Li ni uLptrnhtnhtontrongMaplemuntrnhbykinthclptrnh ng dng. S dng phn mm ton hc c sn cng vi ngnng ca n to ra nhng chng trnh tnh ton trong thc t. Maplecng l chng trnh tnh ton k hiu, bn c c th hc c rtnhiu kin thc nh Maple. Mt khc ngi hc cng c th t mnhto ra chngtrnh phcv chomc ch ca mnh. Ti liu phcv t hc v cng l ti liu ging v mn "Lp trnh tnh ton".Phn l thuyt c trnh by thng qua ngay trn cu lnh caMaple v l cc mu ngi hc theo m lm cc chng trnhkhc. Phn ln bi tp l cc d n dnhcho ngihc t lmvu c mu kt qu cui cng ngi lp trnh nh hng.H Ni, ngy 8 thng 8 nm 2007NguynHuinMc lcLiniu . . . . . . . . . . . . . . . . . . . . . . . . . 3Mclc . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Giithiulptrnhtnhtontrongmaple 71.1 Chng trnh n gin. . . . . . . . . . . . . . . . . 71.2 Nhng cu lnh c iu kin . . . . . . . . . . . . . . 151.3 Vng lp vi cu lnh do . . . . . . . . . . . . . . . . 221.3.1 Cu trc for - do . . . . . . . . . . . . . . . . 221.3.2 Gii thiu mng n gin . . . . . . . . . . . 241.3.3 Php lp . . . . . . . . . . . . . . . . . . . . 261.3.4 Tnh tng . . . . . . . . . . . . . . . . . . . . 281.4 Bi tp . . . . . . . . . . . . . . . . . . . . . . . . . 312 Ccphplp 392.1 Phng php nht ct vng v chia i . . . . . . . 392.1.1 Gii thiu nht ct vng. . . . . . . . . . . . 392.1.2 Phng php nht ct vng. . . . . . . . . . 422.1.3 Vng lp while-do . . . . . . . . . . . . . . . 472.1.4 Phng php chia i . . . . . . . . . . . . . 47MC LC 52.2 Phng php New tn v cc phng php lp khc 482.3 Bi tp . . . . . . . . . . . . . . . . . . . . . . . . . 543 MngtrongMaple 583.1 nh ngha mng v s dng n . . . . . . . . . . . . 583.1.1 S dng mng . . . . . . . . . . . . . . . . . 583.1.2 Lnh seq . . . . . . . . . . . . . . . . . . . . . 593.1.3 Gi tr ban u ca mng. . . . . . . . . . . 603.1.4 Cc phn t ca mng. . . . . . . . . . . . . 603.1.5 V d mng cc s nguynt . . . . . . . . . 613.1.6 V d xc nh phn t cc i ca mt mng 633.2 Mt s o thng k . . . . . . . . . . . . . . . . . 643.2.1 Min gi tr . . . . . . . . . . . . . . . . . . . 643.2.2 Sp xp . . . . . . . . . . . . . . . . . . . . . 663.3 Phng php sng . . . . . . . . . . . . . . . . . . . 683.3.1 V d v tm s nguyn t bng phng phpsng . . . . . . . . . . . . . . . . . . . . . . . 693.3.2 V d v nhng s Ulam p . . . . . . . . . 713.4 Mt s d n v mng . . . . . . . . . . . . . . . . . 733.4.1 Tam gic Pascal . . . . . . . . . . . . . . . . 733.4.2 Gi tr trung bnh . . . . . . . . . . . . . . . 753.5 Bi tp . . . . . . . . . . . . . . . . . . . . . . . . . 764 Mphngxcsut 834.1 Khai bo loi d liu cho i s chng trnh . . . . 834.2 Th nghim xc sut. . . . . . . . . . . . . . . . . . 854.2.1 Gii thiu. . . . . . . . . . . . . . . . . . . . 854.2.2 Sinh s thc ngu nhin. . . . . . . . . . . . 866 MC LC4.2.3 Sinh s nguyn ngu nhin trong mt khongno . . . . . . . . . . . . . . . . . . . . . . 864.2.4 V d rt c tr li. . . . . . . . . . . . . . . 874.2.5 V d rt ba qu bng khng tr li . . . . . 884.2.6 V d v tnh xp x ca xc sut . . . . . . . 894.2.7 V dhai imtronghnhvungckhongcch nh hn mt . . . . . . . . . . . . . . . 914.3 Tr chi qun bi. . . . . . . . . . . . . . . . . . . . 934.3.1 Chng trnh rt ngu nhin k qun bi . . . 934.3.2 V d tnh xc xut rt k qun bi c t nhtmt qun t . . . . . . . . . . . . . . . . . . . 954.3.3 V dtnhxcxutrtkqunbi cngmt qun t . . . . . . . . . . . . . . . . . . . 994.4 Mt s v d v m phng xc sut . . . . . . . . . . 1004.4.1 Kim tra nhim HIV. . . . . . . . . . . . . . 1004.4.2 Xc sut nhng ngi c n trong mt nhm1024.5 Bi tp . . . . . . . . . . . . . . . . . . . . . . . . . 1045 Hphngtrnhngin 1055.1 Gii phng trnh . . . . . . . . . . . . . . . . . . . 1055.1.1 Lnh ca Maple . . . . . . . . . . . . . . . . 1055.2 V d v bi ton hn hp phn trm . . . . . . . . 1075.3 ng thng st nht . . . . . . . . . . . . . . . . . 108Danhmctkha . . . . . . . . . . . . . . . . . . . . 110Chng 1GII THIU LPTRNH TNH TONTRONG MAPLE1.1 Chngtrnhngin . . . . . . . . . . . 71.2 Nhngculnhciukin . . . . . . . . 151.3 Vnglpviculnhdo . . . . . . . . . . 221.4 Bitp. . . . . . . . . . . . . . . . . . . . . 311.1 Chng trnh n ginMt bi ton tnh ton thng c ba phn sau:1. D liu tnh ton;2. Phng php tnh ton;8 Giithiulptrnhtnhtontrongmaple3. Kt qu tnh ton.V d sau y ch ra cch thc cc cng vic tnh ton cn philm g: Ta bit c cng thc tnh th tch hnh tr bng din tchy nhn vi chiu cao. Nu ta c hnh tr trn din tch y l mthnh trn c bn knh r, th din tch y c tnh bng cng thcr2.tnhthtchmthnhtrtrncbnknhybngrvchiu caoh, ta c:1. D liu:r vh;2. Phng php tnh:Bc 1. Tnh din tch ys = r2,Bc 2. Tnh th tchv = s h,3. Kt qu: Th tchv = r2h.Mtchngtrnhmytnhthchinphngphptnhtonnh mt nhim v chnh. N khi u l nhn d liu vo ngi tathng gi l u vo, sau thc hin phng php tnh v chuynktqura, ngitagilura.Mtchngtrnhccoinhl mt hp en i vi ngi dng v ngi s dng theo quy trnh:H a d liu vo v chy chngtrnh, khi nhn ra kt qu.Ta c th vit chng trnh trn bng mt trang Maple. vitchngtrnhtrongmaplecnhiudng, khi xungdngnhn[Shift]+[Enter]. Maple khng bo li th ta g ch thch sau dukt thc lnh ;.# nhnghachngtrnh> cylinder_volume:=proc(radius,height)localbase_area,volume;# binaphngbase_area:=Pi*radius^2; # Thchinthuttonvolume:=evalf(base_area*height);print(Thevolume,volume);# ktquaraend;1.1Chngtrnhngin 9c yl i nde r vol ume := proc( radi us, he i g ht )l ocal base area, vol ume ;basearea := t t x r af a2 ;vol ume := e v a l f ( base_area x he i g ht) ;print ( t he volume , vol ume )endTn ca chng trnh ta t lcylindervolume, n nhn hai i snh l d liu vo l radius v height. Cc bin a phng (ngha lcc bin ch dng trong chng trnh ny) lbase_area vvolumec nh ngha. Khi tnh ton ly cc gi tr cabase_area vvolume tnh. Cui cng kt qu c in ra bng lnhprint.Gistacbnknhr=5.5mvchiucaolh=8.2mTachy chngtrnh trong mi trng Maple:>r:=5.5;h:=8.2;r := 5.5h := 8.2>cylinder_volume(r,h);Thevolume, 779.2720578Chng trnh c thit k dng i dng li nhiu ln. Mi lndng n ta cn phi cung cp d liu u vo theo nguyntc cachngtrnh. Trong v d ca ta th phi cung cp bn knh trcv chiu cao sau.Vi bn knh 3 ft v chiu cao 2 ft:>cylinder_volume(3,2);Thevolume, 56.54866777Khi th t ngc li th kt qu s khc>cylinder_volume(2,3);Thevolume, 37.69911185Tm li mt chngtrnh Maple gm cc thnh phn sau y:1. Mnh nh ngha chng trnh di dng10 Giithiulptrnhtnhtontrongmapleprogramname:=proc(argument_list)Ngi dng s dng tn chng trnh programname chychng trnh. Danh sch bin c lit k trong argument_list.Ch ktthcdng lnhkhai bo chngtrnh khngdngdu ;.2. Khai bo bin a phng di dnglocal variable_list;Khai bo ny l mt phn ca cc mnh nh ngha chngtrnh.3. Nhng mnh ca chng trnh Maple l phng php tnhton s dng cc i s nh d liu u vo.4. C ch u ra, nhiu khi ch n gin l cu lnh in ra.5. Cu lnh kt thc, thng ring mt dngend;6. Cu ch thch: Bao gi cng bt u bng k t #, phn cusau du ny ch l ch thch ca chng trnh v khng c nhhng g n thc hin chng trnh.Mc d ta g trc tip chng trnh vo trong trang Maple, nhngta c th tch qu trnh lp trnh bn bc: (1) son chng trnh,(2)gichngtrnhvo,(3)chythv salichngtrnh,(4)thc hin chngtrnh.1. Son tho chng trnh: Ta c th son chng trnh trong mttpring, chbngmtchngtrnhkhngcktiukhin nh tp "file.txt" trong chng trnh notepad.exe v ghili chng trnh vi tn v phn m rng nh: tinhthetich.mwsl tp ca Maple.2. Gi chng trnh vo: Trong Maple ti du con tr >, g vo1.1Chngtrnhngin 11read( c:/hocmaple/tinhthetich. txt );Maplebtuctpvnbnvo,nucsaistMaplesthng bo. Ngc li chng trnh s c gi vo. Ch du c ly phm cao nht bn tri trong bn phm.3. Chy th v sa cha chng trnh: Nu Maple thng bo lihoc ta pht hin kt qu chy chng trnh sai. Ta phi quayli chngtrnh Notepad sa chng trnh chnh.4. Chychngtrnh:Gtnchngtrnhvccdaliuvoi s thc hin chngtrnh.V d 1. Gi tr tnglai catin tr cp:Gi s ta ngtintitkimchomtchuk nm$Rvoti khontrvi li xutichochukny, khigitrcnhnc$Strongti khonsaun chu k (nm) s lS=R((1 +i)n1)i.Hy vit chng trnh tnh gi tr cho k hoch v hu nu k hochtrtinli nmlA, i hi thunhpmi thngphi ngvo$R, trongynm.Sau dngchngtrnh tnhgi tr tintrongtng lai nu mi thng ng vo l $300, tin li theo nm l 12%trong 30 nm.Solution.Trc tin thy tnh ton cn cc i lng c avoR l tin giA tin li hng nmn l s nm cn tnh.Ta g chng trnh sau y vo tp future.txt:#Chngtrnhtnhgitrthunhpkhivhu#Input:R:Tinnphngthng(\$)12 Giithiulptrnhtnhtontrongmaple# A:Lisuthngnm(.12vi 12\%)# y:Snmfuture:=proc(R,A,y) # ttnchngtrnh"future"# vis R, I,ylocali,n,S; #nhnghabinaphngi:=A/12; # lixuthngthng#1/12calixutnmn:=y*12; # tngsthngS:=R*((l+i)\symbol{94}n-l)/i;#tnhtinhuprint(Thefuturevalue,S); #printra ktquend; # dngktthcchngtrnhBy gi ta c chngtrnh vo Maple> read(a:future.txt);future := proc(R,A,y)locali,n,S;i := 1/12 A; n := 12 y; S:= R (A+i)n 1)/i);print(Thefuturevalue,S)endNhp u vo:>Deposit:=300;AnnualRate:=0.12;Years:=30;Deposit:= 300AnnualRate := .12Years := 30By gi ta c th chychngtrnh xc nh trn. Ch cci sR, A, yl cc bin c thay th bi cc bin mang d liu.