Problem 34 - Calculus 3

Question

Find the domain of the following functions. $$ z=\ln \left(x-y^{2}\right) $$

Step-by-Step Solution

Answer

The domain is all \((x, y)\) such that \(x > y^2\).

Step 1: Understanding the Logarithm Domain

The natural logarithm function, \( ln(x) \), is only defined for inputs where \(x > 0\). This means for \(\ln(x-y^2)\) to be defined, \(x-y^2 > 0\) must hold true.

Step 2: Set the Inequality

Based on the definition described in Step 1, set up the inequality: \(x - y^2 > 0\).

Step 3: Solve the Inequality

Rearrange \(x - y^2 > 0\) to find the values of \(x\) in terms of \(y\). This leads to \(x > y^2\).

Step 4: Define the Domain in Intervals

The domain involves all pairs \((x, y)\) such that \(x > y^2\). This can be expressed as: \((x,y) \in \{(x,y) | x > y^2\}\).

Key Concepts

Domain of a Function

In the world of multivariable calculus, the domain of a function represents all the possible input values for which the function is defined. For the function \( z = \ln(x - y^2) \), the domain consists of all pairs \((x, y)\) that keep the expression inside the logarithm positive.

• Why positivity? Because logarithms, particularly the natural logarithm \( \ln(x) \), require their arguments to be greater than zero to produce a real number.
• So, the task is to find conditions that allow the term \(x - y^2\) to be positive.

To identify the domain, we examine the expression \(x - y^2\) and determine where it remains greater than zero. This gives us the requirement \(x > y^2\). With this condition in place, the domain includes every possible \((x, y)\) pair where this inequality is true. Understanding the domain is crucial as it tells us where the function behaves properly and holds any meaningful output.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a fundamental concept in calculus and mathematics as a whole. It is specifically the logarithm to the base \(e\), where \(e\) is approximately equal to 2.718. This logarithm measures the length of time it would take to reach a certain level of continuous growth.

• Recall that for \( \ln(x) \) to be defined, the value inside the logarithm (its argument) must be positive.
• If the argument is zero or negative, the result is not a real number, making it undefined within the scope of real-valued functions.

For our function \( z = \ln(x - y^2) \), we need \(x - y^2 > 0\). This ensures the natural logarithm accepts the argument, resulting in a real and defined function output. Logarithms help us model exponential growth and decay scenarios, which appear in various scientific and engineering contexts.

Inequalities in Calculus

In calculus, inequalities are essential in solving problems involving conditions like domains, limits, and optimization. They allow us to locate the range of values over which functions are defined or optimal. For the given problem, we needed to work with the inequality \(x - y^2 > 0\).

• This inequality loosely translates to "the value of \(x\) must exceed the square of \(y\)".
• By solving inequalities, calculus practitioners can determine safe zones where mathematical expressions are valid, ensuring no math errors occur due to undefined operations.

Solving such inequalities involves simple algebraic manipulation, identifying where the expressions hold true. It helps us pinpoint precise intervals or regions on a graph, detailing where functions are positive, negative, or even zero. Hence, mastering inequalities is a critical skill in analyzing real-world scenarios with mathematical models.