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kafedra

w m m f

wARIANT 18

w y s { a q matematika

sBORNIK INDIWIDUALXNYH DOMA[NIH ZADANIJ

DLQ STUDENTOW

TEHNI^ESKIH SPECIALXNOSTEJ tpu

tABLICA \KWIWALENTNYH BESKONE^NO MALYH

eSLI (x) ! 0, TO SPRAWEDLIWO:

1: sin (x) (x)
2: arcsin (x) (x)
3: tg (x) (x)
4: arctg (x) (x)							2
5: 1	; cos (x)			( (x))
5: 1	; cos (x)			2
6: ln [1 + (x)]				(x)
				(x)
7: loga [1 + (x)]				ln a
(x)				ln a
8: e (x) ; 1			(x)
9: a		; 1	(x) ln a
	n				(x)
10: q1 + (x) ; 1						n

1: sin (x) (x) ;

( (x))3

2: arcsin (x) (x) +

( (x))3

3: tg

(x) (x) +

( (x))3

4: arctg (x)

(x)

;

( (x))3

5: 1 ; cos (x)

( (x))2

;

( (x))4

6: ln [1 + (x)] (x) ;

( (x))2

7: e (x) ; 1 (x) +

( (x))2

8: n

(x)

1 ; n

( (x))2

1 + (x)

;

2n2

wTOROJ ZAME^ATELXNYJ PREDEL

(

)

lim 1 +

= e

lim 1 +

= e

lim

(1 + (x)) (x) = e

n!1

x!1

e = 2 7182818284590:::

sUMMA n ^LENOW ARIFMETI^ESKOJ PROGRESSII

Sn = a1 + a2 + : : : + an =

a1 + an

sUMMA n ^LENOW GEOMETRI^ESKOJ PROGRESSII SO ZNAMENATELEM q

Sn = b1 + b1q + b1q2 + : : : + b1qn;1 = b1(1

; qn)

pRI jqj < 1

S =

1 ; q

fAKTORIALY

0! = 1

1! = 1

n! = 1 2 3

4 : : : (n ; 1) n

2! = 1

2 = 2

3! = 1

3 = 6

4! = 24

5! = 120 :::

(2n)! = 1 2 3 : : : n (n + 1) : : : (2n ; 1) 2n

(2n)!! = 2 4 6 : : :(2n ; 2) 2n

(2n+1)! = 1 2 3 : : : n (n+1) : : : 2n (2n+1)

(2n+1)!! = 1 3 5 : : : (2n;1) (2n+1)

fORMULA sTIRLINGA

p2 n

pRI BOLX[IH ZNA^ENIQH n

n! e

= k Uk;1 U0

eSLI C{KONSTANTA, A

U(x) I V (x) { DIFFERENCIRUEMYE FUNKCII, TO

oSNOWNYE PRAWILA DIFFERENCIROWANIQ

1: (

C )

6: [y(U(x))] = yu

2: ( C

U )

3: (

V ) =

4: (

)

7: x

(y) =

yx0 (x)

;

U V

V 2

8: y0(x) = y(x) (ln y(x))0

9: UV 0 = V UV ;1 U0 + UV ln U V 0

10:

(

x = x(t)

y0(x) =

y0(t)

y00(x) =

y00(t)x0(t)

; x00(t)y0(t)

y = y(t)

)

(

( ))

tABLICA PROIZWODNYH

1: Uk 0

= ;

4: aU 0

= aU ln a U0

5: eU 0

= eU U0

10: (tg U)

cos2 U

11: (ctg U)

= ;

sin2 U

12: (arcsin U)0

= p

;

13: (arccos U)0 = ;p

;

14: (arctg U)

1 + U2

6: (logaU)

15: (arcctg

= ;

U ln a

1 + U2

16: (sh U)0

= ch U U0

7: (ln U) =

8: (sin U)0

= cos U U0

17: (ch U)0

= sh U U0

9: (cos U)

= ;sin U U

18: (th U)

ch2 U

oSNOWNYE NEOPREDEL<NNYE INTEGRALY

					k+1
1. Z	Uk dU =			U			+ C
				k + 1
			(k =			1)
2. Z			6 ;
	dU = U + C
	dU
3. Z			= 2pU + C
	p
		U

4. Z	dUU2	= ;	1	+ C
			U
5. Z	dUU = ln jUj + C


6. Z		aU dU =			aU
						+ C
					ln a
7. Z eU dU = eU + C
8.	sin U dU = ;cos U +C
9.	ZZ	cos U dU = sin U +C
10. Z			dU	= tg U + C
			cos2 U
11. Z			dU	= ;ctg U +C

			sin2 U

12.

tg U dU = ;ln jcos Uj + C

13.

ctg U dU = ln

sin U

+ C

14. Z

= ln

+ C

sin U

15.

