Hoxoht epe aaio koohht

apairiema OCH opLHHaT

OCH a6cUHcc OCH a6cUHcc

113 TeOpeMlil JIaHFpaH)Ka cJ1eyeT, TO B HHTBJI (a;b) HaHeTc5I TOqKa C TaKa5I, qTo

f(b) - f(a) b—a

f(b)+f(a)

b—a

f(b) - f(a)

b+a

= f’(c)

- f()

= f’(c)

K 4JyHKUH5{MJ(x) H g(x) eopea KOIIIH HpHMeHHMa, ecylH

f(x) H g(x) Ha (a;b) H LHcxjepeHuHpyeMbI Ha (a;b)

—f(x) H g(x) HCHCh[BHh[ Ha [a;b] H g’(x) 0 B HHTCBI1C (a;b)

f(x) H g(x) HCHCh[BHM a [a; b], Hc1xepeHuHpyeMM Ha (a;b) H g’(x) 0 B HHTCBI1C

(ab)

f(x) H g(x) HCHCbIBHh[ Ha (a;b), Hc1xepeHuHpyeMM Ha (a;b) H g’(x) 0 B HHTCBI1C

(ab)

Ecirn 4JyHKUIWJ(x) Fi g(x) HIIEIBHM Ha OT3K [a; b], .LwjxjepeHuMpyeMbI B (a;b) 11

g’(x) 0 B HHTBI1 (a;b), To, corllacHo TeopeMe KomH, B HHTBJT (a;b) HaMeTc5I TO’-IKa

C TaKa5l, qTO

f(b) - f(a)

f’(c)

-

g(b) — g(a) g’(c)

f(b) + f(a)

f’(c)

-

g(b) + g(a) g’(c)

f(b) - f(a)

f(c)

-

g(b)—g(a) g(c)

f(b) - f(a)

g’(c)

=

g(b) — g(a) f’(c)

HpaBFLTIO JTOHFITa.T151 HF1MH51TC51 K HeOHpe.LeJ1eHHOCTF1 BHLa

—O.cD

HpaBIfflOJIOHFITa.T151 HF1MH51TC51 K HeOIIpeLeJ1eHHOCTH BHLa
—O.cc

c/D-c/D

HyCm cl?yHKWWJ(x) H g(x) HHMBHM B (x0,a],,LwxepeHuHpyeMbI B (x0,a),HpHT-IeM

i_no

g’(x) 0, urn f(x) = H Tim g(x) = CYWCTBYT KOHHbI I 111111 6ecKoHeqHjjcl peei

i—n0

f’(x)

urn TO

X+X0 g’(x)

u • rn

f(x)

-

-

hrn

f”(x)

x—>x0 g(x) x—>x0 g (x)

• f(x)

f’(x)

=

g(x) g’(x)

u • rn

x—>x0

f(x)

g(x)

f(x)

-

- urn

x—>x0

f’(x) g (x)

urn

x—>x0 g(x)

= const

HyCTb 4yHKUHHJ(x) Fi g(x) HHhIBHhI B (x0,a],,LwxepeHLuipyeMbI B (x0,a),npne

g’(x) 0, Tim f(x) =0 Fi lirng(x) =0; CYCTBYT KOHeHM1I 111111 6ecKoHeHbIi pee

X—*xo x—*xo

urn

f’(x)

TO

X>X0 g’(x)

h • m

f(x)

-

- hm

f’(x)

x—>x0 g(x) x—>x0 g (x)

-,

—

hm • f(x)

f’(x)

x—>x0 g(x) g (x)

T • im

f(x)

—c

x—>x0 g(x)

u • rn

f(x)

-

-

urn

f”(x)

x—>x0 g(x) x—>x0 g (x)

HpHMeHHMa iiii ‘reopervia PoJu15I K YHKUHH f(x) = 2 + /(x _1)2 Ha oTpe3Ke[1;2]HeT, y=f(x) 3MBH Ha OT3K [1;2]

— a, c=1

HeT, y=f(x) He,THc1X1ePeHUHPYeMa B HHTBJ1 (1;2)

HeT, f(1) f(2)

HpHMeHHMa IIH eopea JlarpaH)Ka K 4YHKUHH f(x) = + 2x +1 Ha OT3K [0;2]

HeT, d2yHKUH5IJ(x) 3MBH Ha [0;2]

HHMHHM

— HeT, cIyHKuH51J(x) He.LH4xepeHuHpyeMa B (0;2)

HeT, f(0) f(2)

HpHMeHHMa IIH eopea KOIIIH K 4YHKU1151M f(x) = 2x +3 H g(x) = Jx —1 Ha OT3K [0;2]15

c=———

HeT, f(0) f(2)

HeT, 4YHKUH5 g(x) He opeeiiea HH x e [o;1)

HeT, 4JyHKUH5{ g(x) HeHc1c1epeHuHpyeMa Ha (0;2)

ECIIH 1JYHKUH5I y=fx). LHc1Jc1JepeHuHpyeMa B HHTBI1 (a;b), TO JI51 Bo3pacTaHH5{f(x,) B (“a;b,)

Heo6xoHMo H)ocTaToqHo, TO6bI)I151 Bcex x e (a; b) BMHOJIH5IJIOCb

f’(x) >0

f’(x) =0

f’(x) <0

—f”(x) O

ECIIH 4YHKUH5I y=f(x,),rHc1Jc1JepeHuHpyeMa B HHTBJI (‘a;b,), TO JI51 y6bIBaHwlf(x,) B (“a;b,)

