ÐÑÏËÏÃÏÓ
Ôï âéâëßï áõôü áðåõèýíåôáé óôïõò öïéôçôÝò ôçò Áíþôáôçò Åêðáßäåõóçò. ¢íåôá üìùò ìðïñåß íá ÷ñçóéìïðïéçèåß êáé áðü üëïõò åêåßíïõò ðïõ åíäéáöÝñïíôáé ãéá Ýíá áíáëõôéêü âéâëßï óôïí ÷þñï ôùí "Áíþôåñùí Ìáèçìáôéêþí". Ç ýëç ôïõ êáëýðôåé ôïí Äéáíõóìáôéêü Ëïãéóìü, ôçí ÁíáëõôéêÞ Ãåùìåôñßá, ôç ÓõíáñôçóéáêÞ ÁíÜëõóç, ôïí Äéáöïñéêü Ëïãéóìü, ôïí Ïëïêëçñùôéêü Ëïãéóìü êáé ôá ÉóôïñéêÜ Óôïé÷åßá óå Ýêôáóç êáé âÜèïò ðïõ ðñïâëÝðïíôáé ãéá áõôÞí ôç âáèìßäá åêðáßäåõóçò. Προστέθηκε ένα ολόκληρο μέρος με το γενικό μαθηματικό πακέτο MATLAB το οποίο επιλύει γρήγορα και με ακρίβεια όλα τα προβλήματα των θεμάτων του βιβλίου. Ç óõããñáöÞ åíüò ôÝôïéïõ âéâëßïõ ðáñïõóéÜæåé ïðùóäÞðïôå óçìáíôéêÝò äõóêïëßåò óôçí åðéëïãÞ, ôç äéÜôáîç êáé ôçí ðáñïõóßáóç ôçò ýëçò ôïõ. Èá ðñÝðåé, ðÝñá áðü ôéò ìáèçìáôéêÝò ãíþóåéò ðïõ ðñïóöÝñåé, íá áðïêáëýðôåé ðáñÜëëçëá êáé ôç ÷ñçóéìüôçôá ôçò ìáèçìáôéêÞò óêÝøçò áëëÜ êáé ôéò äõíáôüôçôåò åöáñìïãÞò ôùí ìáèçìáôéêþí ìåèüäùí óôçí ÅðéóôÞìç êáé ôçí Ôå÷íïëïãßá. Ï óõíäõáóìüò ôùí ðáñáðÜíù åðéäéþîåùí, óå óõó÷åôéóìü ìå ôç äéáðßóôùóç üôé ïé óçìåñéíïß áðüöïéôïé ôùí Ëõêåßùí, ðáñÜ ôï áîéüëïãï ìáèçìáôéêü õðüâáèñï ðïõ äéáèÝôïõí, áíôéìåôùðßæïõí ìå ôçí áëëáãÞ âáèìßäáò åêðáßäåõóçò éäéáßôåñá ðñïâëÞìáôá ðñïóáñìïãÞò óôïí ðñïóáíáôïëéóìü ôçò äéäáóêáëßáò, ïäÞãçóáí óôç óõãêåêñéìÝíç äïìÞ ôïõ âéâëßïõ. Åßíáé óêüðéìï íá äéåõêñéíéóèåß üôé ôï âéâëßï Ý÷åé ó÷åäéáóèåß ãéá íá äéäá÷èåß ó' Ýíá åîÜìçíï ìå ôåôñÜùñç Ýùò åîÜùñç äéäáóêáëßá ôçí åâäïìÜäá êáé íá ÷ñçóéìåýåé ðáñÜëëçëá ùò âéâëßï áíáöïñÜò Áíþôåñùí Ìáèçìáôéêþí. Ãé’ áõôü, ç ðáñïõóßáóç ôçò ýëçò ôïõ Ýãéíå ìå ôñüðï áðëü êáé êáôÜ ôï äõíáôüí óýíôïìï. Ïé êáèáñÜ èåùñçôéêÝò ðñïôÜóåéò óõíïäåýïíôáé ðÜíôïôå áðü ðáñáäåßãìáôá ðïõ áíáëýïõí ôï íüçìÜ ôïõò, åíþ ïé äýóêïëåò áðïäåßîåéò ðáñáëåßöèçêáí Þ ãñÜöôçêáí ìå ìéêñüôåñç ãñáììáôïóåéñÜ þóôå íá äéäá÷èïýí åðéëåêôéêÜ, ÷ùñßò áõôü íá âëÜðôåé ôç óõíÝ÷åéá êáé ôç óôÜèìç ôïõ âéâëßïõ. Åî' Üëëïõ Ýíáò ìåãÜëïò áñéèìüò áðü ðñïôåéíüìåíåò ãéá ëýóç áóêÞóåéò ìáæß ìå ôéò áðáíôÞóåéò ôïõò, äßíïõí ôçí åõêáéñßá óôïõò öïéôçôÝò ãéá Üóêçóç êáé åìðÝäùóç ôçò ýëçò.
