91Ó°ÊÓ

Partial derivatives represent the derivatives of a multivariable function with respect to one variable at a time, keeping the others constant. Consider the function given in our exercise, \( f(x, y, z) = x^2 + xy + yz + \frac{1}{4}z^2 \). To find the linearization, we need to calculate the partial derivatives with respect to each variable: \( x \), \( y \), and \( z \).

• For \( x \), the derivative \( \frac{\partial f}{\partial x} = 2x + y \)
• For \( y \), the derivative \( \frac{\partial f}{\partial y} = x + z \)
• For \( z \), the derivative \( \frac{\partial f}{\partial z} = y + \frac{1}{2}z \)

Evaluating these at point \( P_0(1, 1, 2) \), we find:

• \( \left.\frac{\partial f}{\partial x}\right|_{P_0} = 3 \)
• \( \left.\frac{\partial f}{\partial y}\right|_{P_0} = 3 \)
• \( \left.\frac{\partial f}{\partial z}\right|_{P_0} = 2 \)

These derivatives inform us how the function varies in each direction, helping to formulate our linear approximation.

Error estimation in linearization approximates how far an approximation \( L(x, y, z) \) might deviate from the actual function \( f(x, y, z) \). As we linearize a function, some information is lost, primarily concerning curvature beyond the local region around the point of tangency \( P_0 \).
To estimate the error,

• identify the second derivatives
• determine their maximum values within the region of interest

In this problem, the maximum second derivative in region \( R \) is \( 2 \).
Using the error estimate formula: \ \[ E \leq \frac{1}{2} \times \text{max second derivative} \times \text{sum of max changes in } x, y, z \]
The changes \( \Delta x = 0.01 \), \( \Delta y = 0.01 \), and \( \Delta z = 0.08 \) sum up to \( 0.1 \), leading to an error estimate \( E \leq 0.05 \). This gives a bound within which the true value may deviate from the linear approximation.

Second partial derivatives provide information about the curvature of a function. They measure how the rate of change, given by the first derivative, changes. For our function, we find second partial derivatives for each variable combination:

• \( \frac{\partial^2 f}{\partial x^2} = 2 \)
• \( \frac{\partial^2 f}{\partial y^2} = 0 \)
• \( \frac{\partial^2 f}{\partial z^2} = \frac{1}{2} \)
• \( \frac{\partial^2 f}{\partial x\partial y} = 1 \)
• \( \frac{\partial^2 f}{\partial x\partial z} = 0 \)
• \( \frac{\partial^2 f}{\partial y\partial z} = 1 \)

Particularly, the self-second derivatives (like \( \frac{\partial^2 f}{\partial x^2} \)) inform us about concavity or convexity along each axis at the point of interest. In this context, they are crucial for our error estimation since linear approximation requires understanding how the function curves beyond the tangent plane.

Multivariable calculus extends the concepts of single-variable calculus into functions of several variables. In this exercise, we're dealing with a function of three variables \( f(x, y, z) \). Calculus allows us to find tangent planes via derivatives and assess how close approximations are to the true function values through techniques like linearization.

• To linearize, you compute the function's partial derivatives and evaluate them at a specific point.
• The approximated function \( L(x, y, z) \) gives a flat representation of the function's behavior near that point.
• Error estimation considers higher-order derivatives like second partials to gauge the precision of our linear model.

This exploration of multivariable calculus enhances our understanding of complex systems, allowing for more accurate and practical approximations in real-world applications.