Phm Minh Hong

Maple v cc bi ton ng dng

Nh xut bn Khoa Hc v K Thut

Li ni uTi cn nh cch y khng lu, mt hc sinh lp 9 t cho ti mt bi ton nh sau: Hai i cng nhn lm chung mt cng vic trong 2g24'. Nu mi i chia nhau lm na cng vic th thi gian hon tt l 5g. Hi thi gian mi i lm xong cng vic ca mnh?

Loay hoay mt lc ti mi tm ra phng trnh ca bi ton v phi kh khn lm ti mi ct ngha c cho em, ri li phi mt thm mt t thi gian mi c th t mnh tm c phng trnh ca bi ton. Nhng n khi thay s vo em li lm sai, phi i n khi em dng my tnh th mi chuyn mi xong. Bt cht ti t cu hi: tm ra c phng trnh bi ton l co nh i c 3/4 on ng, phn cn li ch l thay s, vy m em hc sinh ny li lm sai, tht ung. Nu l ngi chm im, ti c th chm chc v cho em 7/10, nhng nu ch cn c vo kt qu hoc thi bng trc nghim c th em b 0 im. Vy th r rng my tnh thay i tt c. Ngy nay, Vit Nam tt c cc k thi cp trung hc tr ln u c php dng my tnh (khng lp trnh), iu c ngha l x hi chp nhn cho cc em min lm tnh bng tay, m chng ai t vn ''mt t duy'' Ton hc c. Ln n bc i hc, cng c ny cn tr ln ti cn thit hn cho sinh vin khoa hc t nhin. Bt sinh vin tnh cc bc trong phng php Runge-Kutta khng my tnh l cc em chu thua ngay mc d tt c u hiu bi. V ln cao hn na, cc nh nghin cu cn phi hon ton da vo nhng my tnh siu mnh vi nhng phn mm thch ng h tr h trong cc bi ton phc tp. Nh th, h ''khon'' tt c nhng tnh ton tm thng cho my tnh v ht tm tr ca mnh vo nhng ch chuyn su ca h. V tr v vai tr ca my tnh ngy mt tr nn quan trng, nht l trong lnh vc gio dc. Tin hc hu nh l mt mn bt buc cho cc sinh vin ngay t bc u vo i hc, v cng ln cao, cng i su vo mt lnh vc no , con ngi bt buc phi dng n my tnh. Bc ra ngoi i, bc vo k ngh, cng nghip, vai tr ca my tnh li tr nn quan trng hn, n ni chng ta c th khng nh rng: ngy nay nu khng c my tnh, con ngi s khng lm g c c. Khng my tnh, ngy hm nay con ngi khng th xy dng nhng cy cu hin i, khng th d on c thi tit, khng th v c cc v tu, cnh my bay, khng th o rung ca mt chic tn la... vn l nhng vt dng, nhng phng tin gn gi vi chng ta. L do l my tnh c mt kh nng tnh ton v mt b nh gn nh v hn. Tuy nhin, cho d c mnh n u i chng na, tt c cc my tnh n tnh ton trn cc con s. Chng c th tnh mt triu s l ca s trong nhy mt nhng khng tai no tm 1)n ra 8 (2n+1 = . iu ny gy t nhiu kh khn cho nhng nh khoa hc vn quen vi n=0 4 cc k hiu nh x, BBx f (x, s), ln[sin(x2 + 1)] . . . nn h vn ao c c mt cng c thch ng lm vic, mt phn mm khng ch thao tc cc con s, m phi lm c iu ny trn cc k

hiu quen thuc. l mt phn mm tnh ton hnh thc1 . V phi i ti nm 1980 i hc Waterloo (Canada) mi hon tt cng trnh s ca mnh v cho ra i Maple. Maple c vit ra t mc ch . Vo nm 1867, nh thin vn Delaunay b ra 20 nm ng ng thit lp v tnh ton qu o ca mt trng di tc dng ca mt tri. Biu thc hon ton bng k t (hnh thc) ny di gn 2000 trang giy. Mt th k sau, nm 1970, nh ton hc A. Deprit ch mt 9 thng vit mt chng trnh tnh ton li2 . Ngy nay, c l chng ta ch mt khong na gi! Maple qu l tit kim cho ngi dng mt khong thi gian khng l. Nhng Maple c th cn lm nhiu hn th. Ti cn nh mt trong nhng ''kinh nghim xng mu'' ca mnh hi hc lp 12. Thy dy chng ti tnh din tch hnh trn bn knh r bng cch chia nh n ra thnh tng mnh nh v xem nh l nhng hnh ch nht ri cng din tch chng li c c kt qu l r2 . Nhng ti mi ln cn ci chuyn phi xem nh l nhng hnh ch nht. V nu ''xem nh'' th r rng c sai s, v nu cng ht cc hnh ch nht l cng ht c cc sai s th u th ra mt ci g trn tra nh r2 . Chnh ci ln cn y lm im ton ca ti st gim nghim trng. Mi n khi c c Maple ti mi nghim ra ''chn l'' ca vn khi v tht nhiu hnh ch nht thy rng r rng l n tin v din tch hnh trn. n y , ti ngh c nhiu thy (thm ch c c cc bn sinh vin) ph ci cho rng ti thuc loi ''chm tiu''. Ti ngh iu y khng sai, nhng ring ti, ti li nhn vn cch khc. Ci g lm cho mnh hiu ra vn ? cu tr li l hnh nh. Ngy xa ti khng ''tiu'' c chng qua l v thy khng sc v tht nhiu nhng hnh ch nht nh Maple. Vy ti sao chng ta khng tn dng nhng kh nng vt tri ca my tnh tit kim thi gian?. Ti ngh khng cn di dng thuyt phc v u im ca my tnh trong mt bi thuyt trnh (ch khng ring g vic hc). Mt din gi ngi c l th s khng cun hut bng chiu cng mt ni dung y ln mn hnh. M khng cun ht th kh a ni dung ''vo u'' thnh gi. c bit nu nhng ni dung y l nhng tru tng nh ton th li cng phi c th ha, sinh ng ha. Ti cn nh khi dy ton bng Maple vo mt ngy khng c my chiu. Sinh vin ngi nhn bng en m ti c ngh tm hn cc bn ang ln l ''chn bng lai'' no (v cc khi nim y cc em u hc qua). Nhng khi c my chiu, ti thy cc em ho hc mng l ra mt. Cp mt l khi ny bng sinh ng khc thng, c mi khi thay slide l khun mt cc em cng thay i theo. Ri n khi thc hin nhng bi tp ln cui hc k, rt nhiu bn lm nhiu hn nhng g ch i hi. L do l cc bn y hiu r hin tng m khng ngn ngi s dng sc mnh ca my tnh khai trin xa hn. iu l vic ''xa nay him''. Vy th r rng Maple gip ch cho vic hc ton. Trn y ti va nhc n nhng hnh ch nht xp x hnh trn. iu y nu l mt ngi c ''hoa tay'', thy ti c th v c. Nhng khi l nhng hnh trong khng gian, nhng hnh co-nic 3 chiu th khng d dng v. Ti cn nh khi dy phng php ng dc nht (steepest descent) trong mn Ti u ha hm nhiu bin, ct ngha phng php, ti c phi lin tc lm nhng ng tc mt ngi ang lao xung vc cc bn hiu ngha hnh hc ca gradient. Nhng n khi hin th bng my tnh th ti chc chn cc bn sinh vin hiu ti sao phng php steepest ascent (ngha l dc ln) m ti khng cn phi lm ng tc no khc. Maple khng ch gip bng hnh nh m cn kch thch c sng to. Chng ta dy cho hc sinh lm th no vit phng trnh mt ng thng qua hai im; vy th cc em c th1 2

Ting Anh l formal computation ting Php l calcul formel. v tm thy ch mt ch sai trong 2000 trang ca Delaunay! Phm Minh Hong

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dng Maple xc nh c ng cao, ng trung tuyn, sau xc nh c trc tm, trng tm ri vit phng trnh ng thng Euler. Tt c cc cng on y lm bng my tnh u c lm ''nht'' t duy ton hc ca cc em u, tri li n lm cho cc em c c hi s dng mt v kh sc bn ca tr tu l tr tng tng3 . Ri bc i hc, chng ta dy cho sinh vin iu kin cho ha mt ma trn M v p dng n tnh M n . Nu lm bng tay s rt ''oi'', d chn, thm ch mi ch l ma trn bc 3; nhng vi Maple, sinh vin c th ''vui chi'' v t to cho mnh nhng trng hp cc k phc tp v nh th cc bn s hiu r vn . Cc th d nh th cn rt nhiu v trong mi lnh vc nh l, ha, sinh, k thut, kin trc... R rng l n gip chng ta hc hiu qu hn. Ti thc s cha bao gi ngh rng Maple c th thay th ngi thy v nh mt lnh Maple tnh din tch hnh trn th ai cng c th lm c, thm ch l mt hc sinh cp II! nhng nu hiu c ngha hnh hc ca nguyn hm (v cc vn sau sa hn) th khng th thiu thy c. Maple ch cung cp cho chng ta mt cng c hiu r vn v khi ngun sng to m thi. Nhng li l yu t rt cn trong cuc i sinh vin k c khi ra trng. Vi tt c nhng tm tnh , ti vit cun Maple v cc bi ton ng dng ny. Sau ln xut bn th nht tc gi b i nhng ch phc tp ng thi thm mt s chng ch li hn cho vic hc Maple, trong c mt chng ni v c php dnh cho cc c gi cha c kinh nghim vi phn mm ny. Tc gi cng chn thnh xin li bn c v nhng s st mc phi trong ln pht hnh u tin. Mi kin ng gp xin chuyn v a ch: Nh xut bn Khoa Hc v K Thut, 28 ng Khi, phng Bn Ngh, qun I, TPHCM. T: 822.50.62-829.66.28 c mong ca tc gi l cun sch nh b ny s gip bn c c mt ci nhn mi v v tr v tn ca Ton hc. Si Gn Xun Mu T 2008 Phm Minh Hong email: pmhoang@hcmut.edu.vn

Vi dng v tc gi:Sinh nm 1955 ti Si Gn, u t ti v i du hc Php nm 1973, tt nghip Cao hc C Hc ng Dng ti i hc Pierre & Marie Curie(Paris VI) v i lm nhiu nm v tn hc qun l v tin hc k ngh ti Paris. Nm 2000 tr v Vit Nam v hin cng tc ti B Mn Ton ng Dng, Khoa Khoa Hc ng Dng, Trng i hc Bch Khoa TPHCM.

Kin thc khng quan trng bng tr tng tng. Kin thc th gii hn nhng tr tng tng c th vy quanh nhn loi (Albert Einstein) Phm Minh Hong

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Mc lcTrang Li ni u Chng 1. C php Maple 1.1 Tng quan . . . . . . . . . . . . . 1.2 Cc thao tc trn mt biu thc . . 1.2.1 Lnh simplify: n gin . 1.2.2 Lnh expand: khai trin . 1.2.3 Lnh factor: tha s . . . 1.2.4 Lnh combine: gom . . . 1.2.5 Lnh convert: bin i . . 1.3 Mnh v hm mi tn . . . . . 1.4 Cc thao tc trn mt dy . . . . . 1.5 Gii tch . . . . . . . . . . . . . . 1.6 th hai chiu . . . . . . . . . . 1.7 Gii phng trnh . . . . . . . . . 1.7.1 Phng trnh i s . . . . 1.7.2 Phng trnh quy np . . . 1.8 Phng trnh vi phn . . . . . . . 1.8.1 Cch gii gii tch . . . . 1.8.2 Cch gii s . . . . . . . . 1.9 i s tuyn tnh . . . . . . . . . 1.10 Lp trnh trong Maple . . . . . . . 1.10.1 Khai thc sau khi bin dch 1.11 Nguyn hm . . . . . . . . . . . . 1.12 Bi tp . . . . . . . . . . . . . . . 1.12.1 Cc lnh c bn . . . . . . 1.12.2 i s . . . . . . . . . . . 1.12.3 Phng trnh vi phn . . . 1.12.4 Nguyn hm . . . . . . . 1.12.5 Lp trnh . . . . . . . . . 1.13 Bi c thm: Thals . . . . . . . i 1 1 3 3 5 5 6 7 8 8 10 11 13 13 14 14 14 16 18 20 21 23 26 26 26 27 27 27 32

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Mc lc

Chng 2. Bi ton cc tr 2.1 Tit kim nhm . . . . . . . . . . . . 2.2 on ng gn nht . . . . . . . . . 2.3 Gc nhn ca phi hnh gia . . . . . . 2.4 Hnh nn v hnh cu . . . . . . . . . 2.4.1 Tnh bng th tch . . . . . . 2.4.2 Tnh bng din tch . . . . . . 2.5 Khc cua gt . . . . . . . . . . . . . 2.5.1 Vn 1 . . . . . . . . . . . 2.5.2 Vn 2 . . . . . . . . . . . 2.6 Ellipsoid . . . . . . . . . . . . . . . . 2.6.1 Hnh v cho bi ton Ellipsoid 2.7 Cc tr ca hm hai bin: Th d 2 . . 2.8 Cc tr ca hm ba bin . . . . . . . . 2.9 Bi c thm: Pythagore . . . . . . . Chng 3. th ba chiu 3.1 Th d 1 . . . . . . . . 3.2 Th d 2 . . . . . . . . 3.3 Th d 3 . . . . . . . . 3.4 Th d 4 . . . . . . . . 3.5 Bi c thm: Euclide

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33 33 34 36 37 37 40 42 43 44 47 48 48 50 53 54 54 57 60 63 67 69 69 71 73 74 77 77 81 81 81 84 85 88 89 89 90 90 v

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Chng 4. Hnh hc gii tch 4.1 Tm v v tip tuyn chung ca hai vng trn 4.2 Din tch phn giao ca hai vng trn . . . . 4.3 Qu tch 1 . . . . . . . . . . . . . . . . . . . 4.4 Qu tch 2 . . . . . . . . . . . . . . . . . . . 4.5 Qu tch 3 . . . . . . . . . . . . . . . . . . . 4.5.1 Cch gii th nht . . . . . . . . . . 4.5.2 Cch gii th hai . . . . . . . . . . . 4.6 Gii hn ca Maple . . . . . . . . . . . . . . 4.6.1 Mt th d thun hnh thc . . . . . . 4.6.2 Mt khc mc... . . . . . . . . . . . 4.6.3 Mt th d in hnh . . . . . . . . . 4.7 Bi c thm: Archimde . . . . . . . . . . .

