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Chapter 7: Problem 20

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

Short Answer

Expert verified
The function is \(f(x, y) = x^{2}y.\)

Step by step solution

01

- Understand Level Curves

Level curves of a function are the set of points \((x, y)\) that satisfy the equation \(f(x, y) = k\) for some constant \(k\). For the given level curve \(y = 2/x^{2}\), the goal is to find a function \(f(x, y)\) such that when \(f(x, y) = k\), it results in \(y = 2/x^{2}\).
02

- Express the Required Equation

Rewrite the given curve \(y = 2/x^{2}\) as \(x^{2}y = 2\). This step converts the curve equation into a form where it can be used as an implicit equation involving \(x\) and \(y\).
03

- Set Up the Function

Identify a function \(f(x, y)\) that equals a constant for all points on the curve. Here, use \(x^{2}y\) and set this expression equal to 2: \(f(x, y) = x^{2}y\).
04

- Generalize the Function

Since the original curve corresponds to some level curve where \(f(x, y) = 2\), the function can be generalized by stating \(f(x, y) = x^{2}y\). When \(f(x, y) = 2\), it matches the given curve exactly.

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Key Concepts

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

Level Curves
Level curves are a way to visualize functions with two variables. They represent sets of points \((x, y)\) where the function \(f(x, y)\) achieves the same value. Think of them like contour lines on a map, where each line indicates the same elevation. For example, given \(y = 2 / x^2\), you can transform it into \(x^2 y = 2\). Here, the curve represents points where the product \(x^2 y\) equals 2. To find the right function that includes the curve \(y = 2 / x^2\) as a level curve, we look for a function where \(f(x, y) = k\) simplifies to our equation.
Implicit Functions
Often, we deal with implicit functions in multivariable calculus. An implicit function is one where the relationship between variables is not isolated on one side of the equation. For instance, the curve \(y = 2 / x^2\) can be transformed into \(x^2 y = 2\). Here, the function \(f(x, y) = x^2 y\) is implicit because it includes both variables intertwined. To find a function whose level curve is \(y = 2 / x^2\), we use this implicit form. The level curve \(x^2 y = 2\) tells us our function should look like \(f(x, y) = x^2 y\).
Function Analysis
Function analysis in multivariable calculus often involves examining specific properties, like level curves or implicit forms. Understanding these properties helps us find and generalize functions. From the exercise, once we rewrote the curve \(y = 2 / x^2\) to \(x^2 y = 2\), we set our function to be \(f(x, y) = x^2 y\). This tells us that for \(f(x, y) = k\), we obtain curves like \(x^2 y = k\). Therefore, we match the given curve exactly by setting \(k = 2\). You can generalize this by noting any curve of the form \(x^2 y = k\) will be a level curve for \(f(x, y)\).

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Most popular questions from this chapter

Solve the following exercise by the method of Lagrange multipliers. Find the values of \(x, y\) that minimize $$ x^{2}+x y+y^{2}-2 x-5 y $$ subject to the constraint \(1-x+y=0\).

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Let \(f(x, y)\) be any production function where \(x\) represents labor (costing $$\$ a$$ per unit) and \(y\) represents capital (costing $$\$ b$$ per unit). Assuming that $$\$ c$$ is available, show that, at the values of \(x, y\) that maximize production, $$ \frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}}=\frac{a}{b} $$ Note: Let \(\quad F(x, y, \lambda)=f(x, y)+\lambda(c-a x-b y) .\) The result follows from (L-1) and (L-2).
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