Краткое справочное пособие по школьному курсу математики Определения; Теоремы; Свойства; Формулы; Алгоритмы

M 79

MOP.llKOBH'IA.r.

Kparxoe cnpaso-nroe rrocooae lIO lllKOJIbHOMY xypcy MaTeMaTHI<H: Orrpezte-

JIeHIDl; Teopexsr; CBOHCTBa; cI>0PMYJIbl; AnroPHTMbI. -

M.: HOBaH urxona, 1994.

:~

f~

-

48 C.-- ISBN 5-7301-0056-6

3a.ll)'MbiBaJI3TOcnpasosuoe noc06He, aBTOp CTaBHlI nepea co6oH clIenylOlUHe 3a,lla'lH:

1. B lIaKOHH'IHOA4'opMe,!laTh '!maTeJJlOBee OCHOBHble onpezteaenaa, reopexse, «f!OPMYlIbI,

npaauna, CBoliCTBa MaTeMaTH'IecKHx06"eKToB, xoropsre acrpesascrca B llIKOllbHOM Kypce

MaTeMaTHKH.

2. PaCnOllOlK!ITb MaTepHall TaK,

'IT06h\ nOHCK HylKHOli

'1HTaTeJJlO HH<fIoPMaJ.(HH 6b1J1

nOCTaTO'lHOnpocr.

HacKollbKO y,!laJlOCb peunrrs 3TH 3a,lla'lH,cY.lUfTb BaM, '!maTeJIlIM.Mbi aaaeexca, 'ITO3Ta

He60Jlblllall KHHlKHUa craaer Ba1lIHM Ha,nelKHblM nOMOI1lHHKOM: B nepaozt 06Y'leHHlIB uncone H

noziroroaxn K BcrynmeJJbHbIM 3K3aMeHaM B By3,

npa "OBTOpeHHH llIKOllbHOro xypca xareaa-

THKH, a TaKlKe BO MHOruX ztpyrax CJJY'IalIX.

IiIiK 221 JI 72

AJIeKCaHAP Iparopsesax MOPAKOBH1J

Kparxoe cnpaaoxnoe nocoriae lIO llIKOJIbHOMYKypcy MaTeMaTHKH: Onpeaeneaas;

Teopexsr: CBOHCTBa; cI>opMyJIbl; AnrOPHTMhI

XYllOX<HHK 06JIO:>KKH B.A: Bopxonor

Penaxrop E.E.EJlHHKHHa

KOMllbIOTepHaH BepCTKa C.B.CyxapeB

Koppexrop A. E. YIBaHoBa

JIHl{eH3Hl1 JIP lW061967 OT 28.12.92

C,naHO B Ha60p 5.05.94. Iloan. K nenara 27.05.94

<I>0PMaT 60x90 1/16 oyMara raseraas. Ileears 04JceTHalI.

Fapmrrypa ..Ilerepeypr. YClI. nes. JI. 3

THpaJK 50000 3K3. 3aKa3 NO 356.

Ilena noroaopaas.

f

Hanarenscrso "HOBalI uncoaa". 107258, MOCKBa,

Kpacaooorarsipcxaa, 75, xopn.z

..~

,'1]

MOCKOBCKU mnOrp8lPHSI NO 6 KOMHTeT8 p~ no neq8TH,

I:

109088, MOCKB8, JK-88, IOlICHonopToB8S1 yn., 24.

ISBN 5-7301-0056-6

©

MopnKOBWI, 1994

)~-.

,-i

AJIfEBPA

.

6

1. TO:>KneCTBeHHble npe06pa30BaHlHI . . . . .

6

1.1. <DopM}'JIbI pa3JIO:>KeHH.lI na MHO:>KHTeJlH

6

1.2. MOnYJIb 'IHCJla

.

6

1.3. Kopens n-o CTeneHH .

. • . . . . .

7

1.4. Creneas C paUHOHaJlbHbIM noaaaarenesr

7

1.5. JIorapHlpMbI. . . .

8

2. <DyHKUHH . . . . . . . . . . . .

9

2.1. JIHHeuHa.lI epyHKUH.lI . . . . .

9

2.2. 06paTHa.lI lIponOpUHOHaJIbHOCTb

9

2.3.

Kaanparaaaaa epyHKUH.lI

10

2.4. <DyHKUH.lI Y = "-.IX . . . . . .

10

2.5. Crenennaa epyHKUH.lI y =x r . .

10

2.6. Iloxaaarensnaa H norapadisoorecsaa epyHKUHH

12

2.7. CBOOCTBa epyHKUHH

.

