Problem 25 - Applied Calculus

Question

Table 1.20 gives the revenues, \(R\), of General Motors, for12 merly the world's largest auto manufacturer. (a) Find the change in revenues between 2003 and 2008 . (b) Find the average rate of change in revenues between 2003 and \(2008 .\) Give units and interpret your answer. (c) From 2003 to \(2008,\) were there any one-year intervals during which the average rate of change was negative? If so, which? $$\begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 \\ \hline R & 184.0 & 192.9 & 193.1 & 205.6 & 181.1 & 149.0 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

(a) -35.0 billion dollars; (b) -7.0 billion dollars per year; (c) 2006-2007 and 2007-2008.

Step 1: Calculate Change in Revenues

To find the change in revenues between 2003 and 2008, we will subtract the revenue in 2003 from the revenue in 2008. Given revenues: - 2003: \( R_{2003} = 184.0 \) billion dollars - 2008: \( R_{2008} = 149.0 \) billion dollars The change in revenues is calculated as follows: \[ R_{2008} - R_{2003} = 149.0 - 184.0 = -35.0 \text{ billion dollars} \] So, the revenues decreased by 35.0 billion dollars from 2003 to 2008.

Step 2: Calculate Average Rate of Change

The average rate of change in revenues is calculated by taking the change in revenues over the interval and dividing by the number of years. Using the previous result for the change in revenues: \[\text{Average Rate of Change} = \frac{R_{2008} - R_{2003}}{2008 - 2003} = \frac{-35.0}{5} = -7.0 \text{ billion dollars per year}\] This means that, on average, revenues decreased by 7.0 billion dollars per year from 2003 to 2008.

Step 3: Identify Negative One-Year Intervals

We now examine the revenue changes for each year interval to determine if any experienced a negative rate of change: - 2003 to 2004: \(192.9 - 184.0 = 8.9 \) (positive)- 2004 to 2005: \(193.1 - 192.9 = 0.2 \) (positive)- 2005 to 2006: \(205.6 - 193.1 = 12.5 \) (positive)- 2006 to 2007: \(181.1 - 205.6 = -24.5 \) (negative)- 2007 to 2008: \(149.0 - 181.1 = -32.1 \) (negative)Therefore, the intervals with negative changes are from 2006 to 2007 and from 2007 to 2008.

Key Concepts

Average Rate of Change

The average rate of change is a key concept in understanding how something varies over a period. In essence, it tells us how much one quantity changes, on average, over the change in another. In this specific exercise, we're looking at revenue, so it's the change in revenue over time.
The formula for average rate of change is straightforward:

• \[\text{Average Rate of Change} = \frac{\text{Change in Revenue}}{\text{Change in Time}}\]

Here, the change in revenue between 2003 and 2008 is calculated as the revenue in 2008 minus the revenue in 2003. This results in a decrease of \(35 billion. The time period spans 5 years, so the average rate of change is obtained by dividing \(-35 \text{ billion dollars} \) by 5. Hence, it results in \(-7 \text{ billion dollars per year}\).
This negative value indicates a decrease, meaning on average, the revenue fell by \)7 billion each year over that timeframe.

Revenue Analysis

Analyzing revenue helps in gaining insights about a company's financial health over a given period. In this exercise, we explore how General Motors' revenue changed from 2003 to 2008.
Revenue analysis involves looking at both total and average changes over time. By inspecting the data given:

• Revenues in billions of dollars were: 184.0 (2003), 192.9 (2004), 193.1 (2005), 205.6 (2006), 181.1 (2007), 149.0 (2008).

From these figures, we observe overall trends and make specific calculations:

• The total change in revenue from 2003 to 2008 was a decrease of $35 billion.
• The average rate of change, as calculated earlier, was \(-7 \text{ billion dollars per year}\).

These calculations help provide a bird's-eye view of how economic factors or company strategies might have influenced financial outcomes during these years.

Negative Growth Intervals

Negative growth intervals indicate periods where a company or quantity experienced a decline. They're important for identifying phases of downturn, helping companies develop strategies to mitigate future risks.
In the case of General Motors from 2003 to 2008, key negative intervals are:

• 2006 to 2007: Revenue fell from 205.6 to 181.1 billion dollars, a significant dip of 24.5 billion dollars.
• 2007 to 2008: Revenue further decreased from 181.1 to 149.0 billion dollars, marking a drop of 32.1 billion dollars.

These intervals show where the company encountered the most significant drops. Analyzing these declines provides critical insights into what might have contributed to these downturns, such as economic recessions, market competition, or shifts in consumer demand.