Μαθηματικά 1 2010-2011-2012-2013 (Με Λύσεις)

, , : , 2/2/10

ONOMA FOITHTH: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Odhgec

1. Sumplhrste to nom sac nw, kai paradste to parn me tic lseic.

2. Dirkeia extashc: 2 WRES.

3. Apagoreetai h anaqrhsh ap thn ajousa prin thn sumplrwsh 30lptou.

4. Apagoreetai h qrsh upologist qeirc. Apagoreetai h qrsh kinhto, kai wc upologist

qeirc.

5. Oi lseic prpei na grafon apokleistik sthn pareqmenh klla.

6. Oi lseic prpei na enai to kat dunatn analutikc. Prpei na fanontai la ta en-

dimesa bmata stouc upologismoc. Topojetste ta telik apotelsmata

entc plaisou.

7. Xekinste ap autc tic askseic pou xrete kai/ dnoun pollc mondec.

8. Proqwrste kje skhsh so mporete! Ja dojon mondec gia askseic lumnec en mrei.

Jmata

1. (Melth troqic) (Mondec: 2) 'Ena antikemeno kinetai pnw sto eppedo xy tsi ste thqronik stigm t na brsketai sth jsh (x(t), y(t)) pou dnetai ap tic exisseic

x(t) = cos t+ t sin t, y(t) = sin t t cos t.H knhsh sumbanei metax twn qronikn stigmn t0 = 0 kai t1 = 2pi. Na apantsete staaklouja:

(a) (Mondec: 1) Poio enai to sunolik mkoc thc troqic tou antikeimnou?

(b) (Mondec: 0.2) Poia qronik stigm, msa sto disthma (0, 2pi), to antikemeno ja qeikalyei to mis mkoc?

(g) (Mondec: 0.2) Poia qronik stigm, msa sto disthma (0, 2pi), ja qei th mgisth stigmiaataqthta (kat mtro)?

(d) (Mondec: 0.4) Poia qronik stigm, msa sto disthma (0, 2pi), ja brsketai sthn elqisthdunat apstash ap to shmeo (1, 1)?

(e) (Mondec: 0.2) Sqediste, pol proseggistik, thn kamplh, upologzontac merik shmea

ap ta opoa dirqetai kai ennontc ta.

2. (Kataqrhstik Oloklhrmata) (Mondec: 2) Dnetai h sunrthsh

f(x) =log x

x2, x > 0.

(Paratrhsh: me log x sumbolzoume ton fusik logrijmo.)

1

(a) (Mondec: 0.6) Brete to aristo oloklrwma thc f .

(b) (Mondec: 0.2) Poia pargousa thc f dirqetai ap to shmeo x = 1, y = 1?

(g) (Mondec: 1.2) Upologste ta aklouja kataqrhstik oloklhrmata: 10

f(x) dx,

+1

f(x) dx,

+0

f(x) dx.

3. (Idithtec Orwn) (Mondec: 1.5)

(a) (Mondec: 1) 'Estw sunrthsh f(x) ttoia ste limxx0

f(x) = +, kai stw sunrthsh g(x)suneqc sto x0, me g(x0) = c > 0. Poio enai to rio thc f(x)g(x) sto x0? Apodexte toqrhsimopointac ton orism tou katllhlou orou, ton orism thc sunqeiac, kai gnwst

jewrmata/lmmata/ktl.

(b) (Mondec: 0.2) Poio enai to rio thc f(x)g(x) sto x0 an isqoun ta nw me th diafor tig(x0) < 0? Exhgste me lgia, qwrc apdeixh.

(g) (Mondec: 0.3) An isqoun oi upojseic tou prtou sklouc, all g(x0) = 0, ti mporomena pome gia to rio lim

xx0f(x)g(x)? Exhgste me lgia, qwrc apdeixh.

4. (Diaforik Exswsh) (Mondec: 1.5) Na breje h genik lsh thc diaforikc exswshc

y + (1

x+ 1)y =

ex

x.

sto disthma x > 0, kajc kai h merik lsh gia thn opoa y(1) = 0.

5. (Kwnik Tom) (Mondec: 1.5) Dnetai o gewmetrikc tpoc

4x2 + xy + 4y2 = 56.

(a) (Mondec: 0.6) Metasqhmatste thn nw exswsh se exswsh thc morfc Ax2+Cy2+Dx+Ey + F = 0, me th bojeia katllhlou metasqhmatismo suntetagmnwn.

(b) (Mondec: 0.3) Anagnwrste to edoc thc kwnikc tomc, kai prosdiorste tic exisseic twn

eujein ep twn opown brskontai oi axnwn thc (an prkeitai gia lleiyh) o xonac thc

(an prkeitai gia uperbol parabol).

(g) (Mondec: 0.6) Brete tic exisseic lwn twn eujein pou enai orizntiec kai efaptomenikc

ston nw gewmetrik tpo (ennoetai sto arqik ssthma suntetagmnwn xy).

6. (Seirc) (Mondec: 1.5) Na exetsete, qrhsimopointac katllhla jewrmata, kat pso oi

akloujec seirc sugklnoun:

(a) (Mondec: 0.8)

+n=1(n log n)/(n+ pi

e),

(b) (Mondec: 0.7)

n=1 (n/(2n+ (log n)

5))n.

2

Tupolgio

cos(y x) = cosx cos y + sinx sin y, sin(x2

)=

1 cosx

2sin 2x = 2 sinx cosx, cos 2x = cos2 x sin2 x = 2 cos2 x 1 = 1 2 sin2 x

sinx+ sin y = 2 sin(x+ y2

)cos(x y2

), cosx+ cos y = 2 cos

(x+ y2

)cos(x y2

)sinx sin y = 2 cos

(x+ y2

)sin(x y2

), cosx cos y = 2 sin

(x+ y2

)sin(x y2

)sinx sin y =

12[cos(x y) cos(x+ y)] , cosx cos y = 1

2[cos(x+ y) + cos(x y)]

sinx cos y =12[sin(x+ y) + sin(x y)] , cosx < sinx

x 0.

(Paratrhsh: me log x sumbolzoume ton fusik logrijmo.)

(a) (Mondec: 0.6) Brete to aristo oloklrwma thc f .

(b) (Mondec: 0.2) Poia pargousa thc f dirqetai ap to shmeo x = 1, y = 1?

(g) (Mondec: 1.2) Upologste ta aklouja kataqrhstik oloklhrmata: 10

f(x) dx,

+1

f(x) dx,

+0

f(x) dx.

4. (Diaforik Exswsh) (Mondec: 1.5) Na breje h genik lsh thc diaforikc exswshc

(x+ 1)y + y = x2 1sto disthma x > 1, kajc kai h merik lsh gia thn opoa y(0) = 1

3.

5. (Efaptmenec Eujeec) (Mondec: 1.5) Dnetai o gewmetrikc tpoc

4x2 + xy + 4y2 = 56.

(a) (Mondec: 0.6) Metasqhmatste thn nw exswsh se exswsh thc morfc Ax2+Cy2+Dx+Ey + F = 0, me th bojeia katllhlou metasqhmatismo suntetagmnwn.

(b) (Mondec: 0.3) Anagnwrste to edoc thc kwnikc tomc, kai prosdiorste tic exisseic twn

eujein ep twn opown brskontai oi axnwn thc (an prkeitai gia lleiyh) o xonac thc

(an prkeitai gia uperbol parabol).

(g) (Mondec: 0.6) Brete tic exisseic lwn twn eujein pou enai orizntiec kai efaptomenikc

ston nw gewmetrik tpo (ennoetai sto arqik ssthma suntetagmnwn xy).

