mijolovic_belec.sagews

Zadatak 1

Ispišite u dekadskom zapisu broj $2^{255}$.

print 2^255

57896044618658097711785492504343953926634992332820282019728792003956564819968

Zadatak 2

Fibonaccijev niz zadan je rekurzivnom formulom $$F_{n+2}=F_{n+1}+F_{n},$$ uz $F_1=1$ i $F_2=2$. Ispišite $F_4$, $F_9$ i $F_{57}$.

F = [0,1,2]
for i in range(60):
    F+= [F[-1] + F[-2]]
print F[4], F[9], F[57]
    

5 55 591286729879

Zadatak 3

Neka su početne vrijednosti varijabli x i y redom 1.0 i 1.0.

Implementirajte python while petlju koja u svakoj iteraciji broj y dijeli s 7.0, dok god je x+y!=x, te u svakoj iteraciji ispisuje trenutni broj y.

Zašto ta petlja staje?

x = 1.0
y = 1.0

while x+y!=x:
    y = y/7.0
    print(y)

0.142857142857143 0.0204081632653061 0.00291545189504373 0.000416493127863390 0.0000594990182661986 8.49985975231409e-6 1.21426567890201e-6 1.73466525557430e-7 2.47809322224900e-8 3.54013317464144e-9 5.05733310663062e-10 7.22476158090089e-11 1.03210879727156e-11 1.47444113895937e-12 2.10634448422766e-13 3.00906354889666e-14 4.29866221270952e-15 6.14094601815646e-16 8.77278002593779e-17

Zadatak 4

Neka su $a$, $b$, $c$, $d$ i $e$ realni parametri. Pomoću naredbe solve nađite rješenja jednadžbi:

• $ax^4=0$,
• $ax^4+bx^3=0$,
• $ax^4+bx^3+cx^2=0$,
• $ax^4+bx^3+cx^2+dx=0$,
• $ax^4+bx^3+cx^2+dx+e=0$,

u ovisnosti o parametrima $a$, $b$, $c$, $d$ i $e$.

x = var('x')
a = var('a')
b = var('b')
c = var('c')
d = var('d')
e = var('e')

solve(a*x^4 == 0, x)
solve(a*x^4 + b*x^3 == 0, x)
solve(a*x^4 + b*x^3 + c*x^2 == 0, x)
solve(a*x^4 + b*x^3 + c*x^2 + d*x == 0, x)
solve(a*x^4 + b*x^3 + c*x^2 + d*x + e == 0, x)

