Question

When a disease appears in a population, health authorities record the number of infected people. One function that describes the number of infected people Is y(t) = 120t^2 e^- It where to is the time in days and y is the number (in units of a thousand) of infected people.

Compute the derivative of y.

Answer 1

The derivative of the function y(t) = 120t^2 e^-1t, which represents the number of infected people over time, is found using the product and chain rules of differentiation. The derivative or rate of change of infected individuals with respect to time is dy/dt = (240t - 120t^2)e^-1t.

Step-by-step explanation:

To compute the derivative of the given function y(t) = 120t2e-1t, we'll use the product rule and the chain rule of differentiation. The product rule states that the derivative of a product of two functions is the derivative of the first function multiplied by the second function plus the first function multiplied by the derivative of the second function. The chain rule is used to differentiate composite functions.

We can then express the derivative dy/dt as:

dy/dt = d/dt(120t2) · e-1t + 120t2 · d/dt(e-1t)

This simplifies to:

dy/dt = (240t)e-1t + 120t2(-e-1t)

The final step is to combine like terms:

dy/dt = (240t – 120t2)e-1t

This gives us the rate of change of the number of infected people over time in a population.