maple tqbao
TRANSCRIPT
TRNG I HC CN TH KHOA S PHM B MN TON
BAI GIANG
NGON NG LAP TRNH
M A PL EBien soan Th.S Trng Quoc Bao
NM 2004
LI GII THIUNgy nay, cng vi nhng thnh tu tuyt vi trong lnh vc cng ngh thng tin, ngi ta xy dng nhiu phn mm h tr cho cng tc hc tp v nghin cu. Mt thc tin c bit t lu l nhng bi ton t ra trong thc tin thng khng c gii quyt bng nhng mo tnh ton th cng m phI dng n nng lc tnh ton ca my tnh in t. Phn mm tnh ton ra I nhm p ng yu cu ca thc tin, a cc tnh ton phc tp (c ph thng ln cao cp) tr thnh cng c lm vic d dng cho mi ngi. Ton hc l thng nht nn cc phn mm tnh ton cng c cu trc c bn ging nhau. V vy, nu bit s dng phn mm ton hc no th cng d dng s dng c cc phn mm khc. Phn mm tnh ton Maple lm cho vic gii cc bi ton tr nn n gin v nhanh chng gp phn lm tng hiu sut lm vic ca chng ta trong hc tp, nghin cu v ging dy. Maple l phn mm do mt nhm cc nh khoa hc ca Canada thuc trng i hc Waterloo lm ra vi mc ch gii quyt mi cng vic lin quan n tnh ton. Tp ti liu ny ch cp n nhng vn c bn, ri t chng ta c th khm ph ra nhng kh nng tnh ton v biu din v cng phong ph ca Maple. iu cn lu l vic s dng cc phn mm tnh ton hin i khng i hi ngi dng phi c k nng lp trnh cao cp m ch yu cu ngi s dng nm vng cc kin thc l thuyt c bn.Vi Maple ta ch cn thc hin nhng cu lnh n gin ch khng phi nh lp trnh cc ngn ng khc trong tnh ton. Thng qua hm tnh ton trong mi trng Maple, chng ta rn luyn k nng s dng my tnh gii quyt vn c th v ton hc. Maple c kh nng tnh ton trn s thc ln s phc, ngoi cc hm ton hc dng sn trong Maple v mi lnh vc: Lng gic, gii tch, hnh hc, i s tuyn tnh, l thuyt s, thng k, th, phng trnh vi phn v o hm ring, Maple cng cho php thit lp thm cc hm hoc th tc chuyn dng theo mc ch ca ngi s dng. Nhng yu cu ti thiu khi s dng Maple: - Bit s dng my tnh (Tt, m my, g vo lnh) - Bit cch gii cc bi ton - Bit ting Anh ti thiu. Yu cu v cu hnh my: Bi ging ny gii thiu Maple Version 6.0 i hi my c dung lng RAM t 8MB tr ln, dung lng a cng dng ring cho n khong 70MB i v i cc my chy trn mi trng Windows v c th chy trn mi trng mng NT, UNIX. Bi ging ny c vit da trn cc sch hng dn s dng Maple t cc ti liu ting Anh cng nh ting Vit, nhng ch yu c dch t phn Help ca chnh chng trnh ny. Do nu c g cha r chng ta c th tham kho phn Help ca chng trnh (bm t hp phm ALT + H) hiu r hn.
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CHNG I
MI TRNG LM VIC CA MAPLEKhi khi ng Maple c mn hnh giao din nh sau:
Cum x l
Biu thc lnh
Kt qua tnh
th
I.Giao din v mi trng tnh ton Mt trang lm vic (Worksheet) ca Maple bao gm nhng thnh phn c bn: 1. Cm x l (Execution group) Nm trong ngoc vung bn tri ca du nhc lnh, mi tnh ton u c thc hin trn cc cm x l ny. N c th cha cc lnh ca maple, kt qu tnh ton, th... Mun a vo Worksheet mt cm x l sau an vn bn ang cha con tr ta thc hin nh sau: + Insert / Execution group / apter cursor hoc + Click chut vo nt c biu tng [> trn thanh cng c. 2.Vn bn (Text) Ta c th nhp vo vn bn text trong worksheet. Ta g ting Vit trong maple tng t nh g ting vit trong cc phn mm ng dng khc nh: Word, Excel........ Mun a vo Worksheet mt an vn bn mi sau con tr ta thc hin lnh: + Insert / Paragraph / Apter cursor. Hoc + Click chut vo nt c biu tng ch T trn thanh cng c. 3. th (Graph) Maple c kh nng ha trc tip c ngha l cho php v th ngay trong trang worksheet.Trang 2
4. Siu lin kt: (Hyperlink) L mt mu vn bn m nu ta kch vo th s dn ta n mt mc khc trong worksheet hin hnh hoc mt worksheet khc. Mun to siu lin kt ta chn chui k t cn click vo khi lin kt ri thc hin nh sau: Format / Convert to / Hyperlink Sau khi hin ra hp thoi ta a a ch cn lin kt vo 5. Lnh v kt qu trong Maple (Command and Output) Lnh ca maple c a vo worksheet sau du nhc lnh trong cm x l. Kt thc dng lnh bng du hai chm : hoc du chm phy ; + Nu kt thc dng lnh bng du : th kt qu tnh ton khng hin th ra mn hnh. + Nu kt thc dng lnh bng du ; th kt qu s hin th dng pha di pha sau cu lnh. Thng thng lnh ca Maple c hin th bng Font ch Courier mu v kt qu c hin th bng Font ca Maple Output mu xanh. (y l nh dng mc nh v chng ta c th thay I bng cc chc nng nh dng ca Maple). V d: [> sin(Pi/3): [> sin(Pi/3);1 3 2
Mun thc hin dng lnh no th a con tr v dng lnh nhn phm Enter Nu c nhiu dng lnh trong cm x l th khi ta nhn phm Enter tt c cc lnh trong cm x l u c thc hin. Khi cn xung dng vit cc lnh trong cng mt cm x l (Khng phi thc hin cc lnh trong cm x l) ta dng Ctrl + Enter. Cn thc hin dng lnh theo th t t trn xung di, v mt s tnh ton trong cc bc sau c th ly kt qu t bc trc, ngc li th khng th c. Lnh ca maple c hai loi: Lnh tr v lnh trc tip + Lnh trc tip: Cho ta bit ngay kt qu ca lnh. V d: [> sum(k,k=1..n);1 n (n + 1) 2
+ Lnh tr: Khi s dng lnh tr ta ch thu c biu thc tng trng v mun bit tr s ca biu thc ta dng thm lnh Value( ) V d: [> S:=Sum(k,k=1..n);S :=k=1
k
n
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[> value(S);1 n (n + 1) 2
Thng th gia lnh tr v lnh trc tip khc nhau k t u l +Lnh tr k t u ch hoa: Sum(k,k=1..n), Int(expr,x),... +Lnh trc tip k t u l ch thng: sum(k,k=1..n), int(expr,x),... 6. Maple qui nh cc php ton bng cc k t sau a. Cc php ton s hc Php cng Php tr Php nhn Php chia Php ly tha Php giai tha b. Cc php ton quan h Ln hn Nh hn Nh hn hay bng Ln hn hay bng Bng Khc > < = = + * / ^ !
c. Cc ton t logic: and, or, not. Kt qu ca cc php ton quan h l: True(ng), False(sai), FAIL (khng so snh c). Ch : Cc thnh phn trn c th c xp vo nhng mc (section) cho d tm hoc mc con (subsection) trong worksheet II. Cc nhm lnh trong mt chng trnh: 1. Bin (Variable) a. Tn bin: C th l mt chui k t, s hoc ng gch di (_), c th ch thng hoc ch hoa, tn bin di ti a l 524.271 k t i vi chun 32 bit, v 34,359,738,335 i vi chun 64 bit. Khng c khong cch gia cc k t. V d: Dathuc:=2x+1; Dathuc:=`dathuc` Phuongtrinh_12:=x2-x+2; Khng nn bt u tn bin bng du gch di (_) v n s trng vi tn bin ton cc trong maple.Trang 4
Khng nn kt thc bng du ng (~) v n s trng vi tn bin trong maple khi bin b rng but bi iu kin. Cc tn bin c th c ghp vi nhau bng ton t || hoc bng hm cat() Th d: Phuongtrinh:=2x+1: Delta:=45: Kt hp hai bin trn bng ton t || phuongtrinh||delta phuongtrinhdelta Kt hp hai bin trn bng hm cat( ) cat(phuongtrinh,delta) phuongtrinhdelta Ch : Maple phn bit k hiu hoa v k hiu thng A a, B b nn khi s dng bin ta phi ch n vn ny. Sau khi thc hin xong cc lnh trong mt c m x l ngi ta thng dng lnh restart khi ng li gi tr cho bin. b. Khai bo bin + Bin cc b: Th d: + Bin ton cc: Th d: 2.Lnh gn C php: V d 1: := ; a:=3; b:=2; b:= a; local ,[],..; local x, y, tong; global ,[],..; global a, b, tongcong;
V d 2: tamthuc:=x^2-3*x+1; x2-3x+1 delta:= discrim(tamthuc,x); 5 cat(delta cua tam thuc,tamthuc,la,delta); delta cua tam thuc x2-3x+1 la 5 V d 3: hamtich:=proc(x,y) local a,b; global c;Trang 5
a:=x*y; b:=a*x*y c:=a*b end proc; 3. Lnh iu kin r nhnh if then ; else ; end if : V d: Th tc n gin kim tra s dng hay m > g:=proc(so) if so>0 then print(`So duong`) else if so=0 then `So khong` else `So am` end if end if end proc: 4. Cclnh vng lp: a. Vng lp FOR (vng lp c s ln xc nh) C php: [for ] [from ] [by ] [to ] do end do; Hoc: [for ] [in ] do end do; Vi expr l mt chui cc gi tr, hay mt chui cc biu thc no b.Vng lp While ( vng lp c s ln khng xc nh) C php: While do < Nhm lnh> end do; c. Vng lp hn hp FOR v While C php: [for ] [from ] [by ] [to ] [while] do end do; V d: a. Tnh tng t 6 n 14 vi mi ln tng haiTrang 6
> for i from 6 by 2 to 14 do print(i) end do;6 8
1012 14
b. Tnh tng t 7 ti 9 vi bc tng l 1 > restart: t := 0; for a from 7 by 1 while a < 10 do t:= t+a; x:=t^2; end do;t := 0t := 7 x := 49
t := 15x := 225 t := 24
x := 576
c. Tnh tng ca cc s trong mt dy c lit k sn > restart: V := 0; for z in x, y, a, 3 do V := V+z; end do;
V := 0 V := x V := x + y
V := x + y + aV := x + y + a + 3
d. Ch dng vng lp while > restart: a:=10; while a>5 doTrang 7
x:=a^2; a:=a-1; end do;
a := 10
x := 100 a := 9x := 81 a := 8
x := 64 a := 7x := 49 a := 6
x := 36 a := 5
III. Hm v th tc 1. Hm trong maple Hm trong maple l mt hnh thc c bit ca th tc, c hai loi hm dng sn v hm do ngi dng xy dng. a.Hm dng sn: Trong maple c rt nhiu hm dng sn, mt s c np sn trong b nh khi chng trnh chy, khi dng ta ch cn goi tn hm, mt s hm khng c np sn vo b nh m n c cha trong nhng gi cng c (package) hoc trong th vin khi s dng nhng hm ny th ta phi np n vo b nh ri sau mi gi tn hm s dng. Khi s dng nhng hm m maple np sn trong b nh nh sin, cos, exp, int ta ch cn gi trc tip vo cm x l V d: A:= sin(x)+tan(x)-x^2; B:= exp(34); C:=int(x^2-1,x=1..4); Khi s dng hm nm trong gi cng c th ta np hm vo b nh ri sau mi s dng c. C php np hm nm trong gi cng c vo b nh: with(Gi cng c): V d: Dng hm slope trong gi cng c student, ta thc hinwith(student): slope(y=2*x+5,y,x);
Mun xem tn cc hm trong mt gi cng c th sau with(Gi cng c) ta dng du chm phy ;Trang 8
Mt s gi cng c ta thng s dng: student, DEtools, PDEtools, LinearAlgebra, geometry, linalg, plottools, plots, + student: Gi cng c cha cc lnh cho tnh ton tng bc bao gm tch phn tng phn, quy tc Simpson, tm cc tr, + DEtools: Gi cng c cha cc lnh lm vic vi phng trnh vi phn. + DEPtools: Gi cng c cha cc lnh cho php lm vic vi phng trnh o hm ring. + LinearAlgebra: Cha cc cng c lin quan n i s tuyn tnh. + geometry: Gi cng c cha cc lnh lin quan vi hnh hc Euclide 2 chiu. + linalg: Gi cng c cha cc lnh lin quan vi ma trn v vect. + stats: Gi cng c cha cc lnh dng trong thng k. + plots: Cc lnh cho php v hnh trong khng gian 2 v 3 chiu. + plottools: Gi cng c cha cc lnh cho php lm vic vi cc i tng hnh nh. Khi s dng hm cc trong th vin th ta cng np hm cn s dng vo b nh trc ri mi dng. C php np hm trong th vin vo b nh: readlib(rationalize): rationalize(1/sqrt(3)+3/sqrt(7)); Mun xem ngha cng nh tp lnh ca gi cng c hay th vin no ta dng c php: ? ; hoc ? ; b. Hm do ngi dng xy dng Maple cho php chng ta to thm hm hay th tc mi v lu vo mt file trong th vin nh nhng hm c sn ca maple. Mun xy dng hm mi ta c th dng mt trong cc cch sau y: + Dng php nh x C php: (vars)->expr; Trong : vars: cc bin ca hm expr: l mt biu thc + Dng hm unapply( ) C php: unapply(expr,vars); expr: biu thc hoc phng trnh vars: cc bin ca hm sTrang 9
readlib(tn hm)
Th d: Dng hm kh cn mu s trong th vin
+ Dng th tc C php: proc(vars) expr end proc; vars: cc bin ca hm expr: biu thc xc nh hm V d 1: Xy dng hm hai bin y=3x3+ey-sin(x)+2 + Dng php nh x: > f:=(x,y)-> 3*x^3+exp(y)-sin(x)+2;f := ( x, y ) 3 x3 + e y sin( x ) + 2
+ Dng hm unapply > g:=unapply(x^3+sqrt(y)+7,x,y);g := ( x, y ) x3 + y + 7
Sau khi nh ngha hm xong, ta c th tnh gi tr ca hm s ti bt k im no nu n xc nh ti im . > f(2,3);26 + e 3 sin( 2 )
>g(a,1);a3 + 8
