Problem 45
Question
Draw a contour map of the function showing several level curves. $$f(x, y)=\sqrt{y^{2}-x^{2}}$$
Step-by-Step Solution
Verified Answer
Draw hyperbolas for \(k > 0\) and lines \(y = \pm x\) for \(k = 0\).
1Step 1: Define Level Curves
Level curves of a function are given by setting the function equal to constant values. For this exercise, we define the level curves by solving \(f(x, y) = k\) where \(k\) is a constant. Therefore, \(\sqrt{y^2 - x^2} = k\) implies \(y^2 - x^2 = k^2\), which can be rewritten as \(y^2 = x^2 + k^2\).
2Step 2: Check Viability of Level Curves
The equation \(y^2 = x^2 + k^2\) gives us a family of hyperbolas when \(k^2 \geq 0\). Notice that when \(k = 0\), the equation becomes \(y^2 = x^2\), which implies \(y = \pm x\) - these are actually lines (not hyperbolas) where \(f(x, y) = 0\).
3Step 3: Plot Level Curves for Different Values of k
Choose several positive values for \(k\), such as \(k = 1, 2, 3, \ldots\) and plot the corresponding hyperbolas \(y^2 = x^2 + k^2\). When \(k = 0\), plot the lines \(y = x\) and \(y = -x\). These plots should resemble hourglass shapes centered along the lines \(y = \pm x\).
4Step 4: Draw the Contour Map
Draw a coordinate plane. For each value of \(k\), sketch the hyperbolas \(y^2 = x^2 + k^2\). Include the lines \(y = x\) and \(y = -x\) for \(k = 0\). Label each curve according to its \(k\) value. This map visually represents the behavior of the function \(f(x, y)\) as the values change.
Key Concepts
Level CurvesHyperbolasCoordinate PlaneFunction Behavior
Level Curves
Level curves are a fascinating way to visualize functions of two variables. Think of them as slices of a three-dimensional surface, similar to how a topographic map shows elevation levels.
Each slice—or level curve—represents all the points where the function has the same value. To find these curves for a given function, we set it equal to a constant. In this exercise, we analyze the function \(f(x, y) = \sqrt{y^2 - x^2}\) and set it equal to \(k\): \(\sqrt{y^2 - x^2} = k\).
By squaring both sides, we derive the equation \(y^2 - x^2 = k^2\). This represents the level curves for different values of \(k\). They help us understand how the function behaves across the coordinate plane.
Each slice—or level curve—represents all the points where the function has the same value. To find these curves for a given function, we set it equal to a constant. In this exercise, we analyze the function \(f(x, y) = \sqrt{y^2 - x^2}\) and set it equal to \(k\): \(\sqrt{y^2 - x^2} = k\).
By squaring both sides, we derive the equation \(y^2 - x^2 = k^2\). This represents the level curves for different values of \(k\). They help us understand how the function behaves across the coordinate plane.
Hyperbolas
A hyperbola is a type of conic section that looks like two opposite-facing, mirrored arcs. They are crucial in describing the behavior of our function's level curves. The equation \(y^2 - x^2 = k^2\) is characteristic of hyperbolas, except when \(k = 0\).
For positive values of \(k\), these curves are indeed hyperbolas, opening along the \(y\)-axis. As we change \(k\), we are effectively tracing out different hyperbolic paths on the coordinate plane.
Interestingly, when \(k = 0\), the function gives us \(y^2 = x^2\), which simplifies to the lines \(y = x\) and \(y = -x\). These act as the borders or asymptotes for our hyperbolas.
For positive values of \(k\), these curves are indeed hyperbolas, opening along the \(y\)-axis. As we change \(k\), we are effectively tracing out different hyperbolic paths on the coordinate plane.
Interestingly, when \(k = 0\), the function gives us \(y^2 = x^2\), which simplifies to the lines \(y = x\) and \(y = -x\). These act as the borders or asymptotes for our hyperbolas.
Coordinate Plane
The coordinate plane is the fundamental backdrop where these mathematical stories unfold. It consists of horizontal \(x\)-axis and vertical \(y\)-axis, dividing the plane into four quadrants.
This two-dimensional plane makes it possible to plot level curves and visualize how the function behaves.
In our exercise, by drawing the coordinate plane and plotting curves for different \(k\), we can see the hourglass-like hyperbolas. The process involves graphing both hyperbolas and lines, giving a comprehensive view of how the function behaves across the plane.
This two-dimensional plane makes it possible to plot level curves and visualize how the function behaves.
In our exercise, by drawing the coordinate plane and plotting curves for different \(k\), we can see the hourglass-like hyperbolas. The process involves graphing both hyperbolas and lines, giving a comprehensive view of how the function behaves across the plane.
Function Behavior
Understanding function behavior through level curves and their plots is essential. For our function, \(f(x, y) = \sqrt{y^2 - x^2}\), level curves illustrate how function values spread over the coordinate plane.
The shape and arrangement of these curves (hyperbolas and lines) tell us how the function increases or decreases in value.
For instance, for \(k>0\), the hyperbolas convey regions where the function value is constant; while on \(k=0\), it touches the lines \(y = x\) and \(y = -x\), marking points where \(f(x, y)\) equals zero.
By mapping these insights, the contour map gives us a visual grasp of the function's behavior across different input values.
The shape and arrangement of these curves (hyperbolas and lines) tell us how the function increases or decreases in value.
For instance, for \(k>0\), the hyperbolas convey regions where the function value is constant; while on \(k=0\), it touches the lines \(y = x\) and \(y = -x\), marking points where \(f(x, y)\) equals zero.
By mapping these insights, the contour map gives us a visual grasp of the function's behavior across different input values.
Other exercises in this chapter
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