91Ó°ÊÓ

When discussing calculus, the concept of a definite integral plays a vital role. It refers to the integral of a function within a set range of values, also referred to as bounds. In the context of this exercise, the definite integral is used to calculate the total height gain of a child between the ages of 4 and 9.

The definite integral is denoted as \( \int_{a}^{b} f(x) \, dx \), where \( f(x) \) is the function to be integrated and \( a \) and \( b \) are the lower and upper bounds, respectively. It essentially sums up the infinitesimally small changes in the function \( f(x) \) over the interval \([a, b]\).

• In this exercise, the function \( f(x) = 6x^{-1/2} \) represents the growth rate of the child.
• The bounds \( x=4 \) and \( x=9 \) define the age range over which the child’s growth rate is evaluated.

Understanding definite integrals as the area under the curve of the function between two points can help find cumulative values, such as total growth or distance.

A growth rate function describes how a variable changes concerning another, often representing real-world growth, like biological growth over time. In this example, the growth rate function \( f(x) = 6x^{-1/2} \) describes the increase in height (in inches per year) of an average child.

This particular function, \( 6x^{-1/2} \), is more commonly seen in growth models where growth decreases as time progresses, which is typical in childhood development. In our problem:

• The value of \( f(x) \) decreases as \( x \) (age) increases, meaning younger children have a higher annual growth rate than older children.
• This diminishing growth rate captures the slowing growth pattern as children grow older.

Understanding the function's behavior is crucial for accurately predicting overall changes and making decisions based on these predictions.

The idea of an antiderivative is central to solving integrals. An antiderivative of a function \(f(x)\) is a function \(F(x)\) such that \(F'(x) = f(x)\), i.e., differentiating \(F(x)\) yields \(f(x)\). For this exercise, finding the antiderivative helps evaluate the definite integral.

When working with a function like \(6x^{-1/2}\), we find its antiderivative as part of the integration process:

• First, recognize that \(x^{-1/2}\)'s antiderivative is \(2x^{1/2}\).
• Therefore, the antiderivative of \(6x^{-1/2}\) is \(12x^{1/2}\).

Antiderivatives allow one to go backwards, from the rate of change back to the original function representing the cumulative value, such as total height increase over time.
Calculating the value of a definite integral with an antiderivative involves substituting the boundary values into the antiderivative and taking the difference, providing a practical application in determining total growth or changes.