Problem 45 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(x\) and \(f(x)\) represent the given quantities. Fix \(x=a\) and let \(h\) be a small positive number. Give an interpretation of the quantities $$ \frac{f(a+h)-f(a)}{h} \text { and } \lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} $$ \(x\) denotes altitude and \(f(x)\) denotes atmospheric pressure.

Step-by-Step Solution

Answer

In short, the difference quotient \(\frac{f(a+h)-f(a)}{h}\) represents the average rate of change of atmospheric pressure with respect to altitude over an interval of \(h\), while the limit of the difference quotient \(\lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}\) represents the instantaneous rate of change of atmospheric pressure with respect to altitude at a specific altitude \(a\). These quantities help us understand how atmospheric pressure changes as altitude changes in both average and specific contexts.

Step 1: Define the variables and given quantities

Let \(x\) represent the altitude and \(f(x)\) represent the atmospheric pressure at altitude \(x\). Fix the altitude at \(x=a\) and let \(h\) be a small positive number. #Step 2: Interpret the Difference Quotient#

Step 2: Interpret the Difference Quotient

The difference quotient is given by the expression: $$ \frac{f(a+h)-f(a)}{h} $$ This represents the average rate of change of atmospheric pressure with respect to altitude over an interval of \(h\). Specifically, it shows how much the pressure changes from altitude \(a\) to altitude \(a+h\), divided by the change in altitude \(h\). In other words, it describes the average decrease or increase in atmospheric pressure as we move from one altitude to another. #Step 3: Interpret the Limit of the Difference Quotient#

Step 3: Interpret the Limit of the Difference Quotient

The limit of the difference quotient is given by the expression: $$ \lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} $$ As \(h\) approaches 0, this limit approaches the instantaneous rate of change of atmospheric pressure with respect to altitude at the point \(x=a\). This quantity represents the rate at which atmospheric pressure is changing at a specific altitude \(a\), rather than over a range of altitudes. In other words, it tells us how the atmospheric pressure is changing right at the specific altitude we are focusing on. #Conclusion# The difference quotient \(\frac{f(a+h)-f(a)}{h}\) represents the average rate of change of atmospheric pressure with respect to altitude over an interval of \(h\), while the limit of the difference quotient \(\lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}\) represents the instantaneous rate of change of atmospheric pressure with respect to altitude at a specific altitude \(a\). These quantities help us understand how atmospheric pressure changes as altitude changes in both average and specific contexts.

Key Concepts

Difference Quotient

The concept of the difference quotient is fundamental in calculus. It is represented by the formula \( \frac{f(a+h)-f(a)}{h} \), where \( h \) is a small positive number. This calculation helps us understand how a function changes its value when its input changes by a small amount. In the context of atmospheric pressure and altitude, the difference quotient

• Measures the average rate of change in pressure as altitude changes from \( a \) to \( a+h \).
• Shows how atmospheric pressure differs across an interval of altitude.
• Demonstrates how change in altitude influences changes in atmospheric pressure over specified intervals.

By averaging these changes, the difference quotient provides insights into how quickly or slowly atmospheric pressure alters with a change in altitude, but over a finite difference rather than at an exact point. This concept serves as a stepping stone to more precise calculations, like those of the instantaneous rate of change.

Instantaneous Rate of Change

The instantaneous rate of change takes the concept of the difference quotient further by considering limits. It uses the expression \( \lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} \), where the idea is to let \( h \) approach zero. When \( h \) is extremely small, the difference quotient effectively provides the rate at which atmospheric pressure is changing at an exact altitude. This allows us to:

• Calculate the precise rate of atmospheric pressure change at a specific point of altitude, \( a \).
• Understand immediate fluctuations in pressure, not just over a range, but at a singular point.
• See how quickly or slowly pressure changes as we rise or descend at altitude \( a \).

The instantaneous rate of change provides a deeper understanding, especially in atmospheric studies, as it helps predict how pressure reacts in immediate circumstances, such as predicting weather patterns or conducting experiments at various altitudes.

Atmospheric Pressure

Atmospheric pressure is a critical concept when studying altitude and related changes. It refers to the pressure exerted by the weight of the atmosphere, and the function \( f(x) \) considers how this pressure varies with altitude \( x \). Here are some important points to understand about atmospheric pressure:

• Atmospheric pressure decreases with increasing altitude because there is less air above exerting downward force.
• Understanding this relationship is crucial for various applications, including aviation, weather forecasting, and even health sciences.
• The change in pressure with altitude can affect human comfort and physiological responses.

The relationship between altitude and pressure is non-linear and varies depending on atmospheric conditions and temperature. Accurately determining pressure changes demands a solid grasp of both the difference quotient and instantaneous rate of change, helping in predicting and managing changes effectively in scientific and practical settings.