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Chapter 3: Problem 25

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ f(x, y)=x y-x ; c=-2,0,2 $$

Short Answer

Expert verified
The level curves for \(c = -2, 0, 2\) are the hyperbolas \(y = \frac{-2}{x} + 1\), line \(y = 1\) excluding \(x = 0\), and hyperbola \(y = \frac{2}{x} + 1\), respectively.

Step by step solution

01

Substitute c into the function

The level curve of the function is given by setting the function equal to a constant value. We begin by substituting the given values of \(c\), one at a time, into the function. The equation to solve for each level curve is:\[ f(x, y) = xy - x = c \] Substituting \(c = -2, 0, 2\), we get three different equations: \[ xy - x = -2, \quad xy - x = 0, \quad xy - x = 2. \]
02

Rearrange the equation for c = -2

For \(c = -2\), the equation is \(xy - x = -2\). Rearrange the equation to solve for \(y\):\[ xy - x = -2 \implies x(y-1) = -2 \implies y = \frac{-2}{x} + 1. \] This is a hyperbola translated up by 1 unit.
03

Rearrange the equation for c = 0

For \(c = 0\), the equation is \(xy - x = 0\). Rearrange to solve for \(y\):\[ xy - x = 0 \implies x(y-1) = 0. \] This simplifies to \(y = 1\) if \(x eq 0\). This is a horizontal line at \(y = 1\), excluding where \(x = 0\).
04

Rearrange the equation for c = 2

For \(c = 2\), the equation is \(xy - x = 2\). Rearrange the equation:\[ xy - x = 2 \implies x(y-1) = 2 \implies y = \frac{2}{x} + 1. \] This is another hyperbola also translated up by 1 unit.
05

Graph the level curves

Graph each function of \(y\) versus \(x\): For \(c = -2\), plot the hyperbola \(y = \frac{-2}{x} + 1\). For \(c = 0\), plot the line \(y = 1\) excluding \(x = 0\). For \(c = 2\), plot the hyperbola \(y = \frac{2}{x} + 1\). Each curve shows how the function behaves for different constant values.

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Key Concepts

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

Functions of Several Variables
Functions of several variables involve an equation where more than one input determines the output. For our example, the function is given as \(f(x, y) = xy - x\).

In this case:
  • \(x\) and \(y\) are the two variables (or inputs).
  • The function describes a surface in three-dimensional space.

Level curves help visualize functions of several variables by examining the surface at different heights. Each curve corresponds to setting the function equals to a constant, \(c\). These curves transform a three-dimensional surface into two-dimensional lines or curves on the \(xy\)-plane.

This can simplify analysis, helping us understand more about the function's tendencies over different regions.
Hyperbola
A hyperbola is a type of curve on a graph defined by equations like \( y = \frac{k}{x}\). In our context, this is seen when we rearrange the function \(xy-x=c\) to see how \(y\) changes with respect to \(x\) for given values of \(c\).

In our example:
  • For \(c = -2\), the equation \(y = \frac{-2}{x} + 1\) forms a hyperbola that is shifted upwards by 1 unit.
  • For \(c = 2\), the equation \(y = \frac{2}{x} + 1\) also results in a hyperbola, again shifted up by 1 unit.

Hyperbolas have two distinctive curves that extend towards infinity. The shape visually depicts how values of \(y\) change drastically as \(x\) moves away from zero. Each hyperbola on its own level shows the behavior of the function for that fixed constant value of \(c\).
Graphing Equations
Graphing equations is crucial for visualizing mathematical relationships and functions. For the function \(f(x, y) = xy - x\), we focus on plotting the level curves for given \(c\) values. This gives insight into the function's behavior.

Steps to graph:
  • For \(c = -2\), plot \(y = \frac{-2}{x} + 1\), displaying a hyperbola.
  • For \(c = 0\), graph the line \(y = 1\), noting this happens when \(x eq 0\).
  • For \(c = 2\), sketch \(y = \frac{2}{x} + 1\), showing another hyperbola.

Each graph helps to examine how the function's resulting shape is influenced by the values \(x\) and \(y\) take, particularly highlighting changes in direction or curvature based on \(c\). Graphs are thus a fundamental tool for illustrating these transformations and summarizing how functions of several variables behave across different domains.

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Most popular questions from this chapter

The electrical resistance \(R\) produced by wiring resistors \(R_{1}\) and \(R_{2}\) in parallel can be calculated from the formula \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}} .\) If \(R_{1}\) and \(R_{2}\) are measured to be \(7 \Omega\) and \(6 \Omega\), respectively, and if these measurements are accurate to within \(0.05 \Omega\), estimate the maximum possible error in computing \(R\). (The symbol \(\Omega\) represents an ohm, the unit of electrical resistance.)

Find the equation for the tangent plane to the surface at the indicated point. $$ z=x^{2}-2 x y+y^{2}, P(1,2,1) $$

For the following exercises, find all first partial derivatives. $$ u(x, y)=x^{4}-3 x y+1, x=2 t, y=t^{3} $$

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample. The linear approximation to the function of \(f(x, y)=5 x^{2}+x \tan (y)\) at \((2, \pi)\) is given by \(L(x, y)=22+21(x-2)+(y-\pi)\)

Find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, \(P_{0}\left(x_{0}, y_{0}, z_{0}\right)\), and a vector \(\mathbf{n}=\langle a, b, c\rangle\) that is parallel to the line. Then the equation of the line is \(\left.x-x_{0}=a t, y-y_{0}=b t, z-z_{0}=c t .\right)\) $$ z=5 x^{2}-2 y^{2}, P(2,1,18) $$
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