In 1960, H. von Foerster suggested that the human population...

In $1960, \mathrm{H}$. von Foerster suggested that the human population could be measured by the function $$ P(t)=\frac{179 \times 10^{9}}{(2027-t)^{0.99}} $$ Here $P$ is the size of the human population. The time $t$ is measured in years, where $t=1$ corresponds to the year 1 A.D., time $t=1973$ corresponds to the year 1973 A.D., and so on. (a) Use a graphing utility to graph this function. You will have to be very careful when choosing a graphing window! (b) Use the graph you found in part (a) to approximate $\lim _{t \rightarrow 2027^{-}} P(t)$ (c) This population model is sometimes called the doomsday model. Why do you think this is? What year is doomsday, and why? (d) In part (b), we considered only the left limit of $P(t)$ as $x \rightarrow 2027$. Why? What is the real-world meaning of the part of the graph that is to the right of $t=2027$ ?

In $1960, \mathrm{H}$. von Foerster suggested that the human population could be measured by the function
$$
P(t)=\frac{179 \times 10^{9}}{(2027-t)^{0.99}}
$$
Here $P$ is the size of the human population. The time $t$ is measured in years, where $t=1$ corresponds to the year 1 A.D., time $t=1973$ corresponds to the year 1973 A.D., and so on.
(a) Use a graphing utility to graph this function. You will have to be very careful when choosing a graphing window!
(b) Use the graph you found in part (a) to approximate $\lim _{t \rightarrow 2027^{-}} P(t)$
(c) This population model is sometimes called the doomsday model. Why do you think this is? What year is doomsday, and why?
(d) In part (b), we considered only the left limit of $P(t)$ as $x \rightarrow 2027$. Why? What is the real-world meaning of the part of the graph that is to the right of $t=2027$ ?

Key Concepts

Model Extrapolation and Real-World Interpretation

Extrapolation involves extending the results of a mathematical model beyond the domain of observed data to predict future outcomes. In real-world contexts, this practice requires caution since models may predict impossible or non-physical behavior, such as a 'doomsday' scenario, which implies a breakdown or radical shift in the system being studied.

Vertical Asymptotes

A vertical asymptote occurs in a function when the function values grow without bound as the input approaches a certain value. This concept is essential for understanding scenarios where a model predicts increasingly extreme or infinite behavior, highlighting limitations of the model or indicating a transition point in the phenomenon being modeled.

One-Sided Limits

One-sided limits focus on the behavior of a function as the variable approaches a particular point from one specific side. This concept is particularly important when dealing with points where the function behaves differently from the left than from the right, such as near discontinuities or vertical asymptotes.

Graphing Functions

Graphing functions is a crucial tool in mathematics that provides a visual representation of how a model behaves over a range of input values. It helps in identifying key features such as trends, intercepts, and behaviors near critical points, which aids in the interpretation and validation of the mathematical model.

Mathematical Modeling

This concept involves developing and analyzing equations that represent real-world phenomena, such as predicting population changes over time. It requires formulating a function based on assumptions and data so that the model can be used to understand behavior, make predictions, and consider practical implications.

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