> future(Deposit,AnnualRate,Years);Thefuturevalue,.1048489240107Hoc ta c th a trc tip i s bng s> future(300,.12,30);Thefuturevalue,.1048489240107Vd2.Lpmtchngtrnhtnhhainghimcaphngtrnhbchai tngqutax2+ bx + c=0bngcngthcnghimtng1.1Chngtrnhngin 13qut. Dng chng trnh ny gii3x25x + 2 = 0 vx23x + 2 = 0.Li gii. Nhp cc dng lnh sau y>quad:=proc(a,b,c)# ttnchngtrnh"quad"#withargumentsa,b,clocalsoil,sol2;# nhnghabinaphng#tnh soil,sol2soil:=(-b+sqrt(b^2-4*a*c)) /(2*a);sol2:=(-b-sqrt(b^2-4*a*c))/(2*a);print(Thesolutions,soil,sol2);# Inraktquend;#Ktthcchngtrnhquad := proc( a , b , c )l ocal s o i l , s ol 2 ;s o i l :=(b + s q r t ( b^2 4ac ) ) /( 2 a ) ;s ol 2 :=(b s q r t ( b^2 4ac ) ) /( 2 a ) ;print ( The s ol ut i ons , s o i l , s ol 2 )end proc gii phng trnh3x25x + 2 = 0, ta ch rnga = 3, b = 5vc = 2:>quad(3,-5,2);Thesolution,1,23 gii phng trnhx23x + 2 = 0:>a:=1;b:=-3;c:=2;a := 1b := 3c := 2>quad(a,b,c);Thesolution,2, 114 GiithiulptrnhtnhtontrongmapleVd3. Nguyn l b stress c cho biSmax= A+B, Smin = A B, yA = E(1 K)Q1 Q22(1 )B=E(1 +k)_(Q1 Q2)2+ (Q2 Q3)22(1 +).Hy vit mt chngtrnh vi u voE, , K, Q1, Q2, Q3v in rakt qu stress. Vi d liu c thE K Q1Q2Q39.3 106 0.32 0.05 138 -56 786Ligii. Ta vit chng trnh>principal_stress:=proc(E,mu,K, Q1,Q2,Q3)localSmax,Smin,A,B;A:=E*(1-K)*(Ql+Q3)/(2*(1-mu));B:=E*(1+K)*sqrt((Q1-Q2)^2+(Q2-Q3)^2)/(2*(1-mu));print(Smax,Smin);print(A+B,A-B);end;s t r e s s := proc (E, mu, K, Q1, Q2, Q3)l ocal Smax , Smin , A, B;A:=E(1K) (Q1+Q3)/(22mu) ;B:=E(1+K) s q r t ( ( Q1Q2)^2+(Q2Q3)^2)/(2 2mu) ;print (Smax , Smin ) ;print (A+B, AB)end proc> stress(9.3*10^6,0.32,0.05,138.0,-56.0,786.0);Smax, Smin1.220668212 1010, 2.01476241 1081.2Nhngculnhciukin 151.2 Nhng cu lnh c iu kinVd1.1. Cho mt hm tng khcf(x) =___(x 1)2+ 2 x < 0,2x + 1 0 x 25e((x2)2)x > 2Ta c th tnh c hm bng Maple>func:=proc(x)# nhnghahm#Khngcbinaphngifx=445thenprint(borderlineA); fi;elifpoints>=350thenprint(C);ifpoints> 395thenprint(borderlineB);fi;elifpoints>=300thenprint(D );ifpoints> 345thenprint(borderlineC);fi;elseprint(F);18 Giithiulptrnhtnhtontrongmapleif points>295thenprint(borderlineD); fi;fi;end;grade := proc( poi nt s )i f 450 type(3.5,numeric); true>type(queen,string); false>type(queen,string); true>Value:=6:type(Value,numeric);true>a:="king":type(a,string); true#Chngtrnhchos qunbit1-52#input:mtgitrtrong#ace,2,310,jack,queen,king#tngngvidaysau#"heart","spade","diamond","club"#output:scaqunbi1-52.>cardnum:=proc(value,suit)localorder,number,err;err:=false;iftype(value,numeric)thenorder:=value;eliftype(value,string)thenifvalue="ace"thenorder:=1;elifvalue="jack"thenorder:=11;elifvalue="queen"thenorder:=12;20 Giithiulptrnhtnhtontrongmapleelifvalue="king"thenorder:=13;elseprint("Err: wrongvalueforthefirstargument");err:=truefielseprint("Err:wrongdatatypeforfirstargument");err:=true;fi;ifsuit="heart"thennumber:=order;elifsuit="spade"thennumber:=13+order;elifsuit="diamond"thennumber:=26+order;elifsuit="club"thennumber:=39+order;elseprint("Err:wrongvalueforthesecondargument");err:=true;fi;iferr=falsethenprint("Thecardnumber",number);fi;end;cardnum := proc( val ue, s u i t )l ocal order, number , er r ;er r := f a l s e ;i f type( val ue, numeric ) thenorder := v al uee l i f type( val ue, s t r i ng ) then1.2Nhngculnhciukin 21i f v al ue =" ace " thenorder := 1e l i f v al ue =" j ac k " thenorder := 11e l i f v al ue ="queen" thenorder := 12e l i f v al ue =" ki ng " thenorder := 13el seprint (" Err : wrong v al ue f or t he f i r s t argument ") ;er r := t r ueend i fel seprint (" Err : wrong dat a type f or f i r s t argument ") ;er r := t r ueend i fi f s u i t =" he ar t " thennumber := ordere l i f s u i t =" spade " thennumber := 13 +ordere l i f s u i t ="diamond " thennumber := 26 +ordere l i f s u i t =" c l ub " thennumber := 39 +orderel seprint (" Err : wrong v al ue f or t he second argument ") ;er r := t r ueend i fi f er r = f a l s e then print ("The card number " , number ) end i fend proc>cardnumC,"diamond");Thecardnumber,29>cardnum("king","club");22 GiithiulptrnhtnhtontrongmapleThecardnumber,52> cardnum(queen,club);Error: wrongdatatypeforfirst argumentError: wrongvalueforthe secondargument> cardnum("queen","spade");Thecardnumber,251.3 Vng lp vi cu lnh do1.3.1 Cu trc for - doVic lp li mt thao tc dn n thit k vng lp. Nu ta munbnh phng mt s nguyni, i = 1, 2, ..., 10 v cc cu lnh sau ygi vng lp v lm c vic :>forifrom1to10doprint(i^2);od;149162536496481100Cu trc vng lp n gin lf or l oop i ndex froms t ar t v al ue to l as t v al ue dob l oc k s t at ement sod;1.3Vnglpviculnhdo 23Nguyn tc s dng vng lp ny nh sau:1. loopindex l tn bin nh li, j.2. start value v last valuec thl ccs, tnbinchashoc mt biu thc c kt qu l s nh l 1, 10, k1, k2, 2n, ...Cc s ny l nhngs nguyn,nhng s thp phn v phns cng chp nhn.3. Vng lp thc hin theo cch sau y:(a) Gnstart value voloopindex,(b)Sosnhloopindexvi last value, nuloop index>last valuethnhyra ngoivnglpsaulnhod;.Ngcli th i vo thn vng lp thc hin.(c) Sau khi xc nh cloop index < last value thblock statements thc hin.(d) Thm 1 voloopindex v quay li (b).4. Khng nn t thay i gi tr caloopindex, hy chngtrnh t thc hin.5. Nustart value=1, thfromstartvalue c th b qua, khi vng lp c dngf or i to 10 doi ^2;od;6. ngtduchmphysau lnhdo m hytduchmphy hoc hai chm sau lnhod.7. Chsloopindexmcnhbcl1.cccbckhci ta thm lnhby nh v d24 Giithiulptrnhtnhtontrongmaplef or i from20 to 2 by2 doi ^2;od;cho kt qu202, 182, 162, ..., 42, 22.Vd1.4. Vng lp sau y tnh gi tr ca hmsin x vcos x t0 nvi bc5:fortfrom0by evalf(Pi/5)toevalf(Pi)dos1:=evalf(sin(t));s2:=evalf(cos(t));od;s1 := 0.s2 := 1.s1 := 0. 5877852524s2 := 0. 8090169943s1 := 0. 9510565165s2 := 0. 3090169938s1 := 0. 9510565160s2 := .3090169952s1 := 0. 5877852514s2 := .80901699501.3.2 Gii thiu mng n ginMng dng cha d liu, v d ta chia on[0, 1]thnh100onnhvidi bngnhau0.01bngccim(gi nlccnt).x0= 0, x1= 0.01, x2= 0.02, ..., x99= 0.99, x100= 1.00Ta c th dng mng cha cc nt ny.1.3Vnglpviculnhdo 251. Trc tin m mt mng rng>nod:=array(0..100);nod:=array(0..100,[])2. Cha d liu vo cc nt bng vng lp vi mi i cha d liuvonod[i]>forifrom0to 100donod[i]:=i*0.01od:3. C th xem li kt qu bng lnhseqseq(nod[i],i=0..100);0 , . 01 , . 02 , . 03 , . 04 , . 05 , . 06 , . 07 , . 08 ,. 09 , . 10 , . 11 , . 12 , . 13 , . 14 , . 15 , . 16 , . 17 ,. 18 , . 19 , . 20 , . 21 , . 22 , . 23 , . 24 , . 25 , . 26 ,. 27 , . 28 , . 29 , . 30 , . 31 , . 32 , . 33 , . 34 , . 35 ,. 36 , . 37 , . 38 , . 39 , . 40 , . 41 , . 42 , . 43 , . 44 ,. 45 , . 46 , . 47 , . 48 , . 49 , . 50 , . 51 , . 52 , . 53 ,. 54 , . 55 , . 56 , . 57 , . 58 , . 59 , . 60 , . 61 , . 62 ,. 63 , . 64 , . 65 , . 66 , . 67 , . 68 , . 69 , . 70 , . 71 ,. 72 , . 73 , . 74 , . 75 , . 76 , . 77 , . 78 , . 79 , . 80 ,. 81 , . 82 , . 83 , . 84 , . 85 , . 86 , . 87 , . 88 , . 89 ,. 90 , . 91 , . 92 , . 93 , . 94 , . 95 , . 96 , . 97 , . 98 ,. 99 , 1. 004. Mi phn t c th ch ra gi tr ca n:nod[19];0.19nod[87];0.8726 Giithiulptrnhtnhtontrongmaple1.3.3 Php lpVd1.5. Vit chng trnh lit kn s u tin ca dyxk=xk12+1xk1, x0= 1, k= 1, 2, ..., n.Dy s ny tin dn ti 2.Ligii. Chng trnh i hi nhp vo sn.#Chngtrnhsinhradyshitticn2#inputn chs cuicngcady>sqrt2:=proc(n)localx,k;x:=array(0..n);# khaibodyx[0]:=l;# khiuphnt0 cady#vnglpsinhra phntdforkfrom1ton dox[k]:=evalf(x[k-l]/2+l/x[k-l]);od;print(seq(x[k],k=0..n)) #uraend;s q r t 2 := proc ( n)l ocal x , k ;x := array(0 . . n ) ;x [ 0 ] := l ;f or k to n dox [ k ] := e v a l f (1/2 x [ kl ]+ l /x [ kl] )end do;print ( seq( x [ k ] , k =(0 . . n ) ) )end proc> sqrt2(10);1, 1.500000000, 1.416666667, 1.414215686, 1.414213562,1.3Vnglpviculnhdo 271.414213562, 1.414213562, 1.414213562, 1.414213562, 1.414213562,1.414213562.Ta thy rng dy tin dn ti 2 rt nhanh.Vd1.6. Vit chng trnh sinh ran s hng ca dy FibonacciF0= 1, F1= 1, Fk= Fk1 +Fk2vik= 2, 3, ..., n.Ligii.#Chngtrnhsinhran shngucadyFibonacci# uvo:n (n>=2).>Fibonacci:=proc(n)localk,F;F:=array(0..n);F[0]:=0;F[l]:=1;forkfrom2to ndoF[k]:=F[k-l]+F[k-2]od;seq(F[k],k=0..n)end;Fi bonacci := proc( n)l ocal k , F;F := array(0 . . n ) ;F[ 0 ] := 0;F[ l ] := 1;f or k from2 to n doF[ k ] :=F[ k l ] +F[ k 2]end doseq(F[ k ] , k =(0 . . n) )end proc28 Giithiulptrnhtnhtontrongmaple> Fibonacci(30);0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418,317811, 514229,832040.1.3.4 Tnh tngV d1.7. Vitchngtrnhtnhtngbnhphngcanslu tin trong dy s nguyn dng. Ngha l tnh tng12, 32, 52, ..., n2.Li gii 1.Phntch:Slutinl1.Nuskllthsltip theo lk + 2:#Chngtrnhtnhtngn slnguyndng#uvo:n#ura:tng.>oddsuml:=proc(n)locali,oddnum,s;s:=0;#Khitotngoddnum:=l;# Khitosl utinfori from1 tondos:=s+oddnum^2;#tnhtngcngthmphnt mioddnum:=oddnum+2;# chunbslsauod;print("Thesumis",s);end;oddsuml := proc ( n)l ocal i, oddnum , s ;s := 0;oddnum := l ;f or i to n do1.3Vnglpviculnhdo 29s := s+oddnum^2;oddnum := oddnum+2end do;print ("The sum i s " , s )end procLigii2. Phn tch: cc s l biu din nh2k 1, k = 1, 2, ...