Heo6xoHMo H)ocTaToqHo, TO6M)I15I Bcex x e (a; b) BBIHOJIH5LJIOCb

f’(x) >0

f’(x) =0

—f”(x) O

f’(x) <0

aa c1yHKuw1 f(x) = 2x4 + x3 +1, Tor,La

x=O 51BI151TC51 TOT-1K011 MHHHMM cyHKuHHf(x)

X = —— 51BI151TC51 TOT-IKOH MHHHMM c1yHKuHHf(x)

c1yHKuHHf(x) He HMT 3KCTMMOB

X = —— 51BI151TC51 TOT-IKOH MKCHMM 4yHKuHHf(x)

YHKUH51 f(x) = — 4xBO3CTT Ha (— co;+co)

BO3CTT Ha (—2:2)

BO3CTT Ha (— co—2) u (2;+co)

BO3CTT Ha [—1 2]

YHKUH51 f(x) =

— 4x

y6bIBaeT Ha (—2:2)

—

y6MBaeT Ha (— c+c)

y6bIBaeT Ha [—a2)

y6bIBaeT Ha (— c—2) u (2;+c)

YHKUH51 f(x) = 2Vx —3 BbIHKI1 Ha HHTBI1 (—c.c;3) BOFHT Ha HHTBJ1 (3;+c.c)BMHKJ1 Ha HHTBJ1 (3;+c.c)

BOFHT Ha HHTBJ1 (3;5)

HyCTb 4JYHKUH5I y=f(x) B (a;b), x0 — BHTHH5{5{ TOK 3T0F0 HOM)KTK H

f’(x0) = 0 (HIm f’(x0) He cywecTByeT), TO

— o63aTeJmHo TOqKa MHHHMM

— o63aTeJmHo TOqKa MKCHMM

— o6M3aTeIlbHo TOqKa HeperH6a

B TOK X0 3KCTMM MO)KT cyluecTBoBam, a MO)KT H He CYWCTBOBTh

K cl?yHKUHH y=f(x) Ha OT3K [a; b] eopea PoJu15I HpHMeHHMa, ecirn

—f(x) Ha [a;b],. Lw1x1epeHuHpyeMa B (a;b) Hf(a)f(b)

—f(x) Ha [a;b]Hf(a)=f(b)

—f(x) w1J4JePeHuMpYeMa B (a;b)

—f(x) B (a;b),,LW1X1?ePeHUHPYeMa B (a;b) Mf(a)f(b)

113 TOMM JIarpaiDRa cJ1eyeT, qTo

rno6a5l K yHKUHHf(x) B (a;b) apauiema xop,Le, CT5JFHBa}omeHKOHLUil yrHf(x) Ha OTpe3Ke [a;b]

K yHKuHHf(x) B (a;b) apauieiia rno6oIi xop,Le B DTOM

HHTBI1

xopa, CT5JFHBIOW5I KOHU )iyrHf(x) a [a; b], napaeiia OCH OY

B FIHTBI1 (a,b) HaMeTc51 KaCaTeJThHa5I, napa.riileJmHa5I xope, CT5JFHBaIOWeH KOHLIbI

yrFIf(x) Ha OTpe3Ke [a; b]

EcjmToqKa x0 51BI15JTC5J TO’-1K011 HeperH6a rpaHKaf(x) C BepTHKaJmHOH KaCaTeJThHOH, TO

f (x)=O

f’(x0) =

f’(x0) =0 H f”(x0) =0

ECJ1M TOT-wa 5IBJI5JeTC5J TO’-IKOH HeperH6a rpacrnKaf(x) c HaKJIOHHOH KaCaTeJmHOH, TO

f(x0)=0M f(x0)=0

T0-wa x0 H3MBTC5J TO’-IKOH Heperll6a rpa4HKaf(x) C ropw3oHTaJmHoH KacaTeJmHoH, eCJIH

f’(x0) =0 H f”(x0) =0

f (x)=O

HpMMeHHMa JIll eopea Poiiiii K 4YHKLIHH f(x) = 3 + J2 — X Ha OT3K [0;2]

a, c=2

HeT, cl?yHKUH5{f(x) He oHpeLeI1eHa HH x e [0;2] HeT, 4JYHKUH5If(X) He Hc1c1epeHuHpYeMa B (0;2) HeT, f(0) f(2)

ilpilMellilMa 1111 eopea JlarpawKa K 4JyHKUHH f(x) = 2— Ji + X Ha OT3K [—1;0]

HeT, c1JyHKuH5{f(x) pa3pbIBHa Ha [—1;0]

HHMHHM

HeT, c1JyHKuH5If(x) He Hc1xepeHuHpYeMa B

(—1;O)

HeT, f(-1) f(0)

TOKaMH HeperFl6a c1yHKLwH y = — 6x2 5JBJI5JIOTC5J

— TOKH = 2J — 2J

— TOJThKO TOK xO

— TOT-IKH = —2 Fi = 2

— y 4yHKwW y = — 6x2 HT TOT-IeK HeperH6a

HpHMeHHMa iwi eopea KOhl K YHKUH51M f(x) = 2x +1 Fi g(x) = —2 Ha OT3K [0;3]