6. ('Orio Ginomnou) (Mondec: 1.5)

(a) (Mondec: 1) 'Estw sunrthsh f(x) ttoia ste limxx0

f(x) = +, kai stw sunrthsh g(x)suneqc sto x0, me g(x0) = c > 0. Poio enai to rio thc f(x)g(x) sto x0? Apodexte toqrhsimopointac ton orism tou katllhlou orou, ton orism thc sunqeiac, kai gnwst

jewrmata/lmmata/ktl.

(b) (Mondec: 0.2) Poio enai to rio thc f(x)g(x) sto x0 an isqoun ta nw me th diafor tig(x0) < 0? Exhgste me lgia, qwrc apdeixh.

(g) (Mondec: 0.3) An isqoun oi upojseic tou prtou sklouc, all g(x0) = 0, ti mporomena pome gia to rio lim

xx0f(x)g(x)? Exhgste me lgia, qwrc apdeixh.

2

Tupolgio


)=

1 cosx



)cos(x y2


(x+ y2

)cos(x y2

)sinx sin y = 2 cos

(x+ y2

)sin(x y2


(x+ y2

)sin(x y2

)sinx sin y =




x 0.

(: log x .)

2

() f .

() pi f pi x = 1, y = 1;

() pi : 10

f(x) dx, +1

f(x) dx, +0

f(x) dx.

:

() log xx2

=

(x1) log x dx = log xx

+x1x1 dx = log x

x+x2 dx

= log xx x1 + C = log x+ 1

x+ C, C R.

, pi pi pi :( log x+ 1

x

)= x

1x (log x+ 1) 1x2

= 1 log x 1x2

=log xx2

.

() pi , pipi pi

1 = log 1 + 11

+ C C = 2 f(x) = 2 log x+ 1x

.

() pi pi , pi pi 0. : 1

0

log xx2

dx = limh0+

1h

log xx2

dx = limh0+

[1 + log x

x

]1h

= limh0+

[1 + log 1

1+

1 + log hh

]= .

, pi pi 1 + log h , pi 0+. , pi pipi . pi pi pi, pi pipi.

, pi : +1

log xx2

dx = limh+

h1

log xx2

dx = limh+

[1 + log x

x

]h1

= limh+

[1 + log h

h+

1 + log 11

]= limh+

[1 1

h log h

h

]= 1.

, pi

limh+

log hh

= 0,

pi pipi LHospital. , pi pi pi: +

0

f(x) dx = 10

f(x) dx+ +1

f(x) dx = + 1 = .

3

0 1 2 3 4 5 6 7 8 9 101

0.8

0.6

0.4

0.2

0

0.2

2: 2.

pi pi f(x), pi pi pi:

f (x) =x1x2 2x log x

x4=

1 2 log xx3

,

f (x) =2x1x3 + (2 log x 1)3x2

x6=

6 log x 5x4

.

pi pi x =e, pi ,

, f(e) = (2e)1 ' 0.1839. pi pi pi, pi, pi

pipi pi . 2.

3. ( )

() f(x) limxx0

f(x) = +, g(x) x0, g(x0) =c > 0. f(x)g(x) x0; pi pi , , //.

() f(x)g(x) x0 g(x0) < 0; , pi.

() pi pi , g(x0) = 0, pi pi limxx0

f(x)g(x);

, pi.

:

() g(c) > 0, pi , pi 1 > 0

0 < |x x0| < 1 g(x) > c2 .

pipi M > 0. 2Mc > 0, , pi pi limxx0f(x) = +

pi 2 > 0

0 < |x x0| < 2 f(x) > 2Mc.

pi = min{1, 2}, pi, pi:

0 < |x x0| < f(x) > 2Mc, g(x) >

c

2 f(x)g(x) > M,

4

, pi +, pipi pi

limxx0

f(x)g(x) = +.

() pi pi, pi pi

limxx0

f(x)g(x) = .

pi : pi pi , pi pi . pi pi , .

() pipi pi pi :

i. +. pi, x0 = 0, f(x) = x4, g(x) = x2. f(x)g(x) = x2, pi +.

ii. . pi, x0 = 0, f(x) = x4, g(x) = x2. f(x)g(x) = x2, pi .

iii. pi c 6= 0. pi, x0 = 0, f(x) = x4,g(x) = x4. f(x)g(x) = 1, pi 1.

iv. 0. pi, x0 = 0, f(x) = x4, g(x) = x5. f(x)g(x) = x, pi 0.

v. pi. pi, x0 = 0, f(x) = x4, g(x) = x. f(x)g(x) = x3, pi .

4. ( )

y + (1x

+ 1)y =ex

x.

x > 0, pi y(1) = 0.

: pi (1x

+ 1)

= log x+ x+ C,

pi, pi , pipi

elog x+x = elog xex = xex,

:

xexy + xex(1x

+ 1)y = 1 (xexy) = (x) xexy = x+ C y(x) = ex + C ex

x.

. pipi pi y(1) = 0,

0 = e1 + Ce1/1 C = 1,

pi

y(x) = ex[1 1

x

].

5. ( )

(x+ 1)y + y = x2 1

x > 1, pi y(0) = 13 .

5

: x > 1, pi x+ 1 :

y + (1

x+ 1)y = x 1.

pi 1

x+ 1= log(x+ 1) + C,

pi, pi , pipi

elog(x+1) = (x+ 1).

pi . :

(x+ 1)y + y = x2 1 (y(x+ 1)) =(x3

3 x) y(x+ 1) = x

3

3 x+ C y(x) = x

3

3(x+ 1)+C xx+ 1

.

. pipi pi y(0) = 13 ,

13

=03

+C

1 C = 1

3.

pi

y(x) =x3 + 1x+ 1

+13 xx+ 1

=x3 3x+ 1

3(x+ 1).

6. () pi

4x2 + xy + 4y2 = 56.

() Ax2 + Cy2 + Dx + Ey + F = 0, .

() , pi pi pi ( pi ) ( pi pi pi).

() pi pi pi( xy).

:

() pi , pi

=12

arccot4 4

1=

12

arccot0 =pi

4.

x = u cos v sin =

22

(u v), y = u sin + v cos =

22

(u+ v),

pipi:

2(u v)2 + 12

(u2 v2) + 2(u+ v)2 = 56 2u2 + 2v2 4uv + u2

2 v

2

2+ 2u2 + 2v2 + 4uv = 56

9u2 + 7v2 = 112 u2

112/9+v2

16= 1.

() pi , pipi pi , v, u = 0x+ y = 0, u, v = 0 x y = 0. 3.

6

5 0 55

4

3

2

1

0

1

2

3

4

5 y

(b)

(c)

(d)

S/4

x=y

x=y

(a)

x

3: 6.

() y = c. pi pi (x, y) pipi , y = c . y = c,

4x2 + xc+ 4c2 = 56.

x, pi pi , , . pi ,

= c2 16(4c2 56) = 63c2 + 16 56, pi pi c. pi :

i.

c =

16 5663

= 8

1463,

, , pi-. pi c, , pi 3 ( (a), (b)).

ii.

|c|

16 5663

= 8

1463,

, pi , , pi (d) .

7. () , pi , pi :

()+n=1(n log n)/(n+ pi

e),

7

()n=1

(n/(2n+ (log n)5)

)n, ( )

()n=1(4

n + n5)/(n!). ( )

:

() pi +. ,

limn+

n log nn+ pie

= limn+

log n1 + pie/n

= +,

pi, pi . , pi Cauchy, pipi .

