Problem 33 - 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. $$z=\sqrt{x^{2}+4 y^{2}} ;[-8,8] \times[-8,8].$$

Step-by-Step Solution

Answer

Question: Graph the level curves of the function \(z=\sqrt{x^{2}+4 y^{2}}\) within the window \([-8,8] \times [-8,8]\) and label at least two level curves with their z-values. Answer: The level curves for \(z=2\) and \(z=4\) can be represented by the equations of ellipses: For \(z=2\), the level curve is given by the ellipse \(1 = \frac{x^{2}}{4}+\frac{y^{2}}{1}\) with semi-major axis along the \(y\)-axis of length 1 and semi-minor axis along the \(x\)-axis of length 2. For \(z=4\), the level curve is given by the ellipse \(1 = \frac{x^{2}}{16}+\frac{y^{2}}{4}\) with a semi-major axis along the \(y\)-axis of length 2 and the semi-minor axis along the \(x\)-axis of length 4.

Step 1: Derive the equation for level curves

To find the level curves of the given function, we set \(z\) equal to a constant value \(c\). So we have: $$c=\sqrt{x^{2}+4 y^{2}}$$ Solve for an equation depending only on \(x\) and \(y\): $$c^{2}=x^{2}+4 y^{2}$$ This is the equation representing the level curves of the given function.

Step 2: Select z-values for level curves

For simplicity, let's select two z-values to plot the level curves. We can choose \(c = 2\) and \(c = 4\).

Step 3: Find the equations of the level curves for the chosen z-values

Plug in the chosen \(c\) values into the equation of the level curves to get the equations for the selected level curves. For \(c=2\): $$2^{2}=x^{2}+4 y^{2}$$ $$4=x^{2}+4 y^{2}$$ $$1= \frac{x^{2}}{4}+\frac{y^{2}}{1}$$ This is the equation of an ellipse with the semi-major axis along the \(y\)-axis of length 1, and semi-minor axis along the \(x\)-axis of length 2. For \(c=4\): $$4^{2}=x^{2}+4 y^{2}$$ $$16=x^{2}+4 y^{2}$$ $$1= \frac{x^{2}}{16}+\frac{y^{2}}{4}$$ This is the equation of another ellipse with the semi-major axis along the \(y\)-axis of length 2 and the semi-minor axis along the \(x\)-axis of length 4.

Step 4: Graph the level curves

Using graphing software or drawing by hand, graph the level curves using the equations obtained in step 3 within the given window \([-8,8] \times[-8,8]\): - \(1= \frac{x^{2}}{4}+\frac{y^{2}}{1}\) is an ellipse centered at the origin with a semi-major axis of 1 unit in the \(y\)-direction and a semi-minor axis of 2 units in the \(x\)-direction. - \(1= \frac{x^{2}}{16}+\frac{y^{2}}{4}\) is an ellipse centered at the origin with a semi-major axis of 2 units in the \(y\)-direction and a semi-minor axis of 4 units in the \(x\)-direction.

Step 5: Label the level curves with their z-values

Label the level curve corresponding to \(c = 2\) with "\(z = 2\)" and the level curve corresponding to \(c = 4\) with "\(z = 4\)".

Key Concepts

Ellipse

An ellipse is a shape that looks like a stretched-out circle. Think of it like an oval, an elongated form that has two focal points. Mathematically, an ellipse can be described by an equation. For instance, in the form \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), where \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes, respectively.

Ellipses can occur naturally in different scenarios within mathematics, especially when dealing with level curves. In the context of the given exercise, level curves of functions can take the shape of ellipses. By setting the function values \(z\) equal to constants, you can derive equations that represent these geometric shapes. It is fascinating how graphical representation helps in visualizing complex mathematics concepts.

Applications of ellipses are vast. From planetary orbits to architectures, many forms mirror these beautiful and symmetric curves. They are fundamental in understanding conic sections, a key concept in geometry.

Graphing Functions

Graphing functions involves plotting points on a coordinate grid to create a visual representation of a mathematical relationship or equation. This process helps in analyzing and understanding the behavior of the function, identifying trends and patterns, and interpreting the data effectively.

When graphing level curves like ellipses, we first need to derive the function equations, as seen in the exercise. By changing the values of one of the variables (like \(z\)), you can graph different curves within the same function. Each level curve on the graph indicates places where the function has the same constant value. This is particularly useful in fields like physics and engineering, where visualizing a system's behavior under different parameters is crucial.

Visual tools such as graphing calculators or software can aid in accurately plotting these equations, especially within defined plots or windows, like \([-8,8] \times [-8,8]\), to ensure the graph is both clear and informative.

Semi-Major Axis

The semi-major axis is the longest diameter of an ellipse. It's half the length of the major axis, which is the longest line that runs through the center and both foci of the ellipse. In simple terms, if you were to draw an ellipse, the semi-major axis would run down the middle of its length.

In the context of level curves that are ellipses, knowing the semi-major axis allows us to understand the orientation and size of the curve. For example, in the exercise, the semi-major axis changes depending on the value of \(z\) chosen for the level curve.

Understanding the semi-major and semi-minor axes is essential for graphing these ellipses accurately. For \(z=2\), the semi-major axis is in the \(y\)-direction with a length of 1, while for \(z=4\), it is in the \(y\)-direction with a length of 2. This visualizes how the size and orientation of the ellipse change based on the constant \(z\) value, offering deeper insights into the function's behavior in different scenarios.