Math 112 - Review Worksheet 4 - 4/22/2016
Fun fact: Which positive integers can be written as a sum of two or more consecutive positive
integers?
3=1+2
5=2+3
6=1+2+3
7=3+4
9=4+5
10 = 1 + 2 + 3 + 4
11 = 5 + 6
12 = 3 + 4 + 5
13 = 6 + 7
14 = 2 + 3 + 4 + 5
15 = 7 + 8
A: Any number that is not a power of 2 can be written as a sum of two or more consecutive
positive integers.
Instructions:
•
•
•
•
Pair up with one or two (preferably two).
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 final exam. 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.)
I’m trying to have at least one question of each possible “question type”. This means that the
distribution of questions on the final need not be the same as the distribution of questions on
this worksheet.
1
2
You will get the most value if you actually do the problems on this worksheet. For this reason,
I will not make solutions to this worksheet available until Wednesday, April 27.
1. Express the area under y = x2 + x, 1 ≤ x ≤ 2 as a limit of sums. Evaluate this limit.
2. Use the FTC to compute
lim
n X
3i
n→∞
i=1
r
n
3. Express the area under f (x) = sin(x2 ), 0 ≤ x ≤
limit.
R 1+x2 p
4. What is the derivative of cos(x) ln(t) dt?
5. Evaluate the following integrals:
R
(a) a2a−1 da.
R1 √
(b) −2 b b + 3 db.
R
(c) 2c dc.
R
(d) log5 (d) dd.
R de
(e) 4+e
2.
R 3 −f
(f) f e df .
R
(g) sin−1 (g) dg.
R 2π
(h) 0 sin2 (h) cos2 (h) dh.
R
(i) √idi
2 −1 .
R
dj
(j) j 2 −2j−8
.
R
dk
(k) k(k+1)
2 +4k+5 .
R∞
(l) 1 ln(l)
dl.
l2
Re
(m) 0 ln(m) dm.
Rx√
6. Let f (x) = 0 x4 + 1 dx.
(a) Show that f has an inverse function.
(b) Compute the derivative (f −1 )0 (0).
7. Find the exact value for tan cos−1
1
3
.
3i 3
· .
n n
√
π as a limit of sums. Do not evaluate the
3
8. An ice cube that is 0◦ C is put outside on a day when the outside temperature is 40◦ C. After
one hour, the ice cube has melted and the water is 20◦ C. What will the water temperature
be after another hour sitting outside? (This problem is slightly unrealistic from a physical
perspective.)
9. Evaluate the following limits:
(a) limx→0
ex −1−x
.
x2
(b) limx→∞ x (tan−1 (x) − π/2).
6
(c) Use power series to compute
1/x
.
(d) limx→0 π2 cos−1 (x)
tan−1 (x2 )−x2 + x3
limx→0
x10
.
10. Find a formula for the sequence
2 −5 10 −17 26 −37 50 −65
,
, ,
,
,
,
,
,....
3 9 27 81 243 729 2187 6561
11. Define a sequence by a1 = 1 and an =
1
.
8−an−1
(a) Show that the sequence is monotonic.
(b) Show that the 0 < an ≤ 1 for all n.
(c) Does limn→∞ an exist? If so, what is it?
12. For each of the following series, determine if it converges or diverges. If it converges, what
does it equal?
P
(−1)n 2n
.
(a) ∞
n=1
3n
P∞
(b) n=1 ln(2n2 + 1) − ln(n2 ).
P
−1
−1
(c) ∞
n=1 tan (n) − tan (n + 1).
P
1
(d) ∞
n=3 n2 −4 .
13. For each of the following series, indicate whether it converges or diverges. Indicate which
test you use to make your conclusion.
P
n−1
(a) ∞
n=1 n5/2 .
P
n−1
√(−1)
(b) ∞
.
n=3
ln(ln(n))
(c)
P∞
(d)
P∞
(e)
P∞
n=1
1
n
(1+ n1 )
.
(3n)!
n=1 26n (n!)3 .
1
n=1 n
− tan−1
1
n
.
4
(f)
P∞
(g)
P∞
sin(sin(n))
.
n2
(h)
P∞
cos(sin(1/n))
.
n
n=1
n=1
n=1
4
n3 e−n .
14. What is the interval of convergence of
∞
X
(−2)n (x − 1)n
√
?
n
n=1
15. Find a formula for the numbers cn for which
∞
x+3 X
=
cn x n .
x + 4 n=0
16. What is the Maclaurin series for (1 + x2 )3/2 ?
17. Evaluate the following series:
P
(−1)n (π/6)2n
(a) ∞
.
n=1
(2n)!
P
(1/2)n
(b) ∞
.
n=1
n
P∞
1
(c) n=1 n(n+1)2
n.
18. Find numbers A and B so that
Z
A≤
1
sin(x2 ) dx ≤ B
0
and A − B < 10−4 .
19. Find the area of the region bounded by x = y + 2 and x = y 2 − 4.
20. p
Find the volume obtained by revolving the region bounded by x = 0, y =
y = sin(x) about the x-axis.
p
cos(x) and
21. Find the volume obtained by revolving the region bounded by y = x − 1 and y = x − x2
about the line x = −1.