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Chapter 7: Problem 8

For each function, find the domain. $$ f(x, y, z)=\frac{\sqrt{x} \ln y}{z} $$

Short Answer

Expert verified
Domain: \( \{ (x, y, z) \mid x \geq 0, y > 0, z \neq 0 \} \)

Step by step solution

01

Determine Conditions for Square Root

For the square root \( \sqrt{x} \) to be defined, \( x \) must be greater than or equal to zero. This gives us the condition \( x \geq 0 \).
02

Determine Conditions for Natural Logarithm

For the natural logarithm \( \ln y \) to be defined and real, \( y \) must be greater than zero. This gives us the condition \( y > 0 \).
03

Determine Conditions for Division by Zero

The function involves division by \( z \), which means \( z \) cannot be equal to zero to avoid division by zero. This gives us the condition \( z eq 0 \).
04

Combine All Conditions for Domain

Combine all the conditions derived from the previous steps. The domain of the function consists of all points \( (x, y, z) \) such that: \( x \geq 0 \), \( y > 0 \), and \( z eq 0 \).

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Key Concepts

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

Square Root Function
The square root function is one of the fundamental mathematical concepts you'll encounter. It is expressed as \( \sqrt{x} \), which asks the question: "What number, when multiplied by itself, gives \( x \)?". For a square root function to be defined and result in a real number, the value under the square root sign, \( x \), must be zero or a positive number, as negative values would result in complex numbers.
  • Condition: \( x \geq 0 \)
  • Reason: Ensures the result is a real number
Let's see an example. Consider \( \sqrt{4} \). This equals 2 because \( 2 \times 2 = 4 \). But if you attempt \( \sqrt{-1} \), it becomes undefined in the realm of real numbers and instead introduces imaginary numbers.
Natural Logarithm Function
The natural logarithm function, denoted as \( \ln y \), is the logarithm to the base of the mathematical constant \( e \), which is approximately 2.71828. This function is crucial because it helps us understand exponential growth and decay, among other things. To calculate \( \ln y \), the value of \( y \) must be greater than zero.
  • Condition: \( y > 0 \)
  • Reason: Logarithms of non-positive numbers are not defined in the real number system
For example, \( \ln 1 = 0 \) because \( e^0 = 1 \). However, trying to compute \( \ln 0 \) will lead to inconsistency as there is no real number that \( e \) can be raised to that will yield zero.
Division by Zero
Division by zero is one of the most prohibited operations in mathematics. Imagine the function \( \frac{\text{anything}}{z} \). If \( z \) were to be zero, then the expression would become undefined because division by zero does not provide a meaningful or finite result.
  • Condition: \( z eq 0 \)
  • Reason: Avoids undefined behavior in mathematical calculations
Think about it this way: if you try to divide 5 cookies among zero children, the concept becomes impractical. Similarly, division by zero leaves us without a sensible answer, necessitating the exclusion of zero from the denominator.

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Most popular questions from this chapter

Find the total differential of each function. $$ f(x, y, z)=x y+y z+x z $$

Evaluate each iterated integral. $$ \int_{-3}^{3} \int_{0}^{3} y^{2} e^{-x} d y d x $$

For the given function and values, find: a. \(\Delta f \quad\) b. \(d f\) $$ \begin{array}{l} f(x, y)=e^{x}+x y+\ln y, \quad x=0, \quad \Delta x=d x=0.05, \\ y=1, \quad \Delta y=d y=0.01 \end{array} $$

Solve each using Lagrange multipliers. (The stated extreme values do exist.) Maximize \(f(x, y, z)=x+y+z\) subject to \(x^{2}+y^{2}+z^{2}=12\)

Which of the following is an iterated integral and which is a double integral? a. \(\iint_{R} f(x, y) d x d y\) b. \(\int_{a}^{b} \int_{c}^{d} f(x, y) d x d y\)
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