a contagious and fatal virus has tragically struck the city of plaguesville, which has been quarantined…

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)= )

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)= )

Answer

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})