CC THAO TC C BN TRN MAPLEMaple c 2 mi trng lm vic l ton v vn bn.
Sau khi khi ng, Maple t ng bt mi trng ton. Mun chuyn sang mi trng
vn bn, kch chut vo biu tng T trn thanh cng c hay vo trnh
Insert->Text. Ngc li, t mi trng vn bn, kch chut vo du "[>"
trn thanh cng c hay vo Insert chuyn sang mi trng ton.
* Cc php ton:+, -, *, /, ^, !, , =, =, :=Sin, cos, tan, * Lnh ca
Maple (Maple Input). Lnh ca Maple c a vo worksheet ti du nhc lnh.
Theo mc nh du nhc lnh l ">" v lnh ca Maple hin th bng Font ch
Courier mu . Kt thc lnh bng du (;) kt qu s hin th ngay, khi ta kt
thc lnh bng du (:) th Maple vn tin hnh tnh ton bnh thng nhng kt qu
khng hin th ngay. Lnh c thc hin khi con tr trong hoc cui dng lnh m
ta nhn Enter. Lnh ca Maple c hai loi lnh tr v lnh trc tip: Lnh tr v
lnh trc tip ch khc nhau ch ci u tin ca lnh tr vit in hoa, lnh trc
tip cho kt qu ngay, cn lnh tr ch cho ta biu thc tng trng.V d 2: Tnh
tng cc bnh phng ca n s t nhin u tin. Lnh trc tip cho ta kt qu ngay
khi nhn Enter.
> sum(k^2,k=1..n);
Lnh tr s cho ta biu thc.
> Sum(k^2,k=1..n);
* Kt qu ca Maple (Maple Output). Sau khi nhn phm Enter cui hoc
trong dng lnh trong mt cm x l th kt qu tnh ton s c kt xut (mu xanh
c ban).II. MAPLE VI CC TNH TON TRONG S HC Bt u cng vic tnh ton ta
dng lnh khi ng chng trnh [> restart:, lnh ny c cng dng xo i tt c
cc bin nh ca cc cng vic tnh ton trc . Vi cc php ton s hc nh php
cng(+), php tr(-), php nhn(*), php chia(/), php lu tha (^), cc php
ton ly phn nguyn,phn d,...1. Tnh gi tr biu thc. > 18*(25^9 +
7^11)-(12+6^8); > 55!; > length(%);Th d2: Biu thc
>b:=sqrt(2+(3+(4+(5+(6+(7+(8+(9+(10+(11+(12+(13)^(1/13))^(1/12))^(1/11))^1/10)^(1/9))^(1/8))^(1/7))^(1/6))^(1/5))^(1/4))^(1/3)):>
evalf(b);
2. Tnh ton vi chnh xc theo yu cuLnh evalf- C php 1:
evalf(bieu_thuc) - tnh ton chnh xc gi tr ca biu thc v biu din kt
quvi mc nh l 10 ch s.- C php 2: evalf(bieu_thuc, k) - tnh ton chnh
xc gi tr ca biu thc v biu din kt quvi k ch s. > 22/7: >
evalf(%); > evalf(Pi,500);3. Cc thao tc vi s nguyn t - Phn tch
mt s n thnh tha s nguyn t:
ifactor(n); - Kim tra mt s n c phi l s nguyn t khng?:
isprime(n); - Tm s nguyn t ng sau mt s n cho trc:
nextprime(n); - Tm s nguyn t ng trc mt s n cho trc:
prevprime(n); - Tm c s chung ln nht ca 2 s nguyn dng a, b:
gcd(a,b); - Tm bi s chung nh nht ca 2 s nguyn dng a, b: lcm(a,b); -
Tm s d khi chia a cho b: lnh
irem(a,b); - Tm thng nguyn khi chia a cho b: lnh
iquo(a,b); 4. Gii phng trnh nghim nguyn Lnh isolve: - C php 1:
isolve(phuong_trinh/he_phuong_trinh); - C php 2:
isolve(phuong_trinh / he_phuong_trinh, );
> isolve({x+y=36,2*x+4*y=100}):
> isolve(x+y=5,{a,b,c}):5. Gii cng thc truy hi, gii dy s Lnh
rsolve: - C php: rsolve(pt/he_pt_truy_hoi, ten_day_so);
> rsolve({f(n)=f(n-1)+f(n-2),f(0)=1,f(1)=1},f(n)):
> rsolve({f(n)=2*f(n-1)},f(n)):
> rsolve({g(n)=3*g(n/2)+5*n},g):
> rsolve(f(n)-f(n-1)=n^3,f):
> simplify(%):
> eqn:=f(n)=f(n-1)+4*n:
> rsolve(eqn,f):
> simplify(%):6. Khi nim bin s, hng s - Trong Maple, bin s c
s dng thoi mi m khng cn khai bo, nh ngha trc - Bin s, hng s c t tn
tha mn mt s quy tc sau:
+ Khng bt u bng ch s
+ Khng cha khong trng v mt s k t c bit nh:
%,^,&,*,$,#,...
