Problem 23 - Thomas Calculus

Question

Find a parametrization for the curve. the lower half of the parabola \(x-1=y^{2}\)

Step-by-Step Solution

Answer

Parametrization: \(x(t) = t^2 + 1, y(t) = t\) for \(t \leq 0\).

Step 1: Understand the equation

We are given the equation \(x - 1 = y^2\), which is the equation of a parabola. Our task is to find a parametrization for the lower half of this parabola.

Step 2: Express x in terms of y

Rewrite the given equation \(x - 1 = y^2\) to express \(x\) in terms of \(y\). So, we have \(x = y^2 + 1\). This expression will help us define the x-component of our parametrization.

Step 3: Choose a parameter

To parametrize the curve, choose a parameter \(t\). Since the equation involves \(y^2\), let \(y\) be the parameter itself, i.e., \(y = t\), since any real number \(t\) can be used as \(y\) in the equation.

Step 4: Define the parameterization

Now, define the parametrization using \(y=t\). Substitute \(y = t\) into the expression for \(x\): \(x = t^2 + 1\). This gives us a parameterization of the curve as \(x(t) = t^2 + 1, y(t) = t\).

Step 5: Apply the condition for the lower half

To restrict to the lower half of the parabola, we need \(y \leq 0\), since the parabola opens symmetrically with respect to the x-axis. Thus, the parameter \(t\) should satisfy \(t \leq 0\).

Step 6: Finalize the parametrization

The parametrization for the lower half of the parabola is \(x(t) = t^2 + 1, y(t) = t\) where \(t \leq 0\).

Key Concepts

Parabola

A parabola is a type of curve that appears frequently in mathematics, particularly in algebra and calculus. It is a symmetrical curve that looks like an arch, and it can open upwards or downwards (or even sideways) depending on how it is oriented. One of the most common ways to represent a parabola is the quadratic equation, which in its simplest form looks like: \[ y = ax^2 + bx + c \] In our exercise, the parabola is given as \(x - 1 = y^2\). This is a special form of a parabola where the variables \(x\) and \(y\) switch roles, making it open sideways. Specifically, this equation defines a parabola that opens to the right. To focus on the lower half, we take the \(y\)-values that are non-positive (i.e., \(y \leq 0\)). This means we are tracing only one side of this mirror-like shape. Parabolas are important for studying both theoretical and practical problems as they describe multiple real-world paths, like the trajectory of thrown objects.

Parametric Equations

Parametric equations involve expressing the coordinates of the points on a curve as functions of a third variable, generally referred to as the parameter. This allows us to represent complex curves, including circles, ellipses, and of course, parabolas, in a more flexible way.To parametrize a curve, the idea is to choose a parameter, often denoted by \(t\), and express both \(x\) and \(y\) as functions of \(t\). In the case of our parabola:

• The equation \(x = t^2 + 1\) defines the \(x\)-component
• The equation \(y = t\) gives us the \(y\)-component

By using \(t = y\), we transform a polynomial equation into a simpler form that can be easily traced or graphically plotted. Parametric equations have advantages in various fields such as physics, computer graphics, and engineering because they simplify the computation of complex shapes and movements.

Curve Tracing

Curve tracing involves sketching or plotting a curve based on its mathematical description, such as an equation or a set of parametric equations. It's a vital skill in both mathematics and applied sciences, providing visual insight into the behavior and characteristics of a curve.With parametric equations, tracing a curve becomes more intuitive, as we can vary the parameter to understand how the curve develops. More specifically, for our parabola we use the parametrization:

• \(x(t) = t^2 + 1\)
• \(y(t) = t\)

The parameter \(t\) here determines the direction and coverage of the curve. By setting \(t \leq 0\), we ensure that we only plot the lower half of the parabola. This restriction is necessary to accurately illustrate the part of the curve we are interested in.Traced curves not only present the trajectory of equations but also help identify key features like intercepts, maxima, minima, and symmetry, assisting in comprehending the complete geometric picture.