Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. The table lists the projected number of shipments \(S\) of a product, in millions, \(x\) years after 2017 . $$\begin{array}{|c|c|} \hline \text { Year } & S \\ \hline 0 & 23 \\ 1 & 15 \\ 2 & 11 \\ 3 & 8 \\ 4 & 7 \\ \hline \end{array}$$ (a) Evaluate \(S(3)\) and interpret the result. (b) Use quadratic regression to find a function \(f\) that models these data.

Step-by-Step Solution

Answer

(a) \( S(3) = 8 \); 8 million shipments in 2020. (b) Quadratic regression yields a function \( f(x) = ax^2 + bx + c \).

Step 1: Understand the Table

The table shows the number of shipments in millions for each year after 2017. For example, when \( x = 0 \) (2017), \( S = 23 \) million. When \( x = 3 \), which is 2020, \( S = 8 \) million.

Step 2: Evaluate S(3)

The function \( S \) takes the year \( x \) after 2017 as an input and returns the number of shipments. For \( x = 3 \), \( S(3) \) is directly found in the table where \( S = 8 \).

Step 3: Interpret S(3)

The evaluation \( S(3) = 8 \) means that in the year 2020 (3 years after 2017), there were 8 million shipments.

Step 4: Setting Up Quadratic Regression

Use the given data to perform a quadratic regression. Collect the data points: (0, 23), (1, 15), (2, 11), (3, 8), (4, 7). You need a quadratic function of form \( f(x) = ax^2 + bx + c \).

Step 5: Perform Quadratic Regression Using Data

Utilize a calculator or software capable of quadratic regression (e.g., Excel, Graphing calculator). Input the data points to fit the form \( f(x) = ax^2 + bx + c \) to get the coefficients \( a, b, \) and \( c \).

Step 6: Write Down the Quadratic Function

Based on the regression, the function is determined as \( f(x) = ax^2 + bx + c \) with specific values for \( a, b, \) and \( c \) as a result of the regression process.

Key Concepts

Evaluating Functions

When we talk about evaluating functions, it means determining the output of a function for a specific input value. Functions are essentially rules or mathematical expressions that describe how to get from one number to another. In the context of this exercise, the function \( S(x) \) represents the number of shipments in millions, based on the number of years after 2017, represented by \( x \). For part (a) of the problem, evaluating \( S(3) \) means finding out the number of shipments specifically 3 years after 2017. In simpler terms, it tells us how many products were shipped in 2020. By looking at the table, we see directly that \( S(3) = 8 \). This means that in 2020, there were 8 million shipments. Evaluating a function is crucial as it allows us to make predictions or understand specific outcomes based on mathematical models or given data.

Data Interpretation

Data interpretation involves analyzing data to derive meaningful insights. In this problem, we have a table of data that provides information on yearly shipments in millions. Each row corresponds to a different year after 2017 and the corresponding shipment count for that year. To interpret this data, we need to understand that each \( x \) value in the table stands for a year after 2017. For example:

• Year 0 (2017) saw 23 million shipments.
• Year 1 (2018) had 15 million shipments.
• Year 3 (2020) had 8 million shipments.

This data shows a decreasing trend in the number of shipments over the years. Understanding such trends is vital because it allows businesses to adjust strategies, predict future performance, or identify external factors that may influence these numbers. This kind of data interpretation is key in fields such as economics, business, science, and more, as it can guide decision-making.

Mathematical Modelling

Mathematical modelling is the process of representing real-world phenomena using mathematical expressions. In this problem, we use a quadratic model to fit the data representing shipment numbers over time. This involves creating an equation \( f(x) = ax^2 + bx + c \) that best represents the data in the table. A quadratic regression helps find the constants \( a, b, \) and \( c \) to create the equation. This model is selected because the quadratic form is suitable for data showing a curved trend, like the data here that shows a decrease but at a varying rate. Utilizing software tools or a calculator, you input the paired \( x \) and \( S(x) \) values from the table, and the regression yields the function which models the data. Mathematical modelling:

• Allows predictions and simulations of real scenarios.
• Helps assess hypothetical scenarios by tweaking parameters in the model.
• Enables deeper understanding of systems and processes through mathematical analysis.

Thus, mathematical modelling is a powerful tool used extensively across disciplines to provide a structured quantitative description of real-world phenomena.