Problem 7 - APEX Calculus

Question

In Exercises \(5-8,\) find the total differential \(d z\). $$ z=5 x-7 y $$

Step-by-Step Solution

Answer

The total differential is \( dz = 5dx - 7dy \).

Step 1: Understand the Problem

Given the function \( z = 5x - 7y \), we are tasked with finding the total differential \( dz \). This will involve taking partial derivatives with respect to each variable \( x \) and \( y \).

Step 2: Find the Partial Derivative with respect to x

To find the partial derivative of \( z \) with respect to \( x \), denoted as \( \frac{\partial z}{\partial x} \), differentiate \( z = 5x - 7y \) with respect to \( x \) while treating \( y \) as a constant. Thus, \( \frac{\partial z}{\partial x} = 5 \).

Step 3: Find the Partial Derivative with respect to y

To find the partial derivative of \( z \) with respect to \( y \), denoted as \( \frac{\partial z}{\partial y} \), differentiate \( z = 5x - 7y \) with respect to \( y \) while treating \( x \) as a constant. This gives \( \frac{\partial z}{\partial y} = -7 \).

Step 4: Form the Total Differential dz

The total differential \( dz \) of a function \( z = f(x, y) \) is given by \( dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy \). Substitute the derivatives from Steps 2 and 3: \( dz = 5dx - 7dy \).

Step 5: Verify the Expression

Check that the total differential \( dz = 5dx - 7dy \) is consistent by ensuring it aligns with the initial formulation of \( z \) as a linear combination of \( x \) and \( y \). It is a correct linear expression as expected for a linear function.

Key Concepts

Partial Derivatives

Partial derivatives provide a way to measure how a function changes as one of its variables changes, while keeping other variables constant. Let's consider the function from our exercise: \( z = 5x - 7y \). If we want to see how \( z \) changes as \( x \) changes, we compute the partial derivative of \( z \) with respect to \( x \), denoted as \( \frac{\partial z}{\partial x} \). Here, \( y \) is treated as a constant, so the derivative simplifies to \( \frac{\partial z}{\partial x} = 5 \).

• This process helps us understand the sensitivity or rate of change of \( z \) with regard to \( x \).
• Similarly, \( \frac{\partial z}{\partial y} = -7 \) indicates the rate of change of \( z \) when \( y \) changes while \( x \) remains constant.

Partial derivatives are foundational in calculus, especially when examining functions of several variables.

Differentiation

Differentiation, a core process in calculus, involves finding the derivative of a function. When dealing with functions of one variable, we want to know how that one variable can affect the function's output. For functions with more than one variable, the concept of partial differentiation is used to find the rate of change with respect to one variable at a time.
The total differential derived in the exercise \( dz = 5dx - 7dy \) uses differentiation to show how small changes in \( x \) and \( y \) affect the change in \( z \). This is crucial for modeling real-world systems where inputs vary slightly and we need to predict changes in outputs.

• Total differentials provide linear approximations of changes in the function.
• It allows us to predict how small increments in variables propagate through the function's relationship.

Overall, differentiation is a critical tool in understanding and predicting behaviors of functions.

Multivariable Calculus

Multivariable calculus extends calculus concepts to functions with more than one variable.
In the provided exercise, we looked at a linear function of two variables: \( x \) and \( y \). This type of calculus is essential for analyzing situations where changes in multiple inputs affect an outcome.
In multivariable calculus, partial derivatives help in understanding how each variable independently influences the function. With these partial derivatives, the total differential \( dz = 5dx - 7dy \) gives a complete approximation of changes in \( z \) from simultaneous small changes in both \( x \) and \( y \).

• This is particularly valuable in fields like physics, engineering, and economics to evaluate systems with multiple influences.
• Total differential equations can be utilized extensively to solve real-world problems by considering multiple variables simultaneously.

Multivariable calculus, therefore, broadens the scope of calculus by dealing with more complex scenarios involving several changing inputs.