[x == 0] [x == -b/a, x == 0] [x == -1/2*(b + sqrt(b^2 - 4*a*c))/a, x == -1/2*(b - sqrt(b^2 - 4*a*c))/a, x == 0] [x == -1/18*(-I*sqrt(3) + 1)*(b^2/a^2 - 3*c/a)/(-1/27*b^3/a^3 + 1/6*b*c/a^2 - 1/2*d/a + 1/6*sqrt(-1/3*b^2*c^2 + 4/3*a*c^3 + 9*a^2*d^2 + 2/3*(2*b^3 - 9*a*b*c)*d)/a^2)^(1/3) - 1/2*(I*sqrt(3) + 1)*(-1/27*b^3/a^3 + 1/6*b*c/a^2 - 1/2*d/a + 1/6*sqrt(-1/3*b^2*c^2 + 4/3*a*c^3 + 9*a^2*d^2 + 2/3*(2*b^3 - 9*a*b*c)*d)/a^2)^(1/3) - 1/3*b/a, x == -1/18*(I*sqrt(3) + 1)*(b^2/a^2 - 3*c/a)/(-1/27*b^3/a^3 + 1/6*b*c/a^2 - 1/2*d/a + 1/6*sqrt(-1/3*b^2*c^2 + 4/3*a*c^3 + 9*a^2*d^2 + 2/3*(2*b^3 - 9*a*b*c)*d)/a^2)^(1/3) - 1/2*(-I*sqrt(3) + 1)*(-1/27*b^3/a^3 + 1/6*b*c/a^2 - 1/2*d/a + 1/6*sqrt(-1/3*b^2*c^2 + 4/3*a*c^3 + 9*a^2*d^2 + 2/3*(2*b^3 - 9*a*b*c)*d)/a^2)^(1/3) - 1/3*b/a, x == 1/9*(b^2/a^2 - 3*c/a)/(-1/27*b^3/a^3 + 1/6*b*c/a^2 - 1/2*d/a + 1/6*sqrt(-1/3*b^2*c^2 + 4/3*a*c^3 + 9*a^2*d^2 + 2/3*(2*b^3 - 9*a*b*c)*d)/a^2)^(1/3) + (-1/27*b^3/a^3 + 1/6*b*c/a^2 - 1/2*d/a + 1/6*sqrt(-1/3*b^2*c^2 + 4/3*a*c^3 + 9*a^2*d^2 + 2/3*(2*b^3 - 9*a*b*c)*d)/a^2)^(1/3) - 1/3*b/a, x == 0] [x == -1/2*sqrt(3/2*a*(b^3/a^3 - 4*b*c/a^2 + 8*d/a)/sqrt((36*a^2*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(2/3) + 4*c^2 - 12*b*d + 48*a*e + 3*(3*b^2 - 8*a*c)*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3))/(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3)) - 1/9*(c^2/a^2 - 3*(b*d - 4*a*e)/a^2)/(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3) - (1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3) + 1/2*b^2/a^2 - 4/3*c/a) - 1/4*b/a - 1/12*sqrt((36*a^2*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(2/3) + 4*c^2 - 12*b*d + 48*a*e + 3*(3*b^2 - 8*a*c)*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3))/(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3))/a, x == 1/2*sqrt(3/2*a*(b^3/a^3 - 4*b*c/a^2 + 8*d/a)/sqrt((36*a^2*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(2/3) + 4*c^2 - 12*b*d + 48*a*e + 3*(3*b^2 - 8*a*c)*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3))/(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3)) - 1/9*(c^2/a^2 - 3*(b*d - 4*a*e)/a^2)/(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3) - (1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3) + 1/2*b^2/a^2 - 4/3*c/a) - 1/4*b/a - 1/12*sqrt((36*a^2*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(2/3) + 4*c^2 - 12*b*d + 48*a*e + 3*(3*b^2 - 8*a*c)*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3))/(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3))/a, x == -1/2*sqrt(-3/2*a*(b^3/a^3 - 4*b*c/a^2 + 8*d/a)/sqrt((36*a^2*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(2/3) + 4*c^2 - 12*b*d + 48*a*e + 3*(3*b^2 - 8*a*c)*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3))/(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3)) - 1/9*(c^2/a^2 - 3*(b*d - 4*a*e)/a^2)/(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3) - (1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3) + 1/2*b^2/a^2 - 4/3*c/a) - 1/4*b/a + 1/12*sqrt((36*a^2*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(2/3) + 4*c^2 - 12*b*d + 48*a*e + 3*(3*b^2 - 8*a*c)*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3))/(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3))/a, x == 1/2*sqrt(-3/2*a*(b^3/a^3 - 4*b*c/a^2 + 8*d/a)/sqrt((36*a^2*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(2/3) + 4*c^2 - 12*b*d + 48*a*e + 3*(3*b^2 - 8*a*c)*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3))/(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3)) - 1/9*(c^2/a^2 - 3*(b*d - 4*a*e)/a^2)/(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3) - (1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3) + 1/2*b^2/a^2 - 4/3*c/a) - 1/4*b/a + 1/12*sqrt((36*a^2*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(2/3) + 4*c^2 - 12*b*d + 48*a*e + 3*(3*b^2 - 8*a*c)*(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3))/(1/27*c^3/a^3 - 1/6*(b*d - 4*a*e)*c/a^3 + 1/2*(b^2*e + (d^2 - 4*c*e)*a)/a^3 + 1/6*sqrt(9*a^2*d^4 - 256/3*a^3*e^3 + 2/3*(2*b^3 - 9*a*b*c)*d^3 - 1/3*(b^2*c^2 - 4*a*c^3)*d^2 + 1/3*(27*b^4 - 144*a*b^2*c + 128*a^2*c^2 + 192*a^2*b*d)*e^2 + 2/3*(2*b^2*c^3 - 8*a*c^4 + 3*(a*b^2 - 24*a^2*c)*d^2 - (9*b^3*c - 40*a*b*c^2)*d)*e)/a^3)^(1/3))/a]

Zadatak 5

Pomoću modula propcalc, ispišite tablicu istinosnih vrijednosti logičkog izraza

$$((a\rightarrow b)\wedge (b\rightarrow c)) \lor c.$$

import sage.logic.propcalc as propcalc
f = propcalc.formula("((a->b)&(b->c))|c")
print f.truthtable()

a b c value False False False True False False True True False True False False False True True True True False False False True False True True True True False False True True True True

Zadatak 6

Pomoću naredbe find_root numerički riješite jednadžbu $$e^x=2x+2.$$

x = var('x')
find_root(exp(x) == 2*x + 2, -1, 1)

-0.7680390470132458