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Chng 5. Bi ton m phng 5.1 Cnh tranh tay i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Gii bng hm t hp . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Gi bng ma trn . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Phm Minh Hong

Mc lc

5.2 5.3

5.4 5.5 5.6 5.7

5.8

5.1.3 Gii bng hm quy np rsolve 5.1.4 Biu din trong khng gian 3-D Kinh t ASEAN . . . . . . . . . . . . . Li sut ngn hng . . . . . . . . . . . 5.3.1 Lp trnh . . . . . . . . . . . . 5.3.2 Hm s hp . . . . . . . . . . . 5.3.3 Dy truy hi (quy np) . . . . . 5.3.4 Phng trnh vi phn . . . . . . Nui tm . . . . . . . . . . . . . . . . Bn khuy nc u . . . . . . . . . . Bi ton cn bng mi sinh . . . . . . . S.A.R.S . . . . . . . . . . . . . . . . . 5.7.1 Gii vi cc i lng ri rc . 5.7.2 Gii vi cc i lng lin tc . Bi c thm: Eratosthene . . . . . . .

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91 91 92 93 93 94 94 95 97 102 103 105 105 108 112 115 115 115 116 117 118 118 119 120 120 121 122 122 122 125 127 127 128 129 129 131 131 137 137

Chng 6. Bi ton kch thc hnh xoay 6.1 Din tch, th tch ellipse v ellipsoid . . . . . . . . . . . . . . 6.1.1 Din tch mt ellipse . . . . . . . . . . . . . . . . . . 6.1.2 Th tch mt ellipsoid . . . . . . . . . . . . . . . . . 6.1.3 Din tch mt ellipsoid . . . . . . . . . . . . . . . . . 6.2 Th tch sinh ra bi php quay quanh trc Ox ca mt hm . . 6.3 Th tch sinh ra bi php quay quanh trc Oy ca mt hm . . 6.4 Trng hp mt hm ni suy . . . . . . . . . . . . . . . . . . 6.5 Tm th tch sinh ra bi php quay ca phn giao ca hai hm . 6.5.1 Xoay quanh Ox . . . . . . . . . . . . . . . . . . . . . 6.5.2 Xoay quanh Oy . . . . . . . . . . . . . . . . . . . . . 6.6 Mt trng hp phc tp . . . . . . . . . . . . . . . . . . . . 6.6.1 Xoay quanh Ox . . . . . . . . . . . . . . . . . . . . . 6.6.2 Xoay quanh Oy . . . . . . . . . . . . . . . . . . . . . 6.7 Bi c thm: Galile . . . . . . . . . . . . . . . . . . . . . . Chng 7. Bi ton sc bn vt liu 7.1 Ti trng u . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Hai u gi n . . . . . . . . . . . . . . . . . . . . . 7.1.2 Ngm mt u, u kia t do . . . . . . . . . . . . . . 7.1.3 Ngm hai u . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Ngm mt u, u kia gi n (h siu tnh) . . . . . 7.1.5 Ngm mt u, u kia gi n mt im bt k u . 7.2 Ti trng tp trung . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Ngm mt u, lc tp trung u kia. [Hnh 7.14 (a)] vi

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Phm Minh Hong

Mc lc

7.3

7.2.2 Ngm mt u, lc tp trung x = u l [Hnh 7.14 (b)] . . . . . . . . 138 7.2.3 Hai gi n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Bi c thm: Kpler - Thi Dng h . . . . . . . . . . . . . . . . . . . . . . 142 145 145 148 149 149 151 154 156 157 165 168 168 170 171 174 177 180 183 183 184 185 188 189 191 193 195 198 198 199 204 209 209 209 210 212 vii

Chng 8. Bi ton n o 8.1 Mi trng khng c ma st khng kh . . . . . 8.2 Mi trng c ma st khng kh . . . . . . . . 8.2.1 Th d 1: nghim gii tch . . . . . . . 8.2.2 Th d 2 : nghim bng phng php s 8.2.3 Tm gc bn xa nht . . . . . . . . . . 8.2.4 Ni di tm bn . . . . . . . . . . . . . 8.2.5 Sc cn trong trng hp phc tp . . . 8.2.6 Dng Do C th k XXI! . . . . . . . 8.3 Bi c thm: Shwerer Gustav . . . . . . . . .

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Chng 9. Bi ton dao ng 1: L xo 9.1 L xo nm ngang . . . . . . . . . . . . . . . . . . . . 9.1.1 Trng hp 1 : Khng c lc gim xc, = 0 9.1.2 Trng hp 2 : Lc gim xc, 0 . . . . . 9.1.3 Kho st hin tng cng hng . . . . . . . . 9.2 H ba l xo nm ngang . . . . . . . . . . . . . . . . . 9.3 Bi c thm: Cu Tacoma . . . . . . . . . . . . . . .

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Chng 10.Bi ton dao ng 2: Con lc ton hc 10.1 Con lc n . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Trng hp 1: gc quay nh . . . . . . . . . . . 10.1.2 Trng hp 2: gc quay ln khng ma st . . . . 10.1.3 Trng hp 3: gc quay ln vi ma st . . . . . 10.2 Con lc kp . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Trng hp 1: gc quay nh, tnh ton hnh thc 10.2.2 Trng hp 2: gc quay ln - Tnh ton s . . . 10.2.3 Kim chng . . . . . . . . . . . . . . . . . . . . 10.3 Con lc n n hi . . . . . . . . . . . . . . . . . . . . 10.3.1 V hnh . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Tnh ton . . . . . . . . . . . . . . . . . . . . . 10.4 Bi c thm: Lch s s . . . . . . . . . . . . . . . . Chng 11.S hc v ng dng 11.1 Tm tt l thuyt . . . . . . . . . . . . . 11.1.1 S hc m-un . . . . . . . . . . 11.1.2 Php chia Eculide trong Z /mZ . 11.1.3 ng dng ca php tnh ng d .Phm Minh Hong

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Mc lc

11.1.4 nh l Trung Quc . . . . . . . . . . . . 11.2 Mt m . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 M Csar . . . . . . . . . . . . . . . . . . 11.2.2 M Khi . . . . . . . . . . . . . . . . . . 11.2.3 M RSA . . . . . . . . . . . . . . . . . . . 11.3 Bi c thm B kha RSA: Con ng chng gai Chng 12.X l hnh ng 12.1 Chuyn ng n gin . . . . . . . . . . 12.1.1 Th d . . . . . . . . . . . . . . . 12.1.2 Th d 2 . . . . . . . . . . . . . . 12.1.3 Th d 3 . . . . . . . . . . . . . . 12.2 Chuyn ng phc tp . . . . . . . . . . 12.2.1 Th d 1 . . . . . . . . . . . . . . 12.2.2 Th d 2 . . . . . . . . . . . . . . 12.3 Chuyn ng c s thay i vn tc . . . 12.3.1 Thay i u . . . . . . . . . . . 12.3.2 Thay i khng u - Th d 1 . . 12.3.3 Thay i khng u - Th d 2 . . 12.4 Chuyn ng vi mt hay nhiu hnh tnh 12.4.1 Th d 1 . . . . . . . . . . . . . . 12.4.2 Th d 2 . . . . . . . . . . . . . . 12.5 ng lp ln bi hnh ng . . . . . . . 12.5.1 Vin bi ln theo ng thng . . . 12.5.2 Vin bi ln theo mt ng bt k 12.5.3 Cycloid . . . . . . . . . . . . . . 12.5.4 im ng hc . . . . . . . . . . 12.6 Bi c thm . . . . . . . . . . . . . . . Ti liu tham kho Ch mc

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213 216 217 219 222 229 231 231 231 232 233 234 234 238 240 240 241 242 245 245 246 248 248 250 252 253 256 278 280

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viii

Phm Minh Hong

Danh mc hnh minh haHnh 1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 4.1 4.2 (a) Ba li gii v (b) khi v chung vi tp hp cc li gii Li gii phng trnh vi phn phng php gii tch . . . Li gii phng trnh vi phn v phng php s . . . . S to v s dng tp tin thc thi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trang . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 16 18 22 34 35 36 38 39 41 43 44 45 47 49 49 50 55 56 58 59 60 61 62 63 65 67 70 71

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Hnh nn ni tip (b) ng biu din ca th tch theo h khi R = 3. . . (a) hnh nn ngoi tip (b) ng biu din ca th tch theo h khi R = 3. . th ca din tch theo h khi R = 3: (a) trng hp ni tip; (b) ngoi tip Hnh nn ni tip v ngoi tip hnh trn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hnh hp ni tip trong mt ellipsoid . . . . . . . . . . . . . . . . . . . . . Hnh khi cc i trong mt ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ng ng mc ca hm f (x, y) . . . . . . . . . . . . . . . . . . . Cc tr hm nhiu bin v hnh chiu ca n . . . . . . . . . . . . . . Cc im dng ca f (x, y) . . . . . . . . . . . . . . . . . . . . . . . im cc i v cc tiu ca f (x, y) . . . . . . . . . . . . . . . . . . im yn nga ca hm f (x, y) . . . . . . . . . . . . . . . . . . . . th gradient, ng ng mc v chuyn ng ca Pk . . . . . . . Chuyn ng Pk trong khng gian . . . . . . . . . . . . . . . . . . . Biu din tham s ca hm rng buc g(x, y) trn f (x, y) . . . . . . . (a) ng ng mc v ellips 2D, (b) Vc-t gradient ti im cc tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Danh mc hnh minh ha

4.3 . . . . . . . . . . . . . . . . . . . . . . . . 4.4 . . . . . . . . . . . . . . . . . . . . . . . . 4.5 . . . . . . . . . . . . . . . . . . . . . . . . 4.6 . . . . . . . . . . . . . . . . . . . . . . . . 4.7 (a), (b) V tr tng i ca H; (c) Qu tch H 4.8 Qu tch ca H vi cc v tr M . . . . . . . . 4.9 Vng trn trc giao . . . . . . . . . . . . . . 4.10 . . . . . . . . . . . . . . . . . . . . . . . . 4.11 . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

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74 75 77 78 79 80 83 84 87 91 92 95 96 98 101 102 104 105 107

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thng v vi th 3-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sai bit gia php gii ri rc v lin tc . . . . . . . . . . . . . . . . . . (a), (b) Pht trin n nh sau 30 thng v (c) pht trin khng n nh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 ng biu din ca lng mun (a) trng hp v (b) trng hp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ly lan ca bnh dch khi khng c v khi c thuc cha . . . . . . . . . 1 1 5.11 Ly lan ca bnh dch vi b = v b = . . . . . . . . . . . . . . . . 10 2 5.12 Ly lan ca bnh dch trng hp c) v d) . . . . . . . . . . . . . . . . . 5.13 Thut ton "Sng Eratosthene" v cch o chu vi tri t. . . . . . . . . . 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

. . . 109 . . . 110 . . . 114 116 119 120 121 122 123 124 125 127 128 130 131 132 133 134

ellipse v ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 th y = f (x) = x3 v hnh xoay quanh Ox . . . . . . . . . . . . . . . . . 2 1 (y) v hnh xoay quanh Oy . . . . . . . . . . . . . . . . . . . . th x = f th mt hm ni suy v hnh xoay quanh Ox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phn giao ca hai hm v hnh xoay quanh Ox, Oy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phn giao ca hai hm trong mt trng hp phc tp v (b),(c) cch v tnh th tch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chuyn v, gc quay v moment trng hp hai gi n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chuyn v, gc quay v moment trng hp ngm mt u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chuyn v, gc quay v moment trng hp ngm hai u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1 7.2 7.3 7.4 7.5 7.6 7.7 x

