Question

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. \(x^{2 / 3}+y^{2 / 3}=1\) (Astroid or hypocycloid with four cusps)

Short answer

Answer: The critical points and cusps of the curve are located at the points (0,1) and (0,-1).

Step-by-step solution

Rewrite the equation in a more convenient form

Since we are given \(x^{2 / 3}+y^{2 / 3}=1\), we first rewrite it in a more convenient form: $$x^{2 / 3} = 1 - y^{2 / 3}.$$ Now let's get rid of the fractional exponents by raising both sides to the power of 3: $$\left(x^{2 / 3}\right)^3 = \left(1 - y^{2 / 3}\right)^3$$ $$x^2 = (1 - y^{2 / 3})^3.$$

Find the first derivative of the curve using implicit differentiation

Differentiate both sides of the equation with respect to \(y\) (in order to get \( \frac{dx}{dy}\)) using the chain rule and implicit differentiation: $$\frac{d}{dy}\left(x^2\right) = \frac{d}{dy}\left((1 - y^{2 / 3})^3\right)$$ $$2x\frac{dx}{dy} = 3(1 - y^{2 / 3})^2\cdot\frac{-2}{3}y^{-1/3}$$ Then, $$\frac{dx}{dy} = \frac{-y^{-1/3}(1 - y^{2 / 3})^2}{2x}$$

Find the critical points

To find the critical points of this curve, let's first analyze the numerator of the derivative we found in the previous step. Notice that the numerator will be zero if \(1 - y^{2 / 3}=0\). Thus, $$y^{2/3}=1\Rightarrow y=\pm1.$$ Now we substitute these values to the original equation to find the corresponding \(x\) coordinates of the critical points. When \(y=1\): $$x^2 = (1 - 1^2)^3 = 0 \Rightarrow x=0$$ Similarly, when \(y=-1\): $$x^2 = (1 - (-1)^{2 / 3})^3 = 0 \Rightarrow x=0$$ Therefore, the critical points are \((0,1)\) and \((0,-1)\).

Find the cusps of the curve by setting the first derivative to be undefined

We look at the denominator of the first derivative to find the cusps: $$2x = 0 \Rightarrow x=0$$ Since we already found the critical points, the cusps of the curve are also located at the points \((0,1)\) and \((0,-1)\).

Use a graphing utility to graph the curve

Now that we have gathered all the necessary details about the curve, it's time to graph the curve. Use a graphing utility (such as Desmos, Geogebra, or a graphing calculator) to plot the equation \(x^{2 / 3}+y^{2 / 3}=1\). Remember to highlight the critical points and cusps that we found during our analysis, and make sure that the graph accurately represents the equation. From the graph, we will see that the curve is symmetric with respect to both the x and y axes and has a shape resembling a four-pointed star, which is why it's called an astroid or hypocycloid with four cusps.

Key concepts

Implicit differentiation

In calculus, implicit differentiation is a powerful tool that helps us find derivatives of equations not easily expressed as a function of one variable. For instance, the equation of an astroid, a curve given by \(x^{2/3} + y^{2/3} = 1\), is not easily rewritten in terms of \(x\) or \(y\) alone.

Let's consider how implicit differentiation works in our case. We start by differentiating both sides of the equation with respect to \(y\). Since \(x\) is also a function of \(y\), we utilize the chain rule:

• Differentiate \(x^2\) with respect to \(y\), applying the chain rule: \(2x\frac{dx}{dy}\).
• Differentiating \((1-y^{2/3})^3\) involves the power rule followed by the chain rule, resulting in \(-2\cdot3(1-y^{2/3})^2y^{-1/3}\).

By setting these derivatives equal, we solve for \(\frac{dx}{dy}\), giving us a relationship between the rates of change of \(x\) and \(y\). Implicit differentiation thus allows us to explore changing aspects of astroid curves.

Critical points

Critical points on a curve are instances where the derivative is zero, indicating possible maxima, minima, or points of inflection. For the given astroid \(x^{2 / 3}+y^{2 / 3}=1\), we find the critical points by assessing where the derivative \(\frac{dx}{dy}\) becomes zero.

Our task is to find values of \(y\) that make the numerator of \(\frac{dx}{dy}\) equal to zero: \(1 - y^{2 / 3} = 0\). Solving this leads us to \(y = \pm 1\).

To determine the corresponding \(x\) values at these points, substitute back into the original equation:

• When \(y = 1\), \(x^2 = (1 - 1)^3 = 0\), hence \(x = 0\) leads to the point \((0, 1)\).
• Similarly, for \(y = -1\), \(x^2 = 0\), resulting in \((0, -1)\).

The curve's critical points here, \((0, 1)\) and \((0, -1)\), indicate where the slope of the tangent is flat, an essential feature in understanding the curve's behavior.

Cusps

Cusps are noteworthy points on a curve where the first derivative is undefined—essentially, where the curve's tangent line is abruptly\/discontinuously undefined or vertical. In the case of the astroid \(x^{2/3} + y^{2/3} = 1\), cusps can be found by analyzing when the denominator of \(\frac{dx}{dy}\) is zero.

From our derivative expression, the denominator \(2x = 0\) tells us \(x = 0\) is the condition for cusps. We have already identified that when \(x = 0\), corresponding \(y\) values are \(+1\) and \(-1\), positioning cusps at \((0, 1)\) and \((0, -1)\).

The sharp 'turning' and peaks of the astroid shape at these points give it its characteristic star-like appearance, showing an intersection of tangent lines forming this uniquely distinct feature on the graph. Recognizing cusps is crucial for comprehending curve behavior, especially where traditional calculus tools for smooth curves expand to include non-smooth junctions.

Graphing utility

A graphing utility is an excellent tool for visualizing complex mathematical functions and their behaviors. For graphing the astroid defined by \(x^{2/3} + y^{2/3} = 1\), utilize graphing calculators or online platforms like Desmos or GeoGebra.

To ensure your graph provides all the needed details, include the following:

• Plot the curve accurately, reflecting its symmetry about both the x and y axes.
• Highlight the critical points \((0, 1)\) and \((0, -1)\) and mark these distinctly as they represent flat points on the curve.
• Identify and emphasize the cusps at the same points—these showcase the curve's unique sharp transitions.

When properly rendered, the graph reveals the astroid's four 'cushion-like' lobes, each centered on one of the coordinate axes. Analyzing graphs with utility programs not only reinforces analytical understanding but also enhances conceptual clarity to aid deeper comprehension of intricate curves like the astroid.