+ Khng c trng vi tn mt s hm v lnh ca Maple: sin, cos, ln, min,
max, - Mt bin s s tr thnh hng s ngay khi n c gn cho mt gi tr no . -
Nu mun bin mt hng s tr li bin s, ta dng php gn:
ten_bien:='ten_bien';
> isolve({x+y=36,2*x+4*y=100}):
> x:=2:
> isolve({x+y=36,2*x+4*y=100}):
> x:='x':
> isolve({x+y=36,2*x+4*y=100}):7. Tnh tng v tch Tnh tng: s
dng lnh sum (tnh trc tip ra kt qu) hoc Sum(biu din dng cng thc)C
php: sum(bieu_thuc_trong_tong, bien :=gia_tri_dau ..
gia_tri_cuoi);
Sum(bieu_thuc_trong_tong, bien :=gia_tri_dau ..
gia_tri_cuoi);Tnh tch: s dng lnh product (tnh trc tip ra kt qu) hoc
Product (biu din dng cng thc)C php: product(bieu_thuc_trong_tong,
bien :=gia_tri_dau .. gia_tri_cuoi);
Product(bieu_thuc_trong_tong, bien :=gia_tri_dau ..
gia_tri_cuoi);
Lu : gi tr v cc c biu din bng t kha infinity >
Sum(x^2,x=1..5): > value(%): > sum(x^2,x=1..5): >
Sum(1/(x^2),x=1..infinity): > value(%): >
Product((i^2+3*i-11)/(i+3),i=0..10): > value(%): >
product((i^2+3*i-11)/(i+3),i=0..10):V d: Tnh tng hu hn. > F =
Sum((1+n)/(1+n^4),n=1..10); > F = sum((1+n)/(1+n^4),n=1..10);
> F = evalf(sum((1+n)/(1+n^4),n=1..10));
V d: Tnh tng v hn: > F = Sum(1/k^2,k=1..infinity); >F =
sum(1/k^2,k=1..infinity);
V d: Tch hu hn. > F = Product((n^2+3*n-11)/(n+3),n=0..10); F
= product((n^2+3*n-11)/(n+3),n=0..10);
V d: Tch v hn. > F = Product(1-1/n^2,n=2..infinity); F =
product(1-1/n^2,n=2..infinity);
8. Tm s nh nht, s ln nht trong mt dy s ta dng lnh min(); v
max(); > max(3/2,1.49,Pi/2); > min(3/2,1.49,Pi/2);
9. Tnh ton vi s phc V d:
> (3+5*I)/(7+4*I);
Ta c th chuyn s phc trn v dng to cc
> convert((3+5*I)/(7+4*I),polar);
III. MAPLE VI CC TNH TON TRONG I S1. Khai trin biu thc i s (bng
lnh expand). V d: Khai trin biu thc (x+y)^3,(x+y)^9 ta a vo biu thc
sau
> expand((x+y)^3);
> expand((x+y)^9);
2. Phn tch a thc thnh nhn t (bng lnh factor). V d: Phn tch a thc
thnh nhn t
> factor((b-c)^3 + (c-a)^3 + (a-b)^3);
> factor(x^8+x^4+1);
3. Tm bc ca a thc (bng lnh degree); V d: Tm bc ca a thc:
> degree(x^12-x^10+x^15+1);
4. Vit a thc di dng bnh phng ca tng ( bng lnh
completesquare()).Trc tin ta khai bo th vin student V d: Vit da thc
di dng bnh phng ca tng
> with(student):
completesquare(9*x^2 + 24*x +16);
5. Sp xp a thc theo bc ( bng lnh collect()). V d: Sp xp a thc
theo bc ca x v bc ca a:
> collect(a^3*x-x+a^3+a,x);
> collect(a^3*x-x+a^3+a,a);
6. n gin (rt gn) biu thc (bng lnh simplify). V d: n gin biu
thc
>
simplify(1/(a*(a-b)*(a-c))+1/(b*(b-a)*(b-c))+1/(c*(c-a)*(c-b)));
7. Ti gin phn thc (bng lnh normal). V d:
> normal((x^8+3*x^4+4)/(x^4+x^2+2));
8. Kh cn thc mu s ( bng lnh readlib). Mun kh cn thc mu s trc tin
ta khai bo th vin readlib(rationalize): V d
> readlib(rationalize):
1/(sqrt(5)-sqrt(2))+1/(sqrt(5)+sqrt(2));
rationalize(1/(sqrt(5)-sqrt(2))+1/(sqrt(5)+sqrt(2)));
9. Tm thng v phn d khi chia a thcC php: irem(m,n) hay
irem(m,n,'p'); trong p l thng nguynS cho ta kt qu l d ca php chia m
cho n, nu mun bit thng ta tip tc dng lnh xut p nh sau (khi dng lnh
th 2):
[>p;
iquo(m,n) hay iquo(m,n,'p'); trong p l d nguynS cho ta kt qu l
thng ca php chia m cho n, nu mun bit d ca php chia tap tip tc xut p
nh sau (khi dng lnh th 2):
[>p;
V d:
> Thuong = rem(x^3+x+1,x^2+x+1,x);
> Du = quo(x^3+x+1,x^2+x+1,x);
10. Thay gi tr cho bin trong biu thc C php: subs(bien = gia_tri
, bieu_thuc); > bt := x^2-1; > subs(x=2,bt): > bt :=
x^2-1; bt := x2K1 > subs(x=2,bt);11. nh ngha hm sCch 1: s dng
ton t -> C php: ten_ham := bien -> bieu_thuc_ham_so;
> f := x->x^2+1/2:
> f(a+b):Cch 2: s dng lnh unapply C php: ten_ham :=
unapply(bieu_thuc, bien);
> g:=unapply(x^3+2,x):
> g(4):nh ngha hm tng khc C php: ten_ham := bien ->
piecewise(k_1, bt_1, k_2, bt_2, ..., k_n, bt_n); ngha: nu k_i ng th
hm nhn gi tr l bt_i
> f:=x->piecewise(x PT:=x^3-a*x^2/2+13*x^2/3 =
13*a*x/6+10*x/3-5*a/3;
Sau ta gii phng trnh bng lnh solve();
> solve(PT,{x});
* Gii h phng trnh. Trc tin ta nh ngha cc phng trnh:
> Pt1:=x+y+z-3=0:
> Pt2:=2*x-3*y+z=2:
> Pt3:=x-y+5*z=5; Sau ta dng lnh gii phng trnh solve.
