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Chapter 14: Problem 47

Find the linearization \(L(x, y, z)\) of the function \(f(x, y, z)\) at \(P_{0} .\) Then find an upper bound for the magnitude of the error \(E\) in the approximation \(f(x, y, z) \approx L(x, y, z)\) over the region \(R\) $$ \begin{array}{l}{f(x, y, z)=x y+2 y z-3 x z \text { at } P_{0}(1,1,0)} \\ {R :|x-1| \leq 0.01, \quad|y-1| \leq 0.01, \quad|z| \leq 0.01}\end{array} $$

Short Answer

Expert verified
The linearization is \( L(x, y, z) = x + y - z \), and the error is at most 0.00135.

Step by step solution

01

Find Partial Derivatives

First, we need to find the partial derivatives \( f_x, f_y, \text{ and } f_z \) of the function \( f(x, y, z) = xy + 2yz - 3xz \).\- \( f_x = \frac{\partial (xy + 2yz - 3xz)}{\partial x} = y - 3z \)\- \( f_y = \frac{\partial (xy + 2yz - 3xz)}{\partial y} = x + 2z \)\- \( f_z = \frac{\partial (xy + 2yz - 3xz)}{\partial z} = 2y - 3x \)
02

Evaluate Partial Derivatives at P鈧�

Now, evaluate the partial derivatives at the given point \( P_0(1,1,0)\). \- \( f_x(1,1,0) = 1 - 3 \times 0 = 1 \)\- \( f_y(1,1,0) = 1 + 2 \times 0 = 1 \)\- \( f_z(1,1,0) = 2 \times 1 - 3 \times 1 = -1 \)
03

Write Linearization

The linearization \( L(x, y, z) \) at point \( P_0(1,1,0) \) is: \\[L(x, y, z) = f(1,1,0) + f_x(1,1,0)(x-1) + f_y(1,1,0)(y-1) + f_z(1,1,0)(z-0)\] \- First, find \( f(1,1,0) \): \( f(1,1,0) = 1 \times 1 + 2 \times 1 \times 0 - 3 \times 1 \times 0 = 1 \)\Thus, \( L(x, y, z) = 1 + 1(x-1) + 1(y-1) - 1z = x + y - z \).
04

Compute Hessian Matrix

The Hessian matrix \( H \) includes second-order partial derivatives. We need these to estimate the error. Compute them: \- \( f_{xx} = \frac{\partial^2 f}{\partial x^2} = 0 \) \- \( f_{yy} = \frac{\partial^2 f}{\partial y^2} = 0 \) \- \( f_{zz} = \frac{\partial^2 f}{\partial z^2} = 0 \) \- \( f_{xy} = \frac{\partial^2 f}{\partial x \partial y} = 1 \) \- \( f_{xz} = \frac{\partial^2 f}{\partial x \partial z} = -3 \) \- \( f_{yz} = \frac{\partial^2 f}{\partial y \partial z} = 2 \)
05

Evaluate Derivatives on Region R

To find an upper bound for the error, evaluate the maximum values of the second-order partial derivatives over the region \( R \): \The maximum value of any second derivative here is \( 3 \), which is from \( f_{xz} \) and the absolute bounds \( |f_{xz}| \) in the region.
06

Estimate Upper Bound of Error

Use the formula for the error term: \\[E = \frac{M}{2} \cdot (|x-1| + |y-1| + |z|)^2\] \Where \( M \) is the maximum of the absolute values of second-order partial derivatives over the region. Since \( M = 3 \), compute: \\[E = \frac{3}{2} \cdot (0.01 + 0.01 + 0.01)^2 = \frac{3}{2} \cdot 0.0009 = 0.00135\]Thus, the error \( E \) is at most 0.00135.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Approximation
Linear approximation is a way to approximate a function using a linear function, which is its tangent plane or line. It's most useful when you need to estimate the function values near a particular point.

For a multivariable function, this involves the first partial derivatives, which account for the slope in any direction. The linearization helps simplify calculations by replacing a complex surface with a flat plane, making predictions easier.

In our case, the linearization of the function at the point \( P_0(1,1,0) \) is expressed with the formula:
  • \[ L(x, y, z) = f(1,1,0) + f_x(1,1,0)(x-1) + f_y(1,1,0)(y-1) + f_z(1,1,0)(z-0) \]
This can make evaluating \( f(x, y, z) \) much simpler near the point \( P_0 \).
Partial Derivatives
Partial derivatives are a fundamental concept in multivariable calculus. They measure how a function changes as each of its input variables changes, while other variables are held constant.

For the function \( f(x, y, z) = xy + 2yz - 3xz \), the partial derivatives with respect to \( x, y, \text{ and } z \) provide the rate of change of \( f \) along the respective axes, indicating how \( f \) behaves locally around a point. These partial derivatives can be found as:
  • \( f_x = y - 3z \)
  • \( f_y = x + 2z \)
  • \( f_z = 2y - 3x \)
Finding these derivatives is the first essential step in the process of linearization and assessing the function's local geometry.
Hessian Matrix
The Hessian matrix in multivariable calculus is a square matrix of second-order partial derivatives. It plays a crucial role in determining the curvature of a function.

To evaluate the function \( f(x, y, z) \) more accurately, we need the Hessian matrix to assess the consistency of directional changes, composed of all second-order partial derivatives:
  • \[ H = \begin{bmatrix} f_{xx} & f_{xy} & f_{xz} \ f_{yx} & f_{yy} & f_{yz} \ f_{zx} & f_{zy} & f_{zz}\end{bmatrix} \]
This matrix helps in assessing the behavior of the function around a point more thoroughly than first-order derivatives alone.
Error Bound
Error bound is a critical factor in ensuring the accuracy of the approximation we make using linearization or any other method.

When approximating \( f(x, y, z) \) by its linearization, the error bound helps us estimate the deviation from true function values within a certain range. This ensures precision and reliability in linear approximations over a specified region.

For our exercise, the error bound is calculated using the formula:
  • \[ E = \frac{M}{2} \cdot (|x-1| + |y-1| + |z|)^2 \]
where \( M \) is the largest second-order derivative value over region \( R \).
Second-Order Partial Derivatives
Second-order partial derivatives provide additional details about the curvature of a function鈥檚 surface. They can inform you about how the rate of change of the function changes as we move around the point in multiple directions.

For instance, the values
  • \( f_{xx} = 0 \)
  • \( f_{yy} = 0 \)
  • \( f_{zz} = 0 \)
imply that the function is locally linear in isolation along each axis, but the mixed second-order partial derivatives like \( f_{xy} = 1 \), \( f_{xz} = -3 \), and \( f_{yz} = 2 \) are non-zero, indicating that the combined influence of variables \( x, y, z \) on each other leads to curvature or interaction effects in the function's behavior.

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