Page 167 - Calculus Workbook For Dummies

151
Chapter 8: Using Differentiation to Solve Your Problems

m s t = t + t - t
4
2
^ h
3
a. Find the zeros of v t ^ h: v t = l ^h s t = t 4 + t 2 - 1
h
^
You’ll need your calculator for this:
3
Graph y = 4 x + 2 x - and locate the x-intercepts. There’s just one: x . .385. That’s the only
1
l ^
zero of s t = ^h v th.
Don’t forget that a zero of a derivative can be a horizontal inflection as well as a local
extremum. You get a turnaround point only at the local extrema.
1
Because v 0 = - (a leftward velocity) and v 1 = 5 (a rightward velocity), s ^ .385h must be
^ h
^ h
a turnaround point (and it’s also a local min on the position graph). Does the first derivative
test ring a bell?
Thus, the sloth is going left from t = 0 sec to t = .385 sec and right from .385 to 5 sec. He
4
2
h
turns around, obviously, at .385 sec when he is at s ^ .385 = .385 + .385 - .385 or –.215
meters — that’s .215 meters to the left of the trunk. I presume you figured out that there
must be another branch on the tree on the other side of the trunk to allow the sloth to go left
to a negative position.
b. There are two legs of the sloth’s trip. He goes left from t = 0 till t = .385, then right from t = .385
till t = 5. Just add up the positive lengths of the two legs.
length leg 1 = s ^ .385 - ^h s 0h
= - .215 - 0
= .215 meters
length leg 2 = s 5 - ^ .385h
s
^ h
2
4
5
= 5 + 5 - - - .215h
^
= 645 .215 meters
The total distance is thus .215 + 645.215, or 645.43 meters. That’s one big tree! The branch
is over 2000 feet long.
His average speed is 645.43 / 5, or about 129.1 meters/second. That’s one fast sloth!
Almost 300 miles/hour!
^
c. Total displacement is s 5 - ^h s 0h, that’s 645 – 0 = 645 meters. Lastly, his average velocity
is simply total displacement divided by total time — that’s 645/5, or 129 meters/second.
n s t = t + 1
^ h
2
t + 4
a. Find the zeros of v t ^ h:
1 l
2
2
1 _
^ t + h _ t + i t + h t + 4i l
4 - ^
l ^
h
s t = ^h v t = 2
2
_ t + 4i
2 2
4
t + - _ t 2 + t 2 i
= 2
2
_ t + 4i
2
t - - t 2 + 4
= 2
2
_ t + 4i
Set this equal to zero and solve:
2
t - - t 2 + 4
2 = 0
2
_ t + 4i
2
4
t + t 2 - = 0
- 2 ! 4 - - 16h
^
t =
2
. - . 3 236 or . 1 236