Page 166 - Calculus Workbook For Dummies

150 Part III: Differentiation

j s t = t 3 - t 5 + - 6
3
4
t
^ h
4
$
5 2 +
a. s 2 ^ h gives the platypus’s position at t = 2; that’s 3 2 - $ 3 2 - 6, or 4 feet, from the
back of the boat.
2
3
^
b. v t = l ^h s t = 12 t - 15 t + 1. At t = 2, the velocity is thus 37 feet per second.
h
c. Speed is also 37 feet per second.
feet / second
2
h
d. a t = l ^h v t = m ^h s t = 36 t - 30 t. a 2 ^ h equals 84 .
^
second
k s t = 1 t + t 8 3 - 3
^ h
1
1
a. At t = 2, s 2 ^ h equals 2 + 1 - 3, or –1 ⁄2 feet. This means that the platypus is 1 ⁄2 feet behind
1
the back of the boat.
4
-
2
-
h
b. v 2 = l ^h s 2 = - 2 + - 24 2 ^ h k
a
^
1 24
= - -
4 16
3
= - 1 feet / second
4
A negative velocity means that the platypus is swimming “backwards,” in other words away
from the back of the boat.
3
c. Speed= Velocity , so the platypus’s speed is 1 feet/second.
4
h
^
d. a t = l ^h v t = m ^h s t = t 2 - 3 + 96 t - 5 or 2 3 + 96 5 . a 2 ^ h is therefore 2 + 96 ,
t t 8 32
1 feet / second
or 3 .
4 second
Give yourself a pat on the back if you figured out that this positive acceleration with a negative
velocity means the platypus is actually slowing down.
l s t = t 2 - t + t 8 - 5
2
3
^ h
a. Find the zeros of the velocity:
2
v t = l ^h s t = t 6 - t 2 + 8
h
^
2
0 = t 6 - t 2 + 8
2
= t 3 - + 4
t
No solutions because the discriminant is negative.
2
The discriminant equals b - 4 ac.
The fact that the velocity is never zero means that the sloth never turns around. At t 0= ,
v t = 8 ft / sec which is positive, so the sloth moves away from the trunk for the entire
^ h
interval t = 0 to t = 5.
b. and c. Because there are no turnaround points and because the motion is in the positive
direction, the total distance and total displacement are the same: 265 feet.
5 =
s 5 - ^h s 0 = 260 - - h 265
^
h
^
Whenever the total displacement equals the total distance, average velocity also equals
average speed: 53 ft/sec.
total displacement s 5 - ^h s 0h 265
^
total time = 5 - 0 = 5 = 53 ft/sec.