Question

The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots.\) c. Use the functions found in part (b) to graph the given equation. \(y^{3}=a x^{2}\) (Neile's semicubical parabola)

Short answer

Based on the given equation and solution, the short answer is: The derivative of the function is \(\frac{dy}{dx} = \frac{2ax}{3y^2}\), and the function can be rewritten as \(y = \sqrt[3]{ax^2}\) or \(y = ax^\frac{2}{3}\). The function has the shape of a semicubical parabola, and its graph depends on the value of \(a\). The function opens upwards if \(a\) is positive and downwards if \(a\) is negative.

Step-by-step solution

1. Differentiate the given equation with respect to \(x\)

We are given the equation \(y^3=ax^2\). To find \(\frac{dy}{dx}\), we will differentiate both sides with respect to \(x\) using the chain rule. \(\frac{d}{dx} (y^3) = \frac{d}{dx}(ax^2)\)

2. Apply the chain rule

By applying the chain rule, we get: \(3y^2 \frac{dy}{dx} = 2ax\) Now, we can solve for \(\frac{dy}{dx}\): \(\frac{dy}{dx} = \frac{2ax}{3y^2}\)

3. Solve the equation for \(y\)

We are given \(y^3=ax^2\). To solve for \(y\), we can take the cube root of both sides: \(y = \sqrt[3]{ax^2}\) Since there is only one equation and one function, we don't have multiple implicitly defined functions \(f_1(x), f_2(x), \ldots\).

4. Graph the equation

To graph the equation \(y^{3}=a x^{2}\), we will plot the function \(y = \sqrt[3]{ax^2}\). It may be helpful to rewrite the function as \(y = ax^\frac{2}{3}\) to more easily see how the function behaves. Notice that the function has the shape of a semicubical parabola, as mentioned in the exercise. To create the graph, you can use graphing software or a calculator with graphing capabilities. Plot the function for a chosen value of \(a\) and observe the shape of the curve. The equation produces a parabola-like shape opening either upwards or downwards depending on the sign of \(a\). If \(a\) is positive, the curve will open upwards, and if \(a\) is negative, the curve will open downwards.

Key concepts

Neile's Semicubical Parabola

Neile's semicubical parabola is an intriguing geometric curve first studied by the mathematician William Neile. It’s given by the equation \(y^3 = ax^2\). Unlike ordinary parabolas, Neile’s parabola has a unique curve due to the cubical power of \(y\) and the square of \(x\). This connection makes it especially interesting in mathematics.

• The curve is called 'semicubical' because it involves \(y^3\), which is a cubic term.
• The term \(ax^2\) is quadratic in nature, linking it back to classical parabolas.

This curve is one of the earliest applications of calculus. It's also an example of an implicit function where \(y\) is defined by an algebraic equation involving \(x\) without being isolated as a function of it. Understanding this curve helps students grasp key concepts like implicit differentiation and behavioral analysis of curves in different equations.

Chain Rule

The chain rule is a fundamental rule in calculus used to find the derivatives of composite functions. In simple terms, if a function is made up of another function, the derivative of the original function can be determined through the chain rule.In our example with the equation \(y^3 = ax^2\), the chain rule is used for the term \(y^3\). Here's how:

• First, recognize \(y^3\) as a composite function where \(y\) is itself a function of \(x\).
• Differentiate \(y^3\) with respect to \(x\), knowing that \(d(y^3)/dx = 3y^2(dy/dx)\).

This step is crucial because it allows us to find \(dy/dx\), the derivative of \(y\) with respect to \(x\). The chain rule simplifies dealing with nested functions and is essential for implicit differentiation. It helps unravel complex equations where direct differentiation is impractical.

Implicit Functions

Implicit functions represent relationships where a function is not expressed explicitly, like \(y = f(x)\). Instead, both variables are mixed in an equation, such as \(y^3 = ax^2\) in the case of Neile’s semicubical parabola.

• In implicit differentiation, you differentiate each term as usual but treat \(y\) as a function of \(x\) (even though it's not directly expressed).
• You apply the chain rule to terms involving \(y\), adding a \(dy/dx\) factor to account for \(y\)’s change relative to \(x\).

For instance,- Differentiating \(y^3\) by \(x\) yields \(3y^2(dy/dx)\).- Differentiating \(ax^2\) by \(x\) directly gives \(2ax\).Solving for \(dy/dx\) provides the rate of change of \(y\) with respect to \(x\) even when \(y\) isn’t isolated initially. Implicit functions show up throughout mathematics, emphasizing the chain rule’s importance and developing deeper understanding of function interactions.