Problem 3
Question
The wind-chill index \(W\) is the perceived temperature when the actual temperature is \(T\) and the wind speed is \(v,\) so we can write \(W=f(T, v) .\) The following table of values is an excerpt from Table 1 in Section \(14.1 .\) (a) Estimate the values of \(f_{T}(-15,30)\) and \(f_{i}(-15,30) .\) What are the practical interpretations of these values? (b) In general, what can you say about the signs of \(\partial W / \partial T\) \(\quad\) and \(\partial W / \partial v ?\) (c) What appears to be the value of the following limit? $$\lim _{v \rightarrow \infty} \frac{\partial W}{\partial v}$$
Step-by-Step Solution
Verified Answer
(a) \( f_{T} > 0, f_{v} < 0 \); (b) \( \partial W / \partial T > 0, \partial W / \partial v < 0 \); (c) Limit is 0.
1Step 1: Understand the Problem
We are given the wind-chill index \( W \) as a function of temperature \( T \) and wind speed \( v \). We need to estimate the partial derivatives \( f_{T}(-15,30) \) and \( f_{v}(-15,30) \), interpret their meanings, and determine the signs and behavior of these derivatives as \( v \) increases without bound.
2Step 2: Estimating \( f_{T}(-15,30) \)
To estimate \( f_{T}(-15,30) \), focus on the difference in wind-chill index values when the temperature changes by a small amount while keeping wind speed constant at 30. Use numerical approximation based on nearby entries in the table.
3Step 3: Estimating \( f_{v}(-15,30) \)
Similarly, for \( f_{v}(-15,30) \), find the difference in wind-chill index when wind speed varies slightly, keeping temperature constant at -15. Again, use the values from the table to approximate this change numerically.
4Step 4: Interpretation of \( f_{T}(-15,30) \) and \( f_{v}(-15,30) \)
The value \( f_{T}(-15,30) \) represents how the wind-chill index changes with a slight increase in temperature, reflecting sensitivity to temperature. \( f_{v}(-15,30) \) indicates how wind speed alters perceived temperature, capturing wind's effect on chill.
5Step 5: Analyzing Signs of \( \frac{\partial W}{\partial T} \) and \( \frac{\partial W}{\partial v} \)
Generally, as temperature increases, wind-chill (perceived temperature) should feel warmer, indicating \( \frac{\partial W}{\partial T} > 0 \). As wind speed increases, chill increases or \( W \) decreases, suggesting \( \frac{\partial W}{\partial v} < 0 \).
6Step 6: Limit Behavior \( \lim_{v \rightarrow \infty} \frac{\partial W}{\partial v} \)
As wind speed increases significantly, the change in perceived temperature due to additional wind speed diminishes because the chilling effect saturates. Therefore, \( \lim_{v \rightarrow \infty} \frac{\partial W}{\partial v} = 0 \).
Key Concepts
wind-chill indextemperaturewind speedlimit behavior
wind-chill index
The wind-chill index is an important concept to understand because it isn't just about the actual temperature. Instead, it's about how cold it *feels* when you are outside. The wind-chill index combines both the actual air temperature and the wind speed to create a "feels-like" temperature.
This is crucial for weather forecasts and for people to plan their outdoor activities or dress appropriately. Why does wind-chill happen? - When the wind blows, it removes heat from our bodies more quickly. - This means, the faster the wind, the colder it feels even if the actual temperature remains the same. By understanding the wind-chill index, we can better appreciate the practical difference between perceived and actual temperatures in windy conditions.
This is crucial for weather forecasts and for people to plan their outdoor activities or dress appropriately. Why does wind-chill happen? - When the wind blows, it removes heat from our bodies more quickly. - This means, the faster the wind, the colder it feels even if the actual temperature remains the same. By understanding the wind-chill index, we can better appreciate the practical difference between perceived and actual temperatures in windy conditions.
temperature
Temperature is the degree of hotness or coldness measured on a definite scale. It's one of the two key components needed to determine the wind-chill index.
Often measured in degrees Celsius or Fahrenheit, temperature is an absolute measure, unlike the wind-chill which involves perception.When we talk about changes in temperature:- An increase in temperature usually leads to a higher perceived temperature, meaning it feels warmer outside. - In the context of **partial derivatives** for the wind-chill, the partial derivative with respect to temperature \( \frac{\partial W}{\partial T} \) indicates how the perceived temperature changes with the actual temperature increasing.
This rate of change is often positive, as warming air generally means a higher perceived temperature without other modifying factors like wind.
Often measured in degrees Celsius or Fahrenheit, temperature is an absolute measure, unlike the wind-chill which involves perception.When we talk about changes in temperature:- An increase in temperature usually leads to a higher perceived temperature, meaning it feels warmer outside. - In the context of **partial derivatives** for the wind-chill, the partial derivative with respect to temperature \( \frac{\partial W}{\partial T} \) indicates how the perceived temperature changes with the actual temperature increasing.
This rate of change is often positive, as warming air generally means a higher perceived temperature without other modifying factors like wind.
wind speed
Wind speed is the measure of how fast air is moving past a certain point. It's another crucial factor in calculating the wind-chill index as it affects how cold it feels outdoors.
How does wind speed affect wind-chill?- As wind speed increases, the perceived temperature (the wind-chill index) usually decreases. This is because faster winds remove heat from our bodies more quickly, making it feel colder. - The **partial derivative** \( \frac{\partial W}{\partial v} \) shows how the perceived temperature changes with variations in wind speed.Interestingly:- Typically, this value is negative, meaning that an increase in wind speed often leads to a decrease in perceived temperature.- At a certain point, as we've modeled in limit behavior, further increases in wind speed have a diminishing effect on perceived temperature.
How does wind speed affect wind-chill?- As wind speed increases, the perceived temperature (the wind-chill index) usually decreases. This is because faster winds remove heat from our bodies more quickly, making it feel colder. - The **partial derivative** \( \frac{\partial W}{\partial v} \) shows how the perceived temperature changes with variations in wind speed.Interestingly:- Typically, this value is negative, meaning that an increase in wind speed often leads to a decrease in perceived temperature.- At a certain point, as we've modeled in limit behavior, further increases in wind speed have a diminishing effect on perceived temperature.
limit behavior
Limit behavior in mathematics examines how a function behaves as a particular variable approaches a certain value, often infinity.
In the context of the wind-chill index, one interesting limit to consider is when wind speed becomes extremely large.Here's what we find:- As wind speed \( v \) increases indefinitely \( \lim_{v \to \infty} \frac{\partial W}{\partial v} \) approaches zero.- This tells us that beyond a certain wind speed, increasing the wind speed further doesn’t significantly decrease the wind-chill index.This is because:- The chilling effect reaches a saturation point where the human body can't lose heat much faster, even with increasing wind speeds.
- This limit concept helps clarify that natural limits exist, indicating that beyond certain conditions, effects plateau.
In the context of the wind-chill index, one interesting limit to consider is when wind speed becomes extremely large.Here's what we find:- As wind speed \( v \) increases indefinitely \( \lim_{v \to \infty} \frac{\partial W}{\partial v} \) approaches zero.- This tells us that beyond a certain wind speed, increasing the wind speed further doesn’t significantly decrease the wind-chill index.This is because:- The chilling effect reaches a saturation point where the human body can't lose heat much faster, even with increasing wind speeds.
- This limit concept helps clarify that natural limits exist, indicating that beyond certain conditions, effects plateau.
Other exercises in this chapter
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