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APEIROSTIKOS LOGISMOS I · 2013-12-18 · APEIROSTIKOS LOGISMOS I 4. Ori'o kai Sunèqeia Sunart...
Transcript of APEIROSTIKOS LOGISMOS I · 2013-12-18 · APEIROSTIKOS LOGISMOS I 4. Ori'o kai Sunèqeia Sunart...
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APEIROSTIKOS LOGISMOS I
1 OktwbrÐou 2012
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APEIROSTIKOS LOGISMOS IKwdikìc Maj matoc: 101 (U)
'Etoc didaskalÐac: 2012-2013, Qeimerinì Ex�mhno
Hmèrec didaskalÐac: Deut. - Tet. - Par. , 11:00-13:00
Did�skontec
Tm ma 1o (AM pou l gei se 0,1,2) Amf 21,
BasÐleioc NestorÐdhc,
GrafeÐo: 307, thl. 210-7276304,
e-mail: [email protected]
Tm ma 2o (AM pou l gei se 3,4,5,6) Amf 23,
Le¸nh Euaggel�tou-D�lla,
GrafeÐo: 207, thl. 210-7276375,
e-mail: [email protected]
Tm ma 3o (AM pou l gei se 7,8,9) Amf 24,
MarÐa PapatriantafÔllou,
GrafeÐo: 201, thl. 210-7276349,
e-mail: [email protected]
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APEIROSTIKOS LOGISMOS I
IstoselÐda Maj matoc:
http://eclass.uoa.gr/courses/MATH130/
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APEIROSTIKOS LOGISMOS I
Perigraf Maj matoc:
1 PragmatikoÐ arijmoÐ
2 AkoloujÐec pragmatik¸n arijm¸n
3 Sunart seic
4 Sunèqeia kai ìrio sunart sewn
5 Par�gwgoc
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APEIROSTIKOS LOGISMOS I
1. PragmatikoÐ arijmoÐ:
Axiwmatik jemelÐwsh twn Pragmatik¸n arijm¸n
FusikoÐ, Akèraioi kai RhtoÐ arijmoÐ
AxÐwma plhrìthtac
'Uparxh tetragwnik c rÐzac
'Arrhtoi arijmoÐ
Akèraio mèroc
Puknìthta twn rht¸n kai twn arr twn stouc
pragmatikoÔc arijmoÔc
Klassikèc anisìthtec
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APEIROSTIKOS LOGISMOS IRhtoÐ arijmoÐ
Η διαγώνια μέθοδος του Cantor και η « 1- 1 » αντιστοιχία με τουςφυσικούς αριθμούς
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APEIROSTIKOS LOGISMOS I'Arrhtoi arijmoÐ
Το σοκ του Πυθαγόρα: Υποτείνουσα ορθογωνίων τριγώνων
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APEIROSTIKOS LOGISMOS I'Arrhtoi arijmoÐ
Qrus tom : φ = 1+√5
2 = 1.6180339887...
Gewmetrik� prokÔptei mèsw thc analogÐac: a+ba = a
b ≡ φ
Algebrik� prokÔptei wc jetik rÐza thc exÐswshc:
x2−x−1 = 0
Ti ekfr�zei pragmatik� autìc o arijmìc?
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APEIROSTIKOS LOGISMOS I'Arrhtoi arijmoÐ
Qrus tom : φ = 1+√5
2 = 1.6180339887...
Gewmetrik� prokÔptei mèsw thc analogÐac: a+ba = a
b ≡ φ
Algebrik� prokÔptei wc jetik rÐza thc exÐswshc:
x2−x−1 = 0
Ti ekfr�zei pragmatik� autìc o arijmìc?
![Page 10: APEIROSTIKOS LOGISMOS I · 2013-12-18 · APEIROSTIKOS LOGISMOS I 4. Ori'o kai Sunèqeia Sunart sewn: H ènnoia tou orÐou sun rthshc - Sunèqeia. Arq thc metafor c Sunèqeia gnwst¸n](https://reader034.fdocumentos.tips/reader034/viewer/2022050611/5fb29eb33dae601996484c72/html5/thumbnails/10.jpg)
APEIROSTIKOS LOGISMOS I'Arrhtoi arijmoÐ
Αρμονία στην αρχιτεκτονική του Παρθενώνα, 438 π.Χ.
