18
1.2. THE NOTION OF LIMIT
(c) Determine the instantaneous velocity of the object when t = 3. Include units
on your answer.
⊳
The closing activity of this section asks you to make some connections among average
velocity, instantaneous velocity, and slopes of certain lines.
Activity 1.6.
For the moving object whose position s at time t is given by the graph below, answer
each of the following questions. Assume that s is measured in feet and t is measured in
seconds.
1
3
5
1
3
5
t
s
Figure 1.8: Plot of the position function y = s(t) in Activity 1.6.
(a) Use the graph to estimate the average velocity of the object on each of the
following intervals: [0.5, 1], [1.5, 2.5], [0, 5]. Draw each line whose slope
represents the average velocity you seek.
(b) How could you use average velocities or slopes of lines to estimate the instantaneous velocity of the object at a fixed time?
(c) Use the graph to estimate the instantaneous velocity of the object when t = 2.
Should this instantaneous velocity at t = 2 be greater or less than the average
velocity on [1.5, 2.5] that you computed in (a)? Why?
⊳
Summary
In this section, we encountered the following important ideas:
• Limits enable us to examine trends in function behavior near a specific point. In
particular, taking a limit at a given point asks if the function values nearby tend to
1.2. THE NOTION OF LIMIT
(c) Determine the instantaneous velocity of the object when t = 3. Include units
on your answer.
⊳
The closing activity of this section asks you to make some connections among average
velocity, instantaneous velocity, and slopes of certain lines.
Activity 1.6.
For the moving object whose position s at time t is given by the graph below, answer
each of the following questions. Assume that s is measured in feet and t is measured in
seconds.
1
3
5
1
3
5
t
s
Figure 1.8: Plot of the position function y = s(t) in Activity 1.6.
(a) Use the graph to estimate the average velocity of the object on each of the
following intervals: [0.5, 1], [1.5, 2.5], [0, 5]. Draw each line whose slope
represents the average velocity you seek.
(b) How could you use average velocities or slopes of lines to estimate the instantaneous velocity of the object at a fixed time?
(c) Use the graph to estimate the instantaneous velocity of the object when t = 2.
Should this instantaneous velocity at t = 2 be greater or less than the average
velocity on [1.5, 2.5] that you computed in (a)? Why?
⊳
Summary
In this section, we encountered the following important ideas:
• Limits enable us to examine trends in function behavior near a specific point. In
particular, taking a limit at a given point asks if the function values nearby tend to