> solve({Pt1,Pt2,Pt3},{x,y,z});
13. Gii bt phng trnh v h bt phng trnh.* Gii bt phng trnh. V
d:
> Bpt:=sqrt(7*x+1)-sqrt(3*x-18) solve(Bpt,{x});
Hoc ta c th a trc tip bt phng trnh vo trong cu lnh.
> solve(sqrt(7*x+1)-sqrt(3*x-18) Bpt1:=x^3-11*x^2+10*x0;
Sau dng lnh gii h ny:
> solve({Bpt1,Bpt2},x);
Hoc ta c th a trc tip bt phng trnh vo trong cu lnh nh sau:
> solve({x^3-11*x^2+10*x0},x);
IV.CC TNH TON TRONG I S TUYN TNH Trc tin ta hy khi ng chng trnh
bng lnh restart: v np gi cng c chuyn ngnh nilalg:1. To ma trn C hai
cch to ma trn: bng lnh matrix hoc bng lnh array (to mng).
V d:
> matrix([[5,4],[6,3]]);
V d 2
> B:=array([[4,1,3],[2, 2,5]]);
2. So snh hai ma trn bng lnh equal Mun so snh hai ma trn xem
chng c bng nhau hay khng ( tc l tt c cc phn t cng v tr tng ng ca
chng phi bng nhau), ta dng lnh equal. Ch : Hai ma trn phi cng s
chiu nh nhau mi c th so snh c.
Th d:
> restart:
with(linalg):
Warning, the protected names norm and trace have been redefined
and unprotected
> A := array( [[2,1],[1,2]] );
> B := array( [[2,1],[1,2]] );
> equal(A, B);
> C := matrix(2,2, [2,2,1,2]);
> equal(A, C);
So snh A vi F
> F := array( [[2,1],[2,1]] );
> equal(A, F);
3. Tnh tng ca hai ma trn bng lnh evalm hoc bng lnh add V d:
> A:=array([[1,-3,2],[3,-4,1]]);
> B:=matrix(2,3,[2,5,6,1,2,5]); Tnh tng ca A v B bng lnh
evalm
> evalm(A+B);
4. Nhn ma trn bng lnh multiply hoc bng lnh evalm V d:
> A:=array([[2,-1,3,4],[3,-2,4,-3],[5,-3,-2,1]]);
> B:=matrix(4,3,[7,8,6,5,7,4,3,4,5,2,1,1]); Nhn A vi B bng
lnh multiply
> multiply(A,B);
5. Tnh tch trong ca ma trn v vc t bng lnh innerprod Hm innerprod
tnh tch trong ca mt dy cc ma trn v vc t. Chiu ca ma trn v vc t phi
tng thch vi nhau trong php nhn. V d:
> restart:
with(linalg):
Warning, the protected names norm and trace have been redefined
and unprotected
> u := vector(2, [1,2]);
> A := matrix(2,3, [1,1,1,2,2,2]);
> innerprod(u, A);
> w := vector(2, [3,2]);
> innerprod(u,w);
6. Tnh tch vc t (tch trc tip) bng lnh crossprod Tch vc t ca hai
vc t l mt vc t c to l:
( u[2]*v[3]-u[3]*v[2],
u[3]*v[1]-u[1]*v[3],u[1]*v[2]-u[2]*v[1])
> v1 := vector([1,2,3]);
> v2 := vector([2,3,4]);
> crossprod(v1,v2);
7. Tnh tch v hng ca hai vc t bng lnh dotprod Theo nh ngha, tch v
hng ca hai vc t trn trng s phc l tng ca u[i]*lin hp ca v[i].
