-
, . ,
.
: 1.) < . , .
2.) = , . 1
3.) , . ∞ - .
: , , 1 .
1. D . . - ¹0, 1
2. D . =0, Dx Dy ¹0, . . .
3. D=Dx=Dy=0, ∞ -
:
x=(x1+lx2)/(1+l)-
Cosl=ax/SQRT(ax2+ay2+az2)-. . .
. - : a()b()=|a()|×|b()|×Cos(a()^b())
1.) a()b()=b()a() 2.) l(a()b())= (la())b() 3.) (a()+b())c()=a()c()+b()c()
4.) a()b()=0, a()^b()
. . - - . : a()b()=axbx ii +axby ij ++azbz kk =axbx+ayby+azbz
- : a() ´ b()=| i j k|=i(aybz-byaz)-j(axbz-azbx)+k(axby-bxay)
SABCD=|AB() ´ AC()| |ax ay az|
SABC=0.5|AB() ´ AC()| |bx by bz| i() ´ j()=k(), j() ´ i()= -k(), j() ´ k()=i()..
1.) a() ´ b()= -b() ´ a() 2.) l ( a() ´ b() ) = ( la() )´ b()
3.) a() ´( b()+c() ) =a() ´ b()+a() ´ c() 4.) . -=0, 1 , ||.
. . - : a() ´ b()=axbx i () ´ i ()+axby i () ´ j ()++azbz k () ´ k ()=D
- : abc=|ax ay az|= |ay az|cx-|ax az|cy+|ax ay|cz = V ( . )
abc=bca=cab= -bac= -acb= -cba |bx by bz| |by bz| |bx bz| |bx by|
|cx cy cz|
:
A(x-x1)+B(y-y1)=0, A,B-. .
Ax+By+C=0 . -
(x-x1)/m=(y-y1)/n, m,n-. . .
y=k(x-x1)+y1 . -
(x-x1)/(x2-x1)=(y-y1)/(y2-y1)-- ., . 2 .
tgj=(k2-k1)/(1+k1k2)- . . (k1=-1/k2- .)
d= (Ax0+By0+C)/SQRT(A2+B2)-. . . M(x0,y0), A,B-. . .
:
(x-x0)2+(y-y0)2=R2- ((x0,y0)- )
(x2/a2)+(y2/b2)=1 - (y=b*SQRT(a2-x2)/a- 1 ;) e=/a, c2=a2-b2, e2=1-b2/a2)
(x2/a2)-(y2/b2)=1- (y= a*SQRT(x2-a2)/b- 1 ;)(e=/a, c2=a2+b2, y=bx/a-)
y2/b2-x2/a2=1 (x-x0)2+(y-y0)2+(z-z0)2=r2-, (x0,y0,z0)
x2/a2+y2/b2=1-. .; x2/a2-y2/b2=1-. .
x2/a2+y2/b2-z2/c2=0-;x2/a2+y2/b2+z2/c2=1
x2/a2-y2/b2+z2/c2=1-
y2=2px - (x= -p/2-) x2/a2-y2/b2+z2/c2=-1 -
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x2/p+y2/q=2z-. ; q, p>0-
x2/p-y2/q=2z- ; q, p>0-
:
x=rCosj, y=rSinj- . .
r=SQRT(x2+y2), Sinj=y/SQRT(x2+y2), Cosj=x/SQRT(x2+y2)- . .
:
A(x-x1)+B(y-y1)+C(z-z1)=0- - -, . .
Ax+By+Cz+D=0 - - . N=Ai()+Bj()+Ck() ( -)
x/a+y/b+z/c=1-- - (a,b,c- . - )
D = |x-x1 y-y1 z-z1| =Ax+By+Cz+D=0-- - ., 3
|x2-x1 y2-y1 z2-z1|
|x3-x1 y3-y1 z3-z1|
xCos l +yCos b +zCos g -p=0-. .(l,b,g- . . . -, - ., . (000) -)
. .: . - . . - N= 1/SQRT(A2+B2+C2),
, , D . -
:
Cos a= N1()N2()/|N1()||N2()|=(A1A2+B1B2+C1C2)/SQRT((A12+B12+C12)(A22+B22+C22))
A1\A2=B1\B2=C1/C2- - - :
A1A2+B1B2+C1C2=0- - A1x+B1y+C1z+D1=0 (. .
