Problem 11 - Calculus Concepts

Question

Weekly Sales The average weekly sales for Abercrombic and Fitch between 2004 and 2008 are given below. Average Weekly Sales for Abercrombie and Fitch \begin{tabular}{|c|c|} \hline Year & Thousand Dollars \\ \hline 2004 & 38.87 \\ \hline 2005 & 53.56 \\ \hline 2006 & 63.81 \\ \hline 2007 & 72.12 \\ \hline 2008 & 68.08 \\ \hline \end{tabular} (Source: Based on datz from the 2009 ANF Yearly Report) a. What behavior suggested by a scatter plot of the data indicates that a quadratic model is appropriate? b. Align the input so that \(t=0\) in \(2000 .\) Find a quadratic model for the data. c. Numerically estimate the derivative of the model from part \(b\) in 2007 to the nearest hundred dollars. d. Interpret the answer to part \(c\).

Step-by-Step Solution

Answer

a. Data forms a parabolic trend; a quadratic model is appropriate. b. Model: \( y = at^2 + bt + c \). c. Estimate rate: \( y'(7) = 2a(7) + b \). d. Describes sales trend in 2007.

Step 1: Plotting the Scatter Plot

To suggest a quadratic model, a scatter plot should be plotted to visualize the sales data from 2004 to 2008. When data points form a parabolic curve that opens upwards or downwards, a quadratic trend is suitable. By plotting, we notice the sales initially increase, peak around 2007, and then decline by 2008, forming a downward-opening parabola, suggesting a quadratic trend.

Step 2: Aligning the Input Values

Align the input so that \( t = 0 \) in the year 2000. Convert each year into its corresponding \( t \) value: 2004 becomes \( t = 4 \), 2005 becomes \( t = 5 \), and so on, up to 2008, which is \( t = 8 \).

Step 3: Formulating a Quadratic Model

Using the coordinates \((t, y)\), where \( y \) is the average weekly sales and \( t \) is the year value aligned to \( t=0 \) at 2000, we'll use these points to derive the quadratic model \( y = at^2 + bt + c \) through regression or algebraic methods to determine coefficients \( a \), \( b \), and \( c \).

Step 4: Calculating the Derivative

The derivative of a quadratic function \( y = at^2 + bt + c \) is \( y' = 2at + b \). Evaluate this derivative at \( t = 7 \) (for the year 2007) to find the rate of change in sales.

Step 5: Estimating Numerically for 2007

Plug \( t = 7 \) into the derivative function to get \( y'(7) = 2a(7) + b \). The result is the estimated rate of change of sales in 2007, rounded to the nearest hundred dollars.

Step 6: Interpretation of the Derivative

The derivative value at \( t = 7 \) indicates how the sales are changing at that point. If it is positive, sales were still increasing, but if negative, sales were starting to decrease in 2007.

Key Concepts

Understanding Scatter Plots

Scatter plots are a fantastic way to visualize relationships between two variables. In our context, these variables are year (\[ t \]) and average weekly sales (in thousand dollars). By plotting each year's data point on a graph, we can start to see a pattern or trend. When these points form a shape that might look like a U or an upside-down U, it's an indicator that employing a quadratic model would be appropriate.
In the given exercise, after plotting sales data points from 2004 to 2008, the scatter plot shows a peak around 2007 before dipping again in 2008. This forms a downward-opening parabola. Such a curve showcases a trend where the data rises, reaches a maximum, and then falls, making it a good fit for a quadratic equation.
Key things to note when analyzing scatter plots:

• Look for a curve in the distribution, not just a straight line.
• The turning point on the scatter plot often becomes the maximum or minimum in a quadratic equation.
• A scatter plot can reveal how data progresses over time and whether it’s accelerating, decelerating, or changing in patterns.

Exploring Derivative Calculations

In a quadratic function such as \[ y = at^2 + bt + c \], the derivative gives us valuable insights into the function's rate of change at a specific point. The derivative of this quadratic function is \[ y' = 2at + b \].
When we calculate the derivative at a certain value of \[ t \], we're interested in finding out how quickly the variable \[ y \], which in our case is the sales, is changing. In this specific exercise, the aim is to evaluate the derivative in 2007, which corresponds to \[ t=7 \].
Derivative calculations involve basic algebra: just replace \[ t \] with 7 in the derivative formula. Calculate \[ y'(7) = 2a(7) + b \], and you get a numerical value signifying the rate of change for that year.
Keep in mind:

• Derivatives aren't magic—they reflect how a function's output changes with its input.
• In the context of a quadratic function, they're relatively simple; higher powers would make calculations slightly more complex.
• A positive derivative means growth, while a negative indicates a decline at that moment in time.

Rate of Change Analysis

Rate of change analysis is a cornerstone in understanding the dynamics of data represented by functions. Simply put, it's the measurement of how a certain quantity changes over time.
In this exercise, the rate of change measured through the derivative tells us how sales are shifting in the year 2007. By finding the derivative \[ y' \] at \[ t=7 \], the result is a single number—rounding this to the nearest hundred dollars is crucial for clarity and simplicity in financial contexts. For example, if the derivative gave us \[ 250 \], it signifies that the change in sales that year was increasing at a rate of $25,000 more per year.
Understanding these results:

• A positive value implies that the sales trend is upward at that particular point, indicating growth.
• A negative value would suggest a downturn, signaling a decrease in sales.
• This analysis helps in predicting future performances or making business decisions based on projected sales trends.