Math 121
Midterm Exam #3
December 4, 2013
This is a closed-book examination. No books, notes, calculators, cell phones, communication
devices of any sort, or other aids are permitted.
You need not simplify algebraically complicated answers. However, num
Math 12
Midterm Exam #3
Compact Version
April 28, 2010
This is a closed-book examination. No books, notes, calculators, cell phones, communication
devices of any sort, or other aids are permitted.
You need not simplify algebraically complicated answers.
Math 121
Midterm Exam #3
December 5, 2012
Answer Key
1.
[15 Points] Find the Interval and Radius of Convergence for the following power series.
Analyze carefully and with full justication.
n=1
(1)n (3x 1)n
n 7n+1
Use Ratio Test.
L = lim
n
an+1
= lim
n
an
Midterm Exam #3
Math 121
December 4, 2013
Answer Key
1.
[15 Points] Find the Interval and Radius of Convergence for the following power series.
Analyze carefully and with full justication.
n=1
(1)n (2x + 3)n
(n + 1) 4n
Use Ratio Test.
L = lim
n
an+1
= lim
Midterm Exam #3
Math 121
April 25, 2014
Answer Key
1.
[15 Points] Find the Interval and Radius of Convergence for the following power series.
Analyze carefully and with full justication.
n=1
(1)n (4x 3)n
n2 7n
Use Ratio Test.
L = lim
n
an+1
= lim
n
an
(1)
Midterm Exam #3
Math 121
Compact Version
December 5, 2012
1.
[15 Points] Find the Interval and Radius of Convergence for the following power series.
Analyze carefully and with full justication.
n=1
2.
[15 Points]
(1)n (3x 1)n
n 7n+1
Find the MacLaurin ser
Math 12
Midterm Exam #3
April 28, 2010
Answer Key
1.
[15 Points] Find the Interval and Radius of Convergence for each of the following power
series. Analyze carefully and with full justication.
(a)
n=0
lim
n
xn
Use Ratio Test.
(2n)!
xn+1
(2n)!
|x|
xn+1
(2
Review Packet for Final Exam
Material since Exam #3
Math 121-D. Benedetto
Polar Coordinates: For each problem, sketch the polar curve(s) and answer the related question(s).
1. Find the area bounded by r = 2 sin .
First note that this is just a circle of r
Math 12, Fall 2006
Solutions to the Final Exam
4
dx
1
3 sec2 d
2
1(a).
=
=
(x = 3 tan , dx = 3 sec d) =
cos d
2 3/2
27 sec3
9
0 (9 + x )
4
4
1
x
4
1
= sin =
= 0= ,
9
9
45
9 + x2 0 9 25
where the third-to-last equality is by a right triangle with angle
Math 12, Fall 2006
Final Exam, Tuesday, December 19, 2006
No books, notes, calculators, cell phones, iPods, etc.
Express yourself clearly and legibly. Explain your reasoning and show your work.
Little or no credit may be awarded if you fail to justify you
Midterm Exam #3
Math 121
April 25, 2014
This is a closed-book examination. No books, notes, calculators, cell phones, communication
devices of any sort, or other aids are permitted.
You need not simplify algebraically complicated answers. However, numer
Basic Polar Curves Handout
Math 121-D. Benedetto
Recognize a few of the following basic polar curves. Try and understand the sketches.
r = sin
r = 2 sin
circle of radius
1
2
centered at 0, 1
2
circle of radius 1 centered at (0, 1)
Note: These circles