The mass of a colony of bacteria, in grams, is modeled by the function p given by p(t)=2+5tan−1(t2), where t is measured in days. what is the instantaneous rate of change of the mass of the colony, in grams per day, at the moment the colony reaches a mass of 6 grams?

P'(1.015)=4.93 grams

Explanation:

Instantaneous Rate Of Change

If one variable y is a function of another variable x, the rate of change can be measured by changing x by a small amount (dx) and computing the change in y (dy). The rate of change is

When dx tends to zero, we call it the instantaneous rate of change and is easily computed as the first derivative of y

Note: We'll be using the standart notation atan for the inverse tangent

We are given the relation between the mass of a colony of bacteria in grams P(t) and the time t in days

Let's find its first derivative, recalling that

We need to know the value of t, so we use the provided condition P=6 gr

We use this value in the derivative:

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