91Ó°ÊÓ

Evaluating a function means determining its output at a specific point in its domain. In our exercise, we have the function:\[f(x, y) = \frac{2x}{x^2 + y^2}\]and we are asked to find its value at the point \((2, 3)\). To do this, we replace the variables \(x\) and \(y\) with 2 and 3 respectively.
Plugging these values into the function, we transform it into:\[f(2, 3) = \frac{2(2)}{2^2 + 3^2}\]This substitution gives us a new expression that needs to be evaluated to find the specific output of the function at the given point. This is an essential skill in multivariable calculus because it allows us to assess the behavior of functions across different dimensions and variables, essential for applications in fields like physics and engineering.

The numerator and denominator are essential components of any function expressed as a fraction. In our function:\[f(x, y) = \frac{2x}{x^2 + y^2}\]the numerator is the term above the fraction line, which is \(2x\), and the denominator is the term below, \(x^2 + y^2\). Calculating these parts separately helps streamline the problem-solving process.
For this function:- **Numerator**: Replace \(x\) with 2. Therefore, \(2(2) = 4\).- **Denominator**: Substitute \(x = 2\) and \(y = 3\). Squaring these values gives us \(2^2 = 4\) and \(3^2 = 9\). Sum these results: \(4 + 9 = 13\).
By individually determining the numerator and denominator, you can then easily assemble the entire expression, which is invaluable in making your work orderly and reducing potential mistakes.

Substitution is a fundamental method in calculus to simplify complex problems. It involves replacing variables with actual values to make the expression solvable. In our exercise, we substitute into:\[f(x, y) = \frac{2x}{x^2 + y^2}\]by setting \(x = 2\) and \(y = 3\). This allows us to transform the equation to a state where we can evaluate it numerically:\[f(2, 3) = \frac{4}{13}\]
This substitution eliminates the variables, converting an abstract function into a concrete numerical expression, simplifying the process of finding its value. Without substitution, analyzing functions or solving equations involving multiple variables would be significantly more challenging.

Simplification refers to the process of reducing an expression to its simplest form. Once our function is expressed at a specific point due to substitution:\[f(2, 3) = \frac{4}{13}\]we need to simplify to ensure it is in its most reduced form for interpretation.For this exercise, simplification involves ensuring the fraction \(\frac{4}{13}\) is reduced to its simplest form. Since 4 and 13 have no common factors (besides 1), this fraction is already simplified.
Simplification is crucial in calculus to present results in a clear and concise manner. It ensures that our solutions are accurate and allows us to further manipulate or compare them efficiently, paving the way for deeper analytical insights.