Problem 27 - APEX Calculus

Question

Form a function \(z=f(x, y)\) such that \(f_{x}\) and \(f_{y}\) match those given. $$ f_{x}=\sin y+1, \quad f_{y}=x \cos y $$

Step-by-Step Solution

Answer

The function is \(f(x, y) = x \sin y + x + C\), where \(C\) is a constant.

Step 1: Integrate with respect to x

Start by integrating the partial derivative \(f_x = \sin y + 1\) with respect to \(x\). The integral of \(1\) with respect to \(x\) is \(x\), and \(\sin y\) is treated as a constant with respect to \(x\), leading to an integration constant that may be a function of \(y\), \(C(y)\). Thus, we obtain \(f(x, y) = x (\sin y + 1) + C(y)\).

Step 2: Differentiate the found expression with respect to y

Differentiate the expression \(f(x, y) = x (\sin y + 1) + C(y)\) with respect to \(y\). This gives \(f_y = x \cos y + C'(y)\).

Step 3: Match the derivative with given \(f_y\)

Given that \(f_y = x \cos y\), we equate this to the expression found in Step 2: \(x \cos y + C'(y) = x \cos y\). To satisfy this equation, we must have \(C'(y) = 0\), indicating that \(C(y)\) is a constant.

Step 4: Formulate the final function

Since \(C(y)\) is a constant, it contributes no variability with respect to \(y\). Therefore, the final function is \(f(x, y) = x \sin y + x + C\), where \(C\) is a constant.

Key Concepts

Partial Derivatives

Partial derivatives are an essential part of multivariable calculus, making it possible to understand how a function changes in one direction while keeping other variables constant.
This is particularly useful when dealing with functions of more than one variable. When we have a function like \(z = f(x, y)\), which depends on both \(x\) and \(y\), partial derivatives help in analyzing the sensitivity of \(z\) to changes in \(x\) and \(y\) separately.

• The partial derivative with respect to \(x\), denoted as \(f_x\), is the derivative of \(f(x, y)\) treating \(y\) as a constant.
• Similarly, the partial derivative with respect to \(y\), denoted as \(f_y\), involves treating \(x\) as a constant.

In the exercise, \(f_x = \sin y + 1\) and \(f_y = x \cos y\) tells us how the function changes as \(x\) and \(y\) vary. By understanding partial derivatives, we know how isolated changes in one variable affect the overall function.

Integration with Respect to a Variable

Integration is a fundamental concept in calculus and involves finding the original function given its derivative. In the context of partial derivatives, we often integrate with respect to one variable while treating others as constants. This helps in reconstructing a multivariable function from its partial derivatives.
In our exercise, the partial derivative \(f_x = \sin y + 1\) means that to find \(f(x, y)\), we integrate \((\sin y + 1)\) with respect to \(x\).

• The integration results in \(x(\sin y + 1)\) because \((\sin y + 1)\) acts as a constant multiplier while integrating with respect to \(x\).
• During this process, we include an arbitrary function \(C(y)\), since the constant of integration might depend on \(y\).

This technique of integration allows us to build our function step by step, ensuring that we accurately account for any variable dependencies through constant functions or arbitrary functions.

Function of Two Variables

Functions of two variables, such as \(f(x, y)\), are pivotal in multivariable calculus, combining two independent variables to produce a single output. This type of function is visually represented as a surface in three-dimensional space.
The main goal in working with such functions is to find smooth relationships and understand how changes in input variables affect the output.

• For example, in our exercise, the function \(f(x, y) = x \sin y + x + C\) demonstrates how an expression can involve products of variables and constants.
• The constant \(C\) represents any fixed value that must be added to accommodate integration results without introducing variability with respect to \(y\).

This type of analysis extends further into differential equations, optimization problems, and modeling dynamic systems where understanding and manipulating functions with multiple inputs is critical. Functions of two variables offer a solid foundation for these advanced mathematical explorations.