Question

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

Short answer

Level curves for the function \(f(x, y)=xy\), given the \(c\)-values ranging from \(\pm1\) to \(\pm6\), are hyperbolas in the xy-plane given by \( y=\pm1/x, \pm2/x, \ldots, \pm6/x \)

Step-by-step solution

Understanding Level Curves

Level curves of a function of two variables are the curves of constant value. For the given function \( f(x, y)=x y \), level curves are found by setting the function equal to a constant value, \(c\), and then solving for \(y\) in terms of \(x\), or vice versa.

Solving for y

By setting \( f(x, y) = c \), we can solve for \( y \) in terms of \( x \). So, \( x y = c \) becomes \( y=c/x \). Now, for given \(c\)-values, the equation becomes \( y=\pm1/x, \pm2/x, \ldots, \pm6/x \).

Sketching the Level Curves

Once the \( y \) is expressed in terms of \( x \) for each \( c \) value, these relationships are sketched in the xy-plane. The graph will show several hyperbolas, which are the level curves of the function.

Key concepts

Multivariable Calculus

Multivariable calculus extends the field of calculus to functions of multiple variables. It's a branch that deals with the differentiation and integration of functions that have more than one input. For example, the function in our exercise, f(x, y) = xy, depends on two variables, x and y. When dealing with such functions, it's important to consider the interactions between the variables and how the function's value changes in the multi-dimensional space.

One of the key concepts in multivariable calculus is a level curve, which is a graphical representation that helps visualize such functions. As you may already know, in single-variable calculus, we often look at graphs of functions in two dimensions, plotting the input on the x-axis and the output on the y-axis. But with functions of two variables, if we tried to plot both inputs and the output, we'd need a three-dimensional graph, which can be hard to draw and interpret. Level curves simplify this by showing slices of the function at constant output values, which allows us to see patterns and behaviors on a familiar two-dimensional plane.

Level Curve Sketching

Level curve sketching is an essential skill in multivariable calculus. It allows us to visualize how a function behaves across a range of values. The process involves drawing curves where the function has the same output value, the 'level' value, which is designated by c. In essence, you're taking an imaginary horizontal slice through a 3D surface and flattening it down into a 2D diagram.

To sketch level curves, you first set the function equal to different constants, as shown in the exercise with f(x, y) = xy. By assigning various values to c, from c = -6 to c = 6, and solving for one variable in terms of the other, you obtain equations like y = c/x. Then, plot each equation on the same graph, ensuring to consider the behavior of the function around important key points and asymptotes. The resulting graph reveals how the value of the function changes in the xy-plane and provides a visual context that can assist in understanding the interaction between x and y for different values of c.

Hyperbolas in Calculus

In calculus, a hyperbola is a type of conic section that can often appear as a level curve for certain functions. The level curves in our exercise turn out to be a family of hyperbolas because, for each constant value of c, the equation f(x, y) = xy simplifies to y = c/x. This is the equation of a hyperbola centered at the origin, with asymptotes along the x and y axes.

Each pair of c values, positive and negative, gives us two branches of a hyperbola, which reflect each other across the origin. As c increases in magnitude, the hyperbolas become wider, representing the increasing value of the function f(x, y) at those levels. Understanding the shape and orientation of these hyperbolas is crucial for visualizing how the function behaves over its domain, and thus, mastering hyperbolas is a valuable part of calculus involving level curves.