The number of visitors $P$ to a website in a given week over a 1-year period is given by $P(t)=119+(t-87) e^{0.02 t}$, where $t$ is the week and $1 \leq t \leq 52$.
a) Over what interval of time during the 1 -year period is the number of visitors decreasing?
b) Over what interval of time during the 1-year period is the number of visitors increasing?
c) Find the critical point, and interpret its meaning.
a) The number of visitors is decreasing over the interval $(1,37)$.
(Simplify your answer. Type integers or decimals rounded to three decimal places as needed. Type your answer in interval notation.)
b) The number of visitors is increasing over the interval
(Simplify your answer. Type integers or decimals rounded to three decimal places as needed. Type your answer in interval notation.)
Answer
Final Answer: \(\boxed{\text{b) The number of visitors is increasing over the interval }(37,52)}\)
Step 1 ：The number of visitors to a website is given by a function of time, \(P(t)\). To find out when the number of visitors is increasing or decreasing, we need to find the derivative of this function, \(P'(t)\), and find out when it is positive or negative.
Step 2 ：The derivative of a function tells us the rate of change of the function. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.
Step 3 ：So, we need to find the derivative of \(P(t)\), set it equal to zero to find the critical points, and then determine the sign of the derivative on the intervals determined by the critical points.
Step 4 ：\(t = t\)
Step 5 ：\(P = (t - 87)*exp(0.02*t) + 119\)
Step 6 ：\(P_prime = 0.02*(t - 87)*exp(0.02*t) + exp(0.02*t)\)
Step 7 ：The critical point is at \(t = 37\). The derivative is negative on the interval \((1, 37)\) and positive on the interval \((37, 52)\). This means that the number of visitors is decreasing on the interval \((1, 37)\) and increasing on the interval \((37, 52)\).
Step 8 ：Final Answer: \(\boxed{\text{a) The number of visitors is decreasing over the interval }(1,37)}\)
Step 9 ：Final Answer: \(\boxed{\text{b) The number of visitors is increasing over the interval }(37,52)}\)
