Problem 11 - APEX Calculus

Question

A function \(z=f(x, y)\) is given. Find \(\nabla f\). $$ f(x, y)=x^{2}+2 y^{2}-x y-7 x $$

Step-by-Step Solution

Answer

\( \nabla f = (2x - y - 7, 4y - x) \)

Step 1: Identify the Function and Variables

The function provided is \( f(x, y) = x^2 + 2y^2 - xy - 7x \). We are dealing with a function of two variables, \( x \) and \( y \). The task is to find the gradient \( abla f \), which involves partial derivatives of \( f \) with respect to \( x \) and \( y \).

Step 2: Derive Partial Derivative with respect to x

To find \( \frac{\partial f}{\partial x} \), differentiate \( f(x, y) \) with respect to \( x \):\[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^2 + 2y^2 - xy - 7x) \]This results in:\[ 2x - y - 7 \]

Step 3: Derive Partial Derivative with respect to y

Now, find \( \frac{\partial f}{\partial y} \) by differentiating \( f(x, y) \) with respect to \( y \):\[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(x^2 + 2y^2 - xy - 7x) \]This results in:\[ 4y - x \]

Step 4: Formulate the Gradient Vector

The gradient of the function, \( abla f \), is the vector consisting of the partial derivatives found previously:\[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (2x - y - 7, 4y - x) \]

Key Concepts

Partial Derivatives

Partial derivatives are used when dealing with functions of multiple variables. In this context, for a function \( f(x, y) \), the partial derivative with respect to one variable measures how the function changes as only that particular variable changes, while keeping all other variables constant. This is a crucial concept in multivariable calculus that helps analyze how a surface (or curve in higher dimensions) behaves at any given point.

• Mathematical Notation: The partial derivative of \( f \) with respect to \( x \) is denoted by \( \frac{\partial f}{\partial x} \). Similarly, the partial derivative with respect to \( y \) is denoted by \( \frac{\partial f}{\partial y} \).
• How to Compute: To compute {\

Function of Two Variables

A function of two variables, like the one given as \( f(x, y) = x^2 + 2y^2 - xy - 7x \), involves inputs that have two independent variables, \( x \) and \( y \). This type of function maps pairs of values from these two variables to a single output value. Understanding functions of two variables is important for studying relationships in physics, engineering, and economics, among many other fields.

• Visualization: The output of such a function can be visualized as a surface in a three-dimensional space. For instance, the graph of \( f(x, y) \) can be seen as a surface above the \( xy \)-plane. This surface provides insights into how the values of the function change across different coordinates on the plane.
• Dependence on Variables: The function \( f(x, y) \) depends on both \( x \) and \( y \), and changing either variable will affect the outcome of the function. The interaction between these two variables and the function's dependency is fundamental in calculating derivatives and analyzing the function's behavior.

The complexity of functions of two variables can be expanded further as you introduce more variables, leading to functions of multiple variables. However, the concepts learned with two variables are a stepping stone to understanding functions that depend on many variables.

Gradient Vector

The gradient vector is a crucial tool in multivariable calculus. For a function of two variables \( f(x, y) \), the gradient, denoted as \( abla f \), is a vector field that points in the direction of the greatest rate of increase of the function. It combines all the partial derivatives of the function with respect to its variables.

• Components of the Gradient: The gradient vector \( abla f \) consists of two components: \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \). This forms a vector \( (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}) \).
• Interpretation: At any point \((x, y)\), the gradient vector tells us the direction in which the function \( f \) increases most rapidly. Its magnitude gives the rate of increase in that direction.
• Applications: Gradient vectors are used extensively in optimization problems, helping to find maxima and minima of functions. They are also vital in physics for representing fields like the gravitational field or the electric field.

In summary, understanding the gradient vector not only helps in finding extremes of functions but also provides insights into the function's behavior in a multidimensional space.