91Ó°ÊÓ

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.

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 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.