() pi :

limn+(an)

1n = lim

n+

(n

2n+ (log n)5

)= limn+

(1

2 + (log n)5/n

)=

12 + lim

n+(log n)5/n

=12,

pi , . , pi LHospital pi :

limn+

(log n)5

n= lim

5(log n)4 1n1

= limn+

5(< logn)4

n= . . . = lim

5!n

= 0.

() :

limn+

[4n+1 + (n+ 1)5

(n+ 1)!

]/

[4n + n5

n!

]= limn+

1n+ 1

4 4n + (n+ 1)54n + n5

= limn+

1n+ 1

4 + (n+ 1)5/4n

1 + n5/4n

= limn+

1n+ 1

limn+

4 + (n+ 1)5/4n

1 + n5/4n= 0

4 + limn+(n+ 1)

5/4n

1 + limn+n

5/4n= 0 4 = 0.

limn+n

5/4n pipi 5 LHospital:

limn+

n5

4n= limn+

5n4

(log 4)4n= . . . = lim

n+5!

(log 4)54n= 0.

pipi limn+(n+ 1)

5/4n.

8

OIKONOMIKO PANEPISTHMIO AJHNWN, TMHMA PLHROFORIKHS

MAJHMATIKA I

KAJ: STAUROS TOUMPHS

TELIKH EXETASH, 7/9/10

ONOMA FOITHTH: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Odhgec

1. Sumplhrste to nom sac nw, kai paradste to parn me tic lseic.



4. Apagoreetai h qrsh upologist qeirc. Apagoreetai h qrsh kinhto, kai wc upologist

qeirc.


6. Oi lseic prpei na enai to kat dunatn analutikc. Prpei na fanontai la ta en-

dimesa bmata stouc upologismoc. Topojetste ta telik apotelsmata

entc plaisou.


8. Proqwrste kje skhsh so mporete! Ja dojon mondec gia askseic lumnec en mrei.

9. KALH EPITUQIA!!

1

Jmata

1. (Pleurik rio) (1 monda) Na dsete ton orism tou orou limxc+

f(x) = +. Akolojwc,na dexete ti an lim

xc+f(x) = +, tte lim

x0f(c+ x2) = +, qrhsimopointac apokleistik toucorismoc twn antstoiqwn orwn.

2. (Dipl paragwgsimh sunrthsh) (1 monda) 'Estw sunrthsh suneqc sto [a, b], kaidipl paragwgsimh sto (a, b). 'Estw epshc pwc to eujgrammo tmma pou ennei ta shmea(a, f(a), (b, f(b)) tmnei to grfhma thc sunrthshc se na shmeo (c, f(c)) pou c (a, b). Nadexete ti uprqei na shmeo t (a, b) ttoio ste f (t) = 0. (Updeixh: parathrste ti oiklseic twn do eujugrmmwn tmhmtwn ap to (a, f(a)) sto (c, f(c)) kai ap to (c, f(c)) sto(b, f(b)) enai sec.)

3. (Oloklrwma) (1.5 monda) Upologste thn tim tou akloujou kataqrhstiko oloklhr-

matoc:

I =

11

|x|1 x2 dx.

(Updeixh: qwrste to oloklrwma se do mrh.)

4. (Mkoc kamplhc) Na upologsete to mkoc twn akloujwn kampuln:

(a) (1 monda) H kamplh pou dnetai ap tic parametrikc exisseic

x(t) = t3, y(t) = 3t2/2, pou 0 t 3.(b) (1 monda) To grfhma thc sunrthshc y(x) =

x0

2 + 2 d, pou 5 x 10.5. (Diaforik Exswsh) (1 monda) Gia th sunrthsh y(x) dnetai ti

y(x) =ex

yy2 + 1

kai ti h y(x) dirqetai ap to shmeo (0,8). Lste thn diaforik exswsh. Dhlad, breteexswsh pou na ikanopoietai ap thn y(x) kai na mhn periqei thn pargwg thc. (Den qreizetaina lsete thn exswsh pou ja brete wc proc y(x).)

6. (Seirc) Na apofanjete, qrhsimopointac gnwst kritria kai idithtec twn seirn kai twn

orwn, kat pso sugklnoun apoklnoun oi akloujec seirc:

(a) (1 monda)

n sin (1/n).

(b) (1 monda)

[n! log n]/[n(n+ 2)!].

(Updeixh: An limx0+

f(x) = L, tte kai limn+

f(1/n) = L.)

7. (Kwnik tom) (1.5 monda) 'Estw h kwnik tom me exswsh

3

2x2 +

3

2y2 + 5xy + 4

2x+ 4

2y = 0.

Na prosdioriste to edoc thc (parabol, lleiyh, uperbol, eidik perptwsh). Na prosdiori-

ston, wc proc to ssthma suntetagmnwn xy, to kntro kai loi oi xonec summetrac.

2

Tupolgio


)=

1 cosx



)cos(x y2


(x+ y2

)cos(x y2

)sinx sin y = 2 cos

(x+ y2

)sin(x y2


(x+ y2

)sin(x y2

)sinx sin y =




x 1:(x 1)3y(x) + 4(x 1)2y(x) = x+ 1.Akolojwc, brete thn eidik lsh pou dirqetai ap to shmeo x = 3, y = 1.

7. (Seirc) Na brete an sugklnoun apoklnoun oi seirc

(a) (Mondec: 0.5)

n2en,

(b) (Mondec: 0.5)

(13

)n | cos pin211888|.

2

Tupolgio


)=

1 cosx



)cos(x y2


(x+ y2

)cos(x y2

)sinx sin y = 2 cos

(x+ y2

)sin(x y2


(x+ y2

)sin(x y2

)sinx sin y =




x 0,

, pi , (

1/n2) .

5. ( pi) pi pi pi :

3x2 23y2 + 263xy + (12 + 104

3)x+ (184 + 52

3)y = 208

3.

() Au2 + Bv2 +Du + Ev + F = 0 - uv.

2

6 4 2 0 2 410

8

6

4

2

0

(2,4)

(0.6,2.5)

(4.6,5.5)

(1.1,2.2)

(5.1,5.8)

1: 5.

() pi (, pi pi) pi pi - ( , , , , / ) xy.

:

() pi , pi pi

cot 2 =A CB

=3 + 23

263

=13 = pi

6.

pi :{x = u cos v sin =

32 u 12v

y = u sin + v cos = 12u+32 v

}{

u = x cos + y sin =32 x+

12y

v = x sin + y cos = 12x+32 y

}(1)

pi :

3(

3

2u 1

2v)2 23(1

2u+

3

2v)2 + 26

3(

3

2u 1

2v)(

1

2u+

3

2v) + (12 + 104

3)(

3

2u 1

2v)

+ (184 + 523)(

1

2u+

3

2v) = 208

3

4u2 9v2 + (16 + 83)u+ (18 36

3)v = 125 52

3

(u+ 2 +3)2

9 (v 1 + 2

3)2

4= 1.

() , pi pi a = 3 b = 2. uv v = 123, (23, 123), (13, 123), (53, 123). pi (1), pipi pi, xy, pi 3y x = 2 43, (2,4), (33/2 2,5/2), (4 23,11/2). pi

1.

3


MAJHMATIKA I, KAJ: STAUROS TOUMPHS


ONOMA FOITHTH: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Odhgec

1. Sumplhrste to nom sac nw, kai paradste to parn me tic

lseic.



4. Apagoreetai h qrsh upologist qeirc. Apagoreetai h qrsh kinhto, kai wc

upologist qeirc.


