A biology field study records the mass (in grams) of a bacterial culture at various times
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A biology field study records the mass M (in grams) of a bacterial culture at various times. Instead of directly measuring M, the researchers estimate the derivative M′(t) at discrete time points t (in hours) as shown in the table below.
Time t (h) | 1 | 3 | 5 | 7 | 9 | 11
:--|:--|:--|:--|:--|:--|:--
M′(t) (g/h) | 2.4 | 1.8 | 0.9 | 0 | −0.8 | −1.3
Use the data to analyze the growth of the bacterial culture.
(a) Using the tabulated values, identify all consecutive time intervals on which the mass M is increasing. Explain your reasoning.
(b) Identify all consecutive time intervals on which M is decreasing. Provide justification.
(c) Estimate the time at which M attains its greatest value on the interval 1≤t≤11. Support your answer with evidence from the table and appropriate calculus reasoning.
(d) Suppose laboratory protocol requires harvesting the culture at the instant its growth rate first becomes non-positive. Using the data, estimate the harvest time and explain your choice in context.