Worksheet ----MS125 Calculus I Dr. Y. Kim
2.1 Limits, Rates of Change, and Tangent Lines
Suppose the following graph represents traveling distance (s) of a car with respect to time (t).
)(tfs
=
s
B
Q1:
What is the average velocity over [a, b]?
(distance/ -------
average rate of change
position)
= slope of the line joining A and B
A
Q2
: How do we measure the instantaneous velocity at t=a?
-------
If we continue to decrease the size of interval, the average
velocity gets closer to the instantaneous velocity at x = a.
a
b
h
t
(time)
1.
Average Velocity over
[
a
,
b
]
=
a
b
afbf
−
−
)()(
h
afhaf )()(
−
+
=
= the slope of the line joining two points A and B
= average rate of change
2.
Instantaneous Velocity at
a
x
=
=
a
b
afbf
ab
−
−
→
)()(
lim
=
h
afhaf
h
)()(
lim
0
−
+
→
= the slope of the tangent line to the curve at t = a
= the slope of the curve at t = a
Ex1) Let
2
)( xxf =
be a position function of a moving particle. Find the average velocity of the
function
)(xf
over [1, 5].
Ex2) Let
2
)( xxf
=
be a position function of a moving particle. Find the instantaneous velocity of the
function
)(xf
at x = 1 over several small time intervals.
HOMEWORK: 2.1----1, 2, 9, 11, 13, 15