Матан Лекции
.PDFcom.neevia.http://www version trial Converter Personal Neevia by Created
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(1) |
f(x) = f(x0) + A(x - x0) + a(x)(x - x0) |
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(2) |
g(x) = g(x0) + B(x - x0) + b(x)(x - x0), |
&( A = f¢ (x0) # B = g¢ (x0). |
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1 * ( 2 */ 4 |
0 +#*# !" + 2( #/. ) * * ! f + g. 2# + * + (1) # (2) # )# |
(3) |
f(x) + g(x) = f(x0) + g(x0) + (A + B)(x - x0) + (a(x) + b(x))(x - x0). |
-#, ) 3 & ( / ( # ' ( $ %## 3 * ) ! $ %## 3 ( # 2 *+ '* + #, ) # 3 * ) ! * ( + * #:
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A + B - )#*, a + b - 3 * ) / # x® x0 $ %#/, + * + (3) ( / f + g
(#$$ %# + ) x0 |
$ %#, #) (f + g)¢ (x0) = A + B = f¢ (x0) + g¢ (x0), ) ( 0!+ + ) * ! 1. |
) +!) + * + (2) #0 (1) # )# |
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(4) |
f(x) - g(x) = f(x0) - g(x0) + (A - B)(x - x0) + (a(x) - b(x))(x - x0). |
A - B - )#*, (-1)b 3 * ) , 1 # a - b - 3 * ) / # x® x0 $ %#/, + * + (4)
( / f - g (#$$ %# + ) x0 $ %#, #) (f - g)¢ (x0) = A - B = f¢ (x0) - g¢ (x0), ) ( 0!+ + ) * ! 1.
) 2# + * + (1) # (2) # )#
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(5) |
f(x)g(x) = f(x0)g(x0) + (Ag(x0) + Bf(x0)) + |
[f(x0)b(x) + g(x0)a(x) + (AB + Ab(x) + Ba(x) + a(x)b(x))(x - x0)](x - x0).
3 0 )#
g(x) = f(x0)b(x) + g(x0)a(x) + (AB + Ab(x) + Ba(x) + a(x)b(x))(x - x0).
2( * & + * g(x) * # */ 0 # x® x0, 1 g - 3 * ) / # x® x0 $ %#/. Ag(x0) + Bf(x0) - )#*, + * + (5) ( / fg (#$$ %# + ) x0 $ %#, #)
(fg)¢ (x0) = Ag(x0) + Bf(x0) = f¢ (x0)g(x0) + g¢ (x0)f(x0),
) ( 0!+ ) * ! 1.
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(6) |
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g (x) |
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g (x0 ) + B(x - x0 ) + b (x)(x - x0 ) |
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f (x0 ) |
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f (x0 ) + A(x - x0 ) +a(x)(x - x0 ) |
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g (x0 ) |
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g (x0 ) + B(x - x0 ) + b (x)(x - x0 ) g (x0 ) |
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f (x0 ) |
+ |
Ag (x0 ) +a(x)g (x0 ) - Bf (x0 ) - b (x) f (x0 ) |
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g (x0 ) |
g (x0 )(g (x0 ) + B(x - x0 ) + b (x)(x - x0 )) |
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3 0 )#
g (x) = |
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f ' |
(x ) = |
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2. ! % # y = f(x) %% # x0, % # z = g(y) %% # y0, y0 = f(x0). " % # z = g(f(x)) %% # x0 %
(g(f))¢ (x0) = g¢ (y0)f¢ (x0).
. 6 * + + # * 2 ' $ %## # * +#/" ! 2 ! 3* (# # + ( 0 * + !
!+ * # * 2 ' $ %##. ** 2( #/ * " / *#, 3 $ %## f # g !+ ! + ) x0.
$ %#/ f (#$$ %# + ) x0, $ %#/ g (#$$ %# + ) y0, * 6 * + )#* A # B # 3 * ) / # x® x0 $ %#/ a = a(x) # 3 * ) / # y® y0 $ %#/ b = b(y) #, ) * + ( #+! $!
(1) |
f(x) = f(x0) + A(x - x0) + a(x)(x - x0) |
# |
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(7) |
g(y) = g(y0) + B(y - y0) + b(y)(y - y0), |
&( A = f¢ (x0) # B = g¢ (y0). |
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. & / y = f(x), (* +# +! 2 # (1) + + * + (7) # )#
(8) g(f(x)) = g(f(x0)) + B(A(x - x0) + a(x)(x - x0)) + b(f(x))(A(x - x0) + a(x)(x - x0)).
