Problem 25

Question

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

Step-by-Step Solution

Verified

Answer

For c = -2, curve is $ y = 1 - \frac{2}{x} $. For c = 0, curves are $ x = 0 $ and $ y = 1 $. For c = 2, curve is $ y = 1 + \frac{2}{x} $.

1Step 1: Understanding Level Curves

Level curves of a function $ f(x, y) $ are curves in the $ xy $-plane where the function has a constant value $ c $. We find these curves by solving the equation $ f(x, y) = c $ for given values of $ c $. In this exercise, we'll determine level curves for $ f(x, y) = xy - x $ at $ c = -2, c = 0, \text{ and } c = 2 $.

2Step 2: Level Curve for c = -2

Set the function equal to -2: $ xy - x = -2 $. Rearranging gives $ x(y - 1) = -2 $. Therefore, the level curve for $ c = -2 $ is $ y = 1 - \frac{2}{x} $, assuming $ x eq 0 $. This is a hyperbola in the $ xy $-plane.

3Step 3: Level Curve for c = 0

Set the function equal to 0: $ xy - x = 0 $. Factoring gives $ x(y - 1) = 0 $. This implies either $ x = 0 $ or $ y = 1 $. Therefore, for $ c = 0 $, the level curves are the lines $ x = 0 $ and $ y = 1 $.

4Step 4: Level Curve for c = 2

Set the function equal to 2: $ xy - x = 2 $. Rearranging gives $ x(y - 1) = 2 $. Thus, the level curve for $ c = 2 $ is $ y = 1 + \frac{2}{x} $, again assuming $ x eq 0 $. This is another hyperbola in the $ xy $-plane.

Key Concepts

Multivariable FunctionsHyperbolasCurve Sketching

Multivariable Functions

A multivariable function involves more than one input variable, usually denoted as $ f(x, y) $, where $ x $ and $ y $ are the variables. These functions represent relationships that depend on two independent variables, often used to describe surfaces in three-dimensional space.
For example, the function $ f(x, y) = xy - x $ calculates an output value by performing operations on $ x $ and $ y $.

For any pair $(x, y)$, substituting into the function provides a result that can represent a height or some other quantity depending on the context.
Visualizing these functions often involves computing level curves, which are the places in the $ xy $-plane where the function attains a constant value.

These curves help us understand the shape and behavior of the surface that the function represents.

Hyperbolas

A hyperbola is a type of curve on the $ xy $-plane, defined by its two branches, which are mirror images of each other. This curve can emerge as a level curve of a multivariable function. In our context, when solving for the function $ f(x, y) = xy - x $ for different values of $ c $, we sometimes encounter hyperbolas.
Consider the equations:

$ y = 1 - \frac{2}{x} $
$ y = 1 + \frac{2}{x} $

Each of these represents a hyperbola where the relationship between $ x $ and $ y $ satisfies the equation for all points on the curve.
Key characteristics of hyperbolas include:

Two separate branches, each in different parts of the plane, unless $ x = 0 $.
The center is often a point of symmetry. In our cases, it's around the line $ y = 1 $.
These equations illustrate that changes in $ x $ and $ y $ are inversely proportional due to the presence of a fraction.

Curve Sketching

Curve sketching is the process of drawing a graph based on understanding mathematical functions and equations. This skill allows us to visualize how functions behave over a range of values. In this particular exercise, sketching helps illustrate the nature of level curves for varied values of $ c $.
Let's break it down:

For $ c = -2 $ and $ c = 2 $, rearranging the function leads to hyperbolas. Sketching these helps in understanding their orientation and spread.
For $ c = 0 $, the result is two straight lines, revealing where the function's impact is minimized or neutralized.

Successful sketching requires attention to details like where curves intersect axes, their symmetry, and any asymptotes they might have.
This visual technique strengthens comprehension of the mathematical relationships within multivariable functions and their level curves.

Problem 24

Problem 26

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