+ Dng th tc > h:=proc(x,y) x^3+sqrt(y)+7 end proc; > h(1,4); 10 + Xy dng hm s a tr C php: piecewise(k 1,biu thc1, k 2,Biu thc 2, ..., k N,Biu thc N, Biu thc cho iu kin cn li); V d: Xy dng hm du ca x -1 y= 0 1 x y:=x->piecewise(x y(1/2); 1 c. Th tc trong maple Ta c th to nhng th tc ring cho mnh gii quyt mt cng vic no , v lu vo trong th vin dng nh nhng hm c sn trong th vin ca maple. Cu trc ca mt th tc: proc (tn bin) [local ;] [global ;] end proc; V d 1: Th tc v th> restart: with(plots): f := proc (hamso) plot(hamso,x=10..10); end proc:
Dng th tc trn v hm x2-x+1> f(x^2-x+1);
V d 2: Th tc tnh tng v hiu ca hai s c s dng bin ton cc v bin cc b > f:=proc(x,y) global a,b; local k,m; k:=x+y: m:=x*y: a:=k+m: b:=k*m: print(['a','b','k','m']); [a,b,k,m]; end proc: Tnh gi tr ca a,b,k,m ti x=2,y=1; > f(2,1);[ a, b, k, m ][ 5, 6, 3, 2 ]
Gi gi tr ca a t b nh. > a;5
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Gi gi tr ca b t b nh > b;6
Gi gi tr ca k t b nh > k;k
Gi gi tr ca m t b nh > m;m
Nhn xt:Ta nhn thy rng gi tr ca k v m khng cn c lu trong b nh v y l bin cc b ch c hiu lc lc th tc thi hnh, khi th tc kt thc th cc bin cc b ny b xa khi b nh. Cc gi tr ca a v b vn cn c lu trong b nh khi th tc kt thc. V d: Th tc v hnh c s dng bin cc b, th tc ny c dng nhng hm trong hai gi cng c plottools v plots> restart: with(plottools): with(plots): dothi := proc( a, b, leq, req ) local r, M, v1, v2, wid, bar, Axesplot, leftpt, rightpt, pt_color, leftend, rightend, LT, RT, textpos; r := b-a ; M := evalf( 1.2 * max( b-a, abs(a), abs(b) ) ); v1 := [a, 0]; v2 := [b, 0]; wid := ceil( M/5) ; bar := arrow( v1, v2,wid, 0, 0, color = red ): textpos := max(wid, 2) ; LT := textplot( [a, textpos, a] ): RT := textplot( [b, textpos, b] ): leftend := min( a - 2*wid, 0); rightend := max( b + 2*wid, 0); Axesplot := plot( 0, x = leftend..rightend , y = (-wid)..(2*wid), axes = none, thickness = 2,scaling = constrained, tickmarks = [1,1] ): if leq=1 then pt_color := red; else pt_color := white; end if; leftpt := disk([ a, 0], wid/2, color=pt_color): if req=1 then pt_color := red; elseTrang 12
pt_color := white; end if; rightpt := disk([ b, 0], wid/2, color=pt_color ): display(leftpt, rightpt, bar, Axesplot, LT, RT); end proc:
Gi th tc thi hnh dothi(0,20,0,1);
+ C php lu hm hay th tc vo trong th vin: savelib(name1, name2, name3...); name1,name2...: tn cc hm, th tc hay bin cn lu vo th vin V d: xy dng hm tnh tng hai s ri lu vo th vin - Xy dng hm: > tinhtong:=proc(x,y) local a: a:=x+y: end proc: - Lu vo th vin > savelib(tinhtong): Khi s dng cc hm lu vo th vin th ta thc hin c php nh cc hm c sn trong th vin ca maple > readlib[tinhtong]: tinhtong(28,43); 71 Mun xem a ch ca th vin th dng hm libname; V d: libname;"C:\\PROGRAM FILES\\MAPLE 6/libIV. CC HM S CP C BN V HM TON HC THNG DNG
- sin(x), cos(x), tan(x), cot(x), arctan(x), arcsin(x), arccos(x), arccot(x). - exp(x) - ln(x) - log[b](x) : hm m c s e : hm logarithm c s e ca x : hm logarithm c s b ca xTrang 13
- log10(x) - sqrt() - root[n](x) - round(x) - trunc(x)
: hm logarithm c s 10 ca x : hm cn bc 2 ca x : hm cn bc n ca x : hm lm trn, ly tr nguyn gn nht ca x : hm ct ly phn nguyn ca mt s
- max(x1,x2,..), min(x1,x2,...): hm cho gi tr cc i v cc tiu ca mt dy cc s c lit k. - abs - Pi - infinity Lu : - Cp du mc n ( ) dng nhm cc phn t ca cng thc, phc ha cc bin ca hm. - Cp du mc vung [ ] c ngha nh mt danh sch (list). - Cp du mc nhn { } c ngha nh mt tp hp (set). - Khi cn khi ng li b nh ta dng lnh restart: - Ghi ch cho chng trnh bng # : hm ly tr tuyt i ca mt s. : : (-infinity: - )
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CHNG II
CC LNH TRC TIP CA MAPLEI. Tnh ton vi s nguyn Maple l mt cng c mnh, cho php tnh ton vI nhng s nguyn ln. 1. Hm m V d: >P:=2^63;
P := 9223372036854775808
2. Hm tnh giai tha V d: > 13!;
6227020800
3. Hm tm c s chung ln nht (gcd) V d: > gcd(157940,78864);212
4. Hm tm bi s chung nh nht (lcm) V d: > lcm(24,15,7,154,812);267960
5. Phn tch mt s ra tha s nguyn t (ifactor) V d: > ifactor(32160);( 2 )5 ( 3 ) ( 5 ) ( 67 )
6. Cc hm lin quan n s nguyn t a. Tm cc s nguyn t ng trc mt s cho trc (prevprime) V d: > prevprime(1234);1231
b. Tm cc s nguyn t ng sau mt s cho trc (nextprime) V d: > nextprime(1234);1237
c. Xt xem mt s c phi l s nguyn t khng (isprime) V d: > isprime(12);false
7. Tm thng v phn d ca php chia nguyn a. Php chia ly phn d nguyn irem(m, n) irem(m, n,q) b. Php chia ly phn thng nguynTrang 15
iquo(m, n) Trong : V d:
iquo(m, n,r) m, n l cc biu thc r, q l cc tn. > irem(23,4,'q');3 5 3 5
> q;
> irem(23,4,'r'); > r;
Ta cng c th dng cc hm ny tm phn thng v s d ca php chia hai a thc V d: > quo(x^3-4,x+2,x,'r');x2 2 x + 4
> r;
-12 4
> rem(3*x^2+1,x-1,x,'q'); > q;3x+3
8. Tnh ton vi cng thc truy hi Maple cho php ta tnh gi tr ca cc biu thc theo cng thc truy hi chng hn tnh s hng tng qut ca dy Fibonacci bng lnh rsolve C php: rsolve (eqns, fcns) eqns: l phng trnh hay tp cc phng trnh fcns: l tn hm hoc tp cc tn hm m hm fsolve phi tm. V d: Tm s hng f(n) ca dy Fibonacci: f (n ) = f (n 1) + f (n 2) Vi iu kin ban u f(1) = f(2) = 1 > restart: rsolve({f(n)=f(n-1)+f(n-2),f(1)=1,f(2)=1},{f}); f( n ) = 1 1 + 1 5 2 5 1 5 + 1 5( n )
n
n 1 1 5 1 2 5 1+ 5 1+ 5
> simplify(%); 4 f( n ) = 5
5 2 n ( ( 1 + 5 ) ( -1 )n + ( 5 1 ) ( 5 1) (1 + 5 )
( n )
)
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Mun c cng thc tng minh ca biu thc truy hi ta phI thc hin hai bc sau y: + Bc 1: Dng mt bin gn tn cho biu thc truy hi. + Bc 2: Tm cng thc tng qut bng lnh rsolve. V d: Tm dng tng minh ca cng thc truy hi f (n + 1) = 3 f (n ) 2 f (n 1) Vi iu kin ban u f(1) = 2, f(2) = 3 > restart: reqn:=f(n+1)=3*f(n)-2*f(n-1); rsolve({reqn,f(1)=2,f(2)=3},f(n));reqn := f( n + 1 ) = 3 f( n ) 2 f( n 1 )1 n 2 +1 2
Maple c th gii h phng trnh truy hi V d: Gii h phng trnh truy hIn+ y (n + 1) + f (n ) = 2 + n vi iu kin ban u: y (k = 1L5) = 2 k 1 , f (5) = 6 n f (n + 1) y (n ) = n 2 + 3
> restart: rsolve({y(n+1)+f(n)=2^(n+1)+n,f(n+1)-y(n)=n2^n+3,y(k=1..5)=2^k-1,f(5)=6},{y,f});{ y( n ) = 1 + 2 n, f( n ) = n + 1 }
II. Tnh ton vi cc s thp phn 1. Tnh gn ng cc gi tr s hc Ta c th thc hin cc php tnh s hc trn cc s thp phn (vi du chm ng) vi chnh xc theo mun. Trong thc t, Maple c th x l cc s vi chnh xc hng trm nghn ch s thp phn bng hm evalf (f, m) V d: Ta tnh gi tr ca s vi chnh xc 20 ch s thp phn > evalf(Pi,20);3.1415926535897932385
Tnh gi tr ca hm GAMMA ti im 2.5 > evalf(GAMMA(2.5));1.329340388
2. Tnh tng ca cc s hng C php: sum(f, k) hoc Sum(f, k) Tnh tng t f(0)..f(k-1) Tnh tng t f(m)..f(n) sum(f, k=m..n) hoc Sum(f, k=m..n) sum(f, k=alpha) hoc Sum(f, k=alpha) sum(f, k=expr) hoc Sum(f,k=expr)Trang 17
f: Biu thc cn ly tng k: ch s ly tng m..n: cn cn ly tng alpha:L nghim ca mt phng trnh no . alpha = RootOf(expr=0) expr: mt biu thc no khng ph thuc vo ch s ly tng v dng thay vo ch s ly tng.k1
V d: Tnh tng sau : a. Dng lnh tr > Sum(k,k); Xem gi tr ca tng trn > value(%);k1
k=1
kkk
1 2 1 k k 2 2
Ch :
k=1
k=kk
b. Dng lnh trc tip > sum(k,k); Hoc
1 2 1 k k 2 2
> Sum(k,k)=sum(k,k);k
k = 2 k2 2 kHay ta c th vit > Sum(k,k):%=value(%);1k
1
1
k = 2 k2 2 kc. Tng cc s nguyn t 4 n 6 > s:=Sum(x,x=4..6);s :=x=4
1
x15
6
> value(%);
d. Tnh tng v hn sau:
i=1
( i + 1 )2
1
> Sum(1/(i+1)^2,i=1..infinity): %=value(%);i=1
( i + 1 )2 = 1 + 6 2Trang 18
1
1
e. Tnh tng sau (ch cn ca tng) > Sum(K/(k+1),k=3*x^2+1):%=value(%);2 k=3x +1
K K = k + 1 3 x2 + 2
f. Tnh tng bnh phng nghim ca c c phng trnh x^2-5*x+6=0 > sum(x^2,x=RootOf(x^2-5*x+6=0));13
3. Tnh tch ca cc tha s C php: product(f, k) hoc Product(f, k) Tnh tch t f(0)..f(k-1) product (f, k=m..n) hoc Product (f, k=m..n) Tnh tch t f(m)..f(n) product (f, k=alpha) hoc Product (f, k=alpha) product (f, k=expr) hoc Product (f,k=expr) f: Biu thc cn ly tch k: ch s ly tch m..n: cn cn ly tch alpha:L nghim ca mt phng trnh no . alpha = RootOf(expr=0) expr: mt biu thc no khng ph thuc vo ch s ly tch v dng thay vo ch s ly tch. V d: Vit k hiu ca tch sau:
k=1
(k + 1) (k + 1)5
5
> Product( k+1, k=1..5 );k=1
Tnh tch sau:
k=a
kb
b
> Product(k,k = a .. b)=product(k,k=a..b);k=a
k=
( b + 1 ) ( a )
( t ) ( z 1 ) dt GAMMA l hm s sau: ( z ) := e t 0
> product( k+1, k=1..4 );120
> product( k^2, k=1..n );( n + 1 )2
> product( k^2, k );Trang 19
( k ) 2
> product( A[k], k=0..4 );A0 A1 A2 A3 A4n
> product( A[k], k=0..n );k=0
Ak
Tnh tch cc nghim ca phng trnh 2x3+7x+5=0 > restart: product( k, k=RootOf(2*x^3+7*x+5) );-5 2
III. Tnh ton vi s phc Maple cho php thc hin tnh ton vi s phc. Ch I c dng lm k hiu n v o. V d: Tnh3 + 5I 7 + 4I 41 23 + I 65 65
> (3+5*I)/(7+4*I);
Bng lnh bin i f v dng ta cc convert (f, polar) ta c th bin I s phc f v dng ta cc (r , ) trong r l mun cn l argument ca s phc trong biu thc V d: > convert((3+5*I)/(7+4*I),polar);1 23 polar 65 2210 , arctan 41
IV. Tnh ton theo Modul 1. Cc tnh ton Modul thng thng a. Tnh modul m trn tp s nguyn C php: e mod m vi cc dng ring: + modp (e, m): Ly biu din dng ca e theo modul m (trong tp gi tr t 0 n m 1 ). + mods (e, m): Ly biu din i xng ca e theo modul m (trong tp gi tr t m 1 m n ). 2 2
Vi
+ e l biu thc i s + m l mt s nguyn khc 0.
Trang 20
Ton t mod tnh gi tr biu thc e trn tp s nguyn modul m. N hp nht vic tnh ton trn trng s hu hn v cc php ton s hc i vI a thc, ma trn trn trng hu hn, k c php phn tch ra tha s. Vic n nh modp hay mods c thc hin thng qua bin mi trng mod (gi tr modp c xem l mc nh). Khi ta cn tnh q mod m vi q l mt s nguyn th khng nn s dng c php hin nhin nh q^n mod m, bi v php ly tha s chuyn s th nht thnh s nguyn (c th l rt ln) trc khi rt gn theo modul m. Thay vo nn dng ton t tr &^ ngha l q&^n mod m. Trong dng ly tha s c bin I kho lo theo php ly mod. Tng t, Powmod (a,n,b,x) mod m tnh Rem (a^n,b,x) mod m (a v b l nhng a thc ca x) khng cn tnh a^n mod m. Nhng php ton modul s hc khc c biu din di dng t nhin ca chng: j + i mod m; j - i mod m; j * i mod m; i / j mod m; V d: Tnh > 12 mod 7; > modp(12,7); > mods(12,7); > 5*3 mod 7; > 11+5*3 mod 7; j^(-1) mod m
5 5 -2 1 5 3 3 5 9
> (11+5*3)^(-1) mod 7; > 1/(11+5*3)mod 7; > 1/3 mod 7; > 5&^1000 mod 23; b. Tnh modul khi e khng l s Khi biu thc e khng l mt s m l mt a thc th php ly modul ca n c hiu l php ly modul ca tt c cc h s ca a thc. V d: Tnh > a:=15*x^2+4*x-3 mod 11;Trang 21
a := 4 x2 + 4 x + 8
V php ly modul mc nh s dng biu din dng (modp). Mun chuyn sang dng i xng th ta dng lnh: > `mod`:=mods: b:=3*x^2+8*x+9 mod 11;b := 3 x2 3 x 2
i vi cc php ton khc nh tm c chung ln nht, phn tch ra tha s nguyn t (vi c s v a thc), cng c thc hin theo phng thc thng thng ngoi tr mt khc bit nh l cc lnh trong php tnh modul c bt u bng ch hoa. V d: > Gcd(a,b) mod 11;x+5
> Factor(x^3+2) mod 5; > Expand(%) mod 5; 2. Gii phng trnh vi modul C php: Trong :
( x2 + 2 x 1 ) ( x 2 )x3 + 2
msolve (eqns, vars, q) hoc msolve (eqns, q) + eqns: Tp cc phng trnh. + vars: Tp cc bin. + q: S nguyn.
