .
1. sin a+ sin b = 2 sin(a+b
2
)cos(ab
2
)2. cos a+ cos b = 2 cos
(a+b
2
)cos(ab
2
)3. cos a cos b = 2 sin
(a+b
2
)sin(ba
2
)1.3.7. 1 tanh2(x) = 1
cosh2(x).
1.3.8.
1. sinh(x y) = sinh x cosh y coshx sinh y
2. cosh(x y) = cosh x cosh y sinhx sinh y
3. tanh(x y) = tanhxtanh y1tanhx tanh y
1.3.9.
1. cosh(2x) = cosh2 x+ sinh2 x = 2 cosh2 x 1
2. tanh(2x) = 2 tanhx1+tanh2 x
.
1.3.10.
1. sinhx sinh y = 2 cosh(x+y
2
)sinh
(xy
2
)2. coshx+ cosh y = 2 cosh
(x+y
2
)cosh
(xy
2
)3. coshx cosh y = 2 sinh
(x+y
2
)sinh
(xy
2
).
1.3.11.
1. sinh (ix) = i sinx
2. sin (ix) = i sinhx
3. cosh (ix) = i cosx
4. cos (ix) = i coshx
1.3.12.
1. arc sinh(x) = ln(x+x2 + 1
)( < x
-
.
1. 12
4. arc coth(x) = 12
ln(x+1x1
)(x > 1 x < 1),
5. arc sech(x) = ln(
1x
+
1x2x
), 0 < x 1
6. arc csch(x) = ln(
1x
+
1+x2
|x|
), x 6= 0.
1.3.13.
1. arcsin(x) = i ln(ix+
1 x2
),
2. arccos(x) = i ln(x+x2 1
),
3. arctan(x) = i2
ln(
1ix1+ix
).
-
. 2
, .
2.1
2.1.1. f (x) f0 ( f0) x x0 :
> 0 : > 0 : 0 < |x x0| < |f (x) f0| < . (2.1)
limxx0
f (x) = f0.
2.1.2. (2.1) : f (x) f0 ( ), x x0 ( 0 < |x x0| < ). (2.1) x = x0 (0 < |x x0|).
2.1.3. f (x) ( ) x x0 :
M > 0 : > 0 : 0 < |x x0| < f (x) > M. (2.2)
f (x) ( ) x x0 :
M < 0 : > 0 : 0 < |x x0| < f (x) < M. (2.3)
limxx0
f (x) =, limxx0
f (x) = .
13
-
.
2. 14
2.1.4. f (x) f0 ( f0) x :
> 0 : M > 0 : M < x |f (x) f0| < . (2.4)
f (x) f0 ( f0) x :
> 0 : M < 0 : x < M |f (x) f0| < . (2.5)
limx
f (x) = f0, limx
f (x) = f0.
2.1.5.
limx
f (x) =, limx
f (x) = , limx
f (x) =, limx
f (x) = .
2.1.6. f (x) f0 x x0 :
> 0 : > 0 : 0 < x0 x < |f (x) f0| < . (2.6)
f (x) f0 x x0 :
> 0 : > 0 : 0 < x x0 < |f (x) f0| < . (2.7)
limxx0
f (x) = f0, limxx+0
f (x) = f0.
.
2.1.7.
limxx0
f (x) =, limxx+0
f (x) =, limxx0
f (x) = , limxx+0
f (x) = .
2.1.8. f (x) limxx0 f (x)
limxx0
f (x) = limxx0
f (x) = limxx+0
f (x) .
, limxx0 f (x) = limxx+0 f (x) limxx0 f (x) .
2.1.9. , -
n R : limxx0
xn = xn0 ( xn0 )
-
.
2. 15
limxx0 f (x) = f0limxx0 g (x) = g0
}
limxx0 (kf (x)) = kf0limxx0 (f (x) g (x)) = f0 g0limxx0 (f (x) g (x)) = f0 g0limxx0
(f(x)g(x)
)= f0
g0( g0 6= 0)
limxx0 (f (x))n = fn0 ( fn0 )
2.1.10. limxx0 f (x) = f0 limxf0 g (x) = g0 , limxx0 (g (f (x))) = g0.
2.1.11. limxx0 f (x) = f0, limxx0 g (x) = g0, limxx0 h (x) = h0 f (x) g (x) h (x), f0 g0 h0.
2.1.12. limxx0 f (x) = limxx0 g (x) = , limxx0 (f (x) + g (x)) =,limxx0 (f (x) g (x)) =.
2.1.13. limxx0 f (x) = limxx0 g (x) = , limxx0 (f (x) + g (x)) =, limxx0 (f (x) g (x)) =.
2.1.14. f (x) x0
limxx0
f (x) = f (x0) . (2.8)
limxx0 f (x) f (x0).
2.1.15. (2.8) , f (x) x0. limxx0 f (x), f (x0), limxx0 f (x) 6= f (x0).
2.1.16. f (x) A R x0 A. f (x) ( A), f (x) .
2.1.17. f (x), g (x) x0, kf (x), f (x) g (x), f (x) g (x),
(f(x)g(x)
)( f (x0) 6= 0).
2.1.18. .
2.1.19. ex, sin (x), cos (x) . .
2.1.20. [a, b] R f (x) [a, b], f (a) < f (b). f0 [f (a) , f (b)] x0 [a, b] f0 = f (x0).
-
.
2. 16
2.2
2.2.1. limx2 (2x 1) = 3..
> 0 : > 0 : 0 < |x 2| < |2x 1 3| < .
> 0. = 2 x |x 2| < =2
|2x 1 3| = |2x 4| = 2 |x 2| < 2 = 2
2=
.
2.2.2. limx0 (x2 + 1) = 1..
> 0 : > 0 : 0 < |x 0| < x2 + 1 1 < .
> 0. = x |x 0| =
|x| < = x2 + 1 1 = x2 = |x|2 < 2 = ()2 =
.
2.2.3. limx1 x23x+2(x2)2 = 0.
.
> 0 : > 0 : 0 < |x 1| < x2 3x+ 2(x 2)2
< . > 0. = 1+ x |x 1| < =
1+ x2 3x+ 2(x 2)2
= (x 1) (x 2)(x 2)2 = x 1x 2
., |x 1| <
|x 2| > ||x 1| 1| = |1 |x 1|| > 1
( |x 1| < ) x 1x 2 < 1 .
, , 1 < .
1
< < (1 ) <
+ < (1 + ) < 0 : M > 0 : 0 < |x x0| < M f (x) > M.
, limxx0 f (x) = M ( ) M , x x0 (. |x x0| < M ) f (x) M . ., , f (x) , x x0. (2.3) .
2.2.5. limx0 1x2 =..
M > 0 : M > 0 : 0 < |x 0| < M 1
x2> M.
M > 0. M = 1M
|x| = |x 0| < M =1M x2 < 1
M
1x2 = 1x2 > 11
M
= M
.
2.2.6. (2.4) (2.5);. limx f (x) = f0 (2.4) .
> 0 : M > 0 : M < x |f (x) f0| < .
, limx f (x) = f0 ( ) M , x M f (x) f0 ( ) . ., , f (x) f0 , x . (2.5) .
2.2.7. limx 1x = 0 limx1x
= 0..
< 0 : M > 0 : M < x1x < .
> 0. M = 1 x x > M =1> 0
1x = 1x < 1M = 11 =
. .
-
.
2. 18
2.2.8. limx x2 =..
M > 0 : NM > 0 : NM < x x2 > M.
M NM =M .
2.2.9. limx x3 = ..
M < 0 : NM < 0 : x < NM x3 < NM .
M < 0 NM = M1/3.
2.2.10. limx0 1x = ..
M < 0 : M > 0 : 0 < 0 x < M x < M.
M < 0. M = 1M x 0 < 0 x 0 : M > 0 : 0 < x 1 < M x+ 1
x2 1> M.
M > 0. M = 1M x 0 < x 1
11M
= M
.
2.2.12. , (x), (x), limxx0 (x) = (x0) limxx0 ( (x) + (x)) = (x0) + (x0).
. (x) = a0 + a1x+ ...+ aNxN .
limxx0
(x) = limxx0
(a0 + a1x+ ...+ aNx
N)
= limxx0
(a0) + limxx0
(a1x) + ...+ limxx0
(aNx
N)
= a0 + a1 limxx0
x+ ...+ aN limxx0
xN
= a0 + a1x0 + ...+ aNxN0 = (x0) .
-
.
2. 19
2.1.9. limxx0 (x) = (x0). , 2.1.9
limxx0
( (x) (x)) =(
limxx0
(x)
)(
limxx0
(x)
)= (x0) (x0) .
2.2.13.
1. limx1 (5x2 + 3x 4).
2. limx1 x2x24x+1
3. limx4x4.
4. limx1
5x2 + 3x 4.
. (1) , 2.2.12,
limx1
(5x2 + 3x 4
)= 5 (1)2 + 3 (1) 4 = 2.
(2)
limx1
x 2x2 4x+ 1
=limx1 (x 2)
limx1 (x2 4x+ 1)=
1 212 4 1 + 1
=12
=1
2
2.1.9, - . (3) , 2.1.9 n = 1/4, limx4
x4 =
44 = 16. (4) g (x) =
x, f (x) = 5x2 + 3x 4
, 2.1.10,
limx1
5x2 + 3x 4 =
limx1
(5x2 + 3x 4) =
4 = 2.
2.2.14. , limxx0 f (x) = f0 limxx0 g (x) = g0,
limxx0
(f (x) + g (x)) = f0 + g0.
.
1 > 0 : 1 > 0 : 0 < |x x0| < 1 |f (x) f0| < 1, (2.9)2 > 0 : 2 > 0 : 0 < |x x0| < 2 |g (x) g0| < 2. (2.10)
. (2.9)-(2.10) 1 = 2 = 2 = min (1, 2).
0 < |x x0| < < min (1, 2){|f (x) f0| < 1 = 2|g (x) g0| < 2 = 2
} |f (x) + g (x) (f0 + g0)| |f (x) f0|+ |g (x) g0| 0n : n n |fn | < .
fn limn fn = .
11.1.5. {fn}n=1 + ( ) M > 0nM : n nM fn > M (M < 0nM : n nM fn < M ). fn + ( ) limn fn = + (limn fn = .).
11.1.6. .
11.1.7. {fn}n=1 {gn}n=1 . limn fn = limn gn =
. .
1. k R : limn (k fn) = k .
2. limn (fn + gn) = + .
3. limn (fn gn) = .
101
-
.
11. 102
4. limn(fngn
)=
, 6= 0.
11.1.8. a |a| < 1 fn = an. limn fn= 0.
11.1.9. a a > 1 fn = an. limn fn=.
