VARIANT-22

kafedra

w m m f

wARIANT 22

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))
							2
6: ln [1 + (x)]							(x)
							(x)
7: loga [1 + (x)]							ln a
(x)
8: e (x) ; 1						(x)
9: a			; 1			(x) ln a
	n							(x)
10: q1 + (x) ; 1									n

1: sin (x) (x) ;

( (x))3

6

2: arcsin (x) (x) +

( (x))3

6

3: tg

(x) (x) +

( (x))3

3

4: arctg (x)

(x)

;

( (x))3

3

5: 1 ; cos (x)

( (x))2

;

( (x))4

2

24

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

( (x))2

2

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

( (x))2

2

8: n

(x)

1 ; n

( (x))2

1 + (x)

;

1

+

n

2n2

q

wTOROJ ZAME^ATELXNYJ PREDEL

n

x

1

n

x

(

)

0

lim 1 +

1

= e

lim 1 +

1

= e

lim

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

n!1

x!1

x

!

e = 2 7182818284590:::

sUMMA n ^LENOW ARIFMETI^ESKOJ PROGRESSII

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

a1 + an

n

2

sUMMA n ^LENOW GEOMETRI^ESKOJ PROGRESSII SO ZNAMENATELEM q

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

; qn)

pRI jqj < 1

S =

b1

1 ; q

1 ; q

fAKTORIALY

0! = 1

1! = 1

n! = 1 2 3

4 : : : (n ; 1) n

2! = 1

2 = 2

3! = 1

2

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

n

n

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 )

0

=

0

0

0

0

0

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

Ux

2: ( C

U )

0

=

C

U

0

0

0

3: (

U

V ) =

U

V

1

0

0

0

0

4: (

U

V

)

=

U

V

+

U

V

7: x

(y) =

yx0 (x)

0

0

0

y

5:

U

=

U

V

;

U V

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)

x0

t

)

(

x0

t

3

(

( ))

tABLICA PROIZWODNYH

1: Uk 0

0

pU

1

0

2:

=

2p

U

0

U

1

1

0

3:

= ;

U

U

U2

4: aU 0

= aU ln a U0

5: eU 0

= eU U0

0

1

U

0

10: (tg U)

=

cos2 U

0

1

U

0

11: (ctg U)

= ;

sin2 U

12: (arcsin U)0

= p

1

U0

1

U2

;

13: (arccos U)0 = ;p

1

U0

1

;

U2

1

0

0

14: (arctg U)

=

U

1 + U2

0

1

0

0

1

0

6: (logaU)

=

U

15: (arcctg

U)

= ;

U

U ln a

1 + U2

0

1

U0

16: (sh U)0

= ch U U0

7: (ln U) =

U

8: (sin U)0

= cos U U0

17: (ch U)0

= sh U U0

9: (cos U)

0

= ;sin U U

0

18: (th U)

0

=

1

U

0

ch2 U

5

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.

Z

tg U dU = ;ln jcos Uj + C

13.

ctg U dU = ln

sin U

j

+ C

Z

dU

jU

14. Z

= ln

tg

+ C

sin U

2

U

dU

15.

Z

= ln tg

+

4 + C

cos U

2

16. Z

dU

1

U

= a arctg a

+ C

a2 + U2

17.

Z

dU

=

1

ln

U

; a

+ C

U2 ; a2

2a

U + a

18. Z

p

dU

U

= arcsin a

+ C

a2

;

U2

dU

19. Z

p

= ln jU

+pU2 a2j+C

U2

a2

20.

Z

sh U dU = ch U + C

21. Z

ch U dU = sh U + C

22. Z

dU

= th U + C

ch2 U

23. Z

dU

= ;cth U + C

sh2 U

1

24. Z pU2 a2 dU =

U p

U2 a2

a2 ln jU U+ pU2 a2j +C

12

25. Z pa2 ; U2 dU =

2

U pa2 ; U2 + a2arcsin a ! + C

26. Z e U sin U dU =

e U

( sin U ; cos U) + C

2

+ 2

27. Z e U cos U dU =

e U

( cos U + sin U) + C

2

+ 2

rQDY mAKLORENA \LEMENTARNYH FUNKCIJ

1: ex = 1 + x + x2

+ x3 + : : : + xn

+ : : : = 1 xn

2!

