Problem 28 - Advanced Engineering Mathematics

Question

Find a vector that gives the direction in which the given function decreases most rapidly at the indicated point. Find the minimum rate. $$ f(x, y)=x^{3}-y^{3} ;(2,-2) $$

Step-by-Step Solution

Answer

The vector of maximum decrease is \([12, -12]\) with a rate of \(-\sqrt{288}\).

Step 1: Find the Gradient of the Function

The first step to determine the direction of maximum increase or decrease of a function at a point is to find the gradient. The gradient of a function \(f(x, y)\), denoted as \(abla f(x, y)\), is a vector containing the first partial derivatives of the function with respect to \(x\) and \(y\). For the function \(f(x, y)=x^{3}-y^{3}\), we calculate: \[ \frac{\partial f}{\partial x} = 3x^2 \] \[ \frac{\partial f}{\partial y} = -3y^2 \]Thus, the gradient is: \[ abla f(x, y) = \langle 3x^2, -3y^2 \rangle \].

Key Concepts

Partial Derivatives

Partial derivatives are an essential concept in multivariable calculus. They represent the rate at which a function changes with respect to one variable while keeping other variables constant. In simpler terms, it's like observing how a mountain slope changes elevation when moving horizontally or vertically. For the given problem, function \(f(x, y)=x^{3}-y^{3}\), the partial derivative with respect to \(x\) is \(\frac{\partial f}{\partial x} = 3x^2\). This tells us how much the function changes when \(x\) changes, assuming \(y\) is constant. Similarly, \(\frac{\partial f}{\partial y} = -3y^2\) describes the rate of change with respect to \(y\) with \(x\) constant. These derivatives help us form the gradient vector that gives the direction of steepest ascent or descent.

Rate of Change

The rate of change is a measure of how a function increases or decreases as its input variables change. In our exercise, the function \(f(x, y)=x^{3}-y^{3}\) changes at the point \((2, -2)\) according to its gradient. The gradient vector \(abla f(x, y) = \langle 3x^2, -3y^2 \rangle\) evaluated at this point becomes \(\langle 12, -12 \rangle\). Since the rate of maximum decrease is the opposite of this gradient, it becomes \(\langle -12, 12 \rangle\). This vector indicates the precise direction where the function decreases most rapidly. The magnitude of this vector is called the minimum rate of change, calculated as \(\sqrt{(-12)^2 + (12)^2} = 12\sqrt{2}\). Thus, at the point \((2, -2)\), the function decreases most rapidly in the direction \(\langle -12, 12 \rangle\), with a minimum rate of \(12\sqrt{2}\).

Function Optimization

Function optimization involves finding the maximum or minimum value of a function, which is frequently guided by the direction provided by the gradient vector. In contexts such as economics or engineering, optimizing functions might mean minimizing cost or maximizing efficiency. The gradient vector \(abla f(x, y) = \langle 3x^2, -3y^2 \rangle\) pinpoints the direction of the steepest increase. Conversely, its negative points to the steepest decrease. For our task, identifying the direction \(\langle -12, 12 \rangle\) and rate in our earlier solution allows optimization by recognizing where the function is minimized or reduced significantly. Understanding optimization through gradients can aid in minimizing functions efficiently over varied dimensions.