The amplitude of a signal in a circuit, S(t)S(t)S(t), can be modeled by the function
S(t)=(2−t4)8 S(t) = \left(2 - \frac{t}{4}\right)^8 S(t)=(2−4t)8where t t\,t represents time in milliseconds.
Find the first 4 terms, in ascending powers of ttt, of the binomial expansion of S(t)S(t)S(t), giving each term in its simplest form.
A noise-cancelling filter F(t)F(t)F(t) is applied such that the resulting signal is given by the product P(t)=F(t)S(t)P(t) = F(t)S(t)P(t)=F(t)S(t). Given that
F(t)=(3+2t)2 F(t) = \left(3 + \frac{2}{t}\right)^2 F(t)=(3+t2)2determine the constant term in the series expansion of P(t)P(t)P(t).
