Question

What is meant by a linear approximation of \(z=f(x, y)\) at the point \(P\left(x_{0}, y_{0}\right) ?\)

Short answer

The linear approximation to a function of two variables \(z=f(x, y)\) at the point \(P(x_0 , y_0)\) is the equation of the plane tangent to the surface described by \(f(x, y)\) at the point \((x_0, y_0, f(x_0, y_0))\). Its formula is \(L(x, y) = f(x_0, y_0) + f_x(x_0, y_0) \cdot (x - x_0) + f_y(x_0, y_0) \cdot (y - y_0)\), where \(L(x, y)\) is the linear approximation, and \(f_x(x_0, y_0)\) and \(f_y(x_0, y_0)\) are the partial derivatives of \(f\) with respect to \(x\) and \(y\) evaluated at \((x_0, y_0)\), respectively.

Step-by-step solution

Identify the Concept of Linear Approximation

Linear approximation, also known as linearization, refers to approximating the value of a function around a particular input using the equation of the tangent line at that point. In other words, it is a method to closely approximate a complicated function with a simpler function, specifically, a linear one. This principle comes from the observation that near any given point, many functions look like straight lines when zoomed in closely.

Provide the General Formula for Linear Approximation

For a function of two variables \(z = f(x, y)\), the linear approximation or linearization of \(f\) at a point \(P(x_0 , y_0)\) is given by the formula: \[L(x, y) = f(x_0, y_0) + f_x(x_0, y_0) \cdot (x - x_0) + f_y(x_0, y_0) \cdot (y - y_0)\] where \(L(x, y)\) is the linear approximation of \(f\) at \(P(x_0 , y_0)\), \(f_x(x_0, y_0)\) and \(f_y(x_0, y_0)\) are the partial derivatives of \(f\) with respect to \(x\) and \(y\) evaluated at \((x_0, y_0)\), respectively.

Interpreting the Formula

The term \(f(x_0, y_0)\) in the formula represents the value of the original function at the point of interest. The terms \(f_x(x_0, y_0) \cdot (x - x_0)\) and \(f_y(x_0, y_0) \cdot (y - y_0)\) represent the adjustments to this value made by the linearization based on how far \(x\) and \(y\) are from \(x_0\) and \(y_0\), respectively. Together, they form an equation for a plane that is tangent to the surface described by \(f(x, y)\) at the point \((x_0, y_0, f(x_0, y_0))\). This plane provides our linear approximation of the function around the point \(P(x_0, y_0)\).