Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the first partial derivatives of the function at the given point. \(f(x, y)=x^{2}+x y+y^{2}+2 x-y ;(-1,2)\)

Step-by-Step Solution

Answer

The first partial derivatives of the function \(f(x, y) = x^{2} + xy + y^{2} + 2x - y\) at the point \((-1, 2)\) are: \(f_x(-1, 2) = 2\) \(f_y(-1, 2) = 2\)

Step 1: 1. Find the partial derivative with respect to x

To find the partial derivative of the function \(f(x, y)\) with respect to \(x\), we will only differentiate the terms containing \(x\) and treat \(y\) as constant. The partial derivative of the function with respect to \(x\) is denoted as \(f_x\). So, for the function \(f(x, y) = x^{2} + xy + y^{2} + 2x - y\), the partial derivative with respect to \(x\) is: \(f_x(x, y) = \frac{\partial}{\partial x}( x^2 + xy + y^{2} + 2x - y)= 2x + y + 2\)

Step 2: 2. Find the partial derivative with respect to y

To find the partial derivative of the function \(f(x, y)\) with respect to \(y\), we will only differentiate the terms containing \(y\) and treat \(x\) as constant. The partial derivative of the function with respect to \(y\) is denoted as \(f_y\). So, for the function \(f(x, y) = x^2 + xy + y^2 + 2x - y\), the partial derivative with respect to \(y\) is: \(f_y(x, y) = \frac{\partial}{\partial y}( x^2 + xy + y^{2} + 2x - y)= x + 2y - 1\)

Step 3: 3. Evaluate partial derivatives at the point (-1, 2)

We now have the expressions for \(f_x\) and \(f_y\). We need to evaluate both partial derivatives at the given point \((-1, 2)\). For \(f_x(-1,2)\): \(f_x(-1, 2) = 2(-1) + 2 + 2 = -2 + 2 +2 = 2\) For \(f_y(-1,2)\): \(f_y(-1, 2) = -1 + 2(2) - 1 = -1 + 4 - 1 = 2\) The first partial derivatives of the function at the point \((-1, 2)\) are: \(f_x(-1, 2) = 2\) \(f_y(-1, 2) = 2\)

Key Concepts

Multivariable Calculus

Multivariable calculus, also known as multivariate calculus, is an extension of single-variable calculus to functions with several variables. It involves the study and analysis of functions that depend on more than one independent variable. In practical terms, this means we're looking at systems where multiple factors influence the outcomes or values.

In the realm of multivariable calculus, we often deal with functions like \(f(x, y)\) that depend on two variables, \(x\) and \(y\). The real-world applications of such functions are bountiful, ranging from calculating the temperature at different points in a room, which varies with both horizontal position and height, to more complex systems in engineering and physics.

When analyzing such functions, one of the main tools at our disposal is partial differentiation, which allows us to focus on the change of the function with respect to one variable at a time, while keeping the others constant.

First Partial Derivative

The first partial derivative is a foundational concept within the field of multivariable calculus. It's a measure of how a function changes as one of its input variables varies, while the others remain fixed. In essence, it's similar to the derivative of a single-variable function, but with the extension to multiple dimensions.

The notation for the first partial derivative of a function \(f(x, y)\) with respect to \(x\) is typically written as \(f_x(x, y)\) or \(\frac{\partial}{\partial x}f(x, y)\). Similarly, the partial derivative of the function with respect to \(y\) is written as \(f_y(x, y)\) or \(\frac{\partial}{\partial y}f(x, y)\). These derivatives reveal the slope of the function along the axis of the variable of differentiation. So, if you're interested in the rate of change of the function \(f(x, y)\) as only \(x\) varies, the first partial derivative with respect to \(x\) will provide that information.

Differentiation of Functions

Differentiation of functions is the process of finding the derivative - that is, the rate at which a function is changing at any given point. It's a concept that originates in single-variable calculus but extends into multiple variables in multivariable calculus.

In single-variable calculus, the derivative provides a precise momentary rate of change, like the speed of a car at a specific moment in time. In multivariable calculus, we differentiate with respect to one variable while treating the other variables as constants. This gives us a detailed view of the function's behavior in relation to each individual variable, which helps us understand more complex systems and their sensitivities to changes in input values.

The application of the first partial derivatives in a problem, such as evaluating them at a specific point, gives us valuable information about the function’s behavior at that point. This procedure is exactly what we would do to find the tangent planes, slope along a path or even determine local maxima, minima, and saddle points within the context of a function of two variables, such as \(f(x, y)\).