Math 2250 HW #8
Due 12:30 PM Thursday, October 3
Reading: Hass §3.9–3.11
Problems: Do the assignment “HW8” on WebWork. In addition, write up solutions to the
following problems and hand in your solutions in class on Thursday.
1. Shown below is the graph of the function f (x) = x1/x . At what point does this curve have a
horizontal tangent line? Be sure to find both the x and the y coordinates of the point.
1.2
0.8
0.4
0
10
20
30
40
50
2. After the end of the Cold War, the Rocky Flats nuclear weapons facility in Colorado was
decommissioned and, after years of cleanup, converted into a wildlife refuge. As wildlife reentered the area, burrowing species like prairie dogs were studied with particular interest by
wildlife ecologists since they were likely to redistribute contaminants left in the soil and even
to uncover buried waste.
(The above is all true...what follows is made up.) Ecologists estimated that there were 600
prairie dogs already on the grounds when their study started. After several years of monitoring
the population carefully, they estimated that the prairie dog population t days after the start
of the study could be estimated by the function
t − 1000
p(t) = 5000 arctan
+ 7000.
300
The graph of p is shown below and follows the classic pattern of population growth for species
entering a new environment: slow growth until some critical threshhold is reached, then rapid
growth until the “carrying capacity” of the environment is reached, upon which the growth
tails off.
How fast was the prairie dog population growing after 1000 days (about 33 months)? After
2200 days (about 6 years)?
1
12 600
9600
6600
3600
600
500
1000
1500
2000
2500
3. You’re flying a kite at a constant height of 300 ft and notice that the kite is being carried
horizontally away from you at 25 ft/sec. When the string is 500 ft long, how fast do you need
to be letting out the string to keep up?
2