#Chngtrnhtnhtngns lnguyndng#uvo:n#ura: tng.>oddsum2:=proc(n)localk,s;s:=0;forkfrom1to ndos:=s+(2*k-l)^2;od;print("Thesumis",s);end;oddsum2 := proc ( n)l ocal k , s ;s := 0;f or k to n dos := s +(2kl )^2end do;print ("The sum i s " , s )end procCu trc chng trnh cngn phn t thng ls:=0;for*from1ton do30 Giithiulptrnhtnhtontrongmaple{Chunb phntmi}s:=s+{phntmi}od;s:=s+{thenewterm}lmuchtcatnhtontng, v mi lnlplygitr tngccngvi mtphntmi thmvo. ngdng iu ny ta xt v dVd1.8. Hm ssin c th tnh bng cng thcsin(x) =x1! x33!+x55! x77!+x99! x1111!+ Vit chng trnh u vo l x v n, xp x hm sin ti x bng cchtnh tngn s hng u tin.Ligii. Cc s hng c dngxkk!vi k l s l v chuyn du +/.#ChngtrnhtnhhmsinedngcngthcTaylor#uvo:x,n,#ura:gitrxpxcasin(x)>sine:=proc(x,n)locali,s,k,sgn;s:=0;k:=1;sgn:=1;fori from1 tondos:=s+sgn*x^k/k!;sgn:=-sgn;k:=k+2;od;print("Thesinefunction",s);#uraend;1.4Bitp 31s i ne := proc ( x , n)l ocal i, s , k , sgn ;s := 0; k := 1;sgn := 1;f or i to n dos := s+sgnx^k/ f a c t o r i a l ( k ) ;sgn :=sgn ;k := k+2end do;print ("The s i ne f unc t i on " , s )end proc>x:=evalf(Pi/6);x:=.5235987758>sine(x,3);Thesinefunction, .5000021328>sine(x,4);Thesinefunction, .4999999921>sine(x,5);Thesinefunction, .5000000003Dng chng trnh dng sn>sin(x);Thesinefunction, .50000000021.4 Bi tpBitp1.1. (Bi ton th chp) Cho $A l s tin ti khon thchp, n l tngs phitr,i l lisut trongchuk phitr. Khi tin phi tr cho chu k $R c cho theo cng thcR =Ai1 (1 +i)(n).32 GiithiulptrnhtnhtontrongmapleHyvitchngtrnh,viuvop, r, y, dgimua,li suthng nm, s nm v vn tc tr thp tng ng, hy tnh vin ra s tin tr hng thng.Dng chng trnh tnh tin tr hng thng mua ngi nh$180,000, gim dn 20%, 7,75% li sut hng nm trong vng30 nm.Dngchngtrnhtnhtintrhngthngmuaxet$15,000,gimdn10%, 9,25%li suthngnmtrongvng5 nm.Kt qu mu chy chng trnh:> price:=180000:down:=20:rate:=7.75:year:=30:> mortgage(price,rate,year,down);The monthl y payment i s 1031. 633646Bi tp1.2. Hy vit chngtrnh hinln kch thcbn knh,chu vi v din tch hnh trn khi ta cho ng knh ca n.Kt qu mu nh sau:> diameter:=10:> circle(diameter);The r adi us : , 5. 0The ci r cumf er ence : , 31. 41592654The area : , 78. 53981635Bitp1.3. C ba t in c ni song songR1, R2vR3, intr chungR c cho bi cng thc1R=1R1+1R2+1R3.1.4Bitp 33Hy vit mt chngtrnh tnhR.Kt qu mu chy chngtrnh:>Rl:=20:R2:=50:R3:=100:>resistor(Rl,R2,R3);The combined r e s i s t a nc e i s 12. 50000000Bitp1.4. S cng thng cc i c tnh bng cng thcmax=P___1 +ecr2cos_LPAE2r____Avi cc i lng bng di y.Hyvitchngtrnhtnhvinracngthngcci. Dngchng trnh tnh vi cc mu u vo di y.P A e c r L E4000 29.55 16.6 5.56 4.68 120 3.0107Bitp1.5. Vit chng trnh a vo mt s v xc nh tngng tn qun bi (tham kho v d v in qun bi).Kt qu mu:>numcard(l);ace, heart>numcard(25);queen, spade>numcard(30);4, diamond>numcard(0);Error: the input mustbe1 52>numcard(55);34 GiithiulptrnhtnhtontrongmapleError: the input mustbe1 52> numcard(45);6, clubBitp1.6. (Tnh gi) Mt cng ty nh mun t k hoch thuxe: $20.00mtngycngvi $0.10/1cysn200cysmtngy. Khng cng thm gi cho ngy nu ln hn 200 cy s (nghal3ngy500cysgi$110.00khim3ngyvi800cysthgi ch $120.00).Hy vit chngtrnh u vo s ngy v tng scy s i c, kt qu ra gi tng phi tr thu.Bitp1.7. Mt hng du lch cho gi cho k hoch i Las Vegas.Mt ngii gi $400.Nu ngidu lchc ngii cngth mingi i cng c gim gi 10%. Mt nhm 10 ngi hoc ln hnth c gim 15% cho mi ngi.Vit chngtrnh u vo l skhch du lch v u ra l tng s gi.Bitp 1.8. (Gii nghim thc chng trnh bc hai) Mt phngtrnh bc hai tng qutax2+bx +c = 0, a = 0nh thc = b24acKhi > 0, th phng trnh c hai nghim thc;Khi = 0, th phng trnh ch c mt nghim thc b2a;Trng hp cn li l khng c nghim thc.Hy vit chngtrnhu voa, b, c, in ra s nghimthcnu c,ngc li th thng bo khng c nghim.Cc kt qu mu:> a:-l:b:=0:c:-l:> quadreal(a,b,c);Therearenoreal solutions1.4Bitp 35>a:=4:b:=4:c:l:>quadreal(a,b,c);Thereis onlyonereal solution .5>a:=3:b:=5:c:=2:>quadreal(a,b,c);Therearetworeal solutions .6666666665, 1.000000000Bi tp1.9. Tacthmrnggiiphngtrnhbchai: a=0th phngtrnhl phngtrnh tuyntnhv chc mt nghimduy nht cbkhib = 0. Nu li cb = 0 th phng trnh khng cloi .. Hy vit chngtrnh tnh nghimthc ca phngtrnhbc hai vi cc kh nng c th xy ra.Kt qu mu:>a:=0:b:=0:c:=0:>quadadv(a,b,c);Error: invalid equation>a:=0:b:=5:c:=3:>quadadv(a,b,c);This is a linear equationwith solution35>a:=5:b:=3:c:=4:>quadadv(a,b,c);Therearenoreal solutions>a:=l:b:=2:c:=l:>quadadv(a,b,c);Thereis onlyonereal solution 1.0>a:=2:b:=-9:c:=6:>quadadv(a,b,c);Therearetworeal solutions3.686140663, .8138593385Bitp1.10. (Bng thu) Bng thu 1996 tai Hoa k c chia ralmbnhoncnh.Hyvitchngtrnhtnhtinnpthuuvolhoncnhnovthunhpgitrbaonhiu: (status:hon36 Giithiulptrnhtnhtontrongmaplecnh, TaxableIncome: thu nhp, Tax: Mc thu):Status1 (c thn)Taxablelncome Tax0 - 23,350 0.15*TaxableIncome23,350 - 56,550 3,502.50+0.28*(TaxableIncome-23,350)56,550 - 117,950 12,798.50+0.31*(TaxableIncome-56,550)117,950 - 256,500 31,832.50+0.36*(TaxableIncome-117,950)256,500 - up 81,710.50+0.396*(TaxableIncome-256,500)Status2 (Qu ph hoc mt v)Taxablelncome Tax0 - 39,000 0.15*TaxableIncome39,000 - 94,250 5,850.00+0.28*(TaxableIncome-39,000)94,250 - 143,600 21,320.00+0.31*(TaxableIncome-94,250)143,600 - 256,500 36,618.50+0.36*(TaxableIncome-143,600)256,500 - up 77,262.50+0.396*(TaxableIncome-256,500)Status3 ( c gia nh nhng c thn)Taxablelncome Tax0 - 19,500 0.15*TaxableIncome19,500 - 47,125 2,925.00+0.28*(TaxableIncome-19,500)47,125 - 71,800 10,660.00+0.31*(TaxableIncome-47,125)71,800 - 128,250 18,309.25+0.36*(TaxableIncome-71,800)128,250 - up 38,631.25+0.396*(TaxableIncome-128,250)Status4 (Ngi ch gia nh)Taxablelncome Tax0 - 31,250 0.15*TaxableIncome31,250 - 80,750 4,687.50+ 0.28*(TaxableIncome-31,250)80,750 - 130,800 18,547.50+0.31*(TaxableIncome-80,750)130,800 - 256,500 34,063.00+0.36*(TaxableIncome-130,800)256,500 - up 79,315.00+0.396*(TaxableIncome-256,500)Cc kt qu mu:> status:=2:Taxablelncome:=56890:1.4Bitp 37>taxschdKstatus,Taxablelncome);10859.20>status:=4:Taxablelncome:=35280:>taxschdKstatus,Taxablelncome);5975.90>status:=1:Taxablelncome:=2590000:>taxschdKstatus,Taxablelncome);.1005776500107Bitp1.11. (Tnh cn bc ba ca s21). Dy sxk=20xk1 + 21_1xk1_221, x1= 1, k= 2, 3, ...hi t ti cn bc ba ca 21 rt chm. Hy vit chngtrnh in ran s hng u tin ca dy v nm c bao nhiu bc n tti gii hn trong 5 ch s thp phn.Bitp1.12. Vit chng trnh tnh chui1 13+15 17+ v kim tra n bng4.Bitp1.13. (T hp gia dy Lucas v Fibonacci) Dy s Lucasc nh nghaL0= 2, L1= 1, Lk= Lk1 +Lk2, k= 2, 3, 4, ..., n.Vit chngtrnh in ran s hng u tin caSk= L2k 5F2k, k = 1, 2, ..., nyFklshngthkcadyFibonacci. Hydavodytnh c v a ra gi thit, c chngminh c gi thit a rakhng?38 GiithiulptrnhtnhtontrongmapleBitp1.14. (Tnh hm cosine) Hm cosine c th tnh bngcos(x) = 1 x22!+x44! x66!+x88! x1010!+x1212! Bitp1.15. Vit chng trnh tnh tngn s hng u tin cady Fibonacci.Chng 2CC PHP LP2.1 Phngphpnhtctvngvchiai . 392.2 Phng phpNewtnv cc phngphplpkhc . . . . . . . . . . . . . . . . . 482.3 Bitp. . . . . . . . . . . . . . . . . . . . . 542.1 Phng php nht ct vng v chia i2.1.1 Gii thiu nht ct vngPhng php nht ct vng c th p dng gii cc bi tonti u ca cc hm li. N tng t nh phng php chia i tmnghim khi hm bng khng. Mc ch ca phn ny dng phngphp nhtct vng m t k thutlp trnh,cn phngphpchia i ch gi v nh l mt bi tp.40 CcphplpMt hm gi l li u nu n thc s tng, t n gi tr ccivsaugim.Mtvdinhnhcahmliucvbng Maple nh:>f:=x->-x^2+3*x-2;f:= x x2+ 3 x 2>plot(f,0..2);-1-20-0.5-1.52 1 1.5 0.5 0Hnh 2.1: th trong MapleCho[a, b]lmtonthngcimcbntrong.Vtrcactrong[a, b] cthmtbngcch"cvbnphi caalbaonhiu". V d d dng ni rng c l mt phn ba on thng v phabn phia. Ngha l di ca[a, c] bng13 di ca on thng[a, b]. Mt phn ba ny l t l cac trong[a, b]. Ta c th ni rng2.1Phngphpnhtctvngvchiai 41a c bxHnh 2.2: im t l trong on thngc bt u ta cng thm13 dib a. Ta cc = a + (c a) = a +13(b a) = (1 13)a +13b.Hontontngtmtimd [a, b]vitlbngt,y0 t 1, thd = (1 t)a +tbv t l nht ctt =c ab a= |c a||b a|.Bygichoctrongkhong[a, b]vitlnhtctt,nghalc=(1 t)a +tb. Nu ngc li ta thay i vai tr t v1 t, ta s nhncimkhcd=ta + (1 t)b,nlimi xngcacquatrung im on[a, b]. Cp hai im ny ta gi l lin hp vi nhautrong[a, b]. Tnti mtt lctcbit, cgi lt lctac = (1 t)a +tbbxd = ta + (1 t)bHnh 2.3: Hai im lin hpvng. N c bit v nuc ct on[a, b] vi t l,c = (1 )a +bth im lin hp ca nd ct on thng[a, c] vi cng t s vngny,d = a + (1 )b = (1 )a +c.42 Ccphplpa c bxdHnh 2.4: Nht ct vngT phng trnh trn ta c th tnh c t l nht ct vng= 1 +52Xp x khong0.618.2.1.2 Phng php nht ct vngCho f(x) l hm li u trn on [a, b]. Ta bit tn ti imcc ixcaf(x) trong on[a, b]. Mc ch ca chngta l ctxn on thng ny sao cho n vn cha imx.Nh hnh 2.5, chom

