91Ó°ÊÓ

Linear functions represent some of the simplest types of functions in multivariable calculus. A linear function of two variables, as seen in the exercise, is written as \( f(x, y) = ax + by + c \). Here, \(a\) and \(b\) are the coefficients of \(x\) and \(y\) respectively, while \(c\) is a constant term. This type of function essentially describes a plane in three-dimensional space.

• The coefficients \(a\) and \(b\) determine the slope or tilt of the plane in relation to each axis.
• The constant term \(c\) shifts the plane up or down depending on its value.

In solving the exercise, recognizing the form of a linear function helps in easily determining the partial derivatives and, subsequently, constructing the required function with the given conditions.

Partial derivatives are fundamental in understanding functions of multiple variables. They represent the rate of change of a function with respect to one variable while keeping the others constant.
For the linear function \( f(x, y) = ax + by + c \), the partial derivative with respect to \(x\), denoted as \( \frac{\partial f}{\partial x} \), is equal to \(a\). Meanwhile, the partial derivative with respect to \(y\), \( \frac{\partial f}{\partial y} \), is equal to \(b\).

• In our exercise, knowing that \( \frac{\partial f}{\partial x} = \pi \) instantly tells us that \(a = \pi\).
• Similarly, \( \frac{\partial f}{\partial y} = e \) lets us conclude that \(b = e\).

Calculating these derivatives gives us crucial information for constructing the linear function, illustrating how changes in each variable affect the output of the function.

Functions of two variables are a natural extension of single-variable functions and allow us to explore how changes in two input variables simultaneously alter output. The given function \(f(x, y) = ax + by + c\) considers both \(x\) and \(y\) to determine its value.

• Two-variable functions can be visualized as a surface in three-dimensional space.
• In this type of function, each pair \((x, y)\) is associated with one single output value \(f(x, y)\).

In the exercise, the condition \(f(0,0) = \ln 2\) serves as an anchor point. Substituting \(x = 0\) and \(y = 0\) into the function helps us find the constant term \(c\). This step utilizes the idea of evaluating the function at a specific point, which is a common technique in analyzing two-variable functions. Understanding these concepts allows for precise interpretations of how the function behaves across different values.