Problem 36 - University Calculus: Early Transcendentals

Question

Sketch the surfaces ASSORTED $$16 x^{2}+4 y^{2}=1$$

Step-by-Step Solution

Answer

The surface is an ellipse centered at (0, 0) with semi-major axis \(\frac{1}{2}\) along y and semi-minor axis \(\frac{1}{4}\) along x.

Step 1: Identify the Surface Equation

The given equation is \(16x^2 + 4y^2 = 1\). This resembles the general form of an ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). It suggests that the surface to be sketched is an ellipse.

Step 2: Simplify the Equation

Rewrite the equation in standard ellipse form:\[ \frac{x^2}{\frac{1}{16}} + \frac{y^2}{\frac{1}{4}} = 1 \]This simplifies to:\[ \frac{x^2}{(1/4)^2} + \frac{y^2}{(1/2)^2} = 1 \]

Step 3: Determine the Ellipse's Axes

From the equation \(\frac{x^2}{(1/4)^2} + \frac{y^2}{(1/2)^2} = 1\), we identify the semi-major axis as \(\frac{1}{2}\) (along the y-axis) and the semi-minor axis as \(\frac{1}{4}\) (along the x-axis). This means the ellipse is taller, stretching more along the y-axis than the x-axis.

Step 4: Sketch the Ellipse

On a coordinate plane, sketch the ellipse:- The center is at the origin (0,0).- On the x-axis, draw the vertices at \((\frac{1}{4}, 0)\) and \((-\frac{1}{4}, 0)\).- On the y-axis, draw the vertices at \((0, \frac{1}{2})\) and \((0, -\frac{1}{2})\).- Connect these points smoothly to form the ellipse.

Key Concepts

Conic Sections

Conic sections are fascinating curves obtained by intersecting a plane with a cone. These curves include ellipses, parabolas, hyperbolas, and circles. Each type has unique characteristics and equations in coordinate geometry that define their shapes and properties. In this context, the ellipse stands out as a closed curve symmetric around two axes.

Ellipses are formed when the intersecting plane cuts through both nappes of the cone at an angle, but not parallel to the base. This results in a shape that resembles an elongated circle. The standard form of an ellipse's equation is crucial, displaying how the axes determine its stretch along the x and y directions.

Understanding the basic properties of conic sections, like the ellipse, helps in visualizing and sketching them accurately. Whether you encounter these curves in algebra or calculus, recognizing their specific forms can make complex graphing tasks more intuitive.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses algebraic equations to represent and analyze geometric shapes on a plane. It's a powerful tool that bridges algebra and geometry, allowing detailed exploration of shapes like ellipses through equations.

An ellipse is represented algebraically by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \(a\) and \(b\) are the semi-major and semi-minor axes, respectively. These parameters determine the stretch of the ellipse along the x-axis and y-axis. By inserting these into the formula, we can determine the exact dimensions and orientation of the ellipse.

Coordinate geometry simplifies complex shapes into manageable equations. This analytical approach is not just useful for ellipses but is applicable to all conic sections, enhancing our understanding of geometry through coordinates.

Graphing Ellipses

Graphing an ellipse requires understanding its equation and how it translates into a visual representation on the coordinate plane. The equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) provides all necessary details to sketch it by indicating the center, axes lengths, and orientation.

For the ellipse given by the equation \( 16x^2 + 4y^2 = 1 \), the simplification into \( \frac{x^2}{(\frac{1}{4})^2} + \frac{y^2}{(\frac{1}{2})^2} = 1 \) reveals its dimensions. The center is at the origin (0,0), with the semi-major axis of length \(\frac{1}{2}\) along the y-axis and the semi-minor axis of length \(\frac{1}{4}\) along the x-axis.

Visualizing the ellipse is straightforward by marking the axes' endpoints on the coordinate plane and smoothly connecting them. Graphing helps in appreciating the symmetry and proportions of ellipses, making it easier to understand other conic sections and their graphical characteristics.