Problem 21 - Calculus with Concepts in Calculus

Question

From (1) it follows that the directional derivative of a function \(f\) at a point is smallest in the direction opposite to the gradient of \(f\) at that point. Thus we say that a function decreases most rapidly in the direction opposite the gradient.Find the direction in which the function decreases most rapidly at the given point. $$ f(x, y)=\sin \pi x y ;\left(\frac{1}{2}, \frac{2}{3}\right) $$

Step-by-Step Solution

Answer

The function decreases most rapidly in the direction \((-\frac{\pi}{3}, -\frac{\pi}{4})\).

Step 1: Calculate the Gradient

To find the direction in which the function decreases most rapidly, we first need the gradient of the function \(f(x, y) = \sin \pi x y\). The gradient, denoted as \(abla f\), is a vector of partial derivatives: \(abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)\). Calculate the partial derivatives:\[ \frac{\partial f}{\partial x} = \pi y \cos(\pi x y) \]\[ \frac{\partial f}{\partial y} = \pi x \cos(\pi x y) \]

Step 2: Evaluate the Gradient at the Given Point

Now evaluate the gradient at the point \(\left( \frac{1}{2}, \frac{2}{3} \right)\). Substitute \(x = \frac{1}{2}\) and \(y = \frac{2}{3}\) into the partial derivatives:\[ \frac{\partial f}{\partial x} \bigg|_{(\frac{1}{2}, \frac{2}{3})} = \pi \cdot \frac{2}{3} \cdot \cos\left( \pi \cdot \frac{1}{2} \cdot \frac{2}{3} \right) = \frac{2\pi}{3} \cdot \cos\left( \frac{\pi}{3} \right) = \frac{2\pi}{3} \cdot \frac{1}{2} = \frac{\pi}{3} \]\[ \frac{\partial f}{\partial y} \bigg|_{(\frac{1}{2}, \frac{2}{3})} = \pi \cdot \frac{1}{2} \cdot \cos\left( \pi \cdot \frac{1}{2} \cdot \frac{2}{3} \right) = \frac{\pi}{2} \cdot \cos\left( \frac{\pi}{3} \right) = \frac{\pi}{2} \cdot \frac{1}{2} = \frac{\pi}{4} \]

Step 3: Determine the Direction Opposite the Gradient

The direction of steepest decrease is opposite to the gradient vector. The gradient vector at \(\left( \frac{1}{2}, \frac{2}{3} \right)\) is \(\left( \frac{\pi}{3}, \frac{\pi}{4} \right)\). Thus, the direction of steepest decrease is the negative of this vector:\[ -abla f = \left( -\frac{\pi}{3}, -\frac{\pi}{4} \right) \]

Key Concepts

Gradient

The concept of a gradient is crucial when dealing with functions of several variables. For any given function, the gradient acts like a multi-directional arrow pointing towards the direction of greatest increase of the function. It's a vector composed of all the partial derivatives of the function.
For a function like \( f(x, y) = \sin(\pi x y) \), the gradient \( abla f \) is represented as:

• The first component is the partial derivative concerning \( x \), \( \frac{\partial f}{\partial x} = \pi y \cos(\pi x y) \).
• The second component is the partial derivative concerning \( y \), \( \frac{\partial f}{\partial y} = \pi x \cos(\pi x y) \).

So, the gradient \( abla f \) is a vector \( (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}) \). It emphasizes how the function changes at a particular point. At every point, this vector tells us in which direction the function increases most sharply.

Partial Derivatives

Partial derivatives are the foundation for understanding gradients. They represent the rate at which the function changes as one of the variables changes, keeping the others constant. These are like looking at a function from different perspectives without considering others.
For the function \( f(x, y) = \sin(\pi x y) \), we calculate:

• \( \frac{\partial f}{\partial x} = \pi y \cos(\pi x y) \): This tells us how \( f \) changes as \( x \) varies, with \( y \) fixed.
• \( \frac{\partial f}{\partial y} = \pi x \cos(\pi x y) \): This tells us how \( f \) changes as \( y \) varies, with \( x \) fixed.

Evaluating these derivatives at a point gives insight into directional changes of the function at that specific position, which is central to finding points of steepest ascent or descent.

Function Decrease

Understanding function decrease involves knowing in which direction a function's value declines most rapidly. This direction is opposite to the gradient at any given point.
For our specific function \( f(x, y) = \sin(\pi x y) \), after calculating the gradient at a point \( \left( \frac{1}{2}, \frac{2}{3} \right) \), we found it as \( (\frac{\pi}{3}, \frac{\pi}{4}) \).
To find where the function decreases most quickly, simply take the negative of this gradient vector:

• The direction is \( \left(-\frac{\pi}{3}, -\frac{\pi}{4}\right) \).

This direction indicates moving along a path where the function's value decreases most substantially, giving a practical aspect to utilising gradients in real-world scenarios, such as minimizing energy uses or costs. It provides a powerful tool to analyze and optimize multivariable functions.