APPM 1360
Midterm #1
Fall 2012
On the front of your bluebook, please write: a grading key, your name, student ID, and
section and instructor. This exam is worth 100 points and has 5 questions.
Show all work! Answers with no justication will receive no po
Solution: APPM 1360
Exam 1
Spring 2012
2 3
/2
x2 dx
(b)
2x3 sin(x2 )dx
16 x2
0
0
Solution: (a) Using the trig substitution x = 4 sin() we have x2 = 16 sin2 () and dx = 4 cos() d
and 16 x2 = 4 cos(), so,
1. (10 pts ea.) Evaluate the integrals, show all wor
Solution: APPM 1360
Exam 3
Fall 2011
3n (x 1)n
n
n=1
(a) (10 pts) Where is f (x) absolutely convergent? Show all work.
(b) (10 pts) Where is f (x) conditionally convergent? Show all work.
(c) (5 pts) What is the radius of convergence of f (x)?
1. Consider
Answers: APPM 1360
Final
Spring 2012
1. (a) (7 pts) Find the area of the region that lies inside r = 2 cos(2) and outside r = 1.
(b) (7 pts) Find the length of the polar curve r = e/2 , 0 2.
Answers:
8
2
(a) A =
+ 3=
+ 3.
12
3
(b) L = 5(e 1)
2. (a) (7 p
APPM 1360
Final Exam
Fall 2010
INSTRUCTIONS: Books, notes, and electronic devices are not permitted. Write (1) your name,
(2) 1360/FINAL, (3) instructors name and (4) FALL 2010 on the front of your bluebook. Also
make a scoring table with room for 7 probl
Solution: APPM 1360
Exam 2
Spring 2011
1. (a) (3 pts) Graph the region bounded by y = x 2 and x = y 2 .
(b) (12 pts) Find the volume of the solid created by revolving the region bounded by y = x 2 and
x = y 2 about the line y = 3.
Solution:
(a) The graph
Solution: APPM 1360
Exam 2
Spring 2012
1. (10 pts) Suppose the region enclosed by y = x + ex , the x-axis, the y-axis and x = 1 is rotated about the
line x = 1, nd the volume of the resulting solid.
Solution:
4.8
4
3.2
2.4
1.6
0.8
-2
-1.6
-1.2
-0.8
-0.4
0
Solution: APPM 1360
Exam #3
Fall 2010
1. (25 pts) Show all work, justify your answers.
(a) (5pts) Find the rst three nonzero terms of the Maclaurin series for tan1 (x2 ).
(b) (5pts) Find the rst three nonzero terms of the Maclaurin series for (1 + x)e .
(
Answers: APPM 1360
Final
Fall 2011
1. (a) (8 pts) Find the Cartesian equation of the curve x = 2 cos(), y = 1 + sin()
(b) (8 pts) Sketch the parametric curve x = 2 cos(), y = 1 + sin(), 0 2.
(c) (8 pts) Find the slope of the tangent line to the curve x =
APPM 1360
Final Exam (200 pts)
Spring 2011
INSTRUCTIONS: Books, notes, and electronic devices are not permitted. Write (1) your name,
(2) 1360/FINAL, (3) instructors name/class time and (4) SPRING 2011 on the front of your
bluebook. Also make a scoring ta
Solution: APPM 1360
Exam 3
Spring 2012
1. Determine whether the following series are conditionally convergent, absolutely convergent, or divergent.
(a) (10 pts)
n=10
(1)n ln n
n
(b) (10 pts)
n=6
(1)n (n + 1)5n
n32n+1
Solution:
(a) Checking for absolute co
Solution: APPM 1360
Exam 3
Spring 2011
1. (18 pts) Answer the following questions, justify your answers.
cos(x2 ) 1
x0
x4
(a) (6 pts) Evaluate the limit using series: lim
(b) (6 pts) Find the rst four terms of the Maclaurin series of f (x) =
x2
, simplif
Solution: APPM 1360
Exam 2
Fall 2011
1. (a) (10 pts) Find the volume generated by rotating the region in the rst quadrant bounded by
y = x2 + 1 and x = 1 about the y-axis using the method of cylindrical shells.
(b) (8 pts) Set-up, but do not evaluate, an
Solution: APPM 1360
Exam 1
Spring 2011
1. (30 pts) Determine if the integrals converge or diverge. Each part is worth 10 pts.
ex
x
2
ex dx
(a)
dx
(b)
dx
(c)
2x + 1
e
1 + x6
0
1
1
Solution:
(a) Writing the integral as a limit, we have
0
ex
dx = lim
2x + 1
Solution: APPM 1360
Exam 1
Spring 2014
e
4
ln(x ) dx
1. (a) Evaluate the integrals: (i) (8 pts)
(ii) (8 pts)
1
arctan x
dx
x
(b) (9 pts) Consider the reduction formula
cosn (x) dx =
1
n1
cosn1 (x) sin(x) +
n
n
cosn2 (x) dx for n 2.
/2
cos4 (x) dx.
Apply t
Solution: APPM 1360
Exam 2
1. The following parts are not related, justify your
answers:
(a) (14 pts) Use any method to nd the volume of the
solid generated by rotating the shaded region shown on
the right (i.e the region bounded by y = ln(x)/x and
y = 0
Solution: APPM 1360
Exam 3
Summer 2014
1. (30 pts) Determine whether the following series are conditionally convergent, absolutely convergent, or
divergent. You may NOT use the Ratio or Root Test for this problem. State any tests you use.
2
n +1
(3)n+1
(
Solution: APPM 1360
Exam 2
Summer 2013
1. (20 pts) DO NOT SIMPLIFY YOUR ANSWERS FOR THIS PROBLEM.
Consider the region bounded by y =
1 x2 , x = ey , y = 2 and the y-axis.
(a) Rotate this region about the line x = 1. Set up, but do not evaluate, an integra
Solution: APPM 1360
Exam 2
Summer 2014
1. (a) (12 pts) Let R be the region between the curve y = 1 x, the x-axis and the y-axis. Determine
the volume of the solid whose cross sections in R, perpendicular to the y axis, are squares. Be sure
to evaluate the
APPM 1360
Midterm #2
Fall 2012
On the front of your bluebook, please write: a grading key, your name, student ID, and
section and instructor. This exam is worth 100 points and has 6 questions.
Show all work! Answers with no justication will receive no po
Solution: APPM 1360
Exam 3
Spring 2014
1. The following parts are not related, justify your answers:
(a) (8 pts) For what values of x (if any) is the power series
(1)n x2n+1
absolutely convergent? conditionally
(n + 1)4n
n=0
convergent?
cn xn converges wh
Solution: APPM 1360
Final, Part 2
Summer 2013
sin(x2 )
. What is the radius of convergence?
x
(b) (4 pts) What is the 5th -order Taylor polynomial for f (x) centered at x = 0?
1. (a) (6 pts) Find the Maclaurin series for f (x) =
(c) (6 pts) Use the 3rd -o
APPM 1360
Midterm #3
Fall 2012
1. (24 points) Determine whether the following series are conditionally convergent, absolutely convergent, or divergent. Explain your work and name any tests that you use.
(a)
n
3n + 1
ln
n=1
(b)
n=1
2 4 6 (2n)
(2n)!
(c)
(1)