A population of bacteria in a laboratory culture is modeled by the differentiable function
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A population of bacteria in a laboratory culture is modeled by the differentiable function P(x), where x is measured in days and P(x) is measured in thousands of cells.
A researcher states, "At x=4, the growth rate of the culture is P′(4)=2.6."
(a) Explain, without performing any calculations, what the numerical value 2.6 tells you about the population’s behavior at x=4. Be sure to reference the correct units in your explanation.
(b) The researcher later changes the time variable from days to hours, defining h=24x. Express dhdP at the instant corresponding to x=4 and justify its units by carefully applying the relationship between P, x, and h.
(c) Discuss how the numerical value of the derivative changes (or does not change) when the independent variable’s unit changes from days to hours. Support your answer with reasoning based on unit analysis.