3!

n!

n=0

n!

x3

x5

x2n+1

X

1

x2n+1

2: sh x = x +

3! +

5! +

: : : +

+ : : : = n=0

(2n + 1)!

(2n + 1)!

x

2

x

4

x

2n

1

X2n

x

3: ch x = 1 +

2! +

+

: : : +

+ : : : = n=0

4!

(2n)!

(2n)!

x3

x5

x2n+1

X

1

x2n+1

4: sin

x= x; 3! +

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

+: : :=

(;1)n

(2n + 1)!

n=0

(2n + 1)!

x2

x4

x2n

x2n

1

X

5: cos

x = 1 ;

2! +

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

+ : : : =

(;1)n

(2n)!

n=0

(2n)!

m

m(m

1)

m(m

X

2)

;

1)(m

;

6: (1 + x)m = 1 +

1! x +

2!;

x2 +

3!

x3 + : : :

1

1

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

7:

1 + x

x2

x3

xn+1

1

xn+1

X

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

+: : := (;1)n

n + 1

n + 1

x3

x5

x2n+1

n=0

x2n+1

1

X

9: arctg x= x;

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

+: : :=

(;1)n

(2n + 1)

n=0

(2n + 1)

3

5

7

1

3

+ 1

3

5 x

X

10: arcsin x = x +

1 x

+

x

+ : : :

2 3

22

2! 5

23 3! 7

1

3

2

5

11: tg x = x + 3x

+

15

x

+ : : :

12: th x = x ;

1

3

+

2

5

; : : :

3 x

15

x

7

rQD I INTEGRAL fURXE (OSNOWNYE FORMULY)

1.

rQD fURXE FUNKCII, ZADANNOJ NA INTERWALE [

;

]

a0

1

f(x) =

+

an cos nx + bn sin nx

2

n=1

= 1

= 1

X

1

Z

(x)dx

Z

f(x) cos nx dx

bn =

Z

(x) sin nx dx

a0

f

an

f

;

;

;

2.

rQD fURXE FUNKCII, ZADANNOJ NA INTERWALE [

l

l]

a0

1

n

n ;

f(x) =

+

X

an cos

x + bn sin

2

l

l

x

n=1

a0 = 1l ;Zl f(x)dx

an = 1l ;Zl f(x) cos

n

x dx

bn = 1l ;Zl

f(x) sin

n

x dx

l

l

l

l

l

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

pO SINUSAM

pO KOSINUSAM

1

n

a0

1

n

f(x) =

X

bn sin

x

f(x) =

+

X

an cos

x

l

2

l

n=1

n=1

bn = 2l Zl

f(x) sin

n

x dx

a0 = 2l

Zl f(x)dx

an = 2l Zl f(x) cos

n

x dx

l

l

0

0

0

2

;

l

4. rQD fURXE f(x)

x

(

l l) W KOMPLEKSNOJ FORME

1

1

n

;Zl

X

f(x) =

2

Sn(!n)e

GDE

!n =

l

Sn(!n) =

l

f(x)e

dx

n=;1

5. iNTEGRAL fURXE FUNKCII f(x)

x

2

(

;1

1

)

1

1

0

1

x) dt1 d!

f(x) =

Z

f

(t) cos !(t

;

;1Z

A

0

@

1

1

dLQ ^ETNOJ FUNKCII

f(x) = 2

Z

cos !x d! Z

f(t) cos !t dt

0

1

0

1

dLQ NE^ETNOJ FUNKCII

f(x) = 2

Z sin !x d! Z

f(t) sin !t dt

0

0

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

1

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

;1

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

8

tABLICA IZOBRAVENIJ I ORIGINALOW

f(t)

F (p)