V d:
> u := vector( [1,x,y] );
> v := vector( [1,0,0] );
> dotprod(u, v);
8. Cc php ton cu trc trn ma trn v vc t * Xo dng, xo ct ca ma trn
bng delrows (delcols) > restart: with(linalg): > a :=
matrix(3,3, [1,2,3,4,5,6,7,8,9]); > delrows(a, 2..3); >
delcols(a, 1..1);
* To ma trn con > A := array( [[1,2,3],[4,x,6]] ); >
submatrix(A, 1..2, 2..3); > submatrix(A, [2,1], [2,1]);
9. Hon v dng (ct) ca ma trn V d:
> A := array( [[1,2,x],[3,4,y]] );
> swaprow(A, 1, 2);
> swapcol(A, 2, 3);
10. Nhn mt dng ca ma trn vi mt biu thc V d:
> A := matrix( [[1,2],[3,4]] );
> mulrow(A, 2, 2);
> mulcol(A, 2, x);
11. Tm ma trn chuyn v bng lnh transpose V d:
> P:=array([[1,2,3],[5,6,4]]);
12. Tm ma trn chuyn v bng lnh transpose > transpose(P);
13. Tm vt ca ma trn bng lnh trace V d:
> T:=array([[4,3,-3],[2,3,-2],[4,4,-3]]);
> trace(T);
14. Tm bt bin ca ma trn bng lnh permanent >
P:=array([[1,-2,-3],[2,-4,1],[3,-5,2]]); > permanent(P);
15. Tnh gi tr ring v vc t ring ca ma trn V d:
> M:=matrix(3,3,[1,-3,3,3,-5,3,6,-6,4]);
> eigenvects(M);
Kt qu ca lnh eigenvects c xp xp nh sau: s u tin trong mi mc vung
ca dng l gi tr ring, s th hai l bi i s ca gi tr ring, v cui cng l
tp cc vc t c s ca khng gian ring ng vi gi tr ring . Mi mc vung ng
vi mt gi tr ring ca ma trn.16. Tnh a thc c trng V d: Tm ma trn c
trng bng lnh charmat > C:=array([[3,1,-1],[0,2,0],[1,1,1]]);
> charmat(C,x);
Tm a thc c trng ca ma trn bng lnh charpoly V d:
> A := matrix(3,3,[1,2,3,1,2,3,1,5,6]);
> charpoly(A,x);
17. Tm hng ca ma trn Th d 1.
> A := matrix(3,3, [x,1,0,0,0,1,x*y,y,1]);
> rank(A);
18. Tnh nh thc V d:
> A:=matrix(3,3,[1/2,-1/3,2,-5,14/3,9,0,11,-5/6]); Tnh nh thc
ca ma trn bng lnh det
> det(A);
19. Lp ma trn t phng trnh v ngc li M t: Hm geneqns sinh ra mt h
cc phng trnh t h s ca ma trn. Nu c bin th ba biu th vc t v phi b th
n s c a vo phng trnh. Ngc li th v phi c coi bng 0. Hm genematrix
sinh ma trn t cc h s ca h phng trnh tuyn tnh. Nu c bin th ba"flag"
th vc t "v phi" c a vo ct cui cng ca ma trn. Th d
> eqns := {x+2*y=0,3*x-5*y=0};
> A := genmatrix(eqns, [x,y]);
> geneqns(A,[x,y]);
> geneqns(A,x);
> eqns := {x+2*z=a,3*x-5*y=6-z};
> A := genmatrix(eqns, [x,y,z], flag);
> A := genmatrix(eqns, [x,y,z], 'b');
> print(b);
> geneqns(A,[x,y,z],b);
20. Gii phng trnh i s tuyn tnhGii phng trnh i s tuyn tnh Ax=u,
trong ,
Nhp A
> A:=array([[3,-2,-5,1],[2,-3,1,5],[1,2,0,-4],[1,-1,-4,9]]);
Nhp u
> u:=vector([3,-3,-3,22]); Gii phng trnh Ax=u
> linsolve(A,u);
V. MAPLE VI PHP TNH VI PHN - TCH PHN1. Tnh gii hn tnh gii hn ca
hm s ti a ta dng lnh [>limit(f(x),x=a); V d: Tnh gii hn hn
s:
> F1 = Limit(((sin(2*x))^2-sin(x)*sin(4*x))/x^4,x=0);
> F1 = limit(((sin(2*x))^2-sin(x)*sin(4*x))/x^4,x=0);
> F2 = Limit((2*x+3)/(7*x+5),x=infinity);
> F2 = limit((2*x+3)/(7*x+5),x=infinity);
2. Tnh o hm ca hm mt bin.* Tnh o hm bc nht (bng lnh
[>diff(f(x),x);). V d: Tnh o hm cc hm s sau.