HeT, 4JyHKUWI g(x) He.1rnc1x1epeHuHpyeMa B (0;3) H g’(x) = 0 B (0;3)

— a, c=3

HeT, c1yHKuH51 g(x) 3MBH Ha [0;3]

HeT, g(x) He Hc1x1ePeHuHpYeMa B (0;3)

YHKUH y = — HMT TOK HeperH6a C FOH3OHTbHO B TOK

(2;—2)

—(O;—3)

—(O;O)

Ho HBHJ1 JIOHHTaJI5{ Hpe.LeI[ hn

—0

5

1—cos3x

5x2

Pae

YHKUH5I y = + 2x BO3CTT TOJThKO HH

x (0;+cc)

xE(—3;2)

x (—ci+ci)

x e (—oc;0)

KpHBa y = + 3x2 —5 BOFHT HH

x e (—co;+co)

xel —c’—— lul —;+ci

3) 3

xel —-——;-——

YHKUM5I y = — x y6bIBaeT llH

xe(—1;1)

— x e (—1;O) u (O;1)

x e (—co;—1) u (1;+co)

— xe(—co;O)u(O;+co)

ilpil 111111

O)

Tim (f (x)- g(x)) = lim(f’(x)- g’(x))

x—*xo x—*xo

urn = urn

X—*X g(x) X—*XO g’(x)

urn (1(x). g(x)) = Tim (f’(x).g’(x))

x—*xo x—*xo

urn f(x)

= urn

f’(x)g(x)-f(x)g’(x)

x—*x0 g(x) x—*x0 g2(x)

Ho paiuiy JIOHFITaJI5I 19

e4x_1

— 5x)

Pae

t’yHKuH5{ y = f(x) H3MBTC5I BO3paCTaIOweII B HHTBJ1 (a; b), ec.iill I151 IlIo6bIx

E (a;b) ii x2 E (a;b)

M3 X1 > x2 ciieye f(x1) < f(x2) M3 X1 <x2 ciieye f(x1)> f(x2) 113 X1 <x2 ciieye f(x1) < f(x2)

113 = x2 ciieye f(x1)> f(x2)

cos3x

Ho HpaBH.ny JT0HHTa.T151 “v 2x — r pae

YHKUW1 y = f(x) H3MBTC51 y6bIBa1oweIi B HHTepBa.lle (a;b), ecrni yii rno6blx x1 e (a;b)

H X2 e (a;b)

113 X1 <x2 ceye f(x1) < f(x2)

113 X1 > x2 ceye f(x1)> f(x2)

113 = x2 ceye f(x1) < f(x2)

113 X1 <x2 ceye f(x1)> f(x2)

Ho paiuiy J1oHHTaJ151 1

---

ctg2x gx

pae

—1

HpHMeHHMa 1111 eopea Po111151 K 4YHKU1111 f(x) =

—La, TaKKaK f(—2)= f(2)

x- —1

Ha OT3K [— 2;2]

a, aic KK f(x) Ha OT3K [— 2;2] H f(— 2) = f(2)

a, aic KK f(x) Ha OT3K [— 2;2], H4xepeHuHpyeMa B (— 2;2) H

f(—2)= f(2)

HeT, He BMHOJ1H51TC51 CI1OBH H11MBHOCTH

A6cUHccbI TO’-IeK HeperH6a 4YHKUHH f(x) = 2x4 — 4x2 +3 BHbI

±1HO

1

HpHMeHHMa JIM eopea JTarpawKa K YHKUHH f(x) = Ha OT3K [— 1;1]

— HeT, 4JYHKUH5I He.LH4X1ePeHUHPYeMa B (— 1;1)

a, TK KK f(— 1) = f(1)

a, 4YHKUH5J Ha [— 11] H f(— 1) = f(1)

a, 4JYHKUH5I Ha [— 11],.LHc1x1ePeHuHpYeMa B (— 11) H f(— 1) = f(1)

YCJIOBHe f’(x0) = 0, f”(x) < 0 51BJ15JTC5J CJIOBHMMHHHMM

BOFHTOCTH

MKCHMM

y6bIBaHwJ

YCJIOBHe f’(x0) = 0, f”(x0)> 0 51BJ15JTC5J CJIOBHMMKCHMM

BMHyK.LIOCTH

BO3CTHH51 MHHHMM

TEMA 7. fipilMeHeHile JUIijHjepeHHHaJ1bHO[O HC’IHCJIeHHH B 3KOHOMWICKHX

IICCJIOBHHHX

‘IyHKuM51J(x) B F1HTBJ1 (a, b) y6bIBaeT Bce 6blcTpee, eciTH

f’(x)<O,f”(x)<O

f’(x)<O,f”(x)>O

f’(x)<O,f(x)>O

f’(x)<O,f(x)<O

tyHKuH51J(x) B F1HTBJ1 (a, b) BO3CTT Bce MeJTeHHee, eciTH

f’(x)>O,f”(x)>O

f(x) > O,f’(x) >0

f’(x)>O,f”(x)<O

f(x)<0,f’(x)>O

3.TIaCTHqHOCTb cl?YHKUIW y =J(x) oHpe.LeI15JeTc5I no 4opMyYIe

E(y)=Ly’