= ln tg

4 + C

cos U

16. Z

= a arctg a

+ C

a2 + U2

17.

; a

+ C

U2 ; a2

U + a

18. Z

= arcsin a

+ C

;

19. Z

= ln jU

+pU2 a2j+C

20.

sh U dU = ch U + C

21. Z

ch U dU = sh U + C

22. Z

= th U + C

ch2 U

23. Z

= ;cth U + C

sh2 U

24. Z pU2 a2 dU =

U p

U2 a2

a2 ln jU U+ pU2 a2j +C

25. Z pa2 ; U2 dU =

U pa2 ; U2 + a2arcsin a ! + C

26. Z e U sin U dU =

e U

( sin U ; cos U) + C

+ 2

27. Z e U cos U dU =

e U

( cos U + sin U) + C

+ 2

rQDY mAKLORENA \LEMENTARNYH FUNKCIJ

1: ex = 1 + x + x2

+ x3 + : : : + xn

+ : : : = 1 xn

n=0

x2n+1

2: sh x = x +

3! +

5! +

: : : +

+ : : : = n=0

(2n + 1)!

X2n

3: ch x = 1 +

2! +

: : : +

+ : : : = n=0

(2n)!

x2n+1

4: sin

x= x; 3! +

5! ;: : :+(;1)n

+: : :=

(;1)n

(2n + 1)!

n=0

(2n + 1)!

x2n

5: cos

x = 1 ;

2! +

4! ; : : : + (;1)n

+ : : : =

(;1)n

(2n)!

n=0

(2n)!

m(m

;

1)(m

;

6: (1 + x)m = 1 +

1! x +

2!;

x2 +

x3 + : : :

= 1 ; x + x2 ; x3 + : : : + (;1)n xn + : : : = n=0(;1)n xn

1 + x

xn+1

8: ln (1 + x) = x; 2 + 3 ;: : :+(;1)n

+: : := (;1)n

n + 1

x2n+1

n=0

x2n+1

9: arctg x= x;

3 + 5 ;: : :+(;1)n

+: : :=

(;1)n

(2n + 1)

n=0

(2n + 1)

+ 1

5 x

10: arcsin x = x +

1 x

+ : : :

2 3

2! 5

23 3! 7

11: tg x = x + 3x

+ : : :

12: th x = x ;

; : : :

3 x

rQD I INTEGRAL fURXE (OSNOWNYE FORMULY)

rQD fURXE FUNKCII, ZADANNOJ NA INTERWALE [

;

]

f(x) =

an cos nx + bn sin nx

n=1

= 1

(x)dx

f(x) cos nx dx

bn =

(x) sin nx dx

;

rQD fURXE FUNKCII, ZADANNOJ NA INTERWALE [

n ;

f(x) =

an cos

x + bn sin

n=1

a0 = 1l ;Zl f(x)dx

an = 1l ;Zl f(x) cos

x dx

bn = 1l ;Zl

f(x) sin

x dx

3. rQD fURXE FUNKCII, ZADANNOJ NA INTERWALE [0 l]

pO SINUSAM

pO KOSINUSAM

f(x) =

bn sin

f(x) =

an cos

n=1

bn = 2l Zl

f(x) sin

x dx

a0 = 2l

Zl f(x)dx

an = 2l Zl f(x) cos

x dx

;

4. rQD fURXE f(x)

(

l l) W KOMPLEKSNOJ FORME

;Zl

f(x) =

Sn(!n)e

GDE

!n =

Sn(!n) =

f(x)e

n=;1

5. iNTEGRAL fURXE FUNKCII f(x)

(

)

x) dt1 d!

f(x) =

(t) cos !(t

;

;1Z

dLQ ^ETNOJ FUNKCII

f(x) = 2

cos !x d! Z

f(t) cos !t dt

dLQ NE^ETNOJ FUNKCII

f(x) = 2

Z sin !x d! Z

f(t) sin !t dt

6. pREOBRAZOWANIE fURXE FUNKCII f(x) x 2 (;1 1)

F (!) = Z f(x)e;i!xdx

7. kOSINUS I SINUS PREOBRAZOWANIQ fURXE FUNKCII f(x) x		2	(0	1	)
1	1
Fc(!) = 2 Z f(x) cos !x dx	Fs(!) = 2 Z f(x) sin !x dx
0	0

tABLICA IZOBRAVENIJ I ORIGINALOW

f(t)

