Ch¬ng 2
Thùc hµnh tÝnh to¸n trªn
Maple
2.1. tÝnh to¸n sè häc vµ ®¹i sè th«ng dông.......................................... 14
2.1.1. TÝnh to¸n víi nguyªn...................................................................... 14
TÝnh giai thõa..............................................................................................................14
T×m íc sè chung lín nhÊt (gcd)................................................................................14
T×m béi sè chung nhá nhÊt (lcm) ...............................................................................14
Ph©n tÝch mét sè ra thõa sè nguyªn tè........................................................................15
T×m c¸c sè nguyªn tè tríc vµ sau mét sè cho tríc.................................................15
T×m nghiÖm nguyªn cña ph¬ng tr×nh........................................................................15
T×m th¬ng vµ phÇn d...............................................................................................16
TÝnh theo c«ng thøc truy håi.......................................................................................17
2.1.2. TÝnh to¸n víi c¸c thËp ph©n......................................................... 20
VÒ ®é chÝnh x¸c cña c¸c phÐp tÝnh sè häc..................................................................20
TÝnh gi¸ trÞ cña biÓu thøc sè häc ................................................................................21
TÝnh tæng cña h÷u h¹n vµ cña v« h¹n c¸c sè h¹ng.....................................................21
TÝnh tÝch cña h÷u h¹n hoÆc v« h¹n thõa sè................................................................22
2.1.3. TÝnh to¸n víi phøc.......................................................................... 23
2.1.4. TÝnh to¸n theo Modul .......................................................................... 24
C¸c tÝnh to¸n Modul th«ng thêng.............................................................................24
Gi¶i ph¬ng tr×nh víi Modul......................................................................................26
2.1.5. Khai triÓn, ®¬n gi¶n ph©n tÝch biÓu thøc ®¹i sè.......................... 27
Khai triÓn biÓu thøc ®¹i sè..........................................................................................27
Ph©n tÝch ra thõa sè.....................................................................................................28
PhÐp ®¬n gi¶n biÓu thøc..............................................................................................28
Tèi gi¶n ph©n thøc......................................................................................................29
G¸n tªn cho biÓu thøc vµ g¸n trÞ cho biÕn..................................................................29
ChuyÓn ®æi d¹ng cña biÓu thøc...................................................................................30
2.1.6. §Þnh nghÜa hµm sè.............................................................................. 31
Hµm sè th«ng thêng..................................................................................................31
Hµm tõng khóc............................................................................................................31
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2.1.7. Gi¶i ph¬ng tr×nh ph¬ng tr×nh............................................... 32
Gi¶i ph¬ng tr×nh ®¹i sè (cã hÖ sè b»ng sè hoÆc b»ng ch÷)......................................32
Gi¶i ph¬ng tr×nh v« tû...............................................................................................33
Gi¶i hÖ ph¬ng tr×nh...................................................................................................33
2.1.8. Gi¶i bÊt ph¬ng tr×nh.......................................................................... 34
2.2. VÏ ®å thÞ vµ c¸c vÊn ®Ò liªn quan ..................................................... 35
2.2.1. ®å thÞ trong mÆt ph¼ng................................................................. 35
VÏ ®å thÞ hµm th«ng thêng.......................................................................................35
VÏ ®å thÞ hµm Èn.........................................................................................................37
C¸c tuú chän c¬ b¶n trong lÖnh vÏ ®å thÞ...................................................................38
Mét sè thÝ dô minh häa...............................................................................................41
2.2.2. ®å thÞ trong kh«ng gian 3 chiÒu................................................... 42
VÏ ®å thÞ hµm 2 biÕn...................................................................................................42
VÏ ®êng møc cña c¸c hµm 2 biÕn.............................................................................44
VÏ ®êng èng trong kh«ng gian 3 chiÒu...................................................................46
2.2.3. VËn ®éng cña ®å thÞ........................................................................... 46
2.3. TÝnh to¸n trong ®¹i sè tuyÕn tÝnh ................................................... 48
2.3.1. C¸c phÐp to¸n trªn vect¬ ma trËn.............................................. 48
T¹o vect¬ vµ ma trËn..................................................................................................48
So s¸nh hai ma trËn.....................................................................................................50
TÝnh tæng cña hai ma trËn (lÖnh ®¸nh gi¸ ma trËn tæng)............................................51
Nh©n ma trËn...............................................................................................................51
