Высшая математика. Том 2

+. 1.46. A& + @ @ 6 6& E 9

% 6& E " ! # 3, + # parametric ! A&-x_expr y_expr, C ! 6 # [param, min, max].

! C	' ! # param. 6 A C < D, %
+ < '+" 3	# (x_expr, y_expr) ! + param
min max: [parametric, x_expr, y_expr, t_range].
	)
	cos x = t	.
A& & @ 6 6& E 9 $ ! # 3:
	sin x = t

(%i7) plot2d ([parametric, cos(t), sin(t)], [t, -4, 4],[nticks, 200],[style,[lines,4,0]])$

+. 1.47. A& ! # 3 @ @ 6 6& E 9

61

4 & '

1. # > ! % & ! + ?

2.# C C +6 3 @ # E 3 > -?

3.$ 3 " + & ! + .

4.P # '$ ! %?

5.# C 3 & ! + " " ! "# 9 ! " "# 2-> ! %?

6.# "D '+" 9 ! > &@ @ ! " &?

7.! "#$ ! +&< ! > &@ @ ! " &?

8.! A& & @ 6 ! # @ D ! & Maxima?

-

'

3-

-

. + ' - ,

’ .

62

2.

,

-

,

’ .

– .

, ,

, .

!

! ' " .

#, , "-

, , ! !

( ' ) - -

. $	, -
.
%	. ,

&.', (.', (.), &.*, %.'

. %.'

, , !

, ,

' . *,

, .

% " "

(.( '.$ .

2.1.

) ! ’ -

. #, -

, , ! ! S " S = r2. + ! r , ! S

, .

+ ! x, ! , -

, . , – y=f(x). ) x ! -

.

- y=C, " – , " , -

x ".

. x, "

f(x), # .

63

- , ! " , -

.

" " ,

:

y = xa , y = ax , y = loga x , y = sin x , y = cos x , y = tg x , y = ctg x , y = arcsin x , y = arccos x , y = arctg x , y = arcctg x .

$ " , -

" -

, , , " " -

.

2.2.

/ x = {xn}, ! -

%

N(ε), ! n>N(ε) : xn − a < ε .

– .

, ,

, .

+ ! x = {xn}, , ! xn -

, lim xn = a .

n→∞

0 " ,

.

& x0 ( , b), !

. / x0, x0 -

, x0 , (b– )/2 – -

.

%, ' , ! -

.

−ε < xn − a < ε a − ε < xn < a + ε .

1 , ! n>N(ε)

! (a – %; a + %).

2. 2.1.

64

%, {xn}, !

- % (% – )

N(ε), ! -

n>N(ε) .

, !

lim

a

= a ( N(ε)). a

=

2n3

,

a = 2.

n3 − 2

n→∞

n

n

3 ), ! - % >0 N(ε), !

n > N(ε)

an − a

< ε .

a

− a

=

2n3

− 2

=

2n3 − 2

(n3 − 2

)

=

2n3 − 2n3 + 4

=

4

.

n

n3 − 2

n3 − 2

n3 − 2

n3 − 2

4

4

3

3

4

4

< ε

< n

− 2 n

>

+

2

n

> 3

+

2 .

ε

ε

ε

n3 − 2

- , ! N (ε ) = !#3								4	+ 2 – ( -
								ε
) -							n > N(ε) -
			an − a			< ε . 1 ,	. !
lim	2n3			= 2 .4
		− 2
n→∞ n3

1. - , !

xn = ,

. , - %

|xn – c| = |c – c| = 0 < %.

2. - , ! -

. , , ! xn > xn > b. $ -

- ε < (b − a)/ 2 b % ( . .

2.2.).

# , ,

, , b, !

.

2. 2.2.

65

3. 5 . 6,

,		1	,	2	,1,	1	,	2		. - ,
	x =	3		3		3		3	,1,

! - .

2.3. ! " 2.1

, ’ ,

.

, !

lim a

= a (

n→∞

n

1. a = 3n − 2 ,

a = 3 .

n

2n − 1

2

3. a = 7n + 4 ,

a = 7 .

n

2n + 1

2

5. a = 7n − 1, a = 7.

n

n + 1

7.

a =

9 − n3

, a = − 1.

1+ 2n3

n

2

9. a

=

1− 2n2

, a = − 1.

n

2 + 4n2

2

11.

a

n

=

n + 1

,

a = − 1 .

1 − 2n

2

13. a

= 1− 2n2

,

a = −2.

n

n2

+ 3

15.

a

n

=

n

,

a = 1.

3n

− 1

3

17.

a

n

= 4 + 2n ,

a = − 2 .

1 − 3n

3

19.

a

=

3− n2

,

a = − 1.

1+ 2n2

n

2

21.

a

n

= 3n − 1

,

a = 3.

5n

+ 1

5

23.

a

=

1− 2n2

,

a = − 1.

n

2 +

4n2

2

25.

a

n

= 2 − 2n ,

a = − 1 .

3 + 4n

2

N (ε ) ).

2.

a = 4n − 1, a = 2.

n

2n + 1

4. a = 2n − 5 ,

a = 2 .

n

3n + 1

3

6.

a =

4n2 +1

, a

= 4.

n

3n2 + 2

3

8. a = 4n − 3 , a = 2.

n

2n + 1

10.

a

= −

5n

,

a = −5.

n

n + 1

12.

a

=

2n + 1

,

a = 2 .

n

3n − 5

3

14.

an

=

3n2

,

a = −3.

