Problem 59 - Calculus for Biology and Medicine

Question

Astroid (a) Consider the curve with equation \(x^{2 / 3}+y^{2 / 3}=4\). Find \(\frac{d y}{d x}\) at \((-1,3 \sqrt{3})\) (b) Use a graphing calculator to graph the curve in (a). If the calculator cannot graph implicit functions, graph the upper and the lower halves of the curve separately. To get the left half of the graph, make sure that your calculator evaluates \(x^{2 / 3}\) in the order \(\left(x^{2}\right)^{1 / 3}\). Choose the viewing rectangle \(-10 \leq x \leq 10\), \(-10 \leq y \leq 10\)

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = \frac{1}{3\sqrt[3]{3}} \) at \((-1, 3\sqrt{3})\). Graph using separate expressions for upper and lower halves.

Step 1: Differentiate Implicitly

Given the equation of the astroid \( x^{2/3} + y^{2/3} = 4 \), we need to differentiate both sides with respect to \( x \). The implicit differentiation yields:\[\frac{2}{3}x^{-1/3} + \frac{2}{3}y^{-1/3} \cdot \frac{dy}{dx} = 0.\]

Step 2: Solve for \( \frac{dy}{dx} \)

Rearrange the equation from Step 1 to solve for \( \frac{dy}{dx} \). Subtract \( \frac{2}{3}x^{-1/3} \) from both sides:\[\frac{2}{3}y^{-1/3} \cdot \frac{dy}{dx} = -\frac{2}{3}x^{-1/3}.\]Now, solve for \( \frac{dy}{dx} \):\[\frac{dy}{dx} = -\frac{x^{-1/3}}{y^{-1/3}}.\]This can be simplified to:\[\frac{dy}{dx} = -\left(\frac{x}{y}\right)^{-1/3}.\]

Step 3: Substitute the Given Point

Substitute the point \((-1, 3\sqrt{3})\) into the derivative expression found in Step 2. \( x = -1 \) and \( y = 3\sqrt{3} \) in \( \frac{dy}{dx} = -\left(\frac{x}{y}\right)^{-1/3} \):\[ \frac{dy}{dx} = -\left(\frac{-1}{3\sqrt{3}}\right)^{-1/3}. \]Calculate this expression.

Step 4: Simplify the Expression

Simplify \( -\left(\frac{-1}{3\sqrt{3}}\right)^{-1/3} \).First, evaluate \( \frac{-1}{3\sqrt{3}}\):\( \frac{-1}{3\sqrt{3}} = \frac{-\sqrt{3}}{9} \).Now, evaluate \( \left(\frac{-\sqrt{3}}{9}\right)^{-1/3}\) and simplify:\[ \left(\frac{-\sqrt{3}}{9}\right)^{-1/3} = \left(-1\right)^{1/3} \cdot \left(\frac{9}{\sqrt{3}}\right)^{1/3} = -\frac{1}{3\sqrt[3]{3}}. \]Thus, \( \frac{dy}{dx} = \frac{1}{3\sqrt[3]{3}} \).

Step 5: Graph the Curve

Graph the equation \( x^{2/3} + y^{2/3} = 4 \) using a graphing calculator within the viewing rectangle \( -10 \leq x \leq 10 \), \( -10 \leq y \leq 10 \).If implicit graphing is not possible, consider rewriting the equation by expressing \( y \) in terms of \( x \):For the upper half (\( y \geq 0\)), solve for \( y \):\[ y = \left(4 - x^{2/3}\right)^{3/2}. \]For the lower half (\( y \leq 0\)), solve for \( y \):\[ y = -\left(4 - x^{2/3}\right)^{3/2}. \]Plot each half separately using these equations.

Key Concepts

Astroid

An astroid is a special type of curve that you might encounter when studying implicit functions. The standard equation for an astroid is expressed as \( x^{2/3} + y^{2/3} = a^{2/3} \), where \( a \) represents a constant. In the problem you're working with, the constant is \( a = 4 \). This equation describes a closed, symmetric shape known as a superellipse. It features four cusp-like points at the locations along the axes.The name "astroid" comes from its star-like shape. When plotted, it looks somewhat like a squashed star. It's a great example of how powerful implicit functions can be when describing complex shapes that can't easily be put into standard function form. Astroids can be explored in various applications such as physics, engineering, and even art. Look for its symmetry and the way it nests perfectly within a square when graphed.

Graphing Implicit Functions

Graphing implicit functions might seem tricky at first because the relationship between the \( x \) and \( y \) variables isn't directly solved in terms of one variable. In this exercise, you were given the equation of an astroid \( x^{2/3} + y^{2/3} = 4 \). Since it's not straightforward to solve for \( y \) in terms of \( x \) or vice versa, we use graphing techniques to visualize it.When graphing using a calculator or software, you may sometimes need to consider the function as separate parts or components, especially if you're solving for just the upper or lower halves. For this exercise, you can split the astroid equation into two:

• The upper half: \( y = (4 - x^{2/3})^{3/2} \).
• The lower half: \( y = -(4 - x^{2/3})^{3/2} \).

Plotting these separately makes handling complex or implicit equations more manageable. Always remember to choose an appropriate viewing window to capture the entire shape; for this astroid, the boundaries are \( -10 \leq x \leq 10 \) and \( -10 \leq y \leq 10 \). This ensures you capture all significant parts of the graph, revealing the beautiful symmetry.

Implicit Differentiation Steps

Implicit differentiation is a technique used when you have an equation involving two or more variables where you can't easily solve for one variable in terms of the others. It's particularly helpful with curves like the astroid, which is given as \( x^{2/3} + y^{2/3} = 4 \). The goal of implicit differentiation is to find \( \frac{dy}{dx} \), the derivative that tells you the slope of the tangent to the curve at any point.To use implicit differentiation:

• Differentiate both sides of the given equation with respect to \( x \). When doing this, remember to apply the chain rule to terms involving \( y \).
• For the astroid equation, this differentiation gives you: \( \frac{2}{3}x^{-1/3} + \frac{2}{3}y^{-1/3} \cdot \frac{dy}{dx} = 0 \).
• Next, rearrange to solve for \( \frac{dy}{dx} \). Move terms around to isolate \( \frac{dy}{dx} \): \( \frac{dy}{dx} = -\left(\frac{x}{y}\right)^{-1/3} \).

By substituting specific values of \( x \) and \( y \) into this resulting equation, you can find the slope of the tangent line at points like \( (-1, 3\sqrt{3}) \). Implicit differentiation is a powerful tool, allowing you to understand the behavior of complex curves that can't be easily expressed as functions.