State whether each function is increasing, decreasing, or neither.


c. Height above Earth’s sea level as a function of atmospheric pressure (assumed nonzero)

Verified step by step guidance

1. Understand the relationship between height above sea level and atmospheric pressure: As you ascend in altitude, atmospheric pressure generally decreases. This is because there is less air above you exerting pressure.
2. Consider the function: Height above sea level as a function of atmospheric pressure. We need to determine if this function is increasing, decreasing, or neither.
3. Analyze the behavior of the function: As atmospheric pressure decreases, height above sea level increases. This suggests a negative correlation between the two variables.
4. Determine the nature of the function: Since height increases as atmospheric pressure decreases, the function is decreasing with respect to atmospheric pressure.
5. Conclude the analysis: The function is decreasing because an increase in height corresponds to a decrease in atmospheric pressure.

Key Concepts

Monotonic Functions

A function is considered monotonic if it is either entirely non-increasing or non-decreasing over its domain. This concept is crucial for determining whether a function is increasing, decreasing, or neither. An increasing function has a positive slope, while a decreasing function has a negative slope. Understanding monotonicity helps in analyzing the behavior of functions in relation to their inputs.

Derivative and Its Sign

The derivative of a function provides information about its rate of change. If the derivative is positive over an interval, the function is increasing; if negative, it is decreasing. For the given function, analyzing the derivative with respect to atmospheric pressure will reveal how height changes as pressure varies. This relationship is fundamental in calculus for understanding function behavior.

Inverse Relationships

In some contexts, functions can exhibit inverse relationships, where an increase in one variable leads to a decrease in another. In this case, as atmospheric pressure decreases, height above sea level increases, indicating a negative correlation. Recognizing such relationships is essential for interpreting the behavior of functions in real-world scenarios, particularly in physics and environmental science.