91Ó°ÊÓ

Population growth is a fascinating phenomenon that reflects how populations, such as those of humans, increase in size over time. Imagine a town starting with a small number of families, which gradually expands as more people move in or are born. Over the years, this growth can be represented mathematically, providing a clearer picture of how fast or slow it is occurring.

• Population growth rates are often expressed as a function, like the given function in the exercise, to show changes over time.
• These rates can help plan for resources, infrastructure, and other community needs as the population grows.

In mathematics, understanding how to work with population growth functions is essential. It allows us to anticipate changes, make informed decisions, and handle real-world problems with confidence. By analyzing growth functions, such as the one in the exercise, you can predict future population sizes and understand the impact of various factors on growth.

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. It is quite common when dealing with continuous growth or decay processes. In the case of population growth, exponential functions efficiently describe how a population multiplies over time.
For the function given in the exercise, the term has the form of an exponential function and is written as:\[ p(t) = 1.7 e^{0.03t} \]

• Here, \( e \) is the base of the natural logarithms, roughly equal to 2.71828.
• The coefficient (1.7) implies the initial growth rate or scale of populations.
• The exponent (0.03t) represents the growth factor, with \( t \) indicating time (in years since 2007).

Exponential functions describe rapid changes, so the population grows at an ever-increasing rate. Understanding these functions gives you the power to model and solve issues involving continuous change, such as in biology, finance, or physics.

When dealing with antiderivatives, a fundamental concept is the constant of integration, denoted as \( C \). This constant comes into play anytime you integrate a function, like when finding the antiderivative of an exponential function.

• Integrating involves reversing differentiation, which will yield an infinite set of solutions. The constant \( C \) represents this infinite family.
• Each specific problem or situation often requires additional information to determine what \( C \) should be; however, it encapsulates any initial value or starting condition.

In the solution, after finding the antiderivative of the population growth rate, the expression becomes:\[ 56.67 e^{0.03t} + C \]This additional \( C \) essentially adjusts the function to fit the initial conditions of the specific context you are analyzing, such as the population at a specific year. Integrating these concepts helps bridge the gap between equations and their real-world applications.