Problem 15 - 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. $$ z(x, y)=y^{2}-x^{2}, \quad c=4 $$

Step-by-Step Solution

Answer

The level curve for \(z = 4\) is a hyperbola centered at \((0,0)\) with vertices \((0,2)\) and \((0,-2)\).

Step 1: Understanding the Level Curves

Level curves are curves along which the function is constant, meaning for a given function \(z(x, y)\), we set \(z = c\). Here, the function is \(z(x, y) = y^2 - x^2\) and \(c = 4\). So, the level curve equation we need to solve is \(y^2 - x^2 = 4\).

Step 2: Rewriting the Level Curve Equation

The equation of the level curve is \(y^2 - x^2 = 4\). Notice that this equation resembles the standard form of a hyperbola \(y^2 / a^2 - x^2 / b^2 = 1\). In this case, it simplifies to \(y^2 - x^2 = (2)^2\), which clearly is a hyperbola centered at the origin \((0,0)\) with \(a = 2\).

Step 3: Identifying the Hyperbola

For the hyperbola \(y^2 - x^2 = 4\), the vertices are found at points where \(y = ext{±}a\) on the y-axis (since \(b = 1\) for these coordinates). The vertices are at \((0, 2)\) and \((0, -2)\). This describes a hyperbola that opens upwards and downwards.

Key Concepts

Hyperbola

A hyperbola is a type of conic section that can be visualized as two separate curves. These curves have a characteristic opening direction, being either up-and-down or side-to-side, depending on the equation form. In our context, we are dealing with the equation \(y^2 - x^2 = 4\), which describes a hyperbola.This specific equation is similar to the standard form of a hyperbola, \(y^2/a^2 - x^2/b^2 = 1\). Here, \(a^2 = 4\) and \(b^2 = 1\). Therefore, \(a = 2\) and \(b = 1\), indicating that the hyperbola opens vertically.

• The vertices of a hyperbola lie along the axis of symmetry, found at points \((0, \pm a)\).
• For the equation given, the vertices are located at \((0, 2)\) and \((0, -2)\), positions on the y-axis.

Such a hyperbola is centered at the origin \((0, 0)\) and opens upwards and downward from the center.

Visualizing Functions

Visualizing functions involves representing them graphically to understand their behavior and key features. Level curves offer a useful means of doing this for functions of two variables, such as \(z(x, y) = y^2 - x^2\). Level curves show where the function holds a constant value—the 'height' of the surface at that slice is level. In our example:

• We set \(z(x, y) = c\), where \(c=4\). This transforms our function into the level curve equation \(y^2 - x^2 = 4\).
• The resulting graph will consist of hyperbolas, each representing different constant values of the function.

These curves help us understand not just specific solution points, but the general shape and sweep of the function over the xy-plane by slicing through at constant heights. By mapping out several level curves, you can get a comprehensive picture of how a function behaves in space.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus into functions of several variables. Instead of dealing with just \(f(x)\), we look at functions such as \(z(x, y)\) where output depends on both \(x\) and \(y\). This added complexity requires new mathematical tools and concepts, such as level curves, to analyze these functions.

• Level curves are used here to determine where a multivariable function retains a constant value across a plane, which provides insight into the function's general form and continuity.
• In our example, the level curve \(y^2 - x^2 = 4\) helps to visualize the function \(z(x, y)\) as forming a surface shaped like an intersecting hyperbola in three dimensions.

By working with such curves, students of multivariable calculus can intuitively grasp the behavior of more complex functions, leading to a deeper understanding of both theoretical and practical applications.