Problem 192 - Calculus 3

Question

Find the total differential of the function \(w=e^{y} \cos (x)+z^{2}.\)

Step-by-Step Solution

Answer

The total differential is \(dw = -e^y \sin(x) dx + e^y \cos(x) dy + 2z dz.\)

Step 1: Understand the Total Differential

The total differential of a function with multiple variables is a way to approximate the change in a function based on changes in its input variables. For a function like \(w = f(x, y, z)\), the total differential \(dw\) is given by \(dw = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \frac{\partial f}{\partial z}dz\).

Step 2: Compute Partial Derivative with Respect to x

Compute the partial derivative of \(w = e^{y} \cos(x)+z^{2}\) with respect to \(x\). Hold \(y\) and \(z\) constant: \(\frac{\partial w}{\partial x} = -e^y \sin(x).\)

Step 3: Compute Partial Derivative with Respect to y

Compute the partial derivative of \(w = e^{y} \cos(x)+z^{2}\) with respect to \(y\). Hold \(x\) and \(z\) constant: \(\frac{\partial w}{\partial y} = e^y \cos(x).\)

Step 4: Compute Partial Derivative with Respect to z

Compute the partial derivative of \(w = e^{y} \cos(x)+z^{2}\) with respect to \(z\). Hold \(x\) and \(y\) constant: \(\frac{\partial w}{\partial z} = 2z.\)

Step 5: Construct the Total Differential

Combine the partial derivatives into the total differential: \(dw = \frac{\partial w}{\partial x} dx + \frac{\partial w}{\partial y} dy + \frac{\partial w}{\partial z} dz.\)Substitute the computed partial derivatives:\[dw = (-e^y \sin(x)) dx + (e^y \cos(x)) dy + (2z) dz.\]

Key Concepts

Partial Derivative

A partial derivative is like a regular derivative, but it focuses on one variable at a time. Imagine a function with many variables, like the weather has temperature, humidity, and wind speed. If we only care about how the weather changes with temperature, we find the partial derivative with respect to temperature.

For our function, which is given by \(w = e^{y} \cos(x) + z^2\), taking the partial derivative with respect to \(x\) means treating \(y\) and \(z\) as constants. This gives us \(\frac{\partial w}{\partial x} = -e^y \sin(x)\).

• Focus on one variable, hold others constant.
• Helps isolate the effect of one variable change.

You can do this for each variable: \(y\) and \(z\). This shows how sensitive \(w\) is to changes in each variable. A very useful tool in analyzing functions.

Multivariable Calculus

Multivariable calculus is the branch of mathematics where you deal with functions of more than one variable. It's like regular calculus but expanded to consider multi-dimensional space, like land, sea, and air. Think about graphs not just drawn on paper, but in real 3D or even higher dimensions.

With the function \(w = e^{y} \cos(x) + z^2\), multivariable calculus helps us understand how \(w\) changes when \(x\), \(y\), or \(z\) change. By calculating partial derivatives, we can create a complete picture of how the function behaves.

• Work in two or more dimensions.
• Essential for physics, engineering, and economics.

The total differential uses these partial derivatives to approximate changes in the entire function. It's like having a map to predict how your journey will go as you change direction in one variable or travel up or down in another.

Function Approximation

Function approximation is about estimating how a function behaves. The total differential is a technique for making these estimations. By using the partial derivatives, we can predict small changes in the function's output based on small changes in its inputs.

Approximating the change in \(w = e^{y} \cos(x) + z^2\) involves calculating \(dw = (-e^y \sin(x)) dx + (e^y \cos(x)) dy + (2z) dz\). Each term in this expression gives a snapshot of how sensitive \(w\) is to changes in \(x\), \(y\), and \(z\).

• Use the total differential to estimate function changes.
• Useful in real-world problem solving.

By understanding these concepts, you can better tackle complex problems, anticipate results, and apply this knowledge in various fields. It's a powerful way of breaking down and approximating otherwise complicated equations.