& & + /, $ %#/ z = b(f(x)) + ( , * $ %#/ z = b(y) ( + ) y0. * +# 1 ( *, 2#+
b(y0) = 0.
. # ( ( ## $ %#/ z = b(y) * +# */ !+ ' + ) y0, limy® y0b(y) = 0 = b(y0).
4 &, * 2 / $ %#/ z = b(f(x)) !+ # 3 * ) + ) x0 0#%#/ !+ ' # 3 * )
' + ) y0 |
$ %## z = b(y) # !+ ' # 3 * ) ' + ) x0 |
$ %## y = f(x), y0 = f(x0). |
3 0 )# |
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143 |
com.neevia.http://www version trial Converter Personal Neevia by Created
,%#/ 15 3. .#0+( / 3 ' $ %##
g(x) = Ba(x) + b(f(x))(A + a(x))
# #7 + * + (8) + $
g(f(x)) = g(f(x0)) + BA(x - x0) + g(x)(x - x0).
. / ' + ' 3 2( */, ) limx® x0g(x) = 0, * $ %#/ g 3 * ) # x® x0. BA - )#*, * ( + * + ( / 0#%# g°f (#$$ %# + ) x0 $ %#, #)
(g(f))¢ (x0) = BA = g¢ (y0)f¢ (x0),
) 0 + 7 ( 0 * + ! 2.
3. ( +
!+ ( $ #0+ ( ' 3 ' $ %## # * +##, ) 3 / $ %#/ * 6 * +.
3. ! % # y = f(x) %% # x0, f¢ (x0) ¹ 0, % # x = f -1(y). "
% # %% # y0, y0 = f(x0), %
( f -1 )'( y |
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f ' (x ) = lim |
f (x) - f (x0 ) |
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x®x0 |
x - x |
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. (* +# * ( x = f -1(y) # 0 #, ) 3 & ( / |
!+ * # $ %#' y = f(x) + ) x |
# x = f |
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, %#/ 15 4. , !' 1 *.
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f ' (x0 ) = lim |
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1 + * + 2 # # + * 6 * + + # ( |
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( f |
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f -1 |
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3. |
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4. # " . % .
(# ( # & 1 *.
( # 1. $ , % # f x0 , d > 0 , x, |x - x0| < d,
f(x) ³ f(x0).
) $ %#/ f # + ) x0 !' *#, * # * 6 * + d > 0 , ) ( / +* " x, |x - x0| < d, +! / */ + * +
f(x) £ f(x0).
%#/ f # + ) x0 !' 1 *, * # # + 1 ' ) !' # # # # !'
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f¢ (x0) = 0.
. .( 2#, ) $ %#/ f, (#$$ %# / + ) x0, # + 1 ' ) !' *#. * ) & # # ** 2( #/ *# #) !. * 0 */ $'
f ' (x0 ) = lim f (x) - f (x0 ) .
x®x0
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(9) |
f ' (x0 ) = lim |
f (x) - f (x0 ) |
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lim |
f (x) - f (x0 ) |
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x®x0 -0 |
x - x |
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f (x) - f (x0 ) |
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(10) |
lim |
f (x) - f (x0 ) |
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(11) |
lim |
f (x) - f (x0 ) |
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146
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, %#/ 15 4. , !' 1 *.
f¢ (x0) = 0
# ( 0!+ 4.
4 ( & +!' 7 & + & # 7 #/ 0 ( )# #* & 1 * (#$$ %# ' $ %##. & * 1 ' +!' 3 * * # + 7 ## + #/
f¢ (x) = 0.
) # & 1 * & " (# */ * (# ' 1 & + #/, ! 0!+ */
% # f.
4 # /* & #) * # %#: &$# (#$$ %# ' $ %## + ) x0 & 1 * # * , *# OX.
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#*. 1. #) * / # * %#/ ! . |
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1 *. '* +# , + # |
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(x3)¢ = 3x2 = 0 |
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1. / * ( 0 ) ## |
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" c Î (a, b) , f¢ (c) = 0.
. . + ' ' 7 ** !+ / 0 [a, b] $ %#/ f ( * #& *+ & # # #
*#. 3 0 )#
m = minx Î [a, b] f(x), M = maxx Î [a, b] f(x).
6 * + ) # x1,x2 Î [a, b] #, ) f(x1) = m # f(x2) = M.
149