A researcher is monitoring a bacterial culture whose population P(t) (in thousands of cells) is known to satisfy dtdP=kP, where t is measured in hours and k is a real constant. No numerical data are provided.
(a) Explain how the sign of the constant k affects whether the population grows or decays over time.
(b) For k>0, describe the end behavior of P(t) as t→∞. Support your conclusion with appropriate reasoning.
(c) For k<0, describe the limit of P(t) as t→∞ and interpret this result in the context of the bacterial culture.