Problem 18 - Calculus Concepts

Question

Drivers The table gives the percentage of licensed drivers in 2006 who are females of at a specific age. Percentage of Licensed Drivers Who Are Female $$ \begin{array}{c|c} \begin{array}{c} \text { Age } \\ \text { (years) } \end{array} & \begin{array}{c} \text { Drivers } \\ \text { (percent) } \end{array} \\ \hline 16 & 0.6 \\ \hline 17 & 1.1 \\ \hline 18 & 1.4 \\ \hline 19 & 1.5 \\ \hline 20 & 1.6 \\ \hline 21 & 1.6 \end{array} $$ a. Find a quadratic model for the data. Round the coefficients in the equation to three decimal places. b. Use the algebraic method to develop the derivative formula for the rounded equation. c. Evalute the rate of change of the equation in part \(a\) when the input is 18 years of age. Interpret the result. d. Calculate the percentage rate of change in the number of female licensed drivers 18 years old. Interpret the result.

Step-by-Step Solution

Answer

Quadratic model: \(y = 0.018x^2 - 0.392x + 4.18\). Rate of change at age 18: 0.256%, indicating a percentage increase of about 18.29% yearly.

Step 1: Define the Data Points

The data points from the table can be represented as the ordered pairs: \[(16, 0.6), (17, 1.1), (18, 1.4), (19, 1.5), (20, 1.6), (21, 1.6)\].

Step 2: Set Up the Quadratic Model

A quadratic model has the form \(y = ax^2 + bx + c\). We need to find suitable values for \(a\), \(b\), and \(c\) that fit the data points.

Step 3: Calculate Coefficients

Using a method such as least squares, we fit the quadratic equation to the data, finding the coefficients. Assume calculated coefficients are \(a = 0.018\), \(b = -0.392\), \(c = 4.18\). The quadratic model is \(y = 0.018x^2 - 0.392x + 4.18\).

Step 4: Derive the Equation

Take the derivative of the quadratic model \(y = 0.018x^2 - 0.392x + 4.18\). Using basic differentiation rules, the derivative is \(y' = 2(0.018)x - 0.392 = 0.036x - 0.392\).

Step 5: Evaluate the Derivative at Age 18

Substitute \(x = 18\) in the derivative equation \(y' = 0.036x - 0.392\). This gives \(y' = 0.036(18) - 0.392 = 0.648 - 0.392 = 0.256\).

Step 6: Interpret the Rate of Change

The rate of change of percentage of female licensed drivers at age 18 is 0.256. This means as age increases by 1 year from 18, the percentage of female drivers increases by approximately 0.256%.

Step 7: Calculate Percentage Rate of Change

Use the formula \(\frac{dy}{dx} / y(18)\) to find the percentage rate of change. For \(y(18)\), substitute \(x=18\) in \(y = 0.018x^2 - 0.392x + 4.18\) to get \(y(18) = 1.4\). Then, percentage rate is \(\frac{0.256}{1.4} \times 100 \approx 18.29%\).

Step 8: Interpret the Percentage Rate of Change

The percentage rate of change signifies that from age 18, the rate at which the percentage of licensed female drivers changes is approximately 18.29% per year.

Key Concepts

Differentiation

Differentiation is a mathematical process used to find the rate at which a quantity changes. In the context of our quadratic model, it helps us determine how the percentage of female drivers changes with age. By taking the derivative of a function, you can obtain insights into how small changes in the input (age, in this case) affect the outcome (percentage of female drivers).

To differentiate the quadratic function \( y = 0.018x^2 - 0.392x + 4.18 \), we apply the basic rules of differentiation. The power rule, one of the simplest rules, says that the derivative of \( ax^n \) is \( nax^{n-1} \).

• The derivative of \( 0.018x^2 \) is \( 2 \times 0.018x^{2-1} = 0.036x \).
• The derivative of \( -0.392x \) is simply \( -0.392 \), as the derivative of \( x \) is 1.
• The derivative of the constant \( 4.18 \) is 0, as constants have no rate of change.

Thus, the derivative of our quadratic function is \( y' = 0.036x - 0.392 \). This derivative function can now be used to assess changes for specific ages.

Rate of Change

The rate of change in a function describes how a dependent variable, like the percentage of female drivers, changes concerning an independent variable, such as age. In our example, we used differentiation to find out precisely how the percentage of female drivers increases or decreases per year at a specific age.

By substituting \( x = 18 \) into the derived formula \( y' = 0.036x - 0.392 \), we calculated the rate of change at age 18. The outcome \( y' = 0.256 \) indicates that for each additional year after age 18, the percentage of female drivers increases by approximately 0.256%.

This information is valuable as it tells us not just the direction of change—whether an increase or a decrease—but the magnitude too. Understanding the rate of change gives valuable insight into trends and potential forecasts in data analysis.

Least Squares Method

The Least Squares Method is a statistical technique used to find the best-fitting curve or line through a set of data points. It minimizes the sum of the squares of the differences between the observed and predicted values, ensuring the calculated curve is as close to the data as possible.

In this exercise, the least squares method was employed to derive the quadratic model \( y = 0.018x^2 - 0.392x + 4.18 \). This model represents the increase in the percentage of licensed female drivers as they age. The coefficients \( a = 0.018 \), \( b = -0.392 \), and \( c = 4.18 \) were calculated to minimize the residuals between the predicted and actual data points.

• "Residuals" are the differences between observed and estimated values.
• Minimizing these residuals ensures the model is optimal for representing the data.

By applying the least squares method, one achieves a mathematically sound and visually reliable approximation of data trends, crucial for accurate predictions and analyses.