Problem 85 - Essential Calculus

Question

(a) Use a graphing utility to graph the curve given by \(x=\frac{1-t^{2}}{1+t^{2}}, y=\frac{2 t}{1+t^{2}}, \quad-20 \leq t \leq 20 .\) (b) Describe the graph and confirm your result analytically. (c) Discuss the speed at which the curve is traced as \(t\) increases from -20 to 20

Step-by-Step Solution

Answer

Given parametric equations represent a circle centered at the origin with radius 1. The circle is being traced at a constant pace as \(t\) grows from -20 to 20.

Step 1: Graphing the curve

By inserting the values of \(t\) from -20 to 20 into the equations \(x=\frac{1-t^{2}}{1+t^{2}}\) and \(y=\frac{2 t}{1+t^{2}}\), the graph can be drawn using a mathematical graphical utility. This will yield an image that helps to visualize the curve these parametric equations are defining.

Step 2: Description of the graph

Upon analyzing the graph, it can be observed that the curve is in the shape of a circle with radius 1, centered at the origin. This means that the parametric equations represent a circle.

Step 3: Analytical confirmation

To confirm this observation analytically, boh the \(x\) and \(y\) equations could be squared and then added. Squaring \(x\) and \(y\) individually and then summing them gives \(\frac{(1-t^2)^2}{(1+t^2)^2} + \frac{(2t)^2}{(1+t^2)^2}=\) which simplifies to \(1\), confirming that this is the equation of a circle with radius 1 and centered at the origin.

Step 4: Discuss the speed of curve tracing

The speed at which the curve is traced as \(t\) increases from -20 to 20 will be constant which means that the speed of tracing the curve is independent of \(t\). This is due to the fact that these are parametric equations for a circle, the curvature of which does not change with time.

Key Concepts

Graphing Utility

Graphing utilities are essential tools in visualizing the relationship between variables in mathematical equations. When faced with parametric equations like \(x=\frac{1-t^{2}}{1+t^{2}}\) and \(y=\frac{2t}{1+t^{2}}\), a graphing utility can be indispensable. By inputting these equations into the software, students can obtain a precise visual representation of the curve defined by the parameters.

For the given exercise, inserting the range \(-20 \leq t \leq 20\) into the graphing utility will result in a visual output without the time-consuming effort of plotting points manually. It essentially provides a quick and reliable method to understand the behavior of these equations. Moreover, with advancements in technology, various graphing utilities now offer interactive features that let users manipulate the parameter \(t\) and observe the changes in real-time, enhancing their comprehension of the concept.

Circle Graph

A graph that represents the parametric equations of a circle is commonly referred to as a circle graph. This graph provides a visual confirmation of what the parametric equations represent. To improve understanding, it's crucial to acknowledge that a circle graph is not merely a collection of random points; it's a set of carefully plotted points that, when connected, form a perfect circle.

In the context of the exercise, the parametric equations plotted using a graphing utility will yield a circle graph, indicating that the equations describe a circle. Exploring this further, a circle with a radius of 1 and centered at the origin represents the fundamental graphical appearance of these parametric equations. Such visual depiction is paramount for students to grasp the abstract concept of parametric equations being mapped onto standard geometric shapes.

Analytical Confirmation

Analytical confirmation is the mathematical process of verifying the results obtained from graphical representations. It provides a solid proof that strengthens the visual interpretation from the circle graph. For the given exercise, analytical confirmation comes in the form of squaring both the \(x\) and \(y\) parametric equations and then summing them up. The squared expressions of \(x\) and \(y\) are as follows: \[\left(\frac{1-t^2}{1+t^2}\right)^2 + \left(\frac{2t}{1+t^2}\right)^2\].

Upon simplification, this sum equals to 1, which is the equation for a circle with a radius of 1, centered at the origin. This analytical step is not just a reinforcement of the graphically obtained shape but also serves as a bridge between the numerical and geometrical understanding of the equation. Mastering such methods helps students build a deeper and more comprehensive understanding of mathematical relationships.