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APEIROSTIKOS LOGISMOS I'Arrhtoi arijmoÐ
Αρμονία στο έργο του Da Vinci - Vitruvian Man, 1487 μ.Χ.
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APEIROSTIKOS LOGISMOS I'Arrhtoi arijmoÐ
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APEIROSTIKOS LOGISMOS I'Arrhtoi arijmoÐ
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APEIROSTIKOS LOGISMOS I'Arrhtoi arijmoÐ
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APEIROSTIKOS LOGISMOS I'Arrhtoi arijmoÐ
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APEIROSTIKOS LOGISMOS I'Arrhtoi arijmoÐ
Υπερβατικοί αριθμοί - Ρητή προσέγγιση του π με 999 δεκαδικά ψηφία
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APEIROSTIKOS LOGISMOS I
2. AkoloujÐec pragmatik¸n arijm¸n:
SugklÐnousec akoloujÐec
Monìtonec akoloujÐec
Kibwtismìc diasthm�twn
Anadromikèc akoloujÐec
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APEIROSTIKOS LOGISMOS IAkoloujÐec
Ti eÐnai mia akoloujÐa pragmatik¸n arijm¸n?
EÐnai mia apeikìnish apì to sÔnolo N sto R.Mèsw tÔpou:
1 αn = n√n, limn→∞
n√n = 1
2 αn = (1+1/n)n, limn→∞(1+1/n)n = e = 2,71828...
Mèsw anadromik c sqèshc:
1 αn+1 =√1+αn, n ≥ 1, α1 = 1, limn→∞ αn = φ .
2 Ακολουθία Fibonacci: αn = αn−1+αn−2, α0 = 0, α1 = 1,limn→∞ αn =+∞, limn→∞
αn+1αn
= φ .
AkoloujÐa twn pr¸twn arijm¸n:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53, ...
Up�rqei sun�rthsh anadromikìc tÔpoc pou na dÐnei thn
parap�nw akoloujÐa?
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APEIROSTIKOS LOGISMOS IAkoloujÐec
Ti eÐnai mia akoloujÐa pragmatik¸n arijm¸n?
EÐnai mia apeikìnish apì to sÔnolo N sto R.
Mèsw tÔpou:
1 αn = n√n, limn→∞
n√n = 1
2 αn = (1+1/n)n, limn→∞(1+1/n)n = e = 2,71828...
Mèsw anadromik c sqèshc:
1 αn+1 =√1+αn, n ≥ 1, α1 = 1, limn→∞ αn = φ .
2 Ακολουθία Fibonacci: αn = αn−1+αn−2, α0 = 0, α1 = 1,limn→∞ αn =+∞, limn→∞
αn+1αn
= φ .
AkoloujÐa twn pr¸twn arijm¸n:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53, ...
Up�rqei sun�rthsh anadromikìc tÔpoc pou na dÐnei thn
parap�nw akoloujÐa?
![Page 20: APEIROSTIKOS LOGISMOS I · 2013-12-18 · APEIROSTIKOS LOGISMOS I 4. Ori'o kai Sunèqeia Sunart sewn: H ènnoia tou orÐou sun rthshc - Sunèqeia. Arq thc metafor c Sunèqeia gnwst¸n](https://reader034.fdocumentos.tips/reader034/viewer/2022050611/5fb29eb33dae601996484c72/html5/thumbnails/20.jpg)
APEIROSTIKOS LOGISMOS IAkoloujÐec
Ti eÐnai mia akoloujÐa pragmatik¸n arijm¸n?
EÐnai mia apeikìnish apì to sÔnolo N sto R.Mèsw tÔpou:
1 αn = n√n, limn→∞
n√n = 1
2 αn = (1+1/n)n, limn→∞(1+1/n)n = e = 2,71828...
Mèsw anadromik c sqèshc:
1 αn+1 =√1+αn, n ≥ 1, α1 = 1, limn→∞ αn = φ .
2 Ακολουθία Fibonacci: αn = αn−1+αn−2, α0 = 0, α1 = 1,limn→∞ αn =+∞, limn→∞
αn+1αn
= φ .
AkoloujÐa twn pr¸twn arijm¸n:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53, ...
