Looking at the graph below, let's focus attention on the point x = 1. At x = 1, the graph passes through (1,1). Instead of connecting this point to other points on the graph by a secant line, let's zoom in around the point (1,1) as if we were increasing the magnification on a microscope. You can do this by clicking the "Zoom" button. Notice that with each click, the scales on the x and y axes are halved.
The important thing is that the closer we zoom in, the more the graph looks like a straight line through the point (1,1). We call this line the tangent line to the graph through (1,1). In this example, a simple measurement leads us to think that the slope of the tangent line should be 1/2. Since we also know that the tangent line passes through (1,1), we have all the information we need to completely determine the tangent line. In fact, it is given by the equation: y = (x+1)/2.
Question: Can you think of a function whose graph would not look like a straight line if you zoomed in like this?
Question: Why do you think this is the same line that we constructed in the last section?
Maybe now you're wondering what the point is behind all of this. That's such a good question that we're going to answer it many times and in many ways throughout this course. You see, linear functions are the easiest functions to work with. For instance, you only need to know two pieces of information -- a slope and a point through which the graph passes -- to know everything there is to know about a linear function. A principal theme in this course is to use information which can be easily extracted from the tangent line to deduce information about the original graph, which could be difficult to deduce more directly.
For instance, in our example above, we know that the slope of the tangent line is 1/2. Since this is positive, we know that the tangent line is rising as we move to the right. What is important is that the graph shares this property with the tangent line; that is, a small increase in the x-coordinate will increase the y-coordinate of the point on the graph. Later in the term, we will see the power of this kind of reasoning.
Now that we can construct the tangent line in two different ways, it is important to notice that there is something very similar about these two processes. In the last section, we built a sequence of secant lines which grow "close" to the tangent line without actually becoming the tangent line. In this section, we have built a sequence of snapshots of the graph each of which look more like a straight line. However, if we made very careful measurements, we would find that the graphs in the snapshots are still bent just a little bit. However, we can still imagine what the straight line would look like.
In a way, it's like trying to get all the sand out of your car after a day at the beach. At first, there's lots of sand, but after you clean and clean, your car gets closer and closer to being spotless. And while there's always a few specks of sand hiding someplace so that you can never actually have a perfectly clean car, it's all right to think that your car is perfectly clean.
If you find this confusing, you shouldn't worry too much (at least for now). These kinds of questions bamboozled some of history's greatest thinkers for several thousand years: ancient mathematicians asked questions like these, but it wasn't until the 1600's that people started to formulate useful answers to them.
What we are really talking about is the fundamental concept of Calculus which we'll soon learn to call a limit . In the next few sections, we'll develop the notion of a limit more carefully and find some nifty ways to compute with them.