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Chapter 15: Problem 46

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$g(x, y)=\ln (x-y)$$

Short Answer

Expert verified
Answer: The function \(g(x, y) = \ln(x-y)\) is continuous at all points of \(\mathbb{R}^{2}\) where \(y < x\).

Step by step solution

01

Identify the domain of the natural logarithm function

The natural logarithm function, \(\ln(x)\), is defined for all \(x > 0\). In our case, the input to the natural logarithm function is \((x-y)\), so we need to ensure that \(x-y > 0\) for the function to be defined.
02

Find the points where the function is defined

To find the points \((x, y)\) where the function \(g(x, y) = \ln(x-y)\) is defined, we need to solve the inequality \(x-y > 0\): $$ x - y > 0 \\ y < x $$ So, the function is defined for all points \((x, y)\) in which \(y < x\).
03

Determine continuity of the function

The natural logarithm function is continuous where it is defined. Since our function \(g(x, y) = \ln(x-y)\) is based on the natural logarithm function, and we have found the points \((x, y)\) for which it is defined (i.e., \(y < x\)), we can conclude that it is continuous at these points.
04

State the result

The function \(g(x, y) = \ln(x-y)\) is continuous at all points of \(\mathbb{R}^{2}\) where \(y < x\).

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Key Concepts

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

Natural Logarithm
The natural logarithm function, represented by \( \ln(x) \), is a logarithm to the base \( e \), where \( e \approx 2.718 \). It is a special mathematical function with several key properties and uses. Here, we are focusing on its domain and the conditions under which it is defined.

  • The natural logarithm is only defined for positive values of \( x \). This means \( x > 0 \) must hold true for \( \ln(x) \) to be a real number.
  • The graph of \( \ln(x) \) is a continuous curve that approaches negative infinity as \( x \) approaches zero and increases without bound as \( x \) gets larger.
  • This property of continuity is important because it implies that if a function within a natural logarithm (like \( x-y \) in our problem) is positive, the function is continuous.
These properties are crucial when determining where functions involving natural logarithms, such as \( g(x, y) = \ln(x-y) \), are defined and continuous. Thus, understanding the natural logarithm helps in finding the domain and continuity of complex functions.
Inequality
Inequalities are mathematical expressions involving the symbols \(<, >, \leq, \geq, \) which describe the relative sizes or order of two values. In the context of our function \( g(x, y) = \ln(x-y) \), we encounter the inequality \( x-y > 0 \). Here's how to approach it:

  • This inequality tells us that \( x \) must be greater than \( y \), ensuring the expression inside the logarithm is positive.
  • To solve such inequalities, treat it like an equation at first, but remember the inequality rules; flip the sign when multiplying or dividing by a negative number.
  • In this problem, \( x - y > 0 \) simplifies to \( y < x \), identifying the values where the function is defined and thus continuous.
Understanding and solving inequalities is fundamental in calculus and algebra, especially when determining domains of functions and their definitions.
Domain of a Function
The domain of a function is the complete set of possible input values (usually \( x \)-values) that the function can accept without leading to undefined or non-real numbers. For our function \( g(x, y) = \ln(x-y) \), finding the domain involves ensuring the natural logarithm is defined, adhering to the inequality \( x-y > 0 \).

  • First, we identify that \( x-y \) must be positive for \( \ln(x-y) \) to be a real number, so we solve \( x-y > 0 \).
  • This leads us to the simpler form \( y < x \), which describes the set of \( (x, y) \) pairs where the function is valid.
  • Visualizing this on a graph, any point below the line \( y = x \) works in two dimensions.
  • Thus, the function \( g(x, y) = \ln(x-y) \) is defined over all such pairs, and it is continuous over this domain due to the continuity of the natural logarithm.
Understanding the domain helps you know where you can use a function without running into mathematical inconsistencies or undefined behavior.

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Most popular questions from this chapter

Challenge domains Find the domain of the following functions. Specify the domain mathematically, and then describe it in words or with a sketch. $$g(x, y, z)=\frac{10}{x^{2}-(y+z) x+y z}$$

a. Determine the domain and range of the following functions. b. Graph each function using a graphing utility. Be sure to experiment with the window and orientation to give the best perspective on the surface. $$h(x, y)=\frac{x+y}{x-y}$$

Use what you learned about surfaces in Sections 13.5 and 13.6 to sketch a graph of the following functions. In each case, identify the surface and state the domain and range of the function. $$G(x, y)=-\sqrt{1+x^{2}+y^{2}}$$

Powers and roots Assume \(x+y+z=1\) with \(x \geq 0, y \geq 0\) and \(z \geq 0\) a. Find the maximum and minimum values of \(\left(1+x^{2}\right)\left(1+y^{2}\right)\left(1+z^{2}\right)\) b. Find the maximum and minimum values of \((1+\sqrt{x})(1+\sqrt{y})(1+\sqrt{z})\)

Find the points at which the following surfaces have horizontal tangent planes. $$x^{2}+2 y^{2}+z^{2}-2 x-2 z-2=0$$
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