Math 155
Exam 2
Spring 2007
NAME:
SECTION:
TIME:
INSTRUCTOR:
Instructions: The exam is closed book and closed notes. You may use an
approved calculator, but be sure to show your work on each problem for full
credit. Work that is crossed out or erased will not be graded. Turn in any
scratch paper that you use during the exam. You will have one hour and 45
minutes to work on the exam.
Problem
1
2
3
4
5
6
7
Total
Points
20
16
12
14
8
16
14
100
Score
1
1. (20 pts) Find the derivatives of the following functions. You do not
have to simplify your answers. Be sure to use parentheses to indicate
multiplication where appropriate.
(a) f (x) = (1 + x2 )4
(b) f (t) = 2t · t2
(c) f (x) =
ln(x)
3x + sin(ex )
2
(d) f (x) = (cxb )(eax )
(e) f (t) = sin(cos(4 ln(t)))
2
2. (16 pts) Evaluate the following limits. Show all of your work. If you
use L’Hospital’s Rule, justify why it can be applied each time you use
it. If you use knowledge of leading behaviors, justify your work by
explaining all of your steps.
e5x + 2
x→∞ x2 + 1
(a) lim
(b) lim
x→∞
ln(2x) + 7x
√
x2 + x + 4
√
(c) lim
x→1
x−1
x−1
ln(e3x ) + 4x1/2
x→∞
x2 + 1
(d) lim
3
3. (14 points) Consider the discrete-time dynamical system:
xt+1 =
xt (xt − r)
+ r,
3+r
r>0
(a) Verify that x∗ = r and x∗ = 3 + r are equilibria.
(b) Show that the derivative of the updating function is
1
(2x − r).
3+r
(c) Let r = 2. Use the Slope Criterion/Stability Test to determine
the stability of each equilibrium.
4
4. (12 pts) Let f (x) =
ax3 + x−1
, and a, c > 0.
x2 + c
(a) Find f∞ (x), the leading behavior of f (x) as x → ∞.
(b) Find f0 (x), the leading behavior of f (x) as x → 0.
(c) Let a = 2 and c = 4. Use the method of matched leading
behaviors to sketch a graph of f (x) on the interval x ≥ 0. Label
your axes and indicate where you have graphed f0 (x), f∞ (x),
and f (x).
5
5. (8 points) For each item below, fill in the blank with the appropriate
letter. Each blank gets one letter only, but letters can be used
multiple times.
Statement
Response
(a) f 0 (c) = 0
x = c is an equilibrium if
x = c is a stable equilibrium if
(b) f 0 (c) > 0
(c) f 0 (c) < 0
(d) f 0 (c) undefined
x = c is a critical point
or if
if
(e) f 00 (c) = 0
(f) f 00 (c) > 0
f (c) is a local maximum
if
and
f (x) is increasing at x = c if
f (x) is concave down at x = c
if
(g) f 00 (c) < 0
(h) f (c) = 0
(i) f (c) = c
(j) |f 0 (c)| < 0
(k) |f 0 (c)| > 0
(l) |f 0 (c)| < 1
(m) |f 0 (c)| > 1
(n) |f 0 (c)| = 1
6
6. (16 points) Let f (x) =
2
.
(x+2)
(a) Find the critical points.
(b) Write out the intervals of increase and decrease. Justify your
answers with calculus.
(c) Write the intervals of concave up and concave down. Justify
your answers with calculus.
(d) Use the information above to sketch a graph of f (x).
7
7. (14 pts) Consider the function f (x) on the interval [0, 10] where
f (x) = x3 − 9x2 + 15x − 5
a) Find the critical points of f (x).
b) Classify each critical point as a local minimum, local maximum or
neither. Justify your answer by using either the first or second
derivative test.
c) Find the global maximum and global minimum on [0, 10] and give
both the x and y coordinates of each.