91Ó°ÊÓ

When dealing with functions of multiple variables like \(f(x, y)\), partial derivatives play a crucial role in understanding how the function behaves as each variable changes.
To compute a partial derivative with respect to one variable, we treat other variables as constants. For example, when finding the partial derivative of \(f(x, y)\) with respect to \(x\), denoted as \(f_x\), the variable \(y\) is held constant. Similarly, for \(f_y\), \(x\) is treated as a constant.

• Partial derivatives are used to formulate Taylor approximations, which estimate the value of a function near a given point.
• In this exercise, you're required to find these derivatives at the origin \((0, 0)\) because the Taylor series is being expanded around this point.

For the given function, these derivatives at \((0, 0)\) will help craft the first and second-order approximations that help in estimating values like \(f(0.2, -0.3)\). Understanding how to compute these derivatives is fundamental in building different orders of Taylor approximations.

A second-order approximation offers more accuracy than a first-order approximation by considering how the function curves around a point.
This involves not only the first derivatives but also the second derivatives. For a function \(f(x, y)\), the second-order Taylor approximation around a point \((x_0, y_0)\) is given by:\[T_2(x, y) = f(x_0, y_0) + f_x(x_0, y_0)(x-x_0) + f_y(x_0, y_0)(y-y_0) + \frac{1}{2}f_{xx}(x_0, y_0)(x-x_0)^2 + f_{xy}(x_0, y_0)(x-x_0)(y-y_0) + \frac{1}{2}f_{yy}(x_0, y_0)(y-y_0)^2\]

• \(f_{xx}, f_{xy}, f_{yy}\) are the second-order partial derivatives. These derivatives account for the changes in slope of the function, providing a better fit when estimating values close to \((x_0, y_0)\).
• Using these second derivatives, you capture how the function's curvature affects the estimate, which is what makes it more precise than first-order.

For our example, once you calculate these second derivatives at \((0,0)\), you can plug them into the approximation formula and evaluate the value at \((0.2, -0.3)\). This step is vital for refining the accuracy beyond what a straight-line approximation (first-order) could provide.

The first-order approximation provides a simpler, linear estimate of a function's value near a known point. It uses the function's first derivatives to determine the tangent plane at that point, serving as an initial predictive model.
For the function \(f(x, y)\), the first-order Taylor approximation around \((x_0, y_0)\) is expressed as:\[T_1(x, y) = f(x_0, y_0) + f_x(x_0, y_0)(x-x_0) + f_y(x_0, y_0)(y-y_0)\]

• Here, \(f_x\) and \(f_y\) are the partial derivatives with respective variables, indicating how quickly \(f\) changes as \(x\) or \(y\) changes.
• This approximation is essentially a linearization that simplifies the analysis of changes in a function's value as its inputs vary slightly away from \((x_0, y_0)\).

In the example, since both the first derivatives at \((0,0)\) are zero, the resulting first-order approximation is constant \((0)\). When calculated, \(f(0.2, -0.3)\approx 0\) clearly shows the limitation here—it doesn’t account for curvature, thus isn’t as accurate as higher-order approximations.