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Томский Государственный Университет Систем Управления и Радиоэлектроники

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kafedra

w m m f

wARIANT 25

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

a2 ; U2

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 p

U2 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 25

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

		8	x + 3y + 3z = 1
		<
		a) >	2x + 3y + 5z = 2	b)
		>	3x + 5y + 8z = 3
4	.	:
	.	rE[ITX SISTEMY METODOM gAUSSA

8 x +

< 2x ;

> 4x +

y + 2z = ;1 y + 2z = ;4 y + 4z = ;2

8 x1

+x3

+x4 = 1

= 2

+x2 ; ;x4 = 3

a) > x1

> x1 +x2 +x3

;

= 4

8:2x1

+2x3

+2x5 = 1

+x4

= 0

+x2

+2x4 +x5 = 1

b) > 2x1

3x2

+3x4

= 0

2x1

+x3 +3x4

;x5 = 0

c) 8

+x2

;x4

+x5 = 0

;

2x2

+x3 +5x4

;

3x5

= 0

> x1

;3x2 +2x3 +9x4 ;5x5 = 0

nAJTI SOBSTWENNYE ZNA^ENIQ I SOBSTWENNYE WEKTORY MATRIC

a) A = 0

3 1

7 2

2 4 1

b) B =

4 5

0 0

~a = f;7 10 ;5g

ESLI (~e i)

I b = f0 ;2 ;1g,

nAJTI:

zadanie N 2	wEKTORNAQ ALGEBRA	wARIANT 25
1. dAN PARALLELOGRAMM ABCD W KOTOROM		;!AB = ~a ;!AC = ~c:
tO^KA DELIT DIAGONALX AC W OTNO[ENII j		AM j : j MC j= 5=2.
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(5 3 ;3) B(2 ;2 5) I
jACj : jABj = 5 : 2

3. w TREUGOLXNIKE S WER[INAMI A(3 ;3 2) B(1 ;1 4) C(2 0 ;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 1 ;3) B(1 ;3 0) C(;3 ;1 5): 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 = 4p~+2q~ GDE j ~p j= 2 j ~q j= 5 (~p ^~q) = 120o: oPREDELITX:

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

I BD.

I b = 3p~;2q~,

6. nAJTI EDINI^NYJ WEKTOR ~e, KOTORYJ ODNOWREMENNO PERPENDIKULQ- REN WEKTORAM =2.

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

A(2 ;4 ;3) B(5 ;6 0) C(;1 3 ;3) D(;10 ;8 7)

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

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

OBRAZU@T BAZIS I NAJTI RAZLOVENIE WEKTORA ~x = f2 7 5g W \TOM BAZISE.

zadanie N3

wARIANT 25

aNALITI^ESKAQ GEOMETRIQ NA PLOSKOSTI

1. sOSTAWITX URAWNENIQ PRQMYH, PROHODQ]IH ^EREZ TO^KU M(13 ;8):

8x = 5

a)PARALLELXNO PRQMOJ < y = 2t ; 1

b)PERPENDIKULQRNO PRQMOJ: 4x + y + 10 = 0

c)POD UGLOM 450 K PRQMOJ x3 + y1 = 1

2. dANY WER[INY TREUGOLXNIKA A(1 1) B(;15 11) C(;8 13): sOSTAWITX: a) URAWNENIE STORONY AC,

b)URAWNENIE MEDIANY wm,

c)URAWNENIE WYSOTY sH I NAJTI EE DLINU.

3. dANY DWE PRQMYE l1 : y = x + 12	l2	: 8 x = 3t		nAJTI:
a) TO^KU PERESE^ENIQ PRQMYH,		< y = t	; 2
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 + 7x + 9y = 0

9x2

18x + 16y2 = 0

x =

;y2

6x + 4y

4 = 0

;

y + 5

;

2 ;

;

; 24xy + 16y

+ 2x ; 10y = 0 6): 3xy ; 4x + 5y = 7

5. sOSTAWITX URAWNENIE I POSTROITX LINI@, KAVDAQ TO^KA KOTOROJ ODINAKOWO UDALENA OT TO^KI M(8 ;4) I OT PRQMOJ x + 7 = 0.

6. pOSTROITX LINII, ZADANNYE W POLQRNYH KOORDINATAH:
			3
1) = p'	2) = 2 ; cos 4'	3) =		:

			2 + cos '

7. pOSTROITX LINII, ZADANNYE PARAMETRI^ESKIMI URAWNENIQMI:

x = t2

x = t

cos t

1) 8

8 y = t

> y =

;

sin t

8. pOSTROITX:FIGURU, OGRANI^ENNU@ LINIQMI

y = x2

x = 2(t

; sin t)

y = 3x

; 1:

2) y = 2(1

; cos t) y = 0

;

t 2 :

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