Math 121 Helpful Sequence Tips and Facts
lim
n
ln n
=0
n
ln n
= 0 for r > 0
n nr
lim
1
lim n n = lim
n
n
1
lim x n = lim
n
n
lim
1+
1
n
lim
1+
x
n
lim
n
n
n=1
x=1
for a xed constant number x > 0
n
n!
=0
nn
n
n
n
=e
n
= ex
for a xed constant number
Midterm Exam #1
Math 121
February 21, 2014
Answer Key
1.
[10 Points]
(a) Let y = arcsin x. Use implicit dierentiation to PROVE that
1
dy
=
.
dx
1 x2
If y = arcsin x, then sin y = x. Implicitly dierentiate both sides,
d
d
(sin y) =
(x).
dx
dx
dy
Then cos y
Math 121 Helpful Sequence and Series Information
Please nd below some denitions and tips for sequences and series. We will pay careful attention
to what they are, how they are dened, how to decide convergence of each, as well as how they are
related. We w
Answer Key for Review Packet for Exam #2
Math 12-D. Benedetto
Integrals: Compute each of the following integrals, or else show that it diverges.
e2x ln(e2x + 1)
e4x + 2e2x ex
dx =
+
arctan ex + C
e2x + 1
2
2
1.
3
2.
0
1
dx
3x + 1
3.
1
1
4.
(x2 + 16)
3
5.
Math 121
Extra Partial Fraction Examples
x2 + 7
dx
x2 + 2x + 10
Note that the integrand is an improper rational function because the degree of the numberator
is not strictly less than that of the denominator. We apply long division of polynomials to write
Innite Series Review
Math 121
It is complicated to determine whether a given series
an converges. Some of the main diculties for deciding convergence, are maybe deciding which convergence test to use, how to apply it
correctly, and nally, how to make the
Math 121
Midterm Exam #1
February 22, 2013
This is a closed-book examination. No books, notes, calculators, cell phones, webpages, communication devices of any sort, or other aids are permitted.
You need not simplify algebraically complicated answers. H
Midterm Exam #1
Math 121
1.
[6 Points]
February 22, 2013
Let y = arcsin x. Use implicit dierentiation to PROVE that
dy
1
.
=
dx
1 x2
If y = arcsin x, then sin y = x. Implicitly dierentiate both sides,
d
d
sin y =
x.
dx
dx
dy
= 1.
Then cos y
dx
dy
1
1
1
So
Midterm Exam #1
Math 121
February 21, 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, nu