If we let
Δ
t
denote the total time for the middle part of the trip centered on the I
marker and
denote by
Δ
D
the distance of that portion of the trip, then the distance traveled
Δ
D
is a function of
the time traveled
Δ
t
and as
Δ
t
→
we have
Δ
D
→
. Our de nition of instantaneous speed at the
I
marker is the limit of the average speed as the time for the trip decreases and approaches zero,
i.e., as Δ
t
→
. Symbolically we write:
Instantaneous Speed
=
lim
Δ
t
→
Average Speed
=
lim
Δ
t
→
Δ
D
Δ
t
( )
e concept is intuitive, but the question remains as to how we are to compute this limit. We again
need a precise de nition of the limit of a function.
Calculus I
©

J. E. Franke, J. R. Griggs and L. K. Norris
Last update: June
,
CHAPTER . THE LIMIT
. . INTRODUCTION
Example .
e position of a car traveling due east along a straight line is
s
(
t
)
=
t
+
. In this
example time
t
is measured in hours and position
s
is measured in miles. We will take
s
=
to be
Raleigh and the positive direction to be east along Interstate
. How far east of Raleigh is the car and
how fast is it going a er one hour, that is when t
=
?
Solution:
When
t
=
, the position is
s
( )
=
⋅
+
=
miles. So the car is
miles east of
Raleigh a er
hour. We will apply Equation ( ) to
nd the car’s instantaneous speed at this moment.
e change in position from
t
=
to
t
=
+
Δ
t
is
Δ
D
=
s
(
+
Δ
t
)

s
( )
=
(
+
Δ
t
)
+

=
⋅
Δ
t
+
(
Δ
t
)
=
(
+
Δ
t
)
Δ
t
e average speed from
t
=
to
t
=
+
Δ
t
is
Average Speed
=
Δ
D
Δ
t
=
(
+
Δ
t
)
Δ
t
Δ
t
=
+
Δ
t
e instantaneous speed a er one hour is
Instantaneous Speed
=
lim
Δ
t
→
Average Speed
=
lim
Δ
t
→
Δ
D
Δ
t
=
lim
Δ
t
→
(
+
Δ
t
)
=
mph
. .
e Problem of Tangents
A third way to motivate the idea of the limit of a function can be found in the ancient problem
of de ning tangents to curves in the plane. By
tangent
to a curve at a point on the curve we mean
the best straight line approximation to the curve at that point.
One can start by observing that for
the beautifully symmetric unit circle
x
+
y
=
in the
xy
plane there is a unique way to de ne the
tangent at each point. One simply takes the straight line through the given point that is perpendicular
to the radius of the circle at that point.
Figure
If we consider a general curve in the plane as in Figure
, then there is no longer a concept of a
single “center” as there is for the circle, and hence no concept of radius.
Euclid’s
Elements
(c.
BCE)
Calculus I
©

J. E. Franke, J. R. Griggs and L. K. Norris
Last update: June
,
CHAPTER . THE LIMIT
. . INTRODUCTION
Figure
One solution to this problem is to try to
nd “best
t circles” at each point and then draw the
corresponding radii.
ree such (approximate) best
t circles are shown in Figure
.
Figure
In fact we will use precisely this method in Chapter
of Calculus III as an application of the calculus of
vectorvalued functions. For now we take a simpler approach and recall the
secant line
approximation
of the tangent line.
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 Calculus, Derivative, Continuous function, J. E. Franke, J. R. Griggs