differentiation

Question

(a) The sales of a book publication are expected to grow according to the function S = 300000(1 − e−0.06t),where t is the time,given in days. (i) Show using differentiation that the sales never attains an exact maximum value. (ii) What is the limiting value approached by the sales function?

Answer 1

S = 300000(1 - e^{- 0.06t}). S' = 300000(0 - e^-0.06t)(- 0.06) = 1800e^{- 0.06t} > 0 for any t. So when t grow without bound S(t) is increasing. And because maximum value of sale S never riched is S_max = lim t → ∞ 300000(1 -

e^{- 0.06t}) = 300000