Question

32\. Sketch a particular example of the "witch of Agnesi," the curve given by \(y^{2}=\frac{4(2-x)}{x}\). Show that it is symmetric about

Short answer

Answer: The coordinates of the tangent intersection on the y-axis are (0, y₀), where y₀ is given by the equation \(y₀ = -mx_0\), and m is the slope of the tangent line at the point (x₀, y₀) on the curve, which can be calculated from the derivative of the function with respect to y, given as \(f'(y) = \frac{-8y}{y^4}\).

Step-by-step solution

Rewrite Equation as Function of y

To analyze the problem easier, let's first rewrite the given equation as a function of y or \(f(x)\). \(y^2 = \frac{4(2-x)}{x}\) Multiply both sides by x: \(x y^2 = 4(2-x)\) Divide both sides by 4: \(\frac{1}{4}x y^2 = 2-x\) Now, let's solve for x to get the function of y: \(x = \frac{4(2 - y^2)}{y^2}\) Now, we have the function as \(f(y) = \frac{4(2 - y^2)}{y^2}\).

Check the Symmetry Property of the Curve

To check if the curve is symmetric about the y-axis, we need to ensure that negative x values result in the same y values as positive x values for the function: If x is symmetric, then \(f(-y) = f(y)\). So let's calculate \(f(-y)\) and see if it is equal to \(f(y)\): \(f(-y) = \frac{4(2 - (-y)^2)}{(-y)^2} = \frac{4(2 - y^2)}{y^2}\) Since \(f(y) = f(-y)\), the curve is symmetric about the y-axis.

Find Intercepts and Asymptotes

To find the x-intercepts and y-intercepts of the curve, let's set y = 0 and x = 0 in the equation \(y^2 = \frac{4(2-x)}{x}\), respectively. For x-intercepts: \(y^2 = \frac{4(2 - x)}{x} = 0\) Since the numerator is equal to zero and 2 - x ≠ 0, there are no x-intercepts for the given curve. For y-intercepts: \(y^2 = \frac{4(2 - 0)}{0}\) Since division by zer0 is not allowed, we have vertical asymptotes at x = 0. To find the horizontal asymptotes, we can analyze the behavior of the function as x approaches infinity: \(\lim_{x \to \infty} y^2 = \lim_{x \to \infty} \frac{4(2-x)}{x}\) As x approaches infinity, y² approaches 0. So, we have a horizontal asymptote at y = 0.

Find the Coordinates of the Tangent Intersection on the y-axis

To find the point where the tangent of the curve intersects the y-axis, we need to find the tangent of the function. The tangent line has the equation: \(y = m(x - x_0) + y_0\) Where m is the slope, (x₀, y₀) is the point on the curve, x is the x-coordinate of the tangent intersection on the y-axis, and y₀ is the y-coordinate of the tangent intersection on the y-axis. Since the y-axis is given by x = 0, we will have: \(y = m(0 - x_0) + y_0\) Notice that when x = 0, y = y₀, which means the tangent line intersects the y-axis at the point (0, y₀), where: \(y₀ = -mx_0\) Now we need to find the slope, m, at the point (x₀, y₀) on the curve. To do this, we can take the derivative of the re-written function of y with respect to x: \(f'(y) = \frac{d}{dy}(\frac{4(2 - y^2)}{y^2})\) Use the quotient rule: \(f'(y) = \frac{-8y}{y^4}\) Now, let's calculate the slope at the point (x₀, y₀): \(m = f'(y₀) = \frac{-8y_0}{y_0^4}\) Now substitute the result back into the equation for the tangent line: \(y = (\frac{-8y_0}{y_0^4})(0 - x_0) + y_0\) Since the tangent line intersects the y-axis at the point (0, y₀), we can say that the coordinates of the tangent intersection on the y-axis are (0, y₀).

Key concepts

Curve Sketching

Curve sketching is a fundamental skill in calculus that involves understanding the behavior of a function to create its graph. For the Witch of Agnesi, described by the equation \(y^{2}=\frac{4(2-x)}{x}\), curve sketching can reveal its interesting properties. To begin with, the function is rewritten for clarity and solved for \(x\) in terms of \(y\), setting the stage for analysis.

Key steps in sketching this curve include finding intercepts, asymptotes, and points of symmetry — though for the Witch of Agnesi, we find it has no x-intercepts and a vertical asymptote at \(x=0\). The horizontal asymptote at \(y=0\) signifies the behavior as \(x\) approaches infinity. These features contribute to the overall shape and position of the curve on the coordinate plane.

Further analysis includes determining the curvature and points of inflection, which require the calculation of derivatives and assessing their signs. The Witch of Agnesi, with its unique bell-shaped structure, offers a classic example for these curve sketching techniques.

Symmetry in Mathematics

Symmetry in mathematics refers to a situation where a figure or graph remains unchanged under certain transformations such as reflection, rotation, or translation. For functions, symmetry often simplifies analysis and graphing. The Witch of Agnesi demonstrates symmetry about the y-axis, which means for every point on the curve, there is an equivalent point mirrored across the y-axis.

This property is tested by comparing the original function \(f(y)\) with \(f(-y)\). If both expressions are equal, the function is symmetric about the y-axis. For the equation \(y^{2}=\frac{4(2-x)}{x}\), substituting \(-y\) results in the same expression as \(f(y)\), confirming its symmetry. This bilateral symmetry reduces the workload by half since calculating for one side of the axis suffices to understand the entire curve's behavior.

Calculus

Calculus, especially differential calculus, plays an essential role in analyzing and understanding the properties of curves. Finding the slope of a tangent line to a curve at a given point is a common task in calculus. For the Witch of Agnesi, the slope of the tangent line relates to the first derivative of the function with respect to \(y\), calculated using the quotient rule.

Once the derivative is found—\(f'(y) = \frac{-8y}{y^4}\)—you can determine the slope of the tangent at any point \((x_0, y_0)\) on the curve. Not only does this allow us to find the slope, but we can also locate where the tangent intersects the y-axis by solving for \(y_0\) when \(x=0\). Calculus techniques are indispensable for comprehending and visualizing the detailed behaviors of a curve at specific points.

Asymptotes

Asymptotes are lines that a curve approaches but never actually reaches, acting as boundaries that guide the behavior of the curve at extremes. For the Witch of Agnesi curve, we determine that there is a vertical asymptote at \(x=0\). This is due to the function being undefined at that point, as division by zero is not allowed within the realm of real numbers.

Additionally, we conclude that there is a horizontal asymptote at \(y=0\) by analyzing the limit of \(y^2\) as \(x\) approaches infinity; the values of \(y^2\) tend towards 0. Understanding the role of asymptotes is crucial because they highlight the end behavior of the curve and ensure a more accurate depiction when sketching it. As asymptotes define the ultimate direction and constraints of the curve, they provide insight into its long-term trends and potential bounds.