Конспект лекций по математическому анализу. Шерстнев А.Н. - 122 стр.

UptoLike

ВУЗ: 

КФУ | Казань

Составители: 

Рубрика: 

y 2 E POLU^AEM Ay = lim    1 A(ty) = lim o(ty) =  (t 2 R | ^ISLOWOJ
                       t!0 t         t!0 t
PARAMETR). iTAK, A = 0, TO ESTX L = Lx: >
   4. eSLI OTOBRAVENIE f DIFFERENCIRUEMO W TO^KE x, TO ONO W \TOJ
TO^KE NEPRERYWNO.
  uTWERVDENIE SLEDUET IZ OCENKI (SM. 74.2)
          kf (x + h) , f (x)k  kf 0(x)k khk + ko(h)k (h ! ): >
   5. eSLI f : ! F POSTOQNNO, TO f 0 (x) = 0 (x 2 ).
   6. wSQKOE LINEJNOE OTOBRAVENIE A : E ! F DIFFERENCIRUEMO W
KAVDOJ TO^KE x 2 E , PRI^EM A0(x) = A.
  w SILU LINEJNOSTI A RAWENSTWO (1) PRIOBRETAET WID A(x + h) , Ax =
Ah (x; h 2 E ): >
   7.eSLI f; g DIFFERENCIRUEMY W TO^KE x, TO W \TOJ TO^KE DIFFE-
RENCIRUEMY OTOBRAVENIQ f  g; f ( 2 ), PRI^EM
            (f  g)0(x) = f 0(x)  g0(x); (f )0 (x) = f 0(x):
   8.[dIFFERENCIROWANIE SUPERPOZICII OTOBRAVENIJ]. pUSTX ZADANY
OTOBRAVENIQ f : ! F; g :  ! G (  E;   F; f ( )  ), PRI^EM f
DIFFERENCIRUEMO W TO^KE x 2 , A g DIFFERENCIRUEMO W TO^KE f (x) 2
. tOGDA g  f DIFFERENCIRUEMO W TO^KE x I
                       (g  f )0(x) = g0(f (x))  f 0(x):
 sPRAWEDLIWA WYKLADKA
  g(f (x + h)) , g(f (x)) = g(f (x) + [f (x + h) , f (x)]) , g(f (x))
               = g0(f (x))[f (x + h) , f (x)] + o(f (x + h) , f (x))
                = g0(f (x))(f 0(x)h + o(h)) + o(f 0(x)h + o(h))
                = g0(f (x))  f 0(x)(h) + g0(f (x))(o(h)) + o(h) (h ! ):

tEPERX IZ OCENKI kg0(f (xk))(  hk
                                  o(h))k  kg0(f (x))k  ko(h)k SLEDUET, ^TO
                                                             khk
g  f (x + h) , g  f (x) = g0(f (x))  f 0(x)(h) + o(h) (h ! ): >
                                     122

Страницы