4.1. DETERMINING DISTANCE TRAVELED FROM VELOCITY
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further that the object’s initial position at time t = 0 is s(0) = 1.
(a) Determine the total distance traveled and the total change in position on the
time interval 0 ≤ t ≤ 2. What is the object’s position at t = 2?
(b) On what time intervals is the moving object’s position function increasing?
Why? On what intervals is the object’s position decreasing? Why?
(c) What is the object’s position at t = 8? How many total meters has it traveled to
get to this point (including distance in both directions)? Is this different from
the object’s total change in position on t = 0 to t = 8?
(d) Find the exact position of the object at t = 1, 2, 3, . . . , 8 and use this data to
sketch an accurate graph of y = s(t) on the axes provided at right. How can
you use the provided information about y = v(t) to determine the concavity of
s on each relevant interval?
⊳
Summary
In this section, we encountered the following important ideas:
• If we know the velocity of a moving body at every point in a given interval and the
velocity is positive throughout, we can estimate the object’s distance traveled and in
some circumstances determine this value exactly.
• In particular, when velocity is positive on an interval, we can find the total distance
traveled by finding the area under the velocity curve and above the t-axis on the given
time interval. We may only be able to estimate this area, depending on the shape of
the velocity curve.
• An antiderivative of a function f is a new function F whose derivative is f . That
is, F is an antiderivative of f provided that F ′ = f . In the context of velocity and
position, if we know a velocity function v, an antiderivative of v is a position function s
that satisfies s ′ = v. If v is positive on a given interval, say [a, b], then the change in
position, s(b) − s(a), measures the distance the moving object traveled on [a, b].
• In the setting where velocity is sometimes negative, this means that the object is
sometimes traveling in the opposite direction (depending on whether velocity is positive
or negative), and thus involves the object backtracking. To determine distance traveled,
we have to think about the problem separately on intervals where velocity is positive
and negative and account for the change in position on each such interval.
217
further that the object’s initial position at time t = 0 is s(0) = 1.
(a) Determine the total distance traveled and the total change in position on the
time interval 0 ≤ t ≤ 2. What is the object’s position at t = 2?
(b) On what time intervals is the moving object’s position function increasing?
Why? On what intervals is the object’s position decreasing? Why?
(c) What is the object’s position at t = 8? How many total meters has it traveled to
get to this point (including distance in both directions)? Is this different from
the object’s total change in position on t = 0 to t = 8?
(d) Find the exact position of the object at t = 1, 2, 3, . . . , 8 and use this data to
sketch an accurate graph of y = s(t) on the axes provided at right. How can
you use the provided information about y = v(t) to determine the concavity of
s on each relevant interval?
⊳
Summary
In this section, we encountered the following important ideas:
• If we know the velocity of a moving body at every point in a given interval and the
velocity is positive throughout, we can estimate the object’s distance traveled and in
some circumstances determine this value exactly.
• In particular, when velocity is positive on an interval, we can find the total distance
traveled by finding the area under the velocity curve and above the t-axis on the given
time interval. We may only be able to estimate this area, depending on the shape of
the velocity curve.
• An antiderivative of a function f is a new function F whose derivative is f . That
is, F is an antiderivative of f provided that F ′ = f . In the context of velocity and
position, if we know a velocity function v, an antiderivative of v is a position function s
that satisfies s ′ = v. If v is positive on a given interval, say [a, b], then the change in
position, s(b) − s(a), measures the distance the moving object traveled on [a, b].
• In the setting where velocity is sometimes negative, this means that the object is
sometimes traveling in the opposite direction (depending on whether velocity is positive
or negative), and thus involves the object backtracking. To determine distance traveled,
we have to think about the problem separately on intervals where velocity is positive
and negative and account for the change in position on each such interval.