Tue, Nov 19, 2013
STUDENT NAME:
EXAM 3 – MATH 1330, FALL 2013
READ EACH PROBLEM CAREFULLY! SHOW ALL WORK! NO WORK, NO CREDIT!
Make sure you turn in ALL pages. No graphing calculators allowed!
• Problem 1 [10 pts] Evaluate the following limits using L’Hôpital’s Rule. Justify why it can be
applied and show all work!
(a) lim
x→∞ x2
x
=
+1
ln(x)
=
x→∞ ln(x2 + 1)
(b) lim
(c) lim x2 e−0.1x =
x→∞
• Problem 2 [10 pts] Find the derivatives of the following functions:
3
(a) f (t) = te−t ,
f 0 (t) =
(b) g(x) = (3 + 4x2 )6 ,
(c) h(y) =
1−y
,
(1+y)2
h0 (y) =
(d) H(x) = (x − sin x)2 ,
2
(e) r(p) = e3p−p ,
g 0 (x) =
H 0 (x) =
r0 (p) =
• Problem 3 [10 pts] Consider the function
f (t) =
Compute
(a) lim f (t) =
t→0
(b) lim f 0 (t) =
t→0
1 − e−t
t
• Problem 4 [10 pts] Bubonic plague viral load (in thousands) is given as a function of time (in
months) B(t) = t4 − 16t2 + 100, for the domain 0 ≤ t ≤ 4 months.
(a) Calculate the value of B(t) at the endpoints t = 0 and t = 4.
(b) Find any critical points in the domain [0, 4]. What is the viral load at these points?
(c) The plague can be treated with Gentamicin, but works most effectively when administered at
a time when the viral load is at a minimum. What is the global minimum of B(t) on the domain
0 ≤ t ≤ 4 months? At what time does this occur?
(d) Use the info above and the second derivative B 00 (t) to sketch the graph of B(t)
• Problem 5 [10 pts] Let r(x) be a function giving per capita production as a function of population
size x(in thousands), with the formula
r(x) =
x
,
1 + 3x2
x≥0
(a) Find the population size that produces the highest per capita production. Explain!
(b) Find the highest per capita production.
(c) Determine lim r(x).
x→∞
(c) Sketch the graph of r(x), x ≥ 0