1. jsMath
CRL 1.8 Functions MrG 2011.0920
#1)
var('y,a,z')
f(x)=x^3-10*x^2+31*x-30
print(f(0))
print(f(y))
print(f(2))
print(f(a))
print(f(3))
print(f(5))
print(f(y*z))
print(f(1))
print(f(-3))
print(expand(f(x-2)))
print(f(-1))
print(-6*f(6))
-30
y^3 - 10*y^2 + 31*y - 30
0
a^3 - 10*a^2 + 31*a - 30
0
0
y^3*z^3 - 10*y^2*z^2 + 31*y*z - 30
-8
-240
x^3 - 16*x^2 + 83*x - 140
-72
-72
#3)
phi(x)=x^4-55*x^2-210*x-216

2. print(f(2))
print(phi(-2))
print(f(3))
print(phi(-3))
print(f(5))
print(phi(-4))
print(f(0)+phi(0)+246)
0
0
0
0
0
0
0
#5)
F(x)=x*(x-1)*(x+6)*(x-1/2)*(x+5/4)
print(F(0))
print(F(1))
print(F(-6))
bool(F(1/2)==F(-5/4))
0
0
0
True
#5a)
def F(x):
return x*(x-1)*(x+6)*(x-1/2)*(x+5/4)
F(1/2)==F(-5/4)
True
#7)
var('a,y,z')
phi(x)=a^x
(phi(y)*phi(z)).simplify()

3. a^(y + z)
bool(phi(y)*phi(z)==phi(y+z))
True
#9)
f(x)=cos(x)
show(bool(f(x)==f(-x)))
show(bool(f(x)==-f(pi-x)))
show(bool(f(x)==-f(pi+x)))
True
True
True
#11)
var('n,m')
psi(x)=x^(2*n)+x^(2*m)+1
show(psi(1)) show(psi(0).simplify())
show(psi(a).simplify())
show(psi(-a).simplify())
3
1
a(2n)+a(2m)+1
(−a)(2n)+(−a)(2m)+1