4
1.1. HOW DO WE MEASURE VELOCITY?
object is moving right now. For instance, a car’s speedometer tells the driver what appears
to be the car’s velocity at any given instant. In fact, the posted velocity on a speedometer
is really an average velocity that is computed over a very small time interval (by computing
how many revolutions the tires have undergone to compute distance traveled), since velocity
fundamentally comes from considering a change in position divided by a change in time.
But if we let the time interval over which average velocity is computed become shorter
and shorter, then we can progress from average velocity to instantaneous velocity.
Informally, we define the instantaneous velocity of a moving object at time t = a to be
the value that the average velocity approaches as we take smaller and smaller intervals
of time containing t = a to compute the average velocity. We will develop a more formal
definition of this momentarily, one that will end up being the foundation of much of our
work in first semester calculus. For now, it is fine to think of instantaneous velocity this
way: take average velocities on smaller and smaller time intervals, and if those average
velocities approach a single number, then that number will be the instantaneous velocity
at that point.
Activity 1.2.
Each of the following questions concern s(t) = 64 − 16(t − 1) 2 , the position function
from Preview Activity 1.1.
(a) Compute the average velocity of the ball on the time interval [1.5, 2]. What is
different between this value and the average velocity on the interval [0, 0.5]?
(b) Use appropriate computing technology to estimate the instantaneous velocity
of the ball at t = 1.5. Likewise, estimate the instantaneous velocity of the ball
at t = 2. Which value is greater?
(c) How is the sign of the instantaneous velocity of the ball related to its behavior
at a given point in time? That is, what does positive instantaneous velocity tell
you the ball is doing? Negative instantaneous velocity?
(d) Without doing any computations, what do you expect to be the instantaneous
velocity of the ball at t = 1? Why?
⊳
At this point we have started to see a close connection between average velocity and
instantaneous velocity, as well as how each is connected not only to the physical behavior
of the moving object but also to the geometric behavior of the graph of the position
function. In order to make the link between average and instantaneous velocity more
formal, we will introduce the notion of limit in Section 1.2. As a preview of that concept,
we look at a way to consider the limiting value of average velocity through the introduction
of a parameter. Note that if we desire to know the instantaneous velocity at t = a of a
moving object with position function s, we are interested in computing average velocities
on the interval [a, b] for smaller and smaller intervals. One way to visualize this is to think
of the value b as being b = a + h, where h is a small number that is allowed to vary. Thus,
1.1. HOW DO WE MEASURE VELOCITY?
object is moving right now. For instance, a car’s speedometer tells the driver what appears
to be the car’s velocity at any given instant. In fact, the posted velocity on a speedometer
is really an average velocity that is computed over a very small time interval (by computing
how many revolutions the tires have undergone to compute distance traveled), since velocity
fundamentally comes from considering a change in position divided by a change in time.
But if we let the time interval over which average velocity is computed become shorter
and shorter, then we can progress from average velocity to instantaneous velocity.
Informally, we define the instantaneous velocity of a moving object at time t = a to be
the value that the average velocity approaches as we take smaller and smaller intervals
of time containing t = a to compute the average velocity. We will develop a more formal
definition of this momentarily, one that will end up being the foundation of much of our
work in first semester calculus. For now, it is fine to think of instantaneous velocity this
way: take average velocities on smaller and smaller time intervals, and if those average
velocities approach a single number, then that number will be the instantaneous velocity
at that point.
Activity 1.2.
Each of the following questions concern s(t) = 64 − 16(t − 1) 2 , the position function
from Preview Activity 1.1.
(a) Compute the average velocity of the ball on the time interval [1.5, 2]. What is
different between this value and the average velocity on the interval [0, 0.5]?
(b) Use appropriate computing technology to estimate the instantaneous velocity
of the ball at t = 1.5. Likewise, estimate the instantaneous velocity of the ball
at t = 2. Which value is greater?
(c) How is the sign of the instantaneous velocity of the ball related to its behavior
at a given point in time? That is, what does positive instantaneous velocity tell
you the ball is doing? Negative instantaneous velocity?
(d) Without doing any computations, what do you expect to be the instantaneous
velocity of the ball at t = 1? Why?
⊳
At this point we have started to see a close connection between average velocity and
instantaneous velocity, as well as how each is connected not only to the physical behavior
of the moving object but also to the geometric behavior of the graph of the position
function. In order to make the link between average and instantaneous velocity more
formal, we will introduce the notion of limit in Section 1.2. As a preview of that concept,
we look at a way to consider the limiting value of average velocity through the introduction
of a parameter. Note that if we desire to know the instantaneous velocity at t = a of a
moving object with position function s, we are interested in computing average velocities
on the interval [a, b] for smaller and smaller intervals. One way to visualize this is to think
of the value b as being b = a + h, where h is a small number that is allowed to vary. Thus,