6. Mporete na qrhsimopoiete molbi /kai stul opoioudpote qrmatoc ektc ap

kkkino.

7. Oi lseic prpei na enai to kat dunatn analutikc. Prpei na fanontai

la ta endimesa bmata stouc upologismoc. Topojetste ta

telik apotelsmata entc plaisou.


9. Proqwrste kje skhsh so mporete! Ja dojon mondec gia askseic lumnec

en mrei.

Jmata

1. (2 mondec) ('Oria) 'Estw sunrthsh g(x) : R R me |g(x)| M gia kjex R, me M R stajer. 'Estw epshc h sunrthsh f(x) = (x 1)2g(x). Nadexete ti:

(a) H f(x) enai suneqc sto 1.

(b) H f(x) enai paragwgsimh sto 1.

1

(g) H deterh pargwgoc thc f(x) mpore na mhn uprqei sto 1 gia kpoiec g(x).(Dste pardeigma g(x) gia thn opoa den uprqei h f (x)).

2. (1.5 monda) (Kataqrhstik Oloklrwma) Upologste thn tim tou ka-

taqrhstiko oloklhrmatoc

0

t5 exp(t3) dt. (Updeixh: Gia na lsete thnskhsh ja prpei na knete kai antikatstash, kai met paragontik oloklrw-

sh.)

3. (1.5 monda) ('Ogkoc ek peristrofc) Neterec arqaiologikc anaskafc

epibebawsan ti oi slpiggec thc Ieriqoc eqan sqma pou prokptei ap peri-

strof tou grafmatoc thc sunrthshc f(x) =x/ exp(x) per ton xona twn xmetax tou x = 1 kai tou x = 100. Na prosdiorsete ton gko touc.

4. (1.5 monda) (Diaforik Exswsh) Dnetai h diaforik exswsh

dy

dx= (sin2 y)(1 + x2),

pou x R, y(x) (0, pi).(a) Na prosdiorsete th genik thc lsh dnontac mia (qi diaforik) exswsh pou

prpei na ikanopoie h y(x).

(b) Na prosdiorsete thn merik lsh pou dirqetai ap to shmeo (0, pi/2).

(Updeixh: poia enai h pargwgoc thc sunefaptomnhc?)

5. (2 mondec) (Seirc) Na brete an sugklnoun oi akloujec seirc:

(a)

[tan

(npi

4(n+ 1)

)]/n!.

(b)

n3/

[n5/2 + (cosn)n3 + 2n+ 1

].

6. (1.5 monda) (Kwnik Tom) 'Estw o gewmetrikc tpoc twn shmewn pou ika-

nopoion thn exswsh

5x2 6xy + 5y2 + 2x 14y + 5 = 0.Na dexete ti o gewmetrikc tpoc enai kwnik tom. Na prosdiorsete to edoc

thc (lleiyh, parabol uperbol) kai epiplon to kntro kai touc xonc thc an

enai lleiyh uperbol, kai thn koruf kai ton xon thc, an enai parabol.

2

Tupolgio


)=

1 cosx

2

sin 2x = 2 sinx cosx, cos 2x = cos2 x sin2 x = 2 cos2 x 1 = 1 2 sin2 x

sinx+ sin y = 2 sin

(x+ y

2

)cos

(x y2


(x+ y

2

)cos

(x y2

)sinx sin y = 2 cos

(x+ y

2

)sin

(x y2


(x+ y

2

)sin

(x y2

)sinx sin y =

1



sinx cos y =1

2[sin(x+ y) + sin(x y)] , cosx < sinx

x 0 > 0 : 0 < |x c| < |f(x) L| < . > 0 > 0 : c < x < c+ |f(x) L| < . > 0 > 0 : c < x < c |f(x) L| < . > 0X R : x > X |f(x) L| < . > 0X R : x < X |f(x) L| < . M R > 0 : 0 < |x c| < f(x) > M.

M RX R : x > X f(x) > M. > 0N N : n > N |an L| < .

|f(x) f(x0)| C|x x0|, |f(y) f(x)| C|y x|

(arcsin y)=

11 y2 , (arccos y)

= 1

1 y2 , (arctan y)=

1

1 + y2

f(x0 + (1 )x1) < f(x0) + (1 )f(x1)

L(f, P ) ,ni=1

mi(ti ti1), U(f, P ) ,ni=1

Mi(ti ti1),ni=1

f(xi)(ti ti1){x = r cos ,y = r sin

}{

r =x2 + y2,

cos = xx2+y2

, sin = yx2+y2

}1

2

ba

f2() d, pi

ba

f2, 2pi

ba

xf(x) dx,

ba

A(t) dt,

ba

(f (x))2 + (g(x))2 dx

y + P (x)y = Q(x), y(x) = [S(x) + C] exp[R(x)]

y(x) =

{y0 +

xx0

Q(u) exp

[ ux0

P (t) dt

]du

}exp

[ xx0

P (t) dt

]En(x) =

1

n!

xa

(x t)nf (n+1)(t) dt, |En(x)| M |x a|n+1

(n+ 1)!

sn =

nk=1

f(k), tn =

n1

f(x) dx

S =

k=1

(1)k1ak, sn =n

k=1

(1)k1ak, 0 < (1)n(S sn) < an+1

A =[A BB2

]B, A = AA, (x x0, y y0)(A,B) = 0 Ax+By = Ax0 +By0

x x0 y y0 z z0x1 y1 z1x2 y2 z2

= 0,x x0 y y0 z z0x1 x0 y1 y0 z1 z0x2 x0 y2 y0 z2 z0

= 0,A(x x0) +B(y y0) + C(z z0) = 0 Ax+By + Cz = Ax0 +By0 + Cz0

y2 = 4px,x2

a2+

y2

a2(1 2) = 1x2

a2+y2

b2= 1,

x2

a2 y

2

a2(2 1) = 1x2

a2 y

2

b2= 1

3

{u = (x x0) cos + (y y0) sin ,v = (x x0) sin + (y y0) cos

}(uv

)=

(cos sin sin cos

)(x x0y y0

){x = x0 + u cos v sin ,y = y0 + u sin + v cos

}(xy

)=

(x0y0

)+

(cos sin sin cos

)(uv

)(uv

)=

(x x0y y0

)(xy

)=

(x0 + uy0 + v

), =

1

2arccot

A CB

x0 y0 z0x1 y1 z1x2 y2 z2

= x0 y1 z1y2 z2

y0 x1 z1x2 z2+ z0 x1 y1x2 y2

, x0 y0x1 y1 = x0y1 x1y0.

4




ONOMA FOITHTH: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Odhgec


lseic.




upologist qeirc.



kkkino.






en mrei.

Jmata

1. (2 mondec) ('Oria) 'Estw sunrthsh k(x) : R R me |k(x)| K gia kjex R, me K R stajer. 'Estw epshc h sunrthsh f(x) = (x + 3)2k(x). Nadexete ti

(a) H f(x) enai suneqc sto (3).(b) H f(x) enai paragwgsimh sto (3).

1

(g) H deterh pargwgoc thc f(x) mpore na mhn uprqei sto (3) gia kpoieck(x). (Dste pardeigma k(x) gia thn opoa den uprqei h f (x)).

2. (1.5 monda) (Kataqrhstik Oloklrwma) Upologste thn tim tou ka-

taqrhstiko oloklhrmatoc

0

t3 exp(t2) dt. (Updeixh: Gia na lsete thnskhsh ja prpei na knete kai antikatstash, kai met paragontik oloklrw-

sh.)

