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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

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Step by step solution

01

Understand the Function Definition

The function given is defined as . This means we need to take the value of x and divide it by the difference between y and z.
02

Compute g(2,3,4)

Substitute x = 2, y = 3, and z = 4 into the function definition . Then simplify the expression: .
03

Compute g(7,46,44)

Substitute x = 7, y = 46, and z = 44 into the function definition . Then simplify the expression .

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Key Concepts

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

Function Evaluation
In multivariable calculus, function evaluation involves plugging specific values into a defined function to find its output. Given a function like \(g(x, y, z) = \frac{x}{y - z}\), you need to substitute each variable with the provided values to compute the result. For example, to find \(g(2, 3, 4)\), replace every 'x' with 2, 'y' with 3, and 'z' with 4 in the function definition. This will let you evaluate the function at specific points in its domain. Function evaluation is a crucial skill because it allows you to understand the behavior of multivariable functions at particular points.
Substitution
Substitution is a fundamental step in function evaluation. It involves replacing variables in a function with the given specific values. In our example, \(g(x, y, z)\), we substitute:
  • x = 2, y = 3, z = 4 to find \(g(2, 3, 4)\)
  • x = 7, y = 46, z = 44 to find \(g(7, 46, 44)\)
By substituting these values into the function \(g(x, y, z) = \frac{x}{y - z}\), you proceed to simplify the new expressions generated. This process transforms an abstract mathematical function into something you can compute with real numbers.
Simplification
Simplification is the process of reducing an expression to its simplest form. After substituting the variables in the function, you'll often need to simplify the resulting expression. For example, after substituting x = 2, y = 3, and z = 4 into \(g(x, y, z)\), you get \(g(2, 3, 4) = \frac{2}{3 - 4}\). Simplify the denominator first: \(3 - 4 = -1\). Then, perform the division: \(\frac{2}{-1} = -2\). For \(g(7, 46, 44)\), substituting gives \(\frac{7}{46 - 44} = \frac{7}{2}\), which simplifies directly to \(3.5\). Simplifying makes the function's output clear and concise, which is vital for understanding and using the function.

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Most popular questions from this chapter

A company manufactures and sells two competing products, I and II, that cost \(\$ p_{\mathrm{I}}\) and \(\$ p_{\text {II per unit, respectively, to produce. Let }} R(x, y)\) be the revenue from marketing \(x\) units of product \(\mathrm{I}\) and \(y\) units of product II. Show that if the company's profit is maximized when \(x=a, y=b\), then $$ \frac{\partial R}{\partial x}(a, b)=p_{\mathrm{I}} \quad \text { and } \quad \frac{\partial R}{\partial y}(a, b)=p_{\mathrm{II}} . $$.

Use Lagrange multipliers to find the three positive numbers whose sum is 15 and whose product is as large as possible.

Find all points \((x, y)\) where \(f(x, y)\) has a possible relative maximum or minimum. Then, use the second-derivative test to determine, if possible, the nature of \(f(x, y)\) at each of these points. If the second-derivative test is inconclusive, so state. $$ f(x, y)=x^{3}-y^{2}-3 x+4 y $$

Let \(f(L, K)=3 \sqrt{L K}\). Find \(\frac{\partial f}{\partial L}\).

Find all points \((x, y)\) where \(f(x, y)\) has a possible relative maximum or minimum. $$ f(x, y)=x^{2}-5 x y+6 y^{2}+3 x-2 y+4 $$
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