11.1.10. fn. {sn}n=1 :
s1 = f1, s2 = f1 + f2, s3 = f1 + f2 + f3,..., sn = f1 + ...+ fn,...
s1, s2, s3, ... .
11.1.11. sN =N
n=1 fn limN sN
n=1 fn. ,
n=1 fn .
11.1.12.
n=1 fn
n=1 fn < , ( )
n=1 fn = .
11.1.13.
1.
n=1 fn = a.
n=1 cfn = ca ( a ).
2.
n=1 fn fn+1 ()lim fn = 0.
11.1.16.
n=1 fn
n=1 |fn|
-
.
11. 103
11.1.17. .
11.1.18.
n=1 fn = A R
n=1 gn = B R,
n=1 (fn + gn) = A+B.
11.1.19. a R,
1. |a| < 1N
n=0 an = 1 + a+ a2 + ... = 1
1a .
2. a > 1N
n=0 an = 1 + a+ a2 + ... =.
-
.
11. 104
11.2
11.2.1. ( ) limn 1n = 0. > 0. n > 1 >
1n> 1
n n n.
n n : >1
n=
1n > |fn 0| limn fn = 0.
11.2.2. ( ) limn(
1n2
+ 1)
= 1.
> 0. n >
1 >
(1n
)2>(
1n
)2 n n.
n n : >1
n2=
1n2 + 1 1 > |fn 1| limn fn = 1.
11.2.3. > 0n : n n |fn | < . . , n n, fn
n
fn
+
-
11.1: .
2, .
n n |fn | < .
, (. ). n , . n.
-
.
11. 105
11.2.4. ( ) limn an = 0 |a| < 1. > 0. |a| < 1,
(|a| , 1). n > log|a| = ln ln|| > 0 ( ln < 0,ln |a| < 0).
n > log|a| n ln |a| < ln |a|n < |an 0| <
n n > |an 0| > |an 0|, . limn fn = 0.
11.2.5. limn 1n2+1 .
limn
1
n2 + 1= lim
n
1n2
n2
n2+ 1
n2
=limn
1n2
(limn 1) +(limn
1n2
) = 01 + 0
= 0.
11.2.6. limn n+1n2+1 .
limn
n+ 1
n2 + 1= lim
n
nn2
+ 1n2
n2
n2+ 1
n2
=
(limn
1n
)+(limn
1n2
)(limn 1) +
(limn
1n2
) = 0 + 01 + 0
= 0.
11.2.7. limn n2+n+1n2+1
.
limn
n2 + n+ 1
n2 + 1= lim
n
n2
n2+ n
n2+ 1
n2
n2
n2+ 1
n2
=limn 1 + limn
1n
+ limn1n2
limn 1 + limn1n2
=1 + 0 + 0
1 + 0= 1.
11.2.8. limnsin(n)n2+1
.
0 1. 1
1
xkdx =
[1
1 kx1k
]x=x=1
=1
k 1 1
n=11nk 0. k = 1
n=11n
=, . .
3. k < 1. a
1
xkdx =
[1
1 kx1k
]x=x=a
=1
1 klimM
(M1k a1k
)=
( 1 k > 0) a [0,M). k < 1 n=1
1nk
-
.
11. 108
11.2.20.
n=11n.
n=11n
=
n=11nk
k = 1/2, .
11.2.21.
n=1n
n3+1.
0 ln 0.001ln 0.6
= 13. 523 n 14.
,1
1 0.6= 2.5000,
14n=0
xn = 2.4982.
12.2.19. eix = cosx+ i sinx.
eix = 1 + ix+(ix2)
2!+
(ix3)
3!+
(ix4)
4!+
(ix5)
5!+ ...
=
(1 x
2
2!+x4
4! ...
)+ i
(x x
3
3!+x5
5! ...
)= cosx+ i sinx.
-
.
12. TAY LOR 121
12.3
12.3.1.
n=0 nxn (. (1, 1) ).
12.3.2.
n=0 xn/ (n3 + 1) (. [1, 1] ).
12.3.3.
n=0 (x/n)n (. (,) ).
12.3.4.
n=0 ln (n)xn/n (. [0, 0] ).
12.3.5.
n=0 (1)n+1 xn/n (. (1, 1] ).
12.3.6.
n=0 xn/n(n+ 1) (. [1, 1] ).
12.3.7.
n=0 (nx)n /n! (. [e1, e) ).
12.3.8.
n=0 n!xn (. [0, 0] ).
12.3.9. Taylor f (x) x0: f (x) = ex, x0 = 1 (.e+ e (x 1) +
(12e)
(x 1)2 +(
16e)
(x 1)3 +(
124e)
(x 1)4 + ...).
12.3.10. Taylor f (x) x0: f (x) = coshx, x0 = 0(. = 1 + 1
2x2 + 1
24x4 + ...).
12.3.11. Taylor f (x) x0: f (x) = (x+ 3) sinx,x0 = 0 (. 3x+ x2 12x
3 16x4 + 1
40x5 + ...).
12.3.12. Taylor f (x) x0: f (x) = ex sinx, x0 = 0(. x+ x2 + 1
3x3 + ...).
12.3.13. Taylor f (x) x0: f (x) = sinx1+x , x0 = 0(. x x2 + 5
6x3 5
6x4 + ...).
12.3.14. Taylor f (x) x0: f (x) = sinx2, x0 = 0(. x2 1
6x6 + 1
120x10 + ...).
12.3.15. Taylor f (x) x0: f (x) = sin2 x, x0 = 0(. x2 1
3x4 + 2
45x6 + ...).
12.3.16. Taylor f (x) x0: f (x) = x2 + 2x + 4,x0 = 0 (. 4 + 2x+ x2).
12.3.17. Taylor f (x) x0: f (x) = x2 + 2x + 4,x0 = 1 (. 7 + 4 (x 1) + (x 1)2).
12.3.18. Taylor f (x) x0: f (x) = 11+x , x0 = 0(. 1 x+ x2 x3 + x4 + ...).
12.3.19. Taylor f (x) x0: f (x) = 1(1+x)2 , x0 = 0(. 1 2x+ 3x2 4x3 + 5x4 6x5 + ...)
-
.
12. TAY LOR 122
12.3.20. Taylor f (x) x0: f (x) = 11+x ,x0 = 2(.13 1
9(2 + x) + 1
27(2 + x)2 1
81(2 + x)3+ 1
243(2 + x)4+ ...).
12.3.21. Taylor f (x) x0: f (x) = 11+x3 , x0 = 0(. 1 x3 + x6 ...).
12.3.22. Taylor f (x) x0: f (x) = arctan x, x0 = 0(. x 1
3x3 + 1
5x5 + ...).
12.3.23. Taylor f (x) x0: f (x) = esinx, x0 = 0(. 1 + x+ 1
2x2 1
8x4 1
15x5 + ...).
-
.
.
.1
.1.1. y (x)
y = y, y (0) = 1.
x
1 + x+x2
2!+ ...+
xn
n!+ ... = lim
n
(1 + x+
x2
2!+ ...+
xn
n!
).
, x. x
f (x) = 1 + x+x2
2!+ ...+
xn
n!+ ....
f (x) -.
f (x) =
(1 + x+
x2
2!+x3
3!+ ...+
xn
n!+ ...
)= 0 + 1 + 2
x
2!+ 3
x2
3!
= 1 + x+x2
2+ ... = f (x) .
. f (x) = f (x). ,
f (0) = 1 + 0 +02
2!+ ...+
0n
n!+ ... = 1
f (x) = 1 + x+x2
2!+ ...+
xn
n!+ ....
123
-
.
. 124
. ( .)
.1.2. f (x) exp (x).
.1.3. a R
(exp (ax)) = a exp (ax) exp (a 0) = 1.
y = y, y (0) = 1.
exp (ax) = g (x) = f (h (x)), f (h) =exp (h), h (x) = ax. , ,
[exp (ax)] = g (x) = f (h)h (x) = exp (h) a = a exp (ax) .
, g (0) = exp (a 0) = exp (0) = 1..1.4. a, b R
exp (ax) exp (bx) = exp ((a+ b) x) .
y (x) = exp (ax), z (x) = exp (bx), p (x) = y (x) z (x).
p (x) = (y (x) z (x)) = y (x) z (x) + y (x) z (x)= a exp (ax) exp (bx) + exp (ax) b exp (bx)
= exp (ax) exp (bx) [a+ b] = (a+ b) p (x) .
p (0) = exp (a 0) exp (b 0) = exp (0) exp (0) = 1 1 = 1.
. p (x) = exp (ax) exp (bx)
p (x) = (a+ b) p (x) , p (0) = 1.
exp ((a+ b) x).
p (x) = exp (ax) exp (bx) = exp ((a+ b) x) . (.1).1.5. (.1) x = 1,
exp (a) exp (b) = exp (a+ b) .
exp (x) , .
Ma+b = MaM b
ex = exp (x)
e = exp (1)
( ). exp (x) = ex , . x ex - e.
-
.
. 125
.1.6. , x R,
e = limn
(1 +
1
n
)n, ex = lim
n
(1 +
x
n
)n. (.2)
f (x) = ex
f (x) = f (x) , f (0) = 1. (.3)
x0 R (.3), . f (x) = f(x+x)f(x)
x.
f (x) f (x+ x) f (x)x
= f (x) f (x+ x) (1 + x) f (x) . (.4)
x = x0/n, n, . (.3) :
x {0 x, 1 x, 2 x, ..., n x}
=
{0 =
0 x0n
,1 x0n
,2 x0n
, ...,n x0n
= x0
}.
( )
f(x0n
)(1 + x0
n
)f(
0n
)f(
2x0n
)(1 + x0
n
)f(x0n
)f(
3x0n
)(1 + x0
n
)f(
2x0n
)...
f(nx0n
)(1 + x0
n
)f(
(n1)x0n
)
f (x0) (
1 +x0n
) ...
(1 +
x0n
) f (0) =
(1 +
x0n
)nf (0) .
f (x0) = ex0, f (0) = e0 = 1,
f (x0) (
1 +x0n
)n x 0, . n,
ex0 = f (x0) = limn
(1 +
x0n
)n.
x0 R (.2). x0 = 1
e = e1 = limn
(1 +
1
n
)n(.5)
. (.2). (.5) e. ..
e (
1 +1
1000
)1000= 2. 716 9
e = 2.71828 18284 59045 23536 . . . .
-
.
.1
.1.1. M1,M2,M3 - (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) r1, r2, r3. E M1,M2,M3 r / (x, y, z)
r (u, v) = r1 + u (r2 r1) + v (r3 r1) u, v R. (.1)
.1.2. (.1) x x1 y y1 z z1x2 x1 y2 y1 z2 z1x3 x1 y3 y1 z3 z1
= 0. (.2). M (x, y, z) (.2) E (x1, y1, z1), (x2, y2, z2), (x3, y3, z3).
