Derivative function

Question

The number of people with a certain disease in NJ is described/modeled by the function P(t)=t^2+3t+10, where t measures the years since 2010.P(t) is called the prevalence of the disease at time t. The incidence of the disease is the number of new cases per unit time over a specific time interval. The points are(2,20) and (7,80).

1. What was the prevalence of the disease at the beginning of 2012?
2. Calculate the incidence of the disease from 2012 to 2017?
3. calculate the incidence of the disease at the beginning of 2012. Explain how that is related to the instantaneous rate of change if P(t).

Answer 1

1) What is the prevalence of the disease at the beginning of 2012 ?

Ans: Since we want to know the prevalence 2 years after 2010, time t = 2. So we know

P(2) = (2)²+3(2)+10 = 20. Hence, the (2,20) point.

2) Calculate the incidence of the disease from 2012 to 2017.

Ans: If we are looking for the point (7,20), then we calculate the incidence from P(t) as time t = 7 years from 2010 such that

P(7) = (7)²+3(7)+10 = 80. Hence, the (7,80) point.

3) Calculate the incidence of the disease at the beginning of 2012. Explain how that is related to the instantaneous rate of change if P(t).

Ans: If we are beginning from 2012 then time t = 5 years. Then,

P(5) = (5)²+3(5)+10 = 50.

However, the rate or instantaneous rate of change is

P(7) - P(2) divided by (7-2) = (80 - 20) divided by (5) = 12 new cases per year for 7 years.