Math 141 Homework #7
Due Tuesday, 10/2/07
Extra Problem
Problem #1 If nis a positive integer, define a polynomial function fn(x) by
fn(x) =
n
X
i=0
xi
i!
(#1a) Write down explicit expressions for fn(x) for a few small values of n(say 0 ≤n≤5). (To get you
started, f0(x) = P0
i=0
xi
i!=x0/0! = 1.)
(#1b) Calculate the derivatives f0
n(x) of the functions you wrote down in part (a). What do you notice?
(#1c) Define a new function, with the curious-looking name f∞(x), by
f∞(x) = lim
n→∞ fn(x)
(you may have to think a bit about how to make this definition make sense). Based on your solution to
part (b), what would you expect about f0
∞(x)?
(#1d) By evaluating fn(1) for a few values of n, make a conjecture about the value of f∞(1).
(#1e) Based on your answers to parts (d) and (e), what function does f∞(x) remind you of ? Evaluate
that function and f∞(x) at a few other values of xto see what else the two functions have in common.