Phm Minh Hong

Danh mc hnh minh ha

7.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Chuyn v, gc quay v moment trng hp ngm mt u, u kia gi n . . 7.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Chuyn v, moment ngm mt u v gi n : (a) x = 8 v (b) x = 7 (cch gii th nht) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.13 Chuyn v, moment trng hp ngm mt u v gi n : (a) x = 8 v (b) x = 7 (cch gii th hai) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.14 Cc trng hp ti trng tp trung vi ngm . . . . . . . . . . . . . . . . . . . 7.15 Chuyn v, gc quay v moment trng hp ngm mt u, lu tp trung khi: (a) u = 4 v (b) u = 6 (cc t l c sa i d nhn) . . . . . . . . . . . . 7.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.17 Lc tp trung, hai ni n qua hai cch gii . . . . . . . . . . . . . . . . . . . 7.18 Thi Dng H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qu o trong trng hp khng ma st vi: (a) v0 = 300 v (b) v0 = 900m/s 1 n o vi h s ma st bng : (a) k = 1 (ti a) v (b) k = . . . . . . . . 10 [ ] 7 n o 5 gc bn P , vi ma st . . . . . . . . . . . . . . . . . . . 10 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ni di tm bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a), (b): Cc hm sc cn p(h) v (c) tm bn tng ng vi = . . . . . . 4 n a ca 6 gc bn vi gia tng . . . . . . . . . . . . . . . . . . . . 50 Cc hm ni suy ftd(d), fdt(d) v fad(t) . . . . . . . . . . . . . . . . . . . . . . Cc hm ni suy ftd(t), fad(a) v fda(x) . . . . . . . . . . . . . . . . . . . . . n o ca 6 gc bn vi gia tng . . . . . . . . . . . . . . . . . . . . 60 Cc hm ni suy spline ca fdt(t), fad(a), fda(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hnh 8.14: i bc Schwerer Gustav (hnh mu trng by) . . . . . . . . . . . L xo v khi m trn trc honh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chuyn ng vi gim xc = df rac120 v . . . . . . . . . . . . . . . . . 11 Chuyn ng vi gim xc ln 0 v (b) gim xc tn hn ( = 0) . . Chuyn ng vi nh hng ngoi lc . . . . . . . . . . . . . . . . . . . . . . Chuyn ng khi ngoi lc (a) cng vn tc gc v (b) khng cng vn tc gc H ba l xo trc v sau khi chuyn ng . . . . . . . . . . . . . . . . . . . . Chuyn ng ca h thng . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135 136 137 137 138 138 139 139 140 141 143 145 147 151 153 154 155 157 158 160 161 162 163 164 165 166 169 170 172 173 174 176 177 179 xi

Phm Minh Hong

Danh mc hnh minh ha

9.9 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

Cu Tacoma lc sp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 185 187 189 190 194 195 196

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Con lc n vi gc quay nh . . . . . . . . . . . . . . . Con lc n vi gc quay ln . . . . . . . . . . . . . . . . Con lc n vi gc quay ln v lc ma st . . . . . . . . Con lc kp . . . . . . . . . . . . . . . . . . . . . . . . . Chuyn ng cng chiu vi gc quay nh . . . . . . . . . Chuyn ng ngc chiu vi gc quay nh . . . . . . . . Chuyn ng vi u1 (0) = u2 (0) = 1 radian . . . . . . . . 10.9 Chuyn ng vi u1 (0) = , u2 (0) = 1radian . . . . . . 2 10.10 th ca ng nng, th nng v c nng ca con lc kp 10.11Con lc n n hi . . . . . . . . . . . . . . . . . . . . . 10.12Khai bo con lc n hi v vi chuyn ng . . . . . . . 10.13(a) Qu o con lc v (b) th nng lng . . . . . . . . 10.14Con lc kp n hi . . . . . . . . . . . . . . . . . . . . . 10.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.17Tukey . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . 197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 198 200 201 203 204 205 206 232 233 234 235 236 237 238 240 241 243 244

12.1 Hnh tnh ca hm sin(x) v 4 chuyn ng khc nhau . . . . . . . . . . 12.2 (a) Chong chng v tr u, (b) sau khi quay 30o v (c) 4 chuyn ng 5o 12.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 (a) Chuyn ng tnh tin ca bnh xe v (b) chuyn ng quay ca van . 12.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Chuyn ng ca van xe trong 3 vng quay . . . . . . . . . . . . . . . . 12.7 4 chuyn ng vi khong cch thi gian (a) u v (b) khng u . . . . 12.8 (a) C hai xe ngng cng lc v (b) ln lt ngng . . . . . . . . . . . . 12.9 (a) Chuyn ng vi thay i u v (b) thay i khng u . . . . . . . 12.10Chuyn ng theo nh lut Kpler . . . . . . . . . . . . . . . . . . . . . 12.11Chuyn ng ca mt trng quanh tri t theo nh lut Kpler . . . . . x 2 12.12Tip tuyn ca hm f (x) = ex sin( ) . . . . . . . . . . . . . . . . . . 2 k 1 12.13S hi t ca n=1 [ cos(x)n cos(nx)] . . . . . . . . . . . . . . . . . . n 12.14Chuyn ng thng ca vin bi . . . . . . . . . . . . . . . . . . . . . . . x sin x , (b)f (x) v (c)f ( ) nhn ln 30 ln . . . . . . 12.15 th ca hm (a), x 3 12.16Chuyn ng ca vin bi (a) trc v (b) sau khi chnh vn tc . . . . . . 12.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.19Biu din ca vn tc v gia tc hnh (a) cardiod v (b) hnh c sn . . . xii

. . . 246 . . . 247 . . . 249 . . . 250 . . . . . . . . . . . . 251 252 254 256

Phm Minh Hong

Danh mc hnh minh ha

12.20Babylone . . . . . . . . . . 12.21Pythagore . . . . . . . . . . 12.22Thales . . . . . . . . . . . . 12.23Hippocrates . . . . . . . . . 12.24Euclide . . . . . . . . . . . 12.25Aristote . . . . . . . . . . . 12.26Archimede . . . . . . . . . . 12.27Eratosthene . . . . . . . . . 12.28Apollonius . . . . . . . . . . 12.29Ptoleme . . . . . . . . . . . 12.30Liu Hui . . . . . . . . . . . 12.31Diophante . . . . . . . . . . 12.32H Thp Phn . . . . . . . . 12.33Abu-bin-Musa-al-Khwarizmi 12.34Fibonacci . . . . . . . . . . 12.35Qin Jinshao . . . . . . . . . 12.36Nicolas . . . . . . . . . . . 12.37Copernic . . . . . . . . . . . 12.38Vite . . . . . . . . . . . . . 12.39Kepler . . . . . . . . . . . . 12.40Neper . . . . . . . . . . . . 12.41Cavalieri . . . . . . . . . . . 12.42Descartes . . . . . . . . . . 12.43Desargues . . . . . . . . . . 12.44Pascal . . . . . . . . . . . . 12.45Fermat . . . . . . . . . . . . 12.46Huygens . . . . . . . . . . . 12.47Leibniz . . . . . . . . . . . 12.48Seki Kowa . . . . . . . . . . 12.49Isaac Newton . . . . . . . . 12.50Jacques Bernoulli . . . . . . 12.51Rolle . . . . . . . . . . . . . 12.52Jean Bernoulli . . . . . . . . 12.53De Moivre . . . . . . . . . . 12.54Jacapo Riccati . . . . . . . . 12.55Euler . . . . . . . . . . . . . 12.56Simpson . . . . . . . . . . . 12.57D'Alembert . . . . . . . . . 12.58Lagrange . . . . . . . . . . 12.59Monge . . . . . . . . . . . .Phm Minh Hong

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256 257 257 257 257 258 258 258 258 259 259 259 259 260 260 260 261 261 261 261 262 262 262 262 263 263 263 263 264 264 264 265 265 265 265 266 266 266 266 267 xiii

Danh mc hnh minh ha

12.60Legendre . . . . . . . . . . 12.61Gauss . . . . . . . . . . . . 12.62Fourier . . . . . . . . . . . . 12.63Poisson . . . . . . . . . . . 12.64Laplace . . . . . . . . . . . 12.65Bolzano . . . . . . . . . . . 12.66Navier . . . . . . . . . . . . 12.67Green . . . . . . . . . . . . 12.68Galois . . . . . . . . . . . . 12.69Lobachevsky . . . . . . . . 12.70Cauchy . . . . . . . . . . . 12.71Dirichlet . . . . . . . . . . . 12.72Jacobi . . . . . . . . . . . . 12.73Cayley . . . . . . . . . . . . 12.74Boole . . . . . . . . . . . . 12.75Chebyshev . . . . . . . . . . 12.76Sylvester . . . . . . . . . . 12.77Venn . . . . . . . . . . . . . 12.78Poincar . . . . . . . . . . . 12.79Frobenius . . . . . . . . . . 12.80Lyapunov . . . . . . . . . . 12.81RungeKutta . . . . . . . . . 12.82Carmichael . . . . . . . . . 12.83Borel . . . . . . . . . . . . 12.84Richardson . . . . . . . . . 12.85Turing . . . . . . . . . . . . 12.86George Dantzig . . . . . . . 12.87Shannon . . . . . . . . . . . 12.88Schwartz . . . . . . . . . . 12.89Hall . . . . . . . . . . . . . 12.90Edward Lorenz . . . . . . . 12.91Tukey . . . . . . . . . . . . 12.92Mandelbrot . . . . . . . . . 12.93Adleman, Rivest, and Shamir 12.94Wiles . . . . . . . . . . . .

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267 267 267 268 268 268 268 269 269 269 269 270 270 270 270 271 271 271 271 272 272 272 272 273 273 273 273 274 274 274 274 275 275 275 275

xiv

Phm Minh Hong

Danh mc bng biuBng biu 1.1 1.2 1.3 2.1 5.1 5.2 5.3 5.4 5.5 5.6 7.1 8.1 9.1 Trang 21 22 23 42 100 106 107 110 111 113 Sp xp mt dy s thc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S dng worksheet vi tp tin thc thi l pgm1.m . . . . . . . . . . . . . . . . S dng worksheet vi tp tin thc thi l pgm2.m . . . . . . . . . . . . . . . . Kt qu theo V v theo S ca hnh nn ni tip hnh cu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Thi Dng h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Chng trnh ca 5 gc bn vi gia tng . . . . . . . . . . . . . . . . . 152 30

Cc lnh to hnh ng cho h ba l xo c mt u t do . . . . . . . . . . . . 180

10.1 Cc lnh to hnh tnh chuyn ng con lc n. . . . . . . . . . . . . . . . 186 10.2 Cc lnh to hnh tnh chuyn ng con lc n vi ma st. . . . . . . . . . 188 10.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 11.1 11.2 11.3 11.4 11.5 11.6 Bng hon chuyn mu t s . . . . . . . . . . . . . . . . . . . . . . Chng trnh m Csar . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kt qu m khi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chng trnh m khi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chng trnh m RSA v php bnh phng lin tip . . . . . . . . . . . . Chng trnh m RSA c k tn v lin kt vi php bnh phng lin tip. . . . . . . . . . . . . 217 218 220 221 224 228

12.1 Chng trnh cine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 12.2 Lch s cc k hiu Ton hc . . . . . . . . . . . . . . . . . . . . . . . 277

xv

Chng1C php MapleChng ny tm tt mt s lnh Maple c bn[1 ] v c dng nhiu trong cun sch ny. c thm chi tit cch hay nht vn l tham kho phn tr gip.

