Calculus of Vector-Valued Functions of One Variable

The material in this section requires students to understand the graph of a vector-valued function of one variable (from ) and vector operations, like addition and scalar multiplication, from .

We again remind you to read through the introduction to this chapter, which introduces the Classic Calculus Approach and reviews important concepts from single variable calculus. You may want to have students read this review of single variable calculus so they can recall the various uses of limits, derivatives, and integrals before we start working with these concepts in more general settings.

For the calculus concepts in this section, we will not utilize the ; we will generalize our intuition and ideas from earlier calculus courses to see how to apply limits, derivatives, and integrals to vector valued functions of one variable. This section continues the use of our self driving car company Steer Clear in our motivation and Preview Activities.

In this book, we will use the structure of the to investigate properties of new kinds of functions, equations, and graphs. In this section, we will investigate the meaning of limits, derivatives, and integrals of vector-valued functions of one variable by generalizing the intuition and concepts from single variable calculus. Additionally, we will connect these measurements to properties of curves in space as the graphs of these vector-valued functions.

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

In single variable calculus, we define the derivative, \(f'\), of a given function \(f\) by \[\begin{aligned}\end{aligned}\] provided the limit exists. At a given value of \(a\), \(f'(a)\) measures the instantaneous rate of change of \(f\), and also tells us the slope of the tangent line to the curve \(y = f(x)\) at the point \((a, f(a))\). The definition of the derivative extends naturally to vector-valued functions and curves in space.

We will take a moment here to note that both the single variable calculus definition of derivative and have the form of the classic calculus approach described above. In particular, both definitions are of the form \[\begin{aligned}\end{aligned}\]

Let \(\vr\) be the vector-valued function whose graph is shown in Figure, and let \(h\) be a scalar that represents a small change in time. The vector \(\vr(t)\) is the blue vector in Figure and \(\vr(t+h)\) is the green vector.

If \(\vr(t)\) determines the position of an object at time \(t\), then \(\frac{\vr(t+h)-\vr(t)}{h}\) represents the average rate of change in the position of the object over the interval \([t,t+h]\). We also call this the average velocity of the object on the interval\([t,t+h]\). It may feel a bit surprising to think of a rate of change as a vector quantity. However, describing how the vector quantity position changes with respect to time requires knowing both about the direction in which position changes and the magnitude of that change. Furthermore, when computing the difference quotient, the vector subtraction in the numerator is done componentwise. The factor of \(1/h\) is also multiplied into each component, so we can view the difference quotient's components as each being the average rate of change for that particular component.

The derivative \[\begin{aligned}\end{aligned}\] is the instantaneous rate of change of \(\vr(t)\) at time \(t\) for those values of \(t\) for which the limit exists. Therefore, the instantaneous velocity of the object at time \(t\) is \(\vv(t) = \vr\, '(t)\). allows you to reduce the step size in the difference quotient to approach \(0\). In doing so, you should notice that \(\vr\, '(t)\) is tangent to the graph of \(\vr(t)\) at time \(t\). We will return to this important characteristic later in the section.

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

As we learned in single variable calculus, computing derivatives using the definition is often difficult because the limit involved is an indeterminate form. Fortunately, properties of limits and vectors make it straightforward to calculate the derivative of a vector-valued function. This will be similar to how we developed shortcut differentiation rules in single-variable calculus. If we analyze the steps in , we identify three steps

1. \(\vr(t+h)-\vr(t)\)
2. \(\frac{1}{h}\)
3. \(h\rightarrow 0\)

Our next activity gives you a chance to practice with componentwise differentiation and also asks you to think about when limits of vector-valued functions do not exist.

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

One of the most important ideas in first-semester calculus is that a differentiable function is locally linear: that is, when viewed up close, the graph of a differentiable function looks like a line. Indeed, when we zoom in sufficiently close to a particular point, the curve is indistinguishable from its tangent line. You can look at for an interactive visualization of this important idea.

In the same way, we expect that a smooth curve in three-dimensional space will be locally linear if the graph of the curve looks like a line when you zoom in sufficiently far. In , we investigate how to find the tangent line to such a curve. Recall from our work in Section that the vector equation of a line that passes through the point at the tip of the vector \(\vL_0 = \langle x_0, y_0, z_0 \rangle\) in the direction of the vector \(\vu = \langle a, b, c \rangle\) can be written as \[\begin{aligned}\end{aligned}\]. The line described by \(\vL\) can also be written in parametric form as \[\begin{aligned}\end{aligned}\]. shows a plot of a curve, a point on that curve, and the line tangent to the curve at that point. You can change the point at which the tangent line is shown by moving the Location slider. By moving the Zoom slider to the right, you can show smaller and smaller regions around the highlighted point to see that the tangent line approximates the curve very well on sufficiently small neighborhoods of the point. If you move the zoom slider all the way to the left, the whole curve and axes will be shown.

Our work in Activity can be generalized to give a method for finding an equation of the line tangent to a curve described by a vector-valued function.

Given a differentiable vector-valued function \(\vr\), the tangent line to the curve at the input value \(a\) is given by \[\begin{aligned}\end{aligned}\].

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

In single-variable calculus, you learned that the definite integral of a (scalar-valued) function \(f(x)\) on the interval \([a,b]\) is defined as \[\begin{aligned}\end{aligned}\], where \(\Delta x_i\) is the width of the interval from \([x_{i-1},x_i]\) and \(x^*_i\) is a point in \([x_{i-1},x_i]\). The definite integral measures the net signed area between the graph of \(f(x)\) and the \(x\)-axis on the interval from \(x=a\) to \(x=b\). The definite integral uses our classic calculus approach to measure the accumulation of the output of \(f\) over the interval \([a,b]\).

In practice, you probably did not use this Riemann sum definition to calculate any definite integrals, since the Fundamental Theorem of Calculus allowed you to use associated antiderivatives to efficiently calculate the value of a definite integral. The Fundamental Theorem of Integral Calculus states that if \(f\) is a continuous function on \([a,b]\) and \(F(x)\) is an antiderivative of \(f\) (that is, \(\frac{dF}{dx}=f(x)\)), then \[\begin{aligned}\end{aligned}\]. The family of functions that are antiderivatives of \(f(x)\) is also called the indefinite integral of \(f(x)\) and is denoted \(\int f(x) \ dx\).

We can apply all of the ideas above related to definite and indefinite integrals to vector-valued functions of one variable by performing each operation separately on each component.

If \(\vr(t) = x(t) \vi + y(t) \vj + z(t) \vk\) with \(t\in[a,b]\), then \[\begin{aligned}\end{aligned}\], provided that all three of the integrals on the right-hand side exist. An antiderivative of a vector-valued function \(\vr\) is a vector-valued function \(\vR\) such that \[\begin{aligned}\end{aligned}\]. The indefinite integral \(\int \vr(t) \ dt\) of a vector-valued function \(\vr\) is the most general antiderivative of \(\vr\) and represents the family of all antiderivatives of \(\vr\). The indefinite integral of \(\vr(t)\) is denoted by \[\begin{aligned}\end{aligned}\].

In the next activity, we look at relationships between position, velocity, and acceleration in a situation that is more complicated than the constant acceleration of . The process of using integration and differentiation will remain the same, but the results will not be nearly as simple.

The previous activity shows a case where the motion of our object is driven by forces that either turn the object or change the speed of the object in the \(z\)-coordinate direction. We will spend the rest of this chapter looking at measurements related to motion along curves like this. Key to these measurements will be the combined use of our new calculus for vector-valued functions with the vector tools from the previous chapter.

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