12

2.8. IIoCTpoeHHe rpadmxos epyHKUHit C lIOMOlI.{bIO

npeoopaaosaaajt H3BeCTHblX rparpaxoa.

13

3.

YpaBHeHH.lI . . . . . . . . . . . . . . . .

14

3.1. Kaaaparuste ypaBHeHH.lI

. '.. . . . . . .

14

3.2 AJlroPHTM peurenas ypamreuas 3-0 crenenn

15

3.3

I1ppaUHOHaJIbHble ypamrenas

15

3.4. Iloxasarem.nue ypasnenaa

15

3.5. Jloraparpxasecxae ypasuenas

15

4.

HepaBeHCTBa • ........

16

4.1.

CBOOCTBa 'lHCJIOBbIX HepaBeHCTB

16

4.2

HepaBeHCTBa C MOnYJI.llMH

16

4.3. I1ppaUHOHaJIbHble HepaBeHCTBa

16

4.4. Iloxaaarensnue HepaBeHCTBa

17

4.5. JIorapHepMH'IecKHeHepaBeHCTBa

17

5.

Ilporpeccaa . . . . . . . . . .

17

5.1. Aparpsreravecxaa nporpeccas

17

5.2. Feosrerpa-recxaa nporpeccas .

18

TPl1fOHOMETPI151 . . . . . .

19

1. TpHroHOMeTpH'leCKHeepyHKUHH. .

19

1.1. lJHCJIOBa.lI OKpy:>KHOCTb . . .

19

1.2. TpHroHOMeTpH'leCKHeepyHKUHH

20

1.3. Oriparnsre TpHrOHOMeTpH'leCKHeepyHKUHH

21

1.4.

fpaepHKH TpHrOHOMeTpH'leCKHXepyHKUHO

23

1.5. fpaepHKH 06paTHbIX TpHroHoMeTpH'leCKHXepyHKUHO

24

3

2.

<1>OpMyJIbI TpnrOHOMeTpHH . . . . . . .

25

2.1. <1>opMYJIbI, CBR3bIBaIOIUHe epYHKIJ;HH

oznroro H roro )Ke apryxeirra

25

2.2. <1>OPMYJIbI, CBR3bIBaIOIUHe epYHKIJ;HH apryxeirron,

H3 KOTOPblX OllHH BllBoe 60JIblIIe llPyroro . . .

25

2.3. <1>OPMYJIbI CJIO)KeHHR apryxenroe

.

25

2.4. <1>OPMYJIbI npeoopaaosanaa CYMM B npoaasezienaa

26

2.5. <1>OPMYJIbI npe06pa30BaHH.sI nponaseztenaa B CYMMbI

26

2.6. <1>OPMYJIbI npHBelleHHR . . . . . .

. . . . • . .

26

2.7. II pocreaurae TpHroHOMeTpH'IeCKHeypasuenaa . . .

27

3JIEMEHTbi )l11<1><1>EPEHUI1AJIbHOfO I1Cl.J:I1CJIEHI151

28

1.

II POH3BOllHa.sI . . . . . . . . . .

28

1.1. Onpenenenae rrpoaaeozmof . .

28

1.2. <1>OPMYJIbI llHepepepeHIJ;HpOBaHH.sI

28

1.3. Ilpasana llHepepepeHIJ;HpOBaHH.sI

29

1.4. Feoxerpa-recaaa CMblCJI IiPOH3BOllHOii

29

1.5. YpaBHeHHe xacarensnoa ... . .

29

2.

I1cCJIellOBaHHe epyHKIJ;HIt C nOMOIUbIO rrpouasozmof

30

2.1. I1cCJIellOBaHHe Ha MOHOTOHHOCTb . . . . . .

30

2.2. I1cCJIellOBaHHe Ha 3KcTpeMyM . . . . . . .

30

2.3. Orucxanae naaoom.urero H naaxensnrero 3Ha'leHHH

uenpepsrsnoa epYHKIJ;HH Ha npOMe)KyTKe. . . .

31

3JIEMEHTbll1HTEfPAJIbHOfO I1Cl.J:I1CJIEHI151

32

1. Ilepnooopasnas

.

32

1.1. Onpeaenenae nepaooopasnoa . . . . . . .

32

1.2. Ilpaaana BbI'IHCJIeHH.sIrrepnooopasnoa

32

1.3. <1>OPMYJIbI BbI'IHCJIeHH.sInepaooopasnoa F (x)

llJIR epYHKIJ;HH f (x)

.

32

2. Heonpenenennstf HHTerpaJI . . . . . .

• . .

