{Use of Tech} Cell population The ​population ​of a culture of cells after t days is approximated by the function P(t)=1600 / 1 + 7e^−0.02t, for t≥0.
e. Graph the growth rate. When is it a maximum and what is the population at the time that the growth rate is a maximum? 

Verified step by step guidance

1. Step 1: Identify the function for the population growth rate. The growth rate is the derivative of the population function P(t) with respect to time t. So, find P'(t) by differentiating P(t) = \frac{1600}{1 + 7e^{-0.02t}}.
2. Step 2: Use the quotient rule to differentiate P(t). The quotient rule states that if you have a function \frac{u}{v}, its derivative is \frac{u'v - uv'}{v^2}. Here, u = 1600 and v = 1 + 7e^{-0.02t}.
3. Step 3: Differentiate u and v. Since u = 1600 is a constant, u' = 0. For v = 1 + 7e^{-0.02t}, use the chain rule to find v'. The derivative of e^{-0.02t} is -0.02e^{-0.02t}, so v' = -0.14e^{-0.02t}.
4. Step 4: Substitute u, u', v, and v' into the quotient rule formula to find P'(t). Simplify the expression to get the growth rate function.
5. Step 5: To find when the growth rate is maximum, set the derivative of the growth rate function (P''(t)) to zero and solve for t. This will give the critical points. Evaluate P(t) at this t to find the population at the time when the growth rate is maximum.

Key Concepts

Logistic Growth Model

The function P(t) = 1600 / (1 + 7e^(-0.02t)) represents a logistic growth model, which describes how populations grow in a limited environment. Initially, the population grows exponentially, but as resources become limited, the growth rate decreases and approaches a maximum carrying capacity, in this case, 1600 cells.

Derivative and Growth Rate

To find the growth rate of the population, we need to compute the derivative of P(t) with respect to t, denoted as P'(t). This derivative indicates how the population changes over time, and finding its maximum involves setting P'(t) to zero and solving for t, which reveals when the population growth is at its peak.

Critical Points and Maximum Values

Critical points occur where the derivative P'(t) is zero or undefined. By analyzing these points, we can determine the maximum growth rate of the population. Additionally, evaluating P(t) at these critical points allows us to find the corresponding population size when the growth rate is at its maximum.