1

1

1

p

2

t

1

p2

3

t2

2

p3

4

;at

1

e

p + a

5

;at

1

t e

(p + a)2

6

2

;at

2

t

e

(p + a)3

7

f(t)

0 t

F (p)(1

;

e;p )

( 0 t >

8

sin at

a

p2 + a2

9

cos at

p

p2 + a2

		f(t)					F (p)
10		t sin at					2ap
						(p2 + a2)2
11	t cos at						p2	; a22 2
						2
						(p		+ a )
12		sh at						a
							p2 ; a2
13		ch at						p
							p2 ; a2
14	e;at sin bt							b
					(p + a)2 + b2
15	;at		cos bt				p + a
	e
					(p + a)2 + b2
16	e;atsh bt							b
					(p + a)2 ; b2
17	;at		ch bt				p + a
	e
					(p + a)2 ; b2
18		(t)						1
19	(t ; )						e;p

9

zadanie N 1	lINEJNAQ ALGEBRA	wARIANT 22

1.	wY^ISLITX OPREDELITELI
			2	3		1	5						2	;1		3	;4
	a)		3	4		6	7				b)		1		0	;1	0
			1	;1		;3	;1						2		1	1	;1
			2	3		9	2						0		1	;1	5
2.	nAJTI MATRICU h IZ URAWNENIQ. sDELATX PROWERKU
				0	1	;3 2		1	X =	0	1		5	;5		1
					3	4	1				3	10			0
				B	2	;5 3		C		B	2		9	;	7	C
3.				@				A		@						A
	rE[ITX SISTEMY LINEJNYH URAWNENIJ:
	A) METODOM kRAMERA,											b)		MATRI^NYM METODOM

		8	x + 5y + 3z = 10
		<	2x ; y + z = 12
		a) >		b)
4		>	x ; 4y ; z = 4
	.	:
		rE[ITX SISTEMY METODOM gAUSSA

8 2x ; y ; z

<> x + 4y + 3z

>: 2x + 6y

=14

=;7

=10

8 x1

+x2 +x3 +x4

+x5 = 1

x1

+x3 +x4

+x5 =

;3

a) > x1

+x2 +x3 +x4

= 0

< x1

+x2 +x3

+x5 = 3

> x1

+x2

+x4

+x5 = ;2

: 6x1

+3x2

+14x3

;2x4

+x5

=

2

8

20x1

+5x2

+10x3

+4x4 +11x5

=

20

b) >

13x1

+4x2

+12x3

+x4

+6x5

=

11

<

4x1

+7x2

+46x3

;12x4

;7x5 =

;12

> x1

;2x2 ;16x3

+5x4

+4x5 =

7

:

c) 8

5x1

;8x2 +3x3 +3x4 = 0

<

4x1

;6x2 +2x3 +x4 = 0

5

.

:

nAJTI SOBSTWENNYE ZNA^ENIQ I SOBSTWENNYE WEKTORY MATRIC

a) A = 0 82 41

1

B = 0

5

0

0

1

b)

1 4

;1

@

A

B

1

;

1

4

C

10

@

A

BAZISE.

zadanie N 2	wEKTORNAQ ALGEBRA		wARIANT 22
1. dAN PARALLELOGRAMM ABCD W KOTOROM		;!AB = ~a	;!AC = ~c: tO^KA
DELIT DIAGONALX AC W OTNO[ENII j AM j		: j MC j= 4=5: wYRAZITX

WEKTORY ;!AD ;!BD ;;!MD ;;!MB ^EREZ WEKTORY ~a I ~c.

