Problem 13 - Essential Calculus

Question

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=\sec \theta, \quad y=\cos \theta $$

Step-by-Step Solution

Answer

The rectangular equation corresponding to the given parametric equations is \(1/x = y\). The curve is a hyperbola with orientation in the positive x direction for the upper half and negative x direction for the lower half.

Step 1: Convert to Rectangular Form

To write the equation in rectangular form, we can just use the fact that \(x=\sec \theta = 1/\cos \theta\), so we can rewrite it as \(1/x = \cos \theta\). Now, we can replace \(\cos \theta\) with \(y\). So we now have the equation in rectangular form: \(1/x = y\).

Step 2: Sketch The Curve

Now we have to sketch the curve represented by the rectangular equation \(1/x = y\). The graph of this function is a hyperbola with vertical axis. Since we have \(1/x = y\), it shows that as \(x\) approaches infinity, \(y\) approaches 0 from the positive side. On the left side, as \(x\) approaches negative infinity, \(y\) approaches 0 from the negative side.

Step 3: Indicate Orientation of the Curve

To indicate the orientation of the curve, increase the values of \(\theta\) from 0. As \(\theta\) increases, we start from \(x = 1\) (\(y = 1\)), moving rightward and downward as \(\theta\) approaches \(\pi/2\), then moving to the left and upwards as \(\theta\) approaches \(\pi\). This suggests that the curve is oriented in the positive x direction for the above half of x-axis, and in the negative x direction for the below half of the x-axis.

Key Concepts

Rectangular Equation

Understanding the conversion from parametric equations to a rectangular equation is a crucial skill in mathematics, particularly when working with curves in a coordinate system. A rectangular equation, also known as a Cartesian equation, is an equation which expresses a relationship between x and y coordinates on a standard graph. In our example, we have the parametric equations
\( x = \sec \theta \) and \( y = \cos \theta \). To find the corresponding rectangular equation, we first recognize that \( \sec \theta \) is the reciprocal of \( \cos \theta \), hence \( x = 1/\cos \theta \). This allows us to substitute \( y \) for \( \cos \theta \) and obtain the simpler equation \( 1/x = y \) or \( y = 1/x \). This equation is now solely in terms of x and y, making it easier to graph and analyze. Understanding the relationship and the conversion process between parametric and rectangular forms aids in visualizing and solving more complex mathematical problems.

Hyperbola

A hyperbola is one of the conic sections obtained when a plane intersects a cone at a certain angle. Hyperbolas have two disconnected curves called branches which mirror each other with respect to the center of the hyperbola. The equation \( y = 1/x \) represents a rectangular hyperbola, which is symmetric about the origin.

When sketching a hyperbola, it is useful to note a few key characteristics:

• The branches of the hyperbola are asymptotic to the axes, meaning they approach the axes but never touch.
• The product of the x and y coordinates of any point on a rectangular hyperbola is constant, which explains the equation \( xy = 1 \) for our example.
• The branches will lie in opposite quadrants (first and third, or second and fourth).

The distinctive shape of the hyperbola requires an understanding of these traits, as they will guide the drawing of accurate and properly oriented hyperbolas in curve sketching.

Curve Sketching

The process of curve sketching is a fundamental aspect of calculus and analytic geometry, enabling us to represent functions and their behaviors graphically. By sketching curves, we can visualize mathematical concepts and the relationships between variables. For our hyperbolic equation \( y = 1/x \), here are the basic steps for sketching:

• Identify asymptotes: In this case, the x-axis and y-axis serve as asymptotes since the hyperbola gets infinitely close to them but never crosses.
• Plot key points: Determine a few coordinates by plugging values into the equation, which in this example are easy to find due to the hyperbola’s symmetry.
• Analyze orientation: Increase the parameter \(\theta\) and note how the x and y coordinates change, determining the direction of the curve.
• Sketch the branches: With the asymptotes and points marked, lightly draw the branches of the hyperbola making sure they mirror each other about the origin and approach the asymptotes without touching them.

Remember, the goal of curve sketching isn't to create a perfect graph, but to get a clear visual representation of how the curve behaves, which can be invaluable when studying calculus and applied mathematics.