.1.3. (.1)
x (u, v) = x1 + u (x2 x1) + v (x3 x1) , (.3)y (u, v) = y1 + u (y2 y1) + v (y3 y1) ,z (u, v) = z1 + u (z2 z1) + v (z3 z1) .
.1.4. , E
Ax+By + Cz +D = 0 (.4)
A,B,C,D . , (.4) - . (.4) .
.1.5. (x1, y1, z1), (x2, y2, z2) p = (a, b, c)
x x1 y y1 z z1x2 x1 y2 y1 z2 z1
a b c
= 0. (.5)126
-
.
. 127
r (u, v) = r1 + u (r2 r1) + v p.
.1.6. (x1, y1, z1) ( ) p = (a, b, c) , q = (d, e, f)
x x1 y y1 z z1a b cd e f
= 0. (.6)
r (u, v) = r1 + u p + v q.
.1.7. p = (a, b, c) Ax+By + Cz +D = 0
a
A=
b
B=
c
C. (.7)
.1.8. (x1, y1, z1) p = (a, b, c)
a (x x1) + b (y y1) + c (z z1) = 0. (.8)
.1.9. p = (a, b, c) Ax + By + Cz +D = 0
a A+ b B + c C = 0. (.9)
.1.10. A1x + B1y + C1z + D1 = 0 , A1x + B1y + C1z + D1 = 0
A1A2
=B1B2
=C1C2. (.10)
.1.11. A1x+B1y+C1z+D1 = 0 , A1x+B1y+C1z+D1 = 0
A1A2 +B1B2 + C1C2 = 0. (.11)
.1.12. E : Ax+By+Cz+D = 0 M (x1, y1, z1).
d (M,E) =|Ax1 +By1 + Cz1 +D|
A2 +B2 + C2. (.12)
-
.
. 128
.2
.2.1. (1, 2, 0), (0, 1, 3) (1, 0, 1) .
. . 14.1.
.14.1
(.2) x 1 y 2 z 00 1 1 2 3 01 1 0 2 1 0
= 5x+ y + 2z 7 = 0.2.2.
.
r1 = (1, 2, 0)
r2 = (0, 1, 3)
r3 = (1, 0, 1)
r2r1 = (0 1, 1 2, 3 0) = (1,1, 3)r3r1 = (1 1, 0 2, 1 0) = (0,2, 1)
r (u, v) = (1, 2, 0) + u (1,1, 3) + v (0,2, 1)
r (u, v) =
x (u, v)y (u, v)z (u, v)
= 1 u2 u 2v
3u+ v
.
-
.
. 129
.2.3. (1, 1, 1), (1, 2, 3),(3, 3, 3).
. (.2), x 1 y 1 z 11 1 2 1 3 13 1 3 1 3 1
= 0 4y 2x 2z = 0..2.4. .1.2, . M1 (x1, y1, z1), M2 (x2, y2, z2), M3 (x3, y3, z3)
x x1 y y1 z z1x2 x1 y2 y1 z2 z1x3 x1 y3 y1 z3 z1
= 0.. .14.2. M1,M2,M3 , E
M (x, y, z). M,M1,M2,M3 r, r1, r2, r3.
.14.2
r r1 = (x x1, y y1, z z1)r2r1 = (x2 x1, y2 y1, z2 z1)r3r1 = (x3 x1, y3 y1, z3 z1)
, . {r r1, r2r1, r3r1} , (.2).
.2.5. () Ax+By+Cz+D = 0 () .
-
.
. 130
. (x1, y1, z1), (x2, y2, z2), (x3, y3, z3).
x x1 y y1 z z1x2 x1 y2 y1 z2 z1x3 x1 y3 y1 z3 z1
= 0.
(x x1) y2 y1 z2 z1y3 y1 z3 z1
(y y1) x2 x1 z2 z1x3 x1 z3 z1+(z z1) x2 x1 y2 y1x3 x1 y3 y1
= 0 Ax+By + Cz +D = 0
A =
y2 y1 z2 z1y3 y1 z3 z1 , B = x2 x1 z2 z1x3 x1 z3 z1
, C = x2 x1 y2 y1x3 x1 y3 y1 ,
D = x1 y2 y1 z2 z1y3 y1 z3 z1
+ y1 x2 x1 z2 z1x3 x1 z3 z1 z1 x2 x1 y2 y1x3 x1 y3 y1
., E (x, y, z)
Ax+By + Cz +D = 0. (.13)
A,B,C,D . (.13) (D/A, 0, 0) , (0,B/A, 0), (0, 0,D/C).
x+D/A y 0 z 00 +D/A D/B 0 0 00 +D/A 0 0 D/C 0
= 0D3 + AxD2 +ByD2 + CzD2
ABC= 0
D2
ABC (D + Ax+By + Cz) = 0.
(.13), . (.13) (D/A, 0, 0) , (0,B/A, 0),(0, 0,D/C).
A,B,C,D .
.2.6. (6, 0, 0),(0,3, 0) (0, 0,1).
.
Ax+By + Cz +D = 0
-
.
. 131
6A+ 0B + 0C +D = 00A 3B + 0C +D = 00A+ 0B C +D = 0
A = D/6, B = D/3, C = D.
D
6x+
D
3y +Dz +D = 0
x+ 2y + 6z + 6 = 0
.2.7.
E1 : x+ y + z 1 = 0E2 : x+ 2y + z + 1 = 0
E3 : 3x+ y + 4z = 0
.
x+ y + z 1 = 0x+ 2y + z + 1 = 0
3x+ y + 4z = 0.
Gauss x = 10, y = 2, z = 7. . (10,2, 7).
.2.8.
E1 : x+ y + z 1 = 0E2 : x+ 2y + z + 1 = 0
E3 : 2x+ 3y + 2z = 0
.
x+ y + z 1 = 0x+ 2y + z + 1 = 0
2x+ 3y + 2z = 0
Gauss :
x = 3 t, y = 2, z = t, t R. (.14)
-
.
. 132
. .14.3. , (.14) , .
.14.3
.2.9.
E1 : x+ y + z 1 = 0E2 : x+ 2y + z + 1 = 0
E3 : 2x+ 3y + 2z + 3 = 0
.
x+ y + z 1 = 0x+ 2y + z + 1 = 0
2x+ 3y + 2z = 0
.
-
.
. 133
, .14.4.
.14.4
.2.10. (1, 2, 0), (0, 1, 3) p = (1, 1, 1)
. .(.5) x 1 y 2 z 00 1 1 2 3 0
1 1 1
= 4y 4x 4 = 0
r (u, v) =
x (u, v)y (u, v)z (u, v)
= 12
0
+ u 11
3
+ v 11
1
..2.11. (1, 1, 1), (1, 2, 3) p = (3, 3, 3).
. .(.5) x 1 y 1 z 11 1 2 1 3 1
3 3 3
= 6y 3x 3z = 0.2.12. A (1, 1, 1),B (1, 2, 3) p = (0, 1, 2).
. .(.5) x 1 y 1 z 11 1 2 1 3 1
0 1 2
= 0
-
.
. 134
, , . ,
AB p A,B
p.
.2.13. .1.5, . (x1, y1, z1) , (x2, y2, z2) p.
x x1 y y1 z z1x2 x1 y2 y1 z2 z1
a b c
= 0.. .14.5 (x1, y1, z1), (x2, y2, z2),
p (x, y, z). r1 =(x x1, y y1, z z1), r2 = (x2 x1, y2 y1, z2 z1) p.
.14.5
{r1, r2,p} r1r2p
=x x1 y y1 z z1x2 x1 y2 y1 z2 z1
a b c
0, .
.2.14. (1, 1, 1) p = (1, 2, 3) q = (3, 3, 3).
. .(.6) x 1 y 1 z 1
1 2 33 3 3
= 6y 3x 3z = 0.
-
.
. 135
.2.15. (1, 2, 0) p = (1, 2, 3) q = (1, 1, 1).
. .(.6) x 1 y 2 z 0
1 2 31 1 1
= 2y x z 3 = 0.
r (u, v) =
x (u, v)y (u, v)z (u, v)
= 12
0
+ u 12
3
+ v 11
1
..2.16. .1.6, . (x1, y1, z1) p,q
x x1 y y1 z z1a b cd e f
= 0.. .14.6 (x1, y1, z1), -
p,q (x, y, z). p,q . r =(x x1, y y1, z z1) p,q.
.14.6
{r,p,q} , rpq
=x x1 y y1 z z1a b cd e f
0, .
-
.
. 136
.2.17. p = (1, 2, 1) x+ 2y+ z+ 10 = 0.. A = 1, B = 2, C = 1 D = 10.
1
1=
2
2=
1
1
(.7)
.2.18. p = (1, 2, 1) 2x+4y+2z+3 =0.
. A = 2, B = 4, C = 2 D = 3.
1
2=
2
4=
1
2
(.7).
.2.19. (3,2, 4) p = (2, 2,3).
. Ax + By + Cz + D. (3,2, 4)
3A 2B + 4C +D = 0 (.15)
, ,
A
2=B
2=
C
3. (.16)
, (.15) (.16) A = 15D, B = 1
5D, C = 3
10D, .
D
5x+
D
5y 3D
10z +D = 0
D . D = 10,
2x+ 2y 3z + 10 = 0.
.2.20. .1.9, . p = (a, b, c) E : Ax+By + Cz +D = 0
a
A=
b
B=
c
C.
. .14.7 M1 (x1, y1, z1) , M2 (x2, y2, z2) , M3 (x3, y3, z3)
-
.
. 137
E r1, r2, r3.
.14.7
r2 r1 = (x2 x1, y2 y1, z2 z1)r3 r1 = (x3 x1, y3 y1, z3 z1)
E. q E r2r1, r3r1. - (A,B,C). , (x1, y1, z1) , (x2, y2, z2) E,
Ax1 +By1 + Cz1 +D = 0
Ax2 +By2 + Cz2 +D = 0
A (x2 x1) +B (y2 y1) + C (z2 z1) = 0
. (A,B,C) (r2 r1). (A,B,C) (r3 r1). (A,B,C)E p E p (A,B,C) .
a
A=
b
B=
c
C.
.2.21. (1, 2, 0) p = (1, 2, 3)
. (.8),
1 (x 1) + 2 (y 2) + 3 (z 3) = x+ 2y + 3z 14 = 0.
-
.
. 138
.2.22. .1.8, . (x1, y1, z1) p = (a, b, c)
a (x x1) + b (y y1) + c (z z1) = 0
. .14.8 (x1, y1, z1), p, (x, y, z). r = (x x1, y y1, z z1) p.