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ÌÁÈÇÌÁÔÉÊÇ ÁÍÁËÕÓÇ
Ðéóôåýïõìå üôé ðáñáèÝôïíôáò ìåñéêÜ éóôïñéêÜ óôïé÷åßá ãéá ôïõò ìåãÜëïõò ìáèçìáôéêïýò ðïõ Üöçóáí áíåîßôçëï ôï üíïìÜ ôïõò óôçí åîÝëéîç ôçò ìáèçìáôéêÞò åðéóôÞìçò, âïçèïýìå ôïõò öïéôçôÝò íá ó÷çìáôßóïõí áö' åíüò ìåí åéêüíá ôçò äéá ìÝóïõ ôùí áéþíùí ðïñåßáò ôùí ìáèçìáôéêþí, áö' åôÝñïõ äå íá áéóèáíèïýí ãéá ìéá áêüìç öïñÜ õðåñÞöáíïé ðïõ ãåííÞèçêáí ¸ëëçíåò. Ðëçèþñá éóôïñéêþí óôïé÷åßùí ìðïñåß íá âñåé ï åíäéáöåñüìåíïò óôï Internet. ÔÝëïò, åßíáé âÝâáéï, üôé êÜèå åðéëïãÞ Þ ðñùôïâïõëßá ôïõ äéäÜóêïíôïò, ðïõ èá âïçèÞóåé ôïõò óðïõäáóôÝò íá äþóïõí áðÜíôçóç óôá êñßóéìá åñùôÞìáôá: - Ãéáôß ôüóá ìáèçìáôéêÜ; êáé, - Ðþò óõíäÝïíôáé ôá ìáèçìáôéêÜ áõôÜ ìå ôá Üëëá áíôéêåßìåíá ôçò óðïõäÞò ôïõò; óõìðßðôåé êáé óõìðëçñþíåé ôéò ðñïèÝóåéò ôων óõããñáöÝων. ÌÉ×ÁËÇÓ ÃËÁÌÐÅÄÁÊÇÓ ΑΝΤΩΝΗΣ ΓΛΑΜΠΕΔΑΚΗΣ
Συνοπτικά Ðåñéå÷üìåíá
ÌÅÑÏÓ ÐÑÙÔÏ • ÂÁÓÉÊÅÓ ÅÍÍÏÉÅÓ ÌÁÈÇÌÁÔÉÊÙÍ ÊåöÜëáéï 1: Óõìâïëéóìïß áðü ôç ìáèçìáôéêÞ ëïãéêÞ êáé ôá óýíïëá ....27 ÊåöÜëáéï 2: Óýíïëá áñéèìþí .....................................................................31 ÊåöÜëáéï 3: Ïñßæïõóåò êáé ãñáììéêÜ óõóôÞìáôá .................................. 61 ÊåöÜëáéï 4: ÐñáãìáôéêÝò óõíáñôÞóåéò.....................................................75 ÌÅÑÏÓ ÄÅÕÔÅÑÏ ÄÉÁÍÕÓÌÁÔÉÊÏÓ ËÏÃÉÓÌÏÓ ÊÁÉ ÁÍÁËÕÔÉÊÇ ÃÅÙÌÅÔÑÉÁ Á. ÄÉÁÍÕÓÌÁÔÉÊÏÓ ËÏÃÉÓÌÏÓ................................................................103 ÊåöÜëáéï 5: Ïñéóìïß êáé ìåôñéêÝò ó÷Ýóåéò óôá äéáíýóìáôá .................105 ÊåöÜëáéï 6: ÓõóôÞìáôá óõíôåôáãìÝíùí.................................................131 ÊåöÜëáéï 7: ÅéóáãùãÞ óôïõò äéáíõóìáôéêïýò ÷þñïõò .........................141 Â. ÁÍÁËÕÔÉÊÇ ÃÅÙÌÅÔÑÉÁ.........................................................................161 ÊåöÜëáéï 8: ÃñáììÝò óôï åðßðåäï (R2) ................................................... 165 ÊåöÜëáéï 9: ÃåíéêÜ ðåñß êáìðõëþí 2ïõ âáèìïý ...................................229 ÊåöÜëáéï 10: Óôïé÷åßá áðü ôç ìåëÝôç ôïõ R3 .........................................237 ÌÅÑÏÓ ÔÑÉÔÏ • ΟΛΟΚΛΗΡΩΤΙΚΟΣ ËÏÃÉÓÌÏÓ
ÊåöÜëáéï 11: ¼ñéï óõíÜñôçóçò ..............................................................273 ÊåöÜëáéï 12: ÓõíÝ÷åéá óõíÜñôçóçò ........................................................301 ÊåöÜëáéï 13: ÐáñÜãùãïò óõíÜñôçóçò ...................................................319 ÊåöÜëáéï 14: Ðåñß äéáöïñéêþí ................................................................363 ÊåöÜëáéï 15: ÅöáñìïãÝò ôïõ äéáöïñéêïý ëïãéóìïý ..............................373 ÌÅÑÏÓ ΤΕΤΑΡΤΟ • ΟΛΟΚΛΗΡΩΤΙΚΟΣ ËÏÃÉÓÌÏÓ
ÊåöÜëáéï 16: Θεμελιώδη θέματα .............................................................435 ÊåöÜëáéï 17: Γενικευμένα ολοκληρώματα ..............................................465 ÊåöÜëáéï 18: Εφαρμογές του ολοκληρωτικού λογισμού .......................473 ÊåöÜëáéï 19: Μέθοδοι Ολοκλήρωσης ......................................................497
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ÌÁÈÇÌÁÔÉÊÇ ÁÍÁËÕÓÇ ÌÅÑÏÓ ΠΕΜΠΤΟ • MATLAB
Περιεχόμενα Πέμπτου Μέρους.................................................................567 ÊåöÜëáéï 20: Βασικά στοιχεία Matlab .....................................................573 Κεφάλαιο 21: Τα Μαθηματικά του Matlab ..............................................603 Κεφάλαιο 22: Τα γραφικά του Matlab......................................................669 Κεφάλαιο 23: Τα πολυώνυμα στο Matlab ................................................711 Κεφάλαιο 24: Αρχεία εντολών και συναρτήσεων M-Files.......................719 Κεφάλαιο 25: Προγραμματισμός Matlab..................................................739 Κεφάλαιο 26: Προσεγγιστικές μέθοδοι με Matlab με συμβολικό προγραμματισμό .........................................................................777 Κεφάλαιο 27: Ειδικά θέματα του ML ........................................................833 Κεφάλαιο 28: Διαχείριση αποτελεσμάτων...............................................861 Παράρτημα .................................................................................................881 Πίνακας Ι 883 Πίνακας