E(y)=

X

y.y

y x y

4T06M 4JyHKUW{ y =J(x) 6MIla 3JIaCTHqHOH B TOT-lKe, HOK3TJTh 3JTCTHHOCTH OJDKH 6bIm

6omme HJ15J

MeHJ3me)HHHUM

pae eHHHUe

6oimme eLHHHuM

4To6Jil 4JyHKUH5{ y =J(x) 6MIla He3JIaCTHqHOH B TOK, HoKa3aTeJm 3J1CTHHOCTH LoJDKeH

6Mm

MeHJ3Lue HJ15{

MeHJ3me eLHHHuM 6oimme eLHHHuM pae elwHHue

3.TIaCTHqHOCTh 4YHKUHH 3KOHOMHCKH oaae

OTHOCHTJThHO W3MHHF1 apryea upil OTHOCHTJThHOM W3MHHHH cIYHKLu1H OTHOCHTJThHO W3MHHF1 cIYHKUHH Ha 1% HPH OTHOCHTJThHOM H3MHHHH apryea OTHOCHTJThHO H3MHHH cIYHKUHH HH OTHOCHTJThHOM H3MHHHH apryea OTHOCHTJThHO H3MHHH cIYHKUHH HH OTHOCHTJThHOM H3MHHHH Ha 1%

3.TIaCTHqHOCTh HpOw3Be.LeHH51,LByX cyHKuHn E (uv) paa

— vE(u)+u.E(v)

E(u).E(v)

E(u)+E(v)

— E(u)+E,(v)

3JTaCTHHOCTb CTHOFO BX cyHKuHi E paa

E(u)

E(v)

E(v)

E(u) E,(u)—E(v)

E(u)—E(v)

41115{ HOIly’-IeHH5I MaKCHMaJmHOH HpH6bLrIH Heo6xoLHMo, TO6M HH LaHHOM O6EeMe HOH3BOCTB xo

BMpy’1Ka 6buIa 6oimme rIpeLeJmHbIx u3Lep)KeK

BMpy’1Ka 6MIla MHMJI peeimix H3Lep)KeK

BM1K BH5LTICb npeeJmHMM H3Lep)KKaM

BMpy’1Ka 6buIa HaH6oJmmeH

tyHKuH5I y = f(x) B HHTBJ1 (a;b) Bo3pacTaeT, ecw

f’(x) <0

f’(x)>O

f”(x)>O

f”(x) <0

YHKUH5I y = f(x) B HHTBJ1 (a;b) y6MBaeT, eciiii

f’(x) >0

f’(x)<O

f(x) >0

YHKUM51 y = f(x) B F1HTBI1 (a;b) BO3CTT Bce 6iic’rpee, ec.riH

f’(x)>O,f”(x)<O

f’(x)>O,f”(x)>O

f(x)>O,f’(x)>O

f’(x)>O,f”(x)=O

tyHKuw1 y = f(x) B F1HTBI1 (a, b) y6bIBaeT Bce eeee

f’(x)<O,f”(x)>O

f’(x) < O,f”(x) <0

f(x)<O,f’(x)<O

f(x)>O,f’(x)<O

3iiacmHocm cpoca S(p) OTHOCF1TJThHO UeHJil p no opyiie

p

p SS’(p)

E p

Eciin K(x) — HoIlHJ3Ie H3,LepKKH, TO npeeme H3)KKH oHpeLeJ15noTc5I KK

—K’(x)

lirnK(x)

x—*xo

urn K(x)

fK(x)dx

3JIaCTWIHOCTb HOCTO5IHHOH BJ1HHHM paa HOCTO5IHHOFI einne

HJI[O

ewnue

BM

4II[5{ HoJ1y1-IeHH5I MaKCHMaJmHOH HpH6bIIlH LOcTaTOqHO, TO6bI ilpil LaHHoM o6eMe HOH3BOCTB xo

V”(x0) = K”(x0)

V”(x0) > K”(x0)

V”(x0) <K”(x0)

3KOHOMHecK14 O6yCIIOBJIeHHOM o6iiacmio onpe,LeYIeHwJ YHKUHH HOJIHMX H3)KK K(x)

