Pg 247

When a particle is moving
along a straight line, it is a
common mistake to think
that the particle speeds up
when the acceleration is
positive and that the
particle slows down when
the acceleration is
negative. Keep these
points in mind:
• Speed increases when
the velocity and the
acceleration have the
same signs.
• Speed decreases when
the velocity and the
acceleration have
opposite signs.
For example,
• when a > 0, a car going
forward speeds up and
a car going backward
slows down
• when a < 0, a car going
forward slows down
and a car going
backward speeds up
a(t) v′(t) s′′(t)
d d
v(t) d
[s(t)]
t d
(3t
t
3 40.5t 2 162t)
9t 2 81t 162
d
d
a(t) d
[v(t)]
t d
(9t
t
2 81t 162)
18t 81
a(6) 18(6) 81
27
At 6 s, the velocity is increasing at the rate of 27 m/s 2 .
(b) The velocity is decreasing when a(t) v′(t) < 0.
18t 81 < 0
18t < 81
t < 4.5
Since t ≥ 0, the velocity is decreasing when 0 ≤ t < 4.5.
(c) The velocity is increasing when a(t) v′(t) > 0.
18t 81 > 0
18t > 81
t > 4.5
The velocity is increasing when t > 4.5.
(d) The velocity is not changing when a(t) v′(t) 0. Substituting and solving
as above gives t 4.5. The velocity is not changing at 4.5 s.
CHECK, CONSOLIDATE, COMMUNICATE
1. Explain how to determine the second derivative of a function.
2. What is the relationship between the position of a moving object and its
acceleration?
3. The instantaneous velocity of an object is decreasing. What does this
change imply about the object’s acceleration?
4. The instantaneous velocity of an object is increasing. What does this
change imply about the object’s acceleration?
KEY IDEAS
• The derivative of the derivative of a function is called the second
d
derivative. Symbols for the second derivative are f ′′(x), 2 y 2
d x 2, d
dx 2[ f (x)],
or y′′.
• Use the second derivative to determine the rate of change of the rate of
change of a given function. The most common application of the second
derivative is acceleration.
250 CHAPTER 3 RATES OF CHANGE IN POLYNOMIAL FUNCTION MODELS