Help me for calculs please? (again)?

Question

an advertising agency starts a campaign to promote a new product and determines that t days later the number of people (N(t))) who have heard about the product will be changing at the rate of N'(t). How many peple will learn the product during the third week? What does (a) represent? why?

Answer 1

This is an exponential growth or decay law. Likely

it said that    N'(t) = a N(t)         where "a" is a constant.

So, this is a differential equation

dN/dt = a N

divide both sides by N, and integrate with respect to t.

Int_n(0)^n(t)  dN/N = int_t0^t a dt

ln(n(t)/n(0) ) = a (t - t0)

n(t) = n(0) e^{ a (t -t0) }

n(0) is the number at time t=t0. If  a> 0, then the solution

describes exponential growth, if a< 0 , decay.