Let's start with an example. The picture below shows the graph of
a function we've seen already:
Notice that at
, the graph
passes through (1,1). Now let's pretend that we can look at the graph
under a microscope and that we can continually increase the
magnification of the microscope. Clicking on the "Zoom" button below
increases the magnification by a factor of two.
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).
- What is the slope of the tangent line?
- Since we know that the tangent line passes through the point
(1,1), find the equation of the tangent line.
(Here is a review of
straight lines and
their equations should you feel you
- Can you think of a function whose graph would not
look like a straight line if you zoomed in like this? After you think
about this, have a look at this example.
- Is the absolute value function differentiable at x = 1?
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 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.
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
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 look at the derivative from a few new angles
and find some nifty ways to compute with them. Gradually, the picture
will become clear.