1.1 Tng quanKhi khi ng maple chng ta s c mt mn hnh n gin: trn cng chng ta c mt menu vi nhng chc nng quen thuc ca mt phn mm Windows: File,Edit,View,Insert... Cch s dng nhng chc nng ny cng kh d dng. Phn ln nht ca mn hnh l mt trang trng, l ni ngi s dng nh cc lnh Maple v nhn kt qu. Mt lnh Maple c nh sau du ">" v mc nh c nt ch courier mu , mt kt qu c mu xanh v nt ch times. Th d:> p:=x+3;p := x + 3

Trc khi vo tng cu lnh Maple, mt vi quy tc chung cn nh:X Lnh u tin l restart (khng bt buc), xo sch b nh v chun b cho nhng iu kin lm vic tt nht cho Maple.> restart:

X Maple phn bit ch thng v ch hoa: th d simplify khc vi Simplify. Trong Maple i a s cc cu lnh u l ch thng nhng c mt s rt t c c ch thng ln ch hoa (v d nhin chc nng cng khc). Th d expand v Exphand, thm ch c nhng option ton vit bng ch in. X Trong Maple, gn gi tr vo mt bin phi dng du := Nu ta nh du =, Maple s khng thng bo sai. Th d:> x:=Pi;x :=

> y=sin(x);1

Phin bn Maple c dng trong sch l phin bn 8

Chng 1. C php Maple y=0

Maple hin th y = 0 v n lp li nhng g ta nh nhng trong bin y vn cn trng (ngi ta gi bin y l bin t do, trong khi x c gn cho gi tr , x c gi l bin rng buc). Kim chng:> x,y;, y

X gii phng mt bin rng buc, (ta ly th d trn, x ang rng buc v n bng ):> x:='x'; > x;x

X Mt lnh ca Maple c chm dt bng du (;) hoc du hai chm (:). Nu chm phy, kt qu s hin ra ; nu l hai chm, lnh s c thc hin nhng kt qu khng hin ra (xem th d trn). X Du %: y l mt k hiu quan trng trong Maple. Du % biu tng cho kt qu va thc hin. Th d khi ta ly nguyn hm ca 3sin(x):> int(3*sin(x),x);

y, % biu tng cho 3 cos(x). Nu ly o hm ca n ta s tm li c 3 sin(x). Trong trng hp ny ta s dng du %:> diff(%,x);3 sin(x)

3 cos(x)

X Mapple cho php nh nhiu lnh trn mt dng. Ly li th d trn:> int(3*sin(x),x): diff(%,x)3 sin(x)

Dng trn gm 2 lnh. Lnh nguyn hm 3 sin(x)d(x) chm dt bng du hai chm, kt qu (-3cos(x)) khng c hin th nhng n gn vo bin %. Lnh o hm chm dt bng du chm phy, kt qu c hin ra.X Maple cho php kt hp nhiu lnh vo mt lnh:> diff(int(3*sin(x),x),x);3 sin(x)

X Hy quan st lnh sau:> Int(3*sin(x),x):=value(%);

Lnh Int (vi ch I c vit hoa) s cho ra kt qu l k hiu nguyn hm 3 sin(x)d(x), nhng v lnh ny chm dt bng du hai chm nn k hiu ny n v c gn vo bin %. 2Phm Minh Hong

1.2. Cc thao tc trn mt biu thc

Lnh tip theo bt u bng du %= c ngha l

3 sin(x)d(x) =, tip theo value(%)

c tc dng tnh gi tr ca bin %. V v lnh ny chm dt bng du chm phy nn kt qu ca n s c in ra:

3 sin(x)d(x) = 3 cos(x)

y % biu tng cho kt qu ca c dng trn. c c phn bn tri du bng, ta dng hm lhs v bn phi bng rhs[2 ]:> lhs(%); rhs(%%);[3 ]

3 sin(x)d(x), 3 cos(x)

X Mt vi quy tc cn nh khi khai bo cc hm ton hc: sin(x)2 ch khng phi sin2 (x), tan(x) ch khng phi tg(x). Hm m ca x l exp(x) ch khng phi l ex . K hiu l Pi, s phc l I... X Dng thuyt minh: nh dng thuyt minh (c th dng font ting Vit), nhp chut vo ni mun nh thuyt minh sau nhp vo nt T nm di hng menu. Du > s bin mt v tt c nhng g bn nh s mang mu en v u l nhng dng ch khng c bin dch bi Maple. Sau khi hon tt, dng chut hoc mi tn (trn bn phm) ra khi dng thuyt minh, nhp vo nt > tr li vi cc lnh Maple. X Lu vo cng: Tt c cc cu lnh Maple v kt qu c gi l mt worksheet v c lu li di 2 dng: dng c (MWS) v dng mi (MW, k t phin bn 9). Trong phm vi cun sch ny, chng ta ch lm vic vi dng MWS. X Maple c trn 1500 lnh (phin bn 8), trong c nhng lnh t c dung. trnh phi nhp tt c cc lnh vo RAM ca my mt cch v ch, ngi ta gom nhng lnh c cng mt ng dng vo nhng package (tm dch l gi). Nhng gi thng gp l plots,linalg,geometry,plottools... Khi cn s dng, dng hm with nhp: > with(plots): Ta cng c th dng lnh m khng cn nhp package bng cch nh (th d lnh display trong gi plots): > plots[display](...);

1.2 Cc thao tc trn mt biu thc Lnh simplify: n ginn gin mt biu thc (expression) i s. y c th l mt a thc, mt biu thc lng gic, logarithm, hm m, hm hu t...> p:=1/(a*(a-b)*(a-c))+1/(b*(b-a)*(b-c))+1/(c*(c-a)*(c-b)): > %=simplify(%);right hand side Hm rhs c dng vi 2 du %, v sau khi thc hin lnh lhs(%);,bin % tr thnh.phi thm mt du % th hai c c ng gi tr bn phi du bng.3 2

Phm Minh Hong

3

Chng 1. C php Maple 1 1 1 1 + + = a(a b)(a c) b(b a)(b c) c(c a)(c b) cab > 2*cos(x)3+sin(x)*sin(2*x):=simplify(%); 2(cos(x))3 + sin(x) sin(2x) = cos(2x)

Tuy nhin c nhng trng hp khng "sun s" nh ta ngh:> sqrt(x2):%=simplify(%);

> (x-2)3/(x2-4*x+4):%=simplify(%); (x 2)3 =x2 x2 4x + 4

?

csgn(x) l hm cho ra 1 nu Re(x) 0[4 ],v cho -1 nu Re(x) 0. Ti sao? Trc khi tr li, chng ta ng bao gi qun rng y l mi trng tnh ton hnh thc, iu c ngha l Maple phi lm vic trn cc k t ch khng phi nhng con s. Khi ta vit k t x, trong u mnh ngh ngay n mt con s (thm ch l s dng). Nhng Maple khng ngh nh th, n s hiu y l mt bin bt k, c th l mt s phc, mt ma trn, hay mt th...V trong tt c nhng "gi thuyt" ny, s phc l hp l hn c v cu tr li ca Maple (csgn(x)x) l hon ton chnh xc. kim chng, ta c th "bo" cho Maple rng x l mt s thc bng cch dng hm assume:> assume(x,real): > sqrt(x2):%=simplify(%);x2

x2 = csgn(x)x

?

= |x

|

Du c thm khi mt bin c assume. Tuy nhin k t by gi d c chng ta s b qua v khng hin th k t ny. V khi x 0:> assume(x,real): > sqrt(x2):%=simplify(%);x2 = x

?

Ni tm li, trc khi p dng mt quy tc n gin, Maple phi xt n bn cht ca bin. Nhng trc khi i vo cc th d, chng ta s gii phng x tr li tnh trng ban u ng thi trnh lp i lp li ch simplify, ta s vit tt thnh Sp qua lnh alias:> s:='x': alias(Sp=Simplify): p:=(x-2)3/ sqrt(x2-4*x+4) > p:%=Sp(%) 3 ? (x 2) = (x 2)2csgn(x 2) x2 4x + 4

Maple n gin t v mu cho (x 2) nhng vn t du hi v du ca x 2. V khi x 2 mi chuyn s tt p.> assume (x>2): p:%=Sp(%); 3 ? (x 2) = (x 2)2 x2 4x + 4 > x:='x':p:=ln(x3) - 2*ln(x): p=Sp(p); ln(x3 ) 2 ln(x) = ln(x3 ) 2 ln(x) > assume (x>0): 'ln(x3)-2*ln(x)'=Sp(p);[5 ]4 5

Tng t vi hm ln:

Trong Maple, phn thc ca x k hiu l Re(x), phn phc l Im(x) Du nhy n trnh Maple t ng n gin khi x>0 Phm Minh Hong

4

1.2. Cc thao tc trn mt biu thc

ln(x3 ) 2 ln(x) = ln(x)

Ta c th n gin hn bng cch:> Sp(p,assume=positive;

Tuy nhin, Maple c option symbolic cho php n gin m khng cn assume nhng vn bo ton cc quy tc ton hc thng thng:> x:='x': (x-2)3/sqrt(x2-4*x+4):%=Sp(%,Symbolic); 3 ? (x 2) = (x 2)2 x2 4x + 4 > ln(x3)-2*ln(x):%=Sp(%,Symbolic); ln(x3 ) 2 ln(x) = ln(x)

Lnh expand: khai trinHm expand c mc ch:X Khai trin a thc (ph du ngoc, hiu theo ngha rng) X Biu din cc hm lng gic theo nx thnh hm theo x> sqrt((4+sqrt(3))*(4-sqrt(3))):%=expand(%);b

(4 +

?

3(4

?

3)) =

?

13

> cos(3*x):%=expand(%);

cos 3x = cos(x)3 3 cos(x)

> (a+b)/(a-b):%=expand(%); a+b a b = + ab ab ab Trong trng hp phn s hu t, expand khng khai trin mu s, phi dng hm normal, expanded: > (x+1)/((x+3)*(x+2)):%=expand(%); x+1 x 1 = + (x + 3)(x + 2) (x + 3)(x + 2) (x + 3)(x + 2) > expand(normal(lhs(%),expanded)); x 1 + x2 + 5x + 6 x2 + 5x + 6 Tri vi simplify, expand khng c symbolic: > ln(a*b):%=expand(%,symbolic); ln(ab) = ln(ab) > assume(a>0, b>0): ln(a*b):%=expand(%); ln(ab) = ln(a) + ln(b)

Lnh factor: tha sLnh ny c xem nh ngc li vi lnh expand> p:=x4-3*x2+2: p=factor(p);x4 32 + 2 = (x 1)(x + 1)(x2 2)

> factor(rhs(%))=expand(lhs(%));Phm Minh Hong

5

Chng 1. C php Maple x 1)(x + 1)(x2 2) = x4 32 + 2

Qua kt qu trn, ta thy a thc x2 2 khng c khai trin trong Q . Ngi ta ni a thc ny ti gin trong Q :> irreduc(x2-2);true

Hy quan st cc lnh sau:> factor(x2-4);(x 2)(x + 2) x2 2 (x

> factor(x2-2); > factor(x2-2,sqrt(2));2)(x + 2)

?

?

> factor(x2+2,sqrt(2),I);(x I 2)(x + I 2)

?

?

Qua 4 lnh ny ta c th kt lun l hm factor ch tha s ho mt a thc trong trng hp cc s hu t. Khi nghim khng thuc Q , phi khai bo phn t ny. V khi c trn mt phn t khai bapf, phi tt c trong ngoc nhn(tu). Trong trng hp cc a thc bc cao ( 2) thng phi dng lnh solve (gii phng trnh) tm nghim trc khi tha s ho:> solve(x3+9);

32/3, 1/232/3 3/2I

?6

3, 1/232/3 + 3/2I 6 3

?

C tt c ba nghim, trong c hai nghim phc v u l v t (32/3 , vi lnh sau:> factor(x3+9,3(1/6),I); ? ? 1 (2x 32/3 3I 6 3)(2x 32/3 + 3I 6 3)(x + 32/3 ) 4

?6

3). Chng ta th

Qua kt qu ny chng ta thy Maple rt "thng minh". Ch c 6 3 c khai bo nhng trong qu trnh tha s ho, Maple cng tm ra c?2/3 . Mt chi tit quan trng cn lu l 3 ? 6 6 nghim ca a thc l 3/2 3 nhng ta ch khai bo 3 v phi b qua tha s 3/2:> factor(x3+9,3/2*3(1/6),I);Error, (in factor) 2nd argument, 3/2*3(1/6), I, is not a valid algebraric extension

?

Lnh combine: gom"Gom" nhng hm ton ging nhau (exp,ln, hm lng gic...) v bin i tch cc hm lng gic thnh tng (ngc li vi expand).> exp(x)*exp(y):%=combine(%);ex ey = ex+y

> sqrt(x)*sqrt(y):%=combine(%,sqrt,symbolic);

?x?y = ?xyxa )b = xab

> (xa)b:%=combine(%, power, symbolic);

6

Phm Minh Hong

1.2. Cc thao tc trn mt biu thc

> ln(x)+ln(y):%=combine(%, ln, symbolic);[6 ] ln(x) + ln(y) = ln(xy) > sin(x)*cos(y):%=combine(%);[7 ] 1 1 sin(x) cos(y) = sin(x + y) + sin(x y) 2 2 > Int(f,x)+Int(g,x):%=combine(%,Int);[8 ]

f dx + gdx =

(f + g)dx

Lu l khi dng vi symbolic, cn thit phi xc nh hm ton hc no cn phi gom.

Lnh convert: bin iBin i mt biu thc ton hc sang dng khc. y l lnh phc tp nht(gn 100 cch bin i khc nhau):> convert(12,binary);1100

> exp(I*x):%=convert(%,trig);eIx = cos(x) + I sin(x)

> arcsinh(x):%=convert(%, ln);arcsinh(x) = ln(x + [[1, 2], [3, 4]] x2 + 1)

?

