Question

Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible. \(y^{2}=\frac{x^{3}}{2-x}(\text { Cissoid of Diocles })\)

Short answer

Question: Sketch and analyze the graph of the curve given by the equation \(y^{2}=\frac{x^{3}}{2-x}\). Find the domain, range, and points of horizontal tangents and inflection. Answer: The domain of the function is \(x \in (-\infty,2)\cup(2,\infty)\), and the range is \(y \in (-\infty,\infty)\). The curve has horizontal tangents at \(x=0\) and \(x=2\). To find the inflection points, analyze the second derivative \(\frac{d^{2}y}{dx^{2}}\). The graph of the curve can be sketched using a graphing calculator or an online graphing tool like GeoGebra, Desmos, or Wolfram Alpha, ensuring the asymptote at x=2 is shown correctly.

Step-by-step solution

Analyze the Given Equation

Before proceeding, let's analyze the given equation \(y^{2}=\frac{x^{3}}{2-x}\). We should determine the domain and the range of the function for which the equation is defined. The denominator of the expression cannot be zero, hence \(2-x \neq 0\). Therefore, \(x \neq 2\). The domain of the function is \(x \in (-\infty,2)\cup(2,\infty)\). Since there's no constraint on the values of \(y\), the range is \(y \in (-\infty,\infty)\).

Find the Derivative of y with respect to x

To find the slope and analyze the behavior of the curve at different points, we need to find the derivative of \(y\) with respect to \(x\). Since the equation is not given explicitly as \(y =\) or \(x =\), we will use implicit differentiation. Differentiating both sides of the equation \(y^{2}=\frac{x^{3}}{2-x}\) with respect to \(x\), we get: \(\frac{d}{dx}(y^{2})=\frac{d}{dx}(\frac{x^{3}}{2-x})\) Now, using the chain rule, the left side becomes: \(2y\frac{dy}{dx}\) On the right side, use the quotient rule: \(\frac{d}{dx}(\frac{x^{3}}{2-x})=\frac{(2-x)\times(3x^{2})-x^{3}\times(-1)}{(2-x)^{2}}\) Simplifying the right side, we get: \(\frac{3x^{3}-6x^{2}+2x^{2}+x^{3}}{(2-x)^{2}}\) Now the equation becomes: \(2y\frac{dy}{dx}=\frac{4x^{3}-4x^{2}}{(2-x)^{2}}\) To isolate \(\frac{dy}{dx}\), we divide both sides by \(2y\): \(\frac{dy}{dx}=\frac{2x^{3}-2x^{2}}{y(2-x)^{2}}\)

Analyze the Zero Points of the Derivative and Inflection Point

From the equation of \(\frac{dy}{dx}\), we can see that the numerator becomes zero when \(x=0\) or \(x=2\). Thus, the curve has a horizontal tangent at these points. For the inflection points (the point where the second derivative changes sign), we need to calculate the second derivative. To find \(\frac{d^{2}y}{dx^{2}}\), differentiate \(\frac{dy}{dx}\) with respect to \(x\): \(\frac{d^{2}y}{dx^{2}}=\frac{d}{dx}(\frac{2x^{3}-2x^{2}}{y(2-x)^{2}})\) Using the quotient rule, we get: \(\frac{d^{2}y}{dx^{2}}=\frac{(6x^{2}-4x)y(2-x)^{2}-(2x^{3}-2x^{2})[2y(2-x)^{2}-2y^{2}(2-x)]}{y^{2}(2-x)^{4}}\) Now, we can plug in the values of \(x\) and \(y\) from the original equation and find the sign of the second derivative, giving us inflection points.

Plot the Curve

Using the information from our analysis, like the domain, range, and the derivative, we can now sketch the graph of the curve. We also need to include the points where the tangent is horizontal and the inflection points. The cissoid of Diocles is a classical curve and has a specific shape. To graph it effectively, you can use a graphing calculator or a graphing tool like GeoGebra, Desmos, or Wolfram Alpha, which can handle implicit functions. Ensure that the features such as the asymptote at x=2 are shown correctly.

