Science:Math Exam Resources/Courses/MATH101 B/April 2024/Question 18/Statement

A key statistic for describing how a population changes is the average population over a fixed time interval, such as a season, a year, or a generation.

For any given function $f$, the average value from $t=a$ to $t=b$ is

$${\frac {1}{b-a}}\int _{a}^{b}f(t)\mathrm {d} t.$$

Suppose the population (in thousands) of a species at time $t$ (in months) is given by the function

$$P(t)=1+e^{-(t-4)^{2}}+\cos(\pi t).$$

For which four-month period (i.e., from $a=T$ to $b=T+4$) is the average value of $P$ the greatest? You need to justify your answer.