NAME:
MATH 1170
Midterm III
Do all four problems (points as indicated). Write readable answers on the test, but feel free to use or
hand in additional paper. You can use one page of notes, but no calculators or any quasi-intelligent device
other than your brain.
1. (25 points) Over time, the daily sales of a new mp3 follow the equation
S(t) = 1000
t2
t2 + et
where t is measured in years.
a. Find the leading behavior of this function for t near 0.
b. Find the leading behavior of this function for t near ∞.
c. Use L’Hopital’s rule where appropriate to find the limit (as t→ 0 or t→ ∞).
d. Sketch a graph of this function and explain what is going on.
2
2. (25 points) Suppose the sales of an mp3 follow S(t) = 1000 2 t t as in Problem 1. We want to use
t +e
Newton’s method to find the first time when daily sales will be exactly 100.
a. Write the Newton’s method updating function for this problem.
b. Pick an initial guess and explain your choice.
c. Sketch the first step of Newton’s method on your graph.
d. What would be a bad starting guess?
number
21.0
22.0
23.0
e1.0
e2.0
e3.0
value
2.0
4.0
8.0
2.7
7.4
20.0
3. (25 points) Musicians must decide how to allocate time between touring and recording. Suppose they
spend a fraction p of each year touring, and this gains them F (p) = 106 1 +p 3p fans. The quality of
their recorded music is Q(p) = 1 − p.
a. Graph and explain these two functions.
b. Multiply the functions F and Q to get a single function for record sales. Why might this function
make sense?
c. Show that p = 1/3 is the value that maximizes sales.
d. Extra Credit: Sketch or find the quadratic approximation to sales at the maximum. What is
this good for?
4. (25 points) The number of fans at a performance Ft (measured in thousands) obeys the discrete-time
dynamical system
Ft+1 = 3Ft (1 − ln(Ft )).
One equilibrium occurs at Ft = 0.
a. For what values of Ft does this system produce reasonable values of Ft+1 ?
b. Find the positive equilibrium.
c. Find its stability by using the slope criterion.
d. Find the stability of the equilibrium at 0 by using the slope criterion.
e. Sketch a graph and cobweb starting from a small number of fans.
f. Extra Credit: Show that Ft = 0 is in fact an equilibrium.
number
ln(2)
ln(3)
e1/3
e2/3
value
0.7
1.1
1.4
2.0