Key concepts

Implicit Differentiation

Implicit differentiation is a powerful technique used in calculus when dealing with equations where one variable is not explicitly expressed in terms of another. In our problem, the equation of the Cissoid of Diocles is given as \(y^2 = \frac{x^3}{2-x}\). Here, \(y\) is not isolated, meaning we can't easily differentiate with respect to \(x\) directly.

To tackle this, we apply implicit differentiation by taking the derivative of both sides with respect to \(x\). On the left side, we have \(y^2\), which differentiates to \(2y \frac{dy}{dx}\) using the chain rule. The right side, \(\frac{x^3}{2-x}\), is differentiated using the quotient rule.

• Chain rule for \(y^2\): involves multiplying the derivative of \(y^2\) by \(\frac{dy}{dx}\), yielding \(2y\frac{dy}{dx}\).
• Quotient rule for \(\frac{x^3}{2-x}\): results in finding the derivative of each term in \(\frac{u}{v}\) form.

Through this process, we isolate \(\frac{dy}{dx}\) to understand how the curve changes. Implicit differentiation helps us reveal information about slopes and behavior without needing an explicit function.

Domain and Range

Understanding the domain and range of a function is crucial, especially when dealing with curves like the Cissoid of Diocles, which are defined by implicit equations. The domain specifies all possible \(x\)-values, while the range indicates all possible \(y\)-values.

For the given equation \(y^2 = \frac{x^3}{2-x}\), the domain excludes any \(x\)-value that would make the denominator zero. Therefore, as \(2-x = 0\) must be avoided, \(x eq 2\) is a crucial restriction. This leads to a domain of \(x \in (-\infty, 2) \cup (2, \infty)\).

• The vertical asymptote at \(x = 2\) means the graph doesn't touch or cross this line, reflecting the exclusion from the domain.

The range can be more challenging to determine since \(y\)-values depend on \(x\)-values through the equation, but is not initially restricted. Here, \(y \in (-\infty, \infty)\), given the \(y^2\) form, is unrestricted across real numbers, considering all real financial values produce legitimate \(y\)-outputs.

Inflection Points

Inflection points on a curve are where the shape changes from concave up to concave down, or vice versa. To determine these for the Cissoid of Diocles, one must evaluate the behavior of the second derivative, \(\frac{d^2y}{dx^2}\). These points are crucial for understanding the geometry of the curve.

First, we identify where the first derivative, \(\frac{dy}{dx}\), equals zero or is undefined to locate critical points. Then, the second derivative is calculated to understand curvature changes. For the Cissoid, the formula for \(\frac{dy}{dx}\) was derived as \(\frac{2x^3-2x^2}{y(2-x)^2}\), and we proceeded to calculate \(\frac{d^2y}{dx^2}\) using the quotient rule once more.

• Zero points of \(\frac{dy}{dx}\): suggest potential local maxima, minima, or horizontal tangents at \(x = 0\) and \(x = 2\).
• Check sign changes in \(\frac{d^2y}{dx^2}\): confirms whether the point is indeed an inflection point by changing curvature direction.

This analysis showcases the nuanced surface of the curve and helps in accurately plotting its full shape.

Graphing Utilities

Graphing utilities are essential tools for visualizing complex functions and curves, like the Cissoid of Diocles, which presents some graphing challenges due to its implicit form.

Using these utilities, you can plot not just explicit functions \(y = f(x)\) but also implicit forms, providing a visual exploration of all the features. By inputting the equation into calculators or software like Desmos or GeoGebra, one can observe crucial characteristics such as domain restrictions, inflection points, and asymptotes visually.

• Highlighting the asymptote at \(x = 2\): ensures users understand this critical gap in the graph.
• Visualizing inflection points: enhances comprehension of where the curve changes direction.

These tools provide interactive and dynamic graphs, allowing for zooming and panning to closely inspect notable features or to view the broader curve landscape. They are invaluable in educational settings, aiding in the concrete understanding of abstract mathematical concepts.