v mrl cp im ct lin hp, tng ngvi t s ct vng, ta c th gi im ct vng bn tri v im ctvng bn phi ca[a, b]. Ta s co[a, b] hoc v pha bn tri thnhon[a, mr] hocvphaphi thnhoncon[m

, b]. Tai tmimduynhtcci cahmli uf(x).Nhvytasosnhccgitr f(m

)vf(mr). Nuf(mr)>f(m

)nhlhnh2.5,th on thng[m

, b] chamrl on ta phi chn. Tng t, nuf(m

) > f(mr), ta s co [a, b] v [a, mr]. Nh vy ta c quy tc chncc on thng con nh sau:Nu gi tr bn tri f(m

) ln hn, th chn on con bn tri[a, mr].2.1Phngphpnhtctvngvchiai 43axbxm

mrf(x)f(mr)f(m

)Hnh 2.5: Phng php nht ct vngNugitr bnphi f(mr)lnhn, th chnonconbnphi[m

, b].Ci p ca phng php l trong mi on con c ct hoc theom

hocmru vi t s vng. Do , co nhng on con ta chcn im ct vng t hp v ch mt nh gi hm cng tnh.Phng php nht ct vng c th m t theo gi m sau:u vo:a, b, fv chnh xc;m bol mt s dng;Bt u lm vic vi on thng[, ] = [a, b];tlt ct t s vng;Tm cp im nht ct vngm

vmr;Tnhv

= f(m

) vvr= f(mr);Khi | | > , th lp li vng lp sau:Nuv

> vrthThay th[, ] vi[, mr];44 CcphplpThay thmrbngm

;Thay thvrbngv

;Thay thm

bng(1 ) +;Tnhv

= f(m

);Ngc liThay th[, ] vi[m

, ];Thay thm

bngmr;Thay thv

bngvr;Thay thmrbng(1 ) +;Tnhvr= f(mr);Kt thcTa c th lp trnh nh sau:#Chngtrnhtnhimccicahmli#uvo:f hmli,#e.g.f:=x->-x^2+3*x-2;#a,bccimvuicaonthng[a,b]#tolsaiscnthit> goldsec:=proc(f,a,b,delta)localgoldratio,gratio,alpha,beta,ml,mr,fml,fmr,stepcount,tau,vl,vr;ifdelta=deltado2.1Phngphpnhtctvngvchiai 45ifvl>vr thenbeta:=mr;mr:=ml;vr:=vl;ml:=tau*alpha+(1-tau)*beta;vl:=f(ml);elsealpha:=ml;ml:=mr;vl:=vr;mr:=(1-tau)*alpha+tau*beta;# imgiamivr:=f(mr)# vrmifi;od;print("Thesolutionis");print(0.5*(alpha+beta));#uraend;g ol ds e c := proc( f, a , b , d e l t a )l ocal g ol dr at i o, g r at i o, al pha, bet a, ml , mr , fml,fmr , s t epcount, tau, vl , vr ;i f d e l t a a:=0;b:=2;tol:=0.001;a:=0b:=2tol :=.001> goldsec(f,a,b,tol);Thesolution is1.5001934984462164631Ta c th gim sai s i, ta c >goldsec(f,a,b,0.00002);Thesolution is 1.4999951775621607752> goldsec(f,a,b,0.000000001);Thesolution is 1.49999999995348818662.1Phngphpnhtctvngvchiai 472.1.3 Vng lp while-do thc hin thut ton nht ct vng ta c dng loi vng lpwhile c ondi t i on dob l oc k of s t at ement sod;Trong vng lp nyblock of statements s lp li khi m conditioncnng. V dnhthuttonnhtctvngtrnlmvicvi[, ] vi vovnglp. Nuonthngchanhth nhngcu lnh c thc hin, khi kim tra thy sai th n nhy n dnglnh sau od;.Chkhidngvnglpwhile-do,khixyraiukintrakhngbaogisaic.Trongtrnghpchngtrnhltctvngtachodeltamthvnglpvtnnntaphickimtradeltam hay khngifdelta x:=array(0..10);x := array(0..10, [])> f:=x->x-cos(Pi*x);f:= x x cos(x)> g:=D(f);#findthederivativeoffg:= x 1 + sin(x)> x[0]:=0.5;2.2PhngphpNewtnvccphngphplpkhc 49x0:= .5>x [1]: =evalf(x[0]-f(x[0]) /g(x[0])) ;xi:= .3792734965Hai ch s ng sau mt bc>x [2]: =evalf(x[1]-f(x[1]) /g(x[1])) ;x2:= .3769695051Nm ch s thp phn ng sau hai bc>x [3]: =evalf(x[2]-f(x[2]) /g(x[2])) ;x3:= .3769670094Ch bngbabcnhncnghimngn9chsthpphn.Qu trnh trn c th dng php lp>f:=x->x-cos(Pi*x);g:=D(f);x[0]:=0.5;forkfrom1to 10dodelta:=evalf(f(x[k-1])/g(x[k-1])):x[k]:=x[k-1]-delta:printf("x[\%2d]=\%15.10fdelta=\%15.10f\n",k, x[k], delta):od:x [ 1]= 0. 3792734965 d e l t a = 0. 1207265035x [ 2]= 0. 3769695051 d e l t a = 0. 0023039914x [ 3]= 0. 3769670094 d e l t a = 0. 0000024957x [ 4]= 0. 3769670092 d e l t a = 0. 0000000002x [ 5]= 0. 3769670093 d e l t a = 0.0000000001x [ 6]= 0. 3769670094 d e l t a = 0.0000000001x [ 7]= 0. 3769670092 d e l t a = 0. 0000000002x [ 8]= 0. 3769670093 d e l t a = 0.0000000001x [ 9]= 0. 3769670094 d e l t a = 0.0000000001x [ 10] = 0. 3769670092 d e l t a = 0. 000000000250 CcphplpMt s ch quan trong khi s dng phng php lp Newton:PhplpNewtonlaphng. Nghalncthsai nuim xut phtx0khng gn nghimx.Thmtr NuphplpNewtonkhnghi t, th chngtacng khng kt lun c c nghim hay bao nhiu bc nath dng.Do nhngl do trn ta phit mt s tiu chun phplpdng li:1. n gin l phi cho s bc lp c th , ngha l u vo cn php lp Newton dng li tixn.2. Php lp dng li khi chnh xc ca nghim tt. Phplp Newton ch ra rng chnh xc caxkph thuc vokf(xk1)f

