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Chapter 14: Problem 22

Draw a branch diagram and write a Chain Rule formula for each derivative. \(\begin{array}{l}{\frac{\partial w}{\partial p} \text { for } w=f(x, y, z, v), \quad x=g(p, q), \quad y=h(p, q)} \\ {z=j(p, q), \quad v=k(p, q)}\end{array}\)

Short Answer

Expert verified
Draw branches from \( w \) to \( x, y, z, v \), then apply the chain rule to find \( \frac{\partial w}{\partial p} \).

Step by step solution

01

Identify Function Dependencies

Recognize that the function \( w = f(x, y, z, v) \) depends on four variables \( x, y, z, \) and \( v \), each of which is a function of \( p \) and \( q \).
02

Draw the Branch Diagram

Illustrate a branch diagram showing \( w \) at the top level branching into \( x, y, z, \) and \( v \). Each of these branches further splits into \( p \) and \( q \).
03

Apply the Chain Rule Formula

Write the chain rule expression for \( \frac{\partial w}{\partial p} \) : \[\frac{\partial w}{\partial p} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial p} + \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial p} + \frac{\partial w}{\partial z} \cdot \frac{\partial z}{\partial p} + \frac{\partial w}{\partial v} \cdot \frac{\partial v}{\partial p}\].
04

Understand Each Term of the Chain Rule

Realize that each term in the chain rule formula represents the rate of change of \( w \) with respect to one of its variables, multiplied by the rate of change of that variable with respect to \( p \). For instance, \( \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial p} \) computes how changes in \( p \) affect \( x \) and subsequently \( w \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Derivatives
In multivariable calculus, partial derivatives are an essential tool for analyzing functions with more than one variable. Imagine you are working with a function like \(w = f(x, y, z, v)\). Each variable \(x, y, z, v\) could independently affect \(w\). To see how \(w\) changes with respect to just one of these variables, you need a partial derivative.
The partial derivative of \(w\) with respect to a variable, say \(x\), is written as \(\frac{\partial w}{\partial x}\). This expression shows how \(w\) changes as \(x\) changes, while keeping other variables constant. It鈥檚 like taking a slice of a surface at a certain height and looking only at how the height changes as you move in one specific direction.
When working with partial derivatives:
  • Pick one variable to differentiate with respect to, treating others as constants.
  • Use the rules of differentiation from single-variable calculus, applying them to the chosen variable.
  • Interpreting the result can tell you much about the effect of that variable on the function overall.
Multivariable Functions
Multivariable functions are functions with more than one input variable. These functions map points from a higher-dimensional space to either the real numbers or another space.
For instance, consider a function \(w = f(x, y, z, v)\). This function takes four inputs and outputs a single value or variable, \(w\).
Multivariable functions are common in physics and engineering, where quantities are often influenced by multiple factors. Understanding how each variable affects the end result requires analyzing the relationships in these functions.
Some key points about multivariable functions are:
  • Each variable can represent a different dimension or aspect of the system.
  • Graphing these functions requires higher-dimensional plots or simplifications via cross-sectional views.
  • They are used to describe surfaces, volumes, and other complex relationships in real-world scenarios.
Calculus Branch Diagram
A calculus branch diagram is a visual tool that helps in understanding and structuring the Chain Rule in multivariable calculus. It clearly shows how complex functions depend on multiple variables, which in turn depend on yet more variables.
Taking our example, with \(w = f(x, y, z, v)\), each of \(x, y, z, v\) depends on \(p\) and \(q\). The branch diagram starts with \(w\) at the top and branches out to each of \(x, y, z, v\). Then, these branches further divide, showing their dependence on \(p\) and \(q\).
The branch diagram helps by:
  • Visualizing the structure of dependencies within a complex function.
  • Simplifying the application of the Chain Rule by making each path of dependency clear.
  • Aiding in the identification of partial derivatives needed for finding rates of change across variable paths.

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Most popular questions from this chapter

Suppose that \begin{equation}w=x^{2}-y^{2}+4 z+t \quad \text { and } \quad x+2 z+t=25.\end{equation} Show that the equations \begin{equation}\frac{\partial w}{\partial x}=2 x-1 \quad \text { and } \quad \frac{\partial w}{\partial x}=2 x-2\end{equation} each give \(\partial w / \partial x,\) depending on which variables are chosen to be dependent and which variables are chosen to be independent. Identify the independent variables in each case.

The condition \(\nabla f=\lambda \nabla g\) is not sufficient Although \(\nabla f=\lambda \nabla g\) is a necessary condition for the occurrence of an extreme value of \(f(x, y)\) subject to the conditions \(g(x, y)=0\) and \(\nabla g \neq 0,\) it does not in itself guarantee that one exists. As a case in point, try using the method of Lagrange multipliers to find a maximum value of \(f(x, y)=x+y\) subject to the constraint that \(x y=16 .\) The method will identify the two points \((4,4)\) and \((-4,-4)\) as candidates for the location of extreme values. Yet the sum \((x+y)\) has no maximum value on the hyperbola \(x y=16\) . The farther you go from the origin on this hyperbola in the first quadrant, the larger the sum \(f(x, y)=x+y\) becomes.

In Exercises \(49-52\) , find an equation for and sketch the graph of the level curve of the function \(f(x, y)\) that passes through the given point. $$ f(x, y)=\frac{2 y-x}{x+y+1}, \quad(-1,1) $$

You will explore functions to identify their local extrema. Use a CAS to perform the following steps: a. Plot the function over the given rectangle. b. Plot some level curves in the rectangle. c. Calculate the function's first partial derivatives and use the CAS equation solver to find the critical points. How do the critical points relate to the level curves plotted in part (b)? Which critical points, if any, appear to give a saddle point? Give reasons for your answer. d. Calculate the function's second partial derivatives and find the discriminant \(f_{x x} f_{y y}-f_{x y}^{2}\) . e. Using the max-min tests, classify the critical points found in part (c). Are your findings consistent with your discussion in part (c)? $$\begin{array}{l}{f(x, y)=5 x^{6}+18 x^{5}-30 x^{4}+30 x y^{2}-120 x^{3}} \\\ {-4 \leq x \leq 3, \quad-2 \leq y \leq 2}\end{array}$$

Let $$f(x, y)=\left\\{\begin{array}{ll}{0,} & {x^{2} < y<2 x^{2}} \\ {1,} & {\text { otherwise }}\end{array}\right.$$ Show that \(f_{x}(0,0)\) and \(f_{y}(0,0)\) exist, but \(f\) is not differentiable at \((0,0) .\)
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