ΙΙ.....................................................................................................884 Πίνακας ΙΙΙ....................................................................................................885 Πίνακας ΙV ...................................................................................................886 Πίνακας V.....................................................................................................888 Πίνακας VΙ ...................................................................................................890 Πίνακας VII ..................................................................................................891 Βιβλιογραφία ..............................................................................................903
Ðåñéå÷üìåíá ÅéóáãùãÞ: Áðü ôïí ÈáëÞ ôïí Ìçëßóéï óôïí Êùíóôáíôßíï ÊáñáèåïäùñÞ .................19 ÌÅÑÏÓ ÐÑÙÔÏ • ÂÁÓÉÊÅÓ ÅÍÍÏÉÅÓ ÌÁÈÇÌÁÔÉÊÙÍ ÊåöÜëáéï 1: Óõìâïëéóìïß áðü ôç ìáèçìáôéêÞ ëïãéêÞ êáé ôá óýíïëá......27 1.1 Óõìâïëéóìïß áðü ôç ìáèçìáôéêÞ ëïãéêÞ ..........................................................27 1.2 Óõìâïëéóìïß áðü ôç èåùñßá ôùí óõíüëùí ......................................................28 1.3 Óõìâïëéóìïß áðü ôçí Üëãåâñá .........................................................................29
ÊåöÜëáéï 2: Óýíïëá áñéèìþí......................................................................31 2.1 Ôï óýíïëï ôùí ðñáãìáôéêþí áñéèìþí (R) .......................................................31 2.1.1 Õðïóýíïëá ôïõ óõíüëïõ ôùí ðñáãìáôéêþí áñéèìþí ...........................31 2.1.2 Ç óõìðáãÞò åõèåßá ôùí ðñáãìáôéêþí áñéèìþí....................................32 2.1.3 ÄéÜóôçìá ðñáãìáôéêþí áñéèìþí............................................................33 2.1.4 Ðåñéï÷Þ ðñáãìáôéêïý áñéèìïý ...............................................................34 2.2Ôï óýíïëï ôùí ìéãáäéêþí áñéèìþí C ..................................................................35 2.2.1 Ïñéóìüò êáé Üëãåâñá ôùí ìéãáäéêþí áñéèìþí .......................................35 2.2.2 Ç öáíôáóôéêÞ ìïíÜäá .............................................................................37 2.2.3 Óõæõãåßò ìéãáäéêïß áñéèìïß.......................................................................39 2.2.4 ÃåùìåôñéêÞ ðáñÜóôáóç ôùí ìéãáäéêþí áñéèìþí...................................41 2.2.5 ÃåùìåôñéêÞ ðáñÜóôáóç ôïõ áèñïßóìáôïò êáé ôçò äéáöïñÜò ìéãáäéêþí áñéèìþí...................................................................................42 2.2.6 ÌÝôñï ìéãáäéêïý áñéèìïý........................................................................43 2.2.7 ÔñéãùíïìåôñéêÞ ìïñöÞ ìéãáäéêïý áñéèìïý... ..........................................47 2.2.8 Èåþñçìá: Éóüôçôá Ìéãáäéêþí................................................................51 2.2.9 Èåþñçìá: Ãéíüìåíï Ìéãáäéêþí ...............................................................51 2.2.10 Èåþñçìá: Ðçëßêï Ìéãáäéêþí..................................................................52 2.2.11 Èåþñçìá: Áíôßóôñïöïò Ìéãáäéêïý ........................................................52 2.2.12 Èåþñçìá De Moivre ................................................................................53 2.2.13 Ñßæåò ìéãáäéêþí áñéèìþí .........................................................................54 2.2.14 ÃåùìåôñéêÞ óçìáóßá ôùí ñéæþí ìéãáäéêïý áñéèìïý ...............................54 2.2.15 Í-ïóôÝò ñßæåò ôçò ìïíÜäáò .....................................................................55 2.2.16 ÅêèåôéêÞ ìïñöÞ ìéãáäéêïý áñéèìïý.........................................................55 2.2.17 ËïãÜñéèìïò (íåðÝñéïò) ìéãáäéêïý áñéèìïý..............................................56 2.3 ÁóêÞóåéò.............................................................................................................57
ÊåöÜëáéï 3: Ïñßæïõóåò êáé ãñáììéêÜ óõóôÞìáôá ................................... 61 3.1 Ïñßæïõóåò ..........................................................................................................61 3.1.1 Ïñéóìüò: Ïñßæïõóá äåõôÝñáò ôÜîåùò....................................................61
