# 1. The production of a chemical satisfies the following differential equation (dP/dt )= ((20)/(1 + 4t)^ 2 ), where t is time in days and P is the amount in moles. (a) Use integration by substitution

1. The production of a chemical satisfies the following differential equation (dP/dt )= ((20)/(1 + 4t)^ 2 ), where t is time in days and P is the amount in moles.

(a) Use integration by substitution to find the solution, P(t), starting from the initial condition P(0) = 0.

(b) Sketch the rate of change ( dP dt ) and the solution.

(c) What happens to P(t) as t → ∞?

3. An outbreak of a novel infectious disease is initially growing at a rate of f(t) = 1.5e 0.12t

new cases per day (where t is time in days).

(a) Evaluate a definite integral to find the number of new cases that occur during the first 2 weeks.

(b) What’s the average number of daily new cases in the first 2 weeks?

(c) If the rate was initially given by g(t) = 1.5 + 0.12t new cases per day (where t is time in days), how many fewer cases would occur during the first 2 weeks?

(550 words)

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