> A:=matrix([[1,2],[3,4]]): convert(A,listlist);

Mt trong nhng chc nng quan trng ca hm convert l phn tch mt phn s hu t thnh tng cc phn t n gin:> 1/(x2-4):%=convert(%,parfrac,x); 1 1 1 = + 24 x 4(x + 2) 4(x 2)

Ging nh trng hp factor, nu a thc c nghim v t hoc phc, cn phi khai bo ? 1 cc nghim ny. Gi s mun phn tch a thc p = x2 + 8, phi tha s ho vi I 2 trc p khi phn tch:> p:=x2+8: > factor(1/p,I,sqrt(2)):%=convert(%,parfrac,x); I I ? 1 ? = ? ? (x 2I 2)(x + 2I 2) 8(x + 2I 2) 8(x 2I 2) convert cng c dng tnh tng hoc tch cc phn t trong mt dy: > convert([1,2,3],`+`);6

6 7

Ngc vi Simplify Ging factor, ngc vi expand 8 Ngc vi expand Phm Minh Hong

7

Chng 1. C php Maple

1.3 Mnh v hm mi tnX Cho biu thc ton hc:> p:=x2-3*x+2;p := x2 3x + 2

bit gi tr ti im x = 2, dng hm subs. Trong Maple, p c gi l mt mnh expression:> subs(x=2,p);0

X Cho hm s f (x) = sin(x), gi tr ti x = c vit nh mt k hiu quen thuc: f ( ). 3 3 Trong Maple, f c gi l mt hm (function hay hm mi tn):> f:=xsin(x): f(Pi/3);

?

3 2

X bin mt mnh sang mt hm mi tn:> g:=unapply(p,x): g(2);0

y l mt lnh quan trng v n cho php khai bo mt hm (mi tn) t mt kt qu phc tp.

1.4 Cc thao tc trn mt dyX Mt dy c th c khai bo bng cch khai bo tng phn t hoc bng hm seq. N c th c hoc khng c du ngoc vung:> p:=seq(2*i+1,i=0..8); q:=[sin(-Pi/3),6,-2,Pi2];p := 1, 3, 5, 7, 9, 11, 13, 15, 17 [ ? ] 3 2 2 , 6, 2,

X Cc phn t ca dy v s phn t ca dy:> p[3],q[4];5, 2

> nops([p]),nops([q]);9, 4

Lu cch s dng khi dy khai bo vi ngoc vung hoc khng ngoc vung.X Khai bo dy bt k bng hm rand(random):> k:=rand(-15..15): > X:=seq(k(),i=1..20); Y:=[seq(k(),i=1..17)];

8

Phm Minh Hong

1.4. Cc thao tc trn mt dy X := 8, 14, 11, 0, 14, 14, 6, 9, 4, 2, 2, 10, 0, 13, 15, 14, 8, , 5, 5, 6 Y := [13, 15, 12, 13, 2, 9, 9, 7, 6, 7, 6, 10, 5, 11, 9, 14, 9]

X Tm cc phn t xi

4 v chia chn cho 3, cc s nguyn t trong Y :

> select(i i>4 and i mod 3 = 0, [X]); select(i isprime(i),Y);[15, 6] [13, 7]

X Ghp hai kt qu ny vo mt dy s. Lnh op dng b ngoc vung:> s:=op(%),op(%%);s := 13, 7, 15, 6

X Khi dy c s v t phi dng lnh is (dng li dy q trn):> select(i>0,q);Error,selecting function must return true or false

> select(iis(i>0),q);[6, 2 ]

X Lnh map: y l mt lnh quan trng. Lnh ny p dng mt hm (c th hm Maple hoc hm mi tn) cho tt c cc phn t trong mt dy.> map(ii2,[s]);[169, 49, 225, 36]

> map(ii-3*I,[s]);[13 3I, 7 3I, 15 3I, 6 3I]

> u:=[seq(sin(i*x),i=1..3)];u := [sin(x), sin(2x), sin(3x)]

> map(diff,u,x);[cos(x), 2 cos(2x), 3 cos(3x)]

X option has:> p:=x=sin(t),y=cos(t)2,z=1+cos(u);p := x = sin(t), y = cos(t)2 , z = 1 + cos(u)

Tm nhng phn t trong p khng c cos(t):> remove(has,[p],cos(t));[x = sin(t), z = 1 + cos(u)]Phm Minh Hong

9

Chng 1. C php Maple

1.5 Gii tchX Hm mt bin

Cc lnh tnh gii hn, o hm, nguyn hm...:> p:=exp(-x2): Limit(p,x=infinity):%=value(%);x

lim ex = 0

8

2

> Diff(p,x):%=diff(%);d x 2 2 e = 2xex dx

Trong lnh trn hm diff(p,x) dng ly o hm mnh p theo x. Hm Diff (vi ch D hoa), c gi l dng tnh (inert form) ca hm diff(p,x). N ch c tc dng d vit k hiu p. dx d2 2 Lnh di y tnh 2 p (du nhy c dng hin th ch p thay v ex ) dx> 'Diff('p;,x,x)':%=value(%);d2 2 2 p = 2xex + 4x2 ex dx2

Tnh o hm bc n: diff(p,xn) (di y ta kt hp vi ton t mi tn):> dp:=ndiff(p,xn): > factor(dp(5));

8xex (15 20x2 + 4x4)2

Tnh nguyn hm: Int (vi ch I hoa), l dng tnh ca hm int(p,x):> Int(p,x):%=value(%);

ex dx =2

1? erf (x) 2

Trong trng hp hm mi tn, o hm bc nht v bc 2:> f:=xsin(x2): D(f);x 2 cos(x2 )x

> f2:=(D@@2)(f);f 2 := x 2 cos(x2 ) 4sin(x2 )x2

> f2(sqrt(Pi/2));

2Trong trng hp nguyn hm ca mt mi tn, phi bin i sang dng mnh (f (x) l mnh tng ng vi hm mi tn f ):> Int(f(x),x):%=value(%);

sin(nx)dx =

cos(nx) nPhm Minh Hong

10

1.6. th hai chiu

X Hm nhiu bin> p:=x3+x*y2+y3;p := x3 + xy 2 + y 3

> 'Diff('p,y,y')':%=value(%);d2 p = 2x + 6y dy 2 > f:=unapply(p,x,y);

Theo trn, f l mt hm theo (x, y). Cc cng thc tnh o hm:

Bf l D[1](f), B2f l (D@@2)[2](f), B2f l D[1,2](f) Bx B2x B xB y> D[1](f),(D[2]@@2)(f),D[2,1](f);(x, y) 3x2 + y 2 , (x, y) 2x + 6y, (x, y) 2y

V thy c sc mnh ca Maple, hy quan st bi ton Pavelle sau y:> p:=sin(n*z*sqrt(x2+y2+z2)/sqrt(y2+z2))sqrt(x2+y2+z2); ( a ) nz x2 + y 2 + z 2 a sin y2 + z2 a p= x2 + y 2 + z 2 > 'Diff('p,x')':%=value(%); ( a ) ( a ) nz x2 + y 2 + z 2 nz x2 + y 2 + z 2 a a cos nzx sin x y2 + z2 y2 + z2 d a p= (x2 + y2 + z2)3/2 dx (x2 + y 2 + z 2 ) y 2 + z 2

Kt qu trn cc k phc tp, nhng y mi ch l o hm bc nht. Trong trng hp bn thc s rnh ri, hy chng minh (bng tay) rng:[

B2 ( B2 + B2 + B2 ) + n2 ( B2 + B2 )] p(x, y, z) = 0 Bx2 Bx2 By2 Bz2 Bx2 By2

Cn nu bn khng c thi gi, cu lnh tng ng l:> Diff('Diff('p,x,x')+Diff('p,y,y')+Diff('p',z,z),x2')+ n2*('Diff('p,x2')+Diff('p,y2')'):%=simplify(value(%),symbolic);

1.6 th hai chiu[9 ]X To Descarter

Gi p, q l nhng mnh , f, g l nhng hm mi tn (theo x)> plot(p,x=-3..3): plot([p,q],x=-3..3,-2..6): > plot(f,-4..4,1..3) > plot([f,g],-1..3,thickness=[2,5],color=[blue,green]):9

Tt c cc th trong mc ny khng c in ra.

Phm Minh Hong

11

Chng 1. C php Maple

X Ta cc> plot(1,u=0..2*Pi,coords=polar); > with(plots): > polarplot(1,u=0..2*Pi); polarplot(u,u=0..4*Pi); > polarplot([theta,4*sin(3*theta)], theta=0..2*Pi,color=[red,blue]);

X Hm n> p:=x3+y3-5*x*y+3;p := x3 + y 3 5xy + 3

> implicitplot(p,x=-3..3,y=-3..3,grid=[100,100]); > p1:=x2+y2=9: p2:=x2/25+y2/9=1: > implicitplot(p1.p2,x=-5..5,y=-3..3,scaling=constrained);

X Hm tham s> plot([sint(t),cos(t),t=0..2*Pi]); > plot([[sint(t),cos(t),t=0..2*Pi],[sin(2*t),cos(3*t),t=0..2*Pi]], color=[red,blue]);

X Hm ni im> k:=rand(-20..20); > p:=seq([i,k()],i=1..40); plot([p]);

X th cc hm khc nhau

Cch d nht l gn cc th vo mt bin, ri sau dng display v chung. Khi gn nh chm dt bng du (:).> a:=plot([sin(t),x-1],t=-Pi..Pi,color=tan): > b:=polarplot([sin(t),t]t=-Pi..Pi,color=navy): > c:=implicitplot(x2+y2-1,x=-3..3,y=-3..3,grid=[100,100]): > display(a,b,c,scaling=constrained);

X Hnh ng> plot([sin(x),sin(x-1)],x=0..4*Pi); > plot([seq(sin(x-i),i=1..10)],x=0..5*Pi); > f:=isin(x-i): > seq(plot(f(i),x=0..5*Pi),i=1..10): > display(%,insequence=true); > display(seq(plot(f(i/5),x=0..5*Pi)i=1..50),insequence=true);

12

Phm Minh Hong

1.7. Gii phng trnh

1.7 Gii phng trnh Phng trnh i sX Phng trnh n> p:=x3+x2-2;p := x3 + x2 2

> solve(p,x);1, 1 + I, 1 I

Tuy nhin khng phi lc no kt qu cng n gin. Phng trnh sau nghim gii tch cc k phc tp, Maple ch xut di dng RootOf:> p:=x4-4*x3+1;p := x4 4x3 + 1

> s:=solve(p);s := RootOf ( Z 4 4 Z 3 + 1, index = 1), RootOf ( Z 4 4 Z 3 + 1, index = 2)

RootOf ( Z 4 4 Z 3 + 1, index = 3), RootOf ( Z 4 4 Z 3 + 1, index = 4)

thy gi tr dng l (float), dng evalf ( y hin th 5 s l):> evalf(%,5);.66963, 3.9842, .32691 + .51764I, .32691 .51764I

c dng tng minh gii thch, dng hm allvalues (khng hin th v di)> allvalues([s]);

Nu ch mun ly thc nghim dng phng php gii gn ng:> fsolve(p);0.6696315467, 3.984188231[10 ]

Mt lnh khc l isolate c tc dng ging solve, nhng c thm v bn tri. Vic ny rt tin li khi tip sau l lnh subs:> p:=x-x*ln(x2);p := x x ln(x2 )

> isolate(p,x): subs(%,p);x=

?e, 0

X H phng trnh (phi thm ngoc nhn khi khai bo hoc khi gii bng solve).> p:=x-y=2), x*y=4;p := x y = 2, xy = 4

> solve({p}): (dng RootOf, khng hin th) > s:=allvalues(%);i khi lnh fsolve khng cho ra ht cc nghim thc, lc phi xc nh thm (bng th) khong cch ly nghim. Th d fsolve(f,x=a..b) s cho 1 nghim ca f trong khong [a,b] Phm Minh Hong10

13

Chng 1. C php Maple s := x = 1 +

?

5, y = 1 +

?

5, y = 1

?

5, x = 1

?

5

trch ra gi tr ca x, y c th dng subs hay eval:> eval(x,s[1]),subs(s[2],y);1+

?

5, 1

?

