/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 79 Gives a function \(f(x, y, z)\) ... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 79

Gives a function \(f(x, y, z)\) and a positive number \(\epsilon .\) In each exercise, show that there exists a \(\delta>0\) such that for all \((x, y, z)\) $$\sqrt{x^{2}+y^{2}+z^{2}}<\delta \Rightarrow|f(x, y, z)-f(0,0,0)|<\epsilon$$ Show that \(f(x, y, z)=x+y-z\) is continuous at every point \(\left(x_{0}, y_{0}, z_{0}\right)\)

Short Answer

Expert verified
The function is continuous at every point because for any \(\epsilon > 0\), setting \(\delta = \frac{\epsilon}{\sqrt{3}}\) ensures the continuity condition is met.

Step by step solution

01

Define the function and the point

Given the function \(f(x, y, z) = x + y - z\), we need to show it is continuous at any arbitrary point \((x_0, y_0, z_0)\). For continuity, we must show that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \(\sqrt{(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2} < \delta\), then \(|f(x, y, z) - f(x_0, y_0, z_0)| < \epsilon\).
02

Compute the difference

The difference \(|f(x, y, z) - f(x_0, y_0, z_0)|\) is calculated as follows: \[|f(x, y, z) - f(x_0, y_0, z_0)| = |(x + y - z) - (x_0 + y_0 - z_0)| = |(x-x_0) + (y-y_0) - (z-z_0)|.\]
03

Apply triangle inequality

Using the triangle inequality on the expression in Step 2 gives: \[|(x-x_0) + (y-y_0) - (z-z_0)| \leq |x-x_0| + |y-y_0| + |z-z_0|.\]
04

Estimate the distance

Because we have \(\sqrt{(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2} < \delta\), it implies that \[|x-x_0| + |y-y_0| + |z-z_0| \leq \sqrt{3} \cdot \delta\] by the inequality \(\sqrt{a^2 + b^2 + c^2} \geq \frac{|a| + |b| + |c|}{\sqrt{3}}\). This implies that: \(|x-x_0| + |y-y_0| + |z-z_0| < \epsilon\) if \(\sqrt{3}\delta < \epsilon\).
05

Choose delta appropriately

To ensure our inequality holds, we set \(\delta = \frac{\epsilon}{\sqrt{3}}\). Then for every \((x, y, z)\) such that \[\sqrt{(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2} < \delta, \] we will have \(|f(x, y, z) - f(x_0, y_0, z_0)| < \epsilon\), satisfying the condition for continuity.

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Key Concepts

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

Continuity of Functions
In multivariable calculus, continuity of a function like \(f(x, y, z)\) at a point \((x_0, y_0, z_0)\) implies that small changes in the input \((x, y, z)\) result in small changes in the output \(f(x, y, z)\). For the function \(f(x, y, z) = x + y - z\), to show that it's continuous at any point, we must verify the following:

  • For any small positive number \(\epsilon\), there should exist another small positive number \(\delta\) such that the distance within \(\delta\)-radius of the point \((x_0, y_0, z_0)\) leads to the function value being within \(\epsilon\) of \(f(x_0, y_0, z_0)\).
  • This condition ensures that as \((x, y, z)\) gets arbitrarily close to \((x_0, y_0, z_0)\), \(f(x, y, z)\) approaches \(f(x_0, y_0, z_0)\).
Continuity in this context reaffirms the intuitive behavior of functions being smooth without jumps at a point.
Epsilon-Delta Definition
The \(\epsilon\)-\(\delta\) definition is a formal way to express the continuity of functions. In simple terms:
  • \(\epsilon\) represents how close the function values need to be to each other. It's a measure of your chosen tolerance.
  • \(\delta\) is the permissible distance around \((x_0, y_0, z_0)\) for \((x, y, z)\) such that changes within this distance keep the function value within \(\epsilon\).
To concretely use this definition:
  • Determine \(\epsilon\) 鈥� your desired accuracy.
  • Find a suitable \(\delta\). In our solution, we picked \(\delta = \epsilon/\sqrt{3}\) because it ensures that the change in the input doesn't exceed the tolerance given by \(\epsilon\).
This approach guarantees that when inputs vary minimally, the outputs remain consistently close to the expected value.
Triangle Inequality
The triangle inequality is an essential tool for analyzing distance relationships. It provides a way to estimate how function inputs are related to each other in terms of absolute differences.
In this context:
  • The triangle inequality states that the absolute sum of three values \(|a| + |b| + |c|\) is always greater than or equal to \(|a + b + c|\).
  • Applying this to our function, we have \(|(x-x_0) + (y-y_0) - (z-z_0)| \leq |x-x_0| + |y-y_0| + |z-z_0|\).
  • This simplifies the expression and helps manage the bounding between inputs \((x, y, z)\) and changes to the function's value.
Understanding this inequality allows us to derive bounds and create strategies for choosing \(\delta\) effectively, as it directly relates changes in variables to resulting changes in the function output.
3D Coordinate Systems
Understanding 3D coordinate systems is crucial when dealing with functions of three variables. These systems involve visualizing and analyzing behaviors in a three-dimensional space with coordinates \((x, y, z)\).
Key points include:
  • Each point in the 3D space is described by three numbers corresponding to the \(x\), \(y\), and \(z\) axes.
  • The distance between two points \((x, y, z)\) and \((x_0, y_0, z_0)\) is given by \(\sqrt{(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2}\).
  • Visualizing these concepts helps in understanding how the function \(f(x, y, z) = x + y - z\) behaves at each point in the space.
This awareness enable us to interpret problems and solutions involving spatial relationships, such as establishing continuity through the \(\epsilon\)-\(\delta\) definition, effectively.

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Most popular questions from this chapter

Find the linearization \(L(x, y, z)\) of the function \(f(x, y, z)\) at \(P_{0} .\) Then find an upper bound for the magnitude of the error \(E\) in the approximation \(f(x, y, z) \approx L(x, y, z)\) over the region \(R\) \(f(x, y, z)=x^{2}+x y+y z+(1 / 4) z^{2} \quad\) at \(\quad P_{0}(1,1,2)\) \(R:|x-1| \leq 0.01, \quad|y-1| \leq 0.01, \quad|z-2| \leq 0.08\)

The Wilson lot size formula in economics says that the most economical quantity \(Q\) of goods (radios, shoes, brooms, whatever) for a store to order is given by the formula \(Q=\sqrt{2 K M / h}\) where \(K\) is the cost of placing the order, \(M\) is the number of items sold per week, and \(h\) is the weekly holding cost for each item (cost of space, utilities, security, and so on). To which of the variables \(K\), \(M\), and \(h\) is \(Q\) most sensitive near the point \(\left(K_{0}, M_{0}, h_{0}\right)=(2,20,0.05) ?\) Give reasons for your answer.

Define \(f(0,0)\) in a way that extends \(f\) to be continuous at the origin. $$f(x, y)=\frac{3 x^{2} y}{x^{2}+y^{2}}$$

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

In economics, the usefulness or utility of amounts \(x\) and \(y\) of two capital goods \(G_{1}\) and \(G_{2}\) is sometimes measured by a function \(U(x, y) .\) For example, \(G_{1}\) and \(G_{2}\) might be two chemicals a pharmaceutical company needs to have on hand and \(U(x, y)\) the gain from manufacturing a product whose synthesis requires different amounts of the chemicals depending on the process used. If \(G_{1}\) costs \(a\) dollars per kilogram, \(G_{2}\) costs \(b\) dollars per kilogram, and the total amount allocated for the purchase of \(G_{1}\) and \(G_{2}\) together is \(c\) dollars, then the company's managers want to maximize \(U(x, y)\) given that \(a x+b y=c .\) Thus, they need to solve a typical Lagrange multiplier problem. Suppose that $$U(x, y)=x y+2 x$$ and that the equation \(a x+b y=c\) simplifies to $$2 x+y=30$$ Find the maximum value of \(U\) and the corresponding values of \(x\) and \(y\) subject to this latter constraint.
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