51BI151TC51

X 0

x 0

Jx 0,

[K(x) 0

Jx 0,

K(x) > 0

YHKUM51 HOIIHMX H3,Lep)KeK K(x) B FIHTepBaJIe (a;b) Bo3pacTaeT, ecii

K’(x) <0

— K”(x) > 0

— K”(x) <0

—K’(x)>O

tYHKUM51 HOJIHOFI BMpyT-IKFI V(x) y6MBaeT B HHTBJI (a;b), ec.r1H

V”(x) >0

V”(x) <0

V’(x)<O

V’(x) =0

iYHKUM51 HOJIHMX H3Lep)KeK K(x) B HHTBJ1 (a;b) BO3CTT Bee CY1H

K’(x) > 0,K”(x) > 0

K(x) > 0,K’(x) >0

— K(x) = 0,K’(x) >0

K’(x) > 0,K”(x) <0

tYHKUH5I HOJIHMX H3)KK K(x) B HHTBI1 (a;b) BO3CTT Bce 6iic’rpee, eciu

K’(x) > 0,K”(x) = 0

K’(x) > 0,K(x) >0

— K’(x) > 0,K”(x) >0

K’(x) > 0,K”(x) <0

HoJlHa5I Bh[4K V(x) ripi x0 6yeT MaKCHMaJmH0H, ecii

V(x0) = 0,V’(x0) <0

V’(x0) = 0,V”(x0) >0

V’(x0) = 0,V”(x0) = 0

V’(x0) = 0,V”(x0) <0

Cilpoc S(p) 6yeT 3I1CTF1HMM HM uee p0, ecni noKa3aTeJm 3J1CTHHOCTFT

6oimme HI15J

MHMJ1 MHMUM

6oimme e.LF1HIrnM

pae e.LnHIrnM

Cilpoc S(p) 6yeT He3IIaCTW-IHMM HM uee p0, ecw llOK3TJTh 3JIaCTW-IHOCTH

— MHMJ1 HJ15J

— 6omme e,LHHF1uM

— MHMJ1 MHHUM

— pae e.Lrnwue

3.TIaCTWIHOCTb c1yHKurn1 cpoca S(p) = 4— p OTHOCF1TJThHO UHM p oHpe,Le.]iIeTc51 KK

4

4-p

p 4-p 1

4-p

4-p p

3JIaCTWIHOCTMO cI?yHKLurnf(x) OTHOCF1TJThHO apryea X H3MBTC5I peeii OTHOCF1TJThHOFO HPHPWHH5{ 4YHKuHH HH AX —* 0

peeii OTHOTJIHH5{ OTHOCHTJThHOFO npHpaeHH5{ apryMeHTa K OTHOCIITJJbHOM

ppawerno cl?yHKUHH HH AX —* 0

HpeeI1 cl?yHKUHH HH AX —* 0

peeii OTHOTJIHH5{ OTHOCHTJThHOFO npHpaweHH5{ 4yHKUHH K OTHOCHTJTbHOM

HpFlpaweHHIo apryea HH AX —* 0

3KoHoMF1ecK11 O6yCJlOBneHHOH o6iiacmio YI51 4yHKLiHH cupoca S(p) = 8— 2p 6yeT

P O

p 4

p 4

—O p 4

Cpeiwlle H3Lep)KKH (x) HH X0 6yyT MHHHMaJmHbI, CJTI4

—K(x0)<O

K(x0) = O,K(x0) <0

K(x0) = O,K(x0) >0

HoJTHa5l BMpy’lKa V(p) B F1HTBI1 (a;b) BopacTaeT Bce ec.rrn

V’(p) > 0,V(p) <0

— V’(p) > 0,V(p) = 0

V’(p) > 0,V”(p) <0

V’(p) > 0,V”(p) >0

Ho.nHa5l BbIpyT-IKa V(p) B F1HTBJ1 (a;b) y6MBaeT Bce 6blcTpee, ecii

— V’(p) <0,V(p) >0

V’(p) <0,V”(p) <0

V’(p) <0,V”(p) >0

— V’(p) <0,V(p) = 0

3KOHOMHTIeCKH O6yCIIOBI[eHHOI1 o6iiacmio,LITI5I YHKUHH HOIIHOcI BMKH V(p) = l2p — p2

6yeT

(—co;+co)

— (0;+co)

[0;12]

(12;+co)

3I1aCTMHOCTb cl?yHKurn1 cpoca S(p) = OTHOCF1TJThHO UeHJil p KK

p

p (p+2)3

E(S)= p

p p+2

E (S)=

(p+2)2

HoKa3aTeJm 3J1CTHHOCTII 4YHKUHM y = + X HH X = 1 pae

HoKa3aTeJm 3JICTHHOCTII c1JYHKUHH y = —2 ripi x = 2 pae

—36

HoKa3aTeJm 3J1CTWIHOCTH cpoca S = 8— 2p ripi uee p = 3 pae

HoKa3aTeJm 3.TIaCTFIqHOCTI1 4YHKLUW y = in(x2 + i) HH x=1 pae

in 2

in 2 in 2 2

21n2

Cilpoc 5(p) = 6— p OTHOCF1TJThHO UHM p 6y,LeT 3JIaCTHT-IHMM 11H

p e (3;+)

pe(O;3)

p e (3;6)

p e (— 3)

HoJlHa5 Bipy’-iKa V(p) HH 3HHOM cpoce S(p) = 16— 2p 6yeT HaH6oJmmeH llH uee p, BHOF1

Cilpoc S(p) =8—p OTHOCHTJThHO LeHbIp 6y,LeT H3J1CTHHMM HH

pe(4;8)

p e (O;4)

p (4;+c)

p E (— 4)

HoKa3aTeJm 3I1CTHHOCTH HOI[HOI BMKH V(p) HH 3HHOM cpoce S(p) = 16— 4p HH

uee p = 1 pae

tYHKUM51 HOJIHMX H3,Lep)KeK K(x) = 2x3 — 24x2 + lOOx + 36, r,ie x — o6’beM npow3Bo,LcTBa,

Bo3pacTaeT Bce eiieee B HHTBJ1

(4;+)

(0;4)

(-4)

(0;+co)

Hornible M3KH K(x) = — 6x + 39x +13, re x — o6eM npoH3BocTBa, BO3CT}OT Bce6blcTpee B HHTBJ1

(0;6)

(-6)

(6;+)

(—+)

Ho.nHble H3))KKH K(x) = 2x3 — 24x2 + 120x + 40, re x — o6eM npoH3BocTBa, BO3CT}OT Bce 6iic’rpee B HHTBJ1

