Problem 6 - Calculus and its applications

Question

Find \(f_{x}(x, y), f_{y}(x, y), f_{x}(-2,4),\) and \(f_{y}(4,-3)\) $$f(x, y)=5 x+7 y$$

Step-by-Step Solution

Answer

\( f_x(x, y) = 5, f_y(x, y) = 7, f_x(-2, 4) = 5, f_y(4, -3) = 7. \)

Step 1: Identify Partial Derivative with Respect to x

For the function \( f(x, y) = 5x + 7y \), identify the partial derivative of the function with respect to \( x \). Treat \( y \) as a constant and differentiate only with \( x \).

Step 2: Compute \( f_x(x, y) \)

Differentiate \( f(x, y) = 5x + 7y \) with respect to \( x \):\[ f_x(x, y) = \frac{\partial}{\partial x}(5x + 7y) = 5 \]

Step 3: Identify Partial Derivative with Respect to y

For the same function \( f(x, y) = 5x + 7y \), identify the partial derivative of the function with respect to \( y \). Treat \( x \) as a constant and differentiate only with \( y \).

Step 4: Compute \( f_y(x, y) \)

Differentiate \( f(x, y) = 5x + 7y \) with respect to \( y \):\[ f_y(x, y) = \frac{\partial}{\partial y}(5x + 7y) = 7 \]

Step 5: Evaluate \( f_x(-2, 4) \)

Substitute \( x = -2 \) and \( y = 4 \) into the partial derivative \( f_x(x, y) \):\[ f_x(-2, 4) = 5 \]

Step 6: Evaluate \( f_y(4, -3) \)

Substitute \( x = 4 \) and \( y = -3 \) into the partial derivative \( f_y(x, y) \):\[ f_y(4, -3) = 7 \]

Key Concepts

Multivariable Calculus

Multivariable calculus extends the ideas of differentiation and integration from one-dimensional calculus to functions of several variables. It is especially useful for analyzing real-world phenomena, like temperature changes or elevation differences, that depend on more than one variable.

In the realm of multivariable calculus, our chief concern is understanding how a function changes when its inputs - variables - change. Consider that instead of a curve, we deal with "surfaces". These are graphs of functions involving two or more variables. For example, in two variables, the function \(f(x,y) = 5x + 7y\) forms a plane in a three-dimensional space. This differs from single-variable calculus, where you'd typically represent functions as curves on a plane.

The tools used in multivariable calculus, such as partial derivatives, are extensions of the differentiation concepts learned in single-variable calculus. Though the principles remain similar, they are applied in a context where each variable can be arbitrarily varied independently of the others.

Differentiation

Differentiation, which originates from calculus, is the process of finding the rate at which something changes. In the case of functions of more than one variable, like \(f(x, y) = 5x + 7y\), we perform this process on each variable while considering the others as constants. This is known as taking partial derivatives.

• **Partial Derivative with Respect to \( x \):** Here, you let \( y \) hold steady and differentiate \( f(x,y) \) with respect to \( x \) alone. For our given function, \( f_x(x, y) = \frac{\partial}{\partial x}(5x + 7y) = 5 \).
• **Partial Derivative with Respect to \( y \):** Similarly, hold \( x \) as a constant and differentiate with respect to \( y \): \( f_y(x, y) = \frac{\partial}{\partial y}(5x + 7y) = 7 \).

This approach aids in understanding how each variable individually contributes to the change in the function's outcome. It's like examining a complex system piece by piece, but here each piece is tied to an individual variable.

Functions of Two Variables

Functions of two variables are an extension of the concept of functions in single-variable calculus, where the output depends on a combination of two inputs. These functions notoriously exemplify real-world data sets where results are determined by more than one factor.

For example, the function \(f(x, y) = 5x + 7y\) represents a simple linear relationship between the two variables. For each combination of \( x \) and \( y \), there is a unique value of \( f(x, y) \).

• **Visualizing Functions:** If you were to graph this particular function in a three-dimensional space, it would appear as a plane where height is determined by the combination of \( x \) and \( y \).
• **Evaluating Specific Points:** Calculating partial derivatives at specific points gives insight into how changes in one variable can alter the function's result. For instance, at \( f_x(-2, 4) \), the function deviates at the rate of 5 units per unit change in \( x \) at that specific point.

Through these examples, you learn how complex relationships can be distilled into simpler parts, making it easier to understand fluctuations within a dynamic system.