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Understanding the average growth rate is fundamental when studying population dynamics. It represents the change in population over a specific period of time, giving us an indication of how rapidly a population is increasing or decreasing. Consider the formula for average growth rate: \[ r_{avg} = \frac{P(t_2)-P(t_1)}{t_2-t_1} \]
In the context of our cell culture problem, to find the average growth rate over the first 10 days, we would substitute the population values at day 10 and day 0 into the formula, essentially measuring the slope of the line segment connecting these two points on the population graph. This rate provides an overarching view of the population's change over the entire interval, not accounting for any variability within those 10 days.

Graphing a population function can provide visual insight into the population's behavior over time. When graphing the function \(P(t)=\frac{1600}{1+7 e^{-0.02 t}}\), we expect to see how the population of cells changes with each day. Initially, when dealing with exponential functions, the graph may start off with a slow increase, then ascend rapidly as the effect of the exponential growth kicks in, before finally levelling off as it approaches the carrying capacity—the maximum population the environment can sustain. By using graphing tools like Desmos or GeoGebra, you can accurately plot this function and observe its nuances, making it easier to identify points of interest, such as when the growth rate is maximum.

Differentiation is a powerful tool in calculus for understanding how a function changes at any given point. When we differentiate exponential functions, we reveal the growth rate function, which can tell us how fast the population is growing at any moment in time. In our case, the differentiation of \(P(t)=\frac{1600}{1+7 e^{-0.02 t}}\) using the quotient rule provided us with the growth rate function \(P^{\'}(t)= \frac{224 e^{-0.02 t}}{(1+7 e^{-0.02 t})^2} \). This new function allows us to observe not just how the population is growing on average, but how its rate of change varies at each infinitesimal point in time.

The maximum growth rate is a point of key interest as it indicates when the population is increasing most rapidly. It occurs at a point where the derivative of the population function reaches its maximum value. After obtaining the growth rate function \(P^{\'}(t)\), we can graph this function to visually inspect where it reaches its peak. Analytically, finding the maximum growth rate would involve setting the derivative of the growth rate function to zero and solving for \(t\). However, by graphing, we can estimate this point more intuitively. Once the time at which the maximum growth rate occurs is known, substituting this back into the original population function will provide us with the size of the population at that specific moment.