A researcher is monitoring a bacterial culture whose population (in thousands of…

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

A researcher is monitoring a bacterial culture whose population

P(t)

(in thousands of cells) is known to satisfy

\displaystyle \frac{dP}{dt}=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)

t\to\infty

. Support your conclusion with appropriate reasoning. (c) For

k<0

, describe the limit of

P(t)

t\to\infty

and interpret this result in the context of the bacterial culture.