.14.8
p r = 0, .
a (x x1) + b (y y1) + c (z z1) = 0
.
.2.23. p = (1, 2, 1) x + 2y 5z + 3 = 0.
. (.10), 1 1 + 2 2 + 1 (5) = 0.
.2.24. .1.9, . E : Ax + By + Cz + D = 0 p = (a, b, c)
a A+ b B + c C = 0.
. E q = (A,B,C). p E
q p = Aa+Bb+ Cc = 0
-
.
. 139
.2.25. E1 : 5x + 2y + z + 6 = 0 E2 : 10x + 4y + 2z = 0 .
. , (.10),
5
10=
2
4=
1
2.
.2.26. x + 2y + z + 1 = 0 2x + 4y + 2z 5 = 0
. .(.10),
1
2=
2
4=
1
2.
.2.27. .1.10, . E1 : A1x + B1y + C1z + D1 = 0 ,E2 : A2x+B2y + C2z +D2 = 0
A1A2
=B1B2
=C1C2.
. E1 p1 = (A1, B1, C1) E2 p2 =(A2, B2, C2).
E1E2 p1p2 p1p2 A1A2 =
B1B2
= C1C2. .14.9.
.14.9
.2.28. x+ y+ z+ 1 = 0 2x+ y 3z 5 = 0 .. (.11), 1 2 + 1 1 + 1 (3) = 0.
.2.29. 5x + 2y + z + 6 = 0 x y 3z = 0 .
. (.11), 5 1 + 2 (1) + 1 (3) = 0.
-
.
. 140
.2.30. E : 5x + 2y + z + 1 = 0 M (1, 0, 0)
. .14.13.
.14.13
.(.12)
d (M,E) =|5 1 + 2 0 + 1 0 + 1|
52 + 22 + 12=
630
=
6
5.
.2.31. M (2, 2, 3) E : 8x 4y z8 = 0.
. .(.12)
d (M,E) =|8 (2) 4 2 1 3 8|
82 + 42 + 12=
35
9.
-
.
. 141
.3
.3.1. (1, 0, 0), (0, 1, 0),(0, 0, 1). .
. x+ y + z 1 = 0.
.3.2. (1, 2, 3), (3, 1,1),(1, 1, 1). .
. 4y 2x 2z = 0.
.3.3. (1, 1,1), (2,2, 2),(1,1, 2). .
. 9y 3x+ 6z = 0.
.3.4. (2, 4, 8), (3, 1, 5),(6,2, 7). .
. 42z 17y 15x 238 = 0.
.3.5. .
x+ 2y + z 1 = 04x+ 2y + z + 1 = 0
x+ y + 4z 2 = 0
. x = 23, y = 4
7, z = 11
21.
.3.6. .
x+ 2y + 2z 2 = 03x+ 2y + 4z 1 = 0x+ 2y + 4z + 1 = 0
. x = 1, y = 2, z = 32.
.3.7. .
x+ 2y + 2z 2 = 03x+ 2y + 4z 1 = 04x+ 4y + 6z 5 = 0
. .
.3.8. (1, 2, 3), (3, 1,1) p = (1, 2, 1). .
. 7x 6y + 5z 10 = 0.
.3.9. (2, 2, 3), (1, 1, 2) p = (1, 1, 1). .
. z y 1 = 0.
-
.
. 142
.3.10. (2, 2, 3), (2, 1, 2) y. .
. x+ z 5 = 0.
.3.11. (2, 2, 3) (2, 1, 2) (1, 1, 0). .
. 2y 2x+ z 3 = 0.
.3.12. (1, 1, 3) (2, 1, 2) (1, 1, 0). .
. 2y 2x+ z 3 = 0.
.3.13. (1, 0, 1) (2, 1, 2) (1, 1, 0). .
. 2y 2x+ z + 1 = 0.
.3.14. (1,5, 4) 4x 7z + 6 = 0. .
. 2y 2x+ z + 1 = 0.
.3.15. 2x 2y + z + 1 = 0 (2,2, 1) .
.3.16. x+y+3z+10 = 0 (1, 1, 3) . ;
.3.17. x+ y+ z 3 = 0 (1, 1,2). .
.3.18. x+ y+ z 3 = 0 (2, 1,3). .
.3.19. 2x y + z + 1 = 0 (2,1,5) . .
.3.20. x+ y + z 3 = 0 x+ y 2z = 0. .
.3.21. x+y+z3 = 0 2x+y3z+12 =0. .
.3.22. (2,3, 6) 2x 5y + 7 = 0. .
. 2x 5y 19 = 0.
.3.23. (3, 4,11) z = 0. .
. z + 11 = 0.
.3.24. (1, 2, 4) 2x 3y 5z + 6 = 0. .
. 2x 3y 5z + 28 = 0.
-
.
. 143
.3.25. (4,2, 1) p = (7, 2,3). .
. 7x+ 2y 3z 21 = 0.
.3.26. (3,2, 4) p = (2, 2,3). .
. 2x+ 2y 3z + 10 = 0.
.3.27. (2, 3, 5) (4, 6, 0). .
. 2x+ 4y 13 = 0.
.3.28. (3,5,2) (4,6, 1). .
. 4x 6y + z 40 = 0.
.3.29. (1,2, 2), (3, 1,2) 2x+ y z + 6 = 0. .
. x 12y 10z 5 = 0.
.3.30. (2, 2, 2) (0,2, 0) x 2y + 3z 7 = 0. .
. 7x 6y z 7 = 0.
.3.31. (2, 2, 2) (0,2, 0) x 2y + 3z 7 = 0. .
. 4x y 2z 2 = 0.
.3.32. (2, 1, 1) (3, 2, 2) x+ 2y 5z 3 = 0. .
. 7x 6y z 7 = 0.
.3.33. (2,1, 6) (1,2, 4) x 2y 2z + 9 = 0. .
. 2x+ 4y 3z + 18 = 0.
.3.34. (3,2, 4) 7x 3y + z 5 = 0 4x y z + 9 = 0. .
. 4x+ 11y + 5z 10 = 0.
.3.35. (1,4, 2) 2x+ 5y z 5 = 0 4x 7y + 3z + 9 = 0. .
. 4x 5y 17z + 10 = 0.
.3.36. (3,2, 1) 3y 5z + 1 = 0 x = 0. .
. 5y + 3z + 7 = 0.
-
.
. 144
.3.37. (1, 1, 2) 2x 2y 4z 6 = 0 3x+ y + 6z 4 = 0. .
. x+ 3y z 2 = 0.
.3.38. (2, 2, 3) 8x 4y z 8 = 0.. 35/9.
.3.39. 2x + y 2z 12 = 0 (2, 2, 3).
. 20/3.
-
.
.1
.1.1. (x1, y1, z1) p = (a, b, c) (. p) (x, y, z)
x x1a
=x x1b
=x x1c
, (.1)
(.1) . (.1)
: x (t) = x1 + t a, y (t) = y1 + t b, z (t) = z1 + t c, (.2) : r (t) = r1 + t p. (.3)
( r = (x, y, z), r1 = (x1, y1, z1) p = (a, b, c)).
.1.2. (.1) (.3) : p t . t = 0, (x1, y1, z1).
.1.3. (x1, y1, z1), (x2, y2, z2)
:x x1x2 x1
=x x1x2 x1
=x x1x2 x1
, (.4)
: x = x1 + t (x2 x1) , y = y1 + t (y2 y1) , z = z1 + t (z2 z1) ,(.5)
: r (t) = r1 + t (r2r1) (.6)
( r = (x, y, z) r1 = (x1, y1, z1), r2 = (x2, y2, z2) ).
145
-
.
. 146
.1.4. :
A1x+B1y + C1z +D1 = 0,
A2x+B2y + C2z +D2 = 0.
.1.5. x = x1 + t a1, y = y1 + t b1, z = z1 + t c1 x = x1 + t a2, y = y1 + t b2, z = z1 + t c2.
cos =a1a2 + b1b2 + c1c2
a21 + b21 + c
21 a22 + b
22 + c
22
. (.7)
.
.1.6. .
.1.7.
x = x1 + t a1, y = y1 + t b1, z = z1 + t c1,x = x2 + t a2, y = y2 + t b2, z = z2 + t c2
.
d =
x2 x1 y2 y1 z2 z1
a b cd e f
(bf ec)2 + (af dc)2 + (ae bd)2
. (.8)
-
.
. 147
.2
.2.1. (1, 2, 0) p = (2, 1, 3)
. (.1)
x 12
=y 2
1=z 0
3
x 12
= y 2 = z3
r (t) =
x (t)y (t)z (t)
= 12
0
+ t 21
3
(. x (t) = 1 + 2t, y (t) = 2 + t, z (t) = 3t
.2.2. (1, 2, 0) p = (2, 1, 3)
. (.1)
x 12
=y 2
1=z 0
3.
x 12
= y 2 = z3
.2.3. (0, 2,1) - p = (1, 0, 4).
. .(.3),
x = t, y = 2 0 t = 2, z = 1 + 4t. (.9)
, (.1)
x
1=y 2
0=z + 1
4. (.10)
y20
; (.10) t
x
1=y 2
0=z + 1
4= t. (.11)
y20
= t, t , y 2 = 0, . y = 2.
-
.
. 148
(.9). , xy( .15.1).
.15.1
.2.4. .1.1, . (x1, y1, z1) p = (a, b, c)
x x1a
=x x1b
=x x1c
, (.12)
x (t) = x1 + t a, y (t) = y1 + t b, z (t) = z1 + t c, (.13)r (t) = r1 + t p. (.14)
. .15.2 (x1, y1, z1), p, (x, y, z) . r = (x x1, y y1, z z1) p.
.15.2
-
.
. 149
r = tp, .x x1a
=y y1b
=z z1c
= t
(.12).
x x1a
= t x = x1 + aty y1b
= t y = y1 + btz z1c
= t z = z1 + ct
(.13). r = (x, y, z), r1 =(x1, y1, z1) (.14).
.2.5. (1, 2, 0), (0, 1, 3)..
x 10 1
=y 21 2
=z 03 0
.
1 x = 2 y = z3
r (t) =
x (t)y (t)z (t)
= 12
0
+ t 0 11 2
3 0
= 12
0
+ t 11
3
(. x (t) = 1 t, y (t) = 2 t, z (t) = 3t ).
.2.6. (2, 1, 3) (4, 2,2).
. .(.4)
x+ 2
4 + 2=y 12 1
=z 32 3
.x+ 2
6=y 1
1=z 35
x (t) = 2 + 6t, y (t) = 1 + t, z (t) = 3 5t.
