How fast is the size of the population changing after 1 minute?

Question

Under a set of controlled laboratory conditions, the size of the population P of a certain bacteria cultureat time t (in seconds) is given by the functionP(t) = 3t2 + 3et + 10, t ≥ 0

Answer 1

So we are looking for the rate of change of the function P(t). We know that the derivative P'(t₀) is the instantaneous rate of change at time t₀.

So we want to find P(t) and evaluate at t₀ = 60 (time is in seconds for the P(t) equation).

P(t) = 3t² + 3e^t + 10

P'(t) = 6t +3e^t

P'(60) = 6(60) + 3e⁶⁰

= 3.43 x 10²⁶ (using a calculator)

So the rate of change of the population at 1 minute is 3.43 x 10²⁶