Problem 47 - Essential Calculus

Question

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Witch of Agnesi: } x=2 \cot \theta, \quad y=2 \sin ^{2} \theta $$

Step-by-Step Solution

Answer

The Witch of Agnesi curve represented by the parametric equations \(x = 2 \cot(\theta)\) and \(y = 2 \sin^2(\theta)\) is plotted using a graphing utility. As \(\theta\) varies from \(-\pi\) to \(\pi\), the curve is drawn and direction is determined. No points are identified where the curve is not smooth.

Step 1: Understanding the Parametric Equations

The given parametric equations are \(x = 2 \cot(\theta)\) and \(y = 2 \sin^2(\theta)\). The first equation represents x-coordinates while the second equation represents corresponding y-coordinates of the curve points. The parameter here is \(\theta\), which varies over a specific interval. The functions cotangent and sine squared determine the shape and the points of the curve.

Step 2: Plotting the Curve Using a Graphing Utility

These parametric equations can be graphed using a graphing utility. By entering these equations into the grapher and setting a suitable range for \(\theta\) (e.g., \(-\pi \leq \theta \leq \pi\)), a curve will be drawn. The direction of the curve can be inferred by observing how it is drawn as \(\theta\) increases from \(-\pi\) to \(\pi\)

Step 3: Identify Non-Smooth Points in the Curve

For this particular curve, there are no points at which the curve is not smooth. However, in general, non-smooth points are where the curve has breaks or sharp turns. These can be identified by looking for any abrupt changes in slope or direction of the curve as \(\theta\) varies. For curves defined by parametric equations, these can often occur when the first or second derivative is undefined.

Key Concepts

Cotangent Function

The cotangent function is one of the trigonometric functions that is less commonly discussed, yet it plays a crucial role in mathematics, especially when dealing with parametric equations. Cotangent, typically written as \(\cot\), is the reciprocal of the tangent function, which means \(\cot(\theta) = \frac{1}{\tan(\theta)}\). It represents the ratio of the adjacent side to the opposite side in a right-angled triangle.

Graphing the cotangent function reveals its periodic nature, similar to other trigonometric functions, but with its own distinctive features. It has an undefined value at \(\theta = n\pi\) (where \(n\) is an integer), which corresponds to vertical asymptotes on its graph. As \(\theta\) increases, the cotangent function decreases from positive infinity, passing through zero, and then off to negative infinity before reaching the next vertical asymptote.

When involved in parametric equations, such as \(x = 2 \cot(\theta)\), the cotangent function helps in defining the x-coordinates of the curve, contributing to the overall shape and behavior of the graph.

Sine Squared Function

The sine squared function is denoted as \(\sin^2(\theta)\) and represents the square of the sine function. This is an essential function that appears frequently in various areas of mathematics, including trigonometric identities and calculus. The key feature of the sine function \(\sin(\theta)\) is its oscillation between -1 and 1, and when squared, the resulting values are non-negative and oscillate between 0 and 1.

In the context of parametric equations, like \(y = 2 \sin^2(\theta)\) in the Witch of Agnesi curve, the sine squared function defines the y-coordinates of the curve. It influences the 'height' of points on the curve while maintaining symmetry due to its inherent periodic properties. The squared nature of the function ensures that all values of \(y\) are positive or zero, shaping the curve's appearance in the upper halves of the Cartesian plane.

Graphing Utility

A graphing utility is an indispensable tool for visualizing complex mathematical functions, especially when dealing with parametric equations. It enables students and mathematicians to plot graphs accurately without the tedious process of manual calculations and point plotting. Modern graphing utilities can handle a wide range of functions and present visually appealing and informative graphs.

To utilize a graphing utility effectively for parametric equations, one must input the equations for \(x(\theta)\) and \(y(\theta)\) separately, along with specifying the range for the parameter \(\theta\). As the utility plots the curve, it automatically takes care of variations in \(\theta\) and draws the resulting curve with precision. By observing the continuous formation of the curve, users can interpret the direction of the curve and understand the relationship between the parametric equations and their graphical representation.

Witch of Agnesi

The Witch of Agnesi is a famous curve that has intrigued mathematicians for centuries due to its unique shape and mathematical properties. Named after the Italian mathematician Maria Gaetana Agnesi, this curve is not related to witchcraft but rather comes from a mistranslation of the word 'avversiera,' which refers to a curve.

The Witch of Agnesi is represented by parametric equations \(x = 2 \cot(\theta)\) and \(y = 2 \sin^2(\theta)\), which, when plotted, yield its characteristic bell-shaped graph. It is symmetrical about the y-axis and has a smooth, rounded top that approaches the x-axis but never quite touches it, getting infinitely closer. This curve serves as a classic example of how parametric equations can describe complex shapes in a concise form and demonstrate the interplay between different trigonometrical functions in determining the path and form of a curve.