Lnh msolve thc hin vic gii phng trnh trong Z theo m. Lnh msolve gii cc phng trnh eqns trn cc s nguyn (theo mod q). N gii theo mi n bt nh c trong cc phng trnh. Nu l nghim v nh, th h cc nghim c biu din thng qua cc bin c tn c cho trong tp bin vars, nu nh vars c b qua th c thay th bng cc tn mc nh ton cc _Z1~, _Z2~, _Z3~,Nhng tn ny khng trng vI cc n v nh v c php ly mi gi tr nguyn. V d: > restart: msolve({3*x-4*y=1, 7*x+y=2},19);{ x = 15, y = 11 }
> restart: msolve(8^i=2,17);
{ i = 3 + 8 _Z1~ }
{ x2 = 2 x1 + 2 x6 + 2 x3 + 2 x4 + 2 x5 + 2 x7 + 2 x8 + 2 x9, x3 = x3, x4 = x4, x5 = x5, x6 = x6, x7 = x7, x8 = x8, x9 = x9, x1 = x1 }Trang 22
> msolve(sum(x[i],i=1..9),3);
Nu phng trnh khng c nghim trn cc s nguyn (mod m) th Maple s khng cho kt qu no. > msolve(x^2=3,5); V. Khai trin, n gin, phn tch v bin i mt biu thc i s 1. Khai trin mt biu thc C php: expand (expr, expr1, expr2....); + expr l biu thc cn khai trin + expr1, expr2, l cc dng biu thc m expr khai trin theo nu c th c. V d: > expr:=(x+1)*(x+y)*(x/y); expr1:=x+1;expr := (x + 1) (x + y) x y expr1 := x + 1
> expand(expr);
x3 x2 + x2 + +x y y
> expand(expr,expr1);
( x + 1 ) x2 + (x + 1) x y
Biu thc th hai c phn tch theo tha s dng expr1. 2. Phn tch mt biu thc thnh tch C php: factor (expr); hoc factor (expr, expr1) + expr l biu thc cn phn tch + expr1l cc dng biu thc m expr phn tch theo nu c th c. V d: > Q:=factor(a^3+b^3+c^3-3*a*b*c);Q := ( b + a + c ) ( b2 a b b c a c + a 2 + c2 ) x4 2
> factor(x^4-2);
Phn tch x4 2 thnh dng cha tha s 2 > factor(x^4-2,sqrt(2));
( x2 + 2 ) ( x2 2 )
> factor(x^4-2,root[4](2));( x2 + 2 ) ( x 22( 1/4 )
) (x + 2
( 1/4 )
)
Phn tch x + 2 thnh tch vi cc tha s c dng s phc > factor(x^2+2,complex);( x + 1.414213562 I ) ( x 1.414213562 I )
Gi l nghim ca phng trnh x2- 2 = 0 > alias(alpha = RootOf(x^2-2));
Phn tch biu thc y2 8 thnh tch trong c cc tha s cha s hng > factor(y^2-8,alpha);Trang 23
( y + 2 ) ( y + 2 )
3. n gin biu thc C php: simplify(expr): n gin biu thc expr simplify(expr,{n1,n2,..}): n gin biu thc expr theo cc iu kin n1, n2,....cho trc. simplify(expr,Option): n gin biu thc expr theo dng ch nh trong Option Option:C cc ty chn sau trig Power Ln Exp Radical RootOf Lng gic Ly tha Logarit neber Hm e m Cn thcc
Cc nghim ca phng trnh... V d: n gin biu thc e (a + ln (be )) > expr:=exp(a+ln(b*exp(c)));expr := ec ( a + ln( b e ) )
n gin thnh dng ly tha (power) > simplify(expr,power);b ec ea
n gin thnh dng e m (exp) > simplify(expr,exp);be
(a + c)
n gin ri thay cc gi tr ca a v c vo biu thc > simplify(expr,{y=a,c+1=x});be( 1 + x )
ea
> simplify(a*x^2+b*x+c,{a=2,b^2+c=1,b-c=3});2 x2 + 3 x + ( x + 1 ) c
Da vo tnh cht ny ca simplify ta c th s dng vo tnh tng cc nghim ca phng trnh: V d: Cho phng trnh x 2 4 x + 1 = 0 . Gi s rng phng trnh c 2 nghim 3 x1 v x2. Hy tnh: S = x13 + x 2 Theo nh l Viet ta c x1 + x 2 = 4 v x1 x 2 = 1 . Vy ta thc hin nh sau: > dk:={x[1]+x[2]=4,x[1]*x[2]=1};dk := { x1 + x2 = 4, x1 x2 = 1 }S := x1 + x23 3
> S:= x[1]^3+x[2]^3; > S:=simplify(S,dk);
Trang 24
S := 52
Vy nu ta dng tnh cht n gin biu thc S theo dk th ta s nhn c gi tr S cn tm m khng cn giI phng trnh. 4. Ti gin phn thc C php: normal (expr): Ti gin biu thc expr normal (expr, expanded): Ti gin biu thc expr trnh by kt qu di dng khai trin. Ti gin phn thc cng l a n v dng chun tc (normal), tc l gin c cc tha s chung ca t s v mu s. thc hin iu ny ta dng hm normal V d 1: > normal((x^3-y^3)/(x^2+x-y-y^2));x2 + y x + y2 x+1+y
V d 2: Tnh tng sau:
k=1
k1 1 1 ( n + 1 )2 n 2 2 2
n
> sum(k,k=1..n);
> normal(%,expanded);1 2 1 n + n 2 2
Xem kt qu di dng tch ca cc tha s: > factor(%);1 n (n + 1) 2
V d 3: > f:=(x-1)^2*(x-3)-2*x*sin(x+y);f := ( x 1 )2 ( x 3 ) 2 sin( x + y ) x x3 5 x2 + 7 x 3 2 sin( x + y ) x
> normal(f);
Hoc ta dng c php sau: > normal(f,expanded); 5. Thay th gi tr cho bin C php: vo biu thc expr.
x3 5 x2 + 7 x 3 2 x sin( x ) cos( y ) 2 x cos( x ) sin( y )
subs(x = a, expr): Thay gi tr x = a vo biu thc expr. subs(s1, s2,, sn, expr): Thay th tun t cc gi tr s1, s2,, sn
Trang 25
subs({s1, s2,, sn }, expr): Thay th ng thi cc gi tr s1, s2,, sn vo biu thc expr. V d 1: Thay th ln lt cc gi tr x = a v y = c vo biu thc x3 - y > subs(x=a,y=c,x^3-y);a3 c
V d 2: Thay th y = x ri th x = a vo biu thc x3 y > subs(y=x,x=a,x^3-y);a3 a
V d 3: Thay th ng thi 02 gi tr y = 2*x v x = a vo biu thc x3 y > subs({y=2*x,x=a},x^3-y);a3 2 x a3 2 x
> subs({x=a,y=2*x},x^3-y); V d 4: Cho hm s y = mx 3 4(m 1)x + 5 tnh gi tr ca y tI x = 3m + 1 > restart: y:=m*x^3-4*(m-1)*x+5; y:=subs(x=3*m+1,y); y:=simplify(%);y := m x3 4 ( m 1 ) x + 5 y := m ( 3 m + 1 )3 4 ( m 1 ) ( 3 m + 1 ) + 5 y := 27 m4 + 27 m3 3 m 2 + 9 m + 9
V d 5: Vit phng trnh giao im ca hai ng y = (m + 1)x 3 x + m 2 v ng thng y = 3 x + 5 > restart: y:=(m+1)*x^3-x+m^2; y:=3*x+5; p:=subs(y=3*x+5,y=(m+1)*x^3-x+m^2);y := ( m + 1 ) x3 x + m 2
y := 3 x + 5p := 3 x + 5 = ( m + 1 ) x3 x + m2
> vp:=rhs(p); > vt:=lhs(p);
vp := ( m + 1 ) x3 x + m 2
vt := 3 x + 5
> pt:=vt-vp=0;
pt := 4 x + 5 ( m + 1 ) x3 m2 = 0Trang 26
> collect(pt,x);
( m 1 ) x3 + 4 x + 5 m2 = 0
6. Hm trch ly v tri v v phi ca mt ng thc a. Hm trch ly v tri: b. Hm trch ly v phi: lhs (f) rhs (f)
Trong : f l mt phng trnh, bt phng trnh hay mt ng thc no 7. Hm gom nhm cc thnh phn ca mt biu thc C php: collect (f, x): Sp xp li biu thc f theo bin ch nh trong x. collect (f, x, procedure): Sp xp li biu thc f theo bin ch nh trong x v cc h s c bin i theo dng c ch nh trong procedure. procedure c th c cc gi tr sau: + recursive: u tin gom nhm biu thc theo x1, tip theo trong mi s hng x1 li gom nhm theo x2, + distributed: gom nhm tch bit ra. u tin gom nhm biu thc theo x1 xong ri n gom nhm biu thc theo x2, V d: > f:=m^3*x+x*m-x^4*m+m^4=0;f := m3 x + x m x4 m + m4 = 0
Gom nhm biu thc theo m: > collect(m^3*x+x*m-x^4*m+m^4=0,m);m4 + m3 x + ( x x4 ) m = 0
Gom nhm biu thc theo x, trong cc s hng ca x c phn tch thnh tch: > collect(m^3*x+x*m-x^4*m+m^4=0,x,factor);x4 m + m ( m 2 + 1 ) x + m4 = 0
8. Sp xp thc mt dy cc gi tr hoc mt a thc C php: Trong : sort(L) hoc sort(A) hoc + L l dy cc gi tr + F l kiu sp xp + N u L l dy cc s hng th F l > hoc < + N u L l dy dng chui th F l lexoder + A l a thc hoc mt dy gi tr + V l mt hoc nhiu bin ca A Mc nhin s sp xp theo th t gim dn ca ly tha ca cc bin c ch nh. Nu c nhiu bin th a thc c sp xp theo tng lu tha ca cc bin.Trang 27
sort(L, F) sort(A, V)
Nu ta thm thng s plex th a thc c sp xp theo th t tng bin c lit k trong biu thc. V d: > sort([3,2,1]);[ 1, 2, 3 ] [ 3, 2, 1 ]
> sort([3,2,1],`>`);
> sort(1+x+x^2,plex); > sort(1+x+x^2);
1 + x + x2 x2 + x + 1
> restart: sort([g,c,a,d,e,f],lexorder);[ a, c, d, e, f, g ]
> p := y^3+y^2*x^2+x^3: sort(p,[y,x]);y2 x2 + y3 + x3
> sort(p,[y,x],plex);
y3 + y2 x2 + x3x+y x + y
> sort((y+x)/(y-x),x);
9. Chuyn i dng ca biu thc C php: Trong : conrvert (f, form, vars) + f l mt biu thc bt k. + vars l cc bin mi + form c th c cc dng sau y: Airy,Bessel;D;Ei; GAMMA; Heaviside; Matrix; PLOT3Doptions; PLOToptions POLYGONS; RootOf; StandardFunctions; Vector; and; array; base; binary; binomial; bytes; confrac; convert/'*'; convert/'+'; decimal; degrees; diff; disjcyc; double; equality; erf; erfc; exp; expln; expsincos; factorial; float; fullparfrac; global ;hex; horner; hypergeom; int; list; listlist; ln; local; mathorner; matrix; metric; mod2; multiset; name; numericproc; octal; or; parfrac; permlist; piecewise; polar; polynom; pwlist; radians; radical; rational; ratpoly; set; signum; sincos; sqrfree; std; stdle; string; symbol; table; tan; trig; vector V d: To mt chui theo quy lut xi ^ I > f := seq( x[i]^i, i=1..4 );
f := x1, x2 , x3 , x4Trang 28
2
3
4
Nhn cc s trn li vi nhau: > convert([f], `*`);
x1 x2 x3 x4
2
3
4
Chuyn s thp phn thnh dng phn s > convert( 1.23456, fraction);3858 3125
> f := (x^3+x)/(x^2-1);f :=
x3 + x x2 1
Chia phn thc hu t > convert(f, parfrac, x);x+
1 1 + x1 x+1
> s := series(f,x,4);
s := x 2 x3 + O( x5 )
Loi b v cng b bc cao: > convert(s, polynom); > f := sinh(x)+sin(x); Chuyn i f thnh dng e m: > convert(f, exp);
x 2 x3
f := sinh( x ) + sin( x )
1 x 1 1 1 (I x) 1 I e e (I x) x 2 2e 2 e
Chuyn i t radian sang : > convert(Pi/3,degrees);60 degrees
Chuyn i t sang radian: > convert(135*degrees,radians);3 4
Chuyn i t biu thc dng sin sang dng tan > convert(sin(x),tan);2 1 tan x 2 1 1 + tan x 2 Trang 292
VI. Mt s dng hm khc 1. Trch ly cc phn t trong mt biu thc hay mt danh sch C php: op(i, expr); op(i..j, expr); op(expr); op(list, expr); Trong : + i, j l cc s nguyn xc nh v tr ca cc phn t. + expr l mt biu thc. + List l dy cc s nguyn xc nh v tr c a cc phn t trong biu thc, cc s ny xc nh v tr ca cc phn t xp lng vo nhau c ly t ngoi vo trong. V d: > expr:=sin(x)*y-sin(sqrt(y)+x)+ln(a)-z;expr := sin( x ) y sin( y + x ) + ln( a ) z
Xem cc phn t cu thnh ca biu thc expr: > op(expr);
sin( x ) y, sin( y + x ), ln( a ), z
Trch ly phn t th nht trong biu thc expr: > A:=op(1,expr);A := sin( x ) y
Trch ly phn t th 2 trong phn t u tin ca expr: > op(2,op(1,expr));y
Ta cng c th thc hin lnh sau: > op([1,2],expr); Trch ly phn t th nht ca A: > op([1,1],expr);
y
sin( x )
Trch ly cc phn t t th 2 n th 4 ca expr: > op(2..4,expr);sin( y + x ), ln( a ), z
Trch ly phn t th 3 trong dy sau: > w := [2,5,36,29]; op(3,w);
w := [ 2, 5, 36, 29 ] 36
2. m s cc phn t trong mt biu thcTrang 30
C php: V d:
nops (expr)
> B:=tan(y)*ln(x)+sin(x)-expr(x^2)+2;B := tan( y ) ln( x ) + sin( x ) expr( x2 ) + 2 tan( y ) ln( x ), sin( x ), expr( x2 ), 2
> op(B); > nops(B);
4
3. Tnh bit thc delta ca mt tam thc bc hai C php: discrim (expr) expr l mt tam thc bc hai 4. Trch ly t s v mu s ca mt biu thc hu t a. Trch ly t s: C php: C php: numer (expr) denom (expr) b. Trch ly m u s: 5. Trch ly h s ca mt a thc a. Trch ly tt c cc h s ca mt a thc: C php: Trong : coeffs (P, x, t) + P l mt a thc + x l bin ca a thc + t l tn lu danh sch cc bc ca bin x b. Trch ly h s bc n ca a thc p: C php: V d: > s := 3*v^2*y^2+2*v*y^3; > coeffs( s );s := 3 v2 y2 + 2 v y3
coeff (P, x, n)
3, 22 y3, 3 y2 v, v2
> coeffs( s, v, 't' ); > t;
6. Gn tn cho biu thc hoc hm C php: alias (e1, e2,, en) e1, e2,, en l cc biu thc hoc cc hm m ta mun t tn li.Trang 31
