Question

Represent the plane curve by a vectorvalued function. (There are many correct answers.) $$ y=4-x $$

Short answer

The vector-valued function that represents the plane curve \( y = 4 - x \) is \( \mathbf{r}(t) = \langle t , 4 - t \rangle \).

Step-by-step solution

Make it a Parametric Equation

Any form of equation can be represented parametrically. Typically, we take \( x = t \) as a starting point to derive a parametric equation. So, let's consider \( t = x \). Now, using the given equation \( y = 4 - x \), we replace \( x \) by \( t \) and get a new equation \( y = 4 - t \). Our parametric equations becomes \( x = t \) and \( y = 4 - t \).

Transform into Vector Form

Now, we transform the pair of parametric equations into a vector-valued function. We take as components of our vector function the two parametric equations we derived earlier. The vector function \( \mathbf{r}(t) \) becomes \( \mathbf{r}(t) = \langle t , 4 - t \rangle \). In this vector function, \( t \) serves as the parameter.

Key concepts

Parametric Equations

Parametric equations are a powerful tool used to represent curves in mathematics. Unlike traditional functions that give a direct relationship between x and y, parametric equations introduce a third variable, often denoted as 't', which serves as the parameter. This parameter 't' represents a value that can vary over some interval, and for each value of 't', there is a corresponding point on the curve.

For example, the line given by the equation \( y = 4 - x \) can be expressed in terms of parametric equations by letting \( x = t \). Then, you translate the relation into terms of 't' to find the corresponding 'y' value, which becomes \( y = 4 - t \). These equations define a set of points \( (x,y) \) that trace out the line as 't' varies.

Parametric equations are not only restricted to linear functions; they can represent circles, ellipses, and more complex curves. They are particularly useful for describing motion where 't' can represent time, making them essential in physics and engineering.

Plane Curves

Plane curves are curves that lie flat on a two-dimensional plane. In other words, they only have dimensions in the 'x' and 'y' directions. Plane curves can be simple, like lines or circles, or can be more complex shapes like parabolas or hyperbolas.

When describing plane curves, we often look for a mathematical representation that makes it easy to work with them. Parametric equations provide one such way to describe plane curves by breaking down their x and y coordinates into separate equations related by a common parameter. The example \( y = 4 - x \) can be thought of as a straight path on a plane, whose points are plotted as 't' changes.

By using parametric equations to define plane curves, we gain the ability to calculate various properties such as the length of the curve, the area enclosed, or the curve's behavior at different parameter values. These properties have important applications in fields like computer graphics, where plane curves are used to model shapes and paths.

Calculus

Calculus is a branch of mathematics that deals with rates of change (differential calculus) and accumulation of quantities (integral calculus). It has profound applications in science, engineering, and economics to model and solve real-world problems.

In the context of vector-valued functions, calculus allows us to analyze the properties of curves such as speed, curvature, and arc length. Taking the derivative of the vector function \( \mathbf{r}(t) = \langle t , 4 - t \rangle \), we can determine the velocity of a particle moving along the curve, for instance. As 't' changes, calculus provides the tools to explore how the shape and position of the curve change in response.

Moreover, calculus helps us understand the behavior of plane curves by exploring the effect of infinitesimal changes in the parameter 't'. This enables us to compute tangents, rates of change, areas under the curve, and even volumes of solids generated by revolving the curve around an axis. By applying the principles of calculus to vector-valued functions and parametric equations, we gain a deeper understanding of the dynamic properties of plane curves.