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ÌÁÈÇÌÁÔÉÊÇ ÁÍÁËÕÓÇ
3.1.2 Ãåíßêåõóç: Ïñßæïõóá í-ïóôÞò ôÜîåùò ....................................................61 3.1.3 Êáíüíáò ôïõ Sarrus ..................................................................................63 3.1.4 Éäéüôçôåò ôùí ïñéæïõóþí.........................................................................63 3.2 ÃñáììéêÜ ÓõóôÞìáôá ........................................................................................66 3.2.1 Åðßëõóç ãåíéêþí ãñáììéêþí óõóôçìÜôùí ............................................. 66 3.2.2 ÏìïãåíÞ ãñáììéêÜ óõóôÞìáôá................................................................69 3.3 ÁóêÞóåéò.............................................................................................................70
ÊåöÜëáéï 4: ÐñáãìáôéêÝò óõíáñôÞóåéò......................................................75 4.1 4.2 4.3 4.4 4.5 4.6
Ïñéóìüò óõíÜñôçóçò ........................................................................................75 ÃñáöéêÞ ðáñÜóôáóç óõíÜñôçóçò ....................................................................77 Áñôéåò êáé ðåñéôôÝò óõíáñôÞóåéò......................................................................80 ÐåñéïäéêÝò óõíáñôÞóåéò ....................................................................................80 Ìïíïôïíßá óõíÜñôçóçò ....................................................................................81 Áëãåâñá ôùí óõíáñôÞóåùí ..............................................................................81 4.6.1 Éóüôçôá óõíáñôÞóåùí .............................................................................81 4.6.2 Óýíèåóç óõíáñôÞóåùí ........................................................................... 82 4.6.3 ¢èñïéóìá óõíáñôÞóåùí ..........................................................................82 4.6.4 ÄéáöïñÜ óõíáñôÞóåùí............................................................................82 4.6.5 Ãéíüìåíï óõíáñôÞóåùí ............................................................................82 4.6.6 Ðçëßêï óõíáñôÞóåùí...............................................................................82 4.6.7 Áíôßóôñïöç óõíÜñôçóç ...........................................................................82 4.7 Áêñüôáôá óõíÜñôçóçò......................................................................................85 4.8 Êáôçãïñßåò óõíáñôÞóåùí .................................................................................87 4.8.1 ÁëãåâñéêÝò óõíáñôÞóåéò ..........................................................................87 4.8.2 ÕðåñâáôéêÝò óõíáñôÞóåéò........................................................................88 4.9 ÁóêÞóåéò.............................................................................................................97
ÌÅÑÏÓ ÄÅÕÔÅÑÏ ÄÉÁÍÕÓÌÁÔÉÊÏÓ ËÏÃÉÓÌÏÓ ÊÁÉ ÁÍÁËÕÔÉÊÇ ÃÅÙÌÅÔÑÉÁ Á. ÄÉÁÍÕÓÌÁÔÉÊÏÓ ËÏÃÉÓÌÏÓ................................................................103 ÊåöÜëáéï 5: Ïñéóìïß êáé ìåôñéêÝò ó÷Ýóåéò óôá äéáíýóìáôá ..................105 5.1 5.2 5.3 5.4 5.5
ÂáóéêÝò Ýííïéåò óôá äéáíýóìáôá ....................................................................105 ÓõíôåôáãìÝíåò äéáíýóìáôïò ...........................................................................107 ¢ëãåâñá ôùí äéáíõóìÜôùí.............................................................................109 ÓõíôåôáãìÝíåò óçìåßïõ ðïõ ÷ùñßæåé äéÜíõóìá óå ëüãï ë .............................110 Ç ãùíßá óôá äéáíýóìáôá ................................................................................111 5.5.1 Äéáíýóìáôá óôï åðßðåäï (R2) ...............................................................111 5.5.2 Äéáíýóìáôá óôïí ÷þñï (R3) ..................................................................115 5.6 Áðüóôáóç äýï óçìåßùí ..................................................................................117 5.7 Åóùôåñéêü ãéíüìåíï äýï äéáíõóìÜôùí (áñéèìçôéêü ãéíüìåíï) ..................... 118
