APPM 1350 Final Exam FALL 2008APPM 1350 Final Exam FALL 2008APPM 1350 Final Exam FALL 2008
INSTRUCTIONS: Books, notes, and electronic devices are not permitted. Write your (1) name,
(2) instructor’s name, and (3) when your lecture meets on the front of your bluebook. Also make a
scoring table, with places for 7 problems, plus a total score. This exam has 7 problems, each worth
25 points. Work any 6 problems. Clearly mark the problem you are skipping on the
front of your bluebook, or we will grade problems 1-6. Start each problem on a new page.
Box your answers. A correct answer with incorrect or no supporting work may receive no credit,
while an incorrect answer with relevant work may receive partial credit. SHOW ALL WORK.
1. Evaluate the following limits and derivatives:
(a) lim
x→∞(ln(x2)) 1
5x(b) lim
x→−1
x+ 1
|x+ 1|(c) d
dxxsin x(d) d
dx Zcos x
0
t
√2−t2dt
2. Evaluate the following integrals:
(a) Z−1/2
−1/2
t
√1−t2dt (b) Z1/4
0
dt
√1−4t2(c) Z4
2
dx
x(ln(3x))5
3. For each of the following, determine if the statement is Always True or Not Always True.
If the statement is not always true, explain why or give a counter-example.
(a) There is some function f(x) such that 0 < f 0(x)≤4 for all x,f(0) = −1, and f(2) = 9.
(b) If g00(3) <0 and g0(3) = 0 then g(x) has a local maximum at x= 3.
(c) d
dxe−4xis negative for all x > 0.
(d) If h(x) is continuous and decreasing on the interval [a, b] then the Trapezoidal Rule will
overestimate Zb
a
h(x)dx.
(e) If a function is differentiable at x=athen it is continuous at x=a.
(f) The function y= ln(x) is o(ln(x2+ 1)) as x→ ∞.
4. Optimization:
(a) Find the maximum of y=x3−6x2+9x−2 on the interval [2,4]. Where does the maximum
occur? Justify your answer.
(b) You are designing a poster to contain 50 in2of printing with margins of 4 inches each at top
and bottom and 2 inches at each side. What overall dimensions will minimize the amount
of paper used?
5. Let f(x) = 1
(x−2)2
(a) What is the average value of f(x) on the interval [−2,1]?
(b) Since f(x) is continuous on [−2,1], the Mean Value Theorem for Definite Integrals guaran-
tees the existence of some cin this interval. Find c.
(c) Using the definition of derivative, find f0(x).
(d) Find the linearization of f(x) at x= 0, and use the linearization to estimate f(0.1).
There are two more problems on the back!There are two more problems on the back!There are two more problems on the back!
