Problem 25 - Calculus Volume 3

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

Answer

Level curves are hyperbolas for \(c = -2, 2\) and lines \(x=0\) or \(y=1\) for \(c=0\).

Step 1: Understanding Level Curves

To find the level curves of the function, we set the function equal to a constant value. This means that for each given value of \(c\), we solve the equation \(xy - x = c\) for the points \((x, y)\) that satisfy it. These points will form the level curve in the plane.

Step 2: Level Curve for c = -2

Set the function equal to \(c = -2\): \(xy - x = -2\). Rearrange to \(x(y - 1) = -2\). This describes a hyperbola in the xy-plane, which is the level curve for \(c = -2\).

Step 3: Level Curve for c = 0

Set the function equal to \(c = 0\): \(xy - x = 0\). Factor out an \(x\) to get \(x(y - 1) = 0\). This equation gives the level curves \(x = 0\) (the y-axis) or \(y = 1\) (a horizontal line).

Step 4: Level Curve for c = 2

Set the function equal to \(c = 2\): \(xy - x = 2\). Rearrange to \(x(y - 1) = 2\). This also describes a hyperbola in the xy-plane, but differing from \(c = -2\), it is shifted in position.

Key Concepts

Visualizing Functions

Understanding how to visualize functions can greatly enhance your ability to interpret mathematical concepts. The idea of visualizing functions involves translating an equation into a graph or a set of curves that represent solutions to the function. In the context of level curves, we look at the function by setting it to a constant value and consider the resulting graph on the xy-plane. By analyzing these graphs at various constant levels (denoted as \(c\)), we can better understand the behavior of the function and how changes in parameters affect its shape. The level curves offer insights into the nature of the function, such as where it maintains the same output despite variations in input values.

Hyperbola

When exploring the shape of level curves for certain functions, you might encounter hyperbolas. A hyperbola is a type of curve formed by the intersection of a plane with both halves of a double cone. In the context of our exercises, hyperbolas appear when we solve equations resembling \(x(y - 1) = c\) for different constant values \(c\). These curves are particularly interesting because they open along two directions, reflecting the dynamic nature of the product terms in our function like \(xy\). For instance, level curves for \(c = -2\) and \(c = 2\) are hyperbolas, depicting symmetry based on the constant value. Hyperbolas illustrate how the function behaves in open space and can help identify points of interest or axes of symmetry.

Constant Values

Constant values play a critical role when visualizing level curves. By setting the function equal to these values, we create distinct curves in the xy-plane. Each constant \(c\) reveals peculiar aspects of the function at that point. For example, if \(c = -2\), our function \(xy - x = -2\) results in a hyperbola that reflects how maintaining this specific output affects input conditions \((x, y)\). At \(c = 0\), the function simplifies to linear elements like \(x = 0\) and \(y = 1\), showcasing a drastic change in the curve type. Therefore, the choice of constant values can highlight different characteristics of the function and help students understand how the equation behaves under various conditions.

Function Equations

The manipulation and understanding of function equations are essential when working with level curves. To extract level curves, you alter the function so it equates to a constant value \(c\). The original function \(f(x, y) = xy - x\) transforms into distinct equations for each level curve, depending on \(c\). For instance:

• Setting \(c = -2\), the equation becomes \(x(y - 1) = -2\), revealing a hyperbola.
• With \(c = 0\), the equation is \(x(y - 1) = 0\), simplifying into linear elements.
• For \(c = 2\), it transforms to \(x(y - 1) = 2\), another hyperbola with a modified position.

Understanding these transformations means you're seeing how a static function equation can create various geometric entities against the constant backdrop of \(c\). It helps build a strong conceptual link between algebraic manipulation and geometric interpretation.