Problem 23 - 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 . $$ f(x, y)=x^{2}+2 y^{2}-x y-7 x, P=(4,1) $$

Step-by-Step Solution

Answer

Gradient is zero at P, so there's no maximal increase.

Step 1: Compute the Gradient

The gradient of the function \( f(x, y) = x^2 + 2y^2 - xy - 7x \) is given by \( abla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \). Compute the partial derivatives:\( \frac{\partial f}{\partial x} = 2x - y - 7 \).\( \frac{\partial f}{\partial y} = 4y - x \).

Step 2: Evaluate Gradient at Point P

Substitute the point \( P = (4, 1) \) into the gradient:\( abla f(4, 1) = (2 \times 4 - 1 - 7, 4 \times 1 - 4) \).Calculate each component:\( abla f(4, 1) = (8 - 1 - 7, 4 - 4) = (0, 0) \).

Step 3: Direction of Maximal Increase

The direction of maximal increase is in the direction of the gradient vector \( abla f(4, 1) \). Since \( abla f(4, 1) = (0, 0) \), there is no specific direction of increase at \( P = (4, 1) \).

Step 4: Maximal Value of Directional Derivative

The maximal value of the directional derivative \( D_{\vec{u}} f \) occurs in the direction of the gradient and is given by the magnitude of \( abla f(4, 1) \). Calculate the magnitude:\( \|abla f(4, 1)\| = \sqrt{0^2 + 0^2} = 0 \).Thus, the maximal directional derivative is 0.

Step 5: Direction of Minimal Increase

The direction of minimal increase is the opposite of the direction of the gradient. Since \( abla f(4, 1) = (0, 0) \), there's no specific minimal direction; all directions give zero change.

Step 6: Zero Change Direction

A direction \( \vec{u} \) such that \( D_{\vec{u}} f = 0 \) is any perpendicular direction to the gradient. Given \( abla f(4, 1) = (0, 0) \), every direction is perpendicular, so any \( \vec{u} \) will satisfy \( D_{\vec{u}} f = 0 \) due to the zero gradient.

Key Concepts

Directional Derivative

The directional derivative is a key concept to understand when analyzing how a function changes as you move in a particular direction. Consider it as the derivative of the function along a specified direction. Imagine you're hiking and want to know how steep the trail will get if you walk in a particular direction - that's the directional derivative at work!
It's defined as \( D_{\vec{u}} f(x,y) = abla f(x,y) \cdot \vec{u} \), where \( abla f(x,y) \) is the gradient of the function at a point, and \( \vec{u} \) is the direction vector.
To ensure you're walking correctly, \( \vec{u} \) should be a unit vector, meaning its length is 1. This helps in comparing the rate of change uniformly in different directions.

Partial Derivatives

Partial derivatives are like the building blocks of gradients. Imagine a surface, like a hilly landscape. If you want to understand how this surface changes in the x-direction, you'd look at its partial derivative with respect to x, denoted as \( \frac{\partial f}{\partial x} \). Similarly, changes in the y-direction are captured by \( \frac{\partial f}{\partial y} \).
To calculate partial derivatives, treat the variable you're differentiating against as usual, while holding others as constants.
In our example function \( f(x, y) = x^2 + 2y^2 - xy - 7x \), the partial derivatives are \( \frac{\partial f}{\partial x} = 2x - y - 7 \) for the x-direction, and \( \frac{\partial f}{\partial y} = 4y - x \) for the y-direction. These derivatives inform us about the slope or steepness of the surface along each direction.

Maximal Increase

The maximal increase of a function tells us the most significant increase in elevation or value at a point. This occurs in the direction of the gradient vector \( abla f \).
The gradient points directly uphill, on the steepest path at the point of interest. So, for a peak experience, follow this vector!
In mathematical terms, the value of the directional derivative is greatest along this direction, and it's equal to the magnitude of the gradient: \( \|abla f(x, y)\| \).
If you compute the magnitude and it turns out to be 0, as in our example at point \( P = (4, 1) \), it means there is no particular steep direction; it's all flat around there.

Minimal Increase

Just as maximal increase directs you up the steepest hill, minimal increase guides you down the steepest descent. This direction is the opposite of the gradient vector. Think of it like carefully descending a mountain to avoid slips - you go against the steepest path up.
However, when the gradient is zero at a point, such as in our example \( abla f(4, 1) = (0,0) \), this minimal direction doesn't hold a specific angle because all directions are equally level.
Thus, if you're looking for minimal increase where \( abla f = (0,0) \), every pathway is a good candidate since they all result in zero change in the function's value.