Question

The Eight Curve (a) Find the slopes of the figure-eight-shaped curve $$y^{4}=y^{2}-x^{2}$$ (b) Use parametric mode and the two pairs of parametric equations $$\begin{aligned} x_{1}(t) &=\sqrt{t^{2}-t^{4}}, \quad y_{1}(t)=t \\ x_{2}(t) &=-\sqrt{t^{2}-t^{4}}, \quad y_{2}(t)=t \end{aligned}$$ to graph the curve. Specify a window and a parameter interval.

Short answer

The slope of the curve \(y^{4}=y^{2}-x^{2}\) is given by \(y' = \frac{2x}{4y^3 - 2y}\). To graph the curve using its parametric equations, a suitable range for \(t\) is [-1,1] with window for \(x\) as (-1.5, 1.5) and for \(y\) as (-1.2, 1.2).

Step-by-step solution

Differentiate the equation

First, take the derivative of both sides of the given function, \(y^{4}=y^{2}-x^{2}\) with respect to \(x\). This results in \(4y^3 y' = 2yy' - 2x\), where \(y'\) is the derivative of \(y\) with respect to \(x\). Solve for \(y'\) using algebraic manipulations.

Find the slope

Rearrange the equation and isolate \(y'\) to get \(y' = \frac{2x}{4y^3 - 2y}\). This gives the slope of the curve at any point (x, y) on the curve.

Define the parametric equations

Next, we use the given pairs of parametric equations \(x_{1}(t) =\sqrt{t^{2}-t^{4}}\), \(y_{1}(t)=t\), \(x_{2}(t) =-\sqrt{t^{2}-t^{4}}\) and \(y_{2}(t)=t\) to graph the curve. Notice that \(x_1(t)\) gives the right half of the curve and \(x_2(t)\) gives the left half of the curve.

Identify the window and parameter interval

To graph the curve, identify acceptable values for \(t\). For this curve, the transformation parameter \(t\) has range -1 ≤ \(t\) ≤ 1. Then specify a window that captures all the main points of the graph, such as (-1.5, 1.5) for x and (-1.2, 1.2) for y.

Key concepts

Differentiation

In calculus, differentiation is a fundamental process used to determine the rate at which something changes. When we differentiate a function, we're looking for its derivative, which gives us the slope of the tangent line at any point along the curve of the function. In the context of our eight-shaped curve problem, we start by differentiating the given equation with respect to x. This step reveals how the slope of the curve (y') changes with respect to x and y, allowing us to understand the curve's behavior at any particular point. Solving the resulting equation for y' yields the exact slope of the tangent to the curve at a given point, which is crucial for understanding its geometry and for graphing purposes.

Through differentiation, specific characteristics of the curve, such as maximum and minimum points, inflection points, and intervals of increase or decrease, can be analyzed and understood, which is a critical aspect of understanding complex functions beyond just this particular eight-shaped curve.

Slope of a Curve

The slope of a curve at any point is the gradient or steepness of the curve at that point. It's equivalent to the slope of the tangent line to the curve at that point. When dealing with normal functions (y=f(x)), you would simply differentiate the function to get the slope. However, in the case of implicit functions like our eight-shaped curve (y^4=y^2-x^2), we follow a different approach. To find the slope, we differentiate both sides with respect to x, and we use the chain rule to handle the y terms appropriately, since y is a function of x. Once this is done, we can solve for y', which provides us with a formula that represents the slope of the curve at any valid x and y value. Understanding the slope is essential, especially when it comes to finding areas where the curve might have horizontal or vertical tangent lines, which indicate significant points on the curve.

Graphing Curves

Graphing curves is a crucial aspect of calculus, as it provides a visual representation of functions and their behavior. When graphing an implicit function like y^4=y^2-x^2, challenges arise because it is not expressed as a function of x alone. Instead, we might turn to techniques such as parametric equations. Graphing involves plotting points that satisfy the equation and drawing a smooth curve through them to reveal the structure of the function. In the context of the exercise provided, using the derived slopes, we can more accurately analyze and sketch the behavior of the curve, especially highlighting regions where the function is increasing or decreasing, or where it might have critical points. This graphical approach not only helps in visual learning but also reinforces the understanding of the relationships between algebraic and geometric representations of functions.

Parametric Graphing

Parametric graphing is a method of graphing where we use equations that express coordinates as functions of a single parameter, usually denoted as t. This method is particularly useful when dealing with curves that are difficult to represent with a single function in the Cartesian coordinate system. In our example, we have two pairs of parametric equations that represent the right and left halves of our eight-shaped curve. By plugging different values of t into these equations, we can map out points on the graph that when connected, will render the shape of the curve. The parametric approach allows us to graph more complex curves, including loops and intersections, that might be impossible to describe with a standard y=f(x) function.

The chosen parameter interval and window are determined based on the range and behavior of the functions derived from the parametric equations. By ensuring that the window captures all main points of the graph, we can appreciate the full structure of the curve and create an accurate graphical representation to complement our algebraic findings.