Problem 22 - Calculus an Applied Approach

Question

Evaluate \(f_{x}\) and \(f_{y}\) at the point. $$ \text { Function } \quad \text { Point } $$ $$ f(x, y)=x^{2}-3 x y+y^{2} \quad(1,-1) $$

Step-by-Step Solution

Answer

The partial derivative of the function at the point (1,-1) for \(f_{x}\) is 5 and for \(f_{y}\) is -5.

Step 1: Find the partial derivative with respect to x

The partial derivative of the function \(f(x, y) = x^{2} - 3xy + y^{2}\) with respect to x is found by differentiating the function with respect to x. Treating y as a constant, we get: \(f_{x} = 2x - 3y\).

Step 2: Find the partial derivative with respect to y

Similarly, for finding the partial derivative with respect to y, we treat x as a constant. For the given function, the partial derivative becomes: \(f_{y} = -3x + 2y\).

Step 3: Evaluate the partial derivatives at the given point

We are required to evaluate these at the point (1,-1). So, substitute x=1 and y=-1 in the partial derivative expressions. For \(f_{x}\), we get: \(f_{x} = 2(1) - 3(-1) = 2 + 3 = 5\). Similarly for \(f_{y}\), we get: \(f_{y} = -3(1) + 2(-1) = -3 - 2 = -5\).

Key Concepts

Calculus

The foundation of calculus lies in its ability to analyze changes. It's a branch of mathematics that deals with rates of change (differential calculus) and the accumulation of quantities (integral calculus). When we talk about rates of change, we usually refer to the derivative of a function. In the context of a real-life scenario, understanding calculus can help in predicting how fast or slow a car is moving at any given moment, based on its acceleration or decal velocity curve. Calculus is essential for analyzing and solving problems in physics, engineering, economics, and beyond because it provides the tools to model and control systems.

Multivariable Calculus

When we step up from single-variable calculus to multivariable calculus, the complexity and application of calculus expand significantly. Multivariable calculus refers to the differential and integral calculus of functions with more than one variable. This allows for analysis of functions that depend on several inputs, such as the topography of a hill or the temperature distribution in a room. In multivariable calculus, partial derivatives encompass the concept of a derivative with respect to one variable while holding the others constant, as seen in this exercise. Understanding this concept is critical in fields such as thermodynamics, electromagnetism, and even in finance with the formulation of multiple-variable economic models.

Derivative Evaluation

The process of derivative evaluation involves calculating the derivative of a function at a certain point. In the context of multivariable functions, this translates into finding the rate of change in a specific direction, which is represented by partial derivatives. To evaluate a derivative, one must understand both the rules of differentiation and how to apply them in practice. Evaluation of derivatives is not only a central task in calculus but a powerful tool that allows us to understand the sensitivity of functions to small changes in their inputs, optimize functions by finding maximum or minimum values, and analyze various physical phenomena.

Function Differentiation

Differentiation is a method to find how a function changes infinitesimally as its input changes. When differentiating functions in multiple variables, we look at partial derivatives to examine the change with respect to each variable independently. This is markedly different from dealing with single-variable functions, where there's only one way the function can change. Function differentiation is a cornerstone of calculus that enables us to mathematically model and solve practical problems that involve changes in various dimensions, whether it be in natural science, social science, or any other field that deals with dynamic systems.