Question

assuming you are a researcher tasked to know about some truths in life. you plan to investigate the growth or decay of some things that interest you. you are to report what you have researched and present exponential or logarithmic functions that would model the involved quantities. you also have to present the graph and explain its characteristics or behavior.

Answer 1

$\displaystyle \textbf{Modeling population growth rates}\\ \textsf{To understand the different models}\\ \textsf{that are used to represent population}\\ \textsf{dynamics, let's start by looking at a}\\ \textsf{general equation for the population growth}\\ \textsf{rate (change in number of individuals}\\ \textsf{in a population over time):}\\ \frac{\mathrm{d}N}{\mathrm{d}t} = rN\\ \textsf{is the growth rate of the population}\\ \textsf{in a given instant,}\, N\,\textsf{is population}\\ \textsf{size,}\,t\, \textsf{is time, and} \\ r\, \textsf{is the per capita rate of increase}\\ \textsf{that is, how quickly the population}\\ \textsf{grows per individual already in the}\\ \textsf{population.}\\ \textsf{If we assume no movement of individuals}\\ \textsf{into or out of the population is,}\,r\\ \textsf{just a function of birth and death rates.}\\ \textbf{Derivation of the solution to the}\\ \textsf{differential equation}\\ \frac{\mathrm{d}N}{\mathrm{d}t} = rN\\ \frac{\mathrm{d}N}{N} = r\, \mathrm{d}t\\ \int\,\frac{\mathrm{d}N}{N} = \int\,r\, \mathrm{d}t\\ \ln(N) = rt + C\\ N = Ae^{rt}\\ \textsf{The population increases exponentially}\\\textsf{as the time increases}\\$