Question

Berechne tabellarisch die Punkte der Versiera der Agnesi für \(t=\frac{i}{8} \pi, i=-7, \ldots,-2,-1,0,1,2, \ldots, 7\), und skizziere die Kurve damit (Kreisradius \(2 \mathrm{~cm}\) ).

Short answer

The points of the Witch of Agnesi for the given values of \(t\) are calculated using the formula \(y = \frac{8a^3}{t^2 + 4a^2}\). The curve is sketched by plotting these points and connecting them.

Step-by-step solution

Calculate the Points

First, we need to calculate the points of the Witch of Agnesi for the following 15 values of \(t\): \(-\frac{7\pi}{8}, -\frac{6\pi}{8}, -\frac{5\pi}{8}, -\frac{4\pi}{8}, -\frac{3\pi}{8}, -\frac{2\pi}{8}, -\frac{\pi}{8}, 0, \frac{\pi}{8}, \frac{2\pi}{8}, \frac{3\pi}{8}, \frac{4\pi}{8}, \frac{5\pi}{8}, \frac{6\pi}{8}, \frac{7\pi}{8}\). The points can be calculated by substituting each value of \(t\) into the equation of the Witch of Agnesi \(y = \frac{8a^3}{t^2 + 4a^2}\), where \(a = 2\, cm\) (the given circle's radius).

Sketch the Curve

After calculating the points, we then sketch the Witch of Agnesi. This can be done by plotting the points calculated in the previous step on a graph, and then connecting the points to form the curve. Please note that due to the cubic nature of the Witch of Agnesi, the curve will look like a bell-shape, intersecting the y-axis at the point (0, 2), and floating above the x-axis but never crossing it (asymptote).

Key concepts

Versiera der Agnesi

The Versiera der Agnesi, or Witch of Agnesi, is a classic curve arising from 18th century mathematics.
The term 'Versiera' originates from Italian, and it has an interesting history, often being misassociated with witchcraft due to translation errors.

However, in the world of mathematics, the Witch of Agnesi is known for its distinctive shape, which is based on a simple equation.

• The curve represents the relationship between two variables, x and y, usually written as \( y = \frac{8a^3}{x^2 + 4a^2} \).
• Here, the parameter 'a' is associated with the radius of a circle that is used to construct the curve.
• The Witch of Agnesi is bell-shaped and has a maximum at the y-point corresponding to the value of the parameter 'a'.
• It has an asymptote along the x-axis, meaning that the curve approaches this axis, but never intersects it.

To sketch this curve, it is standard to calculate specific points by determining the y-values for different x-values, which are then plotted on a graph to visualize the curve. In our exercise, the radius is set to 2 cm, which influences the shape and position of the curve on the graph.

Vector Analysis

While vector analysis is not directly applied in the construction or calculation of the Witch of Agnesi, understanding vectors can help deepen our appreciation of the geometric properties of this curve.
Vector analysis is a branch of mathematics the deals with magnitudes possessing both direction and magnitude, known as vectors.

Key insights include:

• Vectors allow for the description of physical quantities like force, velocity, and acceleration, which are important in fields like physics and engineering.
• In the context of curves, vector functions can represent the position of points along the curve as it changes with another parameter, often time or angle.
• With regard to the Witch of Agnesi, hypothetical vectors could describe movement from the center of a generating circle towards the curve or from one point on the curve to another.

If the curve were to be analyzed using vectors, one might describe points on the curve as position vectors and examine the curvature using concepts of vector calculus such as derivatives and integrals.

Function Theory

Function theory is a foundational element in understanding the Witch of Agnesi and other mathematical curves.
Basically, function theory explores the properties of functions which are mathematical relations that uniquely associate values of one variable with values of another variable.

For example:

• The Witch of Agnesi is described by a function that relates x-values to y-values based on the curve's equation \( y = \frac{8a^3}{x^2 + 4a^2} \).
• The function is what provides us with the ability to calculate specific points for graphing the curve, as seen in the exercise by substituting x-values into the equation.
• Moreover, function theory examines the behavior of the function — such as continuity, limits, and asymptotic behavior — which informs us how the curve behaves near the x-axis and its overall shape.

In summary, function theory is essential for both calculating the points necessary to plot the Witch of Agnesi and understanding the mathematical properties that define the curve's shape and behavior.