Конспект лекций по математическому анализу. Шерстнев А.Н. - 59 стр.

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I OBOZNA^AETSQ f 00 ILI ddxf2 . pO INDUKCII OPREDELQETSQ PROIZWODNAQ n-OGO
PORQDKA W TO^KE x0; OBOZNA^ENIE f (n)(x0). eSLI f n RAZ DIFFERENCIRUEMA
W TO^KE x, TO RAWENSTWOM dnf (x)  f (n)(x)dxn OPREDELQETSQ DIFFERENCIAL
n-OGO PORQDKA FUNKCII f W TO^KE x.
    2. [fORMULA lEJBNICA]. pUSTX u; v | FUNKCII, n RAZ DIFFERENCIRU-
EMYE W TO^KE x. tOGDA (S^ITAQ u(0)  u) IMEEM
                                   Xn n!
                         ( n )
                    (uv) (x) =            u(k)(x)v(n,k)(x):
                                   k=0 k
        
zDESX nk  k!(nn,! k)! | BINOMIALXNYE KO\FFICIENTY, 0!  1.
  dOKAZATELXSTWO PO INDUKCII. pRI n = 1 | \TO FORMULA 30.1(B). eSLI
FORMULA WERNA DLQ WSEH NATURALXNYH ^ISEL  n, TO
                                        n  
    (uv)(n+1)(x) = ((uv)(n))0(x) = ( P nk u(k)v(n,k))0(x)
                      n n (k+1) k=0
                      P
                 =             [u (x)v(n,k)(x) + u(k)(x)v(n,k+1)(x)]
                     k=0 k                n    
                 = u(0)(x)v(n+1)(x) + P [ n + n ]u(k)(x)v(n,k+1)(x)
                                                      k,1
                                             k=1 k
                                                           +u(n+1)(x)v(0)(x)
                              
                   = P n+1
                       n+1
                                 u(k)(x)v(n+1,k)(x): >
                       k=0 k
                                                  
   p R I M E R Y. 3. (cos x)(n) = cos x + n     2 .
   4. (x 
 cos x)(100) = x(cos x)(100) + 100(cos x)(99) = x 
 cos x + 100 sin x.

   x32. oSNOWNYE TEOREMY
   1. t E O R E M A [m. rOLLX]. pUSTX f : [a; b] ! R NEPRERYWNA I NA (a; b)
DIFFERENCIRUEMA, PRI^EM f (a) = f (b). tOGDA SU]ESTWUET c (a < c < b)
TAKOE, ^TO f 0(c) = 0:
  tEOREMA O^EWIDNA, ESLI f POSTOQNNA NA [a; b]. pUSTX f 6= const I SU-
]ESTWUET x 2 (a; b) TAKOE, ^TO, NAPRIMER, f (x) > f (a). tOGDA (SM. 24.2(B))


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