Problem 36 - Applied Calculus

Question

Table \(1.20\) gives the sales, \(S\), of Intel Corporation, a leading manufacturer of integrated circuits. \({ }^{41}\) (a) Find the change in sales between 2003 and 2008 . (b) Find the average rate of change in sales between 2003 and 2008. Give units and interpret your answer. $$ \begin{array}{l} \text { Table } 1.20 \text { Intel sales, in millions of dollars }\\\ \begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 \\ \hline S & 30,100 & 34,200 & 38,800 & 35,400 & 38,300 & 37,600 \\ \hline \end{array} \end{array} $$

Step-by-Step Solution

Answer

Change in sales: 7,500 million dollars; Average rate: 1,500 million dollars/year.

Step 1: Identify Sales Figures for Given Years

From the table, note the sales value for 2003 as \( S_{2003} = 30,100 \) million dollars and for 2008 as \( S_{2008} = 37,600 \) million dollars.

Step 2: Calculate Change in Sales

Subtract the sales in 2003 from the sales in 2008 to find the change in sales: \( \Delta S = S_{2008} - S_{2003} = 37,600 - 30,100 = 7,500 \) million dollars.

Step 3: Determine Time Interval

The time period from 2003 to 2008 is \( 2008 - 2003 = 5 \) years.

Step 4: Compute Average Rate of Change

Use the formula for average rate of change: \( \text{Average Rate of Change} = \frac{\Delta S}{\Delta t} = \frac{7,500}{5} = 1,500 \) million dollars per year.

Step 5: Interpret the Average Rate of Change

The average rate of change of sales between 2003 and 2008 is \( 1,500 \) million dollars per year, meaning that on average, sales increased by \( 1,500 \) million dollars each year over this period.

Key Concepts

Sales Data Analysis

Sales data analysis is a critical aspect of understanding business performance and strategizing future growth. In this context, sales data refers to the revenue figures over a set period, providing a basis for analyzing trends and patterns. For example, considering the sales data for Intel Corporation from 2003 to 2008, as shown in the exercise, allows for insights into how the company's sales evolved over these years.

Analyzing sales data involves several key steps:

• **Identifying Sales Metrics:** Start by pinpointing specific sales figures at crucial time points, such as annually in this case.
• **Comparison Over Time:** Note changes between select years to determine overall patterns or fluctuations.
• **Highlighting Anomalies:** Look for unusual spikes or drops, which might indicate external factors affecting sales.

The sales data extracted from 2003 and 2008 from the table reveals critical numbers that help us calculate additional factors, such as the rate of change. Such data guides strategic decisions by showing whether sales are consistently growing, stable, or declining.

Change in Sales

The change in sales quantifies how much the revenue has increased or decreased within a specific period. This knowledge can be pivotal in decision-making processes for businesses. The change in sales from 2003 to 2008 for Intel Corporation can be calculated by subtracting the sales figure in 2003 from the sales figure in 2008.

In this exercise:

• **2003 Sales:** \(S_{2003} = 30,100\) million dollars
• **2008 Sales:** \(S_{2008} = 37,600\) million dollars
• **Change in Sales:** \(\Delta S = S_{2008} - S_{2003} = 7,500\) million dollars

This change is positive, indicating an increase in sales over the given period. Understanding such changes helps in evaluating past strategies and planning future actions. It offers a simple yet effective measure of growth and market dynamics, which can reflect a company's competitive positioning in its industry.

Interpretation of Data in Calculus

In calculus, interpreting data is about understanding both the rate of changes and overall trends within datasets. Calculus concepts, like the average rate of change, help quantify how rapidly or slowly a variable, such as sales, is changing over time.

For the sales data from Intel, the average rate of change helps condense overall trends into a single, comprehensible figure. This is calculated as follows:

• **Average Rate of Change Formula:** \(\text{Average Rate of Change} = \frac{\Delta S}{\Delta t}\)
• Where \(\Delta S = 7,500\) million dollars and \(\Delta t = 5\) years
• So, the average rate of change is \(\frac{7,500}{5} = 1,500\) million dollars per year

This reveals that, on average, Intel's sales increased by \(1,500\) million dollars each year from 2003 to 2008. Such interpretations are crucial for businesses to gauge overall performance effectively. Calculus thereby provides a lens to view how changes accumulate or evolve over time, supporting strategic data-driven decisions.