Problem 5 - Calculus Early Transcendentals

Question

Given that \(w=F(x, y, z),\) and \(x, y,\) and \(z\) are functions of \(r\) and \(s\), sketch a Chain Rule tree diagram with branches labeled with the appropriate derivatives.

Step-by-Step Solution

Answer

Question: Sketch a Chain Rule tree diagram for the multivariable function \(w = F(x, y, z)\), where \(x, y,\) and \(z\) are functions of the variables \(r\) and \(s\). Label each branch with the appropriate derivatives. Answer: The Chain Rule tree diagram is as follows: w /|\ / | \ dw/dx dw/dy dw/dz x y z /| |\ |\ / | | \ | \ dx/dr dx/ds dy/dr dy/ds dz/dr dz/ds r s r s r s Labels: \(dw/dx = \frac{\partial w}{\partial x}\), \(dw/dy = \frac{\partial w}{\partial y}\), \(dw/dz = \frac{\partial w}{\partial z}\), \(dx/dr = \frac{\partial x}{\partial r}\), \(dx/ds = \frac{\partial x}{\partial s}\), \(dy/dr = \frac{\partial y}{\partial r}\), \(dy/ds = \frac{\partial y}{\partial s}\), \(dz/dr = \frac{\partial z}{\partial r}\), and \(dz/ds = \frac{\partial z}{\partial s}\).

Step 1: Understand Chain Rule

The Chain Rule is a formula to compute the derivative of a composite function. In our case, since \(x\), \(y\), and \(z\) are functions of \(r\) and \(s\), we can say \(w\) is a composite function. The Chain Rule will help us find the derivative of \(w\) with respect to \(r\) and \(s\).

Step 2: Setup the Tree Diagram

To set up the tree diagram, we'll first draw a node for \(w\). Then, draw three branches stemming out from the \(w\) node, connecting to three nodes for \(x\), \(y\), and \(z\) respectively. For each of these nodes, create two branches for each variable \(r\) and \(s\). The tree should look like this: w /|\ / | \ x y z /| |\ |\ r s r s r s

Step 3: Apply Chain Rule and Label Branches

Now we will use the Chain Rule and partial derivatives to derive the appropriate terms to label each branch. For the branches connecting \(w\) to \(x, y,\) and \(z\): \(\frac{\partial w}{\partial x}\), \(\frac{\partial w}{\partial y}\), and \(\frac{\partial w}{\partial z}\). For the branches connecting \(x\) to \(r\) and \(s\): \(\frac{\partial x}{\partial r}\) and \(\frac{\partial x}{\partial s}\). For the branches connecting \(y\) to \(r\) and \(s\): \(\frac{\partial y}{\partial r}\) and \(\frac{\partial y}{\partial s}\). For the branches connecting \(z\) to \(r\) and \(s\): \(\frac{\partial z}{\partial r}\) and \(\frac{\partial z}{\partial s}\). The complete labeled tree diagram would look like: w /|\ / | \ dw/dx dw/dy dw/dz x y z /| |\ |\ / | | \ | \ dx/dr dx/ds dy/dr dy/ds dz/dr dz/ds r s r s r s Here, \(dw/dx = \frac{\partial w}{\partial x}\), \(dw/dy = \frac{\partial w}{\partial y}\), \(dw/dz = \frac{\partial w}{\partial z}\), \(dx/dr = \frac{\partial x}{\partial r}\), \(dx/ds = \frac{\partial x}{\partial s}\), \(dy/dr = \frac{\partial y}{\partial r}\), \(dy/ds = \frac{\partial y}{\partial s}\), \(dz/dr = \frac{\partial z}{\partial r}\), and \(dz/ds = \frac{\partial z}{\partial s}\).

Key Concepts

Composite Functions

Composite functions are a central concept in calculus when dealing with functions that are nested within each other. A composite function is essentially a function of another function, which means it is comprised of two or more functions combined together. In the context of the problem, let's consider the function \( w = F(x, y, z) \), where \( x \), \( y \), and \( z \) are each individually a function of \( r \) and \( s \). This forms a composite situation because \( w \) is dependent on the intermediate variables \( x \), \( y \), and \( z \), which in turn depend on other variables, \( r \) and \( s \). This chain of dependencies makes \( w \) a composite function of \( r \) and \( s \) through \( x \), \( y \), and \( z \).

• Nested Dependencies: Understand that each function depends on another function.
• Direct and Indirect Relations: \( w \) is directly related to \( x, y, z \), but indirectly to \( r, s \).

By using the Chain Rule, we can compute how changes in \( r \) and \( s \) affect \( w \) through these composite relationships, which is crucial in multivariable calculus.

Partial Derivatives

Partial derivatives are an extension of the derivative concept to functions of several variables. Instead of describing how a function changes with just one variable, partial derivatives explain how a function changes with respect to each particular variable, holding the others constant. In our scenario, partial derivatives are used to find how \( w \) changes when you only tweak \( x, y, \) or \( z \) individually.
For instance, \( \frac{\partial w}{\partial x} \) tells us the rate at which \( w \) changes with respect to variable \( x \), assuming \( y \) and \( z \) remain unchanged. Similarly, \( \frac{\partial x}{\partial r} \) indicates how \( x \) shifts when \( r \) changes, keeping \( s \) constant.

• Independent Changes: Understand how each variable individually affects the function.
• Fixed Variables: Key is to see one variable change while others stay constant.

Hence, employing partial derivatives in the chain rule helps in managing these changes to derive the full impact on the ultimate function, \( w \). This is particularly useful in multidimensional analysis where variables are interlinked.

Tree Diagram

A tree diagram is a visual tool used to map out relationships in a structured format. In the context of the problem, the tree diagram helps us visualize how changes made to \( r \) and \( s \) will propagate through \( x, y, \) and \( z \) to ultimately affect \( w \). This is useful because it clearly shows the structure and dependencies between the variables, making it easier to apply the chain rule.
At the top of the tree, you'll find \( w \), and branching out from \( w \) are the intermediate variables \( x, y, \) and \( z \). From each of these, additional branches represent their dependence on \( r \) and \( s \).

• Visual Clarity: Provides a clear map of dependencies and relationships.
• Derivation Pathways: Shows the path on how to derive \( w \) through the Chain Rule.

This structure allows you to label each branch with the corresponding derivative (such as \( \frac{\partial w}{\partial x} \), \( \frac{\partial x}{\partial r} \)), simplifying the process of understanding how combined variable changes affect \( w \). Tree diagrams thus serve as a handy guide for unraveling complex variable interactions.