1(4;+c/D)

(0;4)

(-4)

(0;+cID)

Cilpoc S(p) = 24— 4.p OTHOCHTJThHO UeHJil p 6yeT H3J1CTHHMM HH

p e (3;6)

p e (3;+)

p e (0;3)

p e (— 3)

x

HoKa3aTeJm 3J1CTHHOCTH c1JYHKUHH y = HH x = 2 pae

x +9

5

x

Ecirn HOI1HbI M3,LepKKM H BMpyT-lKa COOTBTCTBHHO COCTBJ15I}OT K(x) = — 3x + 12x + 20;

V(x) = — 4x2 + 22x +11, TO llpH6bIIIb Z(x) 6y.LeT MaKCHMaJmHOIi npn O6TeMe HOH3BOCTB X, paBHOM

YBeirneHHe B cpoce llH IIOCTO5JHHOM 11pe,LO)KeHHH yMeHbmaeT BHOBCHIO i1ey

yBe.T1w-IHBaeT BHOBCHIO uey

yMeHbmaeT BHOBCHO KOrnP-IeCTBO oapacoxpaH5leT BHOBCHO KOI1HCTBO oapa

YMeHMJIeHHe B cpoce HH IIOCTO5IHHOM IIJIO)KHHH

yBeJIHqHBaeT BHOBCHIO uey

yBe.nw-IHBaeT BHOBCHO KOJThqCTBO oapayMeHbmaeT BHOBCHIO uey

coxpaH5leT BHOBCHO KOI1HCTBO oapa

YMeHMJIeHHe B HI1O)KHHH HH IIOCTO5IHHOM cupoce

yBeJ1w-IHBaeT BHOBCHIO uey

yBeJIw-IHBaeT BHOBCHO KOI1HqCTBO oapa

yMeHJ7maeT BHOBCHIO uey

coxpaH5{eT BHOBCHO KOI1HqCTBO oapa

YBe.nwIeHHe B HI1O)KHHH HH HOCTO5{HHOM dilpOCe

coxpaH5leT BHOBCHO KOJ1HCTBO oapayBeJIw-IHBaeT BHOBCHIO uey

yMeHJ7uIaeT BHOBCHO KOJ1HCTBO oapa

yMeHJ7uIaeT BHOBCHIO uey

KpHBa 3HreI15I HJU1IOCTHT 3BHCHMOCTB eiciy

UHOH oapa H CHOCOM

ueHoH oapa Fi

H)KHMM LOXO,TOM H KOJ1HCTBOM rlpHo6peTeHHoro oapa 3TTMH H 06’beMOM BbrnyCKaeMOH HPO,TYKUHH

C IIOBMTJIHHM BHOBCHOFI UHM p0

dilpoc Fl BJ1WIHBIOTC5I

dilpoc yBeIrnHBaeTc5J, a MHMJJTC5J

— dilpoc Fl MHMJI}OTC5I

cripoc a BJTHIHBdTC5J

C CHIUKHMM paBHOBeCHOH UHM p0

dilpoc yMeHbTJIaeTC5I, a BdJTH4HBdTC5J

dilpoc Fl llpe,LJ1O)KeHHe MHbIIIIOTC5I

clipoc yBeJ1FlFlBaeTc5J, a npeoee M}fbfflTC5J dilpoc Fl BJWHBIOTC5I

TEMA 8. HeonpeIe.J1eHHbIe IIHT[JIbI

‘DyHKUW{ F(x) 5IBI15ITC5J nepBoo6pa3HoFT,1i5I yHKuHHJ(x) B HKOTOOM npoMe)KyTKe, ecw B

rno6oIi TO’-IKe 3T0F0 HOM)KTK BMHOJ1H5ITC5I

— f’(x) = F’(x) F(x)J(x)dx

F’(x)J(X)

dF(x)J(x)

Ecm J f(x)dx = F(x) + C, TO BM11OJ1H51TC5I

F(x)f’(x)

F(x)=f(x)dx d(F(x)+C)=f(x)dx

F’(x) = f’(x)

IdF(x)paBeH

f’(x)

—J(x)+C

F(x)+C

J(x)

Ec.LIM HeoHpeeI[eHHMii HHTerpaJl HMT BHL f f(x)dx, TO,Lw1xepeHuHaJ1 3T0F0 erpaiia

Pae

F(x)dx

f’(x)

f’(x)dx

—J(x)dx

HpOM3BOHa5 OT HeoHpeLeI1eHHoro erpaiia $ f(x)dx paBHa

F(x)

— F(x)+C

J(x)

f’(x)

HHTeFpHpOBaHHe HO ‘-IaCT5{M B HeOHpeLeI1eHHbIX erpaiiax BMHOJTH5ITC5I HO 4JopMyIle

uv—$vdu uv+$vdu uv—$udv uv+$udv

BbI6epHTe epoe yTBep)IcLeHHe

$ uvdx = $ udx $ vdx

$uvdx = $udx+ $vdx

$uv’dx = uv —$vdu

u fudx

v fvdx

HHTeFpaJI kf(x)dx pae

k+f(x)dx kf(x)dx k2f(x)dx k-$f(x)dx

HHTeFpaJI j (f(x) + p(x))cb paBeH

f(x)q(x)dx - f(x)

