teorija kalkulus 2

8/10/2019 Teorija Kalkulus 2

1/10

2 -

:

: F ( ) f

I F( )=f(x) I.: I I I.

: F(x) f(x) I, ,F(x)+ . , I F(x)+c, c.

: .

: F(x) G(x) f(x) g(x) 1 .:

) 1*f(x)dx=c 1* F(x)+c

) [f(x)+g(x)]dx=F(x)+G(x)+c

) [f(x)-g(x)]dx=F(x)-G(x)+c

: f f.

: F f g .

d/dx(F(g(x)))=F(g(x))*g(x)

F(g(x))*g(x)dx = F(g(x))+c

f(g(x))*g(x)dx = F(g(x))+c : u=g(x); du=g(x)dx

f(u)du=F(u)+c

: f(g(x))*g(x)dx = F(g(x))+ f(u)du=F(u)+c

.: f(k) m n, m


2/10

T: a) = ( )

) 2 = ( ) ( )

) 3

= (( )

)2

: f (a,b) f(x) 0 (a,b), =f(x) (a,b) A= ( k *)x

: f (a,b) y=f(x) (a,b) - A= ( k *)x.

: f [a,b] ( k *)xk

k * .: f [a,b] f e [a.b]

: f, :

) ( ) = 0

) ( ) ( ) = ( ) ( )

: f a,b , :

( ) = ( ) + ( ) , a,b .

: f I I >0 M>f(x)>-M I. f I =- =.

: f [a,b] .

) f [a,b], [a,b] f [a,b]

) f [a,b], f [a,b].

: ) f [a,b] f(x) 0 , ( ) 0


3/10

) f g [a,b] f(x) g(x) , ( ) ( ) .

T: f [a,b] F f (a,b), ( ) = ( ) ( )

T: f e [a,b], [a,b]

( ) = ( )( )

: f e I, f I. I, F(x) ( ) = ( ) F(x) f(x),.. F(x)=f(x) I ( ) = ( )

: g [a,b] f g(x) a


4/10

:

: f(x)>=0 [a,b] f , f, x- x=a x=b : P= ( ) .

* f [a,b], P= ( ) .T: f(x)>= g(x) [a,b]. f g x=a x=b P= ( ( ) ( )) .

: y=f(x) [a,b]. y , y=f(x) x=g(y), g [c,d] c=f(a) d=f(b) . g, y- y=c y=d P= ( ) .

T ( ): -, = x=b [a,b] A(x). : V= ( ) .

: f(), -, x=a x=b x- .

: f(), -, x=a x=b V= ( ) 2 .

T: f [a,b] x=g(y) y=f(x), g, - y=c=f(a) y=d=f(b) . - V= ( ) 2 .

T: f(x)>=g(x) [a,b], f g e V= ( ) ( ) .

T ( ): =f(x) [a,b] y-o. V= 2 ( ) .

T: y=f(x) x=a x=b e L= 1 + ( ) 2 : f(x) f(x) [a,b].

: f [a,b]. f(x) - [a,b] S= 2 ( ) 1 + ( ) .


5/10

: .

: {A n} L >0, , N, |an-L|< n>=N. n=L. , .

: {a n} {b n} L1 L2 . :

) = ;

) n = n = L1;

) ( n+b n)= n+ n=L 1+L 2;

) ( n*bn)= n* n= L 1*L 2;

) ( n /b n)= n / n= L 1 /L 2;

: f(n) {a n}. n , [1,+ ).f(n) f(x). f(x) L x-> , f(n) L n-> . .

: L L.

-: {a n}, {b n} {c n} , an , {b n} L n-> .

: n| = 0 n=0;

: {a n} :

* 1 2 3...

* 1 2 3...

* 1< 2 2>3

* .

: , .

: {a n} , :

1. , anM n LM.


6/10

2. .. n=0;

: {a n} , :

1. , anM n LM.

2. .. n=0;

: : k =u 1+u2++ u k ;

: S n a n n- .

: S e u1+ u 2++ u k +... S S, S S= k. , .

: k =a+ar+ar 2++ar k + r1 1r>0.