V d: phi g li nhiu ln mt tn hm no , ngi ta gn cho n mt tn ngn hn. > restart; alias(g=GAMMA,b=BesselJ,F=fibonacci, a=2*x^3+1);g, b, F, a
T y v sau khi cn dng hm GAMMA ta ch cn g g(), khi cn dng hm BesselJ ta ch cn g b(), V d: > g(1/2);
> evalf(b(1,1)); > F(3);
.4400505857F( 3 )
7. Tnh tng tch dy cc gi tr xc nh a. Tnh tng dy cc gi tr xc nh C php: Trong : add (f, i = m..n) add(f, i = s) + i l ch s ly tng + s l mt biu thc, mt dy s hoc tp cc nghim ca mt phng trnh. V d 1: Tnh tng t 1 n 5 > add(i,i=1..5); V d 2: Tnh tng t 1 n n > add(i,i=1..n);Error, unable to execute add
Tnh tng dy t f(m),, f(n) Tnh tng ca f trn cc ton hng s
+ f l mt hoc mt s biu thc
15
Khng tnh c v n khng xc nh. > add(i,i={1,4,7,5});17
Tnh tng 2 nghim ca phng trnh x 2 5 x + 6 = 0 > f:=solve(x^2-5*x+6,x): add(i,i=f[1..2]);5
Tnh tng vi bin chy l mt biu thc > add( i^2-1,i=2*a+y);
4 a 2 2 + y2
Trang 32
b. Tnh tch dy cc gi tr xc nh C php: Trong : mul (f, i = m..n) mul(f, i = s) + i l ch s ly tch + s l mt biu thc, mt dy s hoc tp cc nghim ca mt phng trnh. V d: > mul(j,j=3..4); > mul(j,j);Error, wrong number (or type) of parameters in function mul
Tnh tch dy t f(m),, f(n) Tnh tch ca f trn cc ton hng s
+ f l mt hoc mt s biu thc
12
> mul(j^2,j=f[1..2]);
361 48
> mul(1/j,j={1,2,6,4});
8. Gn gi tr cho bin C php: assign (a, b) assign (a = b) assign (t) Trong : + a l mt tn bin no + b l mt biu thc bt k. + t l mt danh sch lit k hoc tp hp cc phng trnh. V d 1: Gn gi tr a = 2 > restart: assign(a,2); > a;
2
V d 2: Gn gi tr b = e x 2 , sin (x ) = 1 2 , c = 3a > assign(b=exp(x-2),sin(x)=1/2); assign(c=3*a); > a,b,c;2, e(x 2)
,6
> sin(x);
1 2
V d 3: Gn cc nghim ca h phng trnh cho cc bin tng ng > restart: s := solve( {x+y=1, 2*x+y=3},{x,y} );Trang 33
s := { y = -1, x = 2 }
> assign(s); x,y;
2, -1
9. ng nht cc h s ca 2 a thc C php: match (p(x) =q(x), var, s) + Nu ng nht c th hm cho gi tr true v s l kt qu ng nht cn ngc li cho gi tr false + Mun ly ra cc gi tr ca s ta dng assign(s) V d: > match(2*x^2-3*x+4=(m+1)*x^2-a*x-y,x,'s');true
> assign(s); > a; > m; > y;
3
1-4
> match((a-3)*x^2-(b+3)*x+c=0,x,'s');false
10. Hm cho bc cao nht v thp nht ca a thc a. Hm cho bc cao nht ca a thc: C php: degree (P, x) + P l a thc + x l bin ca a thc b. Hm cho bc thp nht ca a thc: C php: ldegree (P, x)3
V d: > degree(2/x^2+5+7*x^3,x); > degree(x*sin(x),x);
FAIL
> degree(x*sin(x),sin(x));1
> degree((x+1)/(x+2),x);FAIL -2
> ldegree(2/x^2+5+7*x^3,x);Trang 34
11. Kh cn mu s ca mt biu thc v t Hm ny nm trong th vin nn trc khi s dng ta phi dng lnh c t th vin sau mI gi tn hm. C php: readlib (rationalize): rationalize(expr) expr l biu thc cn trc cn mu s V d: Kh cn mu s ca biu thc sau: a = > a:=x/(sqrt(x+2)-3*sqrt(5));a := x x+2 3 5
x x+2 3 5
> realib(rationalize): a:=rationalize(a);a := x (3 5 + x + 2 ) 43 + x
12. Cc hm lin quan n iu kin assume (x, prop): t iu kin cho bin hoc t mI quan h gia cc bin additonally (x, prop): Tng t nh assume t thm iu kin cho bin x is (x, prop): Kim tra iu kin ca bin x cho kt qu true, false, FAIL coulditbe (x, prop): C kh nng tha hay xy ra iu kin khng? about (x): Cho thng tin v x Trong : + x l mt bin ca mt biu thc no . + prop l mt tnh cht no nh positive (dng), negative (m), real, complex, vector, Square Matrix (ma trn vung), fraction (phn s), Gi s hm assume t li iu kin cho bin a v bin a c lu trong b nh trc th n s b xa v c thay bng iu kin c t trong hm assume. Bin no b rng buc iu kin th s hin th di dng du ~ bn phi a a~, x x~, V d: t iu kin cho a> 0 v b assume( a>0,b is(a,positive);true
> is(b,positive);false
Hoc mun xem thng tin v a > about(a);Originally a, renamed a~: is assumed to be: RealRange(Open(0),infinity)Trang 35
Mun xa iu kin trong hm assume ta dng lI lnh gn a cho chnh n > a := 'a';a := aa
> a; > assume(x cos(n*Pi);
> assume(i/2, integer); is(i/2+1, integer);true
> coulditbe(i/3, integer); > assume(z,real); is(z^2 >= 0);true true
> coulditbe(z^2 is(5,RealRange(5,infinity));true
> is(5,RealRange(Open(5),infinity));falseTrang 36
> assume(a,SquareMatrix); assume(n,integer); is(a^n,SquareMatrix);true
> coulditbe(a^n,tridiagonal);true
Trang 37
CHNG III
PHNG TRNH - BT PHNG TRNH H PHNG TRNH - H BT PHNG TRNHI. Tm nghim nguyn ca phng trnh, h phng trnh C php: Trong : insolve (eqns, vars) + eqns l mt phng trnh hay mt h phng trnh + vars l cc bin ca phng trnh hay h phng trnh V d 1: Tm nghim ca cc phng trnh sau: > isolve(3*x-2*y=2,a);{ y = 1 + 3 a, x = 2 a }
> isolve(2*x^2-4*x-6=0,x);{ x = -1 }, { x = 3 }
V d 2: Tm nghim nguyn ca h phng trnh sau: > eqns:={3*x-2*y+z=2,x-y+2*z=5,x+2*y-3*z=-4}; isolve(eqns);eqns := { 3 x 2 y + z = 2, x y + 2 z = 5, x + 2 y 3 z = -4 }
{ x = 1, y = 2, z = 3 }
Lu : Nu s phng trnh t hn s n s th ta cn thm vo cc h s t do cho s bin ca phng trnh. > eqns:={3*x-2*y+z=2,x-y+2*z=3}; isolve(eqns,{a});{ x = 2 + 3 a, y = 3 + 5 a, z = 2 + a }
eqns := { 3 x 2 y + z = 2, x y + 2 z = 3 }
Nu s tham bin ta thm vo d th Malpe s t ng b qua cc tham bin d. > eqns:={3*x-2*y+z=2,x-y+2*z=3}; isolve(eqns,{a,b,c});{ x = 2 + 3 a, y = 3 + 5 a, z = 2 + a }
eqns := { 3 x 2 y + z = 2, x y + 2 z = 3 }
Nu ta khng thm vo cc tham bin t do th Maple s t ng thm vo ln lt cc bin _Z1, _Z2, i vi cc version t 5.1 tr i v cc bin _N1, _N2, i vi cc version c hn. > eqns:={3*x-2*y+z=2,x-y+2*z=3}; isolve(eqns);
eqns := { 3 x 2 y + z = 2, x y + 2 z = 3 }Trang 37
{ x = 2 + 3 _Z1, y = 3 + 5 _Z1, z = 2 + _Z1 }
Nu phng trnh khng c nghim nguyn (hoc Maple khng c kh nng tm nghim nguyn) th my s bo NULL hoc khng tr li. > eqns:={3*x-2*y+z=2,x-3*y+2*z=5,2*x+2*y-3*z=-4}; isolve(eqns);eqns := { 3 x 2 y + z = 2, x 3 y + 2 z = 5, 2 x + 2 y 3 z = -4 }
V d 3: Tm phng trnh tham s ca ng thng trong mt phng Oxy c cho bi phng trnh tng qut sau: 3 x + 4 y 7 = 0 > isolve(3*x+4*y+7=0,{t});{ x = 5 4 t, y = 2 + 3 t }
V d 4: Tm phng trnh tham s ca mt phng trong khng gian Oxyz c cho bi phng trnh tng qut sau: 2 x 5 y + 5 z + 3 = 0 > pt:=2*x-5*y+5*z+3=0; isolve(pt,{t[1],t[2]});
pt := 2 x 5 y + 5 z + 3 = 0
{ y = t1, x = 4 5 t2, z = 1 + t1 + 2 t2 }
II. Tm nghim gn ng C php: Trong : fsolve (eqns, var, options) + eqns l mt phng trnh hay mt h phng trnh. + vars l cc bin ca phng trnh hay h phng trnh. + options l cc ty chn c th l: maxsols = n: Tm ti a n nghim (nu c) complex: Tm cc nghim dng s phc. interval: Tm cc nghim trong khong a ..b no avoid = s: Tm cc nghim khc iu kin s. s c th l mt phng trnh, nghim ca mt phng trnh hay l mt chui cc nghim c lit k. V d: > p:= 23*x^5 + 105*x^4 - 10*x^2 + 17*x: fsolve( p, x, -1..1 );-.6371813185, 0.
> fsolve( p, x, maxsols=2 );
-4.536168981, -.6371813185
> q := 3*x^4 - 16*x^3 - 3*x^2 + 13*x + 16: fsolve(q, x);1.324717957, 5.333333333
> fsolve(q, x, 2..5); Trong khong t 2 n 5 khng c nghim no nn my khng tr li > fsolve(q, x, 2..6);Trang 38
5.333333333
> fsolve(q, x, complex);
-.6623589786 .5622795121 I, -.6623589786 + .5622795121 I, 1.324717957 , 5.333333333
> f:= sin(x+y) - exp(x)*y = 0: g:= x^2 - y = 2: fsolve({f,g},{x,y},{x=-1..1,y=-2..0});{ x = -.6687012050, y = -1.552838698} -6.283185307
> fsolve(sin(x),x,avoid={x=0,x=Pi,x=-Pi}, -10..10); Tm nghim gn ng xp x vi gi tr ca x > fsolve(sin(x),x=7.0);6.283185307
Ta c th khai bo hm f bng th tc sau y: > f := proc(x) 2-x^2 end proc;f := proc (x) 2 x^2 end proc
Tm nghim gn ng xp x vi gi tr 1.0 > fsolve(f,1.0);
-1.414213562, 1.414213562
III. Tm nghim ca h phng trnh - h bt phng trnh C php: solve (eqns, vars) + eqns l mt phng trnh, bt phng trnh hay mt h phng Trong : trnh, h bt phng trnh no + vars l cc bin. Hm Solve gii phng trnh cho c nghim thc v phc V d: Gii phng trnh 2x2-8x+4=0, ta thc hin nh sau: > restart: solve(2*x^2-8*x+4=0);2 + 2, 2 2
Hay trnh by kt qu di dng ca x: > solve(2*x^2-8*x+4=0,{x});{ x = 2 + 2 }, { x = 2 2 }
> restart: solve(x^3+x-2);
1 1 1 1 1, + I 7 , I 7 2 2 2 2
Gii h phng trnh:
2x - y = 1 3x + 2y = 0
> restart: solve({2*x-y=1,3*x+2*y=0});
Trang 39
{y =
-3 2 ,x= } 7 7
Gii bt phng trnh x2 5x + 6 v trnh by kt qu theo khong > restart: solve(x^2-5*x+6 0
> restart: solve({x^2-2*x0},{x});{ x < 2, 0 < x }
Gii phng trnh sau theo tng bc: x + m + x + m = 10 > restart: (x+m+4/(x+m)=10);x+m+ 4 = 10 x+m
4
> y:=normal(%);x2 + 2 x m + m 2 + 4 y := = 10 x+m
> VT:=lhs(y);VT := x2 + 2 x m + m 2 + 4 x+m
> VP:=rhs(y); > TU:=numer(VT);
VP := 10
TU := x2 + 2 x m + m2 + 4
> MAU:=denom(VT);
MAU := x + m
iu kin phng trnh c ngha, mu s phi khc 0 do : > assume(MAU0); Quy ng mu s, chuyn v: > y:=TU-VP*MAU=0;y := x~2 + 2 x~ m~ + m~2 + 4 10 x~ 10 m~ = 0
> y:=collect(y,x);y := x~2 + ( 10 + 2 m~ ) x~ 10 m~ + m~2 + 4 = 0
> delta:=discrim(lhs(y),x); := 84
Xc nh cc h s a, b, c: > a:=coeff(lhs(y),x,2);Trang 40
a := 1
> b:=coeff(lhs(y),x,1); > c:=coeff(lhs(y),x,0);
b := 10 + 2 m~
c := 10 m~ + m~2 + 4
Tm nghim ca phng trnh: > x[1]:=(-b+sqrt(delta))/(2*a);x1 := 5 m~ + 21
> x[2]:=(-b-sqrt(delta))/)(2*a);x2 := 5 m~ 21
Ch : + My thng cho ta tt c cc nghim k c nghim phc i vi phng trnh i s. + Khi gii cc phng trnh lng gic, i khi my ch cho ta mt vi nghim tng qut i din ch khng phi l tt c cc nghim). + Kt qu ca vic gii h s c hin th dng kt qu nu my tm c nghim chnh xc (nghim c th c biu din thng qua cc hm chun c sn ca Maple nh SQRT, EXP, GAMMA, BESSEL,). Nu khng tm c nghim chnh xc, hoc khng tm c nghim th kt qu s khng hin th ln mn hnh. Khi , ta c th tm nghim ca phng trnh bng cch tm nghim gn ng bng hm fsolve(), hm cho gi tr gn ng evalf() hoc mt s phng php gn ng khc. V d: Tm nghim ca phng trnh sau: > y:=2*x+x^x-5=0;y := 2 x + xx 5 = 0
> solve(y,x);RootOf( 2 _Z + _Z _Z 5 )
Hm solve( ) khng tm c nghim chnh xc do nghim ca phng trnh c lu tr di dng hm RootOf(). Mun xem gi tr ca nghim ny ta dng thm hm nh gi evalf( ) > solve(y,x): evalf(%);1.534888777
Hoc ta gii trc tip bng hm fsolve(y, x) th ta c nghim gn ng ca phng trnh trn: > fsolve(y,x);1.534888777
IV. Tm nghim ca h phng trnh bng hm RootOf C php:Trang 41
RootOf (expr) RootOf (expr, x) RootOf (expr, x, c) RootOf (expr, x, a..b)
Tm tt c cc nghim ca phng trnh expr = 0. Tm nghim ca phng trnh expr = 0 theo bin x. Tm nghim ca phng trnh expr = 0 theo bin x v gi tr ca x gn vi gi tr c. Tm nghim ca phng trnh expr = 0 theo bin x v gi tr ca x nm trong khong a...b.
