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# ИДЗ_1 / VAR-25

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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)) 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 + q n 2n2

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 p U2 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 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 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 x2 +x3 +x4 = 1 x3 x4 = 2 < +x2 ; ;x4 = 3 a) > x1 > x1 +x2 +x3 ; = 4 8:2x1 +2x3 +2x5 = 1 < x2 +x4 = 0 +x2 +2x4 +x5 = 1 b) > 2x1 > 3x2 +3x4 = 0 : 2x1 +x3 +3x4 ;x5 = 0 c) 8 x1 +x2 ;x4 +x5 = 0 < ; 2x2 +x3 +5x4 ; 3x5 = 0 > > x1 ;3x2 +2x3 +9x4 ;5x5 = 0 5 . : nAJTI SOBSTWENNYE ZNA^ENIQ I SOBSTWENNYE WEKTORY MATRIC a) A = 0 3 1 0 7 2 ;2 1 2 4 1 b) B = B 4 5 ;2 C @ A 0 0 3 @ A 10
~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.

11

 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 : 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 + 7x + 9y = 0 2) 9x2 18x + 16y2 = 0 3) x = 4 3p 4) x2 ;y2 6x + 4y 4 = 0 ; y + 5 ; ; 2 ; 2 ; 5) 9x ; 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 t 2) 1) 8 2 8 y = t > y = (t ; 3) sin t > 3 : 8. pOSTROITX:FIGURU, OGRANI^ENNU@ LINIQMI y = x2 2x x = 2(t ; sin t) 1) y = 3x ; 1: 2) y = 2(1 ; cos t) y = 0 ; 0 t 2 :

12

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