> f1(x):=(x^2*sqrt(x^2+1));
> print(`Dao ham cua f1(x) la`);
diff(f1(x),x);
> f2(x):=5*x^3-3*x^2-2*x^(-3);
> print(`Dao ham cua ham so f2(x) la`);
diff(f2(x),x);
* Tnh o hm cp cao (bng lnh [>diff(f(x),x$n);). V d: Tnh o hm
cp cao ca cc hm s sau:
> f3(x):=x^4+x*sin(x);
> print(`Dao ham cap hai cua f3(x) la`);
diff(f3(x),x$2);
> print(`Dao ham cap bon cua f3(x) la`);
diff(f3(x),x$4);
3. Php tnh tch phn* Tch phn xc nh Tnh tch phn xc nh ca hm s f(x)
trn on [a,b] (bng lnh [>int(f(x),x=a..b);). V d: Tnh cc tch phn
sau:
> f(x):=Int((x+1)/sqrt(3*x+1),x=0..7/3);
> print(`Tich phan cua f(x) tren doan [0,7/3] la`);
int((x+1)/sqrt(3*x+1),x=0..7/3);
> g(x):=Int(1/(exp(1)^x+5),x=0..ln(2));
> print(`Tich phan cua g(x) tren doan [0,ln(2)] la`);
int(1/(exp(1)^x+5),x=0..ln(2));
* Tch phn khng xc nh Tnh tch phn khng xc nh ca hm s f(x) bng lnh
[>int(f(x),x); V d: Tnh cc tch phn khng xc nh sau:
> h(x):=Int((3*x^2+3*x+3)/(x^3-3*x+2),x);
> print(`Tich phan khong xac dinh cua ham h(x) la`);
int((3*x^2+3*x+3)/(x^3-3*x+2),x);
* Tch phn suy rng > p(x):=Int(x/(x^4+1),x=0..infinity); >
print(`Tich phan khong xac dinh cua ham p(x) la`);
int(x/(x^4+1),x=0..infinity);
4. Tnh din tch hnh thang cong Tnh din tch hnh thang cong c gii
hn bi cc ng sau:
> y:=x^2;
y:=sqrt(x);
Ta v hnh minh ho nh sau:
> restart:
with(plots):
plot({x^2,sqrt(x)},x=0..1.5);
Warning, the name changecoords has been redefined
> print(`Dien tich phan bi gioi han chinh la`);
Int(sqrt(x)-x^2,x=0..1);
print(`Va dien tich do la`);
int(sqrt(x)-x^2,x=0..1);
5. Tnh o hm ca hm nhiu bin tnh o hm ca hm nhiu bin ta dng lnh
[>grad(f,[x,y,z,...]); V d: Tnh o hm ca hm nhiu bin sau:
> f:=4*x*z;
> print(`Dao ham cua f la`);
grad(f,[x,y,z]);
> g:=5*x*y-3*y*z;
> print(`Dao ham cua g la`);
grad(g,[x,y,z]);
6. Tnh vi phn trn hm n tnh vi phn trn hm n ta dng lnh
[>implicitdiff(f,x,y,z); V d: Tnh vi phn ca hm sau:
> f:=x^2/z;
> print(`Vi phan cua ham f theo x la`);
implicitdiff(f,x,z);
> print(`Vi phan cua ham f theo z la`);
implicitdiff(f,z,x);
> print(`Cho ham g nhu sau`);
g:=x^2+z^3=1;
> print(`Vi phan cua ham g theo x la`);
implicitdiff(g,z,x);
> print(`Vi phan cua ham g theo z la`);
implicitdiff(f,x,z);
7. Dy truy hi* Tm dy cc phn t ca dy FibnacciS hng th n ca dy
Fibonacci c tnh theo cng thcTnh s Fibonacci bng cch s dng Maple
> F(0):=1: F(1):=1: n:=2: while n F(0):=a:
F(1):=b:
n:=2:
while n F(0):=144:F(1):=233:n:=2:while n with(plots): >
with(plottools):2. V th trong khng gian 2 chiu OxyV th hm thng
thng:C php: plot(ham_can_ve, x=gt_dau..gt_cuoi, y=gt_dau..gt_cuoi,
cac_tuy_chon);Mt s ty chn thng dng: - t mu cho th:
color = - t dy k cho th: thickness = k - t s im v cho th:
numpoints = k;
> plot(x^3-3*x^2+1,x=-5..5,y=-5..5):
> f:=x->abs(x^3-x^2-2*x)/3-abs(x+1):
> plot(f(x),x=-5..5,y=-5..5):V nhiu th trn cng mt h trcC php:
plot([ham_1, ham_2,...], x=gt_dau..gt_cuoi, y=gt_dau..gt_cuoi,
cac_tuy_chon); > plot([x^2,sin(x)],x=-2..2,color=[red,green]):V
th ca hm s khng lin tc Khi v th ca mt hoc nhiu hm s c im gin on, ta
phi thm tuy chn discont =true th c v chnh xc hn >
g:=x->(x^2-1)/(x-2): >
plot(g(x),x=-10..10,y=-5..15,discont=true,color=blue):V th hm n C
nhng hm s m chng ta khng c c cng thc tng minh y=f(x), khi v c th ca
chng, ta s dng hm implicitplotC php: implicitplot([bt_1, bt_2,...],