$f(x)ç(x) - $ço(x)dx

— ff(x)dx+$ço(x)dx

$f(x)dxf(x)dx

BbI6epMTe HBF1JThHO YTBP)KLHH

dx 3

=—x3+c

$ dx —3x3

$ dx

= 3x +

BbI6epHTe HBHJThHO

$dx=

1

5[J

$dx=+c

$dx=+c

5

HeHpepbIBHa YHKUH5I HMT

TOJThKO OH HepBoo6pa3Hylo 6ecKoHeHoe MHOKeCTBO HepBoo6pa3HMx

Be HepBoo6pa3HbIx

KOHHO ‘111db HepBoo6pa3HhIx

Be a3IEWIHbI nepBoo6pa3}mIe O,LHOH H TOll)Ke YHKUHH

BHM M){L C06011 OT.TIHqaIOTC5I Ha KOHCTHT

— OThH’11OTC51 Ha HeKOTOp1O 4JYHKUHIO

— OThH’11OTC51 Ha HHTdFHOBHH5I

Hc1x1epeHuHaJ1 OT HeoHpe,LeIIeHHoro erpaiia pae

BbI)KdHH1O

HOwJHTeFpaJmHOFI c1yHKuHH

HJI[O

— 6ecKoHe’1Hocm

K c1YHKUH5JM OTHOC5ITC5I Bce BO3PCT1OLUH

HMBHM

HHOCTO51HHM cl?YHKUHH

r dx

HHTeFpaJI j pae

2x+1

1 (2x+1) 2+c

121n2x+1+C

1n2x+1+C

+c

2(2x+1)2

HHTerpaIl $tgxdx pae

—1ncosx+C 1nsinx+C

—1nsinx+C

tg

f

dx

HHTerpaIl 2—3x

pae

1n2—3x+C

13n2—3x+C

! 3 3x+C

+c

(2—3x)2

HHTerpa.ri ctgxdx pae

—incosx+C

—1nsinx+C

ctg

1nsinx+C

HHTerpai[

S(2_x)2

pae

+c

2—x

+c

x—2

+c

2(2—x)

+c

2(x—2)

$

p’(x)dx

HHTerpaIl pae

q(x)

(x) q’(x)

q(x)

1n(x)+C

$

in xdx

HHTerpaIl pae

x

in x +c

x

in2 x+C

1n1nx+C

—in2 x+ C

HHTerpa.r1 $e3x_2dx

ie32 + C

e3X_2 + C

_ie32 +C

1e3x + C

.1

dx

HHTeFpaJI 2 2 pae

a+x

arcsin—+C

a

—arcsin—+C

a a

1 x

—arctg—+C

a a

x

arctg—+C

a

HHTerpa.n

dx

i,ja2 —x2

pae

—arcsin-- + C

a a

—--arcsin--+C

a a

1 x

—arctg—+C

a a

arcsin— + C

a

HHTerpaI[ $ (K + f(x))dx pae

$f(x)dx K+$f(x)dx i+$f(x)dx

1(x)dx

$

arctgxdx

HHTerpaJl pae

1+x2

iarctg2x+C

arctgx + Carctg2x+C 2arctg2x+C

r dx

HHTerpa.r1 I

‘

1+c

in x

1 +C

in2 x

xlnx

pae

1 +C

21n2x

1n1nx+C

HHTerpaJI $cos3xdx pae

1.

—sin 3x + C

sin3x + C1 2

——cos 3x+C

3sin3x + C

HHTerpaJl $ ctg2xdx pae

1nsin2x+C

12insin2x+C

—211nsin2x+C 21nsin2x+C

HHTeFpaJI

rdx

j

pae

lna—x+

—1na—x+C

1 +c

—

(a x) 2

1 +c

2(a—x)2

r dx

HHTeFpaJT j pae

1nx-a+C

(x — a)2

—1nx—a+C

1 +c

2(x—a) 2

xdx

HHTeFpaJI J 2 pae

x+4

ln(x2+4)+C

1 +c

(x2 4)2

iln(x2+4)+C

lnx+— +C

x

Ec.nM F’(x) = f(x), TO HHTFIIOM $ f(x)dx H3MBTC5I COBOKHHOCTb4yHKuMM BM

f(x)+C

F(x)+C F’(x)+C

HHTerpaIl $cos2dx pae COS —

2 3x

——COS

3 2

—(x + sinx)+ C

1(x—sinx)+C 2

HHTerpaJi $tg2xdx pae

— tgx—x+C

—ctgx—x+C

3

ctg2x+C

HHTerpaJl $e’ cosxdx pae

— ecosxsinx+C

— — e’’ + C

+ C

esmflxsinx+C

HHTeFpaJT f edx pae

1 -3x

--e +C

-e —3x +C

e3x + C

3e_3x + C

HHTerpaJl $ sin2 xdx pae

+ sin 2x) + C

—(x——sin2x)+C 2 2

sin3 x +C

cos x +c

HHTerpaIl $

xdx

2

pae

2(4—x2)2

1n4_x2+C

_1n4_x2+C 21n4_x2+C

2x +3

HHTeFpaJT $ 2 dx pae

x +3x+5

1nx2 + 3x + 5+ C

I1x2+3x+5+c

1nx +3x+5+x+C

+C

2(x2 +3x+5)2

dx

HHTeFpaJI $ pae

tgx

1ntgx+C

ctgx+C

—1nsinx+C

1nsinx+C

dx

HHTeFpaJI $ pae

ctgx

1nctgx + Ctgx+C

—1ncosx+C 1ncosx+C

dx

HHTeFpaJI $ 2

tgx

tgx—x+C

—ctgx—x+C

1

tgx

—tgx—x+C

HHTeFpaJI $ dx pae

(3x —2)

