Suppose we start recording the population change at the start time to = 0, and birth...

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Suppose we start recording the population change at the "start time" to = 0, and birth and death rates are given by b(t) = 2 + sin(πt + π/4) and d(t) = 2 = cos(πt + π/4). (a) What is the earliest time T> 0 at which the population starts decreasing? (b) What is the earliest time T> 0 at which the population returns back to its starting numbers? One important piece of information for understanding how populations change is the average population over a fixed time interval, such as a season, year, or generation. For a function, f, we can calculate its average value from t = a to t = b as b 1 - a cb f" f(t)dt. Suppose that after further observation and reassessment, we modeled the population (in thousands) of the species at time t (in months) by the function P(t) = 1 + e-(t-4)² + cos(πt). For which four-month period (i.e. from t = T to t = T+4) is the average value of P(t) the greatest? Hint: How can we find the maximum of a function? Suppose we start recording the population change at the "start time" to = 0, and birth and death rates are given by b(t) = 2 + sin(πt + π/4) and d(t) = 2 = cos(πt + π/4). (a) What is the earliest time T> 0 at which the population starts decreasing? (b) What is the earliest time T> 0 at which the population returns back to its starting numbers? One important piece of information for understanding how populations change is the average population over a fixed time interval, such as a season, year, or generation. For a function, f, we can calculate its average value from t = a to t = b as b 1 - a cb f" f(t)dt. Suppose that after further observation and reassessment, we modeled the population (in thousands) of the species at time t (in months) by the function P(t) = 1 + e-(t-4)² + cos(πt). For which four-month period (i.e. from t = T to t = T+4) is the average value of P(t) the greatest? Hint: How can we find the maximum of a function?