F(x) is a function which is defined as F(x) = (x+1)² * e^2x. The fiftieth derivative of F(x) with respect to x at x=0, i.e. F⁽⁵⁰⁾(0), is expressible in the form k*2ⁿ where k is an odd integer and n is a positive integer.

Determine the respective values of k and n.

Note the similar formula for (a+b)^n and
the nth derivative with respect to x of
[(x+1)^2]*[e^(2x)].

The nth derivative with respect to x of
e^(2x), at x=0, is 2^n.

At x=0, the

  oth derivative of (x+1)^2 is 1,
  1st derivative of (x+1)^2 is 2,
  2nd derivative of (x+1)^2 is 2,
  nth derivative of (x+1)^2 is 0
      for n > 2.

Therefore, the 50th derivative with respect
to x of (x+1)^2*e^(2x), at x=0, is

        50!
    ----------- (1)*(2^50) 
     0!(50-0)!

        50!
  + ----------- (2)*(2^49) 
     1!(50-1)!

        50!
  + ----------- (2)*(2^48) 
     2!(50-2)!

  = (1)*(1)*(2^50)+(50)*(2)*(2^49)+(25*49)*(2)*(2^48)

  = (2^49)*(2 + 100 + 1225)

  = 1327*2^49.

Thus, k = 1327 and n = 49.

Posted by Bractals on 2009-12-18 13:51:20