x=gt_dau..gt_cuoi, y=gt_dau..gt_cuoi, cac_tuy_chon); >
implicitplot(x^2/9+y^2/4=1,x=-4..4,y=-2..2): >
implicitplot(x^2-y^2-x^4=0,x=-1..1,y=-1..1):ng dng: v th ca hm hu t
> f:=x->(x^2-1)/(x-2): > bt:=convert(f(x),parfrac): >
tcx:=x->x+2: >
g1:=plot([f(x),tcx(x)],x=-10..10,y=-5..15,color=[blue,red],discont=true):
> g2:=implicitplot(x=2,x=-10..10,y=-5..15,color=green): >
display({g1,g2}):3. V th trong khng gian 3 chiu OxyzV th hm thng
thngC php: plot3d(ham_can_ve, x=gt_dau..gt_cuoi,
y=gt_dau..gt_cuoi,z=gt_dau..gt_cuoi,cac_tuy_chon); >
plot3d(x*exp(x^2),x=-2..2,y=-2..2,title="Do thi trong khong gian 3
chieu"): >
plot3d(-exp(-abs(x*y)/10)*sin(x+y)-cos(x*y),x=-Pi..Pi,y=-Pi..Pi,grid=[51,51]):V
th hm nC php: implicitplot3d(ham_can_ve, x=gt_dau..gt_cuoi,
y=gt_dau..gt_cuoi,z=gt_dau..gt_cuoi, cac_tuy_chon); >
implicitplot3d(x^2+y^2/4+z^2/9=1,x=-3..3,y=-3..3,z=-3..3):4. S vn
ng ca thC php: animate(ham_co_tham_so,x=gt_dau..gt_cuoi, tham_so =
gt_dau..gt_cuoi);
animate3d(ham_co_tham_so,x=gt_dau..gt_cuoi, y=gt_dau..gt_cuoi,
tham_so =gt_dau..gt_cuoi); ngha: hin th s bin i, vn ng ca th khi
tham s thay i trong khong cho trc >
animate3d(cos(t*x)*sin(t*y),x=-Pi..Pi,y=-Pi..Pi,t=1..5): >
animate(t*x^2,x=-3..3,t=-5..5):VII. HNH HC GII TCH1. Cc tnh ton
trong hnh hc phng: gi geometryKhi to cc hm tnh ton trong hnh hc
phng > with(geometry):Cc hm trn i tng im - nh ngha im:
point(ten_diem, hoanh_do, tung_do); - Hin th ta ca mt im:
coordinates(ten_diem); - Xc nh trung im on thng to bi hai im:
midpoint(ten_trung_diem, diem_1,diem_2); > point(A,2,3): >
point(B,-3,1): > coordinates(A): > coordinates(B): >
midpoint(M,A,B): > coordinates(M):Cc hm trn i tng ng thng - nh
ngha ng thng qua hai im: line(ten_dt, [diem_dau, diem_cuoi],[x,y]);
- nh ngha ng thng c phng trnh cho trc:
line(ten_dt,pt_duong_thang,[x,y]); -Tm giao im gia hai ng thng:
intersection(ten_giao_diem, dt_1, dt_2); -Tm gc gia hai ng thng:
FindAngle(dt_1, dt_2); - Tnh khong cch t mt im ti mt ng thng:
distance(diem, duong_thang); - Xc nh hnh chiu ca mt im ln trn mt ng
thng: projection(ten_hinh_chieu, diem, duong_thang); - Xc nh im i
xng ca mt im qua mt ng thng: reflection(ten_diem_dx, diem,
duong_thang);-Kim tra 3 im thng hng: AreCollinear( P,Q,R,cond), i s
cond c th c hoc khng, tr v gi tr true nu 3 im thng hng (hoc tr v gi
tr l cond nu s dng i s ny) > line(d1,[A,B],[x,y]): >
line(d2,y=x+1,[x,y]): > detail(d1): > detail(d2): >
intersection(K,d1,d2): > coordinates(K): > FindAngle(d1,d2):
> distance(A,d1): > distance(B,d2): > projection(N,B,d2):
> coordinates(N): > reflection(B1,B,d2): >
coordinates(B1):Cc hm trn i tng ng trn- nh ngha ng trn qua 3 im:
circle((ten_duong_tron,[diem1, diem2, diem3],[x,y]);- Xc nh ng trn
ni tip tam gic: incircle(ten_duong_tron_noi_tiep,tentamgiac);- nh
ngha ng trn c tm v bn knh cho trc: circle(ten_duong_tron,[tam,
bk],[x,y]);- Xc nh bn knh ng trn a nh ngha: radius(tenduongtron);-
Xc nh ta tm ng trn a nh ngha: coordinates(center(tenduongtron));-
Xc nh din tch ng trn a nh ngha: area(tenduongtron);- Tm tip tuyn vi
ng trn ti mt im: tangentpc(tentieptuyen,diem,tenduongtron);- Tm tip
tuyn vi ng trn qua mt im:
tangentline(diem,tenduongtron,[tentieptuyen1, tentieptuyen2]); >
point(C,0,0): > circle(c,[A,B,C],[x,y]): > detail(c): >