33

2.1. Orrpenenenae neonpenenennoro anrerpana

33

2.2. Ilpasana HHTerpHpOBaHH.sI .

33

2.3. <1>OpMyJIbI HHTerpHpOBaHHR .

33

3. Orrpeztenennua HHTerpaJI

. . • .

34

3.1. <1>opMYJIa HbIOToHa-JIeH6HHIJ;a

34

3.2. CBoii:CTBa onpenenennoro mrrerpana

34

3.3. BbI'IHCJIeHHenJIOIUaneIt nJIOCKHX epHryp

C nOMoIUbIO anrerpana

35

llJIAHI1METPI151 .

36

1. TpeyrOJIbHHKH . . . .

• .

36

1.1. 0603Ha'leHH.sI . . . .

36

1.2. Paanocroponnaa rpeyronsnax

36

1.3. llP.sIMOyrOJIbHbIii: TpeyrOJIbHHK (C = 90")

36

1.4. llPOH3BOJIbHbIii: TpeyroJIbHHK . . . . .

37

2.2.

l.J:eTbIpe CJIY'la.sIBbICOTbI nHpaMHllbI . . . . . . . . . .

2.3.

Bbl'lHCJIeHHe06'beMaH nJIOIUallH rroaepxaocra napaxansr

3. Ilpaaxa

.

3.1. OnpelleJIeHH.sI

. • . . . . . . . . . . . .

4.4. lllap

.

a2 - b2 =(a - b) (a + b).

a3 - b3 =(a - b) (a2 + ab + b2).

a3 + b3 = (a + b) (a2 - ab + b2).

a2 ± 2ab + b2 = (a ± b)2.

a3 + 3~b + 3ab2 + b3 = (a + b)3.

a3 -

3a2b + 3ab2 - b3 = (a _ b)3.

ax' + bx + C =a (x - Xt) (x - x 2 ) ,

r lie X t '

X 2 -

KOpHH xsaztparnoro TpeXtIJIeHa a~ + bx + c.

p(x) = (x -

Xt) q(x), rzte X t

--- xopens MHOrOtIJIeHa p(x).

1.2. MOA)'Jlb'lHcJla

1.2.1.

Onpeoenenue

I a I={

a, eCJUI a ~ 0;

- a, eCJIH a < O.

1.2.2. Ceoiicmea

Ia I~ O.

2n

=a

2n

Ia 1

, n EN.

Iab 19 a I. I b I.

Ia + b Is Ia I+ I b I.

a I

Ia I

Ii

=m'

n-{dl = Ia I, eCJIH n

---- -rernoe tIHCJIO;

n-{dl =a, eCJIH n ----

HetIeTHOe tIHCJIO.

1.3.3.

Ceoiicmea (aM a > 0, b > 0)

n--Ja n-{b = nWifi.

n"kWi

= nkWi.

nWi

= n« ~

n- rz:

= nk: ,..-,;jk

n..fb

'J b.

»a:

»a: .

ECJIH a -:t 0

H

n E N,

TO a- n =~an.

.

t)

1

ECJIH a > 0

H

r = L..,

TO a- T =-.

q

aT

a'la'2 =a'l + '2.

a'b' =(ab)'.

a'2

_,

.

:: =(~J

a'l

-=U1

2

(a'l)'2

=a'l'2.

1.5. JIorapu<l>MIII

a

logab

b:

logaa'= r.

- ,

1.5.3. Ceoiicmea (l(TI~ b > 0, C > 0)

logabc =logab + logac,

logcb

logab = -1- .

ogca

b

=lo&b - logac.

logab = log,{b'(r -:t; 0).

loga~

c

logab'= rlo&b.

1.5.4. Ceoiicmea (,llJI~

rrpOH3BOJIbHbIX b, c ozmoro 3HaKa)

logabc =logal b I+ logal c I.

b

= logal b I- logal C I.

log,-c

2 ° <1lyHKU,UU

2.1. JIuHdlHaJi <l>YHKlJ,Ha

fpaq.HIKOM JIHHeHHOH

<PYHKl{HH

Y = kx + b

51BJI5IeTGI

npastaa.

k = tga. ---- yrJIOBOH K03<P<PHUHeHT.

Yf

y'

Y

y=kx+-b

I

. /

~

y::.b b

(;;-;;0)

x

......... x

-

x

UI

01

y

!J

~'>OJ

~.