(xk1).Nhvytascho |k|, mt s nhdngphplpNewton.Nhvy c hai ccht phplp dngli.iuny c ththc hin kt hp gia hai vng lp for-do v while-do:forloop-indexfrom...to...whileconditiondoculnhchunbod;Ta c v d sau:delta:=1.0:forkfrom1to 10whileabs(delta)>10.0^(-8)dodelta:=evalf(f(x[k-1])/g(x[k-1])):x[k]:=x[k-1]-delta:2.2PhngphpNewtnvccphngphplpkhc 51printf("x[\%2d]=\%15.10fdelta=\%15.10f\n",k, x[k],delta):od:x [ 1]= 0. 3792734965 d e l t a= 0. 1207265035x [ 2]= 0. 3769695051 d e l t a= 0. 0023039914x [ 3]= 0. 3769670094 d e l t a= 0. 0000024957x [ 4]= 0. 3769670092 d e l t a= 0. 0000000002Vng lp dng li ti bc th 4 v> 10(8)s ng sau .Phng php Newton c th thc hin theo cch gi m sau y:1. u vo:f, im xut phtx0, bc gii hnn, dung saitol;2. Tnh o hm cafv gn n vog;3. To ra mngx vi ch s t0 nn;4. Gn gi tr chox0;5. t gi tr ban u ca caln hntol;6. Dngcutrc"for... from... to... while... do"thchinvng lp;7. Kim tra nu || < tol;8. Nu ng th in ra php lp ny9. Ngc li thot khoi chngtrnh v thng bo sai.Vd2.2. Ta xt v d p dng phng php Newton tm nghimca>f:=x->x^3-5*x^2+2*x-10;f:= x x35x2+ 2x 10>plot(f,-3..7);thcahmsf(x)thycnghimxpx x=5.Bygita p dng php lp Newton.>newton:=proc(f,x0,n,tol)52 Ccphplp080-8040-406 4 2 0 -2Hnh 2.6: V d php lp Newtonlocalg,x,delta,k;g:=D(f);x[0]:=x0;delta:=1.0;forkfrom1ton whileabs(delta)>toldodelta:=evalf(f(x[k-1])/g(x[k-1]));x[k]:=x[k-1]- delta;printf("x[\%2d]=\%15.10fdelta=\%15.10f\n",k,x[k],delta);od;ifabs(delta)newton(f,4.0,10,10.0^(-8));x [ 1]= 5. 8000000000 d e l t a = 1.8000000000x [ 2]= 5. 1652715940 d e l t a = 0. 6347284061x [ 3]= 5. 0092859510 d e l t a = 0. 1559856433x [ 4]= 5. 0000317760 d e l t a = 0. 0092541752x [ 5]= 5. 0000000010 d e l t a = 0. 0000317752x [ 6]= 4. 9999999970 d e l t a = 0. 0000000037Phep l ap Newton den 6 vong l apNghiem cua no l a 4. 9999999970Nu bt u t mt im bt k th php lp Newton c th sai:54 Ccphplp>x0:=-3.0;>newton(f,x0,10,10.0^(-8));x [ 1]= 1.5084745760 d e l t a = 1.4915254240x [ 2]= 0.3447138130 d e l t a = 1.1637607630x [ 3]= 1. 6065726800 d e l t a = 1.9512864930x [ 4]= 0.8521934710 d e l t a = 2. 4587661510x [ 5]= 0. 4039993150 d e l t a = 1.2561927860x [ 6]= 6.0088494810 d e l t a = 6. 4128487960x [ 7]= 3.5470641800 d e l t a = 2.4617853010x [ 8]= 1.8900892560 d e l t a = 1.6569749240x [ 9]= 0.6757701210 d e l t a = 1.2143191350x [ 10] = 0. 7009957210 d e l t a = 1.3767658420Phep l ap Newton khong hi eu qua buoc 102.3 Bi tpBitp2.1. [T s lt ct vng] Chng minh t s vng l1+52.Bi tp2.2. Lymuchngtrnhltctvngvitchngtrnh giif(x) = 0dng phngphp chia i.Kt qu mu: Tm nghim ca phng trnhf(x) = xsin(x)trong on[1, 1].> f:=x->x-sin(Pi*x);f:= x x sin(x)>plot(f,-1..1);Phngtrnhcbanghimtrongcc khong[1, 0.5], [0.5, 0.5], v[0.5, 1]. iuquantronglphngtrnhiduquacconthng. Dotapdngphngphpchiai theo a phng.> bisect(f,-1,-0.5,0.00000001);2.3Bitp 550.50-0.5-111 0 0.5 -0.5 -1Hnh 2.7: Xc nh ba khongNghiem xap xi la -.7364844497>bisect(f,-0.4,0.5,0.00000001);Nghiem xap xi la .372532403109>bisect(f,0.5,1,0.00000001);Nghiem xap xi l .7364844497Bi tp2.3. Dngchngtrnhcabi tptrcgii phngtrnhx = cos(x)trong khong[2, 1].Mu chngtrnh v gi :>g:=x->x-cos(Pi*x);g := x x cos(x)>plot(g,-2..1); Nh hnh 2.8:>plot(g,-1.1..-0.7); Nh hnh 2.9:Do n c ba nghim:>bisect(g,-1.1,-0.9,0.00000001);56 Ccphplp10-1-3-21 0.5 0 -0.5 -1 -2 -1.52Hnh 2.8: Phng phpchia i khong 1-0.05-0.150.050-0.1-0.7 -0.8 -0.9 -1.1 -1Hnh 2.9: Phng phpchia i khong 2Nghiem xap xi la 1.000000003> bisect(g,-0.9,-0.7,0.00000001);Nghiem xap xi la .7898326312> bisect(g,0,1,0.000000001);Nghiem xap xi la.3769670099Hy th vi hm sf(x) = sin(x) e(x)trn on[0, 7].Bi tp2.4. ThuttonEuclid.c s chngcahai s nguynm vn l mt s nguynd sao cho ng thim vn u chia ht.V dhai snguyn60v45ccschungl3, 5v15. V60 = (2)(2)(3)(5)v45 = (3)(3)(5). c s chung cam vn c sln nht ta gi s l c s chung ln nht cam v n k hiu lgcd(m, n).Vygcd(60, 45)=15.Phngphpcintmcs chung ln nht l phng php Euclid:Gi sn < m.1. Lym chia chon v gnr l s d;2. Nur = 0, thn l c s chung,thot khi chngtrnh;3. Ngc li, thay(m, n) bng cp(n, r) v tr v 1.2.3Bitp 57V d tmgcd(13578, 9198)):>m:=13578;n:=9198;m := 13578 n := 9198>r:=mmodn;r := 4380>m:=n;n:=r;r:=mmodn;m := 9198 n := 4380 r := 438>m:=n;n:=r;r:=mmodn;m := 4380 n := 438 r := 0c s chng ln nht l 438 v khi r = 0.Hyvitmtchngtrnhuvolmvnvktqura lgcd(m, n). Dng chngtrnh tnh mu sau:>gcd(13578,9198);Uoc so chunglon nhat la 438>gcdiv(9198,13578);Uoc so chunglon nhat la 438>gcd(60,45);Uoc so chunglon nhat la 15Bitp2.5. Cho s tin th chp l $ A, m li sut tr theo chuk livtngstinphi trln. Khi ti khonphi tr$PMT cho tng chu k c tnh theo cng thcPMT= Ai1 (1 +i)(n).Gi s ta mua mt cn nh gi $ 120,000 gim 20%. Ta c kh nngtr $713/thng trong 30 nm. Hi li sut cn phi tnh mua nhl bao nhiu?Chng 3MNG TRONG MAPLE3.1 nhnghamngvsdngn . . . . . 583.2 Mtsothngk . . . . . . . . . . . 643.3 Phngphpsng. . . . . . . . . . . . . . 683.4 Mtsdnvmng . . . . . . . . . . . 733.5 Bitp. . . . . . . . . . . . . . . . . . . . . 763.1 nh ngha mng v s dng n3.1.1 S dng mngMt mng dng cha dy d liu. V d ta mun lu 11 phnt u tin ca dy FibonacciF0, F1, ..., F10, u tin phi thit lpmt mng vi ch s t0 n10.>fib:=array(0..10);# definean(empty)arrayfib := array(0..10, [])3.1nhnghamngvsdngn 59Gn cc gi tr vo mi phn t ca mng>fib[0]:=0;fib[1]:=1;fib0= 0 fib1= 1>fori from2 to10dofib[i]:=fib[i-1]+fib[i-2];od;fib2= 1 fib3= 2 fib4= 3 fib5= 5 fib6= 8 fib7=13 fib8= 21 fib9= 34 fib10= 55Tng qut, mng c nh ngha>{Tnmng}:=array({Chsu}..{Chscui});Mi thc th ca mng c th c ra nh {Tnmng}[{Chs}].Ta c th dng?array tra cu ti liu trong Maple.>fib[6]8>fib[10]553.1.2 Lnh seqLnhseqrtcchltoradydliubit. V dmtphn ca dy Fibonacci f0, f1, ..., f10 l dy nh mt mng fi, i =0, 1, .., 10. D dng ghi d liu vofib[0], fib[1], ..., fib[10]hoc cra nhseq(fib[i],i=0..10);0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.Mt cch tng qut seq(f_i,i=m..n) sinh ra dy fvi i t mnn. V d sau y sinh ra cc s l:>seq(2*i-1,i=1..20);1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39Torady bnh phngs t nhin60 MngtrongMaple> seq(i^2,i=1..15);1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 2253.1.3 Gi tr ban u ca mngMng c th c gi tr ban u bng:> t:=array(1..5,[2,6,8,10,3]);t := [2, 6, 8, 10, 3]Tacthgngitrmngbnglnhseq.Vdsinhramngcha cc s chn> even_number:=array(1..20,[seq(2*i,i=1..20)]);even_number = [2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24,26, 28, 30, 32, 34, 36, 38, 40]Saukhimngcgntacthtnhtontrnmngny.Vd ta tnh tng s hng th 8 v th 12 trong mng s chn trn>even_number[8]+even_number[12];40V d mng cc s chnh phng> squ:=array(1..50,[seq(j^2,j=1..50)]);squ := [1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256,289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961,1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849,1936, 2025, 2116, 2209, 2304, 2401, 2500]3.1.4 Cc phn t ca mngTacthnhnghami phnttrongmngcloi dliumMaplechophp, nhccmngtrongccngnnglptrnhkhc.> s:=array(1..3);s := array(1 .. 3, [])> s[1]:=5;s[2]:=JohnDoe;s[3]:=array(1..3,[Mark,Bob,Jack]);3.1nhnghamngvsdngn 61s1:= 5s2:= John Does3:= [Mark, Bob, Jack]>s[3][2];Bobnhnghamngchbntn, mi tnngi chahvtn.Khi khai bo mng 4 phn t, mi phn t li l mng ca 2 phnt khc.>Name:=array(1..4,[seq(array(1..2), i=1..4)]);Name := [[?[1], ?[2]], [?[1], ?[2]], [?[1], ?[2]], [?[1], ?[2]]]>Name[1][1]:=Bill:Name[1][2]:=Clinton:> Name[2][1]:=A1:Name[2][2]:=Gore:> Name[3][1]:=Bob:Name[3][2]:=Dole:> Name[4][1]:=Jack:Name[4][2]:=Kemp:>eval(Name);[[Bill, Clinton], [A1, Gore], [Bob, Dole], [Jack, Kemp]]Phn t th 3 l tn ca Bob Dole>eval(Name[3]);[Bob, Dole]>Name[4][1];Jack3.1.5 V d mng cc s nguyn tS nguynt l mt s nguyndng m n ch chiaht cho1v cho chnh n. Maple c hm s "isprime" xc nh mt s cphil s nguynt khng?V dta bitrng 11l s nguyntcn 15 khng phi l s nguyn t th:>isprime(11);true>isprime(15);false62 MngtrongMapleTacthvitmtchngtrnhnginansnguyntvo mng.#Chngtrnhsinhmngsnguynt#uvo:ns nguyntmunsinhra#ura:p mngchasnguynt>primearray:=proc(n,p)locali,k,count;p:=array(1..n);count:=0;k:=2;whilecountseq(p[i],i=1..50);2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151,157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229p[25]*p[12];3589Ch.Chngtrnhtrntakhnginranghimcabiton,m ta dng i s p ly kt qu ra s dng vi mc ch sau ny.3.1.6 V d xc nh phn t cc i ca mt mngTm s ln nht trong mng cc phn t:2, 5, 9, .#Tmchscaphntccitrongmngs#Input:mchsucamng#n(>m)chscui#slmngs#Output:index_maxchsphntcci#value_maxgitrphntcci#value_maxgitrphntcci>find_max:=proc(m,n,s)locali,value_max,index_max;value_max:=evalf(s[m]);#phntthnhtlmaxindex_max:=m;forifromm+1tondoifevalf(s[i])>value_maxthen#nugitrlnhntmthyvalue_max:=evalf(s[i]);# cpnhtgitrmaxindex_max:=i;#cpnhtchsmaxfi;64 MngtrongMapleod;print("Themaximumvalue:");print(value_max);print("Theindexofthemaximumvalue");print(index_max);end;find_max := proc(m, n , s )l ocal i, value_max , index_max ;value_max := e v a l f ( s [m] ) ;index_max :=m;f or i fromm+1 to n doi f value_max < e v a l f ( s [ i ] ) thenvalue_max := e v a l f ( s [ i] ) ;index_max := iend i fend doprint ("The maximum v al ue : " ) ;print ( value_max ) ;print ("The i ndex of t he maximum v al ue ") ;print ( index_max )end procfind_max(1,7,data);Themaximumvalue : 9.Theindexofthemaximumvalue33.2 Mt s o thng k3.2.1 Min gi trChox1, x2, ..., xnl dy d liu. Min gi trR c nh nghal maxiximinixi. Hy vit chng trnh tnh min gi tr vi uvon v dy d liu.3.2Mtsothngk 65#Chngtrnhtmmingitr camngs#uvo:mchsucamng#n (>m)chscuicamng#x mngccs#ura:mingitr>Range:=proc(m,n,x)locali,value_max,value_min;value_max:=evalf(x[m]);#khiugiscvalue_min:=evalf(x[m]);#mtlmaxvmin# tmmaximumforifrommton do#kimtraphntcnli# nugitrlnhntmthyifevalf(x[i])>value_maxthenvalue_max:=evalf(x[i]);# gnligitrccifi;od;#tmcctiufori fromm tondo #kimtraphntcnliifevalf(x[i])< value_minthenvalue_min:=evalf(x[i]);fi;od;value_max- value_min#aramingitrend;Range := proc(m, n , x )l ocal i, value_max , val ue_min ;value_max := e v a l f ( x [m] ) ;val ue_min := e v a l f ( x [m] ) ;f or i frommto n doi f value_max < e v a l f ( x [ i ] ) thenvalue_max := e v a l f ( x [ i ] )end i f66 MngtrongMapleend dof or i frommto n doi f e v a l f ( x [ i ] ) a:=array(1..10,[2,45,-23,25,43,-26,89,-19,56,9]);a := [2, 45, 23, 25, 43, 26, 89, 19, 56, 9]> rang:=Range(1,10,a);rang := 115.3.2.2 Sp xpGi s ta c dy xm, xm+1, ..., xn v ta mun sp xp chng tngdn. Ta dng m thay cho 1 nn ch s ca mng c khi u mmdo hn. Ta p dng thut ton sau y:Bcm: tm phn t nh nht giaxm, xm+1, ..., xnv i chn vixm.Bcm + 1:tmphntnhnhttrongxm+1, xm+2, ..., xnvi ch n vixm+1Bcm + 2:tmphntnhnhttrongxm+2, xm+3, ..., xnvi ch n vixm+2.......Bcn 1: tm phn t nh nht trongxn1, xnv i ch nvixn1Tng qut, ta c vng lpk = m, m+ 1, ...., n 1Bck:tmphntnhnhttrongxk, xk+1, ..., xnvichn vixkTrongbck,licncvnglptmphntnhnhttrong3.2Mtsothngk 67dy. Do ta c chng trnh:#Chngtrnhspxpdyd liu#x[m],x[m+l]x[n] theothttng#uvo:mchsucamng#n(>n)chscuicady#xmngd liuvichs [m,n]>sort_ascend:=proc(m,n,x)localk,j, tmp,value_min,index_min;ifn evalf(x[j])thenvalue_min:=evalf(x[j]);index_min:=j;fi;od;# i phntth knucnif index_min>k thentmp:=x[index_min];x[index_min]:=x[k];x[k]:=tmp;fi;od;print("ascendingsortingfinished");end;sort _ascend := proc(m, n , x )68 MngtrongMaplel ocal k , j, tmp , value_min , index_min ;i f n seq(a[i],i=2..9);4, , 0,13, e, 3, 23, 83.3 Phng php sngPhng php sng l mt qu trnh gch cc phn t trong dyxm, xm+1, ..., xn3.3Phngphpsng 69m n c tnh cht (hoc khng c tnh cht) no .C th tng tng cch sng lc nh sau: Ta gn vo dysm, sm+1, ..., snban u vi chui "sng". Nu xk b b i, th gn sk:="cht". Cuicng vic sng lc l kim tra li sm, sm+1, ..., sn cn sng hay cht.Nuphnt,vdnhsjl"sng", thtngngxjcntrnsng.3.3.1 V dvtmsnguyntbngphngphpsngBiton:Tmttcsnguyntnmgia1vnbngcchxa i cc b ca s nguyn t.S nguyn t u tin l2, vy tt c s bi ca 2 b xa, nghal ta nh du cc s2 3, 2 4, 2 5, ... nh l phn t "cht".S cn "sng" tip theo l s nguyn t 3, vy ta li b i tt ccc bi ca3, ta nh du3 3, 3 4, 3 5, ... nh "cht" (ch khng nh du3 2 v n cht ri).S cn "sng" tip theo l s nguyn t 5, vy ta li b i tt ccc bi ca5, ta nh du5 5, 5 6, 5 7, ... nh "cht" (ch khng nh du5 2, 5 3, 5 4 v chng cht ri).Tngqut, snguynthinthilk,vytalibittcccbicak,tanhdunh"cht"(chkhngnhduk 2, k 3, k (k 1) v chng cht ri). Sau tm s cn sngsauk v cng vic lp li.#Chngtrnhtmttcccsnguyntnhhnnbngsng#uvo:ncntrncatmkim#ura:p mngchattcccsnguyntnn>prime_sieve:=proc(n,p)locals,i,k,count,flag;70 MngtrongMaple#Khitomngsngs:=array(1..n,[seq("alive",i=1..n)]);#1 khngphilsnguynts[1]:="killed";#thotnu n nforkfrom2to nwhilek^2eval(p);[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73,79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157,163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239,241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331,337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421,431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509,521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613,617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709,719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821,823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919,929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997]3.3.2 V d v nhng s Ulam pT danh sch cc s nguyn dng 1, 2, 3, ... ta b i mi s ngthhai, ch cnccssng1, 3, 5, 7, 9, ... v 3lssngutinvt2 m n khngb "killer", ta b i mi s ngth ba trongcc s sau , nn c1, 3, 9, 13, 15, 19, 21, ... by gi lo n mi sth7cnsngcbi,...Ccskhngbaogibbigils "p". Vit chng trnh in ra cc s p nn.#SpUlam:chody1,2,3,...,n72 MngtrongMaple#bmisth2,ncn:#1,3,5,7,9, 11,13,...#sauddosbs th3. Tngqut,#saubmisth,nu m lsutincn#mlnhnk.cpnhtk chom#vlibmisthk, qutrnhtiptc#nkhiklnhnccady.#iuvo:ncntrnphitm#ura:mngchaccspnn>ulamluck:=proc(n,lucky)locali,count,u,k,flag,survive;#initializethearrayu.#u[i]=alive: isurvives#u[i]=killed: iremoved.u:=array(1..n);fori from1 tondou[i]:="alive"od;survive:=n;#cn phntsngtuforkfrom2ton whilesurvive>k do#bmisthk nuk sngifu[k]="alive"thencount:=0;forifrom1ton doifu[i]="alive"thencount:=count+1;ifcount=kthenu[i]:="killed";count:=0;survive:=survive-1;fi;fi;od;3.4Mtsdnvmng 73fi;od;#Ghilinhngphntsplucky:=array(1..survive);# khitomngspcount:=0;fori tondoifu[i]="alive"thencount:=count+1;lucky[count]:=i;fi;od;print("TheUlamsluckynumbers");seq(lucky[i],i=1..count);# inraend;>ulamluck(100,lucky);"TheUlamsluckynumbers"1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49,51, 63, 67, 69, 73, 75, 79, 87, 93, 993.4 Mt s d n v mng3.4.1 Tam gic PascalH s ca a thc(a +b)nc th tnh c nh s tam gicPascal74 MngtrongMaplen (a +b)nc0, c1, ..., cn0 1 11 a +b 1 12 a2+ 2ab +b21 2 13 a3+ 3a2b + 3ab2+b31 3 3 14 a4+ 4a3b + 6a2b2+ 4ab3+b41 4 6 4 15 a5+ 5a4b + 10a3b2+ 10a2b3+ 5ab4+b51 5 10 10 5 1 Ta c nhn xt sau trong bng nyvi min c(n + 1) s hngc0, c1, ..., cn;c0= cn= 1;Chod0, d1, ..., dk1l cc h s ca(a +b)k1thc0= 1, c1= d0+d1, c2= d1+d2, ..., ck1= dk2+dk1, ck= 1.Hy vit chng trnh a von, a ra (1) h s mngc v (2) inra tam gic Pascal. Mu chy chngtrnh nh sau:>Pascal_triangle(10,c);11, 11,2,11, 3, 3, 11, 4, 6, 4, 11, 5, 10, 10, 5, 11, 6, 15, 20, 15, 6, 11, 7, 21, 35, 35, 21, 7, 11, 8, 28, 56, 70, 56, 28, 8, 11, 9, 36, 84, 126, 126, 84, 36, 9, 11, 10, 45, 120, 210, 252, 210, 120, 45, 10, 13.4Mtsdnvmng 75>seq(c[j],j=0..10);1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 13.4.2 Gi tr trung bnhCho dy s thcxm, xm+1, xm+2, ..., xnc nhng gi tr trung bnh khc nhau nh:(i) Trung bnh cngAA =n

