Вступ до аналізу. Ч. 2
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16. . .
. y = f (x) ,
A , y = g ( x) ,
B . , C = A ∩ B ≠ . C
y = f ( x ) + g ( x) . x0 C
f ( x0 ) + g ( x0 ) . |
! " # C |
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f ( x) − g ( x) , $ |
f ( x) g ( x) |
f ( x) |
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g ( x) |
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( , g ( x) = 0 ).
A y = f ( x) , %
B . & C z = g ( y ),
% G . , H = B ∩ C ≠ . -
A0 A -
z = F ( x ) = g ( f ( x )), % G0
G . $ " % '" -
% % # '. % -
g f .
($ A0 – $" % " z = F (x) = = g ( f ( x )), $ ’ , " % x A
y = f (x) " % $" z = g ( y ). # " ,
% " . )" ’ , ", $ ’ -
. #"- % ' "% "#$ .
* #" ". y = f ( x ) = 4 − x2 .
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= (− ∞; + ∞), B = E f = (− ∞; 4] . & z = g ( y ) = |
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A = Df |
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y . |
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C = Dg |
= [0; + ∞), G = Eg = [0; + ∞) . H = B ∩ C = [0; 4] |
≠ . |
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+ # f g " : z = F ( x ) = g ( f ( x )) = 
4 − x2 .
$" % A0 = DF . , "% Dg = [0; + ∞),
% % 4 − x2 ≥ 0 , x [−2; 2] . $ A0 = [−2; 2] . - % " : G0 = EF = [0; 2] G .
$ . * #" y = f ( x) ,
X , % Y . 57
, " # y0 Y
x0 X , f ( x0 ) = y0 . $ # -
% . , "% # # -
x0 X |
y0 Y , f ( x0 ) = y0 , "- |
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X Y " %. - |
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Y x = ϕ( y) : |
y0 Y : |
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ϕ( y0 ) = x0 , |
f ( x0 ) = y0 . $ |
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y = f (x) " % # ( , # , % ). . x = ϕ( y) % -
y = f (x) . / X = D f , Y = E f , Y = Dϕ , X = Eϕ . - ", # y = f (x) $ $ x = ϕ( y) . ($ $ " y = f (x) $ f (x) = y
’ x , ’ .
/ #" # , $ ? , "% -( x, y ) y = f ( x) x = ϕ( y) , # -
x = ϕ( y ) # y = f ( x) . "%
y = f ( x) x " % y ( . 29 ),
x = ϕ( y ) y " % x ( . 29 $).
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*. 29.
, x = ϕ( y ) x y ,$ #" y = ϕ( x ) . M1 ( x0 , y0 )
y = f ( x) M 2 ( y0 , x0 ) y = ϕ( x ) . , "% - 58
Oxy M1 M 2 y = x , #
$ , $ $ -
1–# 3–# ( . 30).
*. 30. |
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1. * #" y = 3x − 2 . : x = |
y + 2 |
– $ |
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3 |
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. x y , : y = x + 2 . $ -
3
# y = 3x − 2 y = x + 2
3
% , y = x ( . 31).
*. 31.
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2. ) |
y = ex (D |
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= , E |
y |
= + = (0; + ∞)) $ |
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y = ln x ( " x y ). )" |
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D |
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= + , E |
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( . 32). |
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*. 32.
3. $ $ y = sin x . , $ "-
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$" % # (" ).
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% y = sin x . $ -
y = arcsin x . )" Dy = [−1;1], |
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π |
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( .33). |
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*. 33.
4. & % , $ $ .
, " ’ y = f ( x) #" x = f ( y ) . ",
y = k (k = const ) , y = a − x (a = const ). 0
x
y = x . . 34 $ #
y = 1 . x
*. 34.
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17..
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% $ $# ', ". "
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π − arccos x 3 |
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0.5 |
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x + 1 |
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y = arctg |
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− 2 |
sin x3 |
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ctg |
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log |
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( x − 1) |
2log7 (x+2) −1 |
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" ( $ ). !
