2
LIMITS
2.1 Limits, Rates of Change, and Tangent Lines
Preliminary Questions
1.
Average velocity is defned as a ratio oF which two quantities?
SOLUTION
Average velocity is defned as the ratio oF distance traveled to time elapsed.
2.
Average velocity is equal to the slope oF a secant line through two points on a graph. Which graph?
Average velocity is the slope oF a secant line through two points on the graph oF position as a Function oF
time.
3.
Can instantaneous velocity be defned as a ratio? IF not, how is instantaneous velocity computed?
Instantaneous velocity cannot be defned as a ratio. It is defned as the limit oF average velocity as time
elapsed shrinks to zero.
4.
What is the graphical interpretation oF instantaneous velocity at a moment
t
=
t
0
?
Instantaneous velocity at time
t
=
t
0
is the slope oF the line tangent to the graph oF position as a Function
oF time at
t
=
t
0
.
5.
What is the graphical interpretation oF the Following statement: The average ROC approaches the instantaneous ROC
as the interval
[
x
0
,
x
1
]
shrinks to
x
0
?
The slope oF the secant line over the interval
[
x
0
,
x
1
]
approaches the slope oF the tangent line at
x
=
x
0
.
6.
The ROC oF atmospheric temperature with respect to altitude is equal to the slope oF the tangent line to a graph.
Which graph? What are possible units For this rate?
The rate oF change oF atmospheric temperature with respect to altitude is the slope oF the line tangent to
the graph oF atmospheric temperature as a Function oF altitude. Possible units For this rate oF change are
◦
±
/
Ft or
◦
C
/
m.
Exercises
1.
A ball is dropped From a state oF rest at time
t
=
0. The distance traveled aFter
t
seconds is
s
(
t
)
=
16
t
2
Ft.
(a)
How Far does the ball travel during the time interval
[
2
,
2
.
5
]
?
(b)
Compute the average velocity over
[
2
,
2
.
5
]
.
(c)
Compute the average velocity over time intervals
[
2
,
2
.
01
]
,
[
2
,
2
.
005
]
,
[
2
,
2
.
001
]
,
[
2
,
2
.
00001
]
. Use this to estimate
the object’s instantaneous velocity at
t
=
2.
(a)
Galileo’s Formula is
s
(
t
)
=
16
t
2
. The ball thus travels
1
s
=
s
(
2
.
5
)
−
s
(
2
)
=
16
(
2
.
5
)
2
−
16
(
2
)
2
=
36 Ft.
(b)
The average velocity over [2
,
2
.
5] is
1
s
1
t
=
s
(
2
.
5
)
−
s
(
2
)
2
.
5
−
2
=
36
0
.
5
=
72 Ft
/
s
.
(c)
time interval
[
2
,
2
.
01]
[
2
,
2
.
005
]
[
2
,
2
.
001
]
[
2
,
2
.
00001
]
average velocity
64.16
64.08
64.016
64.00016
The instantaneous velocity at
t
=
2is64Ft
/
s.
A wrench is released From a state oF rest at time
t
=
0. Estimate the wrench’s instantaneous velocity at
t
=
1,
assuming that the distance traveled aFter
t
seconds is
s
(
t
)
=
16
t
2
.
3.
Let
v
=
20
√
T
as in Example 2. Estimate the instantaneous ROC oF
v
with respect to
T
when
T
=
300 K.
T
interval
[
300
,
300
.
01
]
[
300
,
300
.
005
]
average rate oF change
0.577345
0.577348
T
interval
[
300
,
300
.
001
]
[
300
,
300
.
00001
]
average ROC
0.57735
0.57735