In Exercises 5-12, find and sketch the domain for each function. f(x, y)=(1)/(ln(4-x^2-y^2))

Name: Problem 12, Chapter 14: Thomas Calculus
Uploaded: 2020-05-16T23:45:49-08:00
Duration: 4 min 33 s
Channel: Latisha West
Description: In Exercises 5-12, find and sketch the domain for each function. f(x, y)=(1)/(ln(4-x^2-y^2))

In Exercises 5-12, find and sketch the domain for each...

Key Concepts

Undefined Points Due to Division

Whenever a function is expressed as a fraction, it is essential to ensure that the denominator is not zero to avoid division by zero, which is undefined. This adds an extra condition to the domain when the denominator is itself a function involving the variables.

Geometric Interpretation of Inequalities

Inequalities in two variables can define regions in the plane, such as the interior of a circle. Understanding and sketching these regions helps in visualizing the domain of the function by identifying which (x, y) pairs satisfy all the necessary conditions.

Domain of a Function

The domain of a function is the set of all input values for which the function is defined. For multivariable functions, this involves determining which (x, y) pairs produce valid outputs, taking into account any restrictions such as divisions by zero or inputs to functions (like logarithms or square roots) that must lie within a certain range.

Properties of Logarithmic Functions

Logarithmic functions, such as the natural logarithm ln(x), require that their arguments be strictly positive. This means that any expression inside a logarithm must be greater than zero, imposing an inequality constraint on the variables.

Transcript

00:01 On this problem, we're trying to find and sketch the domain of this function f of xy is equal to 1 over the natural log of 4 minus x squared minus y squared.

00:15 Function like this, we need to consider what rules we know exist.

00:22 Our first rule is that we know that 4 minus x squared minus y squared needs to be greater than 0.

00:30 And that's because whatever we take the natural log of needs to be greater than 0.

00:38 That the denominator of this function cannot equal zero because we know that we cannot have a zero denominator.

00:47 The only way that we would end up with a zero denominator is if 4 minus x squared minus y squared was equal to 1.

00:55 So we know that that cannot be true in order for this to work.

00:58 So 4 minus x squared minus y squared cannot equal 1 because that would give us a 0 as our denominator.

01:05 So 4 minus x squared minus y squared cannot equal one.

01:10 Start by thinking about what kind of equation we're looking at here, or inequality that we're looking at here.

01:16 We're looking at something that looks similar to the inequality of a circle.

01:20 So i'm going to rearrange this so it looks like that type of inequality.

01:24 So first i'm going to add to both sides, x squared plus y squared, so that i only have my constant on one side.

01:35 When i do that, i'm left with 4 is greater than x squared plus y squared...

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