Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

The elevation of a mountain above sea level at the point \((x, y)\) is \(f(x, y)\). A mountain climber at \(\mathbf{p}\) notes that the slope in the easterly direction is \(-\frac{1}{2}\) and the slope in the northerly direction is \(-\frac{1}{4}\). In what direction should he move for fastest descent?

Step-by-Step Solution

Answer

The climber should move in the direction \\(\left( \frac{2\sqrt{5}}{5}, \frac{\sqrt{5}}{5} \right)\\) for fastest descent.

Step 1: Understand the Problem

We need to find the direction in which the mountain climber should move to achieve the fastest descent. This involves using the gradient of the function, where the negative gradient direction indicates the steepest descent.

Step 2: Define Gradient of the Function

The slope in a given direction at a point \((x, y)\) is given by the components of the gradient vector \(abla f(x, y)\). Each component represents the slope in one direction: \((\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}) = (-\frac{1}{2}, -\frac{1}{4})\).

Step 3: Identify the Component Functions

From the information given, the partial derivatives of the function are: \(\frac{\partial f}{\partial x} = -\frac{1}{2}\) and \(`\frac{\partial f}{\partial y} = -\frac{1}{4}\).

Step 4: Find the Gradient Vector

The gradient vector \(abla f(x, y)\) is \((-\frac{1}{2}, -\frac{1}{4})\). This vector points in the direction of the greatest increase of \`f\.

Step 5: Determine the Direction of Steepest Descent

To find the direction of steepest descent, we take the negative of the gradient vector: \(abla f(x, y) = (\frac{1}{2}, \frac{1}{4})\). This vector gives the direction in which the elevation decreases most rapidly.

Step 6: Normalize the Descent Direction Vector

Normalize the vector to find the unit vector that points in the direction of steepest descent. The length of the vector \((\frac{1}{2}, \frac{1}{4})\) is calculated: \(\sqrt{(\frac{1}{2})^2 + (\frac{1}{4})^2} = \frac{\sqrt{5}}{4}\). Divide each component by this magnitude: \(\left( \frac{1}{2} \div \frac{\sqrt{5}}{4}, \frac{1}{4} \div \frac{\sqrt{5}}{4} \right) = \left( \frac{2\sqrt{5}}{5}, \frac{\sqrt{5}}{5} \right)\).

Key Concepts

Partial Derivatives

Partial derivatives are crucial in understanding functions of multiple variables, like our elevation function \( f(x, y) \). They measure how the function changes as we slightly vary one variable, keeping the others constant. \( \frac{\partial f}{\partial x} \) represents how the elevation changes as we move east or west, while \( \frac{\partial f}{\partial y} \) shows the change as we move north or south. These derivatives form the gradient vector, which combines these directional changes into one handy vector.

• The partial derivative \( \frac{\partial f}{\partial x} = -\frac{1}{2} \) indicates a decrease in elevation when moving east.
• The partial derivative \( \frac{\partial f}{\partial y} = -\frac{1}{4} \) tells us there's also a decrease moving north, but less steep.

Partial derivatives give us the rate of change in specific directions, providing essential insights into the terrain a climber faces.

Directional Derivative

While partial derivatives tell us about change along the axes, directional derivatives answer a broader question: How does the function change in any direction? Suppose a climber wants to know how the elevation changes if they move 45 degrees between east and north.The directional derivative is calculated using the gradient vector and a unit vector that points in the direction of interest. The gradient vector \( abla f(x, y) = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}) \) is dotted with the unit vector.

• The directional derivative provides a precise rate of elevation change in the chosen direction.
• It utilizes the gradient vector and makes the process of finding exact changes in varied paths possible.

With directional derivatives, climbers can chart their course through a mountainside with detailed insight.

Steepest Descent

The concept of steepest descent is vital when aiming to minimize a function, such as decreasing elevation on a hike down a mountain. It is found by moving in the negative gradient direction. Since the gradient points uphill, the opposite direction indicates the way downwards most rapidly.To achieve the steepest descent, we reverse the gradient vector \( abla f(x, y) = (\frac{1}{2}, \frac{1}{4}) \), ensuring it's a unit vector. This helps maintain consistence in the scale of change.

• Calculate the magnitude of the gradient vector to normalize it.
• This ensures our direction vector remains at unit length, maintaining manageable proportions for descent.

Through this method, mountaineers can identify the path leading to the fastest decline. Understanding the steepest descent provides a systematic approach to journey safely and efficiently downhill.