1.1. HOW DO WE MEASURE VELOCITY?
3
Note well: the units on AV [a,b] are “units of s per unit of t,” such as “miles per hour” or
“feet per second.”
Activity 1.1.
The following questions concern the position function given by s(t) = 64 − 16(t − 1) 2 ,
which is the same function considered in Preview Activity 1.1.
(a) Compute the average velocity of the ball on each of the following time intervals:
[0.4, 0.8], [0.7, 0.8], [0.79, 0.8], [0.799, 0.8], [0.8, 1.2], [0.8, 0.9], [0.8, 0.81],
[0.8, 0.801]. Include units for each value.
(b) On the provided graph in Figure 1.1, sketch the line that passes through the
points A = (0.4, s(0.4)) and B = (0.8, s(0.8)). What is the meaning of the slope
of this line? In light of this meaning, what is a geometric way to interpret each
of the values computed in the preceding question?
(c) Use a graphing utility to plot the graph of s(t) = 64 − 16(t − 1) 2 on an interval
containing the value t = 0.8. Then, zoom in repeatedly on the point (0.8, s(0.8)).
What do you observe about how the graph appears as you view it more and
more closely?
(d) What do you conjecture is the velocity of the ball at the instant t = 0.8? Why?
0.4
0.8
1.2
48
56
64
feet
sec
s
A
B
Figure 1.1: A partial plot of s(t) = 64 − 16(t − 1) 2 .
⊳
Instantaneous Velocity
Whether driving a car, riding a bike, or throwing a ball, we have an intuitive sense that any
moving object has a velocity at any given moment – a number that measures how fast the
3
Note well: the units on AV [a,b] are “units of s per unit of t,” such as “miles per hour” or
“feet per second.”
Activity 1.1.
The following questions concern the position function given by s(t) = 64 − 16(t − 1) 2 ,
which is the same function considered in Preview Activity 1.1.
(a) Compute the average velocity of the ball on each of the following time intervals:
[0.4, 0.8], [0.7, 0.8], [0.79, 0.8], [0.799, 0.8], [0.8, 1.2], [0.8, 0.9], [0.8, 0.81],
[0.8, 0.801]. Include units for each value.
(b) On the provided graph in Figure 1.1, sketch the line that passes through the
points A = (0.4, s(0.4)) and B = (0.8, s(0.8)). What is the meaning of the slope
of this line? In light of this meaning, what is a geometric way to interpret each
of the values computed in the preceding question?
(c) Use a graphing utility to plot the graph of s(t) = 64 − 16(t − 1) 2 on an interval
containing the value t = 0.8. Then, zoom in repeatedly on the point (0.8, s(0.8)).
What do you observe about how the graph appears as you view it more and
more closely?
(d) What do you conjecture is the velocity of the ball at the instant t = 0.8? Why?
0.4
0.8
1.2
48
56
64
feet
sec
s
A
B
Figure 1.1: A partial plot of s(t) = 64 − 16(t − 1) 2 .
⊳
Instantaneous Velocity
Whether driving a car, riding a bike, or throwing a ball, we have an intuitive sense that any
moving object has a velocity at any given moment – a number that measures how fast the