i=mxin m+ 1.(ii) Trung bnh nhnGG =_n

i=mxi_ 1nm+1(iii) Trung bnh iu haH1H=n

i=m1xin m+ 1Vit chng trnh u vom, n, x v in ra ba gi tr trung bnh.Mu thc hin chngtrnh nh sau:>x:=array(0..10,[2,5,9,12,21,33,18,8,ll,15,17]):>Mean(0,10,x);Thearithmeticmean: 13.72727273Thegeometricmean: 11.05802293Theharmonicmean: 8.03317634476 MngtrongMaple> y:=array(l..9,[3,9,12,43,31,18,24,38,5]):> Mean(l,9,y);Thearithmeticmean: 20.33333333Thegeometricmean: 14.86548789Theharmonicmean: 9.924686319Ta c th tnh c c gi tr trung bnh ca h s Newton:> Pascal_triangle(12,c):11,11,2,11, 3, 3, 11, 4, 6, 4, 11, 5, 10, 10, 5, 11, 6, 15, 20, 15, 6, 11, 7, 21, 35, 35, 21, 7, 11, 8, 28, 56, 70, 56, 28, 8, 11, 9, 36, 84, 126, 126, 84, 36, 9, 11, 10, 45, 120, 210, 252, 210, 120, 45, 10, 11, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 11, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1> Mean(0,12,c);Thearithmeticmean : 315.0769231Thegeometricmean: 78.51839612Theharmonicmean: 5.8724985363.5 Bi tpBi tp 3.1. Mng Fibonacci. Vit jchng trnh u vo n, u3.5Bitp 77ra l mng ca dy FibonacciF0, F1, ..., FnNhng khng in ktqu khi chng trnh ang chy m kt qu ara nh mt mng i s v dng lnh seq lit k kt qu.V d mu nh sau:>n:=40:>fibarray(40,f);Fibonaccisequenceisgenerated>seq(f[k],k=0..n);0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584,4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811,514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465,14930352, 24157817, 39088169, 63245986, 102334155Bi tp3.2. Tchvhng. Mtvectorcn thnhphncnh ngha nh mt mng(v1, v2, ..., vn).V d nha = (1, 3, 5, 3, 2) v (9, 5, 2,7, ). Mt tch v hnghai vector c nh nghaa = (a1, a2, ..., an) vb = (b1, b2, ..., bn)a b = a1b1 +a2b2 + anbnHy vit chng trnh u von, a, b v in ra gi tr tch v hng.Kt qu mu nh sau:>n:=5:>a:=array(1..n,[1,3,5,-3,2]);a := [1, 3, 5, 3, 2]>b:=array(1..n,[9,-5,2,sqrt(7),Pi]);b := [9, 5, 2, y/7, n]>dotprod(a,b,n);Thedotproductis 2.34593137578 MngtrongMapleBi tp3.3. Danhschqunbi.Hyvitchngtrnhnhnghamng 52 phn t, mi phnt chas v loi qun bi nhbbichun. Nghal,miphntlmngcahaithnhphn,thnhphnth nhtl s v thnhphn th hai l loi qun bi.B bi c xp theo th t nh v d 3 trong chng2.Mu v d nh sau:> decklist(deck):> eval(deck);[[ace, heart], [2, heart], [3, heart], [4, heart], [5, heart], [6, heart],[7, heart], [8, heart], [9, heart], [10, heart], [jack, heart], [queen, heart],[king, heart], [ace, spade], [2, spade], [3, spade], [4, spade],[5, spade], [6, spade], [7, spade], [8, spade], [9, spade], [10, spade],[jack, spade], [queen, spade], [king, spade], [ace, diamond],[2, diamond], [3, diamond], [4, diamond], [5, diomond], [6, diamond],[7, diamond], [8, diamond], [9, diamond], [10, diomond],[/ocA, diamond], [queen, diamond], [king, diamond], [ace, club],[2, club], [3, club], [4, ciu6], [5, club], [6, club], [7, club], [8, club],[9, club], [10, club], [jack, club], [queen, club], [king, club]]> eval(deck[18]);[5, spade]> eval(deck[37]);[jack, diamond]Bitp 3.4. Gitrcctiu. Hy vit chng trnh xc nhphn t cc tiu trong mng bng cch in ra gi tr cc tiu v chs ca n. Dng d liu ca v d 2 kim tra chng trnh.Bitp3.5. Phngsaichun.s =_n

i=1(xi )2n3.5Bitp 79 yl trung bnh cng cax1, x2, ..., xn, ngha l=n

i=1xin .Bi tp3.6. Gi tr trung gianv gi tr phnt. Choxm, xm+1, ..., xnldyn m + 1sthctheothttngdn.Ch s dn um c th l0, 1 hoc mt s nguyn bt k. Gitrtrunggian ca dy ny c nh ngha nh sau:Trng hp 1. Nun m + 1 l s l, khi gi tr trung gianl phn t gia c ch s ln+m2.Trnghp2. Nun m + 1 l s chn,th gitr trung gianl trung bnh cng ca phn t bn tri v phn t bn phi c chs tin+m12vn+m+12tng ng.Gi tr trung gian cn c gi l phntthhai.Phntthnht vphntPhntthbacnhnghanhsau:Trng hp 1. Nun +m1 l chn, khi dy s c th chiai na u tin ca dy conxm, xm+1, ..., xn+m12v na phn th haixn+m+12, ..., xnPhntthnhtl gitr trunggiancadyconnau,phnt th ba l gi tr trung gian ca dy con na th hai.Trnghp2.Nun m + 1ll,thtntiphntgiaxn+m2. Phn t th nht c nh ngha nh gi tr trung gian cady conxm, ..., xn+m2180 MngtrongMaplev phnt thbacnhnghanhgitr trunggiancadyconxn+m2+1, ..., xn.Hy vit chng trnh cho mt dyxm, xm+1, ..., xninrabaloi phnttrn. Chngtrnhphi tipnhnuvom, n v mng x v sp xp li theo th t tng dn v tm cc phnt.Gi : Ta xt cc bc sau y:u vo:m, n, x1. Sp xp li mngx theo th t tng dn;2. Theo gi tr chn/l sn m+ 1:- Tm gi tr trung gian (Phn t th hai).- Tm ch s khongm1 vn1 cho dy con th nht.- Tm ch s khongm2 vn2 cho dy con th hai.3. Theo tnh chn/lcan1 m1 + 1 vn2 m2 + 1 tm phnt th nht v phn t th ba.V d mu:> x:=array(0..10,[23,12,34,87,25,10,5,19,65,29,71]):> quartile(x,0,10);Thesequenceinascendingorder5, 10, 12, 19, 23, 25, 29, 34, 65, 71, 87The1stsubsequence5, 10, 12, 19, 23The2ndsubsequence29, 34, 65, 71, 87The1st,2nd,and3rdquartiles:3.5Bitp 81122565>y:=array(l..14,[22,11,33,44,51,62,12,81,37,19,9,20,18,5]);y:= [22, 11, 33, 44, 51, 62, 12, 81, 37, 19, 9, 20, 18, 5]>quartile(y,l,14);Thesequenceinascendingorder5, 9, 11, 12, 18, 19, 20, 22, 33, 37, 44, 51, 62, 81The1stsubsequence5, 9, 11, 12, 18, 19, 20The2ndsubsequence22, 33, 37, 44, 51, 62, 81The1st,2nd,and3rdquartiles:1221.044Bitp3.7. Cho = (m, m+1, ..., n)l mt vec t nhng s dng sao chon