radius(c): > coordinates(center(c)): > area(c): >
circle(c1,[C,5],[x,y]): > detail(c1): > tangentpc(t1,C,c):
> detail(t1): > Equation(t1): >
TangentLine(t2,point(D,4,5),c,[l1,l2]):Cc hm trn i tng tam gic- nh
ngha tam gic: triangle(ten_tam_giac,[dinh1,dinh2,dinh3],[x,y]);- Xc
nh din tch tam gic: area(ten_tam_giac)- Xc nh ng cao tam gic ng vi
mt nh: altitude(ten_duong_cao,dinh,ten_tam_giac);- Xc nh ng trung
tuyn tam gic ng vi mt nh:
median(tenduongtrungtuyen,dinh,tentamgiac);- Xc nh ng phn gic tam
gic ng vi mt nh: bisector(ten_duong_phan_giac, dinh,
ten_tam_giac);- Xc nh ng phn gic tam gic ng vi mt nh:
ExternalBisector(ten_duong_phan_giac, dinh, tentamgiac);-tm ng thng
i qua mt im cho trc v vung gc vi mt ng thng cho trc:
PerpendicularLine( tn ng vung gc,im,ng thng cha im).- Xc nh trng tm
tam gic: centroid(ten_trong_tam,ten_tam_giac);- Xc nh trc tm tam
gic: orthorcenter(ten_truc_tam, tentamgiac);
> triangle(ABC,[A,B,C],[x,y]): > detail(ABC): >
area(ABC): > altitude(ha,A,ABC): > median(BM,B,ABC): >
detail(BM): > bisector(Ct,C,ABC): > detail(Ct): >
ExternalBisector(Cx,C,ABC): > centroid(G,ABC): >
coordinates(G): > orthocenter(H,ABC): > coordinates(H): >
incircle(cc,ABC): > detail(cc):2. Cc tnh ton trong hnh hc khng
gian: gi geom3dKhi to > with(geom3d):Cc hm trn i tng im- nh ngha
im:
point(ten_diem, hoanh_do, tung_do,cao_do);- Hin th ta ca mt im:
coordinates(ten_diem);- Xc nh trung im on thng to bi hai im:
midpoint(ten_trung_diem, diem_1,diem_2); > point(A,2,3,1): >
point(B,-3,1,3): > coordinates(A): > coordinates(B): >
midpoint(M,A,B): > coordinates(M):Cc hm trn i tng ng thng- nh
ngha ng thng qua hai im: line(ten_dt, [diem_dau, diem_cuoi]);- nh
ngha ng thng c phng trnh tham so cho trc:
line(ten_dt,pt_tham_so_duong_thang, ten_tham_so);-Tm giao im gia
hai ng thng: intersection(ten_giao_diem, dt_1, dt_2);-Tm gc gia hai
ng thng: FindAngle(dt_1, dt_2);- Tnh khong cch t mt im ti mt ng
thng: distance(diem, duong_thang);- Xc nh hnh chiu ca mt im ln trn
mt ng thng: projection(ten_hinh_chieu, diem, duong_thang);- Xc nh
im i xng ca mt im qua mt ng thng: reflection(ten_diem_dx, diem,
duong_thang); > line(d1,[A,B]): >
line(d2,[2+2*t,1-4*t,3*t],t): > detail(d1): Warning, assume that
the parameter in the parametric equations is _t Warning, assuming
that the names of the axes are _x, _y, and _z > detail(d2):
Warning, assuming that the names of the axes are _x, _y, and _z
> intersection(K,d1,d2): intersection: "the given objects do not
intersect" > FindAngle(d1,d2): > distance(A,d1): >
distance(B,d2): > projection(N,B,d2): > coordinates(N): >
reflection(B1,B,d2): > coordinates(B1):Cc hm trn i tng mt phng-
nh ngha mt phng qua 3 im: plane(ten_mat_phang,[diem1, diem2,
diem3],[x,y,z]);- nh ngha mt phng bng phng trnh tng qut:
plane(ten_mat_phang,pt_tongquat,[x,y,z]);- Xc nh giao tuyn ca hai
mt phng: line(ten_giao_tuyen,[mp1,mp2]);- Xc nh khong cch gia mt im
v mt mt phng: distance(ten_diem,ten_mat_phang);- Xc nh gc gia hai
mt phng: FindAngle(ten_mp_1, ten_mp_2); > point(C,0,0,0): >
plane(p,[A,B,C],[x,y,z]): > detail(p): >
plane(p1,2*x-3*y+z=0, [x,y,z]): > line(gt,[p,p1]): >
detail(gt): Warning, assume that the parameter in the parametric
equations is _t > distance(A,p1): > FindAngle(p,p1):
1)TNH TON VI S NGUYN:
-Hm sqrt(a); Cho kt qu cn bc hai ca a.
