Question

The "wind chill factor" is a measurement of how cold it "feels" during cold, windy weather. Let \(W(w)\) be the wind chill factor, in degrees Fahrenheit, when it is \(25^{\circ} \mathrm{F}\) outside with a wind of \(w\) mph. (a) What are the units of \(W^{\prime}(w)\) ? (b) What would you expect the sign of \(W^{\prime}(10)\) to be?

Short answer

(a) \\text{Degrees Fahrenheit per mph}; (b) Negative.

Step-by-step solution

Understanding the Function

The function \(W(w)\) represents the wind chill factor depending on the wind speed \(w\) measured in mph (miles per hour). The output of this function, \(W(w)\), is in degrees Fahrenheit since it measures the perceived temperature.

Analyzing W'(w)

\(W'(w)\) is the derivative of \(W(w)\) with respect to \(w\). This derivative indicates the rate of change of the wind chill factor as the wind speed changes. Therefore, the units of \(W'(w)\) would be in degrees Fahrenheit per mph.

Evaluating the Sign of W'(10)

\(W(10)\) represents the wind chill factor when the wind speed is 10 mph. As wind speed increases, the wind chill factor usually decreases, making it "feel" colder. Therefore, \(W'(10)\) is expected to be negative because increasing \(w\) should result in a lower wind chill factor.

Key concepts

Derivative

A derivative is a core concept in calculus that helps us understand how functions change. It represents the rate at which a quantity changes as another variable changes. In the context of the wind chill factor, the function \(W(w)\) depends on wind speed \(w\). The derivative, \(W'(w)\), tells us how quickly the wind chill factor changes as the wind speed changes.

When you see a derivative expressed as \(W'(w)\), it suggests how much \(W(w)\) will increase or decrease for a small increase in \(w\). Think of the derivative as the speedometer of a car, showing how fast the function's value is changing. This change is essential for understanding environments with varying parameters like wind speed.

In mathematical terms, calculating the derivative can help predict the impact of small changes in conditions, such as wind speed, on how cold it feels.

Rate of Change

The rate of change is a way to describe how one quantity changes in relation to another. It's like asking, "How quickly does the temperature seem to drop when the wind starts blowing faster?"

For the wind chill factor, \(W(w)\), the rate of change is expressed as \(W'(w)\). This shows the degree Fahrenheit per mile per hour (mph) change in perceived temperature. Imagine you are outside, and the wind speed picks up by 1 mph. The rate of change tells you how many degrees cooler it would feel, highlighting the sensitivity of temperature perception to wind speed.

Understanding this rate helps meteorologists predict how weather conditions affect perceived temperature, allowing for better preparation and advice for outdoor activities in cold weather.

Perceived Temperature

Perceived temperature indicates how cold or warm it feels to humans, not the actual air temperature. Elements like wind speed and humidity alter how we experience temperature.

In our exercise, when the wind blows, it lowers the perceived temperature. Thus, \(W(w)\) is a calculation of this perceived chill, which changes from the true air temperature. The wind chill factor quantifies this effect by showing how increased wind speed makes it feel colder than the actual air temperature.

Why does this happen? When wind hits the skin, it removes the layer of warmth our bodies naturally create. That’s why a breezy, cold day can "feel" significantly colder than a calm day at the same temperature. Understanding perceived temperature is crucial for well-being and comfort in cold climates.