RootOf (expr, x, index = i) Tm nghim th i ca phng trnh expr = 0 theo bin x. RootOf (expr, x, label = e) Tm nghim th i ca phng trnh expr = 0 theo bin x v gn cc nghim vi tn trong label phn bit nghim phng trnh ny vi nghim ca phng trnh khc. Cc thng s: + expr +x + a, b, c +i +e Mt biu thc i s hoc phng trnh. Bin Cc gi tr hng s Ch s nghim th i ca phng trnh. Mt biu thc no .
Ch : Hm RootOf khng hin th nghim ca phng trnh do , xem nghim ca cc phng trnh th ta dng thm mt trong cc hm sau: + allvalues (f) + evalf (f) + evalf (f, n) > RootOf(x^3-2);RootOf( _Z 3 2 )
Xem tt c cc nghim ca phng trnh f. Xem nghim gn ng ca f vi 10 ch s c ngha. Xem nghim gn ng ca f vi n ch s c ngha.
V d: Tm nghim ca phng trnh x3 2 = 0
Tm nghim ca phng trnh theo bin c ch nh > RootOf(3*x^3-6=0, x);RootOf( _Z 3 2 )
> RootOf(a*x^2+b*x+c,x);RootOf( a _Z 2 + b _Z + c )
Nu trong phng trnh c nhiu tham s th ta phi ch nh tham s no l bin cn cc tham s cn li c xem nh hng s. Tm nghim ca phng trnh vi bin ca phng trnh c ch nh l b > RootOf(y*b^2-y/x,b);Trang 42
RootOf( _Z 2 x 1 )
Tm nghim ca phng trnh bng hm RootOf ri sau ly modulo theo m > RootOf(2*x-1, x) mod 7;4
Ch : Quy tc tnh modulo a mod b vi a l s hu t + Ly t s ca a cng vi b ri chia cho mu s, nu chia ht th dng. + Ngc li ly t s ca phn s mi tm c cng vi b ri em chia cho mu s ca a. Lp li qu trnh trn cho n khi chia ht th dng. + Phn d ca php chia thng ca phn s va tm c vi b chnh l kt qu cn tm ca a mod b. V d: Tnh gi tr ca + +1+ 7 8 = 3 3 8 + 7 15 = =5 3 3
1 mod 7 3
tip tc dng1 mod 7 = 5 3
+ 5 mod 7 = 5 vy
> (x^3+1/3) mod 7;x3 + 5
Tm nghim ca phng trnh x2 = 2 gn vi gi tr x = 1.4 ta thc hin > RootOf(x^2=2, x, -1.4);RootOf ( _Z 2 2, -1.4 )
xem kt qu ta dng hm evalf( ) > evalf(%);-1.414213562
Tm tt c cc nghim ca phng trnh x3 2 = 0 > RootOf( x^3-2, x): allvalues(%);2( 1/3 ) ( 1/3 ) ( 1/3 ) 1 ( 1/3 ) 1 1 ( 1/3 ) 1 , 2 + I 3 2 , 2 I 32 2 2 2 2
Phng trnh ny c 03 nghim. Mun ly nghim trong khong (1, 2) ta thc hin nh sau: > RootOf(x^3-2,x,1..2); allvalues(%);Trang 43
RootOf( _Z 3 2, 1 .. 2 )
2
( 1/3 )
ly nghim th 2 trong s 03 nghim trn, ta thc hin > r2 := RootOf(x^3-2, index=2): allvalues(r2);( 1/3 ) 1 ( 1/3 ) 1 2 + I 32 2 2
ly kt qu nghim 02 gn ng n 4 s, ta thc hin > evalf(r2,4);-.6300 + 1.091 I
> r3 := RootOf(_Z^4-_Z^2-1, label=1);r3 := RootOf( _Z 4 _Z 2 1, label = 1 )
> evalf(r3);RootOf( _Z 4 _Z 2 1, label = 1 )
> allvalues(r3);1 1 1 1 2+2 5, 2+2 5, 22 5, 22 5 2 2 2 2
> RootOf(x*exp(x)-y, x);RootOf( _Z e _Z y )
> series(%, y);3 8 125 5 y y2 + y3 y4 + y + O( y6 ) 2 3 24
Trang 44
CHNG IV
TH TRONG MAPLEI. th trong khng gian 02 chiu (02 Dimension) 1. th hm thng thng y = f (x) C php: Trong : + expr1,expr2: Biu thc biu din mt hay nhiu hm s hay cc hm dng tham s. + rangeH: Xc nh min v th trn trc honh (x = a ..b) + rangeV: Xc nh min v th trn trc tung (y = c..d) V d: plot(x^2+1,x=-1..3,y=-2..10); + options: Cc ty chn v dng th, gm c cc ty chn sau: a. Dng ca h trc to (axes): C cc ty chn: none, normal, boxed, frame + axes = none: + axes = normal: + axes = boxed: + axes = frame: khng hin th h trc to hin th h trc to dng to cc hin th h trc to dng hnh ch nht hin th h trc to dng frame with (plots): plot([expr1,expr2..], rangeH,rangeV, options)
+ Mc nhin l normal b. T mu cho th (color): color = red, green, blue, black,.... c. Chn kiu ng cho th (linestyle): linestyle = 0, 1, 2, 3, 4 + 1: ng lin (solid) + 2: ng chm (dot) + 3: ng gch (dash) + 4: ng chm gch (dash-dot) + Gi tr mc nhin l 1. d.Chn s lng im v cho mt th: numpoints= Mc nhin l 50 e. Kiu im v th: ng, im hay symbol + style = line, point, patch , patchnogrid style=line: th dng ng style=point: th dng imTrang 45
+ patch: ch dng khi th c cha hnh a gic + patchnogrid: patch v khng c li + Mc nhin l style= line Nu style = point th ta c th chn thm kiu ca im + symbol=circle (trn), cross (gch cho), box(hnh vung), diamond (ht kim cng) + Mc nhin l symbol = diamond Ta cng c th chn kch thc ca cc symbol + symbolsize = + Mc nhin l 10 f. dy ca ng v th (thickness): + thickness = 0,1,2,3 + Mc nhin l 0 Chc nng ny ch c hiu lc khi style = line g. T l co gin trn cc trc to (scaling): + constrained: cc trc c cng t l v di n v + unconstrained: cc trc khng b rng buc v t l di n v + Mc nhin l scaling = unconstrained h. Chn h trc to v th (coords): + coords = cartesian + coords = polar to cc to cc
+ Mc nhin l to cc Mt s cc h to khc: bipolar, cardioid, cassinian, elliptic, hyperbolic, invcassinian, invelliptic, logarithmic, logcosh, maxwell, parabolic, polar, rose, tangent i.Tiu ca th (title): title = tn tiu ca th j. Ch gii cho th (legend): legend = text k.Vit tiu cho trc tung v trc honh: labels=[Tiu trc x,Tiu trc y] V d: labels=[thi gian,qung ng] l. Hng vit ch cho tiu trc tung v trc honh: labeldirections = [x, y] x,y: nhn cc gi tr l horizontal (ngang) hoc vertical (dc) Mc nhin l horizontalTrang 46
V d: vit tiu trc x nm ngang, tiu trc y hng thng ng Labeldirections = [horizontal, vertical] 2. Hin th th hai chiu Khi v th i khi ta khng mun cho hin th ngay th m ta gn th cho mt bin ri khi cn thit th ta s dng hm display( ) hin th cc th , c bit hm ny dng hin th cc th trong gi cng c plottools, v hin th cc ghi ch trn th. C php: Trong : + L: mt th hoc nhiu th + options: cng c cc ty chn nh trong plot + insequence = true: cho hin th nhiu th th cc th s hin th theo th t trong dy lit k L. th c lit k trc s hin th trc, tu chn ny ch c tc dng trong v th ng (animation), cn trong th tnh th n ch hin th th u tin trong dy lit k. + insequence=false: hin th cng mt lc tt c cc th trong dy L. 3. Vit ch trong th C php: Trong : + (x, y): V tr bt u xut chui k t ra mn hnh. + text: Chui cn xut ra mn hnh. V d : textplot([2,3,Do thi ham bac hai]) V d 1:V th hai hm s sau y=x2 v y=sin(x) trong min x [-2,2], trong cng h to cc > with(plots): plot([x^2,sin(x)], x=-2..2,color=[red,blue], linestyle=[1,1], numpoints=50, style=line, labels=["truc x","truc y"], labeldirections=[horizontal,vertical], thickness=3,scaling=constrained); with(plots): textplot ([x, y, text]) with (plottools): display (L, insequence = true, options)
Trang 47
V d 2: V th ng trn dng tham s > restart: with(plots): plot([sin(t),cos(t),t=-Pi..Pi],color=blue, scaling=constrained, thickness=2,axes=frame, labels=["truc y","truc x"], labeldirections=[horizontal,vertical], title="DO THI DUONG TRON");
V d 3: V th hai hm sin(t) v cos(t) trn cng mt h trc to > restart: with(plots): plot([sin(t),cos(t)],t= Pi..Pi, thickness=[1,3],linestyle=[1,2]);
Ch s khc nhau khi t bin t trong ngoc vung cng vi hai hm (trong v d 2) v t bin t ngoi ngoc vung (trong v d 3) V d 4: V th hai hm sin(t) v cos(t) trn cng mt h trc to , min v c xc nh trn trc tung v trc honh > restart: with(plots): plot([sin(t),cos(t)],t=-Pi..Pi,-2..2, thickness=[1,3],linestyle=[1,2]);
Trang 48
> restart: with(plots): plot([sin(t),cos(t)],t=-Pi..Pi,-2..1, view=[-2..2,-1..1]);
V d 5: V th hm y= sin(3x) trong h to cc > restart: with(plots): plot (sin(3*x), x = -Pi..Pi, coords = polar, scaling = constrained);
V d 6: V th hm s gin on c xc nh 1 f( x ) := -1 0 22,1,x with(student): showtangent(x^2+5, x = 2); with (student): showtangent (f(x), x = a)
Trang 50
4. V ng cong theo tham s C php: with (plots): plot ([x (t), y(t), t = a..b], options) V d: V ng xycloide x = t sin(t ) y = 1 cos(t )
> restrat: with(plots): plot([t-sin(t),1-cos(t),t=2*Pi..2*Pi], scaling=constrained);
5. ng cong trong ta cc Mc ch: V ng cong trong ta cc r = f ( ) vi [a, b] C php: with (plots): polarplot ( r(theta), theta = a..b, options) V d: V ng xon c Archimede r = > restart: with(plots): polarplot(theta,theta=0..4*Pi);
6. th hm n Hm n l hm khng c cng thc biu din mt cch tng minh m ch bit c phng trnh biu din mi quan h chng v cc bin c lp f(x,y,z,...) = 0 V d: phng trnh Ellipse:x2 y2 + 1 = 0 9 16Trang 51
Ta dng hm implicitplot( ) trong gi cng c plots v th ca hm n C php: Trong : + x = a..b: min v th trn trc honh + y = c..d: min v th trn trc tung + option: tng t nh khi v th vi hm thng thng m ta s dng phn trn. Khi s dng cc hm trong gi cng c th ta ch np gi cng c (package) mt ln v cc hm ca gi cng c vn cn lu trong b nh cho n khi ta dng lnh restart khi ng li b nh. V d1: V ellipse c phng trnhx2 y2 + =1 9 4
with(plots): Implicitplot (f (x,y) = 0, x = a..b,y = c..d, option)
> restart: with(plots): implicitplot(x^2/9+y^2/4=1,x=-4..4,y=-3..3, scaling=constrained,view=[-5..5,-3..3]);
V d 2: V th hm s c xc nh bi phng trnh
1 2 1 2 x y =1 3 4
> restart: with(plots): implicitplot(x^2/3-y^2/4=1,x=-5..5, y=-5..5,coords=cartesian);
V d 3: v th ca hm s c xc nh bi phng trnh r = 1- cos( ), = 0..2 trong h to cc.Trang 52
> restart: with(plots): implicitplot(r = 1 - cos(theta),r=0..2, theta=0..2*Pi,coords=polar, scaling=constrained);
V d 4: V th ca hai hm s trn cng mt h trc to x2 + y2 = 2 v y = ex > restart: p:=x^2+y^2=2: q:=y=exp(x): with(plots): implicitplot({p,q},x=-Pi..Pi,y=-Pi..Pi, scaling=constrained);
II. th trong khng gian 03 chiu (03 Dimension) 1. th hm thng thng z = f(x, y) C php: Trong : + f,g,h + a, b + c, d + x, y nhng hm v th hng s thc. hng s thc, th tc hoc biu thc theo x tn cc bin ca hm plot3d([f,g,h], x = a..b, y = c..d, options)
+ options: Cc ty chn gm c cc tu chn sau y: a. Trn mu cho th (ambientlight=[r,g,b]): Trn mu cho th theo 3 mu [red, green, blue] b. Kiu hin th ca h trc to (axes): axes = BOXED; NORMAL; FRAME; v NONE. Mc nhin l NONE. c. Mu cho th (color): color =red, green, blue, black,...Trang 53
d. ng vin ca th (contours): contours = n. Mc nhin n =10 e. Chn kiu ca h trc ta (coord): coords = c: h trc to v th. Mc nhin l to cc (cartesian) c c th l mt trong cc h trc to sau cylindrical (tr), cartesian ( cc), spherical (cu), ellipsoidal (elipxoit), ellcylindrical(elip tr), hypercylindrical (hyperbol tr), logcylindrical (logicthm tr), maxwellcylindrical (maxwell tr), oblatespheroidal (cu dt), paraboloidal (paraboloit), paracylindrical (parabol tr), bipolarcylindrical (tr hai cc)... f. T mu cho th (filled): + filled = true: v c t mu + filled = false: v khng t mu g. Chn s im li cho th (grid): grid=[m,n] + m: s ng li trc x + n: s ng li trc y h. Chn dng ca li (gridstyle): gridstyle gm mt trong hai gi tr 'rectangular' (hnh ch nht) ; 'triangular' (hnh tam gic). i. Chn hng vit nhn trn cc trc ta :+ labeldirections = [x,y,z]: hng vit nhn trn cc trc to
+ x,y,z nhn mt trong hai gi tr HORIZONTAL or VERTICAL. Mc nhin l HORIZONTAL. j. t nhn cho cc trc ta : + labels=[x,y,z] : t nhn cho cc trc to + x, y, z: dng chui. k. chiu sng ca th: + light=[phi,theta,r,g,b]: chiu sng cho th. + phi, theta: cc gc trong to cu. + r,g,b: (red, green, blue): nhn gi tr t 0..1 l. Kiu chiu sng: + lightmodel = x: kiu chiu sng + x = 'none', 'light1', 'light2', 'light3', v 'light4'. m. Chn kiu ng v c a th: + linestyle = n: kiu ng v th + n = 0, 1: kiu ng lin nt + n = 2: kiu ng chm chm (dots) + n = 4: kiu ng gch (dashes) n. Xc nh s im v cho th: numpoints=n. Mc nhin l 252 = 625. S im trn mi chiu s l sqrt(n)Trang 54
o. Hng nhn ca th: + orientation=[theta,phi]: hng nhn th. + theta, phi: l cc gc c xc nh trong to cu, gi tr ca theta v phi c tnh theo . Mc nhin l theta=phi=450 p. Php chiu th + projection = r: php chiu ca th + r nhn cc gi tr t 0 n 1, r cng c th nhn cc tn c sn nh l FISHEYE (r = 0, mt c), NORMAL(r = 0.5), ORTHOGONAL (r =1, vung gc). Mc nhin l r = 1. q. T l di n v trn cc trc: + scaling = s: t l di n v trn cc trc + s gm mt trong hai gi tr sau UNCONSTRAINED. Mc nhin l UNCONSTRAINED r. To bng cho th: + shading = s: bng cho th + s c cc gi tr sau XYZ, XY, Z, ZGREYSCALE, ZHUE, NONE s. Kiu th: + style = s: kiu th + s gm mt trong cc gi tr sau POINT, HIDDEN, PATCH, WIREFRAME, CONTOUR, PATCHNOGRID, PATCHCONTOUR, hoc LINE + Mc nhin l style = PATCH t. Chn biu tng cho th: + symbol = s: biu tng cho th kiu point + s c cc gi tr sau BOX, CROSS, CIRCLE, POINT, and DIAMOND. u. Kch thc ca biu tng: + symbolsize = n : kch thc cho symbol + n: s thc. + Mc nhin n = 10 v. B dy ca nt v: + thickness = n: b dy ca ng trong th + n = 0, 1, 2, 3 + Mc nhin n = 0 w. nh s trn cc trc ta : + tickmarks = [k, n, m]: nh s trn cc trc to + k, n, m: l cc s nguyn dng + Mc nhin tickmarks = defaultTrang 55
CONSTRAINED
or
x. Vit tiu cho th: + title=text: vit tiu cho th + Dng k hiu \n bt u cho mt dng mi trong tiu . Mc nhin l khng c tiu . y. Chn font ch cho tiu : + titlefont=font name: font cho tiu ca th + font name: TIMES, COURIER, HELVETICA, and SYMBOL z. Khung nhn ca th: + view =[xmin..xmax, ymin..ymax, zmin..zmax]: Xem mt vng trn th + Mc nhin l xem ton b th 2. Hin th cc th 3D C php: Trong : + L: mt hoc nhiu th. + options: Cc ty chn tng t nh hm plot3d V d 1: V mt phng z = x 2*y trong h to cc. > restart: with (plots): plot3d(x-2*y,x=-2..2,y=-2..2,axes=box); with (plots): display3d (L,options)
V d 2: V th hm s z = e x
2
y2
. Ta thc hin nh sau.
