Problem 20 - Applied Calculus

Question

Do you expect the average rate of change (in units per year) of each of the following to be positive or negative? Explain your reasoning. (a) Number of acres of rain forest in the world. (b) Population of the world. (c) Number of polio cases each year in the US, since 1950 (d) Height of a sand dune that is being eroded. (e) Cost of living in the US.

Step-by-Step Solution

Answer

(a) Negative, (b) Positive, (c) Negative, (d) Negative, (e) Positive.

Step 1: Understanding Average Rate of Change

The average rate of change of a quantity over a period is the change in the quantity divided by the time over which the change occurs. If a quantity increases over time, its average rate of change is positive. If it decreases, the average rate of change is negative.

Step 2: Analyze the Rain Forest Acreage

The number of acres of rain forest in the world is generally expected to decrease due to deforestation and land conversion for agriculture and urban development. Hence, the average rate of change is expected to be negative.

Step 3: Consider Population Growth

The population of the world is observed to increase over the years due to higher birth rates compared to death rates in many regions. Thus, the average rate of change is expected to be positive.

Step 4: Examine Polio Cases in the US

Since 1950, there have been major efforts to eradicate polio, especially with the introduction of vaccines. As a result, the number of polio cases in the US has declined significantly, making the average rate of change negative.

Step 5: Evaluate Sand Dune Erosion

A sand dune that is being eroded is expected to decrease in height over time due to natural processes like wind and water erosion. Thus, the average rate of change in the height of the dune is negative.

Step 6: Assess Cost of Living

Over time, the cost of living in the US tends to increase due to factors like inflation. Therefore, the average rate of change is expected to be positive.

Key Concepts

Deforestation

Deforestation is the process of clearing forests for non-forest uses, such as agriculture, urban development, or logging. This has a significant impact on the environment as it leads to a reduction in biodiversity, disruption of water cycles, and increased greenhouse gases due to the loss of trees that absorb carbon dioxide.
Given our understanding of average rate of change, deforestation typically results in a negative average rate of change for rain forest acreage. The world sees a decline in forested areas each year as trees are cut down faster than they can regrow.
As forests shrink, they contribute less to crucial environmental processes, placing immense pressure on ecosystems and human life.

• Reasons for deforestation include logging, agricultural expansion, urbanization, and industrial purposes.
• Consequences may involve loss of habitat for flora and fauna, increased carbon emissions, and erosion.

Population Growth

Population growth refers to the increase in the number of people in a particular area. It is generally influenced by factors such as birth rates, death rates, immigration, and emigration.
When we calculate the average rate of change, the global population shows a positive trend. Numerous regions experience more births than deaths, accounting for an overall increase in the world population. This sustained growth raises various socio-economic and environmental challenges.
Population growth directly affects resource consumption, economic development, and environmental sustainability.

• High birth rates and decreased mortality rates due to medical advances lead to population growth.
• Implications include increased demand for resources, strain on infrastructure, and environmental degradation.

Disease Eradication

Disease eradication aims to completely eliminate a disease from a population, such as the efforts to eradicate polio globally. Since 1950, the introduction of vaccines and public health initiatives has drastically reduced polio cases in the US, reflecting a negative average rate of change.
This negative trend signifies success in public health strategies and enhances overall societal health.

• Eradication involves significant health campaigns, vaccination programs, and worldwide cooperation.
• Successes include the reduction of preventable diseases, improved public health outcomes, and decreased healthcare costs.

Environmental Erosion

Environmental erosion refers to the gradual wear and reduction of landscapes due to natural processes such as wind and water. Sand dune erosion is a specific example where the height and size of dunes are reduced over time.
The average rate of change for a sand dune facing erosion is negative, as dunes lose more material than they gain. This can alter habitats and landscapes, affecting local ecosystems and biodiversity.

• Erosion can be caused by wind, water, or human activities such as construction or agriculture.
• Consequences include habitat loss, reduced agricultural productivity, and sedimentation in waterways.

Inflation

Inflation is the process whereby the general level of prices for goods and services rises, eroding purchasing power. When considering average rate of change, inflation often results in a positive trend for the cost of living.
In the US, inflation means that over time, the cost of everyday items and services generally increases. This can influence economic decisions at both individual and governmental levels.

• Inflation is driven by factors such as increased production costs, demand-pull inflation, and monetary supply growth.
• Effects include changes in purchasing power, wage adjustments, and interest rates.