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Chapter 7: Problem 3

Let \(g(x, y, z)=x /(y-z) .\) Compute \(g(2,3,4)\) and \(g(7,46,44)\).

Short Answer

Expert verified
g(2, 3, 4) = -2, g(7, 46, 44) = 3.5

Step by step solution

01

Understand the Function

The function given is \( g(x, y, z) = \frac{x}{y - z} \). It takes three variables, namely \(x\), \(y\), and \(z\), and outputs the result of \( x \) divided by the difference between \( y \) and \( z \).
02

Evaluate \( g(2, 3, 4) \)

Substitute \( x = 2 \), \( y = 3 \), and \( z = 4 \) into the function: \[ g(2, 3, 4) = \frac{2}{3 - 4} \] Calculate the denominator: \[ 3 - 4 = -1 \] So, \[ g(2, 3, 4) = \frac{2}{-1} = -2 \]
03

Evaluate \( g(7, 46, 44) \)

Substitute \( x = 7 \), \( y = 46 \), and \( z = 44 \) into the function: \[ g(7, 46, 44) = \frac{7}{46 - 44} \] Calculate the denominator: \[ 46 - 44 = 2 \] So, \[ g(7, 46, 44) = \frac{7}{2} = 3.5 \]

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Key Concepts

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

headline of the respective core concept
Understanding functions of several variables is important in multivariable calculus. Unlike single-variable functions, these functions depend on more than one input. For example, in the exercise given, the function is written as \( g(x, y, z) = \frac{x}{y - z} \). Here, the function \( g \) takes three variables: \( x \), \( y \), and \( z \). Understanding functions of several variables allows us to determine how these variables interact and influence the result.
  • Each variable can change independently.
  • The function maps every combination of variables to a single value.
  • Understanding this mapping helps in visualizing and solving complex problems in higher dimensions.
Knowing how each variable influences the function is fundamental before diving into computations.
headline of the respective core concept
Evaluating functions is the process of determining the value that a function takes at specific inputs. In multivariable functions like \( g(x, y, z) = \frac{x}{y - z} \), we substitute specific values for each variable and then simplify.
  • For instance, to evaluate \( g(2, 3, 4) \), substitute \( x = 2 \), \( y = 3 \), and \( z = 4 \) into the function.
  • The result is \( g(2, 3, 4) = \frac{2}{3 - 4} \).
  • Then, calculate the operation inside the function: \( 3 - 4 = -1 \).
  • Finally, simplify the fraction: \( \frac{2}{-1} = -2 \).
Repeat this process for any set of values. Evaluating the function at different inputs helps to understand the behavior of the function over its domain.
headline of the respective core concept
Substituting values is a technique where we replace the variables in a function with specific numbers. This is the core step in evaluating any function.
  • Consider the function \( g(7, 46, 44) \).
  • Substitute \( x=7 \), \( y=46 \), and \( z=44 \) into the function.
  • Perform the substitution: \( g(7, 46, 44) = \frac{7}{46-44} \).
  • Simplify the operation in the denominator: \( 46 - 44 = 2 \).
  • The simplified function becomes: \( g(7, 46, 44) = \frac{7}{2} = 3.5 \).
By substituting values correctly, we obtain specific outputs from the function, which makes it easier to analyze its behavior and solve problems.

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Most popular questions from this chapter

Find a function \(f(x, y)\) that has the curve \(y=2 / x^{2}\) as a level curve.

Find the values of \(x\) and \(y\) that minimize \(x y+y^{2}-x-1\) subject to the constraint \(x-2 y=0.\)

Both first partial derivatives of the function \(f(x, y)\) are zero at the given points. Use the second-derivative test to determine the nature of \(f(x, y)\) at each of these points. If the second derivative test is inconclusive, so state. $$f(x, y)=\frac{1}{x}+\frac{1}{y}+x y ;(1,1)$$

Both first partial derivatives of the function \(f(x, y)\) are zero at the given points. Use the second-derivative test to determine the nature of \(f(x, y)\) at each of these points. If the second derivative test is inconclusive, so state. $$f(x, y)=2 x^{2}-x^{4}-y^{2} ;(-1,0),(0,0),(1,0)$$

Let \(f(x, y)=x e^{y}+x^{4} y+y^{3} .\) Find \(\frac{\partial^{2} f}{\partial x^{2}}, \frac{\partial^{2} f}{\partial y^{2}}, \frac{\partial^{2} f}{\partial x \partial y},\) and \(\frac{\partial^{2} f}{\partial y \partial x}.\)
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