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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.
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View Full DocumentAt 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 1
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
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 Fall '08
 sc
 Calculus

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