Up�rqei sun�rthsh anadromikìc tÔpoc pou na dÐnei thn
parap�nw akoloujÐa?
![Page 21: APEIROSTIKOS LOGISMOS I · 2013-12-18 · APEIROSTIKOS LOGISMOS I 4. Ori'o kai Sunèqeia Sunart sewn: H ènnoia tou orÐou sun rthshc - Sunèqeia. Arq thc metafor c Sunèqeia gnwst¸n](https://reader034.fdocumentos.tips/reader034/viewer/2022050611/5fb29eb33dae601996484c72/html5/thumbnails/21.jpg)
APEIROSTIKOS LOGISMOS IAkoloujÐec
Ti eÐnai mia akoloujÐa pragmatik¸n arijm¸n?
EÐnai mia apeikìnish apì to sÔnolo N sto R.Mèsw tÔpou:
1 αn = n√n, limn→∞
n√n = 1
2 αn = (1+1/n)n, limn→∞(1+1/n)n = e = 2,71828...
Mèsw anadromik c sqèshc:
1 αn+1 =√1+αn, n ≥ 1, α1 = 1, limn→∞ αn = φ .
2 Ακολουθία Fibonacci: αn = αn−1+αn−2, α0 = 0, α1 = 1,limn→∞ αn =+∞, limn→∞
αn+1αn
= φ .
AkoloujÐa twn pr¸twn arijm¸n:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53, ...
Up�rqei sun�rthsh anadromikìc tÔpoc pou na dÐnei thn
parap�nw akoloujÐa?
![Page 22: APEIROSTIKOS LOGISMOS I · 2013-12-18 · APEIROSTIKOS LOGISMOS I 4. Ori'o kai Sunèqeia Sunart sewn: H ènnoia tou orÐou sun rthshc - Sunèqeia. Arq thc metafor c Sunèqeia gnwst¸n](https://reader034.fdocumentos.tips/reader034/viewer/2022050611/5fb29eb33dae601996484c72/html5/thumbnails/22.jpg)
APEIROSTIKOS LOGISMOS IAkoloujÐec
Φραγμένη - μη συγκλίνουσα ακολουθία πραγματικών αριθμών
απείρων όρων
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APEIROSTIKOS LOGISMOS I
3. Sunart seic:
BasikoÐ orismoÐ
Algebrikèc sunart seic
Trigwnometrikèc sunart seic
Ekjetik sun�rthsh
![Page 24: APEIROSTIKOS LOGISMOS I · 2013-12-18 · APEIROSTIKOS LOGISMOS I 4. Ori'o kai Sunèqeia Sunart sewn: H ènnoia tou orÐou sun rthshc - Sunèqeia. Arq thc metafor c Sunèqeia gnwst¸n](https://reader034.fdocumentos.tips/reader034/viewer/2022050611/5fb29eb33dae601996484c72/html5/thumbnails/24.jpg)
APEIROSTIKOS LOGISMOS IPragmatikèc sunart seic
Γράφημα πολυωνυμικής συνάρτησης 4ouβαθμού
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APEIROSTIKOS LOGISMOS IPragmatikèc sunart seic
Γράφημα τριγωνομετρικής συνάρτησης ημιτόνου
θ
sin θ
0
1
−1
2π
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APEIROSTIKOS LOGISMOS IPragmatikèc sunart seic
Γράφημα αντίστροφης τριγωνομετρικής συνάρτησης ημιτόνου
θ
arcsinθ
0
1
π/ 2
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APEIROSTIKOS LOGISMOS IPragmatikèc sunart seic
Gr�fhma trigwnometrik c sun�rthshc efaptomènhc
θ
tan θ
0
π
2π
![Page 28: APEIROSTIKOS LOGISMOS I · 2013-12-18 · APEIROSTIKOS LOGISMOS I 4. Ori'o kai Sunèqeia Sunart sewn: H ènnoia tou orÐou sun rthshc - Sunèqeia. Arq thc metafor c Sunèqeia gnwst¸n](https://reader034.fdocumentos.tips/reader034/viewer/2022050611/5fb29eb33dae601996484c72/html5/thumbnails/28.jpg)
APEIROSTIKOS LOGISMOS IPragmatikèc sunart seic
Gr�fhma antÐstrofhc trigwnometrik c sun�rthshc
efaptomènhc
θ
arctanθ
0
−π/ 2
π/ 2
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APEIROSTIKOS LOGISMOS IPragmatikèc sunart seic
Γράφημα εκθετικής συνάρτησης για α > 1
x
y
0
y = ax
1
![Page 30: APEIROSTIKOS LOGISMOS I · 2013-12-18 · APEIROSTIKOS LOGISMOS I 4. Ori'o kai Sunèqeia Sunart sewn: H ènnoia tou orÐou sun rthshc - Sunèqeia. Arq thc metafor c Sunèqeia gnwst¸n](https://reader034.fdocumentos.tips/reader034/viewer/2022050611/5fb29eb33dae601996484c72/html5/thumbnails/30.jpg)
APEIROSTIKOS LOGISMOS IPragmatikèc sunart seic
Γράφημα λογαριθμικής συνάρτησης για α > 1
x = logax
1 x
y
0
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APEIROSTIKOS LOGISMOS IPragmatikèc sunart seic
Η συνάρτηση f (x) =
{x , για x ∈Q
−x , για x /∈Q.