.2.7. .1.2, . (x1, y1, z1),(x2, y2, z2)
x x1x2 x1
=y y1y2 y1
=z z1z2 z1
, (.15)
x (t) = x1 + t (x2 x1) , y (t) = y1 + t (y2 y1) , z (t) = z1 + t (z2 z1) , (.16)r (t) = r1 + t (r2 r1) . (.17)
-
.
. 150
. .15.3 (x1, y1, z1), (x2, y2, z2). p = (x2 x1, y2 y1, z2 z1) .
.15.3
(x2 x1, y2 y1, z2 z1) ,
x x1x2 x1
=y y1y2 y1
=z z1z2 z1
(.18)
(.18) t, (.16), .17).
.2.8. x 1
1=y 2
2=z 3
3.
..
x 11
=y 2
2=z 3
3= t
, t
x = 1 + t, y = 2 + 2t, z = 3 + 3t.
.2.9.
x (t) = 1 + t, y (t) = 2 t, z (t) = 1 + 2t.
..
x 1 = t, y 2 = t, z 1 = 2t
x 11
=y 21
=z 1
2= t.
-
.
. 151
.2.10. (2,3, 1), (5, 4,4), (8, 11,9) .. (2,3, 1), (5, 4,4).
x 25 2
=y + 3
4 + 3=
z 14 1
(8, 11,9):
8 25 2
=11 + 3
4 + 3=9 14 1
= 2.
.2.11. x 1
1=y 2
2=z 3
3. (.19)
z = 1.. (.19) z = 1
x 11
=y 2
2=
1 33
= 23.
x 1 = 1 (2
3
) x = 1 2
3=
1
3
y 2 = 2 (2
3
) y = 2 + 4
3=
10
3.
(
13, 10
3, 1).
.2.12.
x (t) = 1 + t, y (t) = 2 t, z (t) = 1 + 2t. (.20)
z = 2.. z = 2 (.20)
2 = 1 + 2t t = 12
x = 1 + t = 1 +1
2=
3
2
y = 2 t = 2 12
=3
2.
(
32, 3
2, 2).
-
.
. 152
.2.13. -
2x 3y + 3z 4 = 0 (.21)x+ 2y z + 3 = 0 (.22)
. (.21) (.22) x = 37z 1
7, y = 5
7z 10
7.
t = z
x = 37t 1
7, y =
5
7t 10
7, z = t
. t -
x+ 1/7
3/7=y + 10/7
5/7=z
1= t.
.15.4.
.15.4
.2.14.
x+ 2y + z + 1 = 0
2x+ y + 3z + 2 = 0
. .,
x (t) = 53t 1, y (t) = 1
3t, z (t) = t.
-
.
. 153
r (t) =
x (t)y (t)z (t)
= 10
0
+ t 5/31/3
1
..2.15.
l1 :x+ 5
2=y 5
1=z 5
4
l2 :x+ 2
3=y 4
1=z 3
2.
.. (5, 5, 5),
x+ 5
2= y 5
y 5 = z 54
x+ 2
3= y 4
y 4 = z 32
.2.16.
l1 :x+ 1
2=y 5
3=z 71
l2 :x+ 4
4=y 1
6
z 32
.. l1 p1= (2, 3,1) l2
p2= (4, 6,2). p1p2 l1l2..2.17.
l1 :7x 15
2=
7y + 34
4=z
1 l2 :
x+ 3
1=y 2
1=z
2
.. l1 p = (2,4, 1) l2
q = (1, 1, 2). p q =0 p,q l1, l2.
.2.18. l1 (2, 4, 3) l2 (1, 3, 4) (2, 2, 3).
. l2 x 12 1
=y 32 3
=z 43 4
.
. l2 p = (3,1,1). l1 x+ 2
3=y 41
=z 31
.
-
.
. 154
.2.19.
x = 1 + t, y = 1 + t, z = 2 + t,
x = 2t, y = 1 + t, z = 0.
.
d =
0 1 1 1 0 1
1 1 20 1 0
1 21 02 + 1 20 0
2 + 1 10 12
=
5
5
( .15.6)
.15.6
.2.20.
x = 2 + 2t, y = 1 + t, z = 1 t,x = 3t, y = 1 + 2t, z = 1.
.
d =
0 2 1 1 1 1
2 1 13 2 0
1 1 1 11 12 + 2 13 0
2 + 2 13 22
=2
5.
.
-
.
. 155
.2.21. A (1, 1,2)
l1 :x 1
2=y + 5
2=z
1.
. l1
x = 1 + 2t y = 5 + 2t z = t.
M (1 + 2t,5 + 2t, t) M A
f (t) = (1 1 2t)2 + (1 + 5 2t)2 + (2 t)2 = 9t2 20t+ 40
f (t) f (t) = 0, . t = 10/9. M A
(1 + 2 10
9,5 + 2 10
9, 10
9
)= (29/9,25/9, 10/9)
d (A,M) =f (10/9) =
2609' 5.3748 -
.
-
.
. 156
.3
.3.1. (0, 2,1) - p = (1,3, 4). .
. x = y23 =z+1
4.
.3.2. (0, 0, 0) p = (1, 1, 1). .
. x = y = z.
.3.3. (2, 4, 3) - (1, 3, 4) (2, 2, 3). .
. x 3y + 14 = 0, y z 1 = 0.
.3.4. (2, 1, 3) (4, 2,2). .
. x+26
= y11
= z35 .
.3.5. (4, 1, 3) (4, 2,2). .
. x40
= y11 =z3
5 x = 4, y = 1 t, z = 3 + 5t.
.3.6. x11
= y23
= z48
. .
. x = 1 + t, y = 2 + 3t, z = 4 + 8t.
.3.7. x = 2 + 3t, y = 1 3t, z = 4 +8t. .
. x23
= y13 =z+4
8.
.3.8.
2x y + t 6 = 0x+ 4y 2t 8 = 0.
, z = 1. .
. x = 329 2
9t, y = 5
9t+ 10
9, z = t x32/92 =
y10/95
= z9. x = 10
3,
y = 53, z = 1.
.3.9.
2x 3y + 3z 4 = 0x+ 2y z + 3 = 0.
.. x = 3
7t 1
7, y = 5
7t 10
7, z = t x+1/73 =
y+10/75
= z7.
-
.
. 157
.3.10. - ; , ;
. x = 35t 1, y = t, z = 7
5t+ 2
.3.11. (2,3, 1), (5, 4,4), (8, 11,9) . - .
.3.12. y, z
x 23
=y + 4
2=z 1
2
x = 3. .. (3,14/3, 5/3) .
.3.13. x 1
1=y 2
2=z 3
3.
z = 6. .. x = 1, y = 4, z = 6.
.3.14.
x (t) = 2 + t, y (t) = 2 t, z (t) = 1 + t.
z = 2. .. x = 3, y = 1, z = 2.
.3.15. y, z
x = 4 3t, y = 1 + 4t, z = 2t 3
z = 5. .. x = 7, y = 5, z = 5.
.3.16.
x+ 1
2=y 5
3=z 71
x+ 4
5=y 13
z 31
. .
.3.17.
7x 152
=7y + 34
5=z
1
x+ 3
1=y 2
1=z
1
. .
-
.
. 158
.3.18.
2x 3y + 3z 4 = 0x+ 2y z + 3 = 0
x = t, y = 2t, z = t. .
.3.19. (2, 4, 3) (3, 1, 1) .
. x 3y + 14 = 0, y z 1 = 0.
.3.20.
x 12
=y 7
1=z 3
4x 6
3=y + 1
2=z + 2
1
. .. .
.3.21.
x 22
=y
3=z + 1
4x 76
=y 2
9=
z
12
. .. .
.3.22.
x = t, y = 2t+ 3, z = t+ 2 x = 2t+ 1, y = 4t, z = 2t+ 766
. .. .
.3.23.
x+ y + z + 1 = 0
x+ 2y + 3z = 0
2x+ 3y 4z 1 = 04x+ 6y = 0
. .. .
-
.
. 159
.3.24.
x 12
=y 22
=z
1x
2=y + 5
3=z 4
1
. .. .
.3.25.
x+ 2y 5z 1 = 0x 2y + 3z 9 = 0
x+ y 3z + 1 = 0x y + z + 3 = 0
. .. .
.3.26. (3,1, 4)
x
3=y
2=z 14
x 1
2=
y
3=z 2
2.
.. x3
8= y+1
14= z4
13.
.3.27. (3,2, 4) ,(5, 3,2) (0, 4, 2). .
. x18/1126
= y22
= z+9/4427
.
-
.
.1
.1.1. .. R (x0, y0, z0). (x0, y0, z0) R,
(x x0)2 + (y y0)2 + (z z0)2 = R2
(. F (x, y, z) = (x x0)2 + (y y0)2 + (z z0)2, c = R2).
.1.2. , . F (x, y, z)
Ax2 +By2 + C z2 +Dxy + E yz + F zx+Gx+H y + J z +K = 0
. x, y, z.
.1.3. , x, y, z
Ax2 +By2 + Cz2 = D, (.1)Ax2 +By2 + Cz = 0. (.2)
.1.4. (.1)
x
2
a2+ y
2
b2+ z
2
c2= 1
x2
a2+ y
2
b2 z2
c2= 1
z2
a2 y2
b2 x2
c2= 1
x2
a2+ z
2
b2= y
2
c2
x2
a2+ y
2
b2= 1
x2
a2 y2
b2= 1
a, b, c ( A,B,C,D).
160
-
.
. 161
.1.5. (.2)
z = x2
a2+ y
2
b2
z = x2
a2 y2
b2
z = x2
a2
a, b ( A,B,C).
.2
.2.1. (1, 1, 1) 4.. (x 1)2 + (y 1)2 + (z 1)2 = 42. .16.1.
.16.1
.2.2. A (1, 2, 3) B (2, 3, 4).
.
d (A,B) =
(2 1)2 + (3 2)2 + (4 3)2 =
3.
(x 1)2 + (y 2)2 + (z 3)2 = 3.
.2.3.
x2 2x+ y2 4y + z2 6z + 13 = 0
.
-
.
. 162
. .
x2 2x+ y2 4y + z2 6z + 13 = 0x2 + 2 1 x+ 12 + y2 2 2 y + 22 + z2 2 3 z + 33 + 13 12 22 32 = 0
(x 1)2 + (y 2)2 + (z 3)2 = 1.
(1, 2, 3) 1.
.2.4.
x2 4x+ y2 2y + z2 4 = 0
.. ,
:
0 = x2 4x+ y2 2y + z2 4= x 2 2 x+ 22 22 + y2 2 y 1 + 12 12 + z2 4= (x 2)2 + (y 1)2 + z2 32
:
(x 2)2 + (y 1)2 + z2 = 32
(2, 1, 0) 3.