-: A2x+B2y+C2z+D2=0 l .):
: M(x1,y1,z1)-; Q=Ax+By+Cz+D=0--;
d=|Ax1+By1+Cz1|/SQRT(A2+B2+C2)- A1x+B1y+C1z+D1+l(A2x+B2y+C2z+D2)=0
3D
A1x+B1y+C1z+D1=0- - (x-x1)/m=(y-y1)/n=(z-z1)/p- -
A2x+B2y+C2z+D2=0 (x-x1)/Cosl=(y-y1)/Cosb=(z-z1)/Cosg- ,
r=r1+ts() s()
x=x1+tm s()=N1() ´ N2()( . -)=| i j k |
y=y1+th - - |A1 B1 C1 |
z=z1+tp |A2 B2 C2 |
(x-x1)/(x2-x1)=(y-y1)/(y2-y1)=(z-z1)/(z2-z1)- - , . 2
Cosl=s1()s2()/|s1()|×|s2()|=(m1m2+n1n2+p1p2)/(SQRT((m12+n12+p12)(m22+n22+p22))-
Sinj=(Am+Bn+Cp)/SQRT((A2+B2+C2)(m2+n2+p2))- . . -(A,B,C-. .
. -, m,n,p-. . . )
z=x+y i -. .(. ); x=Rez, y=Inz; z()=x-y i - z
: :
x=rCosj; y=rSinj; z=r(Cosj+ i Sinj); (r=|z|=SQRT(x2+y2); j=arctg(y/x)) z=re^(i j)
: z1z2 =(x1+y1 i) (x2+y2 i)=(x1x2)+ i (y1y2); z1z2 =(x1+y1 i)(x2+y2 i)=x1x2+y1x2 i +y1x2 i -y1y2=(x1x2-y1y2)+ i (x1y2+y1x2)
z1/z2 =z1z2()/z2z2()=(x1+y1 i)(x2-y2 i)/(x2+y2 i)(x2-y2 i)=(x1x2+y1y2)/(x22+y22)+ i (y1x2-x1y2)/(a22+b22)
z1z2 =r1r2(Cos(j1+j2)+ i Sin(j1+j2)); z1/z2 =r1(Cos(j1-j2)+ i Sin(j1-j2))/r2; zn =r n (Cosj n + i Sinj n)
n . z= n : (r(Cosj+ i Sinj)=( n r)*(Cos((j+2pk)/n)+ i Sin((j+2pk)/n)), k=0,1,2n-1
z1z2 =r1r2e^(i (j1+j2)); z1/z2 =r1e^ i (j1-j2)/ r2; zn =rne^(i nj)
n z=( n r)*e^ i ((j+2pk)/n)
ez=e(x+y i)=exe i y=ex(Cosy+ i Siny)-
z=|z|(Cosj+ i Sinj)=re^ i j
]
...
(" A>0)($ N)(" n>N):|xn|>A, lim(xn∞)xn=∞; ( $ N)(" n>N):lim(xn+∞)xn=+∞-
1. ¹0-. , ×= 2. × = 3. å = 4. /=
5. / (. . ∞/∞). : ) ) )
.M..
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(" e>0)($ N)(" n>N):|xn|<e, lim(xn∞)xn=0;
-:
1 () 2- .
-: ]e>0Þ($ N1)(n>N1):|xn|<e/2 ($ N2)(n>N2):|an|<e/2, Nmax={N1,N2}Þ, .
-Þ|xnan|£|xn|+|an|<e/2+e/2=e : . . . .
2 - 2 .
-: ]e>0Þ($ N1)(n>N1):|xn|<e ]e=1Þ($ N2)(n>N2):|an|<1, Nmax={N1,N2}Þ,
. -Þ|xn×an|=|xn|×|an|<e×1=e : - . .
3 - . - .
-: ]{xn}-., {an}-, .. {xn}-., ($ A>0)(" xn):|xn|£A ., ]e0=e/A(e>0)Þ
($ N)(" n>N):|an|<e/AÞ|xn×an|=|xn|×|an|<Ae/A=e
4 - .
-: ]A-. . ]{xn}-, .: (" (e/A)>0)($ N)(" n>N):|xn|<e/A,ÞA×|xn|<e×A/A=e
5 - ,
6 2- , , , ( 2- . - 0/0)
. - . - . - f (n), - . .
-. -: (" n):xn+1>xn; (" n):xn+1<xn . .
-: : (" e>0)($ N)(" n>N):|A-xn|<e, A- - {xn}
($ M)(" xn):xn£M-. ; ($ m)(" xn):xn³m -.
