Problem 21 - Applied Calculus

Question

Table 1.18 shows the production of tobacco in the US. \(^{35}\) (a) What is the average rate of change in tobacco production between 2003 and \(2010 ?\) Give units and interpret your answer in terms of tobacco production. (b) During this seven-year period, is there any interval during which the average rate of change was positive? If so, when?$$\begin{array}{c|c|c|c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 \\\ \hline \text { Production } & 802 & 882 & 645 & 728 & 788 & 801 & 823 & 719 \\\ \hline \end{array}$$

Step-by-Step Solution

Answer

(a) The average rate of change is -11.86 units per year. (b) Positive change occurred in 2003-2004, 2005-2006, 2006-2007, 2007-2008, and 2008-2009.

Step 1: Identify Relevant Data

From the given table, the production of tobacco in 2003 was 802 units and in 2010 was 719 units. This information will be used to calculate the average rate of change.

Step 2: Calculate Average Rate of Change

The average rate of change is calculated using the formula \( \frac{f(b) - f(a)}{b - a} \), where \( f(b) \) and \( f(a) \) are the production values at years \( b \) and \( a \), respectively. Here, \( a = 2003 \), \( f(a) = 802 \), \( b = 2010 \), and \( f(b) = 719 \). Thus, the average rate of change is \( \frac{719 - 802}{2010 - 2003} = \frac{-83}{7} \approx -11.86 \), with units in tobacco production per year.

Step 3: Interpretation of Average Rate of Change

The calculated average rate of change of approximately \(-11.86\) units per year indicates that, on average, tobacco production decreased by about 11.86 units each year from 2003 to 2010.

Step 4: Identify Positive Rate Intervals

To find intervals where the rate of change is positive, examine the production changes between each consecutive year. Calculate differences: 2004-2003: 80 (positive), 2006-2005: 83 (positive), 2007-2006: 60 (positive), 2008-2007: 13 (positive), 2009-2008: 22 (positive). The intervals with positive changes are 2003-2004, 2005-2006, 2006-2007, 2007-2008, and 2008-2009.

Key Concepts

Tobacco Production

Tobacco production is a vital segment of agriculture in the United States. Understanding its trends and variations can provide insight into larger agricultural and economic patterns. Every year, tobacco leaves are harvested and quantified in units, as showcased in the table provided in the exercise. These figures allow us to trace how production levels change over time, affecting not just the farmers, but also industries reliant on tobacco products. This information is crucial for policymakers, agronomists, and economists who seek to forecast future production and make informed decisions related to agricultural planning and management.

Positive Rate Intervals

A positive rate interval means that within a specific time frame, the quantity or value - in this case, tobacco production - has increased. From the data given in the table, it's clear that not every year saw an increase in production. However, there are certain periods where production rose:

• 2003 to 2004
• 2005 to 2006
• 2006 to 2007
• 2007 to 2008
• 2008 to 2009

These intervals are significant as they could indicate factors that positively influenced production, such as favorable weather conditions, improved farming techniques, or increased demand. By identifying these intervals, students and analysts can focus on understanding the reasons behind these increases, better preparing them to identify similar trends or make predictions for future years.

Applied Calculus

Applied Calculus is essential when analyzing data such as tobacco production. It provides the tools to calculate important figures like average rate of change. This specific calculation shows us how production levels have evolved over a period of time. We use the formula \[ \frac{f(b) - f(a)}{b - a} \] to compute this measure, where \(f(b)\) and \(f(a)\) represent the quantities at years \(b\) and \(a\), respectively.
Applying this to the exercise, we computed an average rate of change of approximately \(-11.86\) units per year, signifying a decrease in tobacco production from 2003 to 2010. This datum is insightful as it helps in understanding whether factors over these years have led to declining production or if the trend is simply part of a longer-term cycle.

Year-on-Year Analysis

Year-on-year analysis involves comparing data from one year to the next. This type of analysis is critical for understanding short-term trends and fluctuations in tobacco production. By evaluating each year's production against the preceding year's, it's possible to detect any abrupt changes or consistent patterns. This method was used in the exercise to discover positive intervals, where certain years showed an uptick in production from the previous year.
Such analysis adds depth to our understanding, revealing how factors like policy changes or environmental conditions might influence tobacco production annually. It highlights the importance of ongoing monitoring and data analysis for recognizing trends or irregularities, allowing for better strategic decision making in industries related to tobacco and agriculture.