TÝnh tÝch trong cña ma trËn vµ vÐc t¬ (lÖnh innerprod)..............................................52
TÝnh tÝch vÐc t¬ (tÝch trùc tiÕp) b»ng lÖnh crossprod...........................................53
TÝnh tÝch v« híng cña hai vÐc t¬ (lÖnh dotprod) ......................................................53
2.3.2. TÝnh gi¸ trÞ riªng vect¬ riªng cña ma trËn.................................... 54
TÝnh ®a thøc ®Æc trng................................................................................................54
TÝnh gi¸ trÞ riªng vµ vÐc t¬ riªng cña ma trËn.............................................................54
TÝnh gi¸ trÞ riªng vµ vÐc t¬ riªng cña ma trËn sè........................................................55
2.3.3. TÝnh h¹ng, tÝnh ®Þnh thøc tÝnh ma trËn ngîc.............................. 56
T×m h¹ng cña ma trËn.................................................................................................56
TÝnh ®Þnh thøc vµ ma trËn ngîc cña ma trËn............................................................57
2.3.4. Gi¶i ph¬ng tr×nh ®¹i tuyÕn tÝnh.................................................... 57
ThiÕt lËp ma trËn tõ ph¬ng tr×nh vµ ngîc l¹i..........................................................57
Gi¶i ph¬ng tr×nh ®¹i sè tuyÕn tÝnh............................................................................59
2.3.5. T×m cho kh«ng gian vect¬....................................................... 60
T×m hä vÐc t¬ c¬ së.....................................................................................................60
T×m c¬ së cho kh«ng gian vÐc t¬ sinh bëi c¸c dßng (cét) cña ma trËn......................61
T×m c¬ së cho h¹ch cña ma trËn.................................................................................61
T×m c¬ së trùc chuÈn cña kh«ng gian sinh bëi mét hä c¸c vÐc t¬.............................61
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2.4. PhÐp tÝnh vi ph©n vµ tÝch ph©n .......................................................... 62
2.4.1. PhÐp tÝnh giíi h¹n................................................................................ 62
TÝnh giíi h¹n cña hµm sè t¹i mét ®iÓm......................................................................62
TÝnh giíi h¹n theo híng (tr¸i hoÆc ph¶i)..................................................................63
2.4.2. TÝnh ®¹o hµm cña hµm mét biÕn ................................................. 63
TÝnh ®¹o hµm bËc nhÊt................................................................................................63
C¸c vÝ dô minh ho¹.....................................................................................................64
TÝnh ®¹o hµm cÊp cao.................................................................................................65
2.4.3. TÝnh vi ph©n hµm nhiÒu biÕn, hµm vect¬ ma trËn hµm............. 66
PhÐp tÝnh ®¹o hµm cña hµm nhiÒu biÕn......................................................................66
TÝnh ®¹o hµm cña mét hµm vÐc t¬..............................................................................67
TÝnh ®¹o hµm cña mét ma trËn hµm...........................................................................67
2.4.4. Hµm Èn ®¹o hµm cña nã.............................................................. 68
Hµm Èn v« híng........................................................................................................68
Hµm Èn vÐc t¬.............................................................................................................70
2.4.5. PhÐp tÝnh tÝch ph©n............................................................................. 72
TÝnh tÝch ph©n x¸c ®Þnh..............................................................................................72
TÝnh tÝch ph©n kh«ng x¸c ®Þnh...................................................................................74
TÝnh tÝch ph©n suy réng..............................................................................................76
2.4.6. Khai triÓn hµm thµnh chuçi........................................................... 76
2.5. Ph¬ng tr×nh Vi ph©n vµ VËt lý to¸n ............................................... 78
2.5.1. Gi¶i ph¬ng tr×nh vi ph©n thêng...................................................... 78
C¸c ph¬ng tr×nh vi ph©n th«ng thêng.....................................................................78
C¸c tuú chän trong gi¶i ph¬ng tr×nh vi ph©n............................................................79
Ph¬ng tr×nh vi ph©n víi c¸c hµm ®Æc biÖt.................................................................80
2.5.2. Gi¶i ph¬ng tr×nh vi ph©n thêng ................................................ 81
Gi¶i hÖ ph¬ng tr×nh vi ph©n thêng..........................................................................81
C¸c tuú chän trong gi¶i hÖ ph¬ng tr×nh vi ph©n.......................................................82
VÏ ®å thÞ nghiÖm cña hÖ ph¬ng tr×nh vi ph©n..........................................................84
2.5.3. Gi¶i ph¬ng tr×nh ®¹o hµm riªng...................................................... 86
Gi¶i ph¬ng tr×nh ®¹o hµm riªng................................................................................86
VÏ ®å thÞ nghiÖm cña ph¬ng tr×nh ®¹o hµm riªng....................................................86
PhÇn nµy giíi thiÖu nh÷ng chñ ®Ò tÝnh to¸n th«ng dông nhÊt. Qua ®©y chóng ta
sÏ mau chãng n¾m b¾t ®îc ph¬ng thøc lµm viÖc víi Maple, ®Ó råi tù m×nh kh¸m
ph¸ vµ t×m hiÓu vÒ kh¶ n¨ng tÝnh to¸n tuyÖt vêi cña Maple. C¸c tÝnh to¸n chuyªn
ngµnh s©u h¬n sÏ ®îc ®Ò cËp trong phÇn tÝnh to¸n thùc hµnh trªn m¸y cña c¸c gi¸o
tr×nh cho tõng bé m«n.
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