Tangent Rates

Rates of Change and Tangent Lines

In this section we are going to take a look at two fairly important problems in the study of calculus. There are two reasons for looking at these problems now.

First, both of these problems will lead us into the study of limits, which is the topic of this chapter after all. Looking at these problems here will allow us to start to understand just what a limit is and what it can tell us about a function.

Secondly, the rate of change problem that we�re going to be looking at is one of the most important concepts that we�ll encounter in the second chapter of this course. In fact, it�s probably one of the most important concepts that we�ll encounter in the whole course. So looking at it now will get us to start thinking about it from the very beginning.

Tangent Lines

The first problem that we�re going to take a look at is the tangent line problem. Before getting into this problem it would probably be best to define a tangent line.

A tangent line to the function f(x) at the point is a line that just touches the graph of the function at the point in question and is �parallel� (in some way) to the graph at that point. Take a look at the graph below.

In this graph the line is a tangent line at the indicated point because it just touches the graph at that point and is also �parallel� to the graph at that point. Likewise, at the second point shown, the line does just touch the graph at that point, but it is not �parallel� to the graph at that point and so it�s not a tangent line to the graph at that point.

At the second point shown (the point where the line isn�t a tangent line) we will sometimes call the line a secant line.

We�ve used the word parallel a couple of times now and we should probably be a little careful with it. In general, we will think of a line and a graph as being parallel at a point if they are both moving in the same direction at that point. So, in the first point above the graph and the line are moving in the same direction and so we will say they are parallel at that point. At the second point, on the other hand, the line and the graph are not moving in the same direction and so they aren�t parallel at that point.

Okay, now that we�ve gotten the definition of a tangent line out of the way let�s move on to the tangent line problem. That�s probably best done with an example.

Example Find the tangent line to at x = 1.

Solution

We know from algebra that to find the equation of a line we need either two points on the line or a single point on the line and the slope of the line. Since we know that we are after a tangent line we do have a point that is on the line. The tangent line and the graph of the function must touch at x = 1 so the point must be on the line.

Now we reach the problem. This is all that we know about the tangent line. In order to find the tangent line we need either a second point or the slope of the tangent line. Since the only reason for needing a second point is to allow us to find the slope of the tangent line let�s just concentrate on seeing if we can determine the slope of the tangent line.

At this point in time all that we�re going to be able to do is to get an estimate for the slope of the tangent line, but if we do it correctly we should be able to get an estimate that is in fact the actual slope of the tangent line. We�ll do this by starting with the point that we�re after, let�s call it . We will then pick another point that lies on the graph of the function, let�s call that point .

For the sake of argument let�s take choose and so the second point will be . Below is a graph of the function, the tangent line and the secant line that connects P and Q.

We can see from this graph that the secant and tangent lines are somewhat similar and so the slope of the secant line should be somewhat close to the actual slope of the tangent line. So, as an estimate of the slope of the tangent line we can use the slope of the secant line, let�s call it , which is,

Now, if we weren�t too interested in accuracy we could say this is good enough and use this as an estimate of the slope of the tangent line. However, we would like an estimate that is at least somewhat close the actual value. So, to get a better estimate we can take an x that is closer to and redo the work above to get a new estimate on the slope. We could then take a third value of x even closer yet and get an even better estimate.

In other words, as we take Q closer and closer to P the slope of the secant line connecting Q and P should be getting closer and closer to the slope of the tangent line. If you are viewing this on the web, the image below shows this process.

As you can see (if you�re reading this on the web) as we moved Q in closer and closer to P the secant lines does start to look more and more like the tangent line and so the approximate slopes (i.e. the slopes of the secant lines) are getting closer and closer to the exact slope. Also, do now worry about how I got the exact or approximate slopes. We�ll be computing the approximate slopes shortly and we�ll be able to compute the exact slope in a few sections.

In this figure we only looked at Q�s that were to the right of P, but we could have just as easily used Q�s that were to the left of P and we would have received the same results. In fact, we should always take a look at Q�s that are on both sides of P. In this case the same thing is happening on both sides of P. However, we will eventually see that doesn�t have to happen. Therefore we should always take a look at what is happening on both sides of the point in question when doing this kind of process.

So, let�s see if we can come up with the approximate slopes I showed above, and hence an estimation of the slope of the tangent line. In order to simplify the process a little let�s get a formula for the slope of the line between P and Q, , that will work for any x that we choose to work with. We can get a formula by finding the slope between P and Q using the �general� form of .

Now, let�s pick some values of x getting closer and closer to , plug in and get some slopes.

So, if we take x�s to the right of 1 and move them in very close to 1 it appears that the slope of the secant lines appears to be approaching -4. Likewise, if we take x�s to the left of 1 and move them in very close to 1 the slope of the secant lines again appears to be approaching -4.

Based on this evidence it seems that the slopes of the secant lines are approaching -4 as we move in towards , so we will estimate that the slope of the tangent line is also -4. As noted above, this is the correct value and we will be able to prove this eventually.

Now, the equation of the line that goes through is given by

Therefore, the equation of the tangent line to at x = 1 is