Problem 42 - Thomas Calculus

Question

If you have a parametric equation grapher, graph the equations over the given intervals. Hyperbola branch \(x=\sec t\) (enter as \(1 / \cos (t) ), y=\tan t\) (enter as \(\sin (t) / \cos (t) ),\) over $$\begin{array}{l}{\text { a. }-1.5 \leq t \leq 1.5} \\ {\text { b. }-0.5 \leq t \leq 0.5} \\ {\text { c. }-0.1 \leq t \leq 0.1}\end{array}$$

Step-by-Step Solution

Answer

Graph each interval separately; they show segments of hyperbola with increasing detail.

Step 1: Understand the given parametric equations

We're given parametric equations for a branch of a hyperbola: \( x = \sec t \) and \( y = \tan t \). In terms of sine and cosine functions, these equations can be expressed as \( x = \frac{1}{\cos t} \) and \( y = \frac{\sin t}{\cos t} \).

Step 2: Identify the intervals

We are asked to graph the equations over three intervals for the parameter \( t \): a) \(-1.5 \leq t \leq 1.5\), b) \(-0.5 \leq t \leq 0.5\), c) \(-0.1 \leq t \leq 0.1\).

Step 3: Graphing over Interval a

For \(-1.5 \leq t \leq 1.5\), plug in values of \( t \) within this range into the parametric equations. Note any restrictions where \( \cos t = 0 \) as these cause the equations to be undefined (e.g., the asymptotes of the hyperbola). Graph the resulting \( x, y \) values, more particularly, note the behavior of \( y \rightarrow \infty \) as \( \cos t \rightarrow 0 \).

Step 4: Graphing over Interval b

For \(-0.5 \leq t \leq 0.5\), repeat the process, substitute values within this smaller interval into the parametric equations. Plot these on the graph and note that the curve is less spread out and closer to the center than in Step 3.

Step 5: Graphing over Interval c

For \(-0.1 \leq t \leq 0.1\), this is an even smaller interval around zero. Evaluate the expressions for points in this interval and plot these coordinates. The resulting graph will show a segment of the hyperbola very close about the origin.

Key Concepts

Hyperbola

A hyperbola is a type of conic section, distinguished by its two separate branches. It is defined as the set of all points for which the difference of the distances to two fixed points, called the foci, is constant. In our exercise, the parametric equations are given by \( x = \frac{1}{\cos t} \) and \( y = \frac{\sin t}{\cos t} \), corresponding to a hyperbolic curve. These equations use trigonometric identities to redefine standard hyperbolic functions:

• \( x = \sec t \) represents the reciprocal of cosine.
• \( y = \tan t \) is the ratio of sine and cosine.

Near the asymptotes (which occur where \( \cos t = 0 \)), the values of \( x \) and \( y \) increase dramatically, leading to points that seem "infinitely" far from the center of the hyperbola. In practical terms, calculating a hyperbola using parametric equations can be helpful, notably when graphing its branches. Understanding these properties is what allows us to interpret the behavior of hyperbolas within different intervals.

Interval Notation

Interval notation is a mathematical notation used to describe a set of numbers lying between two endpoints. It effectively simplifies the expression of ranges or domains of variables involved in calculus and algebra problems. With interval notation:

• Brackets \([ ]\) indicate that an endpoint is included (closed interval).
• Parentheses \(( )\) mean an endpoint is not included (open interval).

In our exercise, three intervals are defined for the parameter \( t \). For example, interval a) is described by \([-1.5, 1.5]\). This means \( t \) can take any value from \(-1.5\) to \(1.5\) inclusive. Similarly, the other intervals, reflected by their respective condition, determine where and how one should evaluate the parametric equations to observe the behavior of the graph. By understanding interval notation, students can pinpoint exactly where critical behaviors like asymptotic points take place.

Graphing

Graphing parametric equations such as \( x = \sec t \) and \( y = \tan t \) involves plotting points derived from these formulas over specified intervals. During graphing:

• Substitute values of \( t \) into the parametric equations to find corresponding \( x \) and \( y \) coordinates.
• Identify points where \( \cos t = 0 \), as these are where the hyperbola's graph is undefined (the asymptotes).
• Note how the graph becomes narrower and closer to the origin when the interval is smaller.

When approaching each interval, ensure that the functionality of each point is considered, focusing on how values change, especially as they approach undefined points. Graphing these points showcases the typical hyperbolic spread and emphasizes the changes in behavior depending on the range of \( t \). Through graphing, the distinct characteristics of the hyperbola are visually represented, making abstract mathematical concepts tangible and easier to understand.