F (p)

;at

p + a

;at

t e

(p + a)2

;at

(p + a)3

f(t)

0 t

F (p)(1

;

e;p )

( 0 t >

sin at

p2 + a2

cos at

p2 + a2

		f(t)				F (p)

10		t sin at				2ap

					(p2 + a2)2
					(p2 + a2)2
11	t cos at					p2	; a22 2
11	t cos at				2		; a22 2
					(p		+ a )
12		sh at					a

						p2 ; a2
						p2 ; a2
13		ch at					p

						p2 ; a2
						p2 ; a2
14	e;at sin bt						b

				(p + a)2 + b2
				(p + a)2 + b2
15	;at		cos bt			p + a
	e
	e			(p + a)2 + b2
				(p + a)2 + b2
16	e;atsh bt						b

				(p + a)2 ; b2
				(p + a)2 ; b2
17	;at		ch bt			p + a
	e
	e			(p + a)2 ; b2
				(p + a)2 ; b2
18		(t)					1
19	(t ; )					e;p

zadanie N 1	lINEJNAQ ALGEBRA	wARIANT 18

1.	wY^ISLITX OPREDELITELI
		;2	2	;3		1				;3		2	;1 5
	a)	1 ;1		4	;3				b)		2	;1	1	4
		3	1	;1		1				;5		1	;2	1
		;4	2	0	;5						2	1	0	1
2.	nAJTI MATRICU		h IZ URAWNENIQ.						sDELATX		PROWERKU
			X 0	6	4	5	1	= 0	1	3	0	1
			X 0	2 1		2	1	= 0	10	;2 7		1
			B	3	3	3	C	B	10	7	8	C
			@				A	@				A
3.	rE[ITX SISTEMY LINEJNYH URAWNENIJ:
	A) METODOM kRAMERA,									b)	MATRI^NYM METODOM

	8	x + 5y ; z = 5
	<
	a) >	2x + 3y + 2z = 2 b)
	>	5x ; y = 15
4.	rE[ITX:SISTEMY METODOM gAUSSA

8 x + 4y ; 13z = ;27

< 4x + 3y ; 12z = ;17

> 11x + y + 6x = 4

		3x1	;2x2	;5x3 +x4 = 3
	a) 8 2x1		;3x2 +x3 +5x4 =					;3
	<		+2x2		;	4x4 =		;	3
	> x1		+2x2			4x4 =			3
	> x1		;x2	;4x3 +9x4 = 22
	:
	8	2x1	+7x2	+3x3 +x4 = 6
	<
	b) >	3x1	+5x2	+2x3 +2x4 = 4
	>	9x1 +4x2 +x3 +7x4 = 2
	:
	8	3x1	+2x2	+x3 +3x4 +5x5 = 0
	<	6x1 +4x2 +3x3 +5x4 +7x5 = 0
	c) > 9x1		+6x2	+5x3 +7x4 +9x5 = 0
	> 3x1 +2x2			+4x4 +8x5 = 0
5.	:
5.	nAJTI SOBSTWENNYE ZNA^ENIQ I SOBSTWENNYE WEKTORY MATRIC
	a) A = 0 4 ;1 1			b)	B = 0		27	;36		26 1
		@ 1	6 A			B	2		2	3	C
		@ 1	6 A			@	2		2	3	A
				10		@					A

zadanie N 2

wEKTORNAQ ALGEBRA

wARIANT 18

1. dAN PARALLELOGRAMM ABCD W KOTOROM

;!AB = ~a ;!AD = b:

tO^KA DELIT STORONU DC W OTNO[ENII j

DM j : j

MC j= 1=5: tO^KA

N DELIT STORONU BC W OTNO[ENII

= 5=4. wYRAZITX

;! ;! ;;! ;! ;;!

WEKTORY AC BD AM AN MN

^EREZ WEKTORY ~a I

2. oPREDELITX KOORDINATY TO^KI

C, LEVA]EJ NA PRQMOJ, PROHO-

DQ]EJ ^EREZ TO^KI A I B, ESLI

A(;3 2 ;4) B(3 ;2 2) I

jACj : jCBj = 0 3 : 2

3. w TREUGOLXNIKE S WER[INAMI

A(;3 2 1) B(;3 0 ;2) C(2 ;4 3):

nAJTI: a) WEKTOR MEDIANY AM,

b)WEKTOR WYSOTY BD,

c)L@BOJ PO MODUL@ WEKTOR BISSEKTRISY UGLA C:

4. dANY TRI WER[INY PARALLELOGRAMMA ABCD:

A(;2 3 0) B(3 2 ;3) C(;3 ;2 1): nAJTI: a) KOORDINATY ^ETWERTOJ WER[INY D,

b) DLINU WYSOTY, OPU]ENNOJ NA STORONU AB

c) KOSINUS OSTROGO UGLA MEVDU DIAGONALQMI AC I BD.

