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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
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()
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
CRL 1.8 Functions

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CRL 1.8 Functions

  • 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