91Ó°ÊÓ

Partial derivatives are a crucial concept when dealing with functions of multiple variables. They allow us to understand how a function changes as we adjust one variable while keeping the others constant. This can be particularly useful in fields like engineering and economics, where systems often depend on multiple factors. In this context, we look at the function:\[ f(x, y) = 12 - 4x^2 - 8y^2 \] To find the partial derivatives, we differentiate the function with respect to each variable separately:

• For the partial derivative with respect to \(x\), we treat \(y\) as a constant: \[ \frac{\partial f}{\partial x} = -8x \]
• For the partial derivative with respect to \(y\), we treat \(x\) as a constant: \[ \frac{\partial f}{\partial y} = -16y \]

This helps us understand how the function's slope changes along the \(x\) and \(y\) axes, respectively. Evaluating these derivatives at a specific point, such as \((-1, 4)\), gives us:

• \( \left.\frac{\partial f}{\partial x}\right|_{(-1,4)} = 8 \)
• \( \left.\frac{\partial f}{\partial y}\right|_{(-1,4)} = -64 \)

These values are essential as they provide the gradient needed to construct the tangent plane.

The tangent plane provides a linear approximation of a function at a given point. Imagine the surface created by a function as a 3D shape. The tangent plane is the flat surface that most closely approximates that shape near the point of tangency.In practical terms, the tangent plane formula is used to estimate values of a function near a certain point using its partial derivatives. For the function we are tackling, and at the point \((-1, 4)\), the expression of the tangent plane is given by:\[ L(x, y) = f(-1,4) + \left.\frac{\partial f}{\partial x}\right|_{(-1,4)}(x+1) + \left.\frac{\partial f}{\partial y}\right|_{(-1,4)}(y-4) \]By calculating \(f(-1,4) = -116\), and substituting the calculated partial derivatives, we form the equation:\[ L(x, y) = -116 + 8(x+1) - 64(y-4) \]This linear function or plane gives us a way to approximate or estimate the values of \(f(x, y)\) for inputs close to \((-1, 4)\). Through this, we get a glimpse of how this function behaves locally around that point.

Function estimation through linear approximation is a powerful method for making predictions about function values without having to compute them directly from the original, possibly complex function. By using the tangent plane equation obtained earlier, we can estimate values of the function near the point of interest.In our exercise, the goal was to estimate \(f(-1.05, 3.95)\). By substituting \((-1.05, 3.95)\) into the linear approximation equation:\[ L(-1.05,3.95) = -116 + 8(-1.05+1) - 64(3.95-4) \]Through simple arithmetic, the result came as:\[ L(-1.05,3.95) = -113.2 \]This essentially informs us that the estimated value of the function at \((-1.05, 3.95)\) is around \( -113.2 \). Such estimations are particularly useful when performing calculations manually, as they provide quick insights without complex computations. Moreover, linear approximations like this are used in real-world applications to make informed predictions based on similar contexts.