Problem 1 - Calculus Early Transcendentals

Question

The temperature \(T\) at a location in the Northern Hemisphere depends on the longitude \(x,\) latitude \(y,\) and time \(t,\) so we can write \(T=f(x, y, t) .\) Let's measure time in hours from the beginning of January. (a) What are the meanings of the partial derivatives \(\partial T / \partial x\) \(\partial T / \partial y,\) and \(\partial T / \partial t ?\) (b) Honolulu has longitude \(158^{\circ} \mathrm{W}\) and latitude \(21^{\circ} \mathrm{N}\) . Suppose that at \(9 : 00\) AM on January 1 the wind is blowing hot air to the northeast, so the air to the west and south is warm and the air to the north and east is cooler. Would you expect \(f_{x}(158,21,9), f_{1}(158,21,9),\) and \(f_{l}(158,21,9)\) to be positive or negative? Explain.

Step-by-Step Solution

Answer

\(f_x\) is positive, \(f_y\) is negative, \(f_t\) is positive.

Step 1: Understanding Partial Derivatives

Partial derivatives represent the rate of change of a function with respect to one variable while keeping the other variables constant. 1. \(\partial T / \partial x\) measures the rate of change of temperature with respect to longitude, keeping latitude and time constant. It informs us how temperature varies as we move east or west. 2. \(\partial T / \partial y\) measures the rate of change of temperature with respect to latitude, keeping longitude and time constant. It shows how temperature changes as we move north or south. 3. \(\partial T / \partial t\) measures the rate of change of temperature with respect to time, keeping longitude and latitude constant. It shows how temperature changes over time at a specific location.

Step 2: Evaluating Partial Derivatives at Honolulu

For Honolulu with coordinates \((x, y) = (158^{\circ} W, 21^{\circ} N)\) at 9:00 AM, we need to consider the given weather conditions:- The wind is blowing hot air to the northeast, implying warmer air is to the south and west, while cooler air is to the north and east.1. \(f_{x}(158, 21, 9)\): Since air to the west (or in increasing longitude direction) is warmer, as we move from east to west, temperature rises, so \(f_{x}(158, 21, 9)\) is **positive**.2. \(f_{y}(158, 21, 9)\): Since air to the south is warmer, moving southwards increases temperature, so \(f_{y}(158, 21, 9)\) is **negative**.3. \(f_{t}(158, 21, 9)\): Without specific information on how temperature changes with time, generally, morning temperatures may rise, suggesting \(f_{t}(158, 21, 9)\) is **positive**, assuming no abrupt weather changes.

Key Concepts

Rate of Change

Understanding the rate of change is crucial in many fields, including meteorology and geography. In the context of this exercise, we look at how temperature changes with respect to different variables like longitude, latitude, and time using partial derivatives. Here, partial derivatives help us understand how a small change in one of these variables affects the temperature.

When we talk about the rate of change of temperature, we are essentially speaking of how quickly the temperature varies as other parameters change. Let's break it down:

• \( \frac{\partial T}{\partial x} \) measures the change in temperature when you vary the longitude, while holding latitude and time constant. So it tells us how moving east or west will affect the temperature.
• \( \frac{\partial T}{\partial y} \) looks at how temperature will change when you move north or south, with longitude and time unchanged.
• \( \frac{\partial T}{\partial t} \) highlights the temperature variation over time at a fixed location.

This approach allows meteorologists to predict weather by understanding the dynamics of temperature changes as they relate to geographic coordinates and time.

Temperature Variation

Temperature variation is influenced by several factors and is significantly analyzed through the tool of partial derivatives in this exercise. Recognizing how temperature differs across various locations and times is essential for understanding climate and weather patterns.

In this problem, the partial derivative \( \frac{\partial T}{\partial x} \) helps us understand the horizontal temperature gradient, showing how moving from one point of longitude to another affects the temperature. If warm air is to the west, as we've considered for Honolulu, moving east to west will increase temperature – indicating a positive rate of change.

Similarly, \( \frac{\partial T}{\partial y} \), the partial derivative with respect to latitude, tells us about the vertical temperature gradient. As you move north to south, and knowing the warm air blows from the south, you would experience a rise in temperature, but moving from south to north lowers it, hence a negative rate of change here.

Lastly, \( \frac{\partial T}{\partial t} \) elaborates on temperature variation over time. Generally, temperatures can rise during the day due to the sun’s influence, suggesting a positive change in the morning unless external conditions like cloud cover or air pressure inversely affect this.

Longitude and Latitude

Longitude and latitude are imaginary lines that map the Earth's surface, providing a coordinate system for locating points worldwide. Understanding these helps in predicting temperature changes observed in different geographical locations.

Longitude (\(x\)) refers to the east-west position, marked from 0° (Prime Meridian) to 180° east or west. Latitudes (\(y\)), on the other hand, stretch from the equator (0°) to the poles (90° north or south).

With this coordinate system, it's easier to map and predict temperature variations since these lines serve as reference points. In our exercise involving Honolulu, which sits at 158°W longitude and 21°N latitude, understanding these coordinates helps in deciphering prevailing weather conditions.

With the knowledge of wind directions and associated air temperature, meteorologists can use partial derivatives to predict and model weather conditions accurately, which is crucial for decision-making in aviation, agriculture, and daily life.