Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

For the monkey saddle $$ z=x^{3}-3 x y^{2} $$ on \(-5 \leq x \leq 5,-5 \leq y \leq 5\), estimate the \(x y\)-coordinates of the point where a raindrop landing above the point \((5,-0.2)\) will leave the surface.

Step-by-Step Solution

Answer

A raindrop landing at \((5, -0.2)\) likely exits near \((5, 5)\).

Step 1: Understand the Problem

We are dealing with a surface defined by the equation \(z = x^3 - 3xy^2\). We need to estimate where a raindrop that lands on this surface at the point \((5, -0.2)\) will end up.

Step 2: Identify Potential Paths

The shape of the surface suggests different paths the raindrop might take. The surface is defined as \(z = x^3 - 3xy^2\), which suggests examining the paths of steepest descent from the starting point \((5, -0.2)\).

Step 3: Calculate Partial Derivatives

To find the direction of the steepest descent, calculate the partial derivatives: \(\frac{\partial z}{\partial x} = 3x^2 - 3y^2\) and \(\frac{\partial z}{\partial y} = -6xy\). This will help in understanding how the surface shape changes in the \(x\) and \(y\) directions.

Step 4: Evaluate Derivatives at Initial Point

Plug the point \((5, -0.2)\) into the partial derivatives. Compute \(\frac{\partial z}{\partial x} = 3(5)^2 - 3(-0.2)^2 = 75 - 0.12 = 74.88\). Compute \(\frac{\partial z}{\partial y} = -6(5)(-0.2) = 6\).

Step 5: Interpret Results

The partial derivative \(\frac{\partial z}{\partial x} = 74.88\) suggests a steeper slope in the positive \(x\)-direction, while \(\frac{\partial z}{\partial y} = 6\) suggests a slope in the positive \(y\)-direction. Together, the raindrop is more likely to move towards increasing \(x\) significantly and increasing \(y\) minimally.

Step 6: Estimate Exit Coordinates

Given the direction of steepest descent, the raindrop will likely move along a path increasing in both \(x\) and \(y\). Considering bounds \(-5 \leq x \leq 5\) and \(-5 \leq y \leq 5\), and the likely exit point due to the slope calculations, one would estimate near the point \((5, 5)\).

Key Concepts

Partial Derivatives

Partial derivatives are like tools that help us understand how a function changes, but with a focus on one dimension at a time. Imagine adjusting the volume and temperature in a mixed drink separately to see how each affects the taste.
In our exercise, the surface is described by the equation \( z = x^3 - 3xy^2 \). To explore how this surface changes, we differentiate with respect to \( x \) and \( y \), treating each variable separately while keeping the other constant. This gives us two partial derivatives:

• \( \frac{\partial z}{\partial x} = 3x^2 - 3y^2 \)
• \( \frac{\partial z}{\partial y} = -6xy \)

These derivatives tell us how the height \( z \) varies as we move along the \( x \) or \( y \) direction. Therefore, by evaluating these at specific points, we can visualize the slope and predict how a raindrop will travel when landing on the surface. It acts like a compass showing us the fastest way to go downhill.

Steepest Descent

The concept of steepest descent is all about figuring out the path where a raindrop travels fastest downhill on a surface. Think of it as finding a shortcut when hiking down a mountain, using gravity's strongest pull.
In the problem, this involves looking at the gradients, given by the partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \). These point towards the quickest descent. By calculating and analyzing these gradients at the starting point \((5, -0.2)\), we discovered:

• The \( x \)-component of the gradient is much stronger with a value of 74.88, indicating the descent is steepest in this direction.
• The \( y \)-component is less significant with a value of 6, suggesting a gentler slope in this direction.

Thus, the path of steepest descent indicates that the raindrop will move primarily in the direction where it can lose height fastest - in both slightly upward \( y \) and steeply upward \( x \) directions.

Surface Equation

The surface equation used here, \( z = x^3 - 3xy^2 \), is key to modeling the environment a raindrop encounters. You can think of it as mapping out the landscape. Here, the surface equation resembles something called a 'monkey saddle', due to its unique curves.
This shape is not your average hill.

• The term \( x^3 \) describes the backbone of the surface that rises and falls.
• The term \(-3xy^2\) introduces twisting and bending, allowing varied paths across the surface.

By analyzing these components, the paths raindrops might take become clearer. This lets us predict movement over the surface using both components' interactions, influenced by the variables \( x \) and \( y \). Understanding these elements helps us model how the landscape curves and undulates, dictating where and how a raindrop might roll.

Coordinate Estimation

Estimating the coordinates where the raindrop moves to helps determine its final destination on the surface. It's like predicting where a ball will end after being rolled uphill and down into valleys.
Starting from the initial point \((5, -0.2)\), partial derivatives guide the direction of steepest descent. These derivatives already pointed that the raindrop will mostly increase its \( x \) and slightly increase its \( y \) coordinates.
Combining this with the constraint \(-5 \leq x \leq 5\) and \(-5 \leq y \leq 5\), you can foresee that the raindrop, while unlikely to drastically travel, has a destined path.
Given the calculations and boundaries considered, the anticipated exit point estimated as nearly \((5, 5)\) aligns with the logical prediction as the path follows the gradient vector directions.