A bacterial culture has population (in thousands of cells) modeled by the function P(t)=t3−3t+5, where t is the time in days since inoculation.
(1) Write an algebraic expression that represents the average growth rate of the population over the time interval [2,2+Δt], where Δt=0. Clearly indicate all variables in your expression.
(2) Describe, without performing explicit calculations, the value that the average growth rate you found in part (a) approaches as Δt approaches 0 from both the positive and negative directions. Explain the physical meaning of that limiting value in the context of the population model.
(3) Justify whether the population growth rate at t=2 days is well defined for the function P. Support your conclusion with appropriate reasoning based on your analysis in parts (a) and (b).