Problem 73 - Calculus Early Transcendentals

Question

Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. The upper half of the parabola \(x=y^{2}\), originating at \((0,0)\)

Step-by-Step Solution

Answer

Question: Write the parametric equations for the upper half of the parabola \(x = y^2\) originating at (0, 0), and the appropriate range for the parameter. Answer: The parametric equations for the upper half of the parabola are: \(x = t^2\) \(y = t\) with the parameter interval \(t \geq 0\).

Step 1: Identify the curve equation.

The given curve is a parabola with the equation \(x = y^2\). Since we are considering only the upper half of the parabola, we will have y ≥ 0.

Step 2: Choose a parameter.

Let's choose parameter t to be equal to y, so \(t = y\). This implies that we are looking for equations in the form \(x = f(t)\) and \(y = t\), where the range of t represents the upper half.

Step 3: Express the curve equation in terms of the parameter t.

Since we have chosen \(t = y\), we can use this to rewrite the curve equation in terms of t: \(x = (t)^2\) or \(x = t^2\)

Step 4: Write the parametric equations and interval for parameter values.

Now, we have the parametric equations for the curve: \(x = t^2\) \(y = t\) As we want to represent only the upper half of the parabola, we need to consider only the non-negative values of y, so our parameter must satisfy \(t \geq 0\). Our parametric equations for the upper half of the parabola \(x = y^2\) originating at \((0,0)\) are: \(x = t^2\) \(y = t\) with the parameter interval \(t \geq 0\).

Key Concepts

Parabola

A parabola is a U-shaped curve that's defined as the set of all points in a plane equidistant from a fixed point (the focus) and a fixed line (the directrix). The standard equation of a parabola with its vertex at the origin is either in the form \(y = ax^2\) for a parabola that opens up or down, or \(x = ay^2\) for a parabola that opens to the right or left. In our exercise, the equation \(x = y^2\) describes a parabola that opens to the right with its vertex at \(0,0\).

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This field allows us to study change and motion, providing the tools to model and analyze physical phenomena and solve practical problems. One of its applications is in finding the slope of a curve at any point, a task that would be deeply related to the study of parabolas, especially if the task involved finding the tangents to the points on a given parabola.

In this exercise, exploring the calculus concepts is not as deeply required, but understanding the basic definition of functions and how to manipulate them is crucial to transition them into parametric form. This is where the connective tissue between calculus and our exercise lies—in knowledge of functions and their representations.

Parametrization

Parametrization is the process of expressing a mathematical object using one or more parameters. In the case of curves, parametrization involves writing the coordinates of any point on the curve as functions of one or more parameters. This is advantageous for various reasons. It simplifies the representation of curves that cannot easily be expressed as functions, allows us to describe the motion of an object along a path, and makes it easier to perform calculus operations like differentiation and integration on these curves.

In our example, we parametrized the parabola \(x = y^2\) using the parameter \(t\), by setting \(t = y\) and rewriting \(x\) as \(t^2\). This gives us a simple, two-equation system where each coordinate depends on the parameter \(t\), describing the entire position of points on the parabola in terms of \(t\).

Function Interval

The interval of a function refers to the range of input values for which the function is defined. It plays an essential role in mathematics, as it helps in setting the domain or scope within which we analyze the function's behavior. In the context of parametric equations, defining an interval for the parameter is crucial as it specifies the portion of the curve we are interested in.

For the given problem, we were asked to find parametric equations for the upper half of the parabola \(x=y^2\), originating at \(0,0\). By choosing our parameter \(t\) to represent \(y\), and knowing that \(y\) must be non-negative for the upper half of the parabola, we define \(t \geq 0\) as the interval for our parameter \(t\). This interval is important because it constraints \(t\) to only the values that correspond to the upper half of the parabola, ensuring that our parametric equations provide an accurate representation of the specified curve.