Problem 17 - Calculus with Concepts in Calculus

Question

Find the direction in which \(f\) increases most rapidly at the given point, and find the maximal directional derivative at that point. $$ f(x, y)=3 x^{2}+4 y^{2} ;(-1,1) $$

Step-by-Step Solution

Answer

The direction of most rapid increase is \((-6, 8)\), and the maximal directional derivative is 10.

Step 1: Compute the Gradient of f

The gradient of a function \( f(x, y) \) is given by the vector of partial derivatives: \( abla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \). Compute each partial derivative. For \( f(x, y) = 3x^2 + 4y^2 \), we have: \( \frac{\partial f}{\partial x} = 6x \) and \( \frac{\partial f}{\partial y} = 8y \). The gradient is then \( abla f(x, y) = (6x, 8y) \).

Step 2: Evaluate the Gradient at the Given Point

Substitute the point \((-1, 1)\) into the gradient. This yields \(abla f(-1, 1) = (6(-1), 8(1)) = (-6, 8)\).

Step 3: Determine the Direction of Most Rapid Increase

The direction in which \( f \) increases most rapidly is given by the direction of the gradient vector \( abla f(x, y) \). Thus, at the point \((-1, 1)\), the direction is \((-6, 8)\).

Step 4: Calculate the Magnitude of the Gradient

The maximal rate of increase (or the maximal directional derivative) is the magnitude of the gradient vector. Compute this by \( \|abla f(-1, 1)\| = \sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \).

Key Concepts

Gradient Vector

In multivariable calculus, the gradient vector is a tool that helps us determine how a function changes at a given point in space. This vector is composed of the partial derivatives of the function, representing the rate of change with respect to each variable. For the function \( f(x, y) = 3x^2 + 4y^2 \), we compute the partial derivatives to form the gradient vector. The partial derivative with respect to \( x \) is \( \frac{\partial f}{\partial x} = 6x \), and with respect to \( y \) is \( \frac{\partial f}{\partial y} = 8y \). Together, these form the gradient vector:

• \( abla f(x, y) = (6x, 8y) \)

The gradient vector points in the direction of the greatest rate of increase of the function, aligning directly with finding increases from any point in its path. Evaluating this at the point \((-1, 1)\), the gradient becomes \((-6, 8)\). This means at the point \((-1, 1)\), the function \( f \) increases most rapidly towards the direction \((-6, 8)\).

Maximal Directional Derivative

The maximal directional derivative is a concept that tells us the greatest rate of increase of a function at a given point and in which direction this occurs. This maximal rate corresponds to the magnitude of the gradient vector. For function \( f(x, y) = 3x^2 + 4y^2 \) at the point \((-1, 1)\), we found the gradient vector to be \((-6, 8)\). The maximal directional derivative at this point is equivalent to the magnitude or length of this gradient vector.
To find the magnitude, apply the formula:

• \( \|abla f(-1, 1)\| = \sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \)

Thus, the maximal directional derivative of the function at \((-1, 1)\) is \( 10 \), meaning that at this point, the function increases at its steepest rate in the direction of the gradient.

Partial Derivatives

Partial derivatives are fundamental when dealing with multivariable functions, as they quantify how a function changes as one independent variable is altered, keeping the others constant. In the given exercise, we have the function \( f(x, y) = 3x^2 + 4y^2 \). Here, the process involves finding:

• \( \frac{\partial f}{\partial x} = 6x \), which indicates how \( f \) changes as \( x \) varies, with \( y \) constant.
• \( \frac{\partial f}{\partial y} = 8y \), which shows the change in \( f \) as \( y \) varies, with \( x \) constant.

These derivatives give direction and rate of change information for the function with respect to each variable individually. The combination of these partial derivatives forms the gradient vector \( abla f(x, y) = (6x, 8y) \), which then aids in understanding the overall behavior of \( f \) at a point like \((-1, 1)\). Partial derivatives play an integral role in crafting the gradient, which in turn reveals the directional derivative at points of interest.