2. oPREDELITX KOORDINATY TO^KI C, LEVA]EJ NA PRQMOJ, PROHO- DQ]EJ ^EREZ TO^KI A I B, ESLI A(2 3 ;3) B(5 ;2 1) I

jACj : jCBj = 1 : 4

3. w TREUGOLXNIKE S WER[INAMI A(;3 1 3) B(1 ;3 4) C(0 2 ;2): 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(0 ;1 ;5) B(2 ;1 4) C(2 ;5 1): nAJTI: a) KOORDINATY ^ETWERTOJ WER[INY D,

b) DLINU WYSOTY, OPU]ENNOJ NA STORONU AB c) KOSINUS OSTROGO UGLA MEVDU DIAGONALQMI AC

5. pARALLELOGRAMM POSTROEN NA WEKTORAH ~a = p~+2q~ GDE j ~p j= 4 j ~q j= 1 (~p ^~q) = 60o: oPREDELITX:

a) KOSINUS UGLA MEVDU EGO DIAGONALQMI ~ b) DLINU WYSOTY, OPU]ENNOJ NA STORONU b.

I BD.

~

I b = 2~p;3~q

6. nAJTI EDINI^NYJ WEKTOR ~e, KOTORYJ ODNOWREMENNO PERPENDIKU-

LQREN WEKTORAM

~a =

14

11 0

~

0

11 8

,

ESLI

f;

;

g

I b =

f

;

g

~

(~e ^ i) =2.

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

A(0 ;2 ;5) B(5 0 ;1) C(2 6 0) D(1 1 1)

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

8. dOKAZATX, ^TO WEKTORY p~ = f2 ;7 1g q~ = f1 2 1g ~r = f2 1 1g

OBRAZU@T BAZIS I NAJTI RAZLOVENIE WEKTORA x~ = f;6 3 6g W \TOM

11

zadanie N3

wARIANT 22

aNALITI^ESKAQ GEOMETRIQ NA PLOSKOSTI

1. sOSTAWITX URAWNENIQ PRQMYH, PROHODQ]IH ^EREZ TO^KU M(5 ;3):

8x = 4t + 7

a)PARALLELXNO PRQMOJ < y = 3t ; 5

b)PERPENDIKULQRNO PRQMOJ: 10x + ;y2 = 1

c)POD UGLOM 450 K PRQMOJ 2y ; 7x = 8

2. dANY WER[INY TREUGOLXNIKA A(;2 1) B(;18 ;11) C(;11 13): sOSTAWITX: a) URAWNENIE STORONY AC,

b)URAWNENIE MEDIANY wm,

c)URAWNENIE WYSOTY sH I NAJTI EE DLINU.

3. dANY DWE PRQMYE l1 : y = 6x + 8	l2 :	x + 6	=	y	. nAJTI:
a) TO^KU PERESE^ENIQ PRQMYH,		;7		1

b)KOSINUS UGLA MEVDU PRQMYMI,

c)SOSTAWITX URAWNENIQ BISSEKTRIS UGLOW MEVDU PRQMYMI.

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

1)	x2 + y2 + 2x + 3y ; 4 = 0					2)	2x2 + 4x + y2 ; 4y = 0
3)	y = 7 ; p						2x2 ; 2y2 + x = 0
		6 ;	2x			4)
	2		2	; 20x + 10y = 50 6)			2	2	+ 10x ; 10y + 1 = 0
5)	x + 4xy + 4y						; 4x + 2xy ; 4y

5. sOSTAWITX URAWNENIE I POSTROITX LINI@, KAVDAQ TO^KA KOTOROJ QWLQETSQ OSNOWANIEM PERPENDIKULQRA, OPU]ENNOGO IZ NA^ALA KOORDI- NAT NA PRQMU@, PROHODQ]U@ ^EREZ TO^KU M(;1 1):

6. pOSTROITX LINII, ZADANNYE W POLQRNYH KOORDINATAH:
		'				5				3
1) = 2;sin 3			2) = 7 cos (' ; 6 )					3) =			:
										2 ; 4 sin '
7. pOSTROITX LINII, ZADANNYE PARAMETRI^ESKIMI URAWNENIQMI:
1)		8 x = 4	; sin 2t		2)	8 x = 3		3
		< y = 1	; cos 2t			< y = 2 sin t
		:				:
8. pOSTROITX FIGURU, OGRANI^ENNU@ LINIQMI
		y = 6x2		x4
1)			;		2)	= p2 sin 2': :
		y = 1:

12