Problem 31 - Calculus Early Transcendentals

Question

Use differentials to approximate the change in \(z\) for the given changes in the independent variables. \(z=2 x-3 y-2 x y\) when \((x, y)\) changes from (1,4) to (1.1,3.9)

Step-by-Step Solution

Answer

Answer: The approximate change in z is -0.1.

Step 1: Calculate partial derivatives

To find the partial derivatives of the function \(z(x, y) = 2x - 3y - 2xy\), we will differentiate with respect to each variable: The partial derivative with respect to \(x\) is: $$ \frac{\partial z}{\partial x} = \frac{\partial (2x - 3y - 2xy)}{\partial x} = 2 - 2y $$ The partial derivative with respect to \(y\) is: $$ \frac{\partial z}{\partial y} = \frac{\partial (2x - 3y - 2xy)}{\partial y} = -3 - 2x $$

Step 2: Evaluate partial derivatives at given point

Now that we have our partial derivative expressions, we need to plug in the given point \((x, y) = (1, 4)\): $$ \frac{\partial z}{\partial x}(1, 4) = 2 - 2(4) = -6 \\ \frac{\partial z}{\partial y}(1, 4) = -3 - 2(1) = -5 $$

Step 3: Calculate the differential \(dz\)

Now that we have our evaluated partial derivatives and the changes in \(x\) and \(y\), \(dx = 0.1\) and \(dy = -0.1\), we can calculate the differential \(dz\): $$ dz = (-6)(0.1) + (-5)(-0.1) = -0.6 + 0.5 = -0.1 $$

Step 4: Interpret the result

The differential \(dz\) is approximately equal to the change in the dependent variable \(z\). Thus, we can say that when \((x, y)\) changes from \((1, 4)\) to \((1.1, 3.9)\), the change in \(z\) is approximately \(-0.1\).

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental aspect of differential calculus. They help us understand how a multivariable function changes when we change one variable while keeping others constant.
For a function like \(z = 2x - 3y - 2xy\), partial derivatives allow us to calculate how \(z\) changes with respect to changes in \(x\) and \(y\) individually.

• The partial derivative with respect to \(x\), \(\frac{\partial z}{\partial x}\), is found by treating \(y\) as a constant and differentiating \(z\) concerning \(x\). In this exercise, this gave us \(2 - 2y\).
• Similarly, for the partial derivative with respect to \(y\), \(\frac{\partial z}{\partial y}\), we treat \(x\) as constant, yielding \(-3 - 2x\).

Evaluating these derivatives at a specific point tells us the rate of change of \(z\) at that point due to changes in \(x\) and \(y\).
For example, at point \((1, 4)\), we find that changes in \(x\) decrease \(z\) sharply (\(-6\)) while changes in \(y\) also decrease \(z\) but less sharply (\(-5\)).

Function Approximation

Function approximation is about estimating how a function behaves when its inputs change slightly. Differential calculus provides tools for linear approximation, which uses tangents to estimate function values around a point.
In our exercise, we express the change in \(z\) when \(x\) changes by \(\Delta x\) and \(y\) by \(\Delta y\) using the differentials:

• \(dz = \frac{\partial z}{\partial x} \cdot dx + \frac{\partial z}{\partial y} \cdot dy\)
• This formula allows us to approximate the change in \(z\) using the evaluated partial derivatives multiplied by the actual changes in \(x\) and \(y\).

Linear approximation simplifies complex changes into manageable predictions.
By applying this to \((x, y)\) changing from \((1, 4)\) to \((1.1, 3.9)\), we estimate \(z\)'s change to be \(-0.1\). This is much easier than recalculating the new value directly without this method.

Chain Rule

The chain rule helps in computing the derivative of a composite function, meaning when a function is built from other functions. It's invaluable in partial derivatives because it considers how changes in one part affect the whole.
In this exercise, though we didn't explicitly use the chain rule, its principles underpin how we combine derivatives.

• The differentials \(dx\) and \(dy\) represent small changes in the inputs.
• The partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) act like multipliers, showing how much \(z\) changes concerning \(x\) and \(y\).

Even though we didn't directly chain multiple functions together, understanding changes in parts' influence on the larger system is foundational for applying the chain rule in complex analyses. It assists in breaking down functions and understanding their connectivity, enriching our comprehension of related rates and changes.