O'0r- x

\10

x

( y=f (k<O)

s.A. MOPAKOBWI

9

rpa<pHKoM KBall.paTHqHOH

<pyHKUHH

y = aXL + bx + C }lBJUlerOI

napario.na

C BeTB}lMH, HanpaBJIeHHbIMH snepx,

eCJIH a> 0, H

BHH3,

.

b

eCJIH a < 0; OCbIO CHMMeTpH~1 napaoonsr CJIy)f(HT npaxas x =-

-2.

.

a

II

y

x

x

2.4. <1>YHKIlIIH Y ="..jX

n - qeTHOe qHCJIO.

n -

HeqeTHOe qHCJIO.

9

y

1

o

x

x

2.5. CTeneHHaH <!»YHKllHH Y = xT

y = x T ;

r = 2n, n E

N.

y = x T ;

r = 2n + 1, n E

N.

y

x

0,

x

y = x';

r = O.

y =x';

r =- 2n, n E N.

9

!I

1

~"--

o

)(

o

x

y =x';

r =E. > 1.

y =x';

r =- (2n - 1), n E N.

..

q

!I

!I

\

x

x

o

y =x';

r =2

0<E.<1.

,

r =-

E.

E. > O.

q'

q

y =x"

q'

q

!J

y

o

)(

o

x

2:

11

lIr06bI nOCTpOUTb rpadnnc epyHKUuu y = I(x) I, Hy)KHO:

OCTaBUTb 6e3 U3MeHeHH>I BeTBU

rparpaxa y = (x),

xoropsre

JIe)KaT nsrure ocu x;

3aMeHUTb BeTBU rpadnrxa y = (x),

xoropsre JIe)KaT HU)Ke OCH x,

CUMMeTpU'lHbIMHUM OTHOCHTeJIbHO oca x.

lITo6bI nOCTpOUTb rpadnnc epyHKUHU y = ( Ix I), Hy)KHO:

OCTaBUTb 6e3 U3MeHeHU>I BeTBU

rparpaxa y = {(x),

xoropsre

JIe)KaT npasee OCH y;

13

OT6poCHfb BerBH rparpaxa Y =f(x),

xoropsre JI~aT nesee OCH Y;

.ll06aBHTb K OCTaBIIIHMOI BeTB}lM CHMMeTpH'IHbleHM OTHOCHTeJIb-

HO OCH y.

4T06bI nOCTpOHTb rparpax ¢>yHKUHH y= kf(x), Hy)l(HO:

ocyutecrmrn, paCT}I)I(eHHe rpadnnca y = f(x) OT OCH x no BepTH-

KaJIH B I k I paa (eCJIH Ik I< 1,

TO ¢>aKTH'IeCKH nOJIY'IHTC}I

B I m I

pa3, (eCJIH I m I<, 1,

TO

nOJIy'lHTC}I paCT}I)I(eHHe C

K03¢>¢>HI.(HeHTOM

I ~ I); eCJIH

npH

3TOM m < 0, . TO C)I(aTbIH

rparpnx

Hy)l(HO

3aMeHHTb CHMMeTpH'lHbIM esry OTHOCHTeJIbHO

OCH y.

3.1.1. (/JoPMYAbl xopneii xeaopamnoeo ypaenenun

ax2 + bx + e =

°(a *" 0)

x t .2 =

-b ± ...J b2 - 4ac

HJIH x t .2 =

-k ± {k1 - ac

, eCJIH b = 2k.

2a

a

4HCJIO b2 -

4ae ---- .llHCKpHMHHaHT KBatlpaTHOro ypasneaaa (0603-

Ha'laeTC516yKBOH D).

3.1.2. Teopeua Buema u ee cneocmeus

ECJIH xl'x 2

--- KOpHH xaazrparaoro ypaBHeHH51 x 2 + px + q = 0, TO:

x t + X2 =- p,

x~ + ~ =- P (p2 - 3q),

Xt x 2 = q,

4

4

22222

Xt

+ x 2 = (p - q) - q.

u

2

+

b

tX + Ct·

xaazrparnsta TpeX'lJIeHax

4. Haiiztare KOpHH x 2'

x 3 ypaBHeHH51 ax2 + btx + c t

= 0.

5. 3anHIlIHTe OTBer: xl'X 2'

x3 --- KOpHH ypaBHeHH51.

3.3 Hppauaoaaxsasre ypaaneaaa

ECJIH n --- nesernoe 'lHCJIO, TO ypasuenae n...jf(x)

= q(x) pasno-

CHJIbHO ypaBHeHHIO f(x) = (q(x))n.

f (X) ~ 0,

f (X) ~ 0,

q(x) ~ 0,

{ q(x) < O.

{ f(x) > ( q(x) )2;