Question

Identifying functions from an equation. 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. $$x^{4}=2\left(x^{2}-y^{2}\right) \text { (eight curve) }$$

Short answer

Question: Use implicit differentiation to find \(\frac{dy}{dx}\), solve the equation for \(y\) to find implicitly defined functions, and graph the functions for the equation \(x^4 = 2(x^2 - y^2)\). Answer: The derivative \(\frac{dy}{dx} = \frac{x(4x^2-4)}{y}\). The implicitly defined functions are \(y = f_1(x) = \sqrt{-\frac{1}{2}x^2(x^2 - 2)}\), and \(y = f_2(x) = -\sqrt{-\frac{1}{2}x^2(x^2 - 2)}\). The graph represents the given equation known as the eight curve, and is valid for the domain \(-\sqrt{2} \leq x \leq \sqrt{2}\).

Step-by-step solution

Apply implicit differentiation on the equation

Differentiate both sides of the equation \(x^4 = 2(x^2 - y^2)\) with respect to x, using the chain rule for the y terms: $$\frac{d}{dx}(x^4) = \frac{d}{dx}\left[2(x^2 - y^2)\right]$$

Calculate the derivatives

Calculate the derivatives to get: $$4x^3 = 2\left[2x - 2y\frac{dy}{dx}\right]$$

Solve for \(\frac{dy}{dx}\)

Now, we can solve for \(\frac{dy}{dx}\): $$4x^3 = 4x - 4y\frac{dy}{dx}$$ $$\frac{dy}{dx} = \frac{4x^3 - 4x}{4y}$$ $$\frac{dy}{dx} = \frac{x(4x^2-4)}{y}$$

Factor the given equation

Now, we solve the given equation for \(y\): $$x^4 = 2(x^2 - y^2)$$ $$x^4 - 2x^2 + 2y^2 = 0$$ $$(x^4 - 2x^2) + 2y^2 = 0$$ $$x^2(x^2 - 2) + 2y^2 = 0$$

Solve for y

Now, solve the equation for \(y\): $$2y^2 = -x^2(x^2 - 2)$$ $$y^2 = -\frac{1}{2}x^2(x^2 - 2)$$ $$y = \pm\sqrt{-\frac{1}{2}x^2(x^2 - 2)}$$ So, the implicitly defined functions are: $$y = f_1(x) = \sqrt{-\frac{1}{2}x^2(x^2 - 2)}$$ $$y = f_2(x) = -\sqrt{-\frac{1}{2}x^2(x^2 - 2)}$$

Graph the functions

Use a graphing tool or software to plot the functions \(f_1(x)\) and \(f_2(x)\) on the same graph. The graph will represent the given equation, also known as the eight curve. Remember that the graph is only valid for the domain where the functions are real, i.e., where \(-\sqrt{2} \leq x \leq \sqrt{2}\).

Key concepts

Differential Equations

Differential Equations are mathematical equations that involve functions and their derivatives. They show the relationship between a function and its rates of change. In the case of implicit differentiation, which is used here, the function is not given in the explicit form of "y = f(x)". Instead, y and x are combined in a single equation.
To find how y changes with respect to x, implicit differentiation becomes handy.When you encounter an equation like the eight curve equation in the entrance exercise, you differentiate both sides with respect to x, applying the chain rule to terms involving y. This approach gives you access to \(\frac{dy}{dx}\), even when y isn't isolated.

• This is fundamental because it helps identify how y will react to changes in x.
• In many scientific fields, knowing this rate of change is essential for predicting behaviors and solving real-world problems.

Implicit Functions

Implicit functions arise when it's not possible to easily express one variable explicitly in terms of another. They are defined by equations where both variables intermix, such as our eight curve example
. Solving such equations requires special techniques since you can't just isolate y as a function of x directly.Thanks to implicit differentiation, you can still understand the behavior of these intertwined variables. For the eight curve:

• We applied implicit differentiation to find \(\frac{dy}{dx}\).
• This shows how changes along the eight curve affect y and x.

When isolating y isn't possible, the functions implicitly defined by the equation still hold valuable information about the system or scenario being analyzed, as they tell you about particular output (y values) for given inputs (x values).

This special understanding aids in complex problem-solving without needing a classic y = f(x) formula.

Mathematical Graphing

Graphing is a powerful way to visualize functions and their characteristics.
The eight curve, for instance, presents a challenge because it is defined implicitly.By solving the implicitly defined equation for y, we derived two functions:

• \( y = f_1(x) \) or \( \sqrt{-\frac{1}{2}x^2(x^2 - 2)} \)
• \( y = f_2(x) \) or \( -\sqrt{-\frac{1}{2}x^2(x^2 - 2)} \)

Graphing these separate functions offers insight into the symmetric nature of the curve around the x-axis. Visual representations help identify any restrictions,

• such as points where the functions are undefined or specific ranges over which the functions remain real (\(-\sqrt{2} \leq x \leq \sqrt{2})\).

Utilizing graphing tools can enhance understanding:

• It simplifies catching patterns or special features of the function
• Solidification of the mathematical concept's grasp.

Ultimately, graphing takes abstract equations and translates them into a visual guide which is essential in conveying the thorough nature of implicit relationships.