Section 2.1:The Tangent and Velocity Problems
The theory of di
ff
erential calculus historically stems from two di
ff
erent
problems  trying to determine the slope of a tangent line from its
equation and trying to find the velocity of a moving object given its
position as a function of time. In this section, we shall explore these
two problems and explain how a solution to one of these problems is
in fact a solution to them both.
1.
The Tangent Problem
Defining the tangent line to a function at a point is di
ffi
cult because it
requires calculus in order to make a formal definition. Therefore, we
start with a naive definition and then develop the theory to make it
formal.
Definition 1.1.
Suppose
f
(
x
) is a function. Then the tangent line to
f
(
x
) at
x
=
a
is the line which passes through the point (
a, f
(
a
)) which
is the closest linear approximation of
f
(
x
) at that point.
With at least a naive definition of tangent line, we can not formally
state the Tangent Problem:
Question 1.2.
How do we find the equation for the tangent line to
f
(
x
) at
x
=
a
?
Answer.
We need a point and the slope.
Since we already know the tangent line passes through the point (
a, f
(
a
)),
we are given a point for free. Therefore, the tangent problem translates
into the following:
Question 1.3.
How do we find the slope of the tangent line to
f
(
x
)
at
x
=
a
?
If we solve this problem, then we have solved the tangent problem, so
we concentrate on this. We start with an example of how we can do
this explicitly.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 JUNGHENN
 Calculus, Derivative, Slope

Click to edit the document details