Problem 78 - Applied Calculus

Question

An average child of age \(x\) years grows at the rate of \(6 x^{-1 / 2}\) inches per year (for \(2 \leq x \leq 16\) ). Find the total height gain from age 4 to age \(9 .\)

Step-by-Step Solution

Answer

The total height gain from age 4 to age 9 is 12 inches.

Step 1: Understand the Problem

We need to calculate the total height gain of a child from age 4 to age 9 years. The rate of growth is given as a function of age: \(6x^{-1/2}\) inches per year.

Step 2: Set Up the Integral for Total Height Gain

The total height gain can be determined by integrating the growth rate function over the interval from 4 to 9 years. The definite integral \(\int_{4}^{9} 6x^{-1/2} \, dx\) will give us the total height gain over this period.

Step 3: Integrate the Growth Function

Integrate the function \(6x^{-1/2}\). The antiderivative of \(x^{-1/2}\) is \(2x^{1/2}\). Therefore, the integral becomes:\[\int 6x^{-1/2} \, dx = 6 \cdot 2x^{1/2} + C = 12x^{1/2} + C\]

Step 4: Evaluate the Definite Integral

Substitute the limits 4 and 9 into the antiderivative to find the total height gain:\[12\left[ x^{1/2} \right]_{4}^{9} = 12\left(9^{1/2} - 4^{1/2}\right) = 12(3 - 2) = 12(1) = 12\]

Step 5: Interpret the Result

The result \(12\) means that from age 4 to age 9, the child has a total height gain of 12 inches.

Key Concepts

Growth Rate Function

In the context of calculus, a growth rate function is a mathematical expression representing how a quantity increases over time. In this exercise, the growth rate function is given as \(6x^{-1/2}\), where \(x\) represents the child's age in years. This function describes how fast the child's height is increasing at any given age.

Key features of a growth rate function include:

• It depends on the independent variable \(x\), which, in our example, is the age of the child.
• A higher value of the function indicates a faster growth rate at that point in time.
• The rate can change over different values of \(x\), illustrating the dynamic nature of growth during different ages.

In practical terms, understanding the growth rate helps us predict future heights based on previous growth patterns. This plays a crucial role in biological studies and health assessments. In mathematics, this concept is pivotal for solving problems involving integration, where a rate of change needs to be translated into a cumulative total.

Definite Integral

A definite integral is a powerful calculus tool that allows us to calculate the total change in a quantity across an interval. In this specific exercise, we use it to determine the total height gain of the child from ages 4 to 9.

The notation for a definite integral is \(\int_{a}^{b} f(x) \, dx\), where \(a\) and \(b\) are the lower and upper bounds of the interval, respectively. This represents the accumulation of growth over the interval from \(x = a\) to \(x = b\).

Benefits of using definite integrals include:

• Computation of total quantities from rates, such as height from growth rate functions.
• Ability to evaluate functions over specific intervals, providing precise numerical results.
• Once evaluated, provides a singular value representing the total accumulation, like the 12-inch height gain in our example.

Understanding definite integrals is fundamental in many fields such as engineering, physics, and economics, where cumulative changes or values are often sought after.

Antiderivative

An antiderivative, or indefinite integral, is essentially the reverse process of differentiation. It helps us find a function whose derivative gives us the original function. In this exercise, finding the antiderivative of the growth rate function \(6x^{-1/2}\) leads us to an expression for total height gain.

Finding the antiderivative involves:

• Recognizing common patterns in derivatives to apply antiderivative rules, like \(x^{-1/2}\) whose antiderivative is \(2x^{1/2}\).
• Including a constant \(C\), which becomes obsolete in definite integrals as limits negate its effect.
• Applying known integration techniques across various functions to simplify linked computations.

In practical applications, antiderivatives allow us to reconstruct original functions from derivative data, which is crucial in determining total change over time in processes modeled by differential equations.