3. (1.5 monda) ('Ogkoc ek peristrofc) Neterec arqaiologikc anaskafc

epibebawsan ti oi slpiggec thc Ieriqoc eqan sqma pou prokptei ap peri-

strof tou grafmatoc thc sunrthshc f(x) =x/ exp(x) per ton xona twn xmetax tou x = 2 kai tou x = 50. Na prosdiorsete ton gko touc.

4. (1.5 monda) (Diaforik Exswsh) Dnetai h diaforik exswsh

dy

dx= (cos2 y)(1 + x3),

pou x R, y(x) (pi/2, pi/2).(a) Na prosdiorsete th genik thc lsh dnontac mia (qi diaforik) exswsh pou

prpei na ikanopoie h y(x).

(b) Na prosdiorsete thn merik lsh pou dirqetai ap thn arq twn axnwn.

(Updeixh: poia enai h pargwgoc thc efaptomnhc?)

5. (2 mondec) (Seirc) Na brete an sugklnoun oi akloujec seirc.

(a)

[cos

(npi

3(n+ 1)

)]/n!.

(b)

n2/

[2n3 + (cosn)n+ 1

].

6. (1.5 monda) (Kwnik Tom) 'Estw o gewmetrikc tpoc twn shmewn pou ika-

nopoion thn exswsh

5x2 26xy + 5y2 + 72x 72y + 216 = 0.Na dexete ti o gewmetrikc tpoc enai kwnik tom. Na prosdiorsete to edoc

thc (lleiyh, parabol uperbol) kai epiplon to kntro kai touc xonc thc an

enai lleiyh uperbol, kai thn koruf kai ton xon thc, an enai parabol.

2

Tupolgio


)=

1 cosx

2


sinx+ sin y = 2 sin

(x+ y

2

)cos

(x y2


(x+ y

2

)cos

(x y2

)sinx sin y = 2 cos

(x+ y

2

)sin

(x y2


(x+ y

2

)sin

(x y2

)sinx sin y =

1



sinx cos y =1




|f(x) f(x0)| C|x x0|, |f(y) f(x)| C|y x|

(arcsin y)=

11 y2 , (arccos y)

= 1

1 y2 , (arctan y)=

1

1 + y2

f(x0 + (1 )x1) < f(x0) + (1 )f(x1)

L(f, P ) ,ni=1


Mi(ti ti1),ni=1


}{

r =x2 + y2,

cos = xx2+y2

, sin = yx2+y2

}1

2

ba

f2() d, pi

ba

f2, 2pi

ba

xf(x) dx,

ba

A(t) dt,

ba

(f (x))2 + (g(x))2 dx


y(x) =

{y0 +

xx0

Q(u) exp

[ ux0

P (t) dt

]du

}exp

[ xx0

P (t) dt

]En(x) =

1

n!

xa


(n+ 1)!

sn =

nk=1

f(k), tn =

n1

f(x) dx

S =

k=1

(1)k1ak, sn =n

k=1

(1)k1ak, 0 < (1)n(S sn) < an+1

A =[A BB2





y2 = 4px,x2

a2+

y2

a2(1 2) = 1x2

a2+y2

b2= 1,

x2

a2 y

2

a2(2 1) = 1x2

a2 y

2

b2= 1

3


}(uv

)=

(cos sin sin cos

)(x x0y y0


}(xy

)=

(x0y0

)+

(cos sin sin cos

)(uv

)(uv

)=

(x x0y y0

)(xy

)=

(x0 + uy0 + v

), =

1

2arccot

A CB

x0 y0 z0x1 y1 z1x2 y2 z2

= x0 y1 z1y2 z2

y0 x1 z1x2 z2+ z0 x1 y1x2 y2

, x0 y0x1 y1 = x0y1 x1y0.

4

1, 2011-2012

Lseic Telikc Extashc Febrouarou Peridou 2011-2012

Omda A

1. () g(x) : R R |g(x)| M x R, M R . pi f(x) = (x 1)2g(x). :() f(x) 1.

() f(x) pi 1.

() pi f(x) pi pi 1 pi g(x). ( pi g(x) pi pi f (x)).

:

() pi f(1) = (1 1)2g(1) = 0. pipi,|f(x)| = (x 1)2g(x) M(x 1)2 M(x 1)2 f(x) M(x 1)2.

, 0 x 1, pi pipi f(x) 1 0, pi f(x) .

() pi

f (1) = limh0

f(1 + h) f(1)h

= limh0

h2g(1 + h) 0h

= limh0

hg(1 + h) = 0.

pipi pi .

()

g(x) =

{sin(

1x1

), x 6= 1,

1, x = 1.

pi |g(x)| M = 1 x R, pi x 6= 1

f (x) = 2(x 1) sin(

1

x 1) (x 1)2 cos

(1

x 1)

1

(x 1)2 = 2(x 1) sin(

1

x 1) cos

(1

x 1).

pipi, x = 1 pi f (x) = 0. f (x) 1, cos

(1

x1) pi f (x) (1, 1)

1. f (x) , pi.

2. ( ) pi 0

t5 exp(t3) dt.

(pi: pipi , pi .)

: pi ( )

I(h) =

h0

t5 exp(t3) dt.

y = t3, dy = 3t2dt, t = 0 y = 0, t = h y = h3. ,

I(h) =1

3

h30

y exp(y) dy = 13

h30

y [exp(y)] dy = 13[y exp(y)]h30

1

3

h30

[exp(y)] dy

= 13h3 exp(h3) 1

3exp(h3) + 1

3=

1

3

[1 exp(h3)(1 + h3)] .

1

pi 0

t5 exp(t3) dt = limh

I(h) =1

3limh

[1 1 + h

3

exp(h3)

]=

1

3 1

3limh

1 + h3

exp(h3)=

1

3 0 = 1

3.

pipi pi pipi LHopital.

3. ( pi) pi pi pi pipi pi pi f(x) =

x/ exp(x) pi

x x = 1 x = 100. pi .

: pi , pi

V = pi

1001

f2(x) dx = pi

1001

x

exp(2x)dx = pi

2

1001

x [exp(2x)] dx

= pi2[x exp(2x)]1001

pi

4

1001

[exp(2x)] dx

= pi2[100 exp(200) exp(2)] pi

4[exp(200) exp(2)] = pi

[3

4exp(2) 201

4exp(200)

].

4. ( )

dy

dx= (sin2 y)(1 + x2),

pi x R, y(x) (0, pi).() pi ( ) pi pipi pi y(x).

() pi pi pi (0, pi/2).

(pi: pi pi pi;)

:

() y(x) (0, pi), sin y 6= 0,

dy

dx= (sin2 y)(1 + x2) dy

sin2 y= (1 + x2)dx

dysin2 y

=

(1 + x2)dx

cot y = x+ x3/3 + C y(x) = arccot [x+ x3/3 + C] .() x = 0, y(x) = pi/2 pipi C = 0

y(x) = arccot[x+ x3/3

].

pi

y(x) = kpi, k Z,

pi . pi pi ;

5. () :

()[

tan

(npi

4(n+ 1)

)]/n!.

()

n3/[n5/2 + (cosn)n3 + 2n+ 1

].

:

2

() an =[tan

(npi

4(n+1)

)]/n!, pi .

, pi pi

limn+

an+1an

= limn+

tan[(n+1)pi4(n+2)

](n+ 1)!

n!

tan[

npi4(n+1)

] = limn+

1n+ 1

tan

[(n+1)pi4(n+2)

]tan

[npi

4(n+1)

]= lim

n+1

(n+ 1) limn+

tan[(n+1)pi4(n+2)

]tan

[npi

4(n+1)

] = 0 11= 0,

.