2 − n2

16.

a

=

3n3

,

a = 3.

n

n3 −1

18.

a

= 5n + 15 ,

a = −5.

n

6 − n

20.

a

=

2n − 1

,

a = − 2 .

n

2 − 3n

3

22.

a

= 4n − 3 ,

a = 2.

n

2n + 1

24.

a

=

5n + 1

,

a = 1 .

n

10n − 3

2

26.

a

= 23 − 4n ,

a = 4.

n

2 − n

66

27.

a

= 1 + 3n ,

a = −3.

28.

a

= 2n + 3 ,

a = 2.

n

6 − n

n

n + 5

29.

a

=

3n2 + 2

,

a = 3.

30.

a

=

2 − 3n2

,

a = −

3.

4n2 −1

4 + 5n2

n

4

n

5

31.

a =

2n3

,

a = 2.

n

n3 − 2

2.4.

6 " y=f(x) . ) , !

x ( . 2.3). 1 , ! ,

. ) x ' . x " -

. . , ! f(x) -

b. # , ! b " f(x) x ' .

2. 2.3. *

7 y=f(x) b x ' , ! -

%, ,

((ε), ! x ' " , !

|x – a| < (, | f(x) – b| < %. + ! b " -

f(x) x ' , : lim f (x)= b f(x)> b x ' .

x→a

) " " ( . 2.4).

% |x – a| < ( | f(x) – b| < %, x ( – (, + () " f(x) (b – %, b + %), ,

% > 0, ((ε), !

x, !	((ε) – , " "
!	2%, = b – % =
b + %.

67

					2. 2.5.
2. 2.4. & "					"

* ( . 2.5), ! "

, ! , lim c = c , !

x→a

x ' " , .

) 1: - " y=2x+1 x ' 1.

3 * " " . 2.5, , ! x ' 1 -

, M(x, ) " M(1; 3),

lim(2x + 1)= 3 . . - % > 0. * ,

x→1

! : |(2x+1) – 3|< % |2x–2| < %, |x– 1| < %. #-

, ! ((ε) = %/2, x, ! -

|x – 1|<((ε), |y – 3| < %. -

, ! 3 "

y=2x+1 x ' 1. 4

) 2: - "

y=ex+1 x ' 0.

		3 $ "
" . 2.6, , !
lim ex+1		= e .4					2. 2.6. " " y=ex
x→0
		) 3: - " , , !:
lim	15x2 − 2x − 1		= 8 (		δ (ε ) ).
		x − 1/ 3
x→1/ 3				15x2	− 2x − 1
		3 # "				x		− 1/ 3 . * -
				x	− 1/ 3

, ! ε > 0 δ (ε ) , ! -

:

68

15x2 − 2x − 1

− 8

< ε ,

(2.1)

x − 1/ 3

!

x − 1/ 3

< δ (ε ). ) x ≠ 1/ 3 (2.1),

:

15x2 − 2x − 1

− 8

=

15( x − 1/ 3)( x + 1/ 5)

− 8

=

15( x + 1/ 5) − 8

=

15x + 3 − 8

=

x − 1/ 3

x − 1/ 3

=

15x − 5

=15

x − 1/ 3

< ε

x − 1/ 3

< ε /15

(2.2)

# , ε > 0 (2.1) , ! (2.2) ( δ (ε ) = ε 15 ). 5 , ! -

" x → 1/ 3 8. $ : δ (ε ) = ε 15 .4

2.5.

2 , " f(x) -

b x ' a x 8 9. 2 , "

y=f(x) .

7 f(x) x ' , -

, ! - ), ,

( > 0, ! :a, |x-a| < (, |f(x)| > M.

+ ! f(x) x'a, lim f ( x) = ∞

x→a

f(x) 89 x'a. + ! f(x) x'a

,

lim f ( x) = +∞

lim f ( x) = −∞ .

x→a

x→a

lim 1

x

= ∞ ,

lim ex2 = +∞ ,

lim ln x = +∞ .

x→0

x→∞

x→+∞

2. 2.7. " " y= ln x	2. 2.8. " "	y = sin 1
		x

69

lim ln x = −∞ ( . . 2.7).

x→0

7 y = sin 1 x ( . 2.8) x>0 -

" .

2.6. # $

7 y=f(x) x'a x'*,
! lim f ( x) = 0	lim f ( x) = 0	, " – " -
x→a	x→∞

, .

) 1: 7 f(x)=(x–1)2

x'1, -

	lim f ( x) = lim		( x − 1)2	= 0 ( .
	x→1	x→1

. 2.9).

) 2: 7 f(x) = tg x

– x'0.

2. 2.9. " " f(x)=(x–1)2

) 3: 7 f(x) = ln(1+x) – x'0.

) 4: 7 f(x) = 1/x – x'*.

% " .

% # 1. 5 , -

" .

% # 2. " (x) " -

f(x) x'a ( x'*) " .

1. + !	lim α ( x) = 0 lim β ( x) = 0 , lim αβ = 0 .
	x→a	x→a	x→a
2. + !	lim α ( x) = 0 c=const, lim cα ( x) = 0 .
	x→a		x→a

% # 3. $ " +(x) " f(x),

, " .

70