Вступ до аналізу. Ч. 2

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# , #

" % # ( . 45).

*. 45.

". ",

" $ ". ", y = 1 x -

" (0,1) , " % $ .

$* '. & ' y = f ( x) -

[a,b] , * * * -

!.

. , "% y = f ( x) [a,b] ,

$ [a,b] , y = f ( x) % -

# %. : M = sup f ( x) . -

[ a,b]

, x [a,b] : f (x) < M . * #" :

M − f ( x)

, "% M − f ( x) ≠ 0 , ϕ( x) [a,b] ,

ϕ( x) $ [a,b] , $ M 0 > 0 , x [a,b] : ϕ( x) ≤ M 0 . ,

M − f ( x) ≥ 1 , M 0

",

f ( x) ≤ M − 1 . M 0

! , M ≠ sup f ( x) . ", c1 [a, b] :

[a ,b ]

f (c1 ) = M , $ y = f ( x) # [a,b] %

#. ! " # %, y = f ( x) # [a,b]

% #.

.

87

". ", -

", $ ". ", y = x

" (0,1) , " # % " % #,

", % #, 1.

) ' + %–& ' ( ). + * ! y = f ( x)

[a,b] , * ! - * , f (a) f (b) < 0 . , c (a,b) : f (c) = 0 .

. " [a,b] ". :

d1 = a + b . 2

/ f (d1 ) = 0 , . / f (d1 ) ≠ 0 ,

[a, d1 ] [d1 , b] $ , # y = f ( x) -

. # [a1, b1 ] . 8 # :

b1 − a1 = b − a . 2

[a1,b1 ] " ". :

d2 = a1 + b1 . 2

/ f (d2 ) = 0 , . / f (d2 ) ≠ 0 ,

[a1, d2 ] [d2 , b1 ] $ , # y = f ( x)

. # [a2 , b2 ] . 8 # :

, , $ -

% dk , f (dk ) = 0 , $ " % " -

[a1,b1 ] [a2 ,b2 ] K [an ,bn ] K, f (an ) f (bn ) < 0 n ,

" % . 5 ( . . 12) c , " % . ,f (c) = 0 . ), , " f (c) > 0 . , "%

y = f ( x) , δ > 0 , x (c − δ, c + δ) f ( x) > 0 . ' # $ , "%

.

88

0 " . 46. 0

, # , -

% Ox $ % # .

*. 46.

x = c % f ( x) = 0 . 3 "% -

"% , " , # .

', , % $". * #" ".

), [0;1] x4 + x3 −1 = 0 %,

% 0,1.

f ( x) = x4 + x3 −1. 2 " ,

[0;1] . , "% f (0) = −1 < 0, f (1) = 1 > 0 , '

6 "%–5 ' [0;1] %. - " x1 = 0, 5 [0;1] "

[0; 0, 5] [0, 5;1] $ , # f ( x ) $ %

. , "% f (0, 5) = −0,81 < 0 , [0, 5;1] . , -

% % ' # . ) "

x2 = 0, 75 " "

[0, 5; 0, 75] [0, 75; 1] $ , # $ -

% . , "% f (0, 75) = −0,12 < 0 , [0, 75; 1] .

, " : x3 = 0,875 ; f (0,875) = 0, 27 > 0

. ,$ " [0, 75; 0,875] . ) % # 0,875- 0,75=0,125. ,, $" #

x4 = 0,8125 , " % [0, 75; 0,875] ", : x4 − c ≤ 0, 0625 < 0,1 , $ $ % #. ,

c ≈ x4 = 0,8125 . f ( x ) : f ( x4 ) ≈ −0, 028 . 89

, % $% -

. !" % % # " , " #-

0,1 ' $ 4 . ! $"%' % " -

# . , $ %

δ > 0 . , "% % n ,

$ " # , "% % %:

10 . ' % ', $"%' -

%, ", %.

+ %–& ' ( ). + * !

y = f ( x) [a,b] , f (a) = A , f (b) = B , A ≠ B .

, C , A B , ! c (a, b)

, ' f (c) = C .

. " , A < C < B . -

ϕ(x) = f ( x) − C . 2 [a,b]

. 5 #, ϕ(a) = f (a) − C = A − C < 0 , ϕ(b) = f (b) − C = B − C > 0 , $ [a,b] ϕ( x)

. # % -

% c (a, b) , ϕ(c) = 0 , $ f (c) = C .

.

2 ,

# ' # $ %. 0 " . 47.

*. 47.

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! ( ). & ' y = f ( x)

[a,b] , f (a) = A , f (b) = B ,

[ A, B] x = g( y) ,

y = f ( x) , [ A, B] .

