Math 112 - Review Worksheet 2 - 3/2/2016
Instructions:
• Pair up with one or two (preferably two) other people in the class (at least one of whom
you don’t know).
• Pick one of the problems below to work on. (You should all work on the same problem).
• Read the problem and discuss possible approaches.
• Agree on an approach as a group. [If you have trouble, ask me.]
• Carry out the approach individually - if you get stuck, ask another member of your
group. [If you don’t get sufficient enlightenment from within you group, ask me.]
• When all the members in your group have finished, check your answers to see if they
agree, and if they are reasonable. Also, check with me to see if your answer is correct.
This worksheet is intended to prepare you for the exam on Friday. The questions on the exam
will involve a range of difficulty levels, and the questions on this worksheet are around the level
of the exam. (In general, when I write review worksheets, I try to make the level of difficulty
a little bit higher on the worksheet than what you’d see on the exam, with the goal of making
sure students are prepared for the exam.)
1. What does L’Hopital’s rule say? How do you apply it? What are all the indeterminate
forms? [I’m not necessarily planning to ask a question like this on the test, but answering this
question may be a helpful way to review this material.]
2. Evaluate the following limits:
(a) limx→0
1−cos(x)
x2
(b) limx→∞
ln(x3 +1)
.
ln(x2 +1)
(c) limx→0+ x
√1
x
(d) limx→∞ x ln
.
x+1
x
.
1
(e) limx→0+ (x − sin(x)) x3 .
3. What is the procedure for doing integration by parts? What are the rules about how you
should select the part to differentiate and the part to integrate?
4. Evaluate the following integrals:
R
(a) x cos(x) dx.
R
(b) x ln(x) dx.
R
(c) tan−1 (x) dx.
p
R
(d) sin(x) cos(x) dx.
1
2
(e)
R
sec3 (x) tan(x) dx.
R
(f) (1 − x2 )3/2 dx.
R
dx
.
(g) x2x+x−2
R 2 −x−1) dx
. (Hint: −1 is a root of x3 + x2 + x + 1.)
(h) (2x
x3 +x2 +x+1
5. Does
Z
∞
dx
(x + 1)(x + 2)
0
converge? If so, what does it converge to?
6. Does
Z
2
−1
dx
x+1
converge? If so, what does it converge to?
7. Find a formula for the sequence {an } where the initial terms are.
−1 4 −7 10 −14 17 −20
, ,
, ,
, ,
,....
3 8 15 24 35 48 63
8. Let {bn }n≥1 be a sequence, and assume that bn = bn−1 − bn−2 . Show that bn+6 = bn for all
n ≥ 1.
9. Define a sequence by c1 = 1, and cn =
1
2+cn−1
for n ≥ 2.
(a) What is c2 ? What is c3 ? What is c4 ?
(b) Show that 0 ≤ cn ≤ 1 for all n. Is the sequence cn monotonic?
(c) Assume that the sequence converges, and compute limn→∞ cn .
10. Determine if the following sequences converge:
√
(a) dn = sin(1/ n), n ≥ 1,
(b) e1 = 2, and en =
2en−1 +1
3
for n ≥ 1.
(c) f1 = 1/3 and fn = fn−1 +
n
2n+1
for n ≥ 1.
P
11. Given a sequence {an } what is the infinite series ∞
n=1 an ? What does it mean for the
series to converge? What is the sequence of partial sums?
12. For each of the series below, determine whether it converges. If it converges, what is the
value of the sum?
P
n2 +1
(a) ∞
n=1 n2 +2 .
P
2n
(b) ∞
n=1 3n+1 .
3
P∞
(n + 1)1/3 − n1/3
P
1
(d) ∞
n
ln
1
+
.
n=1
n
P∞ gn
(e) n=1 10n , where gn = 1 if n has the form 3k + 1 for some non-negative integer k, gn = 2 if
n has the form 3k + 2 for some non-negative integer k, and gn = 3 if n is a multiple of 3.
P
1
(f) ∞
n=2 ln 1 − n2 .
(c)
n=1