ÐÅÑÉÅ×ÏÌÅÍÁ 5.8 5.9 5.10 5.11
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Åîùôåñéêü ãéíüìåíï äõï äéáíõóìÜôùí (äéáíõóìáôéêü ãéíüìåíï) ...................120 Ìéêôü ãéíüìåíï ôñéþí äéáíõóìÜôùí ...............................................................125 ÄéÜöïñåò áíáãêáßåò êáé éêáíÝò óõíèÞêåò ......................................................127 ÁóêÞóåéò...........................................................................................................128
ÊåöÜëáéï 6: ÓõóôÞìáôá óõíôåôáãìÝíùí ..................................................131 6.1 6.2 6.3 6.4
ÃåíéêÜ ...............................................................................................................131 ÁëëáãÞ óõóôçìÜôùí .......................................................................................134 Ìåôáó÷çìáôéóìüò óõóôÞìáôïò óõíôåôáãìÝíùí... ........................................136 ÁóêÞóåéò...........................................................................................................139
ÊåöÜëáéï 7: ÅéóáãùãÞ óôïõò äéáíõóìáôéêïýò ÷þñïõò ..........................141 7.1 ÃåíéêÜ ...............................................................................................................141 7.2 Ïñéóìüò äéáíõóìáôéêïý ÷þñïõ ......................................................................142 7.3 ÃñáììéêÞ áíåîáñôçóßá äéáíõóìÜôùí..............................................................144 7.3.1 Éäéüôçôåò.................................................................................................147 7.4 ÄéÜóôáóç êáé âÜóç äéáíõóìáôéêïý ÷þñïõ ....................................................148 7.5 ÁóêÞóåéò...........................................................................................................150 7.6 ÅöáñìïãÝò ôïõ Äéáíõóìáôéêïý Ëïãéóìïý óôéò ÈåôéêÝò ÅðéóôÞìåò...............152 7.7 ÁóêÞóåéò...........................................................................................................158
Â. ÁÍÁËÕÔÉÊÇ ÃÅÙÌÅÔÑÉÁ.........................................................................161 ÊåöÜëáéï 8: ÃñáììÝò óôï åðßðåäï (R2).................................................... 165 8.1 Ç åõèåßá ...........................................................................................................166 8.1.1 Ç åîßóùóç ôçò åõèåßáò ..........................................................................166 8.1.2 ÓõíôåëåóôÞò äéåýèõíóçò åõèåßáò ..........................................................171 8.1.3 ÓõíåõèåéáêÜ óçìåßá ...............................................................................173 8.1.4 Ãùíßá äýï åõèåéþí .................................................................................174 8.1.5 Ó÷åôéêÝò èÝóåéò äýï åõèåéþí..................................................................176 8.1.6 Ðñïóáíáôïëéóìüò åðéðÝäïõ ùò ðñïò åõèåßá Þ ãñáììéêÞ áíéóüôçôá ùò ðñïò ÷ êáé ø ...............................................178 8.1.7 ÅöáñìïãÞ ôçò ãñáììéêÞò áíéóüôçôáò -Ãñáììéêüò ðñïãñáììáôéóìüò .179 8.1.9 Áðüóôáóç óçìåßïõ áðü åõèåßá.............................................................184 8.1.10 Åîßóùóç åõèåßáò óå ðïëéêÝò óõíôåôáãìÝíåò.........................................185 8.2 Ï êýêëïò ..........................................................................................................187 8.2.1 Ç åîßóùóç ôïõ êýêëïõ.......................................................................... 187 8.2.2 Åîßóùóç êýêëïõ ðïõ ðåñíÜ áðü ôñßá óçìåßá......................................190 8.2.3 Ó÷åôéêÞ èÝóç ôïõ êýêëïõ ìå Üëëç ãñáììÞ............................................192 8.2.4 Åîßóùóç åõèåßáò åöáðôüìåíçò óå óçìåßï êýêëïõ................................193 8.3 Ç Ýëëåéøç .........................................................................................................195 8.3.1 Óôïé÷åßá ôçò Ýëëåéøçò ............................................................................195 8.3.2 Ç åîßóùóç ôçò Ýëëåéøçò ........................................................................197 8.3.3 ÐáñáìåôñéêÝò åîéóþóåéò ôçò Ýëëåéøçò................................................. 200
