Problem 7 - APEX Calculus

Question

In Exercises \(7-12,\) a function \(z=f(x, y)\) is given. Find \(\nabla f\). $$ f(x, y)=-x^{2} y+x y^{2}+x y $$

Step-by-Step Solution

Answer

\(\nabla f = \left( -2xy + y^2 + y, -x^2 + 2xy + x \right)\)

Step 1: Understand the Function

We are given the function \(f(x, y) = -x^2y + xy^2 + xy\). This function involves polynomials of two variables: \(x\) and \(y\). Our task is to find the gradient vector \(abla f\).

Step 2: Find Partial Derivative with Respect to x

To find \(\left(\frac{\partial f}{\partial x}\right)\), we differentiate the function with respect to \(x\), treating \(y\) as a constant:- The derivative of \(-x^2y\) with respect to \(x\) is \(-2xy\).- The derivative of \(xy^2\) with respect to \(x\) is \(y^2\).- The derivative of \(xy\) with respect to \(x\) is \(y\).Thus, \(\frac{\partial f}{\partial x} = -2xy + y^2 + y\).

Step 3: Find Partial Derivative with Respect to y

Next, we find \(\left(\frac{\partial f}{\partial y}\right)\) by differentiating the function with respect to \(y\), treating \(x\) as a constant:- The derivative of \(-x^2y\) with respect to \(y\) is \(-x^2\).- The derivative of \(xy^2\) with respect to \(y\) is \(2xy\).- The derivative of \(xy\) with respect to \(y\) is \(x\).Hence, \(\frac{\partial f}{\partial y} = -x^2 + 2xy + x\).

Step 4: Write the Gradient Vector

The gradient \(abla f\) is a vector composed of the partial derivatives obtained:\[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = \left( -2xy + y^2 + y, -x^2 + 2xy + x \right) \]

Key Concepts

Partial Derivative

When dealing with multivariable functions, like our function of two variables, we often use partial derivatives. A partial derivative tells us how a function changes as one of its variables changes, while keeping the other variables constant. This is crucial in situations where functions depend on more than one variable, which is common in fields such as physics, engineering, or economics. In the exercise, the partial derivative with respect to \(x\) is found by differentiating the function while treating \(y\) as a constant. Similarly, the partial derivative with respect to \(y\) involves treating \(x\) as a constant during differentiation.

• For \(-x^2y + xy^2 + xy\), the partial derivative with respect to \(x\) is \(-2xy + y^2 + y\).
• The partial derivative with respect to \(y\) is \(-x^2 + 2xy + x\).

Understanding partial derivatives is foundational to working with more complex calculus concepts.

Function of Two Variables

A function of two variables is a function that has two independent variables, usually denoted \(x\) and \(y\), and one dependent variable, often denoted \(z\). The relationship is expressed as \(z = f(x, y)\). Such functions can represent surfaces in three-dimensional space, where \(f(x, y)\) gives the height or value of the surface above the point \((x, y)\) on the \(xy\)-plane. In the exercise, the function \(f(x, y) = -x^2y + xy^2 + xy\) is an example of a polynomial function of two variables, meaning it involves terms of \(x\) and \(y\) elevated to positive integer powers.

• Functions like these can describe everything from temperature variations over a map to profit calculations within operational research.

Exploring functions of two variables builds our ability to handle even more complex situations.

Gradient Vector

The gradient vector, denoted as \(abla f\), is a vector that contains all of the partial derivatives of a function. For a function of two variables, the gradient vector is a two-dimensional vector. It gives the direction of the steepest ascent of the function, and its magnitude tells us how steep that ascent is. This property makes the gradient vector extremely useful in optimization problems and in finding the rate of change in specific directions. In mathematical terms, the gradient vector of \(f(x, y)\) is \(abla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})\).

• For our function \(f(x, y) = -x^2y + xy^2 + xy\), \(abla f = (-2xy + y^2 + y, -x^2 + 2xy + x)\).

Hence, the gradient vector provides crucial information about the function's behavior at different points.

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It is the mathematical study of continuous change and is fundamental in many fields of study, providing tools to analyze changes and build mathematical models. The priority concepts in this branch are differentiation and integration:

• **Differentiation** involves finding rates of change, which is what partial derivatives and gradient vectors are a part of.
• **Integration** involves summing areas or cumulative values, though this wasn't directly covered in this exercise.

Our solution focused on differentiation of functions of two variables using partial derivatives. Calculus allows us to model dynamic systems and solve problems across different scientific disciplines, giving insights into real-world situations.