Problem 27 - Calculus Early Transcendentals

Question

(a) Show that a differentiable function \(f\) decreases most rapidly at \(x\) in the direction opposite to the gradient vector, that is, in the direction of \(-\nabla f(\mathbf{x})\) . (b) Use the result of part (a) to find the direction in which the function \(f(x, y)=x^{4} y-x^{2} y^{3}\) decreases fastest at the point \((2,-3) .\)

Step-by-Step Solution

Answer

The function decreases fastest at (2,-3) in the direction (104, 68).

Step 1: Understanding the Gradient Vector

The gradient vector \( abla f(\mathbf{x}) \) of a function \( f \) is defined as the vector of its partial derivatives. It points in the direction of the steepest ascent of the function.

Step 2: Direction of Steepest Descent

To show that the function \( f \) decreases most rapidly in the direction opposite to \( abla f(\mathbf{x}) \), consider a unit vector \( \mathbf{u} \) and calculate the directional derivative \( D_\mathbf{u}f = abla f \cdot \mathbf{u} \). The function decreases fastest when this derivative is most negative. The most negative value occurs when \( \mathbf{u} = - \frac{abla f}{\|abla f\|} \), which is the direction of \( -abla f \).

Step 3: Calculating the Gradient of f(x, y)

Calculate the gradient \( abla f(x, y) \) for the given function \( f(x, y) = x^4 y - x^2 y^3 \). First, find the partial derivatives: \( f_x = 4x^3 y - 2x y^3 \) and \( f_y = x^4 - 3x^2 y^2 \). Thus, \( abla f(x, y) = (4x^3 y - 2x y^3, x^4 - 3x^2 y^2) \).

Step 4: Evaluating the Gradient at a Specific Point

Evaluate \( abla f(x, y) \) at the point \((2, -3)\). Substitute \( x = 2 \) and \( y = -3 \) into the gradient: \(abla f(2, -3) = (4(2)^3(-3) - 2(2)(-3)^3, (2)^4 - 3(2)^2(-3)^2) = (-104, -68) \).

Step 5: Direction of Fastest Decrease

The direction of fastest decrease is the direction opposite to the gradient vector evaluated at the point. Therefore, the direction is given by \( -abla f(2, -3) = (104, 68) \). Normalize this vector by dividing each component by its magnitude to get the unit vector.

Key Concepts

Gradient Vector

The gradient vector is a crucial concept in calculus and optimization. It's a vector consisting of all the partial derivatives of a function. For a function of several variables, such as \( f(x, y) = x^4 y - x^2 y^3 \), the gradient vector \( abla f(x, y) \) is expressed as:

• \( \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \)

The gradient vector points in the direction where the function increases most rapidly. This direction is known as the direction of steepest ascent. Calculating the gradient involves taking the derivative of the function with respect to each variable separately, treating the other variables as constants.
Once you have the partial derivatives, you can construct the gradient vector, which then tells you how to modify each variable to maximize the increase of the function's output. Understanding the gradient helps in optimizing functions and finding optimal solutions in various applications.

Directional Derivative

The directional derivative is the rate at which a function changes in a specified direction. Imagine you're hiking on a mountain represented by a function \( f \). The directional derivative tells you how steep your climb is in a particular direction.

• It's calculated as \( D_\mathbf{u} f = abla f \cdot \mathbf{u} \) where \( \mathbf{u} \) is a unit vector in the direction you're interested in.

The dot product \( abla f \cdot \mathbf{u} \) indicates how aligned the gradient vector is with your chosen direction.
The directional derivative finds its maximum when \( \mathbf{u} \) aligns perfectly with \( abla f \), resulting in the largest increase in function value. Conversely, to find the direction of steepest descent, the directional derivative should be most negative, which occurs when \( \mathbf{u} \) points opposite the gradient vector. This concept is fundamental for understanding how to navigate towards minima or maxima in multidimensional spaces.

Partial Derivatives

Partial derivatives are the building blocks of the gradient vector. They measure the function's sensitivity to changes in one specific variable, holding others constant. Consider the function \( f(x, y) = x^4 y - x^2 y^3 \).

• The partial derivative with respect to \( x \), denoted \( f_x \), is calculated by differentiating \( f \) in relation to \( x \) while keeping \( y \) constant.
• The partial derivative with respect to \( y \), denoted \( f_y \), is found by differentiating in relation to \( y \), keeping \( x \) constant.

For our function, the partial derivatives are:

• \( f_x = 4x^3 y - 2x y^3 \)
• \( f_y = x^4 - 3x^2 y^2 \)

Partial derivatives are essential in multivariable calculus as they facilitate the study of how functions change in each direction independently. They make up the components of the gradient vector, thus helping determine the direction of optimization.

Steepest Descent

The method of steepest descent is a way to find the minimum of a function by moving in the direction of the steepest decrease. For a differentiable function, this means moving in the direction opposite to the gradient vector.

• If \( \mathbf{x} \) is a point in space, and \( abla f(\mathbf{x}) \) is the gradient at that point, the steepest descent direction is \( -abla f(\mathbf{x}) \).

In optimization, you'll often want to minimize cost functions, errors, or energy, and this technique is crucial to that endeavor.
To practically apply this method, one calculates the gradient, determines its opposite, and chooses a step size to navigate the function's landscape effectively. This process leads towards valleys, representing local or global minima of the function. By adjusting movement iteratively, a path to the lowest point becomes clear. The method of steepest descent is foundational in optimization algorithms, including gradient descent used in machine learning.