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, straight lines are the graphs of
functions which have the form
and these are exceptionally easy 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 such a 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 xcoordinate will increase the ycoordinate of the
point on the graph. Later in the term, we will see the power of
this kind of reasoning.
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.
