Problem 22 - APEX Calculus

Question

In Exercises \(19-22,\) use the information provided and the total differential to make the given approximation. $$ \begin{array}{l} f(3,3,3)=5, f_{x}(3,3,3)=2, f_{y}(3,3,3)=0, f_{z}(3,3,3)= \\ -2 . \text { Approximate } f(3.1,3.1,3.1) \end{array} $$

Step-by-Step Solution

Answer

The approximation for \( f(3.1, 3.1, 3.1) \) is 5.

Step 1: Understand the Total Differential

The total differential of a function of three variables, \( f(x, y, z) \), is given by: \( df = f_x \, dx + f_y \, dy + f_z \, dz \). Here, \( f_x, f_y, f_z \) are the partial derivatives with respect to \( x, y, z \), and \( dx, dy, dz \) are infinitesimal changes in \( x, y, z \) respectively.

Step 2: Determine Changes in Variables

We estimate \( f(3.1, 3.1, 3.1) \) using the known values at \( (3, 3, 3) \). The changes in variables are: \( dx = 3.1 - 3 = 0.1 \), \( dy = 3.1 - 3 = 0.1 \), \( dz = 3.1 - 3 = 0.1 \).

Step 3: Apply the Total Differential

Substitute the values into the total differential formula: \( df = (2)(0.1) + (0)(0.1) + (-2)(0.1) \). This simplifies to \( df = 0.2 + 0 - 0.2 \).

Step 4: Simplify the Differential

The resulting \( df = 0.2 - 0.2 = 0 \). This means the small change \( df \) in the function value is 0.

Step 5: Calculate the Approximation

Using the total differential, the approximation for \( f(3.1, 3.1, 3.1) \) is given by \( f(3, 3, 3) + df \). Since \( df = 0 \), we have \( f(3.1, 3.1, 3.1) \approx 5 + 0 = 5 \).

Key Concepts

Total Differential

The total differential is a powerful concept in differential calculus used to approximate changes in a multivariable function. When you have a function like \( f(x, y, z) \), the total differential \( df \) represents the small change in the function in response to small changes in its variables \( x, y, z \). The formula for total differential is: \( df = f_x \, dx + f_y \, dy + f_z \, dz \),where \( f_x, f_y, f_z \) are the partial derivatives of \( f \) with respect to \( x, y, \) and \( z \), and \( dx, dy, dz \) are the infinitesimal changes in each variable.

• The purpose of the total differential is to capture how changes in each variable contribute to the overall change in the function's value.
• This is useful for estimating how a function behaves near a given point, such as approximating \( f(3.1, 3.1, 3.1) \) by starting at \( f(3, 3, 3) \) and considering small changes.

Partial Derivatives

Partial derivatives play a crucial role in understanding how functions behave with respect to each variable independently in a multivariable context. In simpler terms, when a function has more than one variable, a partial derivative tells you how the function changes as you vary just one of those variables, while keeping the others constant. For a function \( f(x, y, z) \):

• \( f_x \) represents the derivative with respect to \( x \), measured by treating \( y \) and \( z \) as constants.
• Similarly, \( f_y \) and \( f_z \) pertain to the changes in the function with respect to \( y \) and \( z \), respectively.

Understanding the partial derivatives helps determine the contribution of each variable's change to the total change in the function, which is especially helpful for computing the total differential.

Function Approximation

Function approximation is a technique used to estimate the value of a function near a known point using relatively simple computations. In the context of differential calculus, we employ tools such as the total differential to make accurate approximations without needing to directly compute the exact function value at the new point.We take a known value, such as \( f(3, 3, 3) = 5 \) from the exercise, and apply the total differential to approximate \( f(3.1, 3.1, 3.1) \) as shown:

• Calculate the infinitesimal changes \( dx, dy, dz \).
• Use the given partial derivatives to find \( df \).
• Adjust the original function value by this small change: \( f(3, 3, 3) + df \).

This process gives us a good approximation, ideal for quick evaluations especially in complex scenarios like the one presented in the exercise, resulting in \( f(3.1, 3.1, 3.1) \approx 5 \).