Question

{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?

Accepted Answer

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 + $$7e^{-0.02t}$$}. 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 + $$7e^{-0.02t}. $$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' = $$-0.14e^{-0.02t}. $$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. 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.

{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?

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Concetti chiave

Logistic Growth Model

Derivative and Growth Rate

Critical Points and Maximum Values