1.1. HOW DO WE MEASURE VELOCITY?
7
t
s
(a, s(a))
(b, s(b))
m =
s(b)−s(a)
b−a
Figure 1.2: The graph of position function s together with the line through (a, s(a)) and
(b, s(b)) whose slope is m =
s(b)−s(a)
b−a . The line’s slope is the average rate of change of s on
the interval [a, b].
velocity of the object on the time interval [a, b] is given by AV [a,b] =
s(b)−s(a)
b−a . Viewing
the interval [a, b] as having the form [a, a + h], we equivalently compute average velocity
by the formula AV [a,a+h] =
s(a+h)−s(a)
h
.
• The instantaneous velocity of a moving object at a fixed time is estimated by considering
average velocities on shorter and shorter time intervals that contain the instant of
interest.
Exercises
1. A bungee jumper dives from a tower at time t = 0. Her height h (measured in feet) at
time t (in seconds) is given by the graph in Figure 1.3.
In this problem, you may base your answers on estimates from the graph or use the
fact that the jumper’s height function is given by s(t) = 100 cos(0.75t) · e −0.2t + 100.
(a) What is the change in vertical position of the bungee jumper between t = 0
and t = 15?
(b) Estimate the jumper’s average velocity on each of the following time intervals:
[0, 15], [0, 2], [1, 6], and [8, 10]. Include units on your answers.
(c) On what time interval(s) do you think the bungee jumper achieves her greatest
average velocity? Why?
(d) Estimate the jumper’s instantaneous velocity at t = 5. Show your work and
explain your reasoning, and include units on your answer.
7
t
s
(a, s(a))
(b, s(b))
m =
s(b)−s(a)
b−a
Figure 1.2: The graph of position function s together with the line through (a, s(a)) and
(b, s(b)) whose slope is m =
s(b)−s(a)
b−a . The line’s slope is the average rate of change of s on
the interval [a, b].
velocity of the object on the time interval [a, b] is given by AV [a,b] =
s(b)−s(a)
b−a . Viewing
the interval [a, b] as having the form [a, a + h], we equivalently compute average velocity
by the formula AV [a,a+h] =
s(a+h)−s(a)
h
.
• The instantaneous velocity of a moving object at a fixed time is estimated by considering
average velocities on shorter and shorter time intervals that contain the instant of
interest.
Exercises
1. A bungee jumper dives from a tower at time t = 0. Her height h (measured in feet) at
time t (in seconds) is given by the graph in Figure 1.3.
In this problem, you may base your answers on estimates from the graph or use the
fact that the jumper’s height function is given by s(t) = 100 cos(0.75t) · e −0.2t + 100.
(a) What is the change in vertical position of the bungee jumper between t = 0
and t = 15?
(b) Estimate the jumper’s average velocity on each of the following time intervals:
[0, 15], [0, 2], [1, 6], and [8, 10]. Include units on your answers.
(c) On what time interval(s) do you think the bungee jumper achieves her greatest
average velocity? Why?
(d) Estimate the jumper’s instantaneous velocity at t = 5. Show your work and
explain your reasoning, and include units on your answer.