91Ó°ÊÓ

In applied calculus, exponential functions are often used to model natural phenomena or human behaviors that change at a constant proportional rate. The general form of an exponential function is \( f(t) = ab^{ct} \), where:

• \( a \) is the initial amount or size;
• \( b \) is the base of the exponential, representing growth if greater than 1 or decay if between 0 and 1;
• \( t \) stands for the time variable;
• \( c \) is a constant that affects the rate of change.

In our problem, the divorce rate is modeled by the function \( f(t) = 1.02 e^{-0.02 t} \). Here, \( 1.02 \) indicates the initial rate in millions, \( e^{-0.02 t} \) signifies a declining exponential function, which fits well for modeling the decrease in divorce rates over time.

Understanding this principle helps us interpret changes in the context of real-world scenarios, making exponential functions invaluable in many fields such as finance, biology, and social sciences.

To find the total number of divorces over a period, one must integrate the rate of divorces per year with respect to time. Integration is a fundamental technique in calculus, often used to determine the total accumulation of a quantity.
There are various integration techniques, but for exponential functions like \( f(t) = 1.02 e^{-0.02 t} \), recognizing the basic form \( e^{ax} \) is key. The integral of \( e^{ax} \) is \( \frac{1}{a} e^{ax} + C \), where \( C \) is the constant of integration. This technique involves using the antiderivative formula, which allows us to calculate areas under curves.

In the problem at hand, integrating \( 1.02 e^{-0.02 t} \) from 0 to \( t \) provides us with the total number of divorces over \( t \) years. By applying this integration technique, one can find practical solutions to real-life predictions about changing behaviors over time.

Definite integrals are used to calculate the total accumulation of a quantity over a specified interval. Unlike indefinite integrals, which provide a family of functions through a constant of integration, definite integrals give a precise value by evaluating the limits.
The formula for a definite integral is:\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]where \( F(x) \) is the antiderivative of \( f(x) \). This operation translates graphically to finding the area under the curve of \( f(x) \) from \( x = a \) to \( x = b \).

In solving the original exercise, we evaluated the definite integral of the function \( f(t) = 1.02 e^{-0.02 t} \) over the interval \( [0, 5] \), which provided an estimate of the total divorces between 2010 and 2015. Understanding definite integrals is crucial for accurately modeling and predicting cumulative quantities in applied calculus scenarios.