Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Describe the level curves of the function. Sketch the level curves for the given c-values. $$ f(x, y)=x y \quad c=\pm 1, \pm 2, \ldots, \pm 6 $$

Step-by-Step Solution

Answer

The level curves of the function \( f(x, y) = xy \) are hyperbolas which run through the origin. For positive 'c' the hyperbolas are in the first and third quadrants. For negative 'c', they are in the second and fourth quadrants. And as the absolute value of 'c' increases, the hyperbolas come closer to the axes.

Step 1: Write down the Level Set form

Firstly, to write the equation for the level curve, it is needed to set the function equal to a constant. In our case it would be \(xy = c\).

Step 2: Rearrange the equation for ease of sketching

Generally, it is easier to sketch the graph when it is in the form of \(y=c/x\) or \( y= -c / x\), as it is more readable and it can be seen as a hyperbola equation.

Step 3: Sketch the curves

Now, sketch the curves for values \(c = \pm 1, \pm 2, ..., \pm 6\). From here, we know that the curves will be hyperbolas going through the origin. When 'c' is positive, the hyperbolas are in the first and third quadrants. When 'c' is negative, the hyperbolas are in the second and fourth quadrants. As the absolute value of 'c' increases, the hyperbolas get closer to the axes. Therefore, we get a family of hyperbolas that get gradually narrower as the value of 'c' (positive or negative) increases.

Step 4: Conclusion

In conclusion, the level curves of the function \( f(x, y) = xy \) are hyperbolas that run through the origin, with the direction depending on the sign of 'c'.

Key Concepts

Level Set Form

Understanding the level set form is pivotal for visualizing the behavior of functions in multivariable calculus. In essence, a level set of a function is a collection of points that produce the same function value, hence the name level. For a function of two variables, like our example function f(x, y) = xy, we set it equal to a constant value c to find its level curves, which leads to the equation xy = c. This equation represents an infinite set of points (x, y) where the product of x and y equals c.

Imagine slicing through a 3D surface with a plane parallel to the base; this slice will reveal one of these level curves. By changing the value of c, we change which slice we're looking at, similar to selecting different floors in a building. In the case of our exercise, we are examining the level curves for various c-values: ±1, ±2, ..., ±6. This set of curves provides a comprehensive picture of how the function behaves across its domain.

Hyperbola Equation

Hyperbolas are a fascinating feature of algebraic geometry and surface well in multivariable calculus when examining level curves of certain functions. They represent the set of points where the difference or the quotient of the distances to two fixed points (foci) is constant. When a level curve is given in the form of xy = c, with c ≠ 0, its graph is a hyperbola.

The standard hyperbola equation for our exercise takes the form xy = c, which can be rearranged to y = c/x or y = -c/x, depending on the sign of c. The asymptotes of these hyperbolas are the x-axis and y-axis, since these are the lines the graph approaches but never actually touches. Each hyperbola has two separate branches, extending indefinitely, and for our function f(x, y), these branches are symmetric across the origin.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions with multiple variables. It's an enchanting world where surfaces, contours, and various shapes come into play, rather than just simple curves. Functions like f(x, y) represent surfaces in three-dimensional space, and we use level curves to explore these surfaces without the complexity of 3D graphing.

In our context, when we talk about the function f(x, y) = xy, we are actually referring to a surface in a 3D space. Each point on this surface corresponds to an x, y, and f(x, y) value. When we fix the value of f(x, y) and find the level curves in the xy-plane, we are effectively understanding how this surface behaves at different heights or levels, providing insight into the structure of the function.

Graph Sketching

Graph sketching is the art of representing functions visually. It's where the theoretical meets the practical in a dance of lines, curves, and shaded areas that bring mathematical concepts to life. When sketching level curves, we illustrate how the function behaves for different constant values of c.

In our exercise, we sketch hyperbolas that correspond to different c-values. Positive c-values generate hyperbolas in the first and third quadrants, while negative c-values yield hyperbolas in the second and fourth quadrants. This pattern is invaluable for visual learners, as it combines the abstract concept of level sets with the concrete imagery of the graphs. As the magnitude of c increases, the hyperbolas become narrower, and this gradual shift in the graphs assists in visualizing the function's behavior at large scales. By comparing these sketches, one gains an intuitive understanding of the function beyond the raw algebraic expressions.