quiz6.html

MCS 320 Quiz 6 Friday 30 September 2006

> restart;

1. Define the following function

{ 1 if x < -1

f(x) = { x^2 if x >=1 and x <= 2

{ 4 if x > 2.
Illustrate your definition by computing the derivative of f, i.e. give the output of the command diff(f,x).

ANSWER :

> f := piecewise(x<-1,1,x>2,4,x^2);

> diff(f,x);

2. Write an efficient recursive procedure to compute polynomials p(x) defined by

p[0](x) = 5, p[1](x) = 1 - x, p[n] = (x-1)*p[n-1](x) + p[n-2](x-1), for n > 1.

The degree of the polynomial must be used as an index.

The argument of the procedure is x.

The procedure should return an expanded polynomial.

What is the coefficient of x^50 in p[50](x)?

ANSWER :

> p := proc(x)

> option remember;

>   local n:
  n := op(procname):
  if n = 0 then
     return 5;
  elif n = 1 then
     return 1 - x;
  else
     return expand((x-1)*p[n-1](x) + p[n-2](x-1));
  end if;
end proc;

> p[0](z); p[1](z); p[2](z);

> p[50](z);

> coeff(%,z,50);

>