Question

Earth's atmospheric pressure decreases with altitude from a sea level pressure of 1000 millibars (a unit of pressure used by meteorologists). Letting \(z\) be the height above Earth's surface (sea level) in \(\mathrm{km}\), the atmospheric pressure is modeled by \(p(z)=1000 e^{-z / 10}.\) a. Compute the pressure at the summit of Mt. Everest which has an elevation of roughly \(10 \mathrm{km}\). Compare the pressure on Mt. Everest to the pressure at sea level. b. Compute the average change in pressure in the first \(5 \mathrm{km}\) above Earth's surface. c. Compute the rate of change of the pressure at an elevation of \(5 \mathrm{km}\). d. Does \(p^{\prime}(z)\) increase or decrease with \(z\) ? Explain. e. What is the meaning of \(\lim _{z \rightarrow \infty} p(z)=0 ?\)

Short answer

Answer: The limit \(\lim_{z \rightarrow \infty} p(z)\) represents that as we approach infinite altitude, the atmospheric pressure approaches zero. This means that the higher we go above Earth's surface, the closer the pressure gets to zero, indicating virtually no atmospheric pressure at extremely high altitudes.

Step-by-step solution

Compute Pressure at the Summit of Mt. Everest

To find the pressure at the summit of Mt. Everest, plug in the elevation (z=10 km) into the given pressure function: \(p(10)=1000 e^{-10 / 10}=1000 e^{-1}.\) Then compare it with the pressure at sea level, which is 1000 millibars.

Compute the Average Change in Pressure in the First 5 km

To find the average change in pressure in the first 5 km above sea level, compute the difference in pressure between 0 and 5 km, then divide it by the range (5): Average change in pressure \(= \frac{p(5)-p(0)}{5 - 0} = \frac{1000 e^{-5/10} - 1000}{5}\).

Compute the Rate of Change of Pressure at an Elevation of 5 km

First, find the first derivative of the pressure function \(p(z)\): \(p'(z) = \frac{d}{dz}(1000 e^{-z/10}) = -100 e^{-z/10}\). Now, plug in the elevation z=5 km to find the rate of change of pressure at that height: \(p'(5) = -100 e^{-5/10}\).

Determine if \(p'(z)\) Increases or Decreases with z

To see if the rate of change of pressure increases or decreases with respect to z, examine the first derivative \(p'(z) = -100 e^{-z/10}\). The exponential term is always positive, so the entire expression is always negative. Since there are no other terms in the expression, and the exponential term itself is decreasing as z increases, the rate of change of pressure is always decreasing with respect to z.

Analyze the Limit \(\lim_{z \rightarrow \infty} p(z)\)

We're asked to find the meaning of \(\lim_{z \rightarrow \infty} p(z)=0\). As the altitude z increases indefinitely, the atmospheric pressure \(p(z)\) approaches zero. This means that the higher we go above Earth's surface, the closer the atmospheric pressure gets to zero. In other words, as we approach infinite altitude, there is virtually no atmospheric pressure.

Key concepts

Exponential Decay

The concept of exponential decay is a critical mathematical model that applies to a variety of real-world phenomena, including atmospheric pressure, radioactive decay, and population decline. It describes a process by which a quantity decreases at a rate proportional to its current value.

In our atmospheric pressure problem, the model for pressure with respect to altitude is described by the function p(z)=1000 e^{-z / 10}. This function is an exemplary case of exponential decay. As altitude increases, the pressure decreases sharply at first and then more gradually, never reaching zero but getting infinitely close to it. The base of the exponential, e, is the mathematical constant approximately equal to 2.71828, often used in natural growth and decay models. The negative sign in the exponent indicates that it is, in fact, a decay rather than growth.

Educators should emphasize that exponential functions are distinctive in that they decrease (or increase if positive) by a given proportion over equal increments of time or distance. This is why such functions are perfect to depict processes like atmospheric pressure changes with altitude—each kilometer up results in the pressure decreasing by a consistent factor, governed by the nature of the exponential function.

Rate of Change

The rate of change in calculus is a fundamental concept that represents how a certain quantity changes with respect to another variable. In physical terms, it tells us how fast or slow something is changing at any point. For the atmospheric pressure model, we express the rate of change of pressure with the altitude z using p'(z), which is derived from the original function p(z).

When we compute p'(5) = -100 e^{-5/10}, we are finding the instantaneous rate of change of atmospheric pressure at 5 km above Earth's surface. This expresses how quickly the pressure decreases per kilometer at that specific height. It's crucial for students to understand that this value changes with the altitude due to its exponential nature; it's not a constant rate. This deviation from constancy is a hallmark of exponential functions, and it's quintessential to highlight this in educational content. Understanding this concept can help students in numerous disciplines such as physics, engineering, and environmental science.

Derivative Calculus

In the realm of derivative calculus, we focus on understanding the fundamentally instantaneous nature of change. The derivative is a measure that provides the slope of the tangent line to a function at any point, representing how the function is changing at that precise instant.

To get a better grasp of how this applies to our example, look at the derivative of the atmospheric pressure function, p'(z) = -100 e^{-z/10}. This formula gives us the rate at which pressure decreases for every small increase in altitude. Derivative calculus shows us the 'moment by moment' behavior of functions. For educators, reinforcing the concept that derivatives represent the immediate rate of change can help students to understand motion, forces, rates of reaction, and even economic changes, transcending the boundaries of just one scientific or engineering discipline.

Another important note is the presence of the negative sign in our derivative, which indicates the direction of change—in this case, that the atmospheric pressure is decreasing as we ascend.

Limits of Functions

Examining the limits of functions is a fundamental aspect of understanding their behavior, especially as certain variables approach extreme values or specific points. In our context, the query \(\lim_{z \to \infty} p(z) = 0\) is examining what happens to the pressure as we go infinitely high above the Earth's surface.

The limit tells us that atmospheric pressure approaches zero as altitude goes to infinity. This mirrors reality: as one goes higher above the Earth, the air becomes thinner until it virtually ceases to exist, thus exerting no pressure. It’s essential for students to recognize that this zero is an asymptotic value—meaning the function approaches it but never actually reaches it.

Limits are also critical in understanding the concept of continuity and in calculating derivatives. They serve as a fundamental tool for determining the behavior of functions beyond the points where they may not be explicitly defined or where they exhibit indeterminate forms. For students, learning about limits is a foundational step that prepares the ground for further study in calculus and mathematical analysis.