A technology startup tracks its total user base (in thousands) as a differentiable…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A technology startup tracks its total user base U (in thousands) as a differentiable function of time t (measured in months since launch). A product manager reports the following information about the instantaneous rate of change U′(t):
• For 0<t<4, U′(t)>0 and is increasing toward a maximum at t=4.
• For 4<t<10, U′(t)>0 but is decreasing and crosses 0 at t=10.
• For 10<t<14, U′(t)<0 and attains its minimum at t=12.
• For t>14, U′(t) is negative but increasing toward 0.
Answer the following questions about the company’s user-growth trajectory.
(a) Identify each time interval on which the total user base U is increasing. Justify your answer using the information about U′(t).
(b) Identify each time interval on which the total user base U is decreasing. Provide a brief verbal interpretation of this trend for the company.
(c) At what time(s) does U attain a local maximum? Explain how the sign changes of U′(t) support your conclusion.
(d) The marketing team can launch only one major advertising campaign. Based on your analysis, recommend a month in which such a campaign would most effectively accelerate user growth, and justify your choice using the behavior of U′(t).