Problem 24 - APEX Calculus

Question

A function \(z=f(x, y)\) and a point \(P\) are given. (a) Find the direction of maximal increase of \(f\) at \(P\). (b) What is the maximal value of \(D_{\vec{u}} f\) at \(P ?\) (c) Find the direction of minimal increase of \(f\) at \(P\). (d) Give a direction \(\vec{u}\) such that \(D_{\vec{u}} f=0\) at \(P\). Note: these are the same functions and points as in Exercises 13 through 18 . $$ f(x, y)=x^{2} y^{3}-2 x, P=(1,1) $$

Step-by-Step Solution

Answer

(a) Direction of maximal increase: (0, 3); (b) Maximal value: 3; (c) Direction of minimal increase: (0, -3); (d) Direction \( \vec{u} \) such that \( D_{\vec{u}} f = 0 \): (1, 0).

Step 1: Find the Gradient of f

To find the direction of maximal increase, we first need to compute the gradient of \( f \), denoted by \( abla f \). The partial derivative of \( f \) with respect to \( x \) is \( \frac{\partial f}{\partial x} = 2xy^3 - 2 \), and with respect to \( y \) is \( \frac{\partial f}{\partial y} = 3x^2y^2 \). Thus, the gradient is given by \( abla f = (2xy^3 - 2, 3x^2y^2) \).

Step 2: Evaluate the Gradient at Point P

To find the gradient at point \( P = (1,1) \), substitute \( x = 1 \) and \( y = 1 \) into the gradient. \( abla f(1,1) = (2(1)(1)^3 - 2, 3(1)^2(1)^2) = (0, 3) \).

Step 3: Determine the Direction of Maximal Increase

The direction of maximal increase is given by the gradient itself. At \( P = (1,1) \), the direction of maximal increase is \( (0, 3) \).

Step 4: Determine the Maximal Value of D_{\vec{u}} f at P

The maximal rate of change of \( f \) at \( P \) is the magnitude of the gradient. The magnitude is computed as \( \|abla f(1,1)\| = \sqrt{0^2 + 3^2} = 3 \).

Step 5: Determine the Direction of Minimal Increase

The direction of minimal increase is the opposite of the gradient. Thus, the direction of minimal increase at \( P = (1,1) \) is \( (0, -3) \).

Step 6: Find a Direction u such that D_{\vec{u}} f = 0

A direction where \( D_{\vec{u}} f = 0 \) is orthogonal to the gradient. At \( P = (1,1) \), a vector orthogonal to \( (0, 3) \) is \( (1, 0) \).

Key Concepts

Directional Derivative

The directional derivative measures how a function changes as we move in a specific direction from a given point. Think of it like probing the slope of a hill in a particular direction. To compute the directional derivative of a function \(f(x, y)\) at a point \(P\) along a vector direction \(\vec{u}\), we first find the gradient of the function \(abla f\), which contains partial derivatives with respect to each variable. The directional derivative in direction \(\vec{u}\) is then:

• \(D_{\vec{u}} f = abla f \cdot \vec{u}\)

The directional derivative tells us the rate at which the function value is changing as we move along vector \(\vec{u}\). It is at its maximum if \(\vec{u}\) is aligned with the gradient vector and at its minimum if \(\vec{u}\) opposes the gradient.
Directional derivatives are central to understanding how functions behave in different directions, especially in multivariable calculus. By examining directional derivatives, we can better understand gradients, maximal and minimal increases, and orthogonal directions.

Maximal Increase

The maximal increase of a function at a point refers to the direction and rate at which the function increases most rapidly. This is directly linked to the gradient vector, \(abla f\). The direction of maximum increase occurs precisely in the direction of the gradient vector.

• The magnitude of the gradient \(\| abla f \|\) represents the steepest slope at that point.
• To find this maximal increase direction, we compute the gradient at the point of interest.

From the solution, we know that at point \(P = (1,1)\), the gradient is \((0, 3)\), indicating the direction of steepest ascent. The magnitude of this gradient vector gives the maximal rate of increase, which is 3 in this case. This means as we move from point \(P\) in the direction of \(abla f\), \(f(x, y)\) increases by this maximal amount.

Minimal Increase

The minimal increase is essentially the direction where the function decreases most rapidly, or equivalently, increases the least. This occurs opposite to the gradient direction.

• The direction of minimal increase is the negative gradient \(-abla f\).
• At point \(P = (1,1)\), the minimal increase direction is \((0, -3)\).

This indicates that by moving from point \(P\) in the direction opposite to \(abla f\), the function decays the fastest or increases the least. This is a crucial concept in optimization and physics, as it helps in finding valleys or minimal points of a function. Note, however, that the actual rate is the negative magnitude of the gradient, indicating a decrease.

Orthogonal Direction

An orthogonal direction to the gradient of a function at any point is one where the directional derivative is zero. This means there is no change in the function value as you move in this direction.

• To find such a direction, we need a vector \(\vec{u}\) such that \(abla f \cdot \vec{u} = 0\).
• For example, at point \(P = (1,1)\), the gradient is \((0, 3)\), so an orthogonal vector is \((1, 0)\).

A vector orthogonal to the gradient does not contribute to increasing or decreasing the function value along that path. This property is highly valuable in various mathematical and engineering applications, as it helps maintain stability or balance at a point without affecting the function output.