> restart: with(plots): plot3d(exp(-x^2-y^2),x=-2..2,y=-2..2, title="MAT HAM MU",axes=frame, gridstyle=triangular, labels=["truc-x","truc-y","truc-z"], scaling=constrained,shading=xyz, light=[30,30,1,0.8,0.5],scaling=constrained, filled=true, contours=30, orientation=[60,30], thickness=1);Trang 56
V d 3: V th hm s z=sin(xy) trong to cc. > restart: with(plots): plot3d(sin(x*y),x=-2..2,y=-2..2, title="MAT HAM SIN(X*Y)",axes=frame, gridstyle=triangular, labels=["truc-x","truc-y","truc-z"], scaling=constrained, orientation=[30,45], thickness=1,grid=[30,10],gridstyle=rectangular, numpoints=1600,projection=1, style=patchcontour,light=[30,45,0.3,0.9,0.8], lightmodel=light4);
V d 4: V hai th trong cng mt h trc to cc > restart: with(plots): plot3d({x/2+y,sin(x-y)},x=-5..5,y=-5..5, axes=box,labels=["truc-x","truc-y","truc-z"], scaling=constrained,orientation=[152,71]);
Trang 57
Th d 5: V th trong to cu (coords=spherical) > restart: with(plots): plot3d({r*sin(phi)},r=0..2*Pi,phi=0..2*Pi, axes=box,labels=["truc-x","truc-y","truc-z"], scaling=constrained,orientation=[119,71], coords=spherical);
V d 6: V th trong to tr (coords=cylindrical) > restart: with(plots): plot3d({2},phi=0..2*Pi,r=0..2,axes=box, labels=["truc-x","truc-y","truc-z"], scaling=constrained,orientation=[119,71], coords=cylindrical);
Trang 58
3. th hm 3D n C php: Trong : + exprs: mt hay nhiu phng trnh v th + x = a..b, y = c..d, z = p..q: min v th. + a, b, c, d, p, q: l cc s thc + < options >: cc ty chn (nh phn th 3D) V d 1: V hai th ca hai hm n x2 + y2 + z2 = 4, y = e-xz trn cng mt h trc to . > with(plots): implicitplot3d({x^2 +y^2 + z^2 = 4,y = exp(-x*z)}, x=-1..2,y=-1..2,z=-1..1,axes=box,orientation=[161,60]); with (plots): implicitplot3d ({exprs}, x =a..b, y = c..d, z = p..q,< options >)
V d 2: V hnh cu trong h to tr > with(plots): implicitplot3d(r^2+z^2 = 9, r=0..3,theta=-Pi..Pi,z=-3..3, coords=cylindrical,scaling=constrained);
4. V th trong h ta tr x = r cos( ) Mc ch: V ng cong trong h ta tr c phng trnh y = r sin ( ) z = z
Trang 59
C php: Trong :
with (plots): cylinderplot (r = a..b, theta = c..d, z = e.. f, options)
+ r, theta, z l cc bin ca phng trnh. +, a..b, c..d, e..f l cc khong bin thin ca r, theta, z. + options: Cc ty chn nh plot3D V d: > restrat:with(plots): cylinderplot(1,theta=0..2*Pi,z=-1..1);
> restrat:with(plots): cylinderplot(1+z^2,theta=0..2*Pi,z=-1..1,style=patch, shading=zhue,thickness=1);
5. V th trong h ta cu Mc ch: V ng cong trong h ta cu c phng trnh x = r cos( )sin ( ) y = r sin ( )sin ( ) z = r cos( )
C php:
with (plots): with (plottools): sphereplot (r = a..b, theta = c..d, phi = e..f, options)
Trang 60
Trong : + r, theta, phi l cc bin ca phng trnh. +, a..b, c..d, e..f l cc khong bin thin ca r, theta, phi. + options: Cc ty chn nh plot3D V d: > restart: with(plots): with(plottools): sphereplot(1,theta=0..2*Pi,phi=0..Pi/2, axes=boxed,style=patch);
> restart: with(plots): with(plottools): display(sphere([0,0,2]),torus([0,0,-2]), style=patch,axes=frame,scaling=constrained);
III. th ng 1. th 02 chiu ng C php: with (plots): animate(expr, x =a..b, t =c..d, options);Trang 61
Trong : + expr: hm v th ng + x = a..b: khong v trn trc honh + t: bin thay i dng ca th + t = c..d: khong thay i ca bin t. + options: Cc ty chn nh th 2D + frames = n: s lng frame dng cho hin th ng Khi mun cho th chuyn ng sau khi v xong ta click chut vo th ta s thy mt dy cc nt dng iu khin cho vic chuyn ng ca th xut hin, click vo nt play th th s chuyn ng. V d: V th hm sin(x) v cho th chuyn ng di ngang mt chu k (t=0..2* Pi), s frame hin th l 100 frame > with(plots): animate(sin(x+t),x=0..2*Pi,t=0..2*Pi,frames=100);
V d 2: V th hm s y=e-x/t bin i > with(plots): animate(exp(-x/t),x=0..2,t=1..100,frames=100);
V d 3: V hai ng trn c bn knh tng dn thay phin nhau hin th > with(plots): a:=animate([t*cos(x),t*sin(x),x=0..2*Pi], t=1..10,frames=10,scaling=constrained): b:=animate([t*cos(x),t*sin(x),x=0..2*Pi],t=1..10, frames=10,scaling=constrained,color=blue): display([a,b],insequence=true);
Trang 62
2. th 03 chiu ng C php: Trong : + F: hm v th ng + x, y: khong v trn trc x v y + t: bin ng + t = e..f: khong gii hn ca frame + options: Cc ty chn tng t nh plot3d Cch cho th chuyn ng tng t nh th ng hai chiu V d 1:V th ng trong to cc, bin ng l u > restart: with(plots): animate3d([x*u,t-u,x*cos(t*u)],x=1..3,t=1..4,u=2..4); with (plots): animate3d (F, x = a..b, y =c..d, t =e..f, options)
V d 2:V th ng trong to cu, bin ng l u >with(plots): animate3d([x,y,(1.3)^x * sin(u*y)], x=1..3,y=1..4,u=1..2,coords=spherical);
Trang 63
V d 3:V ng thi hai th ng trong to tr > p:=animate3d([x*u,u*t,x*cos(t*u)],x=1..3,t=1..4, u=2..4,coords=cylindrical): > q:=animate3d(cos(t*x)*sin(t*y),x=-Pi..Pi, y=-Pi..Pi,t=1..2,color=cos(x*y)): > plots[display](p,q);
IV. Mt s dng th c bit Ta c th v nhng th dng c bit nh ng trn, ellipse, hyperbal...bng nhng hm nm trong gi cng c plottools. 1. Hm v ng trn C php: Trong :+ c: to ca tm ng trn
circle(c,r,options): v ng trn tm c bn knh r
+ r: di bn knh ca ng trn. Mc nhin r = 1 + options: cc ty chn nh hm plot() V d: V ng trn c tm [2,1] bn knh r=3 > restart: with(plottools): dt:=circle([2,1], 3, scalling=constrained, style=line,thickness=2):Trang 64
plots[display](dt);
2. Hm v Ellipse C php: Trong : + c: tm ellipse + a: na di ca trc ellipse trn trc honh + b: na di ca trc ellipse trn trc tung + filled = true: t ellipse (false: khng t). Mc nhin l filled = false V d: v ellipse c t eq := x2 + y2 = 1 >restart: with(plottools): eq := (x-x0)^2/a^2 + (y-y0)^2/b^2 = 1; a := 2: b := 1: x0 := 0; y0:=0; e := ellipse([x0,y0], a, b, filled=true, color=gold): plots[display](e, scaling=constrained);1 eq := x2 + y2 = 1 4 1 4
ellipse(c, a, b, options)
x0 := 0y0 := 0
3. Hm v hnh ch nht C php: Trong : + [x1, y1]: to nh di bn tri + [x2, y2]: to nh trn bn phi + options: tng t nh hm plot( )Trang 65
rectangle([x1, y1],[x2,y2],options);
V d: v hnh ch nht c nh trn bn tri v nh di bn phi ln lt l: A = [1,2] ,C = [4,5] > cn:=plottools[rectangle]([1,1],[4,3],color=blue):plots[display](cn,scaling=constrained,view=[0..5,0..4]);
4. V on thng ni hai im C php: Trong + [x1, y1] l im u ca on thng + [x2, y2] l im cui ca on thng V d: V on thng t A:=[-1,1] n B:=[3,4] > dt:=plottools[line]([-1,1],[3,4],style=line):plots[display](dt,view=[-2..5,0..5], color=red,thickness=2);
with (plottools): line([x1,y1], [x2,y2], options);
5. V hyperbol C php: Trong : + [x0, y0]: to tm i xng + a, b: l mt n a di cc cnh ca hnh ch nht c s a^2+b^2=c^2 + range: min v x = a..b,y = c..d V d: V hyperbol c xc nh bi phng trnh > restart:with(plottools): with(plots):Trang 66
with(plottools): Hyperbola ( [x0,y0], a, b, range);
x2 y2 = 1 4
a := 2: b := 1: x0 := 0: y0 := 0: eq := (x-x0)^2/a^2 - (y-y0)^2/b^2 = 1; h := hyperbola([x0,y0], a, b, -2..2): display(h);
6. V a gic C php: with (plottools): polygon([[x1,y1], [x2,y2], [x3,y3],....], options); V d: V a gic bn cnh > with(plottools): dagiac := polygon([[0,1], [3,4], [3,1],[1,0]], color=red, thickness=2, scaling=constrained): plots[display](dagiac, view=[-1..4,-1..5]);
7. V cung ellipse C php: Trong : + [x0, y0]: to tm ellipse + a, b: di mt na cnh ca hnh ch nht c s + arc1: gc u + arc2: gc cui V d: V cung ellipse t gc u 00 n gc cui 4 /3 > restart: with(plottools): d := ellipticArc([0,0], 4, 2, 0..4*Pi/3, filled=true, color=green): plots[display](d,scaling=constrained, view=[-5..5,-5..5]); with (plottools): ellipticArc([x0,y0], a, b, arc1..arc2, options);
Trang 67
8. V cung trn C php: Trong : + c: to tm cung trn + r: bn knh + a..b: t gc bt u n gc cui V d: V cung trn tm [1,0] bn knh r=1 t gc2
with(plottools): arc(c, r, a..b, options);
n
> with(plottools): a := arc([1,0], 1, Pi/2..Pi): plots[display](a,scaling=constrained);
9. V th theo cc im cho trc C php: with(plots): PLOT(CURVES([[x1,y1],[x2,y2],[x3,y3],..])); V d: V th qua cc im sau > A:=[1,2]:B:=[3,5]:C:=[3,2]:E:=[1,2]: PLOT(CURVES([A,B,C,E]),COLOR(RGB,0,0.5,1));
Trang 68
CHNG V
GII HN - O HM TCH PHNI. Gii hn1. C php
Limit(f, x=a): Hm tr Limit(f, x=a, dir): Hm tr limit(f, x=a): Hm cho gi tr trc tip limit(f, x=a, dir) : Hm cho gi tr trc tip Trong : + f l mt biu thc i s (an algebraic expression) + x l bin ca hm f + a l mt gi tr, hoc infinity ( + ), hoc -infinity (- ) + dir ch hng ly gii hn left - gii hn bn tri right - gii hn bn phi real - gii hn thc complex - gii hn phc V d 1:Vit gii hn di dng hm tr > f:=Limit(sin(x)/x, x=0);f := limx 0
sin( x ) x
Mun tnh gi tr ca gii hn trn ta thc hin > f:=value(f); Hoc ta tnh bng lnh trc tip > f:=limit(sin(x)/x, x=0);f := 1
f := 1
Ta cng c th thc hin nh sau >Limit((x^2-3*x+3)/(x*(5*x-1)),x=infinity):%=value(%);x2 3 x + 3 1 lim = 5 x x (5 x 1)
V d 2:Tnh gii hn lim e xx
Trang 69
> limit(exp(x), x=-infinity);01 khi x 0 x
V d 3: Tnh gii hn ca hm s y = > limit(1/x, x=0);
undefined
V kt qu ny c nhiu gi tr nn my khng tnh c. Nhng nu ta tnh c th cho tng trng hp th my s tnh c. Tnh gii hn phc ca1 x1 =I x
> Limit(1/x, x=0,complex):%=value(%);x 0,complex
lim
Tnh gii hn bn tri im 0 > Limit(1/x, x=0,left):%=value(%);x 0-
lim
1 = x
Tnh gii hn bn phi im 0 > Limit(1/x, x=0,right):%=value(%);x 0+
lim
1 = x
V d 4: Tnh gii hn: limx 1
x +1 2 x1 3
13 2 4
> limit((sqrt(x+1)-sqrt(2))/(x^(1/3)-1),x=1);
> (Limit(1/x,x=0)*Limit(x,x=0)); lim 1 ( lim x ) x 0 x x 0
> combine(%);x 0
lim 1
2. V d ng dng Dng nh ngha vit on chng trnh tnh o hm ca hm s x 2 + x ti im x0 = 0.Trang 70
2 3
> restart: with(student): f:=x->-2/3*x^2+x;
2 f := x x2 + x 3
> (f(x+h)-f(x))/h;
2 2 ( x + h )2 + h + x2 3 3 h
> Limit(%,h=0);
2 2 ( x + h )2 + h + x2 3 3 lim h h04 x+1 3
> value(%);
> kq:=subs(x=0,%);
kq := 1
>print(`Vay ta co:(`,Diff(2/3*x^2+x,x),`x=0)= `,kq);Vay ta co : (, 2 2 x + x , x = 0 ) = , 1 x 3
II. O HM CA HM S 1. o hm ca hm thng thng a. o hm mt bin C php: diff (f,var): hm cho gi tr trc tip Diff (f,var): Hm tr D(f)(var): ton t vi phn Trong + f l hm s cn ly o hm + var l bin ly o hmx2 + x +1 V d 1: Tnh o hm ca hm s f = x 1
> f:=(x^2+x+1)/(x-1);f := x2 + x + 1 x1
> diff(f,x);2 x + 1 x2 + x + 1 x1 ( x 1 )2Trang 71
Rt gn li biu thc trn > normal(%);x2 2 x 2 ( x 1 )2
V d 2: Tnh o hm ca hm s y=xsin(cos(x)) > Diff(x*sin(cos(x)),x):%=value(%); x sin( cos( x ) ) = sin( cos( x ) ) x cos( cos( x ) ) sin( x ) xx 3 + x x 0 V d 3: Tnh o hm ca hm s g ( x) = 2( x 1)2 x > 0
