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Section 1.3 The tangent problem

Now that we have reviewed functions and some important classes of them that we will use often, we will soon introduce one of the most important ideas in calculus - the notion of a limit. To do so, we focus on the two central problems in calculus - the tangent problem and the area problem.

So what is the tangent problem and why is it important? In our review of functions, we thought about functions as models of real processes. Historically, the processes that scientists and mathematicians were thinking about were physical processes - the movement of objects, the orbits of planets, waves, heat diffusion, and many others. They were trying to understand how the world works.

Constructing functions as models of physical processes has challenges - you almost never have enough data or knowledge of the system to easily describe a model function. Scientists have two big tools. On one hand, they collect data about their process - like locations of planets in the sky at different times - and try to use that data to build theories that would point to model functions. On the other hand, they took different laws or principles - like the law of gravitation - that were already known and tried to apply them in novel situations.

Central to many of these problems and questions was motion - balls flying through the air, planets moving in the sky, arrows shot from bows. And central to motion is the notion of change - when will the ball change direction? how fast is it going? is it speeding up or slowing down?

Let's work through this with an example. A general principle in kinematics is that velocity is distance divided by time, or \(v=\frac{d}{t}\text{.}\) But this is really average velocity - if I measure the position of a rolling ball at time zero and then again after three seconds and find that it has travelled a distance of three meters, all I know is that the velocity of the ball averaged one meter per second. It could have sped up or slowed down.

Table 1.3.1.
time (s) 0 1 2 3
distance (m) 0 3 4 3

After puzzling this out, maybe I decide to collect more data and take the measurement once a second. Table 1.3.1 shows what this might look like and I can quickly compute that in the first second the average velocity was three meters per second, in the second it was one meter per second, and in the third second it was negative one meter per second. This is better, but these are still all averages.

What we are really after is the \(instantaneous\) velocity, but how can we find it? It helps to think about this more abstractly. If we model the distance measured at time \(s\) by a function \(d(s)\text{,}\) we can compute the average velocity between times \(a\) and \(t\) where \(a \lt t\) using the formula

\begin{equation*} v_{avg}=\frac{d(t)-d(a)}{t-a}. \end{equation*}

To get at the instantaneous velocity at \(a\text{,}\) we'd somehow want \(t\) to be as close as possible to \(a\) without actually equaling it. This is something of a paradox or at least very confusing, but it is precisely the confusion that the notion of a limit helps us resolve. Imagine that we had distance data for every single value of \(s\text{,}\) we could examine the average velocities for values of \(t\) that are closer and closer to \(a\) and try to find out what (if anything) those values are tending towards.

This is where something a little magical happens - it is a trick the occurs over and over in mathematics. If we look at this problem a little differently, we can see that it is not just about velocities but in fact envelopes a huge number of problems. A nice way to do this here is to translate the velocity problem into a geometric one.

In our discussion of functions, we talked about building graphs to help visualize them. Let's do that here. We first plot our data points and then draw a curve that signifies our model. We are cheating a bit - we don't know the model function, but are creating a plausible interpolation between the points to help think about what is going on. Next, we draw our two points corresponding to times \(a\) and \(t\text{.}\) While we plot these as specific points, we think of them generally - they could be anywhere on this curve. Looking at the average velocity computation, we can recognize it as the difference of the y-values on these axes divided by the x-values. In other words, it is the slope of the line connecting these two points. This is called a secant line on the graph.

As we move \(t\) the line follows along, and as it slides towards \(a\) the secant line converges to what is called the tangent line to the curve at time \(a\text{.}\) The slope of this tangent line is the instantaneous velocity in our original discussion but, more generally, it is the instantaneous rate of change of the function at that point. The animation in Figure 1.3.2 demonstrates this.

Figure 1.3.2. An animation of secant lines converging to tangent lines.

While it might not seem like much, this is a huge jump in abstraction that helps us immensely. Instead of thinking through these steps for every specific model for each physical situation, we now have a general method for finding any instantaneous rate of change!

The process of sliding the value of \(t\) towards \(a\) is the process of taking a limit - we can't directly find the slope of the tangent line because we only know one point on it rather than two. But by building a sequence of secant lines that approach it, we can deduce its value.

This approach highlights two of our overarching themes in this course. First, it emphasizes calculus as a language with which we can describe and solve problems. Second, it shows us one of our central approaches - the idea of first creating something that is close to a solution to a problem and then refining it until we eventually come to an exact solution.