, , : , 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
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.
(: 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.
2. Dirkeia extashc: 2 WRES.
3. Apagoreetai h anaqrhsh ap thn ajousa prin thn sumplrwsh 45lptou.
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.
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
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 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
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,
, 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
OIKONOMIKO PANEPISTHMIO AJHNWN, TMHMA PLHROFORIKHS
MAJHMATIKA I, KAJ: STAUROS TOUMPHS
TELIKH EXETASH, 15/2/12
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. 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.
8. Xekinste ap autc tic askseic pou xrete kai/ dnoun pollc mondec.
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
cos(y x) = cosx cos y + sinx sin y, sin(x2
)=
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
), cosx+ cos y = 2 cos
(x+ y
2
)cos
(x y2
)sinx sin y = 2 cos
(x+ y
2
)sin
(x y2
), cosx cos y = 2 sin
(x+ y
2
)sin
(x y2
)sinx sin y =
1
2[cos(x y) cos(x+ y)] , cosx cos y = 1
2[cos(x+ y) + cos(x y)]
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
OIKONOMIKO PANEPISTHMIO AJHNWN, TMHMA PLHROFORIKHS
MAJHMATIKA I, KAJ: STAUROS TOUMPHS
TELIKH EXETASH, 15/2/12
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. 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.
8. Xekinste ap autc tic askseic pou xrete kai/ dnoun pollc mondec.
9. Proqwrste kje skhsh so mporete! Ja dojon mondec gia askseic lumnec
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
cos(y x) = cosx cos y + sinx sin y, sin(x2
)=
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
), cosx+ cos y = 2 cos
(x+ y
2
)cos
(x y2
)sinx sin y = 2 cos
(x+ y
2
)sin
(x y2
), cosx cos y = 2 sin
(x+ y
2
)sin
(x y2
)sinx sin y =
1
2[cos(x y) cos(x+ y)] , cosx cos y = 1
2[cos(x+ y) + cos(x y)]
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
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
OIKONOMIKO PANEPISTHMIO AJHNWN, TMHMA PLHROFORIKHS
MAJHMATIKA I, KAJ: STAUROS TOUMPHS
TELIKH EXETASH, SEPTEMBRIOS 2012
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. 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.
8. Xekinste ap autc tic askseic pou xrete kai/ dnoun pollc mondec.
9. Proqwrste kje skhsh so mporete! Ja dojon mondec gia askseic lumnec
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
cos(y x) = cosx cos y + sinx sin y, sin(x2
)=
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
), cosx+ cos y = 2 cos
(x+ y
2
)cos
(x y2
)sinx sin y = 2 cos
(x+ y
2
)sin
(x y2
), cosx cos y = 2 sin
(x+ y
2
)sin
(x y2
)sinx sin y =
1
2[cos(x y) cos(x+ y)] , cosx cos y = 1
2[cos(x+ y) + cos(x y)]
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
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
finalAfinalBfinal_solfinalA_sepfinalB_sepfinal_sol_sepfinals10-11finalAfinalfinal_solfinal_sepfinal_sep_solfinals11finalA_11finalB_11final_sol_11final_sep_11final_sol_sep_11
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