Problem 42
Question
Determine the equation of the level curves \(f(x, y)=c\) and sketch the level curves for the specified values of \(c\). \(f(x, y)=x^{2}-y^{2} ; c=0,1,-1\)
Step-by-Step Solution
Verified Answer
The level curves are lines \(y = x\) and \(y = -x\) for \(c = 0\), a hyperbola for \(c = 1\) along the \(x\)-axis, and a hyperbola for \(c = -1\) along the \(y\)-axis.
1Step 1: Understand the Function
The given function is a two-variable function, \(f(x, y) = x^2 - y^2\), which involves both \(x\) and \(y\). The level curves are found where \(f(x, y) = c\).
2Step 2: Set Up the Equation for Level Curves
The equation for a level curve is obtained by equating the function to a constant \(c\): \(x^2 - y^2 = c\). This represents a family of hyperbolas or a pair of intersecting lines depending on the value of \(c\).
3Step 3: Case 1 - Level Curve for \(c = 0\)
For \(c = 0\), the equation becomes \(x^2 - y^2 = 0\). This can be factored as \((x-y)(x+y) = 0\), representing the lines \(y = x\) and \(y = -x\). These are lines through the origin at 45 degrees to each axis.
4Step 4: Case 2 - Level Curve for \(c = 1\)
For \(c = 1\), the equation becomes \(x^2 - y^2 = 1\). This represents a hyperbola with opening along the \(x\)-axis. The vertices are at \((1,0)\) and \((-1,0)\).
5Step 5: Case 3 - Level Curve for \(c = -1\)
For \(c = -1\), the equation becomes \(x^2 - y^2 = -1\), which rearranges to \(y^2 - x^2 = 1\). This represents another hyperbola but opening along the \(y\)-axis. The vertices are at \((0,1)\) and \((0,-1)\).
6Step 6: Sketch the Level Curves
Start by drawing the coordinate system. For \(c = 0\), draw the lines \(y = x\) and \(y = -x\). For \(c = 1\), draw a hyperbola centered at the origin opening along the \(x\)-axis. For \(c = -1\), draw a hyperbola centered at the origin opening along the \(y\)-axis.
Key Concepts
Two-Variable FunctionHyperbolaEquation of Level CurvesCoordinate System Sketching
Two-Variable Function
A two-variable function involves two independent variables, commonly labeled as \(x\) and \(y\). Unlike single-variable functions, which map one input to one output, two-variable functions map a pair of inputs to a single output. For instance, the function given in the exercise is \(f(x, y) = x^2 - y^2\). This means every unique combination of \(x\) and \(y\) yields a particular value of \(f(x, y)\).
When dealing with two-variable functions, we are often interested in how the function behaves in different regions of the \((x, y)\) plane. That's where the concept of level curves comes in. Level curves help visualize these regions by showing where the function has a constant value.
When dealing with two-variable functions, we are often interested in how the function behaves in different regions of the \((x, y)\) plane. That's where the concept of level curves comes in. Level curves help visualize these regions by showing where the function has a constant value.
Hyperbola
A hyperbola is a type of conic section that looks like two mirrored arcs opening in opposite directions. It is defined as the set of all points \((x, y)\) where the difference of the distances to two fixed points (foci) is constant. In our exercise, the equation \(x^2 - y^2 = c\) for different values of \(c\) forms hyperbolas in the coordinate system.
One can think of a hyperbola as an ellipse with imaginary focus distances or as a reflection of the ellipse's properties through its eccentricity. The standard form of a hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) for a horizontally opening hyperbola, and \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\) for a vertically opening one. This difference in orientation is also observed in the exercise's level curves for different values of \(c\).
One can think of a hyperbola as an ellipse with imaginary focus distances or as a reflection of the ellipse's properties through its eccentricity. The standard form of a hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) for a horizontally opening hyperbola, and \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\) for a vertically opening one. This difference in orientation is also observed in the exercise's level curves for different values of \(c\).
Equation of Level Curves
The equation of level curves is crucial in understanding how a function behaves for different constant values \(c\). For a two-variable function like \(f(x, y) = x^2 - y^2\), a level curve equation \(x^2 - y^2 = c\) signifies all the points where the function value is \(c\).
Each level curve can represent a different geometrical shape. In this exercise, when \(c = 0\), the level curve is a set of intersecting lines through the origin, given by \(y = x\) and \(y = -x\). For \(c = 1\), the equation forms a hyperbola opening along the x-axis, while for \(c = -1\), it forms a hyperbola opening along the y-axis. These different shapes for each \(c\) help visualize how the function changes.
Each level curve can represent a different geometrical shape. In this exercise, when \(c = 0\), the level curve is a set of intersecting lines through the origin, given by \(y = x\) and \(y = -x\). For \(c = 1\), the equation forms a hyperbola opening along the x-axis, while for \(c = -1\), it forms a hyperbola opening along the y-axis. These different shapes for each \(c\) help visualize how the function changes.
Coordinate System Sketching
Sketching in a coordinate system involves plotting points and curves based on equations, like the level curves derived from a two-variable function. A good sketch of level curves starts with establishing a grid based on the x and y axes, which serves as a reference.
For the exercise, this involves drawing the intersecting lines and hyperbolas for \(c = 0, 1, -1\).
For the exercise, this involves drawing the intersecting lines and hyperbolas for \(c = 0, 1, -1\).
- For \(c = 0\), draw the lines \(y = x\) and \(y = -x\).
- For \(c = 1\), plot the hyperbola that opens along the x-axis and centers at the origin.
- For \(c = -1\), draw the hyperbola that opens along the y-axis, also centered at the origin.
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