Problem 24 - Calculus 3

Question

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ f(x, y)=x^{2} ; c=4,9 $$

Step-by-Step Solution

Answer

Level curves are vertical lines at \(x = \\pm 2\) for \(c = 4\) and \(x = \\pm 3\) for \(c = 9\).

Step 1: Understanding Level Curves

Level curves for a function of two variables, like \(f(x, y)\), are the sets of points in the \(xy\)-plane where the function takes on a constant value. Here, those values are \(c = 4\) and \(c = 9\).

Step 2: Express the function as an equation in terms of \(x\) and \(y\)

Substitute the constant value \(c\) into the equation \(f(x, y) = c\). For each \(c\), we substitute and solve. Start with \(f(x, y) = x^2 = 4\) and \(f(x, y) = x^2 = 9\).

Step 3: Solve for \(x\) in terms of \(y\) for each constant \(c\)

For \(c = 4\), rearrange the equation \(x^2 = 4\) to get \(x = \pm 2\). For \(c = 9\), rearrange \(x^2 = 9\) to get \(x = \pm 3\). These represent the vertical lines where the function equals these constants.

Step 4: Interpret level curves in the context of the function

The level curves for \(c = 4\) are vertical lines at \(x = -2\) and \(x = 2\), and for \(c = 9\), they are at \(x = -3\) and \(x = 3\). Since \(f(x, y) = x^2\) has no \(y\) term, the level curves do not depend on \(y\).

Key Concepts

Functions of Two Variables

In mathematics, functions of two variables are crucial for describing surfaces in a three-dimensional space. Think of them as functions which take two inputs, usually labeled as \(x\) and \(y\), and produce a single output, \(f(x, y)\). For instance, the function \(f(x, y) = x^2\) does not include \(y\) and only depends on the square of \(x\). This might seem simplistic, but the idea expands to surfaces where both variables play integral roles.
This concept is useful as it allows us to analyze different phenomena in one scenario, such as temperature, altitude, or pressure levels, which often rely on two spatial dimensions. In mathematical terms, you can visualize this as a surface that hovers over or under the flat \(xy\)-plane. Each point \((x, y)\) on this surface corresponds to a height determined by the function. Hence, understanding this kind of function is key to grasping a wide array of mathematical models.

Constant Value

A constant value in the context of functions of two variables denotes a specific output that a function can take. This is often referred to as \(c\) in mathematics, representing the level at which you are slicing through a surface to explore its characteristics. For our example, we have the function \(f(x, y) = x^2\) and we want to analyze it at constant values \(c = 4\) and \(c = 9\).
By setting the function equal to these constant values, \(f(x, y) = c\), we can explore what points \((x, y)\) would give us an outcome of 4 or 9. Specifically, \(x^2 = 4\) becomes \(x = \pm 2\), and \(x^2 = 9\) becomes \(x = \pm 3\).
The focus of setting a function to a constant ensures that we comprehend its configuration and nature at these levels, showing us predefined sections of the complete surface derived from the function.

Visualization in the xy-plane

When visualizing functions of two variables, level curves offer a way to understand the structure of the function in the \(xy\)-plane. Essentially, these curves are the traces that show where the function maintains a constant value - it is like taking a horizontal slice of a surface to reveal a contour map.
For our function \(f(x, y) = x^2\), the level curves for \(c = 4\) are vertical lines at \(x = \pm 2\), and for \(c = 9\), they are vertical lines at \(x = \pm 3\).
These lines signify locations on the plane where the function keeps the same value as specified, even though \(y\) does not alter it because it is missing from the function. In geometric terms, it results in vertical bands, and there's no movement along \(y\), since the function is solely reliant on \(x\). This reiterates the significance of level curves in providing a flat snapshot of cross-sections of a potentially complex surface.