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Chapter 13: Problem 65

Find the value of \(\partial z / \partial x\) at the point (1,1,1) if the equation $$ x y+z^{3} x-2 y z=0 $$ defines \(z\) as a function of the two independent variables \(x\) and \(y\) and the partial derivative exists.

Short Answer

Expert verified
\(\partial z / \partial x = -2\) at the point (1, 1, 1).

Step by step solution

01

Implicit Differentiation Setup

Given the equation \(x y + z^3 x - 2 y z = 0\), we are to find \(\frac{\partial z}{\partial x}\) using implicit differentiation. Treat \(z\) as a function of \(x\) and \(y\), then differentiate both sides of the equation with respect to \(x\) while treating \(y\) as a constant.
02

Differentiate Each Term

Differentiate each term with respect to \(x\). The first term \(x y\) differentiates to \(y\). The second term \(z^3 x\), using the product rule, differentiates to \(z^3 + 3z^2 x \frac{\partial z}{\partial x}\). The third term \(-2yz\), applying the product rule, differentiates to \(-2y \frac{\partial z}{\partial x}\).
03

Combine and Simplify

Combine the differentiated terms: \[ y + z^3 + 3z^2 x \frac{\partial z}{\partial x} - 2y \frac{\partial z}{\partial x} = 0. \]Rearrange the equation to solve for \(\frac{\partial z}{\partial x}\):\[ 3z^2 x \frac{\partial z}{\partial x} - 2y \frac{\partial z}{\partial x} = - (y + z^3). \]
04

Factor and Solve for \(\partial z/\partial x\)

Factor \(\frac{\partial z}{\partial x}\) out:\[ \frac{\partial z}{\partial x} (3z^2 x - 2y) = - (y + z^3). \]Solve for \(\frac{\partial z}{\partial x}\):\[ \frac{\partial z}{\partial x} = -\frac{y + z^3}{3z^2 x - 2y}. \]
05

Evaluate at the Given Point

Substitute the point \((1,1,1)\) into the expression for \(\frac{\partial z}{\partial x}\):\[ \frac{\partial z}{\partial x} = -\frac{1 + 1^3}{3(1)^2(1) - 2(1)} = -\frac{2}{3 - 2} = -2. \]

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Key Concepts

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

Implicit Differentiation
In mathematics, implicit differentiation is a method used when dealing with equations where one variable is not explicitly expressed as a function of another. Here, variables are intertwined, making it difficult to isolate one on one side of the equation.

To tackle this:
  • We differentiate each term with respect to the variable of interest.
  • We treat the other variables as constants unless they are explicitly functions of the main variable.
  • This method is particularly useful when dealing with relations involving multiple variables where direct differentiation is cumbersome.
In our given problem, the function was set implicitly as: \[ x y + z^3 x - 2 y z = 0, \]and we treated \( z \) as a function of \( x \) and \( y \) to find \( \frac{\partial z}{\partial x} \). By implicating \( z \) as dependent, we differentiate and isolate the terms, applying calculus rules like the product rule where necessary.
Partial Derivatives
Partial derivatives are a core element of calculus involving functions of multiple variables. They measure how a function changes as one of the variables changes, with others held constant.

Here is how it works:
  • The notation \( \frac{\partial f}{\partial x} \) denotes the partial derivative of a function \( f \) concerning \( x \), treating other variables like \( y \) as constants.
  • Partial derivatives help describe surfaces and multidimensional shapes, giving insight into how those surfaces behave.
In this exercise, the use of \( \frac{\partial z}{\partial x} \) provided insight into how the function behaves when changing \( x \) alone. This approach simplifies analysis in complex systems by focusing on one variable at a time.
Multivariable Calculus
Multivariable calculus extends the ideas of calculus from single-variable functions to those involving several variables. It is foundational for disciplines that require modeling real-world phenomena involving more than one variable at a time.

Key aspects include:
  • Functions that can depend on multiple inputs, e.g., \( f(x, y, z) \).
  • Techniques like finding critical points using gradients, Jacobians, and Hessians.
  • Applications across physics, engineering, economics, and more due to its ability to handle complex models.
In the provided exercise, a function involving \( x \), \( y \), and \( z \) illustrates how relationships in real-world scenarios often require consideration from multiple angles. By applying concepts from multivariable calculus, we can understand and solve such intricate systems, revealing how one variable influences another in a holistic setting.

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Most popular questions from this chapter

Use a CAS to plot the implicitly defined level surfaces. $$x^{2}+z^{2}=1$$

Maximize the function \(f(x, y, z)=x^{2}+2 y-z^{2}\) subject to the constraints \(2 x-y=0\) and \(y+z=0.\)

If \(f\left(x_{0}, y_{0}\right)=3,\) what can you say about \(\lim _{(x, y) \rightarrow\left(x_{0}, y_{\mathrm{a}}\right)} f(x, y)\) if \(f\) is continuous at \(\left(x_{0}, y_{0}\right) ?\) If \(f\) is not continuous at \(\left(x_{0}, y_{0}\right) ?\) Give reasons for your answers.

a. Maximum on line of intersection \(\quad\) Find the maximum value of \(w=x y z\) on the line of intersection of the two planes \(x+y+z=40\) and \(x+y-z=0\) b. Give a geometric argument to support your claim that you have found a maximum, and not a minimum, value of \(w\)

You plan to calculate the area of a long, thin rectangle from measurements of its length and width. Which dimension should you measure more carefully? Give reasons for your answer.
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