Activity 1.3.4.
A rapidly growing city in Arizona has its population \(P\) at time \(t\text{,}\) where \(t\) is the number of decades after the year 2010, modeled by the formula \(P(t) = 25000 e^{t/5}\text{.}\) Use this function to respond to the following questions.
(a)
Sketch an accurate graph of \(P\) for \(t = 0\) to \(t = 5\) on the axes provided. Label the scale on the axes carefully.
(b)
Compute the average rate of change of \(P\) between 2030 and 2050. Include units on your answer and write one sentence to explain the meaning (in everyday language) of the value you found.
(c)
Use the limit definition to write an expression for the instantaneous rate of change of \(P\) with respect to time, \(t\text{,}\) at the instant \(a = 2\text{.}\) Explain why this limit is difficult to evaluate exactly.
(d)
Estimate the limit in (c) for the instantaneous rate of change of \(P\) at the instant \(a = 2\) by using several small \(h\) values. Once you have determined an accurate estimate of \(P'(2)\text{,}\) include units on your answer, and write one sentence (using everyday language) to explain the meaning of the value you found.
(e)
On your graph, sketch two lines: one whose slope represents the average rate of change of \(P\) on \([2,4]\text{,}\) the other whose slope represents the instantaneous rate of change of \(P\) at the instant \(a=2\text{.}\)
(f)
In a carefully-worded sentence, describe the behavior of \(P'(a)\) as \(a\) increases in value. What does this reflect about the behavior of the given function \(P\text{?}\)