5. tREUGOLXNIK ABC POSTROEN NA WEKTORAH

;!AB = 5p~ + 9q~

;

~q GDE

= 1

= 3

(~p

~q) = 120o: nAJTI:

a) DLINU WYSOTY, OPU]ENNOJ NA STORONU AB,

b) KOSINUS UGLA MEVDU STORONOJ AB I MEDIANOJ AM.

6. nAJTI EDINI^NYJ WEKTOR ~e,	KOTORYJ ODNOWREMENNO PERPENDIKU-
LQREN WEKTORAM ~a = f3 4 1g	~	~
LQREN WEKTORAM ~a = f3 4 1g	I b = f7 6 9g,	ESLI (~e ^ i) =2.

7. w PIRAMIDE ABCD S WER[INAMI W TO^KAH

A(3 4 ;7) B(1 5 ;4) C(;5 ;2 0) D(;12 7 ;1)

NAJTI OB_EM I DLINU WYSOTY, OPU]ENNOJ NA GRANX ABC.

8. dOKAZATX, ^TO WEKTORY p~ = f;2 0 1g q~ = f1 3 ;1g ~r = f0 4 1g

OBRAZU@T BAZIS I NAJTI RAZLOVENIE WEKTORA ~x = f;5 ;5 5g W \TOM BAZISE.

3 : 4:

zadanie N3	wARIANT 18
aNALITI^ESKAQ GEOMETRIQ NA PLOSKOSTI

1. sOSTAWITX URAWNENIQ PRQMYH,	PROHODQ]IH ^EREZ TO^KU M(;5 3):
a) PARALLELXNO PRQMOJ x + 5 = y
0	7
b) PERPENDIKULQRNO PRQMOJ	4x + 5y ; 1 = 0
c) POD UGLOM 450 K PRQMOJ 8 x = 4t + 7
< y = ;3t ; 1
2. dANY WER[INY TREUGOLXNIKA:	A(;5 7) B(7 ;2) C(11 20):

sOSTAWITX: a) URAWNENIE STORONY AC,

b)URAWNENIE MEDIANY wm,

c)URAWNENIE WYSOTY sH I NAJTI EE DLINU.

3. dANY DWE PRQMYE l1 :	x	+	y	= 1 l2 :	x + 5	=	y	. nAJTI:

	;4		3		6		;3
a) TO^KU PERESE^ENIQ PRQMYH,

b) KOSINUS UGLA MEVDU PRQMYMI,

c) SOSTAWITX URAWNENIQ BISSEKTRIS UGLOW MEVDU PRQMYMI.

4. pRIWESTI URAWNENIQ LINIJ K KANONI^ESKOMU WIDU I POSTROITX:

x2 + y2 + 4x + 5y + 4 = 0

9x2 + 8x + 4y2 + 18y = 0

y = 6

;

1:5px2

6x + 13

x = 2y2

;

2y + 3

3x ; 2xy + 3y

; 8 = 0

+ 4xy + 2y

+ 8x + 8y + 1 = 0

5. sOSTAWITX URAWNENIE I POSTROITX LINI@, DLQ KAVDOJ TO^KI KOTO- ROJ RASSTOQNIE OT NA^ALA KOORDINAT I OT TO^KI M(4 ;3) OTNOSQTSQ KAK

6. pOSTROITX LINII, ZADANNYE W POLQRNYH KOORDINATAH:
1) = 2 sin 3'		2) =	1		3) =	2	:
			3 sin ' ; cos '			3 + 2 cos '
7. pOSTROITX LINII, ZADANNYE PARAMETRI^ESKIMI URAWNENIQMI:
1) 8 x = 2 ; sin t			2) 8 x = et
	< y	= 3 + cos t	< y = sin t
	:		:
8. pOSTROITX FIGURU, OGRANI^ENNU@ LINIQMI
	y = x2=3			x = ln(1 + t)
1) y =		;2x + 4	2) y = 5t ; t2
	y = 10 ; x:			y = 0:

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