($ A>0)(" xn):|xn|£ A -; (" A>0)($ xn):|xn|>A -
-
- {xn} -, : lim(n∞)xn=∞
-:
1) - xn ., yn , n∞, lim(xn+yn)=∞; lim(xnyn)=∞; lim(xn/yn)=0; lim(xn-yn)=∞
2) - xn yn , +∞, lim(n∞)(xn-yn)=+∞
3) xn yn , xn +∞, yn -∞, : lim(xn-yn)=+∞, lim(xnyn)= -∞
4) xn , yn ∞, : lim(xnyn)=+∞, a>0 lim(xnyn)=-∞, <0.
5) xn , yn 0, : limxn/yn=+∞, >0 limxn/yn=-∞, <0
, , -, -:
1) ] lim(n∞)xn=a; lim(n∞)yn=b, å - lim(n∞)(xnyn)=ab ( n∞; nn0)
2) lim(xnyn)=limxnlimyn ( n∞; n∞; nn0)
3) lim(xn/yn)=limxn/limyn, limyn¹0
:
1) xn , "R- l: lim(n∞)(xn^l)=(lim(n∞)(xn))^l
2) - xn - lim¹0, ">0: lim(logaxn)=loga(limxn) ( n∞)
3) : lim(a^xn)=a^(limxn) ( n∞)
. - f(x) xx0 , (" e>0)($ d>0)(" xÎ..,x¹x0,|x-x0|<d):|f(x)-A|<e
- . , xx0, lim f (x)=0
- . , lim(xx0) f (x)= ∞
A+e -: 1.) 2- -, -
A2e -: ] u(x)=a(x)+b(x), lim(xa) a (x)=0; lim(xa) b (x)=0, ] : a(x)<e/2
A-e (a-d1)<x<(a+d1) b(x)<e/2 (b-d2)<x<(b+d2).
d d dmin={d1;d2}, Þ . -Þ|u(x)|=|a(x)+b(x)| £|a(x)|+|b(x)|<e/2+e/2=e
x0-dx0 x0+d : . . -, -
2) - - y= f (x) -, - u= f (v), xx0 (x∞) - .
-: xx0. M>0 - x=x0, : |u|<M.
"e>0 -, |y|<e/M. . - : |yu|<eM/M=eÞuy-̷
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3) - y= f (x) . b - a=u(x), lim(xx0 ∞) y =b .
-: .. y=b+a, |y-b|=|a|, ("e>0)( - , ):|a|<e,Þ y, . .
. |y-b|<e, , lim y= b ( xx0 ∞).
4) y= f (x) 0 xx0 ( x∞) 0, y=1/ f (x) .
("M>0) :1/|y|>M, : |y|<1/M; y, ,
.. y= f (x) 0
. :1) lim .å . . -=.å lim -: lim(u1+u2+un)=lim u 1+lim u 2++lim u n
]limu1=a1,limu2=a2, - 3:u1=a1+a1,u2=a2+a2,a1,a2-Þu1+u2=a1+a2+a1+a2Þlim(u1+u2)=a1+a2=limu1+limu2
2) limu1×u2××un=limu1×limu2××limun ]limu1=a1,limu2=a2, - 3:u1=a1+a1,u2=a2+a2,u1u2=(a1+a1)(a2+a2)-Þ
limu1u2=a1a2=limu1limu2 3) limu/v=limu/limv (limv¹0) ]limu=a,limv=b¹0Þu=a+a,v=b+b, ab-.
u/v=(a+a)/(b+b)=a/b+((a+a)/(b+b)-a/b)=a/b+(ab-ab)/(b+b)b; a/b-const,(ab-ab)/(b+b)b-Þlimu/v=a/b=limu/limv
.
lim x+∞: . - f (x) x+∞,
A+e (" e>0)($ N)(" x>N):|f(x)-A|<e
A A-e< f (x)<A+e-. , . - y= f (x) "x>N
A-e . : A-e< f (x)<A+e
lim x-∞: . - f (x) x-∞,
N (" e>0)($ M)(" x<M):|f(x)-A|<e
A-e< f (x)<A+e-. , . - y= f (x) "x<M .
: A-e< f (x)<A+e
b - - y= f (x) xx0 (xx0+0), (" e>0)($ M>x0)(x0<x<M):|f(x)-b|<e
b- - y= f (x) xx0 (xx0-0), (" e>0)($ M<x0)(M<x<x0):|f(x)-b|<e
- . $ lim lim , $ lim xx0.
b- - y=f(x) xx0, (" e>0)($ M N)(" xÎ[M,N], . x0):|f(x)-b|<e
lim -
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