Home > APCALC > Chapter 6 > Lesson 6.2.2 > Problem6-72

6-72.

ASTROID

The graph of $x^{2/3}+y^{2/3}=4$ is called an astroid. .

1. Use implicit differentiation to find $\frac { d y } { d x }$.

$\frac{2}{3}x^{-1/3}+\frac{2}{3}y^{-1/3}y^\prime=0$

$\frac{2}{3}x^{-1/3}=-\frac{2}{3}y^{-1/3}y^\prime$

$y^\prime=-\frac{y^{1/3}}{x^{1/3}}$

$\frac{dy}{dx}=-\sqrt[3]{\frac{y}{x}}$

2. Where does the astroid have a horizontal tangent? Explain.

Look at the derivative equation. Will values of $y$ or values of $x$ make the derivative $0$?

A horizontal tangent should exist when $\frac{dy}{dx}=0$, which happens when $y=0$.

Now look at the graph. There are two coordinate points in which $y=0$. Does the derivative exist at those points?
How about the tangent lines?

3. Where is the derivative undefined? What happens at those points?

Look at the derivative equation. What values of x would make that equation undefined?
Now look at the graph. Describe the shape of the graph at that value of $x$.

Use the eTool below to visualize the problem.
Click the link at right for the full version of the eTool: Calc 6-72 HW eTool