i=mi= 1khi trung bnh cng v trung bnhnhn trng s tngng vivec t c nh ngha(i) Trung bnh cng trng s lA=n

i=mixi= mxm +m+1xm+1 + +nxn.(ii) Trung bnh nhn trng s ngha lG=n

i=mxii= (xmm)(xm+1m+1) (xnn)82 MngtrongMapleCho trng sm, m+1, ..., nc nh ngha nh saum=12, m+1=14, ..., i=i12, ..., n1=n22, n= n1.Hy vit chng trnh u vo lm, n, x tnh hai gi tr trung bnhtrngs trn.Dngchngtrnhtnhhaigitr trungbnhtrngs trong d n trn.Mu chy chngtrnh:> Weighted_means(0,10,x);Theweightedarithmeticmean 5.521484376Theweightedgeometricmean 3.933248559> Weighted_means(l,9,y);Theweightedarithmeticmean 9.54296875Theweightedgeometricmean 6.309771865> Weighted_means(0,12,c);Theweightedarithmeticmean 64.87329102Theweightedgeometricmean 6.551390952Bi tp3.8. Vit chngtrnhu voa, b, n tnh(a + b)ndngtam gic Pascal v kt ca h s a thc. Chng trnh c cu trcsau yBc 1. Tnhc0, c1, ..., cndng tam gic Pascal.Bc 2. Tnh v in rac0an+c1a(n1)b +c2a(n2)b2+c3a(n3)b3+ +cn1ab(n1)+cnbn.Chng 4M PHNG XC SUT4.1 Khai boloi dliuchoischngtrnh . . . . . . . . . . . . . . . . . . . . . . 834.2 Thnghimxcsut . . . . . . . . . . . . 854.3 Trchiqunbi . . . . . . . . . . . . . . 934.4 Mtsvdvmphngxcsut . . . 1004.5 Bitp. . . . . . . . . . . . . . . . . . . . . 1044.1 Khaiboloid liuchoischngtrnhMaple phn loi d liu vo cc dng sau y (c th xem ?type).84 Mphngxcsut! . And NotOr PLOT PL0T3D Point RangeRootOf TEXT algext * + algebraic anything algfun algnumalgnumext anyfunc complex arctrig arrayatomic boolean disjcyc complexcons constantcubic dependent exprseq equation evenevenfunc expanded function facint floatfraction freeof infinity hfarray identicalindexed indexedfun list integer intersectlaurent linear matrix listlist literallogical mathfunc negint minus monomialname negative numeric nonneg nonnegintnonposint nothing polynom odd oddfuncoperator point protected posint positiveprime procedure radext quadratic quarticradalgfun radalgnum radnumext radfun radfunextradical radnum relation range rationalratpoly realcons rgf-seq scalarNhng loi d liu sau y thng c s dng nhtarray,numeric,integer,odd,even,positive,negativeKhi ta vit chng trnh mun mi i s thuc mt d liu nomytnhkimtrakhi adliuvo. V dchngtrnhtm s phn t trong mng sxm, xm+1, ..., xnTacthdngbai sm, n, xnhlsnguyn, snguynvmng tng ng. Khi ta c th khai bo chng trnh nh sau:quartiles:=proc(m::integer,n::integer,x::array)Khi chngtrnhthchin, Mapletngkimtraphntvoxem c tng ng vi loi d liu khai bo khng? nu khng ngMaple s thng bo li.Vd4.1. V d sau tnh cn bc hai ca ca tng ba s thc:4.2Thnghimxcsut 85>sqrt3:=proc(a::numeric,#uvo:sthcb::numeric,#uvo:sthcc::numeric,#uvo:sthcans::evaln#uraphikhaibonhloi"evaln"ans:=sqrt(a+b+c);end;>sqrt3(3,5,8,t);4>sqrt3(s,5,7,ans);Error, sqrt3 expects its 1st argument, a, to be of type numeric, butreceived sCh khi khai bo u ra l mt bin phi l evaln.4.2 Th nghim xc sut4.2.1 Gii thiuTrong xc sut mt th nghim l mt hot ng hoc mt hnhngxy ra vimt kt qu no,nh tri qunbi xungmtbn, ... Mi ln lp li mt th nghimta gi l mt phpth.Vd c baqu bngtrng v nmqubng trong hp.Lymtqu bng t trong hp ngi ta nhn bit mu ca n. Nh vyhnh ng rt bng t trong hp l mt th nghim. Nu ngi tatrlibngsaukhirtkhibngtronghpththnghimcgilrt bngctrli.Nutrnghpnyxyrath tacvhn cuc th nghim.Nu ta t cu hi sc xut ca rt bng c tr li l bao nhiukhi ta rt lin tip ba qu u l mu ? V mt l thuyt cho ktqu lp =_58_3. Ta c th lp chngtrnh m phngth nghimv tnh xc sut ca n.86 Mphngxcsut4.2.2 Sinh s thc ngu nhinTrong Maple c hm s rand() sinh s nguyn dng 12 ch sngu nhin. Chng trnh sau y sinh ra mt s thc trong khong[a, b]:# hmsinhrasngunhintrongkhong[a,b]real_ran:=proc(a::numeric,b::numeric)localx;x:=rand();evalf(a+(b-a)*x/999999999999);end;V chy th> s:=real_ran(-3,2);s := 1.6031247374.2.3 Sinhsnguynngunhintrongmtkhongno Munsinhramtsnguyntrongmtkhongnotaxydng hm sau:int_ran:=proc(m::integer,n::integer)round(evalf(m-0.5+(n-m+1)*rand()/999999999999))end;Kt qu l> k:=int_ran(l,52);k := 50> m:=int_ran(l,52);m := 84.2Thnghimxcsut 874.2.4 V d rt c tr liGi s trong hp c 3 qu bng trng v 5 qu bng . Ta cth nh s t 1 n 3 cho cc qu bng trng v cng nh s t4 n8 chocc qu bng. Rt ngunhinmt qu bngtronghp,tngngvilymtst1n8vnhvysautaxc nh c mu qu bng. Do th nghim rt mt qu bngc th m phng bng chng trnh sau:one_ball:=proc()localk;k:=int_ran(l,8);ifkone_ball();white>one_ball();red88 Mphngxcsut4.2.5 V d rt ba qu bng khng tr liGisvnc3qubngtrngv5qubngtronghp.Nu ta rtkqu bng t hp khng tr li, th th nghim ca tac th m phng bng cch:1.nhsccqubngt1n8.Basumutrngccs cn li l .2. Rtkqu bng tng qu mt:- Rt ln th nht bng t 1 n 8. Khi ta i qu bng rt thnh s 8 (= 8 1 + 1).-Rtlnthhaibngt1n7.Khitaiqubngrt thnh s 7 (= 8 2 + 1).- Rt ln th ba bng t 1 n 6. Khi ta i qu bng rtthnh s 6 (= 8 3 + 1).- Tng qut, rt ln th j bng t 1 n n=8-j+1. Khi ta iqu bng rt thnh s n , cho n khi khng i c na.3. Bin dch cc s thnh mu.draw_k_ball:=proc(k::integer,#sbngcnrtdraw::evaln#ura:muccqubng)localball,i,j,n,m,temp;ball:=array(1..8);#hpchabngforito 3doball[i]:="white";od;forifrom4to 8doball[i]:="red";od;draw:=array(1..k);# chabngrtran:=8;# nls bngcnli4.2Thnghimxcsut 89forjtok do#Rttngqumtm:=int_ran(1,n);#rtst1 nndraw[j]:=ball[m];# lymuifmn andball[m] ball[n]thentemp:=ball[m];# chuynbngxungcuiball[m]:=ball[n];# ivoch rtraball[n]:=temp;fi;n:=n-1;# salinod;end;Th nghim>draw_k_ball(3,ball):>eval(ball);[white, red, red]4.2.6 V d v tnh xp x ca xc sutNumuntnhxcsutcaskinrt3qubngkhngtrlivcbaquul,tacthmphngthnghimnln,m s ln thnh cng rt ra theo cng thcs ln rt ra thnh cng nh nl gi tr xc sut xp x.#Chngtrnhm phngtnhxcsutrtkqubngkhngtrli,tronghpc3bng#uvo:ksbngcnrt#nlsphpthdraw_test:=proc(k::integer,n::integer)locali,j,count,ball,flag;ball:=array(1..k);#chabngrtc90 Mphngxcsutcount:=0;# smfori tondodraw_k_ball(k,ball);# rtkbngkhngtrliflag:=0;#khito flag=0,gisthnhcngforj tokwhileflag=0do#kimtraxemcttckhngifball[j]="white"thenflag:=l;#qubngtrngfi;od;# tnhnhnglnrtthnhcngcbaquu(flag=0)ifflag=0thencount:=count+1;fi;od;evalf(count/n);# tnhxcsutend;draw_k_ball:=proc(k::integer, draw::evaln)localball,i,j,n,m,temp;ball:=array(1..8);fori to3doball[i]:="white";od;fori from4 to8doball[i]:="red";od;draw:=array(1..k);n:=8;fori tokdom:=int_ran(1,n);draw[i]:=ball[m];ifm nandball[m]ball[n]thentemp:=ball[m];4.2Thnghimxcsut 91ball[m]:=ball[n];ball[n]:=temp;fi;n:=n-1;od;end;int_ran:=proc(m::integer,n::integer)round(evalf(m-0.5+(n-m+1)*rand()/999999999999))end;Ta th vi s liu>draw_test(3,300);.1700000000Xc sut chnh xc l>evalf(5)-2/(2!)*(8!)));!.1785714286By gi tng s th ln>draw.test(3,10000);.1758000000Vy cng cho nhiu php th th xc sut cng tin gn ti gitr ca n.4.2.7 V d hai im trong hnh vung c khong cchnh hn mtCho hnh vung c cnh 1 n v. tnh xc sut nm hai hn nhtronghpvungnymkhongcchcachngnhhn1?iunyrt khtnhtonchnhxc.Tacthvitchngtrnhtnh gn ng chng.Mthnhvungcthxemnhtphpim {(x, y)|0 x 1 v0 y 1} trong h ta xy. Nm ngu nhin hn c thbiu din nh im(x, y) trong tp ny. Chng trnh m phng l92 Mphngxcsutsinh ra hai tp s ngu nhinx vytrong khong[0, 1].#Chngtrnhtnhxpxxcsutnmhaiimckhongcch#uvo:num_of_trialssphpth#ura:xpxxcsuttwo_pts:=proc(num_of_trials::integer)localcount,i,x1,y1,x2,y2,distance;ifnum_of_trials> 0thencount:=0;foritonum_of_trialsdox1:=real_ran(0,1);# sinhraimthnhty1:=real_ran(0,1);x2:=real_ran(0,1);# sinhraimthhaiy2:=real_ran(0,1);# Kimtracnhhn1distance:=evalf(sqrt((x2-x1)^2+(y2-y1)^2));ifdistance< 1.0thencount:=count+1;fi;od;evalf(count/num_of_trials);elseprint("invalidinput");fi;end;real_ran:=proc(a::numeric,b::numeric)localx;x:=rand();evalf(a +(b-a)*x/999999999999);end;> two_pts(500);0.97400000004.3Trchiqunbi 93>two_pts(2000);0.9780000000nh gi bng cng thc t hp>(.98*500+0.967*2000)/2500;0.9696000000Php th cng ln th cng chnh xc>two_pts(10000);0.97570000004.3 Tr chi qun bi4.3.1 Chng trnh rt ngu nhin k qun biMcchlmphngrtngunhinkqunbi. Nhphntrc ta phi c ba chngtrnh. Nhng chng trnh ny lm ccvic sau y:1. Chng trnh chnh: lyks ngu nhin khc nhau t 1 n52 bng cch gi hm int_ran nh hm con ca n.2. Mi s nhn c c dch sang tng ng vi tn qun biv gi chng trnh con card lm vic .