5

Phng trnh quy npX Th d 1. Dy tuyn tnh> eq:=u(n)=-3*u(n-1)-2*u(n-2);eq := u(n) = 3u(n 1) 2u(n 2)

> rsolve({eq,u(0)=1,u(1)=1},u);(1)n (2)n

> S:=unapply(%,n): seq(S(i),i=0..10);0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023

X Th d 2. Dy Fibonacci> eq:=f(n)=f(n-1)+f(n-2);eq := f (n) = f (n 1) + f (n 2)

> rsolve({eq,f(1..2)=1},f); ? ( ? )n ? ( ? )n 5 1 5 + 55 1 25 5 2 2 2 > g:=unapply(%,n): seq(simplify(g(i)),i=1..15);1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610

X Th d 3. Dy quy np cho> s:=u(n+1)=u(n)-2*v(n), v(n+1)=u(n)+4*v(n);s := u(n + 1) = u(n) 2v(n), v(n + 1) = u(n) + 4v(n)

> p:=rsolve(s,u(0)=-1,v(0)=-1,u,v);p := u(n) = 33n 42n , v(n) = 33n + 22n

> su:=unapply(eval(u(n),p),n): seq(su(i),i=1..6);1, 11, 49, 179, 601, 1931

1.8 Phng trnh vi phn Cch gii gii tchX Phng trnh bc nht> p:=diff(y(x),x)=-k*y(x); d y(x) = ky(x) dx

14

Phm Minh Hong

1.8. Phng trnh vi phn

> dsolve({p,y(0)=1}); (vi iu kin u, du ngoc nhn bt buc)y(x) = ekx

Chng ta cng c th gii khng iu kin u. V trnh lp i lp li cc k hiu o hm, chng ta s phi khai bo cc k hiu ny nh sau:> y0=y(x): y1=diff(y(x),x): y2:=diff(y(1),x): CB:='color=blue': > eq:=y1-y02+3*y0; dsolve(eq); (lnh dsolve ngoc nhn khng bt buc)d y(x) y(x)2 + 3y(x) dx 3 y(x) = (1 + 3e3x C) eq :=

Ba th ca li gii tng ng vi C1 = -1,0,1:> f:=unapply(rhs(%), C1): > plot([seq(f(i),i=-1..1)],x=-2..2,y=-2..5,CB); (Hnh 1.1(a))

Maple c th v tp hp li gii di dng cc mi tn (khng hin th):> with(DEtools): > gr1:=dfieldplot(eq,y(x),x=-3..2,y=-3..5,CB): display(%):

th ba li gii v chung vi tp hp cc li gii:> gr2:=plot([f(-8),f(3),f(10)],x=-3..2,CB): > display(gr1,gr2); (Hnh 1.1(b))

Hnh 1.1: (a) Ba li gii v (b) khi v chung vi tp hp cc li gii

X Phng trnh bc hai> eq:=y2+5*y1+6*y0;eq :=Phm Minh Hong

d2 d y(x) + 5 y(x) + 6y(x) 2 dx dx

15

Chng 1. C php Maple

iu kin u: k hiu o hm bt buc l ton t D thay v diff:> dsolve({eq,y(0)=0,D(y)(0)=1});y(x) = e3x + e2x

X H phng trnh bc nht> eq:= diff(x(t),t)=-4*y(t)+5, diff(y(t),t)=x(t)-2; d d eq := x(t) = 4y(t) + 5 y(t) x(t) 2 dt dt dsolve trong trng hp ny bt buc phi c ngoc nhn: > s:=dsolve({eq,x(0)=1,y(0)=-1})"

s := x(t) =

9 9 1 5 sin(2t) cos(2t) + 2, y(t) = cos(2t) sin(2t) + 2 4 2 4

*

ng biu din x(t) theo y(t): (Hnh 1.2(a))> plot([eval(x(t),s),subs(s,y(t))],t=0..2*Pi,linestyle=[1,4]);

ng biu din y(t) theo x(t): (Hnh 1.2(b))> plot([eval(x(t),s),subs(s,y(t)),t=0..2*Pi],color=black);

Hnh 1.2: Li gii phng trnh vi phn phng php gii tch

Cch gii sS dng khi cch gii gii tch khng cho ra kt qu hoc kt qu khng khai thc c.X Phng trnh bc nht> restart: with(plots): [11 ] > eq:=diff(y(x),x)-x2*y(x)=1;11

Th mc plots cn thit cho cc lnh display, odeplot c dng di y Phm Minh Hong

16

1.8. Phng trnh vi phn d y(x) x2 y(x) = 1 dx

> dsolve({eq,y(1)=2}): (Khng hin th v qu phc tp) > s:=dsolve({eq,y(1)=2},y(x),numeric);s := proc(x rkf 45) . . . end proc

Mc nh Maple dng phng php Runge-Kutta 4 nt gii phng trnh vi phn [12 ]. Kt qu cch gii s l mt dy nhiu phn t. hin th mt phn t (th d honh x=1.54)> s(1.54);[x = 1.54, y(x) = 5.76389774048564440]

hiu r ngha ca li gii s s, trc tin ta to t s mt dy u gm 5 cp im, mi im cch nhau mt khong cch bng 1 v ni nhng im y li vi nhau ( th g1), sau dng hm odeplot v li gii ca bi ton ( th g2):> f:=t[subs(s(t),x), eval(y(x),s(t))]; > u:=seq(f(i),i=-3..1.2);u := [3., 0.1217], [2., 0.3483], [1., 0.4176], [0., 0.5090], [1., 2.]

> g1:=plot([[u],[u]], style=[point,line], symbol=diamond): > g2:=odeplot(s,[x,y(x)],-3..1.1): display(g1,g2);

Hnh 1.3(a) cho thy hai th khng trng nhau, n gin laf v chia qu ln. Lnh sau chia mn hn 3 ln, ta s thy kt qu tt hn:> seq(f(i/3),i=-9..4.5):

X Phng trnh bc hai

Phng trnh dao ng ca mt l xo cng K, h s ma st B:> eq:=diff(x(t),t,t)+B*diff(x(t),t)+K*sin(x(t));eq := d2 d x(t) + B + K sin(x(t)) 2 dt dt

Php gii gii tch khng cho kt qu. gii bng phng php s, iu quan trng nht l phi gn tr s cho tt c cc bin:> B:=1/4: K:=2: > s:=dsolve({eq,x(0)=1,D(x)(0)=0},numeric): s(.13);[t = 0.13, x(t) = 0.9859, d x(t) = 0.2146] dt

th ca bin d x theo thi gian t l:> odeplot(s,[t,x(t)],0..15,numpoints=200); (Hnh 1.3(b))

Tng t, th ca vn tc x1 theo thi gian t l:> odeplot(s,[t,diff(x(t),t)],0..15,numpoints=200);12

Xem chng trnh Phng php tnh, Gii tch s

Phm Minh Hong

17

Chng 1. C php Maple

Hnh 1.3: Li gii phng trnh vi phn v phng php s

1.9 i s tuyn tnhNhng iu cn nh l: Nhp th vin: lnhlinalg hin th ma trn: lnh eval tnh ton ma trn: lnh evalmX Nhp th vin v khai bo ma trn bng 2 cch:> with(linalg): > A:=matrix(3,3,[-2,2,3,3,7,8,10,-4,-3]): > B:=matrix([[2,4,-1],[3,6,-3],[2,3,9]]): A:=eval(A),B:=eval(B); [ [ ] 2 2 3 ] 2 4 1 7 8 , B = 3 6 3 A= 3 10 4 3 2 3 9 > A+B=evalm(A+B), AB=evalm(A&*B); ] ] [ [ 8 13 23 0 6 2 6 13 5 , AB = 43 78 48 A+B = 2 7 25 12 1 6 > inverse(A):%=1/90*map(ii*90,eval(%)); (ma trn nghch o) 11 1 1 90 [ 18 11 6 5 ] 89 15 5 4 = 1 89 24 25 90 15 41 2 18 90 82 12 20 2 45 15 9 > diag(a,b,c), band([x,y,z],4); (ma trn ng cho, ma trn bng) [a 0 0 ] y z 0 0 y z 0 0 b 0 , x x y z 0 0 0 c 0 0 x y

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Phm Minh Hong

1.9. i s tuyn tnh

X Tch mt ma trn v mt vector> v:=vector([1,2,3]): evalm(A&*v);[11, 41, 7]

> evalm(A&*[a,b,c]);[2a + 2b + 3c, 3a + 7b + 8c, 10a 4b 3c]

X a thc c trng, tr ring v nh l Cayley-Hamilton:> cp:=charpoly(A,x); s:=eigenvals(A);cp := x3 2x2 33x + 90 s := 3, 5, 6

> subs(x=A,cp):%=evalm(%); ] [ 0 0 0 3 2A2 33A + 90 = 0 0 0 A 0 0 0

X Vc-t ring v ma trn chuyn c s P :> v:=eigenvects(A);v := [5, 1, t[1, 13/2, 2]u], [3, 1, t[1, 49/4, 13, 2]u], [6, 1, t[1, 1, 2]u]

Kt qu l mt dy ba phn, mi phn gm ba phn t. Phn t th nht l tr ring, th hai l s bi, th ba l vc-t ring tng ng. T ba vc-t ny c th xy dng ma trn chuyn v:> P:=concat(op(v[1,3]),op(v[2,3]),op(v[3,3])); 1 1 1 13 49 1 P := 2 4 13 2 2 2

(1.9.1)

Ta c P JP 1 = A vi J l ma trn ng cho:> evalm(P*diag(5,3,-6)inverse(P));

Tuy nhin, ta c th tnh ma trn ng cho J v ma trn chuyn c s bng lnh jordan:> J:=jordan(A,'P'): J:=eval(J),P=eval(P); 71 8 4 11 99 [ 9 6 0 0] 71 49 0 3 1 , P := J := 4 52 99 11 0 0 5 142 26 16 99 9 11

(1.9.2)

Lu : P ti (1.1) khc vi P ti (1.2) v hai ma trn ng cho tng ng khng ging nhau. Ti (1.1), ma trn ng cho l diag(5,3,-6), ti (1.1) l (-6,5,3).X Khai bo ma trn bng hm mi tn v kch thc ng.$ &

Cho ma trn M bc n m cc phn t l: mij =

%

0 1

> B:=matrix(4,4),(i,j)if i=j then 0 elif i>j then -1 else 1 fi);Phm Minh Hong

(i = j) (i j) 1 (j i)

19

Chng 1. C php Maple

Khai bo nh trn, kch thc ca ma trn lun l 4. Cch hay nht l to mt hm mi tn vi tham s l n> C:=nmatrix(n,n,(i,j)if i=j then 0 elif i>j then -1 else 1 fi);

Mt th d khc:> E:=nmatrix(n,n,(i,j)(n-1)*(i-1)+j): C(4),E(4); 1 2 3 4 0 1 1 1 7 0 1 1 C(4) = 1 1 0 1 , E(4) = 4 5 6 10 7 8 9 1 10 11 12 13 1 1 1 0

1.10 Lp trnh trong MapleLp trnh Maple rt n gin, ch cn bit c php v nh vi chi tit:X Lnh u tin l tn chng trnh :=proc (tham s). Lnh sau cng l end:. xung hng nhn Shift+Enter X Khng c sa bin u vo. Gi s tham s u vo l u v a l mt bin bt k, ta c th vit u:=a. Trong trng hp ny Maple s xut ra mt thng bo sai:Error, (in pgm) illegal use of a formal parameter

(1.10.1)

X Kt qu ca lnh sau cng trc end: (khc vi lnh print) c th in ra v c th c gn vo mt bin. X Ch c hai lnh cn nh l lnh {if ... fi} v {for ... do ... od} hay {for ... while do ... od}. fi vit tt ca end if, od vit tt ca end do

Chng trnh sau c tn rsort c cng dng sp xp mt dy s thc[13 ]> p:=ln(3),-Pi,exp(-sqrt(2)),sin(Pi/9),-1; ? ln(3), , e 2 , sin( ), 1 9 > q:=rsort([p]);10 q := [, 1, e

?

2 , sin( ), ln(3)]

9

dng 3 nu b lnh u:=u1: v lm vic trc tip trn u1, s nhn c thng bo 1.3. L do l v sa bin u vo u1 dng 10, lnh print(k) xut ra kt qu 10 [14 ] v lnh u: xut ra dy c sp xp. Nhng ch c dy u mi c th c gn v bin q. Kim chng: q[5] = ln(3).13 Trong Maple c lnh sort, nhng lnh ny ch c tc dng trn cc dy c phn t hu t. Cc s dng c thm vo d ct ngha. 14 k = 10 tng ng vi s ln lp = 5 4/2

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Phm Minh Hong

1.10. Lp trnh trong Maple

Mc ch ca lp trnh trc tin l gom cc dng lnh vo trong mt thut ton, nhng cn mt tin ch khc l sau khi bin dch (compiler), chng ta c th lu li trong mt f ile dng v sau m khng cn bin dch li. F ile ny c dnagj nh phn, cn c gi l tp tin thc thi (executable) f ile. Trong Maple, tp tin ny c ui l m. Lnh to ra tp tin ny l:> save xxx,"C:/DIR/pgm.m":

Trong xxx l tn ca thut ton, pgm l tn ca tp tin thc thi. Hai tn ny khng nht thit ging nhau. C:/DIR l ng dn th mc cha tp tin thc thi. Sau khi lu li, ngi ta c th gi ra thi hnh bng lnh:> read "C:/DIR/pgm.m":

Phm Minh Hong

> auto:proc(m) local M,s,mp,lu,,lx,ly,f,i: M:=pmatrix(4,4,[0,0,0,100,0.11,0,0,0,0,0.2,0,0,0,0,0.3,0.4]): s:=ieigenvals(M(i)): mv:=pmax(op(evalf(select(iIm(i)=0,[s(p)])))); for i to 20 do if mv(i/50)1 then break: fi: od: lu:=seq([j/50.,mv(j/50.)],j=i-2..i+3); lx:=seq(lu[i,1],i=1..nops([lu])): ly:=seq(lu[i,2],i=1..nops([lu])): f:=interp([lx],[ly],x); opselect(iis(Im(i)=0 and i0),[solve(f-1)])); end:

Bng 1.1: Sp xp mt dy s thc

Khai thc sau khi bin dch

Tt c c tm tt trong s sau: Trong :X ms l tn ca thut ton (cn gi l chng trnh con) X pgm.m l tn ca tp tin thc thi ca thut ton ms X work.mws l tn ca worksheet to ra thut ton ms X test.mws l tn ca worksheet gi thut ton ms

Th d: Cho ma trn p:> p:=matrix(2,2,[1/2,2/3,3/4,4/5]); 1 2 p := 2 3 3 4 4 5

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Chng 1. C php Maple

.> . ms:=proc(n) > . restart: > . read "C : /DIR/pgm.m" : e . nd: > . save ms, "C : /DIR/pgm.m" : w . ork.mws > . ms(n); t . est.mws

Hnh 1.4: S to v s dng tp tin thc thi

Cc phn t p u l nhng phn s nn cch vit ny kh "tn ch". Di y chng ta s thc hin mt chng trnh c tn msimply t mu s chung. msimply c vit trong worksheetmsimply.mws v tp tin thc thi tng ng l pgm1.m:> msimply:=proc(m) local p: linalg[coldim](m):linalg[rowdim](m); p:=ilcm(seq(seq(denom(m[i,j]),j=1..%%),i=1..%)); evalm(m*p)/p; end: > seve msimply,"D:/My path/pgm1.m": msimply.mws

Bng 1.2: S dng worksheet vi tp tin thc thi l pgm1.m

(Lnh ilcm tnh bi s chung nh nht ca mt dy s, denom(x) l mu s ca phn s x)

Sau khi to pgm1.m trong mt worksheet mi ta lm:> restart: read "D:/My path/pgm1.m": > p:=matrix(2,2,[1/2,2/3,3/4,4/5]); > eval(p)=msimply(p); 1 2 1 [30 40] p := 2 3 = 3 4 60 45 48 4 5

By gi chng ta s vit mt thut ton khc c tn mjor ly tha mt ma trn M bng cng thc M n = P Dn P 1 (D l ma trn ng cho v P l ma trn chuyn c s ).mjor c hai tham s: ma trn v bc ly tha: Sau khi to pgm2.m trong mt worksheet mi ta s ln lt khai bo mt ma trn m, gi hai thut ton msimply v mjor dng mt lc tnh v thu gn m3 :> restart:

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Phm Minh Hong

1.11. Nguyn hm

> mjor:=proc(m,n) linalg[jordan](m,'p'): map(iin,%): simplify(evalm(p&*%&*linalg[inverse](p))): end: > seve mjor,"D:/My path/pgm2.m": mjor.mws

Bng 1.3: S dng worksheet vi tp tin thc thi l pgm2.m

> m:=matrix(2,2,[2,1,-1/3,3/2]); > read "D:/My path/pgm2.m": read "D:/My path/pgm1.m": > eval(m)3:%=msimply(mjor(m,3)); [ ] 1 2 3 1 [ 444 642] p := 1 3 = 72 214 123 3 2

Lu : Th d trn ch c mc ch nu cch to v s dng tp tin thc thi. y chng ta c th tnh m10 d dng, nhng vi mt ma trn vung bc 3 c phn s, dng cc thut ton trn l khng kh thi v kt qu qu phc tp.

1.11 Nguyn hmMaple ly c hu nh mi nguyn hm v tch phn:> Int(x/(x3+1),x):%=value(%); ? (? ) 1 3 3 1 x 2 x + 1) + dx = ln(x arctan (2x 1) ln(x + 1) 3+1 x 6 3 3 3

Kim chng bng o hm:> diff(rhs(%),x):%=normal(%,expanded); 2x 1 1 + (3x 1 3) = x3 x 1 2 x + 1) 2 x + 1) 6(x 2(x + + > Int(x2*sin(x),x=0..Pi):%=value(%);

x2 sin(x)dx = 2 4

0

Ngc li, c nhng dng khng th ly c nguyn hm di dng gii tch hoc kt qu l nhng hm siu vit (transcendant). Trong trng hp bt buc phi ly gi tr gn ng:> Int(sin(x)/x,x):%=value(%); sin(x) dx = Si(x) x > Int(sin(x)/x,x=0..1):%=evalf(%,5);1

sin(x) dx = 0.94608 x

0

> Int(1/ln(x2-1),x):%=value(%);Phm Minh Hong

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Chng 1. C php Maple

1

ln(x2

1)

dx =

1

ln(x2

1) dx

(1.11.1)

Trong kt qu 1.4, du nguyn hm ( ) ca v bn tri mu en, y l k hiu ca lnh tnhInt, cn du nguyn hm v tri mu xanh [15 ]. iu c ngha Maple khng tnh c v cng khng th biu din kt qu di dng cc hm Si, Ei, Elliptic, hypergeom...

Tuy nhin, c nhiu trng hp Maple "b tay" nh 1.4, nhng vi vi bin i, ngi ta vn c th tm ra li gii gii tch. Hai cch bin i thng thng m mi ngi u bit l nguyn hm tng phn v bin i. Di y l mt vi th d n gin (m Maple gii d dng):X Nguyn hm tng phn ( (u1 v) = uv (uv 1 )). s dng hai php bin i, trc

tin phi nhp th vin student sau khai bo nguyn hm dng tnh (khng hin th):> with(student): > p:=Int(x*ln(x),x):

Chn u = ln(x) :> intparts(p,ln(x));1 2 x ln(x) 2

x dx 2

Gi tr ca biu thc trn n gin, c th dng value> p:=value(%);

1 1 x ln(x)dx = x2 ln(x) x2 dx 2 4

Ta cng c th chn u = x> intparts(p,x):expand(%);x2 ln(x)

x2

x ln(x)dx + xdx

Biu thc trn chnh l p, vy ta c th "n gin" bng hm solve tm p:> solve(p=%,p);1 2 1 1 x ln(x) x2 + xdx 2 2 2

n y th ta c th dng value hin th kt qu:> p:%=value(%);

1 1 x ln(x)dx = x2 ln(x) x2 dx 2 4

phng php ny (cng nh phng php i bin) thc s c ngha, ta ch c th p dng lnh value khi biu thc n gin v c th nhn ngay ra kt qu bng cc cng thc quen thuc.15

K t phin bn V9, tt c u mang mu xanh Phm Minh Hong

24

1.11. Nguyn hm

X Phng php i bin.> p:=Int(x*sqrt(1-x4),x);

p :=

x 1 x4 dx

?

i bin bng cch x2 = sin(u):> changevar(x2=sin(u),p,u); 1a 1 sin(u)2 cos(u)du 2 > simplify(%,symbolic); 1 cos(u)2 du 2

1 n y ta c th dng value(%) v cos(x)2 = (cos(2x) + 1): 2> value(%); 1 1 sin(u) cos(u) + u 4 4

Tr li bin x:> simplify(subs(u=arcsin(x2,%),symbolic); 1? 1 1 x4 x2 + arcsin(x2 ) 4 4

Kim chng bng o hm:> simplify(diff(%,x));x 1 x4 [16 ]

?

Trn y ch l nhng th d n gin, trong thc t i khi phi lp li nhiu ln cc lnh intparts hay changevar, thm ch phi lm c hai lnh.X Thit lp cc cng thc quy np.

Cc cch bin i trn c mt ng dng rt hay l tnh cc cng thc tch phn quy np, chng hn nh thit lp cng thc ca x lnn xdx.> p:=Int(x*ln(x)n)

x lnn xdx

> q:=intparts(p,ln(x)n); n 1 n x2 1 ln(x) nx dx q := ln(x) 2 2 ln(x)

Nu t F (n) =

hin th cng thc quy np ta cn khai bo hm F (n), F (n 1) bng alias v dng hm map (trong Maple F [n] c ngha l Fn ):> alias(F[n]=Int(x*ln(x)n,x),F[n-1]=Int(x*ln(x)(n-1),x)): > F[n]=map(simplify,q); 1 1 Fn := ln(x)n x2 nFn1 2 216

x ln(x)n dx, ta thy biu thc c du nguyn hm chnh l F (n 1).

Chng ta s gii bi ton ny bng mt phng php khc (xem mc 1.12.4)

Phm Minh Hong

25

Chng 1. C php Maple

p dng cng thc quy np ta vit mt chng trnh quy (recursive)[17 ]. Chng trnh c tn l F c nh ngha nh sau:$ ' &

F (n) =

x ln (x)dx =

n

' %

1 2 1 x ln(x) x2 (n = 1) 2 4 1 1 ln(x)n x2 nFn1 (n 1) 2 2

1.12 Bi tp Cc lnh c bnCho a thc p = x2 4x3 1 1. V ng biu din ca p. Tm cc nghim trong R v trong C . 2. Tha s ha p v 1 trong R v trong C . p

17

F:=proc(n) if n=1 then 1/2*x2*ln(x)-1/4*x2 else 1/2*ln(x)n*x2-1/2*n*F(n-1):fi: end:> factor(F(3)); int(x*ln(x)-F(3); 1 2 x (4 ln(x)3 6 ln(x)2 + 6 ln(x) 3) 8 0

i s[ 2t 1. Cho M = 2 ] 1 t . Xc nh t sao cho M nhn gi tr ring kp.

? 2I 2 Tnh vc-t ring trong trng hp ny. (S: t = , [1, I 2]). 3 x 0 ? 0 0 y 2. Cho M = ? 3. Tm x, y, z sao cho m nhn ba gi tr ring 4, 8, 12. 0 3 z(S: tx = 4, y = 11, z = 9u, tx = 4, y = 9, z = 11u). 5 1 1 3. Cho M = 1 5 1. 1 1 5 quy l chng trnh gi chnh n. Phm Minh Hong

?

Hng dn: Dng tnh cht det(M ) = det(J) vi J l ma trn ng cho.

26

1.12. Bi tp

Tm an , bn sao cho @n P N , M n = an M + bn In 1 (S: an = 2n 1, bn = (2n+1 + 1)). 3

Phng trnh vi phn1. Cho phng trnh vi phn E : x(x2 1)y 1 (x) (x2 1)y(x) + 2x = 0. Chng minh rng tip tuyn ti honh x = 2 ca cc li gii ca E u i qua mt im c nh A. Tm A, v th. 2. Cng mt cu hi nhng ti honh x = u vi: E(k) : x(x2 1)y 1 (x) (x2 1)y(x) + kx = 0

3. Cng mt cu hi nhng ti honh x =

3 vi: 2 E(k) : x(x2 1)y 1 (x) (xk 1)y(x) + 2x = 0

Nguyn hmI. Tm nguyn hm v kim chng bng o hm (bi c du : kh hn) ln(x)3 1. p = x3 2. p = x arccos(x) 3. p = ex sin(x)2

4. p = x 1 + x4 5. p = 6. p =arcsin(x) x2a

?

x(x2 + 1) ln(x2 1) ax2 1 + bx + c

II. Phn s hu t: Tm nguyn hm v kim chng bng a hm cac

b 1 b2 (Hng dn: i bin bng cng thc: x + = (c ) tan(u)) [18 ] 2a a 4a 1 1 1. p = 2 3. p = 2 3x + 6x + 5 (x 6x + 18)2 2. p = (2x2 1 4x + 10)2 4. p =

?

3x [19 ] 8 2x x2

III. Tm cng thc quy np v vit hm quy tng ng. 1. p := xn ex

2. p := xn sin(x)

Lp trnh#

Dy Syracuse l mt dy c nh ngha nh sau: ui+1 =18 19

ui (ui chn ) 2 3ui + 1 (ui l )

Xem hm completesquare t (x + 1) = 3 sin(u).

Phm Minh Hong

27

Chng 1. C php Maple

Vi u0 l mt s nguyn dng cho sn, ngi ta thy (nhng cha chng minh c) rng t uu 1 1. Vit mt chng trnh c tn orbite c tham s l u0 xut ra dy [i, uni ] vi i P [0, n]. 2. Vit mt chng trnh plotorb(n) v qu o ca tuu. Xem cch v nh hnh bn orbite(20)) ng ni nhng im ca dy ny c gi l qu o ca tuu.