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ÌÁÈÇÌÁÔÉÊÇ ÁÍÁËÕÓÇ
8.3.4 ÐïëéêÞ åîßóùóç ôçò Ýëëåéøçò................................................................201 8.3.5 Åîßóùóç åöáðôüìåíçò ó' Ýíá óçìåßï ôçò Ýëëåéøçò..............................202 8.4 Ç õðåñâïëÞ .....................................................................................................203 8.4.1 Óôïé÷åßá ôçò õðåñâïëÞò ........................................................................204 8.4.2 Ç åîßóùóç ôçò õðåñâïëÞò.....................................................................206 8.4.3 ÐáñáìåôñéêÝò åîéóþóåéò ôçò õðåñâïëÞò ..............................................210 8.4.4 ÐïëéêÞ åîßóùóç ôçò õðåñâïëÞò ............................................................211 8.4.5 Åîßóùóç åõèåßáò åöáðôüìåíçò óå óçìåßï õðåñâïëÞò .........................212 8.5 ÐáñáâïëÞ ........................................................................................................213 8.5.1 Óôïé÷åßá ôçò ðáñáâïëÞò .......................................................................214 8.5.2 Ç åîßóùóç ôçò ðáñáâïëÞò....................................................................215 8.5.3 Óõããåíåßò ðáñáâïëÝò.............................................................................216 8.5.4 ÐáñáìåôñéêÝò åîéóþóåéò ôçò ðáñáâïëÞò .............................................217 8.5.5 ÐïëéêÞ åîßóùóç ôçò ðáñáâïëÞò ...........................................................217 8.5.6 Åîßóùóç åõèåßáò åöáðôüìåíçò óå óçìåßï ðáñáâïëÞò ........................218 8.6 ÁóêÞóåéò...........................................................................................................219
ÊåöÜëáéï 9: ÃåíéêÜ ðåñß êáìðõëþí 2ïõ âáèìïý ....................................229 9.1 9.2 9.3 9.4 9.5 9.6
ÃåíéêÞ åîßóùóç êáìðýëçò 2ïõ âáèìïý...........................................................229 Ó÷åôéêÞ èÝóç äýï êáìðõëþí...........................................................................230 Äéåñåýíçóç ôçò ãåíéêÞò åîßóùóçò êáìðýëçò 2ïõ âáèìïý... .........................231 ÊÝíôñï êáìðýëçò ............................................................................................233 ¢îïíåò êáìðýëçò.............................................................................................233 ÁóêÞóåéò...........................................................................................................234
ÊåöÜëáéï 10: Óôïé÷åßá áðü ôç ìåëÝôç ôïõ R3 ...........................................237
10.1 ÅðéöÜíåéåò êáé êáìðýëåò ôïõ R3 ....................................................................237 10.2 Ç åõèåßá óôïí R3 .............................................................................................238 10.2.1 Åîéóþóåéò ôçò åõèåßáò............................................................................238 10.2.2 Áðüóôáóç óçìåßïõ áðü åõèåßá.............................................................242 10.3 Ôï åðßðåäï.......................................................................................................242 10.3.2 Óõíåðßðåäá óçìåßá ................................................................................247 10.3.3 Ó÷åôéêÞ èÝóç äéáíýóìáôïò êáé åðéðÝäïõ..............................................248 10.3.4 Ó÷åôéêÞ èÝóç äýï åðéðÝäùí ..................................................................249 10.3.5 Ðñïóáíáôïëéóìüò ôïõ R3 ùò ðñïò åðßðåäï.........................................251 10.3.6 Áðüóôáóç óçìåßïõ áðü åðßðåäï ..........................................................252 10.3.7 Áðüóôáóç åðéðÝäïõ áðü åðßðåäï .......................................................253 10.4 Ç óöáßñá. ........................................................................................................253 10.4.1 Åîéóþóåéò ôçò óöáßñáò ..........................................................................254 10.4.2 Åîßóùóç óöáßñáò ðïõ ðåñíÜ áðü 4 óçìåßá ........................................255 10.4.3 Ó÷åôéêÞ èÝóç åðéðÝäïõ ùò ðñïò óöáßñá .............................................256 10.4.4 Åðßðåäï åöáðôüìåíï óöáßñáò .............................................................258 10.5 Ï êýëéíäñïò .....................................................................................................259 10.5.1 ÊõëéíäñéêÞ åðéöÜíåéá.............................................................................259
