Problem 35 - 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. $$z=\sqrt{25-x^{2}-y^{2}} ;[-6,6] \times[-6,6].$$

Step-by-Step Solution

Answer

Question: Graph several level curves of the function \(z = \sqrt{25 - x^2 - y^2}\) within the window \([-6, 6] \times [-6, 6]\) and label at least two level curves with their corresponding \(z\)-values. Answer: The level curves for the given function are concentric circles centered at the origin with equations of the form \(x^2 + y^2 = 25 - k^2\), where \(k\) represents the \(z\)-value. Some examples of level curves within the given window are: 1. For \(z = 0\), the level curve is a circle with radius 5 and centered at the origin, described by the equation \(x^2 + y^2 = 25\). 2. For \(z = 2\), the level curve is a circle with radius \(\sqrt{21}\) and centered at the origin, described by the equation \(x^2 + y^2 = 21\). These level curves and others can be graphed in the window \([-6, 6] \times [-6, 6]\) and labeled with their corresponding \(z\)-values.

Step 1: Understanding Level Curves

Level curves, also known as contour lines, are the curves on a graph that represent all the points where the function has the same value. In this case, we are going to find the level curves for the function \(z = \sqrt{25 - x^2 - y^2}\) for some particular \(z\)-values. To do this, we will set \(z\) equal to a constant value, then solve for \(x\) and \(y\).

Step 2: Finding the Equation for Level Curves

To find the equation representing a level curve of the given function, set \(z = k\), where \(k\) is a constant. In this case, we will have: $$k = \sqrt{25 - x^2 - y^2}$$ Squaring both sides gives, $$k^2 = 25 - x^2 - y^2.$$ Rearranging the equation to express it in the form of a circle, we get: $$x^2 + y^2 = 25 - k^2.$$ This equation represents the level curves of the given function.

Step 3: Finding Level Curves for Different \(z\)-Values

Choose a few different \(z\)-values (\(k\)-values) and find the corresponding equations for the level curves: 1. When \(z = 0\) (or \(k = 0\)), $$x^2 + y^2 = 25.$$ This is a circle with radius 5 and centered at the origin. 2. When \(z = 2\) (or \(k = 2\)), $$x^2 + y^2 = 25 - 2^2 = 21.$$ This is also a circle with a radius of \(\sqrt{21}\) and centered at the origin. 3. Similarly, you can choose a few more \(z\)-values and find the corresponding level curves.

Step 4: Graphing the Level Curves

Plot the level curves found in the previous step within the given window of \([-6, 6] \times [-6, 6]\). You will see that the level curves are concentric circles centered at the origin. Make sure to label at least two level curves with their \(z\)-values. For example, label the level curve for \(z = 0\) as "\(z = 0\)" and the one for \(z = 2\) as "\(z = 2\)". By following these steps, you will successfully graph several level curves of the given function in the specified window and label them with their \(z\)-values.

Key Concepts

Contour Plot

A contour plot, or contour map, is a two-dimensional representation of a three-dimensional surface. It shows the curves or lines (contour lines) where the function remains constant. In this context, each contour line corresponds to a specific function value, just like how a topographical map depicts elevation levels.
To create a contour plot for the function \(z = \sqrt{25 - x^2 - y^2}\), we first derive the level curves. By setting the function equal to constant values like \(z = k\), and solving \(k = \sqrt{25 - x^2 - y^2}\), we find the level curves expressed as circles. Hence:\[x^2 + y^2 = 25 - k^2\]
This indicates that as \(z\) changes, the radius of these circles changes, depicting elevation on the plot. Therefore, a contour plot reveals how drastically values change over the plane. Concentric circles indicate that we're mapping a function similar to a hill, where the height decreases downwards along formed circular rings around the origin.

Graphing Functions

Graphing functions in a multivariable setup involves visualizing level curves which represent different constant values of the function. The chosen window, here \([-6, 6] \times [-6, 6]\), ensures all critical points within the specified domain are plotted.
When graphing the function \(z = \sqrt{25 - x^2 - y^2}\), level curves appear as concentric circles. Each circle represents where the function's output \(z\) remains constant. Using values like \(z = 0\) and \(z = 2\), you graph circles with radii derived from the relation \(x^2 + y^2 = 25 - z^2\).
In graphical representation, it's crucial to highlight these points:

• Each circle's radius decreases as \(z\) increases, illustrating the function's surface dips.
• Proper labeling of level curves emphasizes clarity in function behavior.

By graphing these shapes, students can better interpret how regions in the plane relate to specific values and thus understand multivariable influences.

Multivariable Calculus

Multivariable calculus extends calculus concepts to functions of several variables. In this setting, we explore how functions change with different input variables, often leading to intriguing visual patterns like the ones seen here.
Within multivariable calculus, level curves provide critical insights:

• They allow us to simplify a complex 3D surface into manageable 2D slices.
• They help visualize contours and slopes of terrain which underpin practical applications, like landscape planning or fluid dynamics.

Our function \(z = \sqrt{25 - x^2 - y^2}\) is a classic example of a three-dimensional surface displayed in two dimensions using level curves. By visualizing this using level curves, we transition from complex data to more digestible insights.
This approach roots deeply in multivariable calculus, fostering a powerful understanding of dynamic systems where variables do not work in isolation. Understanding these interactions clarifies how changes in one variable can influence the outcome of a function involving many variables.