. , "% y = f ( x) , x [a, b] -: A ≤ f (x) ≤ B , A = inf f ( x) , B = sup f ( x) . -

# y0 [ A, B] x0 [a, b] , f ( x0 ) = y0 . &-

x0 " . ),

. , x1 [a, b] ,f ( x1 ) = y0 , x1 ≠ x0 . / x1 > x0 , "

y = f ( x) [a,b] $ f ( x1 ) > f ( x0 ) , ", "%

f ( x1 ) = f ( x0 ) = y0 . ! " # " % " % x1 < x0 , $

$ "% x1 = x0 . $ x = g( y) -

. , [ A, B] . * #"

y1, y2 [ A, B], , y1 < y2 . $ , g ( y1 ) < g ( y2 ) .

, , $ g ( y1 ) ≥ g ( y2 ) . : x1 = g ( y1 ) ,

x2 = g ( y2 ) . x1 , x2 [a, b] , x1 ≥ x2 . , "% y = f ( x)

[a,b] , f ( x1 ) ≥ f ( x2 ) , $ y1 ≥ y2 , -

% , y1 < y2 . x = g( y) -[ A, B] . " ' " , [ A, B] .

y0 – "% " ( A, B) . , x = g( y)

y0 . )" % # % " % -

:

x = g( y) . 2 % , [a,b] , - " " " % Eg . ! " # % -

" % # (23.1), $ lim g( y) = g( y0 ) .

y → y0 + 0

$ " %, x = g( y)

A " B .

.

" 1. / y = f ( x) [a,b] , -

x = g ( y) [ A, B] .

" 2. ! " # " % %

$ , " y = f ( x) " $ -

".

& ' y = f ( x) ,

(a, b) , x = g( y) ,

24. # % .

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, , " - # # " . ,,

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$" , , "

$" .

1. 3 "% . . y = x -

, "% y = x → 0 x → 0 " $% # x . -

y = xn (n ) $ # " -

. , "% y = C , C – ", (

y = 0 x ), y = axn , a = const , ,

", # " n

Pn ( x) = an xn + an−1 xn−1 + ... + a1 x + a0

( # " -

).

* "%

R(x) = Pn (x) , Qm (x)

Pn ( x), Qm ( x) – # " n m ,

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, - # " Qm ( x) . $ , R(x) .

2. 3 "% . * #"

y = xr , r . / r , r , y = xr

, $ y = x1 r ,

. r , r . * #" « » -

y = xr , " ' [0, + ∞) .

$ y = x1 r , [0, + ∞) .

. y = x2 k , k , x (− ∞, 0) $ –

y = −2 k x .

* #" " ’ , $-

y = x− n = 1 xn , n .

/{0} . n = 2k + 1 ( k ) $ /{0} ,

n = 2k ( k ) $ (− ∞, 0) (0, + ∞) .

r = mn , m , n . :

(0, + ∞) , m ≥ 0 m < 0 .

y = xr (0, + ∞) , , r ≥ 0 , r < 0 .

% "% $" ,

, "%

$" . , "#$ $" -

.

3. # $ # .

(. ( x (− π2, 0) (0, π2) :

x

. ) ' " x > 0 .

# $ ( . 48).

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$ " 1.

OB % # " x ( ) # $-

, # ".

4 A(1; 0) , " $ ( -

"). OB -

C . " AOB $ sin x , "

!$ :

cos x < sin x < 1. x

94

, "% y = sin x x y = cos x , ' (24.1)

. / x = 0 , % (24.2) . x ≠ 0 . / x (0, π 2) , (24.1) :

sin x < 1,

x

# " (24.2). , "% y = sin x x , -

% (24.2) % | x | < π 2 . ! | x | ≥ π 2 , %-

" "% x , y = sin x

" .

. y = cos x

: cos x = sin( x + π 2) .

.

# 1. - y = tg x /{x = π 2 + πk} ,

y = ctg x /{x = πk} .

), y = tg x = sin xcos x -

, , y = cos x -

", x = π2 + πk . . y = ctg x = cos xsin x

, , y = sin x ", -x = πk .

# 2. - y = arcsin x , x [−1,1] -

y = sin x [− π2, π2] ( . . 16). - y = arccos x ,

x [−1,1] y = cos x [0, π] . - -

y = arctg x y = tg x

(− π 2, π 2 ) . - y = arcctg x

y = ctg x (0, π) .

95

4. " # . ), y = a x (a > 0, a ≠ 1) . * #" a > 1 .

x . x x #": y = a x+Δx − a x = a x (a x − 1) .

/ , lim (a x −1) = 0 , % lim y = 0 , $-

x→ 0 x→ 0

y = a x x . 3 % ( . .10, " 2):

1

a n0 < a x < a n0 ,

1 − ε < a x < 1 + ε ,

$

a x − 1 < ε ,

" "% ε > 0 ", lim (a x − 1) = 0 .

x→ 0

,, " "% x , y = a x a > 1

.

0 < a < 1. 1 a > 1,

, "% y = (1 a)x , (1 a)x > 0

x , y = a x -

.

. y = loga x (a > 0, a ≠ 1) a > 1 (0, + ∞) $ -

y = a x , 0 < a < 1 – -

(0, + ∞) $ y = a x .

96