: k k 1 b1, 2 b2,, k bk ,.

a) , .


7/10

) , .

: k k = k /b k . , , .

: k =lim k-> uk-1 /uk .

) 1, ;

) =1, .

: k =lim k-> = lim k->=uk 1/k .

) 1 =+, ;

) =1, .

: .

: (1)k+1 ak =a 1-a2+a3- (1)k ak =-a 1+a 2-a3+ 1 2 3... k = 0.

: k k | , k | .

: k =lim k->|uk |

) 1 .

)

: f n =0, n-

f : f(0)+f(0)*x/1!+f(0)*x2

/2!++f (n)

(0)*xn

/n!.* n

f n =0;

: f n 0. n- f x=x 0 Pn(x)=f(x 0)+f(x 0)(x-x 0)/1!+f(x 0)(x-x0)

2 /2!++f(n)(x 0)(x-x 0)n /n!


8/10

: Rn=f(x)-Pn(x)=f(x)- ( (k)(x0)(x-x 0)k /k!)

* f n+1 I x0 |f (n+1) (x)| I, .. |f (n+1) (x)| I I, |Rn(x)|(M/(n+1)!)|x-x 0|n+1

T: f x0, f()= (k)(x0)(x-x0)

k /k! n(x)=0

:

) - .

) -

) +

: f(x,y) (,) D ( ) - .

: n 1,2,...x n f(1,2,...x n) (1,2,...x n) n- .

: x=x(t) y=y(t). x0=x(t 0) y0=y(t 0). ( , )( , ) ( , ) =

( ), ( )

: f e f

( 0, 0) ( 0, 0 ).* ( , )( , ) ( , ) = , >0 >0

|f(x,y)-L|< 0< ( x0)2+(y -y 0 )2]L ( , ) (x 0,y0), f(x,y) ->L ( , ) (x 0,y0) .

) ( , )( , ) ( , ) f(x,y) ( , ) (x0,y0) ,

( , )( , ) ( , ) .

: f(x,y) (x 0,y0) f(x 0,y0) ( , )( , ) ( , )= f(x 0,y0).

: a) g(x) x0 h(y) 0, f(x,y)=g(x)*h(y) e (x 0,y0).


9/10

) h(x,y) (x 0,y0) g(u) u=h(x 0,0), f(x,y)=g(h(x,y)) e (x 0,y0).

) f(x,y) (x 0,y0) x(t) y(t) c t0 x(t 0)=x 0 y(t 0)=y 0, f(x(t),y(t)) t0.

: z=f(x,y) (x 0,y0) D f f (x 0,y0) x0 =0 , .

* z=f(x,y) (x0,y0) D f f (x 0,y0) 0 =0 , .

: f . f(x,y) f(y,x) , f(x,y)=f(y,x)

: x=x(t) y=y(t) z=f(x,y) (x,y)=(x(t),y(t)), z=f(x(t),y(t)) t = * + * .

: x=x(u,v) y=y(u,v) (u,v) z=f(x,y) (x(u,v),y(u,v)), z=f(x(u,v),y(u,v)) (u,v) , : = * + * = * + * .

: f , () (x0,y0), (x 0,y0) , f(x 0,y0)f(x,y), . f (x 0,y0) a f(x 0,y0)f(x,y), f.

: f , () (x0,y0), (x 0,y0) , f(x 0,y0)f(x,y), . f (x 0,y0) a f(x 0,y0)f(x,y), f.

: D ( ) (x 0,y0) (x 0,y0) D.

: () D ( ) (x 0,y0) (x 0,y0) D, D.

: D , .

: f(x,y) R, f R.


10/10

: , .

: f (x 0,y0) , f

x(x

0,y

0)=0 f

y(x

0,y

0)=0.

: (x 0,y0) f(x,y) f x(x0,y0)=0 f y(x0,y0)=0 (x 0,y0).

: f (x0,y0) D=f xx(x0,y0) * f yy(x0,y0)-f 2xy(x0,y0)

) D>0 f xx(x0,y0)>0, f (x 0,y0).

) D>0 f xx(x0,y0)

teorija kalkulus 2

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