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. \(y^{3}=a x^{2}(\text { Neile's semicubical parabola })\)

Short answer

Question: Given the implicit equation \(y^{3} = a x^{2}\), find the derivative of \(y\) with respect to \(x\), determine the explicitly defined functions \(y=f_{1}(x), y=f_{2}(x),\) and graph the equation using the functions found in part (b). Answer: The derivative of \(y\) with respect to \(x\) is \(\frac{d y}{d x} = \frac{2 a x}{3 y^{2}}\). The explicitly defined function is \(y=f_{1}(x) = \sqrt[3]{a x^{2}}\). When graphing the equation for various values of \(a\), the graph will resemble a parabola opening to the right if \(a>0\) and to the left if \(a<0\), with the vertex at the origin (0,0). The slope of the curve increases as we move away from the origin.

Step-by-step solution

Find the derivative of y with respect to x using implicit differentiation

To find \(\frac{d y}{d x}\), we'll differentiate both sides of the equation \(y^{3} = a x^{2}\) with respect to \(x\). Differentiating the left side gives us \(3 y^{2}\frac{d y}{d x}\). Differentiating the right side gives us \(2 a x\). Therefore, \[3 y^{2}\frac{d y}{d x} = 2 a x\] Now we can solve for \(\frac{d y}{d x}\): \[\frac{d y}{d x} = \frac{2 a x}{3 y^{2}}\]

Solve the given equation for y

Solving the given equation for \(y\), we have: \[y^{3} = a x^{2}\] Now, take the cube root of both sides: \[y = \sqrt[3]{a x^{2}}\] Since the cube root function can yield real results for positive, negative, and zero values of \(a x^{2}\), we have a real function \(y=f_1(x)\): \[f_{1}(x) = \sqrt[3]{a x^{2}}\]

Graph the given equation using the functions found in part (b)

To graph the equation, we can plot the function \(f_{1}(x)=\sqrt[3]{a x^{2}}\) for various values of \(a\). The graph of Neile's semicubical parabola resembles a parabola opening to one side but with a sharper curve near the vertex. Here are a few points to keep in mind while graphing the given equation: 1. If \(a>0\), the graph will open to the right (positive x-axis). 2. If \(a<0\), the graph will open to the left (negative x-axis). 3. The vertex of the parabola is at the origin (0,0). 4. The slope of the curve increases as we move away from the origin. To graph the equation, plot a series of points \((x, f_{1}(x))\) for various values of \(x\). Then connect these points with a smooth curve to represent the function \(y=\sqrt[3]{a x^{2}}\).

Key concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This mathematical field provides a system for calculating the dynamic changes within various scientific, economic, and engineering disciplines.

One of the fundamental concepts in calculus is differentiation, which deals with the rate at which things change. When we differentiate a function, we're looking for an expression that gives us the rate of change of one variable with respect to another. In our exercise involving Neile's semicubical parabola, this means finding the rate of change of the function's output, or height \(y\), as the input, or horizontal position \(x\), changes.

Implicit differentiation, a technique we used in the exercise, is particularly useful when the relationship between \(x\) and \(y\) isn't expressed as a straightforward function, or when isolating \(y\) is difficult or impractical. Instead of explicitly solving for \(y\), we differentiate both sides of the equation regarding \(x\), and this provides us with a way to describe the slope of the curve at any given point.

Derivatives of Functions

The derivative of a function at a point is the slope of the tangent line to the function at that point. It represents how quickly the function's value is changing at that specific input.

When working with explicit functions, where \(y\) is given as a function of \(x\), finding the derivative is typically straightforward. However, when functions are defined implicitly, as in the equation \(y^3 = a x^2\), we turn to implicit differentiation to find \(\frac{d y}{d x}\). This technique involves differentiating each term concerning \(x\) and solving for \(\frac{d y}{d x}\).

In our exercise, after applying implicit differentiation, we derived the relationship \(3 y^2\frac{d y}{d x} = 2 a x\), which we then rearranged to isolate \(\frac{d y}{d x}\) and find the derivative. It's worth highlighting that derivatives aren't just abstract concepts; they have practical implications. For instance, in physics, the derivative of position with respect to time gives us an object's velocity.

Graphing Equations

Graphing equations is a way to visually interpret the relationship between variables. It converts algebraic expressions into a visual format that can be more easily understood and analyzed.

For equations that define a relationship between two variables, like \(y^3 = a x^2\), the graph in Cartesian coordinates gives us significant insight into the behavior of the function. In our step-by-step solution, we used the implicitly defined function \(f_1(x) = \sqrt[3]{a x^2}\) to graph the curve.

Key Points When Graphing

• Identify the shape of the graph based on the equation.
• Consider the effect of different constants on the graph, such as \(a\) in our exercise, which determines the direction in which the graph opens.
• Locate key features, like the vertex, which in the case of Neile's semicubical parabola is at the origin (0,0).

To graph the function accurately, we plot points for various values of \(x\) and determine their corresponding \(y\) values using the function we've derived. Once a sufficient number of points have been plotted, we can connect them with a smooth curve, reflecting the continuous nature of the function. This visual representation not only allows us to see the overall shape and direction of the graph but also lets us observe how the slope changes at different points along the curve—a concept closely related to the derivative.