() pi n, pipi 2/n2.

1/n2 . pi , an

, an = n3/[n5/2 + (cosn)n3 + 2n+ 1

], bn = 1/n2.

limn+

n3/[n5/2 + (cosn)n3 + 2n+ 1

]1/n2

= limn+

n5

n5/2 + (cosn)n3 + 2n+ 1

= limn+

1

1/2 + (cosn)/n2 + 2/n4 + 1/n5= 2,

.

6. ( ) pi pi pi

5x2 6xy + 5y2 + 2x 14y + 5 = 0.

pi . pi (, pi pi) pipi pi, , pi.

: pi , uv, xy pi xy pi

cot 2 =5 56 = 0.

pi = pi/4. {u =

2

2(x+ y), v =

2

2(x+ y)

}{x =

2

2(u v), y =

2

2(u+ v)

}. (1)

pi,

5

2(u2 + v2 2uv) + 5

2(u2 + v2 + 2uv) 6

2(u2 v2) +

2(u v) 7

2(u+ v) + 5 = 0

2u2 + 8v2 62u 8

2v + 5 = 0 2

(u2 3

2u+

9

2

)+ 8

(v2

2v +

1

2

)= 8

(u 3

2

2

)24

+

(v 2

2

)2= 1.

pi, u = 32/2, v =

2/2, pi, pi (1)

x = 1, y = 2, u = 32/2 x+ y = 3, v = 2/2 x y + 1 = 0.

3

Omda B

1. () k(x) : R R |k(x)| K x R, K R . pi f(x) = (x+ 3)2k(x).

() f(x) (3).() f(x) pi (3).() pi f(x) pi pi (3) pi k(x). ( pi k(x)

pi pi f (x)).

:

() pi f(3) = (3 + 3)2k(3) = 0. pipi,|f(x)| = (x+ 3)2k(x) K(x+ 3)2 K(x+ 3)2 f(x) K(x+ 3)2.

, 0 x 3, pi pipi f(x) (3) 0, pi f(x) .

() pi

f (3) = limh0

f(3 + h) f(3)h

= limh0

h2k(3 + h) 0h

= limh0

hk(3 + h) = 0.

pipi pi .

()

k(x) =

{sin(

1x+3

), x 6= 3,

1, x = 3. pi |k(x)| K = 1 x R, pi x 6= 3

f (x) = 2(x+ 3) sin(

1

x+ 3

) (x+ 3)2 cos

(1

x+ 3

)1

(x+ 3)2= 2(x+ 3) sin

(1

x+ 3

) cos

(1

x+ 3

).

pipi, x = 3 pi f (x) = 0. f (x) (3), cos

(1

x+3

) pi f (x) (1, 1)

(3). f (x) , pi.

2. ( ) pi 0

t3 exp(t2) dt.

(pi: pipi , pi .)

: pi ( )

I(h) =

h0

t3 exp(t2) dt.

y = t2, dy = 2tdt, t = 0 h = 0, t = h y = h2. ,

I(h) =1

2

h20

y exp(y) dy = 12

h20

y(exp(y)) dy = 12[y exp(y)]h20

1

2

h20

[exp(y)] dy

= 12h2 exp(h2) 1

2exp(h2) + 1

2=

1

2

[1 exp(h2)(1 + h2)] .

pi 0

t3 exp(t2) dt = limh

I(h) =1

2limh

[1 1 + h

2

exp(h2)

]=

1

2 1

2limh

1 + h2

exp(h2)=

1

2 0 = 1

2.

pipi pi LHopital.

4

3. ( pi) pi pi pi pipi pi pi f(x) =

x/ exp(x) pi

x x = 2 x = 50. pi .

: pi , pi

V = pi

502

f2(x) dx = pi

502

x

exp(2x)dx = pi

2

502

x [exp(2x)] dx

= pi2[x exp(2x)]502

pi

4

502

[exp(2x)] dx

= pi2[50 exp(100) 2 exp(4)] pi

4[exp(100) exp(4)] = pi

[5

4exp(4) 101

4exp(100)

].

4. ( )

dy

dx= (cos2 y)(1 + x3),

pi x R, y(x) (pi/2, pi/2).() pi ( ) pi pipi pi y(x).

() pi pi pi .

(pi: pi pi pi;)

:

() y(x) (pi/2, pi/2), cos y 6= 0, dy

dx= (cos2 y)(1 + x3) dy

cos2 y= dx(1 + x3)

dy

cos2 y=

(1 + x3)

tan y = x+ x4/4 + C y(x) = arctan [x+ x4/4 + C] .() x = y(x) = 0, pipi C = 0

y(x) = arctan[x+ x4/4

].

pi

y(x) = kpi + pi/2, k Z,pi . pi pi ;

5. () .

()[

cos

(npi

3(n+ 1)

)]/n!.

()

n2/[2n3 + (cosn)n+ 1

].

:

() , pi . , pipi

limn+

an+1an

= limn+

cos[(n+1)pi(n+2)3

](n+ 1)!

n!

cos[

npi(n+1)3

] = limn+

1(n+ 1)

cos[(n+1)pi(n+2)3

]cos[

npi(n+1)3

]= lim

n+1

(n+ 1) limn+

cos[(n+1)pi(n+2)3

]cos[

npi(n+1)3

] = 0 1/21/2

= 0,

.

5

() pi n, pipi 1/(2n).

1/n pi. , an ,

an = n2/[2n3 + (cosn)n+ 1

], bn = 1/n.

limn+

n2/[2n3 + (cosn)n+ 1

]1/n

= limn+n

3/[2n3 + (cosn)n+ 1

]= limn+1/

[2 + (cosn)/n2 + 1/n3

]= 1,

pi.

6. ( ) pi pi pi

5x2 26xy + 5y2 + 72x 72y + 216 = 0.

pi . pi (, pi pi) pipi pi, , pi.

: pi , uv, xy pi pi

cot 2 =5 526 = 0.

pi = pi/4. {u =

2

2(x+ y), v =

2

2(x+ y)

}{x =

2

2(u v), y =

2

2(u+ v)

}. (2)

pi,

5

2(u2 + v2 2uv) + 5

2(u2 + v2 + 2uv) 26

2(u2 v2) + 36

2(u v) 36

2(u+ v) + 216 = 0

8u2 + 18v2 722v + 216 = 0 8u2 + 18

(v2 4

2v + 8

)= 72 u

2

9(v 22)2

4= 1.

pi, pi u = 0, v = 22, pi, pi (2)

x = 2, y = 2, () u = 0 x + y = 0, ( pi)v = 2

2 x y + 4 = 0.

6



TELIKH EXETASH, SEPTEMBRIOS 2012

ONOMA FOITHTH: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Odhgec


lseic.




upologist qeirc.



kkkino.






en mrei.

1

JEMATA

1. ('Oria) (2 mondec) Na upologsete ta aklouja ria:

limx0

x2(1 + sinx)

(2x + sinx)2, lim

xx(

2x + log x2x)2. (Kataqrhstik Oloklhrmata) (2 mondec) Na upologsete

ta

1

16 + x2dx,

1

1x + 1 +

x 1 dx.

3. (Diaforik Exswsh) (2 mondec) Na brete th genik lsh thc

diaforikc exswshc

ex log xy(x) + ex log x(log x)y(x) = 1, x > 0.

Na brete epshc thn eidik lsh pou dirqetai ap to shmeo tou epi-

pdou xy me suntetagmnec (1, 1).

4. (Seirc)

(a) (1 monda) Na exetsete an sugklnei h

n=1

(n

n+1

)n2.

