1
Section 2.1
The Tangent and Velocity Problems
• Goals
– Use two problems to explain the need for the
notion of limit
:
•
The tangent problem
•
The velocity problem
•
What would it mean in general for a line to be
tangent
to a curve?
Tangent Problem
touching once
touching twice
Example
•
Let’s use a specific example to see how
tangent lines in general could be defined:
•
We want to find an equation of the tangent
line to the parabola
y
=
x
2
at the point
P
(1, 1) .
•
Since we know the point
P
, the only
question is that of the slope
m
of the line.
Example (cont’d)
•
Our idea is to use a
nearby point
Q
(
x
,
x
2
)
on the parabola and
find the slope
m
PQ
of
the secant line
PQ
.
•
We choose
x
≠
1
so
that
Q
≠
P
.
Then
2
1
1
PQ
x
m
x
−
=
−
Example (cont’d)
•
Now we allow
x
to
approach
the value
1
without ever being
1 .
– Meanwhile we monitor
m
PQ
.
•
The tables on the next slide give the
values of
m
PQ
as
x
approaches
1
both
from above and below.
•
In both cases it seems that
m
PQ
approaches 2 !
Example (cont’d)

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*