Problem 30 - Calculus Early Transcendentals

Question

Graph several level curves of the following functions using the given window. Label at least two level curves with their \(z\) -values. $$z=x^{2}+y^{2} ;[-4,4] \times[-4,4].$$

Step-by-Step Solution

Answer

Answer: The equation for the level curves is \(x^2 + y^2 = z\). Two level curves obtained by substituting different z-values are a circle centered at (0,0) with radius 1 (z=1), and a circle centered at (0,0) with radius 2 (z=4).

Step 1: Write the equation for level curves

For a function \(z = f(x, y)\), the equation for level curves is given by: $$f(x,y) = z$$ We are given the function: $$z = x^2 + y^2$$ So, the equation for level curves becomes $$x^2 + y^2 = z$$

Step 2: Find two level curves

Let's find and label two level curves by substituting different \(z\)-values. First, let \(z = 1\). The equation becomes: $$x^2 + y^2 = 1$$ Here, we get a circle centered at \((0,0)\) with radius \(1\). Second, let \(z = 4\). The equation becomes: $$x^2 + y^2 = 4$$ Here, we get a circle centered at \((0,0)\) with radius \(2\). We can also observe that we are getting circles with increasing radii for different increase in \(z\) values.

Step 3: Graph the level curves

Using the given window \([-4,4] \times[-4,4]\), let's graph the level curves we obtained in step 2. 1. First, draw the circle centered at origin with radius \(1\) and label it with \(z=1\). 2. Next, draw the circle centered at origin with radius \(2\) and label it with \(z=4\). These two level curves represent different heights or \(z\) values of the given function in the specified window. Analyzing more level curves in this window would lead to a better understanding of the overall shape of the surface represented by the given equation \(z = x^2 + y^2\).

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function that can be represented in the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The characteristic feature of these functions is the term \(x^2\), which gives them a parabolic shape when graphed in the coordinate plane. In the context of level curves, particularly those derived from quadratic expressions like \(z = x^2 + y^2\), these functions represent circular parabolas when visualized in a three-dimensional coordinate space.
In this specific example, each level curve is essentially a slice of the three-dimensional surface described by \(z = x^2 + y^2\). These curves are circles that have different radii corresponding to different \(z\)-values. The unique property of such a quadratic function is that regardless of the level curve's specific \(z\)-value, the mathematical relationship remains consistent—each represents a cross-section of a paraboloid.
Quadratic functions are not just limited to circles; depending on the coefficients and terms included, they might graph as different conic sections, including ellipses, hyperbolas, and parabolas.

Graphing in Coordinate Plane

Graphing in a coordinate plane involves plotting points, lines, or shapes using a grid system defined by two perpendicular number lines: the \(x\)-axis (horizontal) and the \(y\)-axis (vertical). Each point is defined by an \(x\) and a \(y\) coordinate pair \((x, y)\). This systematic approach can be extended to three dimensions by adding a \(z\)-axis, resulting in a three-dimensional graph where points are described using \((x, y, z)\) coordinates.
For the function \(z = x^2 + y^2\), plotting its level curves on the coordinate plane involves setting \(z\) to some constant value, thus transforming the expression into a form like \(x^2 + y^2 = k\). This represents a circle in the \(xy\)-plane. The value of \(k\) gives us the radius squared of the circle, with each increment in \(k\) resulting in larger circles moving outwards concentrically from the origin \((0,0)\).
When constructing such graphs within a given window, such as \([-4, 4] \times [-4, 4]\), it's crucial to ensure that the graph stays within the confines of the specified range. This keeps the visualization accurate and ensures that all relevant level curves can be effectively plotted and observed.

Contour Plots

Contour plots, also known as level plots, offer a method to represent three-dimensional data in two dimensions. They illustrate where a function has constant values, known as contours or level curves, within a specified range of \(x\) and \(y\). This approach is particularly useful in fields such as meteorology, topography, and engineering, among others, where it is vital to understand how a variable behaves across a given surface.
In the example of the function \(z = x^2 + y^2\), each contour line, or level curve, represents a curve where the function maintains constant \(z\)-value. As you move up or down these contour lines, you're essentially traversing through different 'slices' of a three-dimensional object. These lines provide insight into the relative height and steepness of the surface at various points.
Contour plots simplify complex three-dimensional data by focusing on major changes across the \(xy\)-plane. By analyzing the spacing between lines, one can get a sense of the rate of change—closely spaced lines suggest a steep incline, while broadly spaced lines indicate a gentler slope.