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Population growth refers to the increase in the number of individuals in a population. It's often influenced by birth rates, death rates, immigration, and emigration. In this example, we focus on global human population growth, assuming it follows an exponential pattern. - **Exponential growth** means that the population grows by a constant percentage per year rather than a constant number. - **Human population growth** specifically considers factors like fertility, improvements in healthcare, and living standards which have led to rapid increases historically. Understanding population growth is vital for planning resources, infrastructure, and policy-making. In this exercise, the population grows by 1.3% annually, illustrating how even a small growth rate can lead to significant population increases over time.

Calculus problem-solving involves using mathematical models and methods to find solutions. In this specific exercise, we're dealing with exponential growth, which is commonly tackled using calculus principles. **Steps for Solving the Population Growth Problem:** - **Identify Key Variables**: Here, the initial population (6 billion), the growth rate (1.3%), and the target population (7 billion). - **Formulate the Problem**: Use the exponential growth formula to model the population changes over time. - **Set Up the Equation**: Establish an equation that reflects when the population reaches the target (7 billion). - **Solve Mathematically**: We solve the equation using logarithms to isolate the variable. These problem-solving techniques allow you to approach and find solutions for various exponential growth or decay scenarios in calculus.

An exponential function is a type of mathematical expression where a constant base is raised to a variable exponent. It's written in the form \( f(x) = ab^x \), where \( a \) is a constant representing the initial value, and \( b \) is the growth factor.In our problem:- The **initial value** is the population in 2000, 6 billion.- The **growth factor** is determined by the annual rate of 1.3%, converted to the exponential base, \( e \), leading to the expression \( 6 imes e^{0.013t} \).**Properties of Exponential Functions:**- They show how quantities grow or shrink over time.- Characterized by rapid increases or decreases as time progresses.- Used widely in population dynamics, finance, and natural sciences for modeling behaviors and trends.Exponential functions are crucial in understanding how changes compound over time, producing results that initially seem modest but escalate quickly with time.