1

2(3x—2)

1n3x—23 +C

1

6(3x — 2)

1 +c

12(3x — 2)

HHTeFpaJT $

dx

/ 4x

pae

J5 — 4x +c

iln(5—4x)+C 2

1 +c

6-J(5 — 4x)3

25—4x+C

$

xdx

HHTeFpaJT pae

J9-x

arcsin+C

9—x +C

+c

—9—x +C

HHTeFpaJI f x cos xdx pae

—xsinx+cosx+C xsinx—cosx+C xsinx + cosx + C

—xsinx—cosx+C

TEMA 9. OnpeJe.J1eHHbIe, HeCO6CTBeHHbIe H KTHbI HHT[JIbI

EcIIH 4JYHKUH5I Ha OT3K a < b, H rn H M — COOTBTCTBHHO HHMHbTJJ H HaH6oJmmee 3HHWI Ha OT3K [a; b], TO

rn(b-a)f(x)dxM(b-a)

rn(a-b)f(x)dxM(a-b) in(ba)<ff(x)dx<M(ba) M (b-a)<f(x)dx<m(b-a)

YHKUH51Y = f(x)HHTerpHpyeMa Ha OTpe3Ke [a;b], eCJTII oHa

Ha 3TOM OT3KMOHOTOHH Ha 3TOM OT3KHOTPHUTJThH Ha 3TOM OT3K HO.nOKHTeJmHa Ha 3TOM OT3K

3HaeHHe oHpeeI[eHHoro erpaiia 3BHCHTTOJThKO OT OT3K [a;b]

— TOJThKO OT HOwJHTeFPaJmHOH 4YHKUHH f(x)

OT OT3K HHTFHOBHH5J [a; b] H OT HOwilHTeFPaJmHO 4YHKUHH f(x)

OT dlloco6a BbIHCI1HH5{ opeeiieoro erpaiia

ECILH cl?YHKUH5I f(x) H HOTPHUTJThH Ha [a; b], re a < b, TO 3HHH

oHpeeI[eHHoro HHTerpaiia 6yeT

HO.r1OKHTeJmHJ,IM

HOTPHUTJThHMM OTPHUTJThHMM

rno6bIM

TeopeMa 0 cpee 3HHHH oHpeeI1eHHoro HHTeFpaJla BbInoJm5leTc5{, CJTH YHKUH5I

HMT KOHe’-IHOe ‘-IHCJIO TO’-IeK papia llepBOrO poaOFpaHH’-IeHa Ha OT3K [a; b]

HOTPHUTJThH Ha [a;b]

Ha OT3K [a; b]

Heco6cTBeHHbIi HHTFI1 f(x)dx CXOHTC, eCiiH

Lirn$f(x)dx=ci

a

b

Lim f f(x)dx — KOHHO ‘-111db

b—*ca a

b Limff(x)dx=—czD b—*oo a

b

Limff(x)dx He CYWCTBYT

b—*c a

EC.T1H Fx) — HepBoo6pa3Ha5J K cyHKunnftx) Ha [a,b], TO aene onpe,LeJ1eHHoro HHTeT’pajTa

S f(x)dx BHO

F(a)—F(b)

— F(x)+C

— F(b)—F(a)

— F(x)-C

yHKuH51J(x) Ha OT3K [1;8], Sf(x)dx = 13 H Sf(x)dx = 4. Tora

HHTF If(x)dx pae

—17

HHTeFpaJI f(x)dx pae

2f(a) 2a

ECIIH c1:yHKuH5IJ(X) Ha [a,b], ToJ(x) H H [b, a:i H BM11OJ1H51TC51

Sf(x)dx1f(x)dx ff(x)dx =f f(-x)dx

Sf(x)dxSf(-x)dx

ff(x)dxf(x)dx

Heco6cTBeHHbIIi FIHTeFpaJI ff(x)dx pacxowrc5J, ec.rin

Lim ff(x)dx— KOHHO qHC.T1O

a

b

— Limff(x)dx=

b—*cjD

a

b

— Lim$f(x)dx = 0

a

b

Lim f(x)dx — KOHHO OTF1UTJThHO tWCIlO

a

EcIIM cjHrypa o6pa3yeTc5l KHBMMH y = f (x) Fi y = f2 (x) Fl Ha OT3K [a,b], r,ie a = H b = x2 (x1 <x2) — a6cUHccbl TO’-IeK epeceewi,LByX KpnBbIx, f2 (x) f (x), TO HJ1O,Lb 3TOFI cFHrypM oHpe,LeJI5IeTc5I HO copyie

b

S = f(f2(x)-f1(x))dx

S=f(f2(x)+f1(x))dx

S = [(f1(x)f2(x))dx

S = f(f1(x)-f2(x))dx

OHpeeI[eHHMii HHTFJ1 HO T-IaCT5IM BMHCJ15{TC5{ HO opyie

bb

(iiv) + vdu

b b

(uv)+fudv (uv)—1vdu (uv) — d(uv)

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Дата добавления: 2015-10-30