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

The raindrop exits the surface around the point \((-5, -5)\).

Step 1: Identify the Equation of Motion

When a raindrop lands on a surface described by the equation \(z = x^3 - 3xy^2\), gravity will make it move to the point where the slope is steepest. This is done by moving in the direction opposite to the gradient of the surface. Let's find the gradient to determine the direction of motion.

Step 2: Calculate the Gradient of the Surface

The gradient \(abla z\) is calculated using partial derivatives. \(\frac{\partial z}{\partial x} = 3x^2 - 3y^2\) and \(\frac{\partial z}{\partial y} = -6xy\). Therefore, \(abla z = (3x^2 - 3y^2, -6xy)\).

Step 3: Initial Point Gradient Calculation

Evaluate the gradient at the given starting point \((5, -0.2)\). For \(x = 5\) and \(y = -0.2\), \(\frac{\partial z}{\partial x} = 3(5)^2 - 3(-0.2)^2 = 75 - 0.12 = 74.88\), and \(\frac{\partial z}{\partial y} = -6(5)(-0.2) = 6\). This gives the gradient vector \((74.88, 6)\).

Step 4: Direction of Motion

The raindrop will move in the direction opposite to the gradient vector to minimize the height. Thus, its path follows the vector \((-74.88, -6)\) initially.

Step 5: Estimate the Ending Point

Starting at \((5, -0.2)\), move in small increments in the direction of \((-74.88, -6)\). Continue until reaching the boundary of \(-5 \leq x \leq 5\) and \(-5 \leq y \leq 5\). The extreme vector suggests the raindrop exits near approx \((-5, -5)\) given the large value of \(x\) displacement.

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept in calculus, especially when working with functions of two or more variables. They help us understand the change in a function's output concerning one variable while keeping the others constant. In the case of the monkey saddle, represented by the equation \[ z = x^3 - 3xy^2, \] we calculate partial derivatives to analyze the slope of the surface at any given point.

• \(\frac{\partial z}{\partial x}\): Represents the rate of change of \(z\) with respect to \(x\),
• \(\frac{\partial z}{\partial y}\): Represents the rate of change of \(z\) with respect to \(y\).

For the given surface, the partial derivatives are:

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

By determining these partial derivatives, we can calculate the gradient, which points in the direction of steepest ascent or descent on the surface. This is crucial for finding where a raindrop will move once it lands on the surface at a given point.

Direction of Motion

When a raindrop lands on a surface defined by a multivariable function like the monkey saddle, it tends to move in the direction that reduces its height as quickly as possible. This direction is dictated by the gradient of the function, but the raindrop moves in the opposite direction for minimization.In step-by-step calculations, we identified the gradient vector at a specific point as \( (74.88, 6) \). The raindrop's direction of motion is then opposite to this vector:

• Gradient vector: \( abla z = (74.88, 6) \)
• Direction of motion: \(- abla z = (-74.88, -6)\)

By following the opposite vector, we ensure that the raindrop is moving downhill, toward a lower altitude, which is consistent with natural gravitational motion. Understanding this concept allows us to predict the movement path of objects on any gradient-defined surface.

Extreme Value

The concept of an extreme value in calculus refers to the highest or lowest point on a surface. However, when discussing motion directions on surfaces like the monkey saddle, the emphasis is on minimizing the function value \( z = x^3 - 3xy^2 \). The target is to locate where the raindrop is likely to find these lowest points. While the exact calculations require continuing adjustments in the original provided direction, the key is understanding:

• An extreme value represents boundary conditions as well as specific conditions where both partial derivatives equal zero.
• Such points are often critical for understanding where movements will stop naturally.

Knowing where these extremes are can help predict stable locations or stopping points for motion on a surface, assisting in everything from simple physics problems to complex engineering calculations.

Coordinate Estimation

In practical application, estimating coordinates involves determining the likely path and endpoints of motion on a surface when given conditions involve specified boundaries. For this exercise, involving a raindrop and monkey saddle, coordinate estimation follows from an understanding of direction and boundary limits.Starting with the initial point of landing at \((5, -0.2)\), and using the negative gradient direction \((-74.88, -6)\), estimations are based on small incremental movements until reaching defined surface boundaries \(-5 \leq x \leq 5\) and \(-5 \leq y \leq 5\). This predicts that, due to the large movement primarily along the x-axis, the raindrop's endpoint is near \((-5, -5)\).Factors in estimating include:

• The size of each incremental step.
• Consistent checking against boundary exceedances.

These considerations ensure accurate predictions, making coordinate estimation a critical tool in planning for movements and in practical applications across scientific and engineering fields.