Vector-Valued Functions of One Variable

We encourage instructors to read through the , which will be used for many of the new measurements and tools of this chapter (and most other new concepts in this text). Since we will get into calculus elements in this chapter, we offer a quick review of some vital concepts from single variable calculus. Of particular note is the description of integration as measuring the accumulation of a scalar valued function, which is not always covered as an interpretation of integration in single variable calculus.

This section expects students to have familiarity with vectors and the parametric form of a line from . Conic sections (and variations on these in 3D) are used several times as examples of curves we will want to parameterize and describe motion along. The activities in this section offer a variety of depths in working with parameterizations of curves in space with looking at multiple ways to parameterize the same path, looking at variations some common curves, and working to think of traces of a surface as curves in space.

The preview activities throughout will all use our self driving car company Steer Clear and tracking location over time to motivate the topics covered in this chapter. This section introduces the tracking system for our self driving car having a transmitter and receiver and giving an output of a vector from the receiver to the transmitter, which is a concept we will use in Preview Activities throughout this chapter.

As a final note to the instructor, we encourage you to spend more time than typical in this chapter. Many instructors skip most, if not all, of the material in this chapter. The authors have found that the material in this chapter allows students to recall their derivative/integral rules and get used to the many vector tools that will be used in later chapters.

In our previous work, we have seen several examples of curves in space, such as lines in three dimensions and conic sections (in two dimensions). For a line through a fixed point \(\vr_0\) in the direction of vector \(\vv\), we can express the line parametrically through the vector equation \[\begin{aligned}\end{aligned}\]

We call \(\vr(t)\) a vector-valued function of one variable (or simply a vector-valued function, when the the single variable aspect is clear in context) because there is one scalar input (\(t\)) and the output of \(\vr\) is a vector that will vary based on the value of the scalar input. When the output vectors are graphed with initial point at the origin, the terminal points of these vector outputs trace out the line in space. and provide illustrations of this idea.

Similar to lines, other curves in space are classified as one-dimensional objects because there is only one way to move along this object. We refer to this direction as forward/backward depending on whether motion is toward larger (forward) or smaller (backward) values of the input variable.

We will similarly express the coordinates of points on a curve in terms of a single variable. This expression will describe movement along this path. For instance, the graph of an ellipse, such as the one shown in , cannot be expressed with \(y\) as a function of \(x\). However, we still classify an ellipse as a one-dimensional graph because at any point on the graph, there is only one dimension to move along the graph.

Vector-valued functions of one variable are the perfect vehicle for describing curves in general. We use vectors starting at the origin to identify points in space and the terminal points of these vectors trace out a curve in space. This approach allows us to draw a wide variety of graphs in two- and three-dimensional space. We can also describe curves in \(\R^n\) for any \(n\). This same approach to drawing curves will also allow us to represent slices of surfaces in space, which we will discuss more in and many times in .

In the preview activity, we saw how location data for how an object is moving along a curve can be given by vectors with a common initial point. In this section, we will continue to look at the implications of using this as an algebraic description of movement along a curve. In particular, we will look at algebraic rules and definitions for functions using the output vectors as points and we will conclude this section by generating several families of examples.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

As in Preview Activity, we can think of a point on the curve shown in Figure as the terminal point of a vector starting at the origin. As the point we are looking at moves along the curve, the vector changes to terminate at the desired point. shows a curve in space with a vector from the origin to a highlighted point on the curve. Use the slider to change the location on the curve to see how changing the scalar input gives different outputs with terminal points on the curve.

This correspondence between a location and a vector starting at the origin will allow us to think of a curve as a collection of terminal points of vectors emanating from the origin. We can describe an object traveling along this curve by defining a function \(\vr\) whose input is the variable \(t\) and whose output is the vector from the origin to the point on the curve at time \(t\). In so doing, we have introduced a new type of function, one whose input is a scalar and whose output is a vector.

The terminal points of the output vectors of \(\vr\) will trace out the curve in space. From this perspective, the \(x\), \(y\), and \(z\) coordinates of the points on the curve are functions of time, \(t\), \[\begin{aligned}\end{aligned}\], and thus we have three coordinate functions that represent the collection of points on the curve. The variable \(t\) is called a parameter and the equations \(x = x(t)\), \(y = y(t)\), and \(z = z(t)\) are called parametric equations (or a parameterization of the curve). The function \(\vr\) whose output is the vector from the origin to a point on the curve is defined by \[\begin{aligned}\end{aligned}\].

Note that every set of parametric equations determines a vector-valued function of the form \[\begin{aligned}\end{aligned}\] and every vector-valued function defines a set of parametric equations for a curve. Moreover, we can consider vector-valued functions as parameterizations in \(\R^2\), \(\R^3\), or any dimension.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

We now discuss how to parameterize several types of curves in space. An important idea that will be used throughout the rest of the text is the way that different relationships of between coordinates can be expressed or understood, such as expressing one coordinate as a function of the other or relationships such as trig identities between the coordinate values.

Using parametric equations to define vector-valued functions in two dimensions is a more versatile way of describing curves than defining \(y\) as a function of \(x\). In fact, if \(y = f(x)\) is a function of \(x\), then we can parameterize the graph of \(f\) by \[\begin{aligned}\end{aligned}\]. Hence, the graph of every single-variable function can be described parametrically. In addition, as we saw in and , we can use vector-valued functions to represent curves in the plane that cannot be described with \(y\) as a function of \(x\) (or \(x\) as a function of \(y\)).

The next activity has you consider a few examples of curves generated by vector-valued functions, including in three dimensions.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.