In a microbiology experiment

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

In a microbiology experiment, the size of a bacterial population (measured in thousands of cells) after tt hours is modeled by P(t)=(t2+4)sin⁡t (ln⁡t)tP(t)=\bigl(t^2+4\bigr)^{\sin t}\,\bigl(\ln t\bigr)^{t}, t>1.t>1. (a) Derive a simplified formula for the instantaneous relative growth rate 1P(t)dPdt\dfrac{1}{P(t)}\dfrac{dP}{dt}. (b) Compute the instantaneous relative growth rate at t=2t=2 and express your answer to three decimal places. (c) Interpret the numerical value obtained in part (b) in the context of the experiment. What does it tell you about the population at t=2t=2 hours?

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