.2.5. (1, 2, 3) x.
. (x0, y0, z0) x d =y20 + z
20. .16.2.
.16.2
-
.
. 163
R =
22 + 32 =
13
(x 1)2 + (y 2)2 + (z 3)2 = 13.
.2.6. (3, 6,4) E : 2x 2y z 10 = 0.
. E
d =|3 2 6 2 + 4 1 10|
22 + 22 + 12= 4
(x 3)2 + (y 6)2 + (z + 4)2 = 42.
.2.7. (1, 1, 1), (2, 0, 2), (1, 0, 1) (0, 1, 1).
.
x2 + y2 + z2 + Ax+By + Cz +D = 0
12 + 12 + 12 + A 1 +B 1 + C 1 +D = 022 + 02 + 22 + A 2 +B 0 + C 2 +D = 012 + 02 + 12 + A 1 +B 0 + C 1 +D = 002 + 12 + 12 + A 0 +B 1 + C 1 +D = 0.
.
A+B + C +D = 32A+ 2C +D = 8A+ C +D = 2B + C +D = 2
A = 1, B = 1, C = 5, D = 4.
x2 + y2 + z2 x y 5z + 4 = 0
x2 212x+
1
22+ y2 21
2y +
1
22+ z2 25
2z +
52
22+ 4 1
4 1
4 25
4= 0(
x 12
)2+
(y 1
2
)2+
(z 5
2
)2=
(3
2
)2.
. (1/2, 1/2, 5/2) 3/2.
-
.
. 164
.2.8. xy (5, 4, 0) 6. E : 3x+2y+6z1 =0.
.
S : (x 5)2 + (y 4)2 + z2 = 62
z = 0. .16.3.
.16.3
S () : x2 + y2 + z2 10x 8y + 5 z = 0
S () E. . S () d E R () S (). S () 3 5 + 2 4 + 6 2 132 + 22 + 62
=
100 + 64 + 2
4 5
(22 + 3
7
)2=
164 + 2
4 5
: = 32013
= 16. :
S1 : x2 + y2 + z2 10x 8y + 5 + 320
13z = 0
S2 : x2 + y2 + z2 10x 8y + 5 + 16z = 0
-
.
. 165
,
S1 : (x 5)2 + (y 4)2 +(z 160
3
)2=
(178
13
)2S2 : (x 5)2 + (y 4)2 + (z 8)2 = 102.
.2.9. E1 : x 2z 8 = 0,E2 : 2x z + 5 = 0 l1 : x = 2, y = 0.
. E3 E1 E2( .16.4)
.16.4
E3
E3 :1
2 (x 2z 8) + 1
2 (2x z + 5) = 0
.E3 : x z 1 = 0
l1
x = 2, y = 0, z = (2 1) = 3.
(2, 0,3) E1:
d =|1 (2) + 0 0 + (2) (3) 8|
12 + 02 + 22=
45
(x+ 2)2 + y2 + (z + 3)2 =16
5
-
.
. 166
.2.10. x2
a2+ y
2
b2+ z
2
c2= 1
. .16.5.
.16.5
z = c1 [c, c] ( xy ) .
x2
a2+y2
b2= 1 c
21
c2
. . yz zx.
.2.11. ( (0, 0, 0) R) , a = b = c = R.
. , a = b = c = R
x2
R2+y2
R2+z2
R2= 1 x2 + y2 + z2 = R2.
.2.12.
x2
16+y2
9+z2
4= 1
. a = 4, b = 3, c = 2.
.2.13.
2x2 + 3y2 + z2 8x+ 6y 4z 3 = 0
-
.
. 167
.. .
2x2 + 3y2 + z2 8x+ 6y 4z 3 = 02 (x2 2 2 x+ 22
)+ 3
(y2 + 2 y 1 + 12
)+(z2 2 2 z + 22
) 4 1 4 3 = 0
2 (x 2)2 + 3 (y + 1)2 + (z 2)2 = 12(x 2)2
62 +
(y + 1)2
22+
(z 2)2(2
3)2 = 1
(2,1, 2) a =
6, b =2, c = 2
3. 16.6,
.
.16.6
.2.14. x2
a2+ y
2
b2 z2
c2= 1.
-
.
. 168
. .16.7.
.16.7
z = c1 R ( xy ) .
x2
a2+y2
b2= 1 +
c21c2
. x = a1 R ( yz) .
y2
b2 z
2
c2= 1 a
21
a2
. zx.
.2.15.
x2
4+y2
4 z
2
9= 1
. a = 2, b = 2, c = 3.
.2.16.
x2
4 y
2
4+z2
9= 1 (.3)
-
.
. 169
. (.3) . .16.8.
.16.8
, Oy Oz ( .. . 16.7). y, z( Oy Oz). , .
.2.17.
1
4x2 1
2x+ y2 6y 1
9z2 +
33
4= 0. (.4)
. (.4)
(x 1)2
4+ (y 3)2 z
2
9= 1
, (1, 3, 0) a = 2,b = 1, c = 3.
.2.18.
x2
a2 y
2
b2 z
2
c2= 1
. .16.8. xy zx , yz .
-
.
. 170
.2.19.
(x 1)2
22 (y + 1)
2
32+
(z 2)2
22= 1
. 0x 0z (1,1, 2). . 16.9
.16.9
.2.20.
1
9x2 2
9x y2 z2 8
9= 0.
.
(x 1)2
32 y2 z2 = 1
. .
.2.21.
14x2 + y2 1
9z2 +
2
9z 1
9 1 = 0
.
y2 x2
4 (z 1)
2
9= 1
. .
-
.
. 171
.2.22.
x2
a2+y2
b2=z2
c2. (.5)
. .16.10. xy , z = c1 (.5) x
2
a2+ y
2
b2=
c21c2. yz ,
y = b1 (.5) x2
a2 z2
c2=
b21b2
( zx).
.16.10
.2.23.
y2 + 4z2 x2 = 0 (.6)
. (.6)
y2
22+z2
12=x2
22
Ox Oz.
.2.24.
x2 2x+ y2 4y z2 2z + 4 = 0
.
(x 1)2 + (y 2)2 (z + 1)2 = 0
a = b = c = 1 (1, 2,1).
-
.
. 172
.2.25.
x2
a2+y2
b2= 1. (.7)
. .16.11.
.16.11
(.7) ( ). , (x, y, z) ( ) (.7). (x1, y1, 0) (x1, y1, z) . z (.7). xy .
.2.26. 4x2 + 9z2 = 36..
x2
32+z2
22= 1
Oy Oz
-
.
. 173
. .16.12.
.16.12
.2.27.
x2
a2 y
2
b2= 1
. .16.13. xy .
.16.13
-
.
. 174
.2.28.
1
4x2 1
2x 1
9z2 +
2
3z 7
4= 0
. ,
(x 1)2
22 (z 3)
2
32= 1
.
.2.29.
z =x2
a2+y2
b2
. .16.14.
.16.14
z = c1 . x = a1 y = b1 .
.2.30.
4x2 + 9y2 = 36z
.
z =x2
32+y2
22
.
-
.
. 175
.2.31.
x 116y2 +
1
8y 1
9z2 2
9z 25
144= 0
.
x =(y 1)2
42+
(z + 1)2
32
. (0, 1,1)
.2.32.
1
4x2 +
1
2x+ y2 6y z + 37
4
.
z =1
4(x+ 1)2 + (y 3)2
.
.2.33.
z =x2
a2 y
2
b2
. .16.15.
.16.15
z = c1 . x = a1 y = b1 .
-
.
. 176
.2.34.
9x2 4y2 = 36z
.
z =x2
22 y
2
32
.
.2.35. (0, 0, 0), Oy, (1, 1, 1) (1, 2, 3).
. (0, 0, 0) Oy,
y = Ax2 + Cz2.
(1, 1, 1) (1, 2, 3)
1 = A 12 + C 12
1 = A 22 + C 32
A = 85, C = 3
5. .
y =8
5x2 3
5z2
.
.2.36.
y =x2
a2
. .16.16.
.16.16
-
.
. 177
z = c1 .
.2.37. M AMBM
= 23,
A (2, 2,2) B (3,3, 3).. M (x, y, z).
(x+ 2)2 + (y 2)2 + (z + 2)2(x 3)2 + (y + 3)2 + (z 3)2
=2
3
9 ((x+ 2)2 + (y 2)2 + (z + 2)2
)= 4
((x 3)2 + (y + 3)2 + (z 3)2
)
5x2 + 60x+ 5y2 60y + 5z2 + 60z = 05 ((x+ 6)2 + (y 6)2 + (z + 6)2 108
)= 0.
( ). (6, 6,6) 6
3.
.2.38. y z.
. (x, y, z) y x2 + z2
z x2 + y2
x2 + z2 = 3
x2 + y2
8x2 + 9y2 z2 = 0
.
.2.39. y xz.
. (x, y, z) y x2 + z2,
xz , , y. 4y = x2 + z2 .
y =x2
22+z2
22
.
.2.40. M AM +BM = 8, A (2, 3, 4) B (2,3, 4).
. M (x, y, z). (x 2)2 + (y 3)2 + (z 4)2 +
(x 2)2 + (y + 3)2 + (z 4)2 = 8
-
.
. 178
. (x 2)2 + (y 3)2 + (z 4)2 = 8
(x 2)2 + (y + 3)2 + (z 4)2
3y + 16 = 4
(x 2)2 + (y + 3)2 + (z 4)2
9y2 + 96y + 256 = 16 ((x 2)2 + (y + 3)2 + (z 4)2
)
16x2 + 7y2 + 16z2 64x 128z + 208 = 0(x 2)2
7+y2
16+
(z 4)2
7= 1
(2, 0, 4).
.2.41. M (2,1, 3) x.
. (x 2)2 + (y + 1)2 + (z 3)2 = 2
y2 + z2
x2 4x 3y2 + 2y 3z2 6z + 14 = 0
(x 2)2 3 (y 1/3)2 3 (z + 1)2 = 403
(y 1/3)2409
+(z + 1)2
409
(x 2)2
403
= 1
(1/3,1, 2).
.2.42. M AM BM = 6, A (4, 3, 1) B (4, 3, 1).
. ((x+ 4)2 + (y 3)2 + (z 1)2
)2=
(6 +
(x 4)2 + (y 3)2 + (z 1)2
)2
4x 9 = 3
(x 4)2 + (y 3)2 + (z 1)2
16x2 72x+ 81 = 9 ((x 4)2 + (y 3)2 + (z 1)2
)
7x2 9y2 9z2 + 54y + 18z = 153x2
9 (y 3)
2
7 (z 1)
2
7= 1
.
.2.43. x2 + 4z2 = 16 ( xz) Ox.
