Problem 5 - Applied Calculus

Question

Table \(6.3\) shows the cumulative number of AIDS deaths worldwide. \({ }^{6}\) Find the absolute increase in AIDS deaths between 2003 and 2004 and between 2006 and 2007 . Find the relative increase between 2003 and 2004 and between 2006 and 2007 . $$ \begin{array}{c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 \\ \hline \text { Cases } & 30.2 & 33.3 & 35.5 & 37.6 & 39.6 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The absolute increase was 3.1 million (2003-2004) and 2.0 million (2006-2007). Relative increase was approximately 10.26% (2003-2004) and 5.32% (2006-2007).

Step 1: Identify Cases in Each Year

From the table given, identify the number of AIDS deaths (in millions) for the years mentioned. For 2003, the number is 30.2 million, for 2004 it's 33.3 million, for 2006 it's 37.6 million, and for 2007 it's 39.6 million.

Step 2: Calculate Absolute Increase (2003 to 2004)

The absolute increase between two years is found by subtracting the value of the first year from the value of the second year. For 2003 to 2004, this is: \[ 33.3 - 30.2 = 3.1 \text{ million} \]

Step 3: Calculate Absolute Increase (2006 to 2007)

Similarly, for 2006 to 2007, the absolute increase is:\[ 39.6 - 37.6 = 2.0 \text{ million} \]

Step 4: Calculate Relative Increase (2003 to 2004)

The relative increase can be calculated using the formula: \[ \text{Relative Increase} = \left( \frac{\text{Absolute Increase}}{\text{Value of First Year}} \right) \times 100\% \]For 2003 to 2004:\[ \left( \frac{3.1}{30.2} \right) \times 100\% \approx 10.26\% \]

Step 5: Calculate Relative Increase (2006 to 2007)

Using the same formula as in the previous step, for 2006 to 2007, we have:\[ \left( \frac{2.0}{37.6} \right) \times 100\% \approx 5.32\% \]

Key Concepts

Absolute Increase

The concept of absolute increase is a fundamental part of analyzing changes over time in calculus and statistics. It tells us how much a quantity has grown or declined in simple numerical terms without considering the initial amount or percentage changes. Here, we focus on the cumulative AIDS deaths worldwide.

• To find the absolute increase, subtract the earlier year's figure from the later year's figure.
• In the dataset, the number of deaths grew from 30.2 million in 2003 to 33.3 million in 2004. The absolute increase is therefore calculated as: \[ 33.3 - 30.2 = 3.1 \text{ million cases} \]
• Similarly, between 2006 and 2007, the increase was: \[ 39.6 - 37.6 = 2.0 \text{ million cases} \]

This notion provides a straightforward numerical value to show growth over time, regardless of how small or large the base number is.

Relative Increase

While absolute increase gives us the exact number, relative increase offers perspective by showing how much a value has grown relative to its starting amount. It is typically expressed as a percentage, making it easier to compare across different contexts or datasets.

• To calculate the relative increase, use this formula: \[ \text{Relative Increase} = \left( \frac{\text{Absolute Increase}}{\text{Value of First Year}} \right) \times 100\% \]
• For 2003 to 2004, the relative increase would be: \[ \left( \frac{3.1}{30.2} \right) \times 100\% \approx 10.26\% \]
• Between 2006 and 2007, it becomes: \[ \left( \frac{2.0}{37.6} \right) \times 100\% \approx 5.32\% \]

The relative increase helps us understand the significance of the change by contextualizing it within its initial size; a larger percentage indicates a more notable change.

Cumulative Data Analysis

Cumulative data is a running total that aggregates accumulated statistics over time. It allows us to observe the entire progress or trend of a phenomenon like AIDS deaths, from inception to the most recent time point.

• Cumulative datasets are valuable for long-term trend analysis.
• By tracking the total number of AIDS deaths yearly, we can determine increase rates and predict future outcomes.
• This type of data helps in strategic planning and resource allocation by highlighting significant changes over periods.

Analyzing the cumulative figures simplifies the recognition of patterns, peaks, or declines over extended periods.