ÐÅÑÉÅ×ÏÌÅÍÁ
15
10.5.2 Åîßóùóç êõëéíäñéêÞò åðéöÜíåéáò...........................................................259 10.5.3 Åßäç êõëéíäñéêþí åðéöáíåéþí ................................................................261 10.6 ÁóêÞóåéò...........................................................................................................262 ÌÅÑÏÓ ÔÑÉÔÏ • ÄÉÁÖÏÑÉÊÏÓ ËÏÃÉÓÌÏÓ
ÊåöÜëáéï 11: ¼ñéï óõíÜñôçóçò ...............................................................273 11.1 Ïñéóìüò ïñßïõ óõíÜñôçóçò ............................................................................273 11.2 ÐëåõñéêÜ üñéá..................................................................................................278 11.3 Ãåíßêåõóç ôçò óýãêëéóçò .................................................................................281 11.3.1 Óýãêëéóç (üñéï) ãéá ÷ → + ∞, f/(á, ∞) ...................................................282 11.3.2 Óýãêëéóç (üñéï) ãéá ÷ → - ∞, f/(-∞, â) ....................................................282 11.3.3 Óýãêëéóç (üñéï) ãéá → î, f/(á, â) êáé î∈(á, â) .......................................282 11.4 ÃåùìåôñéêÞ åñìçíåßá ôçò óýãêëéóçò ...............................................................285 11.5 ÈåùñÞìáôá - Éäéüôçôåò ôùí ïñßùí.................................................................288 11.5.2 Èåþñçìá: ¼ñéï óå óôáèåñÜ: ÷ = á .............................................289 11.5.3 Ðüñéóìá: ¼ñéï óôáèåñÜò:
c = c...................................................289
11.5.4 Èåþñçìá: ¼ñéï óôáèåñÜò åðß óõíÜñôçóç: (ê · f(÷)) = ê · f(÷), ê ≠ 0.........................................................289 11.5.5 Èåþñçìá: ¼ñéï ñßæáò:
á ≥ 0, ÷ ≥ 0 ..............................290
11.5.6 Èåþñçìá: Áëãåâñá ïñßùí óõíáñôÞóåùí .............................................291 11.6 Ï áñéèìüò e ùò üñéï óõíÜñôçóçò ..................................................................296 11.7 ÁóêÞóåéò...........................................................................................................298
ÊåöÜëáéï 12: ÓõíÝ÷åéá óõíÜñôçóçò .........................................................301 12.1 Ïñéóìüò óõíÝ÷åéáò óõíÜñôçóçò .....................................................................301 12.2 Åßäç áóõíÝ÷åéáò ...............................................................................................306 12.3 ÂáóéêÝò éäéüôçôåò óõíå÷þí óõíáñôÞóåùí......................................................311 12.4 ÓõíÝ÷åéá óõíÜñôçóçò óå äéÜóôçìá ................................................................312 12.4.1 ÐëåõñéêÞ óõíÝ÷åéá óõíÜñôçóçò óå óçìåßï ...........................................312 12.4.2 ÓõíÜñôçóç óõíå÷Þò óå äéÜóôçìá..........................................................313 12.5 ÁóêÞóåéò...........................................................................................................315
ÊåöÜëáéï 13: ÐáñÜãùãïò óõíÜñôçóçò....................................................319 13.1 Ïñéóìüò ðáñáãþãïõ.......................................................................................319 13.2 ÐëåõñéêÝò ðáñÜãùãïé .....................................................................................322 13.3 ÃåùìåôñéêÞ åñìçíåßá ôçò ðáñáãþãïõ............................................................323 13.4 Ó÷Ýóç ìåôáîý óõíÝ÷åéáò êáé ðáñáãþãïõ óõíÜñôçóçò .................................326 13.4.1 Èåþñçìá óõíÝ÷åéáò...............................................................................327 13.5 Êáíüíåò ðáñáãþãéóçò ....................................................................................329 13.5.1 ÐáñÜãùãïò óôáèåñÜò ...........................................................................329
