Конспект лекций по математическому анализу. Шерстнев А.Н. - 212 стр.

eSLI TEPERX jx , x0j < ; (x; x0 2 [a; b]), TO
                    Z (x)                    Z (x0)
jF (x) , F (x0)j = j '(x) f (x; y) dy , '(x0) f (x0; y) dyj
                    Z (x)
                 = j          [fe(x; y) , fe(x0; y)] dy
                      '
                   Z (x)( x)                 Z (x0)
                 +           f (x ; y) dy , 0 fe(x0; y) dyj
                              e   0
                     '(
                   Z (x)x )                   '(x )       Z '(x0)
                  '(x) jf (x; y) , f (x ; y)j dy + j '(x) fe(x0; y) dy
                               e          e   0
                   Z (x0)                    Z (x)                Z (x0)
                 +            f (x ; y) dy + 0 f (x ; y) dy , 0 fe(x0; y) dyj
                               e    0                e  0
                     '(x0 )                     (x ) Z 0           '(x )
                                                        '(x )
                  3( ", ) [ (x) , '(x)] + j '(x) jfe(x0; y)j dyj
                    Z (x)
                 + j 0 fe(x0; y)dyj
                        (x )
                  "=3 + M j'(x0) , '(x)j + M j (x) , (x0)j < ": >
   4.   t E O R E M A [OB INTEGRIROWANII PO PARAMETRU]. w USLOWIQH P. 3
             Zb            Z b Z (x)              ZZ
                F (x) dx = dx        f (x; y) dy = f (x; y) dxdy:
              a             a   '(x)
                 ZbZd                ZdZb
w ^ASTNOSTI, a ( c f (x; y) dy)dx = c ( a f (x; y) dx)dy.
  |TO SLEDSTWIE PZ .d3 I x123. oTMETIM, ^TO IZ 123.1 SLEDUET LI[X, ^TO
FUNKCIQ F (x) = c f (x; y) dy (a  x  b) INTEGRIRUEMA PO x. iZ P. 3
SLEDUET DOPOLNITELXNO, ^TO F NEPRERYWNAZ. >
   5. p R I M E R. wY^ISLIM lim p 1
                                            b
                                              expf,   x22 g dx. nEPOSREDST-
                                 !0+ 2 a          2
WENNO PRIMENITX TEOREMU P. 2 ZDESX NE UDAETSQ. tAK KAK NAS INTERESUET
ZNA^ENIE PREDELA W TO^KE 0, OGRANI^IMSQ PROMEVUTKOM 0 <   1. pUSTX
SNA^ALA 0 < a < b. tOGDA FUNKCIQ
                   8
                   < p 1 expf, x22 g dx; ESLI  > 0; a  x  b,
        f (x; ) = : 2          2
                     0;                   ESLI  = 0; a  x  b,

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