-
.
. 179
. .16.17.
.16.17
M MB xy. ABM AB = y, BM = z, AP 2 = y2 + z2. , , x2 = 16 4AP 2 = 16 4 (y2 + z2) .
x2 + 4y2 + 4z2 = 16 (.8)
. . (.8) (;;).
.2.44. x2 2z2 = 16 ( xz) Oz.
-
.
. 180
. .18.
.16.18
M1 (x1, 0, z) M (0, 0, z) M1 Oz. M1 Oz, . M (x, y, z) . x1 =
x2 + y2
x21 2z2 = 16 x2 + y2 2z2 = 16. (.9)
. . (.9) (;;).
.2.45. 2x+ 3y = 6 ( xy) Oy.
-
.
. 181
. .16.19.
.16.19
x x2 + z2
2x2 + y2 = 3 (2 y) 4
(x2 + z2
)= 9 (2 y)2
(0, 2, 0) .
.2.46.
r (u, v) = (x (u, v) , y (u, v) , z (u, v)) (.10)= (R cosu cos v,R sinu cos v,R sin v) , 0 u 2, 0 v .
x2 + y2 + z2 = R2. (.11)
.. u, v (.10).
x = R cosu cos vy = R sinu cos v
} x2 + y2 = R2
(cos2 u cos2 v + sin2 u cos2 v
)= R2 cos2 v
z2 = sin2 v
x2 + y2 + z2 = R2 cos2 v +R2 sin2 v = R2.
r (u, v) = (x (u, v) , y (u, v) , z (u, v)) (.12)= (R sinu sin v,R cos v,R sinu cos v) , 0 u 2, 0 v
x2 + y2 + z2 = R2 ( u, v).
-
.
. 182
.2.47.
(x x0)2 + (y y0)2 + (z z0)2 = R2.
.
r (u, v) = (x (u, v) , y (u, v) , z (u, v)) (.13)= (x0 +R cosu cos v, y0 +R sinu cos v, z0 +R sin v) , 0 u 2, 0 v .
, (.13)
x x0 = R cosu cos vy y0 = R sinu cos vz z0 = R sin v
, , (x x0)2 +(y y0)2 + (z z0)2 = R2.
.2.48.
(x x0)2 + (y y0)2 + (z z0)2 = R2, z 0.
.
r (u, v) = (x (u, v) , y (u, v) , z (u, v)) (.14)= (x0 +R cosu cos v, y0 +R sinu cos v, z0 +R sin v) , 0 u 2, 0 v /2.
(.13) v [0, /2] [0, ].
.2.49.
(x x0)2
a2+
(y y0)2
b2+
(z z0)2
c2= R2.
.
r (u, v) = (x (u, v) , y (u, v) , z (u, v)) (.15)= (x0 + a cosu cos v, y0 + b sinu cos v, z0 + c sin v) , 0 u 2, 0 v .
u, v (.15).
.2.50.
r (u, v) = (x (u, v) , y (u, v) , z (u, v)) =(a
1 + u2 sin v, a
1 + u2 cos v, cu), (.16)
u R, v [0, 2] .
.
x2
a2+y2
a2=a2 (1 + u2)
a2(sin2 v + cos2 v
)= 1 + u2 = 1 +
z2
c2
(.16) .
-
.
. 183
.2.51.
r (u, v) = (x (u, v) , y (u, v) , z (u, v)) (.17)= (sinhu cos v, a sinhu sin v, c cosh u) , u R, v [0, 2] .
.
x2
a2+y2
a2 z
2
c2=a2
(cos2 v + sin2 v
)a2
sinh2 u c2 cosh2 u
c2
= sinh2 u cosh2 u = 1
.z2
c2 x
2
a2 y
2
a2= 1
.
.2.52. z = x2
a2+ y
2
b2.
. x, y, z x (u, v) , y (u, v)
z =x2 (u, v)
a2+y2 (u, v)
b2.
x (u, v) = au cos v, y (u, v) = b
u sin v, z (u, v) = u.
.2.53.
r (u, v) = (x (u, v) , y (u, v) , z (u, v)) = (av cosu, bv sinu, v) , u [0, 2] . (.18)
. x2
a2+y2
b2= v2
(cos2 u+ sin2 u
)= z2
(.18) .
.2.54.
x2
a2+y2
b2= 1.
. x, y, z x (u, v) , y (u, v)
x2 (u, v)
a2+y2 (u, v)
b2= 1
z (u, v) R.
x (u, v) = a cosu, y (u, v) = b sinu, z (u, v) = v.
-
.
. 184
.2.55.
r (u, v) = (v + 2v cosu, 2v sinu, 5 v) (.19)
(0, 0, 5) (3 + 4 cosu, 4 sinu, 0), . xy , (1, 0) 2.
. (.19) :
r (u, v) = (0, 0, 5) + v ((1 + 2 cosu, 2 sinu, 0) (0, 0, 1)) = r0 + v (r1 (u) r0) (.20)
r0 = (0, 0, 5) , r1 (u) = (1 + 2 cosu, 2 sinu, 0) .
.16.19 (.20).
.16.19
r1 (u) = (4 + 4 cosu, 4 sinu, 0) xy 2 (3, 0, 0). r1 (u) r0 - r0 = (0, 0, 1), . z = 1. , (.20) r0 = (0, 0, 1) - r1 (u) r0. , (0, 0, 1) () (). , .
.2.56. (.18) r0 + v (r1 (u) r0).
(av cosu, bv sinu, v) = (0, 0, 0) + v (a cosu, b sinu, 1) .
-
.
. 185
. r0 = (0, 0, 0), r1 (u) = (a cosu, b sinu, 1), z = 1. , .16.20
.16.20
(0, 0, 0) .
.2.57. x2
a2+ y
2
b2= 1
r (u, v) = r1 (u) + v r2 (.21)
.
r1 (u) = (a cosu, b sinu, 0) , r2 = (0, 0, 1)
(.21)
r (u, v) = (a cosu, b sinu, 0) + v (0, 0, 1) = (a cosu, b sinu, v)
x (u, v) = a cosu, y (u, v) = b sinu, z (u, v) = v
x2
a2+y2
b2= 1 z R.
,
-
.
. 186(x2
a2+ y
2
b2= 1, z = 0
) (0, 0, 1) .16.21
.16.21
.2.58. (0, 1, 1) x2 +z2 = 1 xz. .
.
r (u, v) = (cosu, 0, sinu) + v (0, 1, 1) = (cosu, v, v + sinu) .
.r1 (u) = (cosu, 0, sinu) r2 = (0, 1, 1) .
-
.
. 187
(z = x2, y = 0) (0, 1, 1) . .16.22.
.16.22
.3
.3.1. (2, 3, 1) 4.. (x+ 2)2 + (y 3)2 + (z 1)2 = 16.
.3.2. (1, 0, 4) 1.. (x 1)2 + y2 + (z 4)2 = 1.
.3.3. (1, 1, 1) (1, 1, 2).
. (x 1)2 + (y 1)2 + (z 1)2 = 1.
.3.4. x2 +6x+y24y+z210z11 = 0.. (3, 2, 5) 7.
.3.5. x2 + 2x+ y2 + 8y+ z2 6z+ 22 = 0.. (1,4, 3) 2.
.3.6. (3, 6,4) 2x 2y z 10 = 0.
. (x 3)2 + (y 6)2 + (z + 4)2 = 16.
.3.7. (3,5, 6) (5, 7,1).
. (x 3)2 + (y 6)2 + (z + 4)2 = 16.
-
.
. 188
.3.8. (7, 9, 1), (2,3, 2),(1, 5, 5) (6, 2, 5).
. x2 + y2 + z2 + 8x 14y + 18z 79 = 0.
.3.9. (1, 1, 1), (1, 2, 1),(1, 1, 2), (2, 1, 1)
. x2 + y2 + z2 3x 3y 3z + 6 = 0.
.3.10. (2, 1, 3), (3,2, 1),(4, 1, 1), (1, 1,3).
. 51x2 + 51y2 + 51z2 + 45x+ 37y 33z 742 = 0.
.3.11. (2,5, 8), (8,2, 5),(5,8, 2) (2,8,5).
. x2 + y2 + z2 = 93.
.3.12. (4, 2, 3) 2x y 2z + 7 = 0.
. x2 + y2 + z2 + 8x 4y 6z + 20.
.3.13. (2, 3,4) - (x 2)2 + (y 3)2 + (z 5)2 = 6.
. (x 2)2 + (y 3)2 + (z + 4)2 = 9
6.
.3.14. S
S1 : (x 5)2 + (y + 2)2 + (z 6)2 = 16,S2 : (x 5)2 + (y + 2)2 + (z + 4)2 = 9,
S S1 S2.
. : (x 5)2 + (y + 2)2 + (z 4.5)2 = R2 R2 {2.25, 72.25, 30.25, 20.25}.
.3.15. (6, 3,4) x.
. x2 + y2 + z2 12x 6y + 8z + 36 = 0.
.3.16. (4,2, 3) yz.
. x2 + y2 + z2 + 8x+ 4y 6z + 13 = 0.
.3.17. (2,3, 2) 6x 3 + 2z 8 = 0.
. 49x2 + 49y2 + 49z2 196x+ 294y 196z + 544 = 0.
.3.18. (1, 2, 4) 3x 2y + 4z 7 = 0.
. 29x2 + 29y2 + 29z2 58x 118y 232z + 545 = 0.
-
.
. 189
.3.19. (0, 0, 0) 9x 2y + 6z + 11 = 0.
. x2 + y2 + z2 = 1.
.3.20. (7,2, 4) (9,8, 6) S S. S.
. (x 8)2 + (y + 5)2 + (z 5)2 = 11.
.3.21. x 2z 8 = 0,2x z + 5 = 0 x = 2, y = 0.
. : x2+y2+z2+4x+6z+49/5 = 0 x2+y2+z2+4x+22z+481/5 = 0.
.3.22. (1,3, 4), (1,5, 2),(1,3, 0) x+ y + z = 0.
. x2 + y2 + z2 2x+ 6y 4z + 10 = 0.
.3.23. (11, 6, 5) (16,3, 8) x2 + y2 + z2 = 49;
. , (2, 3, 6)
.3.24. (8, 15, 10) xy 0 7.
. x2 + y2 + (z 17)2 = 338.
.3.25.
x2
16+y2
9+z2
4= 1
. .
.3.26.
2x2 + 3y2 + z2 8x+ 6y 4z 3 = 0
.
.3.27.
2x2 + 3y2 + z2 8x+ 6y 4z 3 = 0
.
.3.28. 2x3y+5z = 0
x2 +y2
4+z2
9= 1.
. : 2x 3y + 5z
265 = 0.