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ÌÁÈÇÌÁÔÉÊÇ ÁÍÁËÕÓÇ
13.5.2 ÐáñÜãùãïò ôçò áíåîÜñôçôçò ìåôáâëçôÞò...........................................329 13.5.3 ÐáñÜãùãïò óýíèåôçò óõíÜñôçóçò.......................................................329 13.5.4 ÐáñÜãùãïò áëãåâñéêïý áèñïßóìáôïò óõíáñôÞóåùí...........................330 13.5.5 ÐáñÜãùãïò ãéíïìÝíïõ óõíáñôÞóåùí ...................................................330 13.5.6 ÐáñÜãùãïò ôïõ áíôéóôñüöïõ ìéáò óõíÜñôçóçò .................................332 13.5.7 ÐáñÜãùãïò ðçëßêïõ äýï óõíáñôÞóåùí ...............................................333 13.5.8 ÐáñÜãùãïò ëïãáñéèìéêÞò óõíÜñôçóçò.................................................335 13.5.9 ÐáñÜãùãïò åêèåôéêÞò óõíÜñôçóçò ......................................................335 13.5.10 ÐáñÜãùãïò óýíèåôçò åêèåôéêÞò óõíÜñôçóçò ......................................337 13.5.11 ÐáñÜãùãïò áíôßóôñïöçò óõíÜñôçóçò.................................................338 13.5.12 ÐáñÜãùãïé ôñéãùíïìåôñéêþí óõíáñôÞóåùí ........................................339 13.5.13 ÐáñÜãùãïé áíôßóôñïöùí ôñéãùíïìåôñéêþí óõíáñôÞóåùí.................342 13.5.14 ÐáñÜãùãïé õðåñâïëéêþí óõíáñôÞóåùí...............................................345 13.5.15 ÐáñÜãùãïò óõíÜñôçóçò ðáñáìåôñéêÞò ìïñöÞò. ................................347 13.5.16 ÐáñÜãùãïò óõíÜñôçóçò ðëåãìÝíçò ìïñöÞò.. .....................................349 13.5.17 ÐáñÜãùãïé áíþôåñçò ôÜîçò ................................................................ 350 13.5.18 Ðßíáêáò ðáñáãþãùí âáóéêþí óõíáñôÞóåùí ......................................352 13.6 ÁóêÞóåéò...........................................................................................................357
ÊåöÜëáéï 14: Ðåñß äéáöïñéêþí.................................................................363 14.1 14.2 14.3 14.4 14.5
Ïñéóìüò............................................................................................................363 ÃåùìåôñéêÞ åñìçíåßá ôïõ äéáöïñéêïý ............................................................364 Êáíüíåò äéáöüñéóçò ........................................................................................368 ÄéáöïñéêÜ áíþôåñçò ôÜîçò ............................................................................368 ÁóêÞóåéò...........................................................................................................371
ÊåöÜëáéï 15: ÅöáñìïãÝò ôïõ äéáöïñéêïý ëïãéóìïý...............................373 15.1 Õðïëïãéóìüò áðñïóäéüñéóôùí ìïñöþí (Êáíüíáò L' Hopital*)......................373 15.2 ÌåëÝôç óõíáñôÞóåùí ìå ÷ñÞóç ðáñáãþãùí................................................381 15.2.1 Èåþñçìá ôïõ Fermat* .......................................................................... 381 15.2.2 Èåþñçìá ôïõ Rolle* ..............................................................................382 15.2.3 Èåþñçìá ôïõ Lagrange* (Þ ðåðåñáóìÝíùí áõîÞóåùí).......................383 15.2.4 Èåþñçìá (Ó÷Ýóç ìïíïôïíßáò êáé ðáñáãþãïõ óõíÜñôçóçò) ...............385 15.2.5 Èåþñçìá (Ó÷Ýóç áêñüôáôùí ìå ôçí ðñþôç êáé ôç äåýôåñç ðáñÜãùãï)...............................................................................388 15.2.6 Óçìåßá êáìðÞò ...................................................................................... 389 15.2.7 ÄéÜôáîç êïßëùí êáé ôñßôç ðáñÜãùãïò..................................................390 15.2.8 Áóýìðôùôåò åõèåßåò ôçò êáìðýëçò ìéáò óõíÜñôçóçò ....................... 393 15.2.9 ÂÞìáôá ìåëÝôçò óõíÜñôçóçò ø =f(÷)...................................................397 15.3 ×ñÞóç ðáñÜãùãùí óå äéÜöïñåò åðéóôÞìåò...................................................407 15.3.1 Óôçí ÉáôñéêÞ - ÐáñÜäåéãìá....................................................................407 15.3.2 Óôéò Åðé÷åéñÞóåéò- ÐáñÜäåéãìá .............................................................407 15.3.3 Óôç ÓôáôéóôéêÞ - ÐáñÜäåéãìá.. .............................................................408 15.3.4 Óôç ÖõóéêÞ êáé óôéò Üëëåò èåôéêÝò åðéóôÞìåò......................................409