Problem 42 - Applied Calculus

Question

The area of Brazil's rain forest, \(R=f(t)\), in million acres, is a function of the number of years, \(t\), since 2000 . (a) Interpret \(f(9)=740\) and \(f^{\prime}(9)=-2.7\) in terms of Brazil's rain forests. \(^{13}\) (b) Find and interpret the relative rate of change of \(f(t)\) when \(t=9\).

Step-by-Step Solution

Answer

(a) In 2009, the rainforest area was 740 million acres and decreasing by 2.7 million acres per year. (b) In 2009, the area was decreasing by 0.365% per year.

Step 1: Understand the function values

First, recognize that the function value at a specific year represents the area of the rainforest. The expression \( f(9) = 740 \) means that in the year 2009 (since \( t=9 \) refers to the year 2009), the area of Brazil's rainforest was 740 million acres.

Step 2: Interpret the derivative

The derivative \( f^{\prime}(9) = -2.7 \) tells us the rate of change of the rainforest area in the year 2009. Specifically, it indicates that the rainforest area was decreasing by 2.7 million acres per year at that time.

Step 3: Calculate the relative rate of change

The relative rate of change can be calculated using the formula \( \frac{f^{\prime}(t)}{f(t)} \). When \( t=9 \), this becomes \( \frac{-2.7}{740} \). Calculate this value as follows: \[ \frac{-2.7}{740} \approx -0.00365. \]

Step 4: Interpret the relative rate of change

The relative rate of change \( -0.00365 \) implies that in 2009, the rainforest area was decreasing at a rate of approximately 0.365% per year relative to its size at that time.

Key Concepts

Derivative Interpretation

In calculus, a derivative provides us with valuable insights into the behavior of a function over time. When we say that \( f^{\prime}(9) = -2.7 \), we are observing the derivative of the rainforest area function, \( f(t) \), when \( t = 9 \). This derivative tells us the rate at which the area of Brazil's rainforest is changing. To put it simply, the rainforest is losing 2.7 million acres per year in 2009. A negative derivative indicates a decrease, so every year, the rainforest becomes smaller by this amount. This insight is essential for understanding not only the extent of the change but also its direction. When analyzing derivatives, pay close attention to:

• The sign of the derivative (negative indicates a decrease, positive indicates an increase).
• The magnitude, which shows how rapidly the change is happening.

Understanding derivatives helps us see the immediate impact on a function, offering a snapshot of what happens over a specific time frame.

Relative Rate of Change

The relative rate of change provides a percentage view of how a function's value changes compared to its current size. For the function \( f(t) \) at \( t = 9 \), we use the relative rate of change formula: \( \frac{f^{\prime}(t)}{f(t)} \). With the given values, this becomes \( \frac{-2.7}{740} \approx -0.00365 \). Essentially, this means that the rainforest area is shrinking by approximately 0.365% per year when compared to its size in 2009. This figure is helpful for gaining perspective on the scale of change. While raw numbers like -2.7 million acres indicate large numerical changes, the relative rate tells us how significant these changes are relative to the total area:

• It shows proportional change, making it easier to grasp the impact.
• This percentage can be more intuitive, giving a broader understanding of the data.

Using relative rates provides an efficient way to compare changes across different functions or time periods.

Function Values

Function values are the backbone of interpreting data using calculus. The expression \( f(9) = 740 \) signifies the state of the Brazil rainforest in 2009, specifically that its area was 740 million acres at that time. Function values anchor our calculations and interpretations. They represent specific points in time where the function takes on a particular value. In the context of this problem, it allows us to:

• Correlate specific years with the corresponding rainforest area.
• Use it as a reference point for calculating and understanding changes described by derivatives or relative rates.

Recognizing function values is crucial as it provides context and baseline data which other interpretations rely upon. Without knowing the value \( f(t) \), calculating or assigning meaning to derivatives and relative rates would be impossible.