Barrons AP Calculus - David Bock

(d) The speed when equals

FIGURE N4–15

(e) P’s motion can be indicated as shown in Figure N4–15. P moves to the right if t < 1, reverses

its direction at t = 1, moves to the left when 1 < t < 2, reverses again at t = 2, and continues to

the right for all t > 2. The position of P at certain times t are shown in the following table:

t: 0 1 2 4

s: −4 1 0 28

Thus P travels a total of 34 units between times t = 0 and t = 4.

EXAMPLE 27
Answer the questions of Example 26 if the law of motion is

s = t^4 − 4t^3.
SOLUTION: Since v = 4t^3 − 12t^2 = 4t^2 (t − 3) and a = 12t^2 − 24t = 12t(t − 2), the signs of v and
a are as follows:

Thus
(a) s increases if t > 3.
(b) v increases if t < 0 or t > 2.
(c) Since v and a have the same sign if 0 < t < 2 or if t > 3, the speed increases on these
intervals.
(d) The speed when

FIGURE N4–16

(e) The motion is shown in Figure N4–16. The particle moves to the left if t < 3 and to the right

if t > 3, stopping instantaneously when t = 0 and t = 3, but reversing direction only when t = 3.

Thus: