Question

a contagious and fatal virus has tragically struck the city of plaguesville, which has been quarantined until the virus has run its course. the function ( p(t)=14 t^{3}-5 t^{4} ) gives the number of people who are newly infected ( t ) days after the outbreak began. (a) find first and second derivatives ( p^{prime}(t) ) and ( p^{prime prime}(t) ). (b) solve ( p(t)=0, p^{prime}(t)=0 ) and ( p^{prime prime}(t)=0 ). then complete the table. (c) use the table to describe the details of the plaguesville tragedy. (a) find first and second derivatives ( p^{prime}(t) ) and ( p^{prime prime}(t) ). ( p^{prime}(t)= ) ( p^{prime prime}(t)= )

Explanation:

Step1: Find the first derivative \(P^{\prime}(t)\)

Use the power rule \((x^n)^\prime = nx^{n - 1}\).
For \(P(t)=14t^{3}-5t^{4}\), we have:
\(P^{\prime}(t)=\frac{d}{dt}(14t^{3}-5t^{4})\)
\(P^{\prime}(t)=14\times3t^{2}-5\times4t^{3}\)
\(P^{\prime}(t)=42t^{2}-20t^{3}\)

Step2: Find the second derivative \(P^{\prime\prime}(t)\)

Differentiate \(P^{\prime}(t)=42t^{2}-20t^{3}\) with respect to \(t\) using the power rule.
\(P^{\prime\prime}(t)=\frac{d}{dt}(42t^{2}-20t^{3})\)
\(P^{\prime\prime}(t)=42\times2t - 20\times3t^{2}\)
\(P^{\prime\prime}(t)=84t-60t^{2}\)

Answer:

\(P^{\prime}(t)=42t^{2}-20t^{3}\)
\(P^{\prime\prime}(t)=84t - 60t^{2}\)