(b) (1.5 monda) Na upologiste to aristo oloklrwma

dx

x(log x)2kai

met na apofanjete gia thn sgklish thc seirc

n=2

1n(log n)2.

5. (Kamplh) (1.5 monda) 'Estw h kamplh

x(t) =(2t + 3)3/2

3, y(t) = t + t2/2, 0 t 3.Na upologiste to mkoc thc. Apodexte pwc den uprqei kpoio shmeo

pou h kamplh tmnei ton eaut thc. An h kamplh perigrfei thn

knhsh enc stereo sunartsei tou qrnou t, poio enai to mtro thc

taqthtac tou stereo sunartsei tou t?

2

Tupolgio


)=

1 cosx

2


sinx+ sin y = 2 sin

(x+ y

2

)cos

(x y2


(x+ y

2

)cos

(x y2

)sinx sin y = 2 cos

(x+ y

2

)sin

(x y2


(x+ y

2

)sin

(x y2

)sinx sin y =

1



sinx cos y =1




|f(x) f(x0)| C|x x0|, |f(y) f(x)| C|y x|

(arcsin y)=

11 y2 , (arccos y)

= 1

1 y2 , (arctan y)=

1

1 + y2

f(x0 + (1 )x1) < f(x0) + (1 )f(x1)

L(f, P ) ,ni=1


Mi(ti ti1),ni=1


}{

r =x2 + y2,

cos = xx2+y2

, sin = yx2+y2

}1

2

ba

f2() d, pi

ba

f2, 2pi

ba

xf(x) dx,

ba

A(t) dt,

ba

(f (x))2 + (g(x))2 dx


y(x) =

{y0 +

xx0

Q(u) exp

[ ux0

P (t) dt

]du

}exp

[ xx0

P (t) dt

]En(x) =

1

n!

xa


(n+ 1)!

sn =

nk=1

f(k), tn =

n1

f(x) dx

S =

k=1

(1)k1ak, sn =n

k=1

(1)k1ak, 0 < (1)n(S sn) < an+1

A =[A BB2





y2 = 4px,x2

a2+

y2

a2(1 2) = 1x2

a2+y2

b2= 1,

x2

a2 y

2

a2(2 1) = 1x2

a2 y

2

b2= 1

3


}(uv

)=

(cos sin sin cos

)(x x0y y0


}(xy

)=

(x0y0

)+

(cos sin sin cos

)(uv

)(uv

)=

(x x0y y0

)(xy

)=

(x0 + uy0 + v

), =

1

2arccot

A CB

x0 y0 z0x1 y1 z1x2 y2 z2

= x0 y1 z1y2 z2

y0 x1 z1x2 z2+ z0 x1 y1x2 y2

, x0 y0x1 y1 = x0y1 x1y0.

4

1, 2011-2012

Lseic Telikc Extashc Septembrou Peridou 2011-2012

1. () pi :

limx0

x2(1 + sinx)

(2x+ sinx)2, lim

xx(

2x+ log x2x)

:

()

limx0

x2(1 + sinx)

(2x+ sinx)2= lim

x0x2 + x2 sinx

4x2 + sin2 x+ 4x sinx= lim

x01 + sinx

4 + sin2 xx2 + 4

sin xx

=1

4 + 1 + 4=

1

9.

pi limx0

sin xx = 1.

()

limxx

(2x+ log x

2x)=x (2x+ log x 2x)2x+ log x+

2x

=x log x

2x+ log x+2x

=

x log x

2 + log xx +2=.

, pi limx

log xx = 0, pi pi

22, , .

2. ( ) pi

1

16 + x2dx,

1

1x+ 1 +

x 1 dx.

: pipi, pi , pi pi , pi pi . , pipi:

()

1

16 + x2dx =

1

16

1

1 + (x/4)2dx =

1

4

1

1 + y2dy

=1

4

(arctan y)dy =

1

4arctan y| =

1

4

(pi2+pi

2

)=pi

4.

, y = x/4.

() pi 1

1x 1 +x+ 1 dx

1

1

2x+ 1

dx =

1

(x+ 1

)dx =,

.3. ( )

ex log xy(x) + ex log x(log x)y(x) = 1, x > 0.

pi pi pi pipi xy (1, 1).

1

: pi

y(x) + (log x)y = ex log x,

pi pi . pilog x dx =

(x) log x dx = x log x

x(log x) dx = x(log x 1) + C,

, pi , pipi ex(log x1),

ex(log x1)y(x) + (log x)ex(log x1)y(x) = ex (ex(log x1)y(x)

)= (ex)

ex(log x1)y(x) = ex + C y(x) = (C ex)ex(log x1).

x = 1 y(x) = 1, pipi pi C = 2e1, pi

y(x) = (2e1 ex)ex(log x1).

4. ()

()

n=1

(n

n+1

)n2.

() pi

dxx(log x)2 pi

n=21

n(logn)2 .

:

() :

a1nn =

((n

n+ 1

)n2) 1n=

(1 1

n+ 1

)n= exp

(n log

(1 1

n+ 1

)).

,

limn+

(n log

(1 1

n+ 1

))= lim

n+

log(1 1n+1

)1n

= limn+

11 1n+1

1(n+1)2 1n2

= 1.

, pi LHopital. pi

limn+

((n

n+ 1

)n2) 1n= e1 < 1,

.

() y = log x dy = dxx :dx

x(log x)2=

dy

y2= y1 + C = 1

log x+ C.

, ,

n=2

1

n(log n)2,

2

1

x(log x)2dx

pi . pi 2

1

x(log x)2dx =

2

( 1log x

)dx = 0 + log 2,

pi pipi pi .

2

5. (pi) pi

x(t) =(2t+ 3)3/2

3, y(t) = t+ t2/2, 0 t 3.

pi . pi pi pi pi pi pi . pi pi t, pi t;

:

() l(C) pi, pi

l(C) =

30

(x(t))2 + (y(t))2 dt =

30

(2t+ 3

)2+ (1 + t)2 dt =

30

2t+ 3 + 1 + t2 + 2t dt

=

30

t2 + 4t+ 4 dt =

30

(t+ 2) dt =

[t2

2+ 2t

]30

=21

2.

pi pi t1, t2

0 t1 < t2 3, x(t1) = x(t2), y(t1) = y(t2).

, x(t), y(t) . , pi pi x(t) pi x, pi pi x. , y(t) pi y, pi pi y. pi, (x(t), y(t)),

(x(t))2 + (y(t))2 = t+ 2.

3

1, 2012-2013

Lseic Telikc Extashc Febrouarou Peridou 2012-2013

Omda A

1. () pi :

limx0

xe1x , lim

xpi2

x pi21 sinx.

: pi , pi pi

limx0+

xe1x = lim

x0+e1x

1x

= limh

eh

h= limh

eh

1=,

limx0

xe1x = lim

x0e1x

1x

= limh

eh

h= 0.

pi pi LHopital. pi pi pi pi. pi pi , pi .

, pi pi

limxpi2

x pi21 sinx = limh0

h1 sin (h+ pi/2) = limh0

h1 cosh = limh0

h1 + cosh1 cosh1 + cosh = limh0

h1 + cosh

| sinh| .

pi h = x pi2 .

limh0+

h1 + cosh

| sinh| = limh0+h

sinh limh0+

1 + cosh = 1

2 =2,

limh0

h1 + cosh

| sinh| = limh0+h

sinh limh0+1 + cosh = (1)

2 =

2,

pi, pi .

2. () f(x) [0, 1] f(0) = f(1). pi pi x [0, 23] f(x) = f

(x+ 13

). (pi: pi g(x) = f

(x+ 13

) f(x) pi .)

