1220-90 Fiiia1 Exam
Fall 2012
Name
Instructions. Show all work and include appropriate explanations when necessary. Please try to do all
all work in the space provided. Please circle your final answer. The last page contains some useful identities.
1. (l5pts) Compute the foIljjrvati
I
(a) (5pts) D(log
x
5
)
2
(b) (5pts) D(3x
)
D((2x)
x
(c) (5pts) 3
•t ,( (z)
)
Qu2.x
y,c
—
‘,cj(2c)
e
2. (l5pts) Compute the following indefinite integrals: Remember: +C!
(a) (5pts)
(b) (5pts)
f
f
1
/1_x2
dx
+ 4 dx
.-
p—
(c) (5pts)
f
xcosx dx
oW:coc v
cvx
1
1
3)c
z->
(3Lt(Zc>3)
3. (l2pts) Find
1
1
J x/x+4
dx by making the rationalizing substitution x
=
2 tan t.
2RAf. =‘
ftq
5
LkCC
x
(ço
2-
(
I
(
4. (lOpts) Use partial fractiolls to find the antiderivative
A
—
+
‘f_
f
x
—
2x
Abc-2-) l-x
—
cCc-2-)
(L?c-2-)
/“
--
_J.
A—l
17
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.
L
D
n=1
Ia
1+n+n
2
D
C
C
5()fl_1
c
n=1
c:::D
n= 1
6. (l2pts) Use either the integral test or the comparison test to determine whether the series
°°
tan —1 n
2
n= 1 1 + n
converges or diverges. Note: You can assume that this series satisfies the hypotheses of both tests;
you do not need to check this.
pcvtTe
)k4tyu
—1
?C
t-4oo
lt)4tI<
‘v
IIww
%)o4c
J
14-)c’
1
4-
i&_.
——
/7
oW-
IT-I
(2-.
—
‘
1
2
i.v
JLA.A)
I)
.:
fLL
6
F
4-,L-t)
2.
.s_
,
1—
—
‘—
—
;.-
,
‘—
I1_-
v’
1¼
*1/
t.bA.YT.r
bw’
c,
--
32-
-<:
3
Lecevis
e-1)
-.-
4,
“
1
3
jt
\.3
\
)(
‘I
1
1
1
NI
lb
-
I.
*
:.
.
‘1
t’
r
‘I
£
to
&
_(I)
CD
Co
0
CD
0
0
CD
CD
CD
0
CD
0
0
CD
0
ct
CD
0
0
0-
Co
D
k
-i.
1—
1)
‘I
-c
x
x
1-
-,-
“a
+
)
p
1
‘(00
t
Cl)
CD
CD
to
CD
—
CD
C
Co
+
H
Co
CD
+
to
H
CD
to
+
H
CD
+
CD
C
to
H
+o
Co
CD
CD
Co
CD
.
-
to
Cl
H
CoM
CD
CD
Co
CO
OH
CD
Co+
CD
•
-t
Co
-t-
00
0
00-
CD
CD
CD
o
o-t
9. (8pts) Consider the ellipse determined by the equation
2
+ 4y
—
(a) (6pts) The center of the ellipse is the point
6x + 8y
(
= 0.
3
,
—
I
.
(b) (2pts) The length of the major diameter if the ellipse is
txx+’)-i
(x3)
+ LIL(y*;)
13
(x-3)
10.
(a) (4pts) Find the polar coordinates of the point with Cartesian coordinates (1,
=r-a.
8-- r
r {S)
“-&
Jj
(b) (4pts) Find the Cartesian coordinates of the point with polar coordinates (3,
x
rco 0
3 c-
).
L%
11. (l4pts) Match the polar equation to the type of curve it determines by writing the letter in the blank
provided. Every answer will be used exactly once.
A
C..
9
P
A. a circle centered at the origin
0
r
r
=1
= 5 + 5 sin 6
C. a horizontal line
r
=
D. an angled line
r
=
sin0
l2sinO
r
=
.
r
B. a circle centered on the y-axis
E. a cardioid
F. a
2sin6 + 5cos6
= 3 cos (26)
C. a spiral
5
12. (24pts) Consider the curve determined by the polar equation r
(a) (5pts) At what angles 6 with 0
=
3 sin (26).
0 < 27r is r equal to zero?
0
)_
-ir)
.
(b) (5pts) Consequently. which of the following is a graph of the curve r
=
3 sin (26)?
\\
A
B
(c) (8pts) Find the area of one loop.
A
I Cct(z))
.
Jo
-fr
0
t
q
“Jo
(l—coS(’19))dfr
p
—I
--
(d) (6pts) Set up an integral which gives the total length of the curve. You do not have to evaluate
it.
24r
L
0
6
Trigonometric Formulas:
•
9
9
sin x+cosx=1
1 +2
tan x = sec
2 x
1
sin x=(1—cos2x)
x
2
cos
sin
2x
(
sin x cos y =
(1 +cos2x)
=
=
2 sin x cos x
sin(x
sinisiny = (cos(x
cos x cos y
(
=
y) +
sin(x
y)
cos(x +y))
—
—
cos(x
—
—
y) +
cos(x
+ y))
+ y))
Inverse Trigonometric Formulas:
9
Dsin_1x=
D
—1<x<1
1 x =
cos
—
1 <x < 1
—
D tan x
=
1 +
2
1
x=
1
Dsec
Calculus with Polar Curve r
=
(x>1
f(s):
b
A=f[f(6)]2 d9
=
L
f
+
2
[f’(6
d9
f’()sin+f()cos
m—
—
f’()cos
7
—
f(O)sino