A bacterial culture obeys the differential equation

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

Problem

\dfrac{dy}{dt}=k\,y\,(C-y)

, where

y(t)

is the number of bacteria at time

t

k

is a positive constant, and

C

is the carrying capacity of the environment. (1) Describe how the sign and magnitude of

\dfrac{dy}{dt}

depend on the value of

y

for

0<y<C

. Focus on whether the rate of change is positive, negative, or zero and on how fast the culture grows at different population sizes. (2) Based on the form of the differential equation, explain whether the culture grows fastest when the population is very small (

y\ll C

), moderate (

y\approx C/2

), or large (

y\approx C

). Provide reasoning without performing any formal differentiation. (3) Identify, in terms of

C

, the population size at which the instantaneous growth rate is greatest, and briefly justify your conclusion.

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