> g:=piecewise(x>=0,x^3+x,2*(x-1)^2);g := { x3 + x 2 (x 1)2
0x otherwise
> diff(g,x); 4x4 undefined 3 x2 + 1 x expr:=x*y-sin(x)-sin(y);expr := x y sin( x ) sin( y )
> diff(expr,[x,x]);sin( x )Trang 72
V d 2: Vit cng thc tnh o hm ca > Diff(f(x,y,z),x,y,y);3 y 2 x
3 f ( x, y , z ) y 2 x
f( x, y, z )
> Diff(y^3*cos(y),y$2):%=value(%);2 y2
y3 cos( y ) = 6 y cos( y ) 6 y2 sin( y ) y3 cos( y )
2. o hm ca hm n Hm n l hm khng c cng thc biu din mt cch tng minh m ch bit c phng trnh biu din mi quan h gia chng v cc bin c lp. a. Hm n v hng: L hm xc nh bi phng trnh f(x,y)=0, trong y l bin ph thuc mt chiu (hm) cn x l bin c lp. C php: implicitdiff(f, y, x): Ly o hm cp 1 ca y theo mt bin x implicitdiff(f,y,x$k): Ly o hm bc k ca y theo mt bin x implicitdiff(f, y, x1,...,xk): Ly o hm ring bc k ca y theo cc bin x1,x2,....,xk.x2 = 0 ta V d 1: Tnh o hm ca hm y c xc nh bi biu thc y z
thc hin > f:=y - x^2/z=0;f := y x2 =0 z
> implicitdiff(f,y,x);2 2x y= x z x z
Vy ta c
x2 =0 o hm ca y theo z c xc nh bi phng trnh y z
> implicitdiff(f,y,z); x2 z2
Vy ta c
x2 y= 2 z zTrang 73
V d 2: Cho x 2 + y 3 = 1 > g:=x^2+y^3=1;g := x2 + y3 = 1
Tnh
y ta thc hin x
> implicitdiff(g,y,x); 2 x 3 y2
Ta c th tnh tun t nh sau: + Rt y theo x > y:=(1-x^2)^(1/3);y := ( 1 x2 )( 1/3 )
+ Ly o hm ca y theo bin x > Diff(y,x):%=value(%);( 1/3 ) 2 ( 1 x2 ) = 3 x
x ( 1 x2 )( 2/3 )
Tnh
x y
> implicitdiff(g,x,y);3 y2 2 x
Tnh
y z
> implicitdiff(g,y,z);0
> implicitdiff(g,y(x),x); 2 x 3 y2
Ch > implicitdiff(f,y(a),x); Error, (in implicitdiff) 2nd argument, y(a), must be a function of, x My bo li v y=y(a) bin theo a cn ly o hm th theo xTrang 74
V d 3 : Tnh o hm cp hai ca hm y theo x ca biu thc g = x 2 + y 3 = 1 > restart: g:=x^2+y^3=1;g := x2 + y3 = 1
> implicitdiff(g,y,x,x);2 3 y3 + 4 x2 9 y5
Hoc ta c th vit nh sau > implicitdiff(g,y,x$2); 2 3 y3 + 4 x2 9 y52y z2 = 0 z
V d 4: Cho x,y,z c xc nh bi biu thc ax 3 y > f := a*x^3*y-2*y/z-z^2=0;f := a x3 y y ( x, z ) x
2y z2 = 0 z
o hm ring
> implicitdiff(f,y(x,z),x);3 a x2 y z a x3 z 2
o hm ring ca hm y(x,z) theo hai bin x v z x, z c xc nh bi phng trnh ax 3 y
2 y ( x, z ) . Trong y, zx
2y z 2 = 0 . Ta thc hin nh sau: z
> implicitdiff(f,y(x,z),x,z);6 a x2 ( 2 y z 3 ) a 2 x6 z 2 4 a x3 z + 4
b. Hm n vect: Hm n vect c xc nh bi h phng trnh {f1, f2,,fm}, trong mi fi l mt phng trnh theo bin ph thuc {y1, y2,,yn} (cc hm n) v cc bin c lp x1,x2,, xk. ly o hm ca hm n vect ta thc hin nh sau: + Tnh o hm ca cc hm thnh phn {u1 , u 2 ,L, u r } trong vect hm {y1 , y 2 ,L, y n } theo bin xTrang 75
implicitdiff({ f1 ,..., fm },{ y1 ,...., yn },{ u1 ,..., ur },x); + Tnh o hm ring cp k theo cm bin x1,x2,...,xk ca cc hm thnh phn {u1 , u 2 ,L, u r } trong vect hm {y1 , y 2 ,L, y n } implicitdiff({ f1 ,..., fm },{ y1 ,...., yn },{ u1 ,..., ur },x1,x2....,xk); Trong : + fi : cc biu thc i s hoc cc phng trnh + yi : cc bin ph thuc (cc hm) + ui : Nhm cc hm m ta s ly o hm + xi : Tn cc bin m ta cn ly o hm ring V d 1: Cho f v g c xc nh nh sau > restart: f:=x^2+2*y^2=z^2+1;g:=y^2-2*x^3-z=0;f := x2 + 2 y2 = z 2 + 1 g := y2 2 x3 z = 0
> implicitdiff({f,g},{y(x),z(x)},{z,y},x);{ D( y ) = 1 x (1 + 6 z x) x (1 + 6 x) , D( z ) = } 1 + z 2 y ( 1 + z )
dng
y x
Hoc ta c th dng lnh sau y t cho biu tng D(y) l k hiu o hm
> implicitdiff({f,g},{y(x),z(x)},{z,y},x,notation=diff);{ 1 x (1 + 6 z x) x (1 + 6 x) y= , z= } x 1 + z 2 y ( 1 + z ) x
V d 2: Cho hai phng trnh f v g c xc nh bi: > f := y^2-2*x*z = 1;g := x^2-exp(x*z) = y;f := y2 2 z x = 1
g := x2 e
(z x)
=y
Trong y=y(x) v z=z(x). Tnh o hm ca y theo x. Bit x, y, z c xc nh theo g > implicitdiff({g},{y(x),z(x)},{y},x,notation=Diff);{(z x) (z x) D( z ) x e z} y=2xe x
Tnh o hm ca y theo x. Bit x, y, z c xc nh trong h {f,g}Trang 76
> implicitdiff({f,g},{y,z},{y},x,notation=Diff);{ x } y=2 (z x) x ye +1
Ta tnh cng mt lc
z v y . Bit x, y, z c xc nh trong h {f,g} x x
> implicitdiff({f,g},{y(x),z(x)},{y,z},x,notation=Diff);(z x) z+z x z = 2 y x + y e , y=2 x (z x) (z x) x x (y e + 1) ye +1
V d 3: Cho 3 phng trnh f, g, h c xc nh nh sau: > f := a*sin(u*v)+b*cos(w*x)=c;f := a sin( u v ) + b cos( w x ) = c
> g := u+v+w+x=z;g := u + v + w + x = z
> h := u*v+w*x=z;h := u v + w x = z
Trong cc hm u=u(x,z), v=v(x,z), w=w(x,z) + u = u(x,z): Ngha l u l hm ph thuc vo hai bin x v z Tnh o ring ca hm u theo bin z xem x l hng s. Bit u, v, w, z ,x c xc nh t h 3 phng trnh {f, g, h} trn.Ta thc hin nh sau: > implicitdiff({f,g,h},{u(x,z),v(x,z),w(x,z)},{u},z, notation=diff);a cos( u v ) u + cos( u v ) a x u b sin( w x ) x + x b sin( w x ) u { u = } z ( a cos( u v ) + b sin( w x ) ) ( u v ) x x
o ring ca cc hm u, v, w theo bin x xem z l hng s. Bit u, v, w, z ,x c xc nh t h 3 phng trnh {f, g, h}.Ta thc hin nh sau: > implicitdiff({f,g,h},{u(x,z),v(x,z),w(x,z)},{u,v,w},x);{ D1( w ) =
w u ( w + x ) v (w x) , D1( u ) = , D1( v ) = } x x (v u) (u v) x
> implicitdiff({f,g,h},{u(x,z),v(x,z),w(x,z)},{u,v,w}, x,notation=Diff); u ( w + x ) w v (w x) { u = , w = , v = } x x x x x (v u) (u v) x z z z
o ring ca cc hm u, v, w theo bin z xem x l hng s. Bit u, v, w, z ,x c xc nh t h 2 phng trnh {g, h}.Ta thc hin nh sau:Trang 77
> implicitdiff({g,h},{u(x,z),v(x,z),w(x,z)},{u,v,w}, x,notation=diff); v = x z u x + u v + w x x x z z ux , w = x z u v u u u + w x x z z ux
,
u = u x x z z
III. TCH PHN 1. Tch phn mt lp a. Tch phn bt nh:(nguyn hm ca hm s) C php: Trong : + f: biu thc di du tch phn. + var: bin ly tch phn. V d 1: Tnh tch phn (nguyn hm) ca hm f (x ) = 2 x 2 + sin x > f:=2*x^2+sin(x);f := 2 x2 + sin( x )
int(f, var); Int(expr, x);
> int(f,x);
2 3 x cos( x ) 3
Ta cng c th thc hin nh sau: > A:=Int(f,x);A := 2 x2 + sin( x ) d x
> value(A);
2 3 x cos( x ) 3
Hoc ta vit nh sau: > Int(f,x):%=value(%);
2 x2 + sin( x ) d x = 2 x3 cos( x ) 3
V d 2: Tnh tch phn
4 x2 + x 2 d x ta thc hin nh sau: 3 x3 + 2 x 5
> f:=(4*x^2+x-2)/(3*x^3+2*x-5);
Trang 78
4 x2 + x 2 f := 3 x3 + 2 x 5
> B:=int(f,x);B :=
3 35 13 1 ln( x 1 ) + ln( 3 x2 + 3 x + 5 ) + 51 arctan ( 6 x + 3 ) 51 51 11 66 187
Rt gn biu thc li ta c: > B:=normal(B);B :=
3 35 13 1 ln( x 1 ) + ln( 3 x2 + 3 x + 5 ) + 51 arctan ( 2 x + 1 ) 51 17 11 66 187
kim tra li ta ly o hm ca B > diff(B,x);
35 78 (6 x + 3) 66 187 3 1 + + 2 11 x 1 3 x + 3 x + 5 3 1 + ( 2 x + 1 )2 17
Rt gi li biu thc ta c: > normal(%);
4 x2 + x 2 ( x 1 ) ( 3 x2 + 3 x + 5 )
V d 3: Tnh tch phn ln( x + 1 + x2 ) d x
> C:=int(ln(x+sqrt(1+x^2)),x);C := ln( x + 1 + x2 ) x 1 + x2
> Int(ln(x+sqrt(1+x^2)),x):%=value(%);ln( x + 1 + x2 ) d x = ln( x + 1 + x2 ) x 1 + x2
V d 4: Tnh tch phn sin( x ) e x dx
> E:=int(sqrt(1-x^2)*sin(x),x);E := 1 x2 sin( x ) d x
My khng tnh c tnh phn ny do n hin ln biu thc nguyn thy, nhng nu ta ch cho n phng php tnh thch hp (i bin, tng phn, nghim chui) th n s tnh c. Ta tm nghim tch phn trn di dng chui bng cch khai trin biu thc tch phn trn theo chui ly tha ca x > E:=series(E,x,6);E := 1 2 1 4 1 6 x x x + O( x7 ) 2 6 180
Trang 79
( x ) ln( x ) d x V d 5: Tnh tch phn e 2
> F:= exp(-x^2)*ln(x);F := e2 ( x )
ln( x )
> P:=int(F,x); ( x2 ) P := e ln( x ) d x
My khng tnh c tch phn ny chnh xc do n khng hin th kt qu, do ta tm nghim gn ng ca tch phn trn di dng chui nh sau Khai hm di du tch phn thnh chui ly tha > F:=series( F, x, 4 );F := ln( x ) ln( x ) x2 + O( x4 )
Tnh tch phn ca hm khai trin ny > P:=int(F,x);
1 1 P := x ln( x ) x ln( x ) x3 + x3 + O( x5 ) 3 9
Hoc ta cng c th lm nh th d 4 l ta khai trin tch phn thnh chui th ta cng tm c kt qu: > P:=series(P,x);1 1 P := ( ln( x ) 1 ) x + ln( x ) + x3 + O( x5 ) 3 9
Ly phn chnh trong biu thc trn: >convert(P,polynom);
1 1 P := x ln( x ) x ln( x ) x3 + x3 3 9
b. Tch phn xc nh C php: Trong : + f: biu thc cn tnh tch phn.(f phi lin tc trn an [a,b] hoc c im gin an loi 1 trn an [a,b]) + x: bin ly tch phn + a..b: khong ly tch phn V d 1: Tnh tch phn sau: 01 2
int(f, x=a..b); Int(f, x=a..b );
cos( x ) dx 1 + sin( x ) 2
Trang 80
> int(cos(x)/(1+sin(x)^2),x=0..Pi/2);1 4
Hoc ta mun nhn dng tng minh ca biu thc tch phn ta dng hm Int(): > p:=Int(cos(x)/(1+sin(x)^2),x=0..Pi/2); p := 01/2
cos( x ) dx 1 + sin( x )2
Mun xem kt qu tnh trn ta dng hm value(): > value(p);1 4
x V d 2: Tnh B:= ( 1 + sin( x ) ) e dx ta thc hin 0
1
> f:=(1+sin(x))*exp(x);f := ( 1 + sin( x ) ) e x
> B:=int(f,x=0..1);1 1 1 B := e e cos( 1 ) + e sin( 1 ) 2 2 2
Mun nhn tr gn ng ca biu thc trn ta thc hin: > evalf(%,4);.7855dx V d 3: Tnh tch phn: 01 1
1 1 + x4
> Int(1/sqrt(1+x^4),x=0..1):%=value(%); 0 1 1+x4
dx =
1 1 EllipticK 2 2 2
Kt qu tnh c c biu din qua hm EllipticK hm Elliptic c nh ngha nh sau: EllipticF(z,k) = 0bz
1 1 t2 1 k2 t 2
dt
EllipticK(k)=EllipticF(1,k) sin( x ) V d 4: Tnh tch phn x dx aTrang 81
> Int(sin(x)/x,x=a..b)=int(sin(x)/x,x=a..b); sin( x ) d x = Si( b ) Si( a ) x ab
Tch phn c biu din qua hm Si(x). Hm ny c nh ngha sin( t ) Si(x)= t dt 01 V d 1: Cho hm s f( x ) := { 2 f := { 1 2 1x Tnh tch phn sau f( x ) d x otherwise 0 1x otherwise2
x
> f:=piecewise(x>=1,1,2);
V ng biu din ca hm s: > plot(f,x=0..2,y=-2..2,discont=true); Ta nhn thy rng x=1 l im gin an loi 1 nn n kh tch trong an [0,2]
> s:=int(f,x=0..2);
s := 3
> plot(f,x=0..2,y=-2..2,discont=false,filled=true);
Nhn xt: s chnh l din tch phn hnh t mu ca th.Trang 82
c. Tch phn suy rng: l tch phn c cc dng sau: c1. Khong ly tch phn l v hn: f( x ) d x , f( x ) d x , f( x ) d x a + Nu khong ly tch phn l [a, ) f( x ) d x = lim f( x ) d x b a a b b
+ Nu khong ly tch phn l (- ,b] f( x ) d x = lim f( x ) d x a ( ) ab b
+ Nu khong ly tch phn l (- ,+ ) th ta chia khong ra lm hai khong (- , a] v [a,+ ) ri ly tch phn nh trn. c2. Hm cn ly tch phn c im gin an v cc trong khang ly tch phn 1 1 gin an v cc ti x = 0 trong on [-1, 2] V d: x dx hm y = x -1 tan( x ) d x hm tan(x) gin an v cc ti x = trong on [0, ] 2 0
2
Nu hm f(x) lin tc trn na on [a,b) v gin on v cc ti x = b. Gi s tn ti gii hn lim f( x ) d x hu hn th tch phn l hi t v: c b- a f( x ) d x = lim f( x ) d x c b- a ab cc
Nu hm f(x) lin tc trn na an (a,b] v gin on v cc ti x = a. Gi s tn ti gii hn lim f( x ) d x hu hn th tch phn l hi t v c a+ c b
f( x ) d x = lim f( x ) d x c a+ a cc d lim f( x ) d x + lim f( x ) d x hu hn th tch phn trn Gi s tn ti gii hn c a+ c b- a c d c d f( x ) d x = lim f( x ) d x + lim f( x ) d x l hi t v c b- c a+ a a c
b
b
Nu hm f(x) lin tc trong khong [a,b), (b,d] v b gin on v cc ti x = b.