-
.
. 190
.3.29. x2 y2 + 2y + z2 6z + 7 = 0 .
.3.30. x2 2x y2 + 2y + z2 1 = 0 .
.3.31. x2 + y2 8y z2 + 6z + 6 = 0 .
.3.32. x2 y2 + 2y z2 + 6z 11 = 0 .
.3.33. x2 y2 + 2y+ z2 6z+ 7 = 0 .
.3.34. x2 y2 + 2y z2 + 6z 10 = 0 .
.3.35. x2 + y2 2y z2 + 6z 8 = 0 .
.3.36. y2 2y + z2 6z + 6 = 0 .
.3.37. y2 + 2y + z2 6z + 12 = 0 .
.3.38. y2 2y + z2 6z x+ 10 = 0 .
.3.39. y2 + 2y + z2 6z + x+ 8 = 0 .
.3.40. x2 + 8x z2 2z+ y 17 = 0 .
.3.41. (0, 0, 0), z, (3, 0, 1) (3, 2, 2).
. z = x2
9+ y
2
4.
.3.42. (0, 0, 0), Oz (2, 0, 3) (1, 2, 3).
. 12x2 + 9y2 16z = 0.
.3.43. (0, 0, 0), Oz (1, 0, 1) (0, 2, 1).
. 4x2 + y2 = 4z.
.3.44. (0, 0, 0), Ox (1, 2, 2) (2, 6, 8).
. z2 2y2 + 4x = 0.
-
.
. 191
.3.45. x2+2x+y2 = 0
.3.46. z2+6z+x7 = 0 .
.3.47. (2, 3, 1) 4.. (x+ 2)2 + (y 3)2 + (z 1)2 = 16.
.3.48. (1, 0, 4) 1.. (x 1)2 + y2 + (z 4)2 = 1.
.3.49. (1, 1, 1) (1, 1, 2).
. (x 1)2 + (y 1)2 + (z 1)2 = 1.
.3.50. x2 +6x+y24y+z210z11 = 0.. (3, 2, 5) 7.
.3.51. x2 +2x+y2 +8y+z26z+22 = 0.. (1,4, 3) 2.
.3.52. (3, 6,4) 2x 2y z 10 = 0.
. (x 3)2 + (y 6)2 + (z + 4)2 = 16.
.3.53. (3,5, 6) (5, 7,1).
. (x 3)2 + (y 6)2 + (z + 4)2 = 16.
.3.54. (7, 9, 1), (2,3, 2),(1, 5, 5) (6, 2, 5).
. x2 + y2 + z2 + 8x 14y + 18z 79 = 0.
.3.55. (1, 1, 1), (1, 2, 1),(1, 1, 2), (2, 1, 1)
. x2 + y2 + z2 3x 3y 3z + 6 = 0.
.3.56. (2, 1, 3), (3,2, 1),(4, 1, 1), (1, 1,3).
. 51x2 + 51y2 + 51z2 + 45x+ 37y 33z 742 = 0.
.3.57. (2,5, 8), (8,2, 5),(5,8, 2) (2,8,5).
. x2 + y2 + z2 = 93.
.3.58. (4, 2, 3) 2x y 2z + 7 = 0.
. x2 + y2 + z2 + 8x 4y 6z + 20.
-
.
. 192
.3.59. (2, 3,4) - (x 2)2 + (y 3)2 + (z 5)2 = 6.
. (x 2)2 + (y 3)2 + (z + 4)2 = 9
6.
.3.60. S
S1 : (x 5)2 + (y + 2)2 + (z 6)2 = 16,S2 : (x 5)2 + (y + 2)2 + (z + 4)2 = 9,
S S1 S2.
. : (x 5)2 + (y + 2)2 + (z 4.5)2 = R2 R2 {2.25, 72.25, 30.25, 20.25}.
.3.61. (6, 3,4) x.
. x2 + y2 + z2 12x 6y + 8z + 36 = 0.
.3.62. (4,2, 3) yz.
. x2 + y2 + z2 + 8x+ 4y 6z + 13 = 0.
.3.63. (2,3, 2) 6x 3 + 2z 8 = 0.
. 49x2 + 49y2 + 49z2 196x+ 294y 196z + 544 = 0.
.3.64. (1, 2, 4) 3x 2y + 4z 7 = 0.
. 29x2 + 29y2 + 29z2 58x 118y 232z + 545 = 0.
.3.65. (0, 0, 0) 9x 2y + 6z + 11 = 0.
. x2 + y2 + z2 = 1.
.3.66. (7,2, 4) (9,8, 6) S S. S.
. (x 8)2 + (y + 5)2 + (z 5)2 = 11.
.3.67. x 2z 8 = 0,2x z + 5 = 0 x = 2, y = 0.
. : x2+y2+z2+4x+6z+49/5 = 0 x2+y2+z2+4x+22z+481/5 = 0.
.3.68. (1,3, 4), (1,5, 2),(1,3, 0) x+ y + z = 0.
. x2 + y2 + z2 2x+ 6y 4z + 10 = 0.
.3.69. (11, 6, 5) (16,3, 8) - x2 + y2 + z2 = 49;
. , (2, 3, 6)
-
.
. 193
.3.70. (8, 15, 10) xy 0 7.
. x2 + y2 + (z 17)2 = 338.
.3.71.
x2
16+y2
9+z2
4= 1
. .
.3.72.
2x2 + 3y2 + z2 8x+ 6y 4z 3 = 0
.
.3.73.
2x2 + 3y2 + z2 8x+ 6y 4z 3 = 0
.
.3.74. 2x3y+5z = 0
x2 +y2
4+z2
9= 1.
. : 2x 3y + 5z
265.
.3.75. x2 y2 + 2y + z2 6z + 7 = 0 .
.3.76. x2 2x y2 + 2y + z2 1 = 0 .
.3.77. x2 + y2 8y z2 + 6z + 6 = 0 .
.3.78. x2 y2 + 2y z2 + 6z 11 = 0 .
.3.79. x2 y2 + 2y+ z2 6z+ 7 = 0 .
.3.80. x2 y2 + 2y z2 + 6z 10 = 0 .
.3.81. x2 + y2 2y z2 + 6z 8 = 0 .
-
.
. 194
.3.82. y2 2y + z2 6z + 6 = 0 .
.3.83. y2 + 2y + z2 6z + 12 = 0 .
.3.84. y2 2y + z2 6z x+ 10 = 0 .
.3.85. y2 + 2y + z2 6z + x+ 8 = 0 .
.3.86. x2 + 8x z2 2z+ y 17 = 0 .
.3.87. (0, 0, 0), z, (3, 0, 1) (3, 2, 2).
. z = x2
9+ y
2
4.
.3.88. (0, 0, 0), Oz (2, 0, 3) (1, 2, 3).
. 12x2 + 9y2 16z = 0.
.3.89. (0, 0, 0), Oz (1, 0, 1) (0, 2, 1).
. 4x2 + y2 = 4z.
.3.90. (0, 0, 0), Ox (1, 2, 2) (2, 6, 8).
. z2 2y2 + 4x = 0.
.3.91. x2+2x+y2 = 0
.3.92. z2+6z+x7 = 0 .
.3.93. x
2
25+ y
2
14= 1 Oy.
.. x
2
25+ z
2
25+ y
2
14= 1.
.3.94. x
2
36+ z
2
16= 1 Oz.
.. x
2
36+ y
2
36+ z
2
16= 1
.3.95. x
2
64 y2
36= 1 Ox.
.. x
2
64 y2
36 z2
36= 1
-
.
. 195
.3.96. x = y2/2 Ox. .
. x = y2
2+ z
2
2.
.3.97. 2x = 3z Ox. .
. x2 = 94 (y2 + z2).
.3.98. x
2
9 z2
4= 1 Oz.
.. x
2
9+ y
2
4 z2
4= 1.
.3.99. x = 3 Oz. .
. x2 + y2 = 9.
.3.100. (0, 3, 0) (0,3, 0) 8.
. 16x2 + 7y2 + 16z2 = 112, .
.3.101. (1, 1,2) (2, 3, 2) 3/4.
. 7x2 + 7y2 + 7z2 68x+ 22y + 100z 57 = 0.
.3.102. (3, 2,4) (3, 2, 4) 10. .
. (x3)2
9+ (y2)
2
9+ z
2
25= 1, .
.3.103. yz (1,2, 2). .
. 3x2 + 4y2 + 4z2 8x+ 16y 16z + 36 = 0, .
.3.104. x (2, 3,3).
. 9x2 + 8y2 + 8z2 36x 54y + 54z + 198 = 0.
.3.105. (0, 0, 3) (0, 0,3) 4.
. 5z2 4x2 4y2 = 20, .
.3.106. -
x+ 4y + 2z = 0
2x y + z = 02x+ y 3z = 0
-
.
. 196
10. .. x2 + y2 + z2 = 10, .
.3.107. (3, 2,4) (3, 2, 4) 10. .
. (x3)2
9+ (y2)
2
9+ z
2
25= 1, .
.3.108. (5, 0, 2) (5, 0, 2) 12. .
. x2
36+ y
2
11+ (z2)
2
11= 1, .
.3.109. Oz xy.
. 2z = x2 + y2, .
.3.110. y = 4 1/4 (2,3, 1). .
. 16x2 + 15y2 + 16z2 64x+ 88y 33z + 208 = 0, .
.3.111. (x 1)2+(y 2)2+(z 1)2 =4.
. r (u, v) = (1 + 2 cosu cos v, 2 + sinu cos v, 1 + sin v).
.3.112. x2
4+ y
2
9+ z
2
25= 1
. r (u, v) = (2 cosu cos v, 3 sinu cos v, 5 sin v).
.3.113. z = x2 + y2.. r (u, v) = (
u cos v,
u sin v, u).
.3.114. x2 = z2 + y2.. r (u, v) = (u, u cos v, u sin v).
.3.115. (0, 0, 1) x2 + y2 = 1 xy. .
. r (u, v) = (0, 0, 1) + v ((cosu, sinu, 0) (0, 0, 1)) = (v cosu, v sinu, 1 v) .
.3.116. (0, 1, 0) x2 = z xz. .
. r (u, v) = (0, 1, 0) + v ((u, 0, u2) (0, 1, 0)) = (vu, 1 v, vu2) .
.3.117. (0, 1, 0) x2 + z2 = 1 xz. .
. r (u, v) = (cosu, 0, sinu) + v (0, 1, 0) = (cosu, v, sinu) .
-
.
. 197
.3.118. (1, 1, 1) x2 + z
2
4= 1 xz. -
.. r (u, v) = (cosu, 0, 2 sinu) + v (1, 1, 1) = (v + cosu, v, v + 2 sinu)