Chn cc hnh trn v ng ni nhng mu khc nhau. Th d: plotorb(20,red,blue);

Bi gii1.12.1 > p:=x4-4*x3-1: plot(p,x=-2..6,-2..2);solve(p); > allvalues([%]); evalc(%); select(iIm(i)=0,%); > factor(p,sqrt(2));factor(1/p,sqrt(2)):%=convert(%, parfrac,x); > factor(1/p,{I,(-10+8*2(1/2))(1/2)}):%=convert(%, parfrac,x); 1.12.2.1 > restart: with(linalg):M:=matrix(2,2,[2*t,1,2,-t]); > s:=eigenvects(M); r:=solve(%[1,1]=%[2,1]); > p:=subs(t=r[1], eval(M)); eigenvects(%); > p:=subs(t=r[2],eval(M);>eigenvects(%); 1.12.2.2 > restart: with(linalg): p:=matrix(3,3,[x,0,0,0,y,sqrt(3),0,sqrt(3),z]); > s:=eigenvals(p);solve(s[2]=8,s[3]=12); > det(p); s[2]+s[3];solve(y*z-3=8*12,y*z=20); > assigs(%[1]): x:=4:map(eval,p);eigenvals(%); > restart:with(linalg):S:=matrix(3,3,(i,j)if i=j then 5 else -1 fi); 1.12.2.3

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Phm Minh Hong

1.12. Bi tp

> J:=jordan(avalm(1/3*S),p); Jn:=map(iin,eval(J)); > Sn:=evalm(P&*Jn&*inverse(P));evalm(Sn-(1/3+2/3*2n)*diag(1,1,1)); > (1/3*2n-1/3)*map(ii/(1/3*2n-1/3),eval(%)): U:=map(simplify,eval(%)); 1.12.3.1 > restart:with(plots): > eq:=diff(y(x),x)*x*x(x2+1)-(x2-1)*y(x)+2*x; > dsolve(eq);f:=subs(rhs(%), C1): > tg:=asubs((x=2,diff(f(a),x)*(x-2))+subs(s=2,f(a)); > plot([f(2),tg(2),tg(2),f(3),tg(3)],x=-3..3,0..12); > solve(tg(u)=tg(v),x);subs(%,tg(u)); > s:=seq(plot([f(i),tg(i)],x=-3..3,0..12,color=[blue,red]),i=0..6): > dr:=plot([2,x,x=0..12]);display(dr,s); 1.12.3.2 > eq:=kdiff(y(x),x)*x*(x2+1)-(x2-1)*y(x)+k*x; > dsolve(eq(k));f:=unapply(rhs(%), C1,k): > tgx:=(a,u,k) subs(x=u,diff(f(a,k),x))*(x-u)+subs(x=u,f(a,k)); > solve(tgx(x1,u,h)=tgx(x2,u,h),x);simplify(subs(%,tgx(x2,u,k))); 1.12.3.3 > eq:=kdiff(y(x),x)*x*x(x2-2)-(xk-1)*y(x)+2*x; > dsolve(eq(3));f:=unapply(rhs(%), C1): > tg:=(a)subs(x=3/2,diff(f(a),x)*(x-3/2)+subs(x=3/2,f(a)); > plot([f()2],tg(2),f(3),tg(3),f1,tg(1)],x=0..3,-1..12); 1.12.4.a.1 > with(student):p:=Int((ln(x)/x)3,x); > intparts(p,ln(x)3);intparts(%,ln(x)2); > intparts(%,ln(x));value(%);simplify(diff(%,x),symbolic); 1.12.4.a.2 > P:=Int(x2*arccos(x),x);intparts(p,x2);expand(%)); > solve(p=%,p);simplify(diff(%,x),symbolic); 1.12.4.a.3 > p:=Int(exp(-x)*sin(x),x);intparts(p,sin(x)); simplify(intparts(%,cos(x))); > solve(p=%,p);simplify(diff(%,x),symbolic); 1.12.4.a.4 > p:=Int(x*sqrt(1+x4),x):%=value(%); > changevar(tan(u)=x2,p,u):%=simplify(convert( sincos),symbolic); > RP:=rhs(%): RP=changevar(cos(u)2=t-t2,RP,t);op(1,2*rhs(%)); > Int(convert(%,parfrac,t),t)/2:%=cionbine(value)(%); > subs(t=sin(acrtan(x2)),rhs(%));simplify(%,symbolic); simplify(diff(%,x));

Phm Minh Hong

29

Chng 1. C php Maple

1.12.4.a.5[20 ] > P:=Int(arcsin(x)/x2,x);intparts(p,1/x2;expand(%); > solve(p=%,p);res:=expand(%);op(1,res),op(2,res);p:=op(3,res); > changevar(x=cos(u),p,u);subs(sqrt(1-cos(u)2)=sin(u),%); > map(simplify,(intparts(%,sin(u))));value(%); > subs(u=arccos(x),%);res3:=expand(%); > diff(%,x);simplify(%,symbolic);rationalize(%); > op(1,res),op(2,res),res3;op(1,res)+op(2,res)+res3;q:=diff(%,x); > simplify(%,symbolic);simplify(op(2,q)+op(3,q)+op(4,q),symbolic); 1.12.4.a.6 > p:=Int(x*sqrt(x2+1)*ln(x2-1),x);value(%); > chargevar(t=sqrt(x2+1),p,t);

intpats(%,ln(-2+t2));s:=map(expand,%) > c:=op(2,s);op(1,op(2,c)); op(1,op(2,c))L%=convert(%,parfrac,t); > r:=op(3,rhs(%));changevar(t=2/sqrt(2)*u,Int(r,t0,u); > value(%);subs(u=sqrt(2)/28t,%); res:=op(1,s)-2/3*(int(t2+2,t)+%); > subs(t=sqrt(x2+1),res); diff(%,x):%=simplify(%,symbolic); > cobine(rhs(%),symbolic);factor(%); 1.12.4.b.1 > p:=Int(1/3*x2+6)*x+5),x):%=conpletesqyare(%,x); > changevar(x+1=sqrt(2/3)*tan(u),rhs(%),u):%=simplify(%); > p=subs(isolate(x+1=sqrt(2/3)*tan(u),u),value(rhs(%))); > simplify(diff(rhs(%),x)); 1.12.4.b.2 > p:=Int(1/(2*x2-4*x+10)2,x):%=conpletesquare(%,x); > char((x-1)=2*tan(u),rhs(%),u):%=simplify(convert(%,sincos)); > p=simplify(subs(isolate((x-1)=2*tan(u,),u),value(rhs(%)))); > p=subs(ioslate) > simplify(diff(rhs(%),x)); 1.12.4.b.3 > p:=Int(1/(x2-6*x+18)3,x):%=completesquare(%,x); > changevar(x-3=3*tan(u),rhs(%),u):%=simplify(convert%,sincos); > combine(rhs(%)):%=value(%); > p=sim;ify(subs(isolate(x-3=3*tan(u),u),value(rhs(%)))); > p1:=simplify(diff(rhs(%),x)); > factor(rationalize(convert(%,exp))); 1.12.4.b.420

Trong bi ny chng ta chp nhn

1 dx l mt trng hp "n gin". cos(x) Phm Minh Hong

30

1.12. Bi tp

> p:=Int(3*x/sqrt(8-2*x-xx2),x):%=completesquare(%,x); > changevar((x+1)=3*sin(u),p,u);simplify(%,symbolic);value(%); > subs(isolate(x+1=3*sin(u),u),%);>simplify(diff(%,x),symbolic); 1.12.4.c.1 > p:=Int(xn*exp(x),x);value(%);intparts(p,xn);simplify(%); > alias(F[n]=Int(xn*exp(x),x),F[n-1]=Int(x(n-1)*exp(x),x)): > map(simplify,%%) > F:=proc(n) if n=1 then exp(x) else factor(xn*exp(x)-n*F(n-1)): fi: end: 1.12.4.c.2 > P:=Int(xn*sin(x),x);u:=unapply(p,n): > alias(F[n]=u(n),F[n-1]=u(n-1),F[n+1],F[n+2]=u(n+2)); > intparts(p,sin(x));x:=intparts(%,cons(x));s:=F[n]=map(simplify,s); > F:=proc(n) if n=0 then-cos(x) elif n=1 then sin(x)-x*con(x) else (-F(n-2)+sin(x)*x(n-1)/(n-1)-cos(x)/(n-1)*x(n)/n*(n-1)*(n): fi: end: 1.12.5 > orbite:=proc(u0) local u,f,i,s; s:=[0,u0];u:=u0; f:=uif u mod 2=1 then 3*u+1 else u/2 u/2 fi; for i to= 10000 while u1 do u:=f(u): s:=s,[i,u]:od: s; end: > with (plots): > plotorb:= proc(n,a,b) local g1,g2: g1:=pointplot([orbite(n)],symbol=circle,symbolsize=20,color=a): g2:=plot([orbite(n)],color=b):display(g1,g2); end: > display(plotorb(33,blue,red),plotorb(68,red,black));

Phm Minh Hong

31

Chng 1. C php Maple

1.13 Bi c thm: ThalsThals (625-547 trc Cng Nguyn) Thals vn l mt thng gia, nhng sau khi pht ti giu c th ng dng tin kim c i chu du, giao lu hc hi, nghin cu. Chng ng u tin ng dng chn l Ai Cp. Tng truyn rng c ngi thch Thals o chiu cao ca kim t thp m khng cn leo ln (m cho d c leo ln nh cng chng gii quyt c g !). Thals lin cm mt cy gy xung t ri ni: Khi chiu di bng cy gy bng cy gy th chiu di bng kim t thp bng chiu cao ca n. C l lc y ng khng th ng rng hn 26 th k sau, cc hc sinh u phi hc qua cu ni bt h ny: nh l Thals. Sau khi i nhiu, giao du rng, hc hi khp ni, ng tr v Milet v ni ting l mt nh thng thi, hiu bit uyn bc nhiu lnh vc: Doanh nghip, Chnh tr, Khoa hc, Thin vn, Ton hc... ng bit v truyn b cho nhng ngi xung quanh rng mt nm c 365 ngy, ngi ta cn n rng vo thi ng hiu v sao c nguyt thc, nht thc. ng c ngi ng thi v hu th tn vinh l nh ton hc u tin ca loi ngi. Quan im ca ng v ngun gc ca mun loi l t nc m ra. Theo Aristore th Thals cng l "nh u c u tin ca loi ngi" v c mt nm, c ma qu -liu, Thals bn b ht tin ra mua sch tt c -liu va thu hoch, sau ng cho bn t t ra th trng. Nm , ng thu c tin nhiu v k. Tht kh kt lun l nhng pht minh tm ti thi c phi tht l ca ng hay khng, nhng c iu gn nh chc chn ng l nh hnh hc u tin ca nhn loi. Tc phm ca ng v Hnh hc s cp nh nhng iu ng pht biu v ng thng, gc, tam gic... u kh r rng tuy c mt s kin thc c bit t thi vn minh Babylone nhng Thals quan tm n chng minh, l iu lm tn gi tr khoa hc ca ng. Nhng nh kho c su tp c nhng bi pht biu ny, chnh l c s nh gi tnh khoa hc ca vic nghin cu ng li cho i sau. Su kt qu quan trng sau y ngi ta cho l cng lao ca Thals:X Cc gc y ca tam gic cn bng nhau. X Nu hai ng thng ct nhau th cc gc i nh bng nhau. X Mt tam gic c xc nh chnh xc nu bit di ca mt cnh v hai gc k vi cnh . X Nhng cnh tng ng ca cc tam gic u th t l vi nhau. X ng knh chia vng trn ra hai phn bng nhau. X Mt tam gic ABC ni tip trong mt vng trn v nu BC l ng knh ca z vng trn th BAC l mt gc vung.

32

Phm Minh Hong

Chng2Bi ton cc trCc tr l mt bi ton c in nhng nhiu ng dng. Ngay t thi xa xa con ngi lun lun thc mc: on ng no gn nht, th tch no ln nht, chu vi no nh nht... Bng cc tnh ton qua kinh nghim, ngi xa thit lp ra cc cng thc thc nghim, nhng phi n khi cc nh ton hc tm ra o hm th mi vic mi c suy lun mt cch cht ch. Bi ton cc tr n thun l gii phng trnh o hm ca i lng mun tm cc tr. Phng php tng qut ca bi ton cc tr l:X Xc nh i lng mun tm cc tr. X Tm s quan h gia cc bin v tm cch a v hm mt hoc hai bin. X Gii phng trnh o hm.

Kh khn khi dng trong mi trng tnh ton hnh thc ch sau khi gii phng trnh phi tm ra cc nghim thch hp.

2.1 Tit kim nhmCho mt hnh tr th tch lt. Tm kch thc hnh tr sao cho din tch ca n nh nht. Gi V l th tch, S l din tch ton phn ca hnh tr. t r l bn knh y, h l chiu cao. Mc ch ca chng ta l tm cc tr ca S theo r (hoc h) bng cch gii phng trnh: dS = 0. Ta c: dr> S:=2*Pi*r2+2*Pi*r*h;S := 2r2 + 2rh

> V:=Pi*r2*h;V := r2 h

loi h ta lm (th tch l lt):> eh:=h=solve(V=Pi,h);

Chng 2. Bi ton cc tr

r .

.

.

h .

.Hnh 2.1:

eh := h =

1 r2

Thay vo biu thc ca S:> subs(%,S); 2 + 2r2 r

Gii

> r=select(iIm(i)=0,solve(diff(%,r))); 1 r = 22/3 2

dS = 0 ta s c ba nghim, trong ta ch ly mt nghim thc: dr

Chiu cao h v din tch S ca hnh tr l:> subs(%,eh);h = 22/3

> 'S'=subs(%%,%,S);S = 321/3

Kim chng vui: Bn knh v chiu cao ca mt lon sa b quen thuc l 36.5 v 75mm. Ta c dung tch ca n l times(36.5)2 75 314cm3 . Hy kim chng rng din tch ca lon sa b 2 36.52 + 2 36.5 75 255cm2 l ti u.

2.2 on n