: pi, g(x) = f(x+ 13

) f(x), pi [0, 23]. pi

g(0) = f

(1

3

) f(0), g

(1

3

)= f

(2

3

) f

(1

3

), g

(2

3

)= f(1) f

(2

3

).

g(0) + g(13

)+ g

(23

)= f(1) f(0) = 0. pipi pi

0, pipi pi x = 0, 13 ,23 . g(x0)

, pi , g(x1), pi, Bolzano x.

3. () pi I =

3dxx(4+x)

. , pi -

I = 10

3dxx(4+x)

. (pi: (arctanx) = 11+x2 .)

: , u =x u2 = x

I =

6du

4 + u2=

3

2

du

1 +(u2

)2 ,1

t = u/2 pipi 3dxx(4 + x)

= 3

dt

1 + t2= 3arctan(t) + C = 3arctan

(x

2

)+ C.

, pi pi 10

3dxx(4 + x)

= limt0+

1t

3dxx(4 + x)

= limt0+

[3 arctan

(x

2

)]1t

= 3arctan

(1

2

)3 arctan(0) = 3 arctan

(1

2

).

4. ( pi) pi pi g(x) = log(1 + sinx) log(1 sinx) (0, pi/2). , pi f(x) = log(cosx) x (0, pi/6). (pi: pi pipi x(t) = t, y(t) =f(t).) pi , x (0, pi/2).: pi pi

[log(1 + sinx) log(1 sinx)] = cosx1 + sinx

+cosx

1 sinx =cosx(1 sinx) + cosx(1 + sinx)

1 sin2 x=

cosx sinx cosx+ cosx+ sinx cosxcos2 x

=2

cosx.

pi,

l(C) =

pi6

0

(y(t))2 + 1 dt =

pi6

0

( sin tcos t

)2+ 1 dt =

pi6

0

1

cos2 tdt

=

pi6

0

dt

cos t=

1

2

pi6

0

[log(1 + sin t) log(1 sin t)] dt = 12

[log

3

2 log 1

2

]=

1

2log 3.

x (0, pi/2), pi pi limxpi2

f(x) =

, pi pipi pi , .5. ( )

y(x) =ey

x(log x)2

x > 1. , pi pi pi (x0, y0) = (e, 0).

: pi

dy

dx=

ey

x(log x)2 eydy = dx

x (log x)2

ey dy =

dx

x(log x)2

ey + C = ((log x)1) dx = 1

log x+ C y(x) = log

(C 1

log x

).

x = e, y = 0, pipi pi C = 2.

6. () :

( nn+ 1

)n2,

1nsin

(1

n2

).

(pi :

1n3 .)

: pi , , pi pi pi . :

limna

1nn = lim

n

(n

n+ 1

)n= en log(1

1n+1 ).

2

,

limnn log

(1 1

n+ 1

)= limn

log(1 1n+1

)1n

= limn

1

1 1n+1 1

(n+ 1)2 (n2) = 1,

pi limna

1nn = e1, .

, pi pi

sin

(1

n2

) 1n2 1

nsin

(1

n2

) 1n3,

1n3 , pi pipi .

Omda B

7. () pi :

limx0

x2e1x , lim

xpi2

x+ pi21 + sinx

.

: pi , pi pi

limx0+

x2e1x = lim

x0+e1x

1x2

= limh

eh

h2= limh

eh

2h= limh

eh

2=,

limx0

x2e1x = lim

x0e1x

1x2

= limh

eh

h2= 0.

pi pi pi pi , LHopital. pi pi pi pi. pi pi , pi .

, pi pi

limxpi2

x+ pi21 + sinx

= limh0

h1 + sin (h pi/2) = limh0

h1 cosh = limh0

h1 + cosh1 cosh1 + cosh

= limh0

h1 + cosh

| sinh| .

pi h = x+ pi2 .

limh0+

h1 + cosh

| sinh| = limh0+h

sinh limh0+

1 + cosh = 1

2 =2,

limh0

h1 + cosh

| sinh| = limh0h

sinh limh0+1 + cosh = (1)

2 =

2,

pi, pi .

8. () f(x) [0, 1] f(0) = f(1). pi pi x [0, 34] f(x) = f

(x+ 14

). (pi: pi g(x) = f

(x+ 14

) f(x) pi .)

: pi, g(x) = f(x+ 14

) f(x), pi [0, 34]. pi

g(0) = f

(1

4

) f(0), g

(1

4

)= f

(2

4

) f

(1

4

), g

(2

4

)= f

(3

4

) f

(2

4

), g

(3

4

)= f(1) f

(3

4

).

g(0) + g(14

)+ g

(24

)+ g

(34

)= f(1) f(0) = 0. pipi pi

0, pipi pi x = 0, 14 ,24 ,

34 .

g(x0) , pi , g(x1), pi, Bolzano x.

3

9. () pi I =

6dxx(9+x)

. , pi -

I = 10

6dxx(9+x)

. (pi: (arctanx) = 11+x2 .)

: , u =x u2 = x

I =

12du

9 + u2=

4

3

du

1 +(u3

)2 , t = u/3 pipi

6dxx(9 + x)

= 4

dt

1 + t2= 4arctan(t) + C = 4arctan

(x

3

)+ C.

, pi pi

10

6dxx(9 + x)

= limt0+

1t

6dxx(9 + x)

= limt0+

[4 arctan

(x

3

)]1t

= 4arctan

(1

3

)4 arctan(0) = 4 arctan

(1

3

).

10. ( pi) pi pi g(x) = log(1 + cosx) log(1 cosx) (0, pi/2). , pi f(x) = log(sinx) x (pi/3, pi/2). (pi: pi pipi x(t) = t, y(t) = f(t).) pi , x (0, pi/2).: pi pi

[log(1 + cosx) log(1 cosx)] = sinx1 + cosx

sinx1 cosx =

sinx(1 cosx) + sinx(1 + cosx)1 cos2 x

= sinx sinx cosx+ sinx+ sinx cosxsin2 x

= 2sinx

.

pi,

l(C) =

pi2

pi3

(y(t))2 + 1 dt =

pi2

pi3

(cos t

sin t

)2+ 1 dt =

pi2

pi3

1

sin2 tdt

=

pi2

pi3

dt

sin t= 1

2

pi2

pi3

[log(1 + cos t) log(1 cos t)] dt = 12

[log

3

2 log 1

2

]=

1

2log 3.

x (0, pi/2), pi pi limx0+

f(x) =

, pi pipi pi , .11. ( )

y(x) =e2y

x(log x)3

x > 1. , pi pi pi (x0, y0) = (e, 0).

: pi

dy

dx=

e2y

x(log x)3 e2ydy = dx

x (log x)3

e2y dy =

dx

x(log x)3

12e2y + C =

(12(log x)2

)dx = 1

2(log x)2+ C y(x) = 1

2log

(C 1

(log x)2

).

x = e, y = 0, pipi pi C = 2.

4

12. () :

( nn+ 3

)n2,

1n2

sin

(1

n3

).

(pi :

1n5 .)

: pi , , pi pi pi . :

limna

1nn = lim

n

(n

n+ 3

)n= en log(1

3n+3 ).

,

limnn log

(1 3

n+ 3

)= limn

log(1 3n+3

)1n

= limn

1

1 3n+3 3

(n+ 3)2 (n2) = 3,

pi limna

1nn = e3, .

, pi pi

sin

(1

n3

) 1n3 1

n2sin

(1

n3

) 1n5,

1n5 , pi pipi .

5

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