Problem 10 - Calculus with Concepts in Calculus

Question

Find the gradient of the function at the given point. $$ f(x, y)=x \cos x y ;(1,-\pi) $$

Step-by-Step Solution

Answer

The gradient at the point (1, -π) is (-1, 0).

Step 1: Understand the Concept

The gradient of a function at a given point is a vector that points in the direction of the greatest rate of increase of the function. It is calculated as the partial derivatives of the function with respect to each variable.

Step 2: Find the Partial Derivative with respect to x

The partial derivative of the function with respect to x, denoted as \( f_x \), is found by differentiating \( f(x, y) = x \cos(xy) \) with respect to x.\[ f_x = \cos(xy) - xy \sin(xy) \]

Step 3: Find the Partial Derivative with respect to y

The partial derivative of the function with respect to y, denoted as \( f_y \), is found by differentiating \( f(x, y) = x \cos(xy) \) with respect to y.\[ f_y = -x^2 \sin(xy) \]

Step 4: Evaluate the Partial Derivatives at the Given Point

Substitute the point \((x, y) = (1, -\pi)\) into the partial derivatives. For \( f_x \):\[ f_x(1, -\pi) = \cos(1 \times -\pi) - 1 \times (-\pi) \sin(1 \times -\pi) \]\[ = -1 - \pi \times 0 \]\[ = -1 \]For \( f_y \):\[ f_y(1, -\pi) = -1^2 \times \sin(1 \times -\pi) \]\[ = 0 \]

Step 5: Construct the Gradient Vector

The gradient vector is given by \( abla f(x, y) = (f_x, f_y) \). Substitute the evaluated partial derivatives into this vector.\[ abla f(1, -\pi) = (-1, 0) \]

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept when dealing with functions of multiple variables. They represent the rate of change of a function with respect to one variable while keeping all other variables constant. For instance, if you have a function \( f(x, y) = x \cos(x y) \), the partial derivative with respect to \( x \), denoted as \( f_x \), is obtained by differentiating \( f(x, y) \) while treating \( y \) as a constant. Similarly, the partial derivative with respect to \( y \), denoted as \( f_y \), involves differentiating with respect to \( y \) while treating \( x \) as a constant.

• \( f_x = \cos(xy) - xy \sin(xy) \)
• \( f_y = -x^2 \sin(xy) \)

Understanding partial derivatives is critical because they give us the tools to examine how changes in each individual variable influence the overall function. When evaluating these at a specific point, such as \((1, -\pi)\), we directly plug in the values to find the specific rates of change at that point.

Directional Derivative

The directional derivative is an extension of the concept of partial derivatives. While partial derivatives give us the rate of change along coordinate axes, the directional derivative tells us the rate of change of a function in any given direction. This can be particularly useful when you're interested in understanding how a function changes along a certain path or trajectory.
The directional derivative in the direction of a vector \( \mathbf{u} = \langle u_1, u_2 \rangle \), for a function \( f(x, y) \), is calculated using the gradient vector \( abla f(x,y) = (f_x, f_y) \) and the dot product:
\[ D_\mathbf{u} f = abla f \cdot \mathbf{u} = f_x u_1 + f_y u_2 \]

• Ensure \( \mathbf{u} \) is a unit vector. If not, normalize it first by dividing by its magnitude.
• This directional derivative provides the best estimation of how the function behaves in the specified direction at a particular point.

Using the gradient not only streamlines computation but also visually demonstrates the geometry of the function's behavior.

Gradient Vector

The gradient vector is a powerful tool in multivariable calculus that combines all of the function's partial derivatives into a single vector. This vector points in the direction of the steepest ascent of the function, making it an important concept in optimization and other areas of analysis.
For a function \( f(x, y) \), the gradient vector \( abla f(x, y) \) is given by:

• \( abla f(x, y) = (f_x, f_y) \)

Evaluating the gradient vector at a specific point, we calculate \( f_x \) and \( f_y \) at that point, resulting in a vector that provides key insights into the function's local behavior. At the point \((1, -\pi)\), the gradient vector is \((-1, 0)\), indicating that:

• The steepest increase is in the negative \( x \)-direction.
• There is no change in the \( y \)-direction at that specific point.

This comprehensive insight into both the direction and magnitude of change reinforces the gradient's role as a guiding vector in multivariate functions.