####chngtrnhchnh####Mphngrtkqunbitrongb52qun#uvo:kls quncnrt#ura:mnchatnqunbirtcdraw_k_card:=proc(k::integer,#squnbicnrthand::evaln#tnqunbi)localdeck,i, m,n,temp;ifk< 1ork >52 then# quickexitif k 52print("isthnhtt 1 n52");94 MphngxcsutRETURN();fi;hand:=array(1..k);deck:=array(1..52,[seq(i,i=1..52)]);n:=52;fori from1 tokdom:=int_ran(1,n);hand[i]:=deck[m];ifmnthentemp:=deck[m];deck[m]:=deck[n];deck[n]:=temp;fi;n:=n-1;od;# Lytnqunbifori from1 tokdom:=hand[i];#lyshand[i]:=card(m);# gichngtrnhconod;end;###chngtrnhconlysngunhin###int_ran:=proc(m::integer,n::integer)round(evalf(m-0.5+(n-m+l)*rand()/999999999999))end;###chngtrnhcontnqunbi####uvo:m lst1 n52#ura:tnqunbicard:=proc(m::integer)localrank,suit;ifm>0andm0andmk_card_get_ace(4,400);.3125000000Xc sut chnh xc tnh theo l thuytla1522552129= .2812632745.>k_card_get_ace(5,200);.3100000000Xc sut chnh xc tnh theo l thuyt la 148!5!43!52!5!47!= .3411580017.>k_card_get_ace(5,500);.3480000000Xc sut chnh xc tnh theo l thuytla1522552129= .2812632745.4.3Trchiqunbi 994.3.3 V d tnh xc xut rt k qun bi c ng mtqun tChng trnh di y phn chng trnh con b qua, tng tnh chng trnh phn trc.>#Chngtrnhtnhxcsutrtk qunbicngmtqunacek_card_get_1_ace:=proc(k::integer,n::integer)localcount,i,j,hand,num_of_ace;count:=0;fori from1 ton dodraw_k_card(k,hand);num_of_ace:=0;forj from1 tok doifhand[j][1]= "ace"thennum_of_ace:=num_of_ace+1;#mstfi;od;ifnum_of_ace=1thencount:=count+1;#msthnhcngfi;od;evalf(count/n);end;Ta xt trng hp rt 4 qun vi s ln khc nhau>k_card_get_l_ace(4,300);.2766666667>k_card_get_l_ace(4,1000);.2480000000Xc sut ng l69184270725= .2555. Nu kt hp 1300 ln th(.2766666667*300+.2480000000*1000)/1300.2546153846100 Mphngxcsut4.4 Mt s v d v m phng xc sut4.4.1 Kim tra nhim HIVGi s s kim tra HIV l 99%. Ngha l s chnh xc c hiunh sau:1. Nungi mangvirusHIV, th ngi nyc0.99xcsutc kim tra l dng tnh v 0.01 xc sut l m tnh.2. Numtngi khngmangvirusHIVth ngi c0.99xc sut kim tra m tnh v 0.01 xc sut kim tra dng tnh.Ngi ta bit rng 0.5% dn s mang virus HIV. Nu mt nginokimtraHIVmdngtnhthxcsutbaonhiungi mang mm bnh HIV?Bn c th ch ra cho ngi ny vi d kin l ngi y ch c13may mn mang HIV. Hy chng minh bng chngtrnh Maple.Chng trnh i hi m phng mt ngi trong cng ng no kim tra HIV dng tnh v tinhsxacs sut ngi c th mangHIV. m phng ngi no kim tra dng tnh:1. Sinhran qunchngm 5%trong chngmc bnhHIV vnhng ngi cn li vn khe.2. Nht tng ngi ra kim tra:(a) Nu ngi mang bnh, 99% kh nng dng tnh, 1% mtnh.(b) Nu ngi khng mang bnh, 99% m tnh v 1% dngtnh lp li cho n khi gp ngi mang HIV dng tnh.#uvo:n slnlplithnghim#uratnhxcsutxpxbnhimHIV>true_hiv:=proc(n)localcount,i,j,k,u,s,result;4.4Mtsvdvmphngxcsut 101count:=0;#Khitothnhcngu:=array(1..200,["sick",seq("healthy",i=2..200)]);fori tondoresult:="negative";#vnglptmcdngtnhwhileresult="negative"do#lymtngitrongdnsk:=int_ran(1,200);#thchinkimtra(nhquays s)ifu[k]="sick"thens:=int_ran(1,100);#lyngingunhinifs =1 thenresult:="negative"# 1out100 bmtnhelseresult:="positive"fi;elses:=int_ran(1,100);#lyngunhinifs =1 thenresult:="positive"# 1out100bdngtnhelseresult:="negative"fi;fi;od;ifu[k]="sick"thencount:=count+1fi;od;evalf(count/n);end;int_ran:=proc(m::integer,n::integer)102 Mphngxcsutround(evalf(m-0.5+(n-m+1)*rand()/999999999999))end;Chy th chngtrnh ny> true_hiv(500);0.002000000000> evalf(99/298);0.33221476514.4.2 Xc sut nhng ngi c n trong mt nhmTrongmtnhmkngi,mingicktnghavinhngngi khcnhlbn. Numtngi khngktnghavi aithngi ny gi l c n. Tm xc sut mt ngi c n?Ta c th dng kt bn mt cch ngu nhin l# uvo:ks ngitrongnhm# nsphpthlonesome:=proc(k::integer,n::integer)localu,i,j,r,s,flag,count;u:=array(1..k);#tokhnggianchomingicount:=0;# msngicnfori tondo# slnthnghimlpliforj tokdo# btuchomingicnu[k]:="lonesome"od;forj tokdo# mingiktbnngunhin# ngiktbnutin: rr:=j;whiler=jdo#khngktbnchnhmnhr:=int_ran(1,k);#ngij ktbnod;u[r]:="happy";# rktbnhtcn4.4Mtsvdvmphngxcsut 103#ktbnthhai: ss:=j;whiles=jors=rdo#lpchonkhigpngimis:=int_ran(1,k)od;u[s]:="happy";# ngi sktbnod;#kimtraxemaicncnflag:=0;#cbnforjtok whileflag=0doifu[j]="lonesome"thenflag:=1fi;od;ifflag=1thencount:=count+1fi;od;evalf(count/n);end;int_ran:=proc(m::integer,n::integer)round(evalf(m-0.5+(n-m+1)*rand()/999999999999))end;Th nhm 5 ngi trong 200 ln th>lonesome(5,200);.06000000000Th nhm 30 ngi trong 500 ln th>lonesome(30,500);.1180000000104 Mphngxcsut4.5 Bi tpBi tp 4.1.Mt ng trn bn knh 1 nm trong mt hnh vung22. Nu ngi ta nm hnh trn vo hnh vung, th xc sut hnhtrnnmtronghnhvungl4. Nungi talpli nlnthnghimthngitatnhcxcsutcannhnvi4gnvis. ylmtcchtnhsnhngkhnghiuqu. Hyvitchngtrnhdngthuttontrntnhgnngvi uvon s th nghim.Bi tp4.2. Vitchngtrnhtnhxcsutrtkqunbi ccng cht. Xc sut chnh xc l413!(52k)!(13k)!52!.Bi tp4.3. Tronghpc4qubngtrngv8qubng.Vit chng trnh tnh xc sut rtk qu bng khng tr li v ttc qu bng u l .Bitp4.4. Dng chng trnh trong HIV tm xc sut m ngib kim tra l m tnh nhng ngi ny li thc s khe.Bi tp4.5. C 15 ngicon trai v 10 ngi con gi. Mi ngicontraislmbnvimtngicongi.Tuynhin, cthmts ngi con trai mun chn bn cng mt ngi con gi. Tm xcsut m t nht mt ngi con gi c n?Bi tp4.6. Ckngi trongmtnhmvcbnnalmttrong s , bn c gi l s # 1. Mi ngi lm bn vi hai ngikhc. Nu bn chn mt ngi bn m ngi li cng chn chnhbnlbnth gi lbntmgiao. Tmxcsutbncthtmthy cp tm giao ca mnh.Chng 5H PHNG TRNHN GIN5.1 Giiphngtrnh . . . . . . . . . . . . . . 1055.2 Vdvbitonhnhpphntrm. . 1075.3 ngthngstnht. . . . . . . . . . . . 108Danhmctkha . . . . . . . . . . . . . . . . . 1105.1 Gii phng trnh5.1.1 Lnh ca MapleTrongMaplehmsolvednggii phngtrnh. Hyxemhng dn bng lnh >?solve. C php chung gii phng trnhl>solve(,);106 HphngtrnhnginV d gii phng trnh5x + = 9 c lnh> solve(5*x+Pi=9,x); 15 +95Hoc mt cch khc>eqn:=5*x+Pi=9:variable:=x:> solve(eqn,variable);15 +95 gii h phng trnh ta c mu>solve({},{})V d gii h phng trnh sau:_3x + 6y = 54x 5y = 3> equations:={3*x+6*y=5,4*x-5*y=3};equations := {3x + 6y= 5, 4x5y= 3}> variables:={x,y};variables := x, y> solve(equations,variables);{y=1139, x =4339}Mun s dng c nghimphi dng lnh gn vo cc binnghim. V d cho h phng trnh>equations:={a*x+b*y+c*z=d,e*x+f*y+g*z=h,i*x+j*y+k*z=l};equations := {ax +by +cz= d, ex +fy +gz= h, ix +jy +kz= 1}variables:={x,y,z};variables := {x, y, z}solutions:=solve(equations,variables);solutions := {z=jedajh+aflfidebl+ibhafkajgebk+jecfic+ibg,y= aglahkeclgid+hic+edkafkajgebk+jecfic+ibg,x =bglbhk+cjhcfl+dfkdjgafkajgebk+jecfic+ibg}assign(solutions);x;y;z;bglbhk+cjhcfl+dfkdjgafkajgebk+jecfic+ibgaglahkeclgid+hic+edkafkajgebk+jecfic+ibgjedajh+aflfidebl+ibhafkajgebk+jecfic+ibg5.2Vdvbitonhnhpphntrm 1075.2 V d v bi ton hn hp phn trmMthachthnhpc11%axittrnvi hnhp4%. Hicn mi loi bao nhiu mi loi hnhp c hn hp axit 700mililit vi 6%. Bi ton tng qut hn, nu ngi ta mun trn hnhpa%vhooanxhpb%chnhpc%vidunglngs.Choxvylslnghnhpcnphitrn.Thbitonivgii phngtrnh sau y:_x +y = sax +by = csTa c th lm chng trnh tnh s lng ca mi loi hn hp.#uvo:a,b,c,s nhmttrn#ura:x khilngcaa#ykhilngcaofbmix_2_solutions:=proc(a::numeric,b::numeric,c::numeric,s::numeric,x::evaln,y::evaln)localequations,variables,solutions;equations:={x+ y= s,a*x+b*y=c*s};#nhnghiaphngtrnhvariables:={x,y };#nhnghabinsolutions:=solve(equations,variables);#giiphngtrnhassign(solutions);#gnnghimvox,yend;Th vi cc s:>mix_2_solutions(11,4,6,700,x,y);108 Hphngtrnhngin> x;y;200 5005.3 ng thng st nhtNgi ta cho mt dy s liu:x:=array(1..5,[1,2,3,4,5]);y:=array(1..5,[4.1,4.4,5.1,5.5,5.7]);Thitlpdyimpoints:=[seq([x[i],y[i]],i=1..5)]:Ta c th v th:plot(points,style=point); Cuhi tralngthngno5.24.45 4 3 2 15.64.8Hnh 5.1:l tt nht i qua cc im ny?Nghaltakhngbity=g(x).ngintachongl mt ng thng c dngy= ax +bv ta c gng tma vb.Ta bit rng phng trnh ng thngy= ax+b vi a l h snghing v s b l im ct trc tung. Ti mi im xi, i = 1, 2, ..., 55.3ngthngstnht 109hiu gia d liu v ng thng lyi (axi +b), i = 1, 2, 3, 4, 5.Ta c th xc nh tt nht sao cho5

i=1[yi (axi +b)]2c gi tr nh nht. Mt cch tng qut vi d liu xi, yi i = m, ..., n,ta c gng tma vb sao cho hm sf(a, b) =n

i=m[yi (axi +b)]2l nh nht.Tabitrngiukincnchomthmtcctiulttccc o hm bcnht ca n bng0. Hmf(a, b) l hm hai binnn n c cc tiu khiafvbphi bng0. Do 2_n

i=m[yi (axi +b)]xi_= 02_n

i=m[yi (axi +b)]_= 0n gin ha cn_n

i=mx2i_a +_n

i=mxi_b =n

i=mxiyi(5.1)_n

i=mxi_a + (n m+ 1)b =n

i=myi(5.2)Bng cch gii phng trnh via vb ta tm c ng st nht.>line_fit(1,5,x,y,a,b);110 Danhmctkha