1, x ≥ 0, y =
−1, x < 0
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1. -# " ( $ "). ( ) n -
% #":
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( x ) = a |
xn + a |
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xn−1 + ... + a x + a . |
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= const . / n = 1, P |
( x ) = a x + a |
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. / n = 2 , P2 ( x) = a2 x2 + a1 x + a0 – ".
2. * "% . 2 , ' # "
( $ ’ # ):
R ( x ) = Pn ( x ) . Qm (x )
":
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3x2 |
− 5x + 4 |
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x5 |
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1 |
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y = |
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y = |
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, y = |
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2x3 + 8x2 − 7 |
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+ 1 |
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(Qm ( x) ≡ 1).
3.& "% 2 , % # -
# " "% -
"% . ":
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x2 − 4 |
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2 x |
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y = 3 |
, y = 5 (x3 + x + 1)7 + |
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x4 + 2x + 5 |
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x3 + 6 |
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* "% "% % . 0
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" " ( % " ' "-
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. . y = f ( x ) % X ,
M > 0 , x X : f ( x) ≤ M .
& ' " $ %
[−M ; M ]. # , # " " %
y = −M , y = M ( . 35).
*. 35.
, $ $ , '. ", y = x2 $ [−2; 2] ( % y ≤ 4 )
# " $% " , " $ -
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" . . y = tg x $ [0; π
4] ( y ≤ 1 ), "
$ " [0; π
2) . . y = sin x $ - " ( y ≤ 1).
2. - .
. . y = f ( x ) % ( )
X , x1 , x2 X , x1 < x2 , %
f ( x1 ) < f ( x2 ) ( f ( x1 ) > f ( x2 )).
$ " $"%' #
$"%' , " $"%' # -
' .
. . y = f ( x ) % ( )
X , x1 , x2 X , x1 < x2 ,
f ( x1 ) ≤ f ( x2 ) ( f ( x1 ) ≥ f ( x2 )).
$ " $"%' #
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$"%' . 3 # - $ . 36 ( – , $ – , – , # – -).
*. 36 ( ). *. 36 ($).
*. 36 ( ). *. 36 (#).
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, " " $ -
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(0; + ∞) .
, , % -
. % %
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/ " . 16, y = f (x) (x X , y Y ) $
"% , " %
X Y . " % %, , # -
. ), " x1 < x2 " %
f ( x1 ) < f ( x2 ) , " x1 < x2 " %
f ( x1 ) > f ( x2 ) . , $% # $ -. %, y = f (x) ( ), $
x = ϕ( y) ( ).
3. .
. . y = f (x) % ( ), . f (−x) = f ( x) ( f (−x) = − f (x))
", $" %
$ ’ x = 0 . 0 -
Oy ( . 37 ( )), – -
( . 37 ($)).
*. 37 ( ). |
*. 37 ($). |
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" , ", y = x2 , y = x126 , y = cos x , y = x3 , y = x5 , y = sin x , y = tg x , y = ctg x ,
y = arcsin x, y = arctg x .
" , , , . &
"% # , " % , . -
" y = arccos x , y = x2 + x, y = 2x .
)" $ # % $
# "% ", $
Oy (" ), $ ("
).
4. .
. . y = f ( x ) % ,
T > 0 , x D f : f ( x + T ) = f ( x) .
4 " T % . ,, $ -
$% " kT (k ) . ' -
% .
" "% #-
. , y = sin x , y = cos x 2π , -
y = tg x, y = ctg x π .
)" $ # , T % $ -
# $% T (" (0;T ) ),
# %, # -
T . /, #, , , % $ -
# " (0;T 
2) .
18..
.
3$ # ", $" .9,
# "% y = f ( x ). % " %
x1 , x2 ,..., xn ,... % # x , $# % " x0 , $
lim xn = x0 . y = f ( x ) $% -
n→∞
" % f ( x1 ), f ( x2 ),..., f ( xn ),...
. 4 " A % y = f ( x ) x → x0
( $ x0 ), " $% " {xn } % # x ,
, xn ≠ x0 , lim xn = x0 , " % %
n→∞
{ f ( xn )} $# % " A , $ lim f ( xn ) = A .
n→∞
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