1 dx V d 1: Tnh tch phn sau: 1 + x2 0Trang 83
> f:=1/(1+x^2);f :=
1 1 + x2
Dng thc tch phn > A:=Int(f,x=0..infinity); 1 A := dx 1 + x2 0
Gi tr ca tch phn ny l > A:=value(A);A :=2
1 2
1 dx V d 2: Tnh tch phn sau: ( x a )2 0
> int( 1/(x-a)^2, x=0..2 ); 0 2 2 0
a 0 0 a < 2 a + 0 2 a ( 2 + a ) a
a 0 a < 2 a2 2 a
Ta nhn thy rng gi tr ca tch phn trn ph thuc vo gi tr ca a: cn tri, cn phi, hay a nm trong on [0,2]. Nu a khng thuc khong ly tch phn th cho biu thc n gin ta b qua im gin an ca hm s ti a. > int( 1/(x-a)^2, x=0..2,continuous );2 1 ( 2 + a ) a
Ch : Kt qu ny ch ng trong trng hp a khng thuc khong ly tch phn. V d 3: Tnh tch phn sau:
x 1 dx0
1
1
> int(1/(1-x), x=0..1);
x 1 dx0
1
1
My khng tnh c tch phn ny. Nhn xt: Hm di du tch phn c im gin an v cc ti im x = 1 nm trong khong ly tch phn. y l tch phn suy rng loi 2, mun tnh tch phn ny th dng cng thc tnh tch phn suy rng tnh. > restart:Trang 84
g:=1/(1-x); f:=Int(1/(1-x), x=0..c):g :=
1 1x
>Limit(f,c=1,left):%=value(%); 1 lim dx = c 1- 1 x 0c
Vy tch phn ny phn k 2. Tch phn bi Ta c th tnh tch phn hai lp, ba lp bng cc hm trong gi cng c student. a. Tch phn 2 lp C php: with(student): Doubleint (f, x, y): tch phn bt nh hoc Doubleint (g, x, y, Domain) hoc Doubleint (g, x = a..b, y = c..d ): tch phn xc nh. Trong : + f:biu thc ly tch phn. + x, y: Cc bin ca f. + a, b, c ,d: Cn ly tch phn. + Domain: Tn min ly tch phn. V d 1:Vit cng thc tnh tch phn bt nh sau: f( x, y ) d x d y
> with(student): Doubleint(f(x,y),x,y);f( x, y ) d x d y
V d 2: Tnh tch phn sau x2 y sin( y ) dx dy
Ta thc hin > f(x,y):=x^2*y-sin(y); with(student): Doubleint(f(x,y),x,y);f( x, y ) := x2 y sin( y ) x2 y sin( y ) d x d y
Mun bit gi tr ca tch phn tnh c ta dng hm value( ) > value(%);Trang 85
1 3 2 x y + x cos( y ) 6
Hoc ta c th vit nh sau: > with(student): Doubleint(x^2*y-sin(y),x,y):%=value(%);x2 y sin( y ) d x d y = 1 x3 y2 + x cos( y ) 6 d b
V d 3: Vit cng thc tnh tch phn sau c
x2 y sin( y ) dx d y a
Ta thc hin > with(student): Doubleint(f(x,y),x=a..b,y=c..d); cd
x2 y sin( y ) dx d y a a
b
2 V d 4: Tnh tch phn sau : x y sin( y ) dx d y 0 1
> with(student): Doubleint(x^2*y-sin(y),x=1..a,y=0..Pi); x2 y sin( y ) dx d y 0 1 a
Kt qu tnh c l: > value(%);1 2 3 1 2 a 2a+2 6 6
V d 5: Vit cng thc tnh tch phn trong min D > with(student): > Doubleint(x-3*y,x,y,D); x 3 y dx dy D
b.Tch phn 3 lp C php: with (student): Tripleint (g, x, y, z) : tch phn bt nh Tripleint (g, x, y, z, Domain) Trong :Trang 86
:
Tripleint(g, x = a..b, z = e..f, y = c..d ) ; : tch phn xc nh.
+ g: biu thc di du tch phn. + x, y, z: Cc bin ly tch phn. + a, b, c , d, e, f : Cc cn ca min ly tch phn. + Domain: Min ly tch phn. V d 1: Vit cng thc tnh tch phn sau f( x, y, z ) d x d y d z Ta thc hin > with(student): Tripleint(f(x,y,z),x,y,z);f( x, y, z ) d x d y d z
V d 2: Tnh tch phn bt nh sau sin( y ) x z d x d y d z > with(student): Tripleint(sin(y)*x-z,x,y,z);sin( y ) x z d x d y d z
> value(%);
1 1 x2 cos( y ) z z 2 x y 2 2
V d 3: Vit cng thc tnh tch phn sau f( x, y, z ) d x dy dz e c a > with(student): Tripleint(f(x,y,z),x=a..b,y=c..d,z=e..f); f( x, y, z ) d x d y d z e c af d b
f
d
b
V d 4: Tnh tch phn sau 1 d x d y d z 0 0 0 > with(student): Tripleint(1,x=0..1,y=0..1,z=0..1); 1 dx dy dz 0 0 01 1 1
1
1
1
> value(%);12 2 2 V d 5: Tnh tch phn sau x + y + z dx d y d z -1 -1 -1 1 1 1
> with(student):Trang 87
Tripleint(x^2+y^2+z^2,x=-1..1,y=-1..1,z=-1..1); value(%); x2 + y2 + z 2 d x d y d z -1 -1 -11 1 1
8
3. Phng php i bin v tch phn tng phn a. Phng php i bin s: C php: with (student): changevar (s, f); changevar (s, f, u); changevar (t, g, v); Trong : + s l mt biu thc dng h(x) = g(u) xc nh x nh hm ca u. + f L biu thc c cho di dng thc nh Int(F(x), x = a...b) + u Tn bin mi. + t - Tp cc phng trnh xc nh php bin i nhiu n. + g K hiu hnh thc ca tch phn bi. + v Danh sch cc bin mi. Ch : Tham s th nht trong cc lnh trn l mt hay nhiu phng trnh xc nh bin mi theo bin c. Nu c nhiu hn 2 bin th bin mi phi c cho v tr ca tham s th ba. Tham s th hai l biu thc cha Int, Sum, Limit hay Doubleint hoc Tripleint. V d 1: Dng phng php i bin s tnh tch phn ( cos( x ) + 1 )3 sin( x ) dx
Ta thc hin i bin x thnh bin u vi u = cos(x) + 1 > with(student): changevar(cos(x)+1=u,Int((cos(x)+1)^3*sin(x),x),u);u 3 d u
Mun bit gi tr ca tch phn trn ta dng hm value() > f:=value(%);1 f := u 4 4
Mun tr v kt qu vi bin x, ta thc hin nh sau: >subs(u=cos(x)+1,f);1 ( cos( x ) + 1 )4 4Trang 88
V d 2: Dng phng php i bin tnh tch phn sau x2 + y2 dx d y
> with(student): assume(r>0); changevar({x=r*cos(phi),y=r*sin(phi)}, Doubleint(x^2+y^2,x,y),[phi,r]);r 3 d d r with assumptions on r
> value(%);1 4 r 4 with assumptions on r
b. Phng php tch phn tng phn C php: with(student): Intparts (f, u); Trong : + f: l biu thc c dng Int (udv, x) + u: l tha s kh vi ca biu thc di du tch phn. Lnh trn s cho kt qu di dng uv-Int (vdu, x) V d 1: Dung phng php tch phn tng phn, tnh tch phn sau x5 cos( x ) dx
> with(student): intparts(Int(x^5*cos(x),x),x^5);x5 sin( x ) 5 x4 sin( x ) d x
V d 2: Tnh tch phn tng phn sau K := x2 e x dx
> restart: with(student): K:=Int(x^2*exp(x),x);K := x2 e x d x Trang 89
Ly tch phn tng phn vi u = x2, dv = exdx > P:=intparts(K,x^2);P := x2 e x 2 x e x d x
v = ex
Trch ly phn th nht ca biu thc P > p1:=op(1,P);p1 := x2 e x
Trch ly phn th hai ca biu thc P > c1:=op(2,P);c1 := 2 x e x d x
Ta tip tc ly tch phn tng phn ca tch phn ny vi u = x, dv = exdx > c1:=intparts(c1,x);c1 := 2 x e x + 2 e x d x
v = ex
Trch ly phn th nht ca biu thc c1> p2:=op(1,c1);p2 := 2 x e x
Trch ly phn th hai ca biu thc c1 c2:=op(2,c1);c2 := 2 e x d x
Tnh gi tr ca tch phn cui cng: p3:=value(c2);p3 := 2 e x
> K:=p1+p2+p3;K := x2 e x 2 x e x + 2 e x
V d 3: Tnh tch phn tng phn sau > with(student): intparts(Int(sin(x)*x+sin(x), x=0..Pi/2), sin(x)); 1 2 1 + 8 2 01/2
1 cos( x ) x2 + x d x 2
> expand(%);1 2 1 1 + 8 2 2
1/2
0
cos( x ) x2 d x 0
1/2
x cos( x ) d x
Trang 90
Ta c th ln lt tnh cc tch phn cn li bng phng php tch phn tng phn c kt qu cui cng.
Trang 91
CHNG VI
I S TUYN TNHMaple cho php ta tnh nh thc ma trn, gii h phng trnh tuyn tnh, Ngoi ra Maple cn cho php ta tm gi tr ring, vect ring, a thc c trng, tm dng chnh tc ca ma trn v tnh rt nhiu ma trn c bit nh Hilbert, Toeplitz, Cc lnh ca i s tuyn tnh c ci t sn trong gi cng c linalg do trc khi s dng ta phi np gi cng c ny vo trong b nh bng lnh: > with(linalg):Warning, the unprotected protected names norm and trace have been redefined and
I. Cc php ton i s trn ma trn v vect 1. Cc lnh to ma trn cp m x n a. matrix (m, n, L) Trong + L: Bng lit k cc phn t ca ma trn theo th t t tri sang phi v t trn xung di. + L = [[a11 , a12 ,L, a1n ], [a 21 , a 22 ,L, a 2 n ],L, [a m1 , a m 2 ,L, a mn ]] + Trong trng hp L c xc nh bi cc phn t c th th ta c th b qua cc ch s m, n. V d: To ma trn 3 dng, 4 ct sau (m = 3, n = 4) > matrix(3,4,[[1,2,1,2],[3,5,4,6],[3,4,0,2]]);1 3 3 2 5 4 1 4 0 2 6 2
Hoc ta c th vit: > matrix(3,4,[1,2,1,2,3,5,4,6,3,4,0,2]);1 3 3 2 5 4 1 4 0 2 6 2
Hoc ta b ch s hng, ct: > matrix([[1,2,1,2],[3,5,4,6],[3,4,0,2]]);1 3 3 2 5 4 1 4 0 2 6 2
Trong trng hp ny ta thy hng ca ma trn c xc nh ngay t s lng vect trong danh mc L (3 vect) v s lng cc phn t trong mi vect (4 phn t). b. matrix (m, n, f) To ma trn cp m x n vi cc phn t ca ma trn l cc gi tr hm ca f xc nh trn cc ch s hng v ct ca ma trn.
Trang 92
V d 1: To ma trn C := > C:=matrix(2,2,f);
f( 1, 1 ) f( 2, 1 )
f( 1, 2 ) f( 2, 2 ) f( 1, 2 ) f( 2, 2 )
f( 1, 1 ) C := f( 2, 1 )
V d 2: To ma trn
x3 x2 A := 2 x3 2 x4 4 5 3 x 3 x
> g:=(i,j)->i*x^(i+j);A:=matrix(3,2,g);g := ( i, j ) i x(i + j)
x3 x2 A := 2 x3 2 x4 4 3 x 3 x5
c. Matrix (m, n, init): To ma trn cp m x n vi cc phn t c xc nh trong init. Matrix (n, init): To ma trn vung cp n vi cc phn t c xc nh trong init. V d: > init:=(x,y)->x-y;B:=Matrix(1..2,1..2,init); init := ( x, y ) x y0 B := 1 -1 0
> B:=Matrix(3,2,init);
0 B := 1 2
-1 0 1
> init:={(1,1)=a,(1,2)=b,(2,1)=2,(2,2)=2};C:=Matrix(2,2,init); init := { ( 1, 2 ) = b, ( 1, 1 ) = a, ( 2, 1 ) = 2, ( 2, 2 ) = 2 }a C := 2 b 2
Phn t no khng c khai bo th n s t ng gn gi tr 0 V d: > init:={(1,1)=a,(2,1)=2,(2,2)=2};C:=Matrix(2,2,init); init := { ( 1, 1 ) = a, ( 2, 1 ) = 2, ( 2, 2 ) = 2 }a C := 2 Trang 93
0 2
> Matrix(2,3,[[6,2],[2,5,1]]);6 2 2 5
0 1
V d: Khai bo ma trn vung cp 3 ca hm f vi f c xc nh bi cng thc sau: f ( j, k ) = j + k &