1220-90 Final Exam
Fall 2013
Name
k’ Y
Instructions. Show all work and include appropriate explanations when necessary. Answers unaccompa
iiied by work may not receive credit. Please try to do all work in the space provided and circle your final
answers. The last page contains some useful formulas.
1. (lOpts) Compute the following derivatives:
(ln
1
(a) (5pts) D
(3
+ x))
(3x
1
(b) (5pts) D(x’)
(.
--
D
(x) sii
x
)
s’c
(x
4
2. (lOpts) Use L’Hôpital’s Rule to compute the following limits:
(a) (5pts) lini
x_1
2
e
11kM
X—,O
X
I
c
L’11
(b) (5pts) lim
—
x-40 Sfl Z
Ii
?(3b
3. (5pts) Find a formula for the inverse of the function
f(x)
‘/3
)(
3
(Y
-
=
1
=
2—
N
4. (24pts) Evaluate the following indefinite integrals. Remember +C!
(a) (6pts) fxe3 dx
LL-
-=?j
)(
uoç?(
.=;,
(b) (6pts)
f
4
sin
cix (I-cIx)cox
x cos
3 .x dx
=
toS
c4
c
(c) (6pts)
f9 x+2
dx
Hint:
2
—3x+2= (x—2)(x— 1).
A-fl+8(x-2.)
-3x2
(2
1-2-Xi)
1.AQA4
(
-‘?—
LA.jiO>AL
Ac-S
-
x
(d)(6Pt5)fx
d
5
x
2
QU4-
)
1.
+4x±5=(x±2)
2
Hint:x
+
C
64
2
5. (l2pts) Determine, by whatever method you wish, whether the following series are convergent or
divergent. Circle ‘C’ if the series is convergent or ‘D’ if the series is divergent. No work is necessary.
00
C
n=1
9
n+1
52
+9
00
fl
D
D
D
sin (n7r)
C®
6. (l2pts) Perform the integral test in two steps:
(a) (6pts) Use integration by parts to find
I lnx
dx.
I
J x
—-
Hint: Set u
=
lnx and dv
1
V—)(
eLLLkt)
-
j
2 dx.
x
x
‘4—’
x +
—.QL4..x
-Jxx
-
*c)
(b) (6pts) Use the above and the integral test to determine whether the following series converges
or diverges
00
lnn
2
n
n=1
(
(
k—” J,
=IILAA
lt
Li
+
IO
1
AT-b
3
-Ai
o-*
I
x)J
I
.—IH
‘iI’—
I
H
rj
+IH
I
DII—
H
—ID
p
H
+
H
71
H
9
II
,lr\
I
CD
C-
-\x
EH
—
I<
CD
—
c
—
÷
cy
•—r•)
‘.—
Uj
•
-
CD
-
(—%.
-
I
‘I
I
I
I—
II
P
0
0
C
-.
..—
0
/ j;;
I
r
‘I
p
CDH
C-
CD
CD
C-
=
C
C
-+,
CD
L+LH
II
><I
h
II
‘—__-
CD)(I
Ci)
c-,-
CD
C
C
CD
C
II
II
—
CD
J
V
-
‘I
ci4
—I
,J
J
X.
r
x
‘I
—
I
I
M8
,
10. (8pts) Consider the conic section
y
+
2
4x
+
8x—6y+4=0.
(a) (2pts) What
type of conic section is
Parabola
this? Circle one answer below:
Hyperbola
E11ipse.
(b) (4pts) The center of the conic section is the point
3
(
(c) (2pts) The shortest distance from the center to a point on the conic section is
+
()-
+_
3
11. (l6pts) Match the polar equation to its graph by writing the letter in the blank povided. Every answer
will be used exactly once.
13
r=cos3
rsin
r=1+2cos
E
r=sin20
F
r=2
r=O
r=2+lcosO
A
r=cos6
A
C
Bç
JJ
E
0
G
0
5
12. (6pts) Match the Cartesian coordinates (x, y) on the left to the corresponding polar coordinates (r, 9)
on the right by writing the letter in the blank provided. Every answer will be used exactly once.
C.
A
(1,1)
A.(1,4)
(-/,i)
B.(/)
(0, —1)
C. (2,
)
13. (Gpts) Match the polar coordinates (r, 9) on the left to the corresponding Cartesian coordinates (x, y)
on the right by writing the letter in the blank provided. Every answer will be used exactly once.
(4, 7r)
A. (—4,0)
2
C-
B. (—1,—)
(/,Z)
C. (1,—i)
14. (l2pts) Consider the curve determined by the polar equation r
between 9 = 0 and 9 = 7t is given a the bottom of the page.
=
. A graph of this polar curve,
3
e°/
(a) (6pts) Find the area of the region inside the curve and above the x-axis.
1
/
(1
(b) (6pts) Find the length of the curve.
s
L \W’j (e)
=+
I/3
11
3
6
L