radionica_uvod -- Sage

Sage kao kalkulator

77^2

77**2

5*6+7/12

18%7

18//7

2==3

2<3

2!=5

simboličke varijable

a+b

var('a b')

a+b

automatic_names(True)

c+2/c

x_1+w

a=6; b=7.3; c=5/12

a+b*c

Oslobađanje varijable

reset('a')

i^2

I^2

i=5

i^2

reset('i')

numerical_approx(e)

n(e)

n=10

n(e)

reset('n')

n(e,digits=90)

help(n)

pi.n(digits=50)

a,b,c

reset('a b c')

a,b,c

izraz=a+b^(1/2)+b*c

izraz.subs(a=1,b=3,c=-2)

izraz.subs({a:1,b:3,c:-2})

a,b,c

izraz.subs(b=a^2)

izraz.subs(b=a^2).simplify()

assume(a>0)

izraz.subs(b=a^2).simplify()

forget()

izraz.subs(b=a^2).simplify()

forget?

True and False

True or False

1/Infinity

Infinity+Infinity

Definiranje funkcije

f(x)=x*sin(x)+(x-1)/(x^2+3)

f(2)

f(2.0)

numerical_approx(f(2),digits=50)

g(n)=expand((x+2)^n)

g(3)

g(n)

def g1(n): return expand((x+2)^n)

g1(3)

g1(n)

def par_nepar(n): if not (n in NN): return 'nije prirodni broj' if n%2==0: return 'paran' else: return 'neparan'

par_nepar(-2)

Skupovi brojeva

NN,ZZ,QQ,RR,CC,SR

GF(3)

2 in NN, -2/3 in ZZ, 5/4 in QQ, 2+I in RR, t in SR

2.0 in GF(3),4 in GF(3)

polje=GF(3)

polje(4)

polje(1)+polje(2)

sqrt(3) in QQ

sqrt(3) in QQbar

polje.list()

polje2=GF(8,a)

polje2.list()

solve(x^2-3*x+5==0,x)

solve(x^2-3*x+5>0,x)

solve(x^2-3*x+5<0,x)

solve(x^2-3*x-5>0,x)

solve([x+y==5,3*x-2/5*y==-8],x,y)

numeričko rješavanje jednadžbi

solve(x^2-3*x-5==0,x)

rj=solve(x^2-3*x-5==0,x,solution_dict=True);rj

map(lambda p:p[x], rj)

map(lambda p:p[x].n(digits=20),rj)

solve(sin(x)==x*cos(x),x)

slika1=plot(sin(x),x,-2*pi,2*pi,color='red') slika2=plot(x*cos(x),x,-2*pi,2*pi,color=(0,0,1)) (slika1+slika2).show(figsize=[5,4])

find_root(sin(x)==x*cos(x),4,5)

deriviranje i integriranje

diff(x*sin(x)+e^(x^3+sin(x)),x)

diff(x*sin(x),x,2)

f(x,y)=x*sin(x^2*y)+log(x,3)

diff(f(x,y),x),diff(f(x,y),y)

diff(f(x,y),x,x)

diff(f(x,y),x,y)

diff(f(x,y),y,y)

f(x,y).hessian()

f(x)=x*sin(x)

plot(f(x),x,-5,5,figsize=[5,3],ymax=3)

f.find_local_maximum(0,4)

f.find_local_minimum(0,4)

f.find_local_maximum(1,1.5)

g(x,y)=x^3*y+x*y

plot3d(g(x,y),(x,-3,3),(y,-3,3))

integrate(x*sin(x)^2,x)

integral(x*sin(x)^2,(x,0,1))

numerical_integral(x*sin(x)^2,0,1)

numerical_integral?

integral(sin(x),(x,0,2*pi))

integrate(x/log(x),x)

Ei?

limit((1-cos(x)^t)/x^2,x=0)

limit(abs(x)/x,x=0,dir='minus')

limit(abs(x)/x,x=0,dir='plus')

limit(abs(x)/x,x=0)

taylor(x*sin(x)^2,x,0,10)

Sume

sum(1/n^2,n,1,oo)

sum(1/k,k,1,n)

sum(1/k,k,1,oo)

sum(k,k,1,n)

Latex

$\LaTeX{}$ radi unutar SAGE worksheeta. Definiramo funkciju $f(x)=\dfrac{\log_2{2x}}{\sin{x}+4^x}$ .

latex(sin(x)/x)

Sudoku

g = Sudoku('1....7.9..3..2...8..96..5....53..9...1..8...26....4...3......1..4......7..7...3..')

g.to_latex()

g.to_matrix()

g.to_ascii()

g.solve().next()

Rubikova kocka

rubik=RubiksCube().scramble()

rubik.show()

rubik.show3d()

rubik.solve()

rubik.plot3d().show(frame=False)

Pravilni poliedri

icosahedron(color='cyan',opacity=0.6,frame=False)

icosahedron(color='cyan',opacity=0.6,frame=False)+dodecahedron(color='red',center=(3,0,0),opacity=0.6,frame=False)

Teorija grafova

graphs.CubeGraph(3).show(layout='circular',vertex_labels=False)

graphs.CubeGraph(4).show(layout='circular',vertex_labels=False)

graphs.CubeGraph(4).show3d()

G=graphs.DesarguesGraph() G.show()

G.is_eulerian()

G.chromatic_number()

G.show(vertex_colors=G.coloring(hex_colors=True))

G.is_bipartite()

G.bipartite_sets()

drugi programi iz SAGE-a

%python 3/2

3/2

%maxima solve(5*x+3=0,x)

solve(5*x+3=0,x)

solve(5*x+3==0,x)