-Tnh ton s thp phn vi chnh xc ty :C php: evalf(P,m);
trong chnh xc ca s P l m (nu khng c m th mc nh s ly 10 s)Lu : Nu
dng lnh evalf(P), ta c th nh trc s ch s thp phn cn ly (khng phi l
10 theo mc nh) bng cch dng lnh >Digits:= m, m l s cc ch s thp
phn cn nh trc, nh vy ta t lnh ny trc lnh evalf(P); nu ch dng lnh
>Digits; ng sau evalf(P) s cho ta bit chnh xc bao nhiu ch s thp
phn.-Phn tch mt s ra tch cc tha s nguyn t :
C php : ifactor(S);
-Tm USCLN, BSCNN:
C php: gcd(ccs);
lcm(ccs);
-Kim tra 1 s c phi l s nguyn t
C php : isprime(S);-Tm s nguyn t trc s cho v sau s choC php :
prevprime(S);
nextprime(S); -Tm phn d nguyn v thng nguynC php: irem(m,n) hay
irem(m,n,'p'); trong p l thng nguynS cho ta kt qu l d ca php chia m
cho n, nu mun bit thng ta tip tc dng lnh xut p nh sau (khi dng lnh
th 2):
[>p;
iquo(m,n) hay iquo(m,n,'p'); trong p l d nguynS cho ta kt qu l
thng ca php chia m cho n, nu mun bit d ca php chia tap tip tc xut p
nh sau (khi dng lnh th 2):
[>p;
-Tm nghim nguyn ca phng trnh:
C php: islove(eqns,vars);
Eqns: tp cc ptrnh cn gii
Vars: tp cc bin t do. Nu khng cung cp th Maple t ng to ra cc bin
t do2) Tng v tch hu hn- v hn:
C php: sum(f(i),i=m..n);
Sum(f(i),i=m..n);value(%);
sum(f(i),i=m..infinity);
Sum(f(i),i=m..infinity);value(%);3) Tch v hn:
C php: product(f(i),i=m..n);
Product(f(i),i=m..n);value(%);
product(f(i),i=m..infinity);
Product(f(i),i=m..infinity);value(%)4) Tnh ton n gin vi biu
thc:
-Khai trin biu thc: C php: expand(expr); expr: biu thc cn khai
trin-Phn tch thnh nhn t:C php : factor(expr);expr: biu thc cn phn
tch
-n gin biu thc:C php : simplify (biu thc);-Ti gin phn thc:C php
: normal(fraction);-Kh muC php : rationalize(biu thc);-Tnh gi tr ca
biu thc:C php : subs(var1=val1,,varn=valn,expr);VD:
expr:=x^2+y^2-2*z^2*x;
subs(x=1,y=1,z=1,expr);-Tm bc ca a thc:
lnh: degree(biu thc);-Chuyn i dng ca biu thc:
C php: convert(chuyn i);5)Lnh iu kin IF:
C php: if iu_kin1 then vic1 elif iu_kin2 then vic2 else vic3
fi;Nu iu_kin1 ng s thc hin lnh sau then (vic1) nu khng th kim tra
iu_kin2, nu iu_kin2 ng th thc hin vic2, c tip tc n khi cc iu_kin 1
v 2 khng tho th thc hin vic3
Nu khng ta c th dng lnh: if-then-else-fi;6)Vng lp FOR:C php: for
name from start by change to finishdo
Cngvic
od;
name: ch s; ch s name bt u t gi tr start, nu gi tr start nh hn
gi tr finish th thc hin cng vic sau do sau ch s name c cng thm
change n v (lc name:=start+change) v tip tc so snh vi gi tr finish
xem c tip tc na hay khng; c nh th n khi name = finish + change, y l
gi tr m name s thc hin cui cng v kt thc vng lp.
Nu: from start hoc by change khng s dng th mc nh ta s c from 1
hoc by 1Cch khc:
for name in expressiondo
Cng vic
od;
Tng t nh trn, ch khc l bin name s ly ln lt cc gi tr ca biu thc
expression7)Vng lp WHILE:C php:
while
Condition
do
Cng vic
od;
Vng lp while kim tra iu kin Condition v s thc hin cng vic sau do
khi Condition ang cn ng, kt thc vng lp khi iu kin Condition
sai8)Lnh break:
Thot ngay khi vng lp while hoc for nu c s dng; lnh break c t
trong cc vng lp while, for (nu t ngoi vng lp s bo li); v trc break
thng l biu thc iu kin if ... then9)Lnh next:Tng t nh cch s dng ca
break, next dng nhy qua ln lp tip theo ca vng lp.Mt s lnh thng
dng:-m mod n: d ca m chia cho n-modp(m, n): d ca m chia cho n (ly
biu din dng)-mods(m, n): d ca m chia cho n (i xng ca modp(m, n)
trong mod n tc modp(m, n) n = mods(m, n))10)Thay th trong biu thc,
lnh: subsC php:subs(var=rep1,expr)
sups(var=rep1,var2=rep2,expr)
C th m rng hn na cc biu thc cn thay th.
var, var1,.... cc biu thc cn c thay th bi rep1, rep2,.... trong
biu thc expr.11)Tnh gii hn:
Hm: Limit(f(x),x=a);
Gii hn ti v cc: limit(f(x),x=infinity);
Gii hn tri ca ti a: limit(f(x),x=a,left);
Gii hn phi ti a: limit(f(x),x=a,right);12)o hm:diff(f(x),x);
Diff(f(x),x); value(%);
nu hm cn cng knh : simplify(%);13)Ly kt qu ca php tnh pha trn:
value(%);14)Cc php ton v tp hp:-Khai bo tp hp:
tn_tp_hp:={cc_phn_t};
-Cc php ton: union (hp ca hai tp hp); intersect (giao);
minus(hiu); tp hp rng k hiu l {};
C php: tm giao ca A v B ta vit: A intersect B;
VD: >
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