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APEIROSTIKOS LOGISMOS I
4. 'Orio kai Sunèqeia Sunart sewn:
H ènnoia tou orÐou sun�rthshc - Sunèqeia.
Arq thc metafor�c
Sunèqeia gnwst¸n sunart sewn
Sunèqeia kai topik sumperifor�
Je¸rhma endi�meshc tim c
'Uparxh megÐsthc kai elaqÐsthc tim c gia suneqeÐc
sunart seic orismènec se kleist� diast mata - Monìtonec
sunart seic
SuneqeÐc kai {1-1} sunart seic
AntÐstrofec trigwnometrikèc sunart seic
Logarijmik sun�rthsh
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APEIROSTIKOS LOGISMOS I
5. Par�gwgoc:
Eisagwg : ParadeÐgmata apì th GewmetrÐa kai th Fusik .
H ènnoia thc parag¸gou
Kanìnec parag¸gishc
Par�gwgoi basik¸n sunart sewn
Je¸rhma mèshc tim c
Je¸rhma Darboux
Krit ria monotonÐac sun�rthshc
Krit ria topik¸n akrot�twn
Genikeumèno je¸rhma mèshc tim c
Kanìnec De L’Hospital
Kurtèc kai koÐlec sunart seic - ShmeÐa kamp c
Melèth sunart sewn
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APEIROSTIKOS LOGISMOS IPar�gwgoc sun�rthshc
Η συνάρτηση f (x) =
{sin(1/x), x 6= 0
0 , x = 0
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APEIROSTIKOS LOGISMOS IPar�gwgoc sun�rthshc
Η συνάρτηση g(x) =
{xsin(1/x), x 6= 0
0 , x = 0
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APEIROSTIKOS LOGISMOS IPar�gwgoc sun�rthshc
Η συνάρτηση h(x) =
{x2sin(1/x), x 6= 0
0 , x = 0
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APEIROSTIKOS LOGISMOS IPar�gwgoc sun�rthshc
Η συνάρτηση f (x) =
{x2, για x ∈Q
−x2, για x /∈Q.
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APEIROSTIKOS LOGISMOS IBibliografÐa
Σ. Νεγρεπόντης, Σ. Γιωτόπουλος, Ε. Γιαννακούλιας:
« Απειροστικός Λογισμός Ι », Εκδόσεις Συμμετρία.
Λ. Τσίτσας: « Εφαρμοσμένος Απειροστικός Λογισμός », Εκδόσεις
Συμμετρία.
M. Spivak: “ Calculus ”, Benjamin (κυκλοφορεί σε Ελληνικήμετάφραση με τίτλο: « Διαφορικός και Ολοκληρωτικός Λογισμός
», Πανεπιστημιακές Εκδόσεις Κρήτης.)
R. Courant and F. John: “ Introduction to Calculus and Analysis ”,Vol. I, Interscience.
G. H. Hardy: “ A Course in Pure Mathematics”, CambridgeUniversity Press.
S. Salas and E. Hille: “ Calculus ”, John Wiley.
R. Bartle and D. Sherbert: “ Introduction to Real Analysis ”, JohnWiley.
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APEIROSTIKOS LOGISMOS I