Question: An epidemiologist working for a healthcare facility was tracking a short-lived outbreak of influenza, which she modeled with the following function, \(f\left( t \right)=18{{t}^{2}}-2{{t}^{3}}\), to determine the number of new cases reported daily; the independent variable, t , represents the number of days into the outbreak. How many new cases were reported on the ( a ) second day, ( b ) fourth day, and ( c ) eighth day? ( d ) This outbreak lasted for how many days? [ COMMENTS & HINTS: An outbreak ends when the number of new cases reported daily returns to zero.] ( e ) On what day did the number of reported cases peak, or reach a maximum? [ COMMENTS & HINTS: Use calculus methods to explain your answer.] ( f ) How many new patient cases were reported on that day? ( g ) Affirm that this extreme point is indeed a maximum for this function. [ COMMENTS & HINTS: Use calculus methods to confirm the concavity of the graph at this point.] ( h ) On what day did the rate of change in the number of reported new patient cases begin to level off? [ COMMENTS & HINTS: Use calculus methods. This does not refer to an extreme point, but rather a point of inflection.]

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