1220-90 Final Exam
Summer 2013
Name_____
Instructions. Show all work and include appropriate explanations when necessary. Answers unaccornpa
nied 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:
(a) (5pts) D(tan’
(2))
2-x
1’
)
D(x
t
(b) (5pts) 7
7?c&Ac
1 &
2
C
7.LA-x# 7?q)
—
-
2. (l5pts) Compute the following indefinite integrals: Remember: -i-C!
—
—
3 dx
1
4
(a) (5pts)f
:
j
(b) (5pts)
3
f
3 x dx
cos
—
fcbx(tcx)9
I
IA 1kX
-
X
J(i)
oosx
LL-j+
-
(c) (5pts)
LA
f
2 lii x dx
x
A
6
t
5-
1
3. (5pts) a formula for the inverse of the function
f(x)
?(
c
=
ln (3x + 8)
M(3jS)
(J-))
4. (lOpts) Use the rationalizing substitution x
=
f
tant to find the antiderivative
(x ±1)3/2
dx
cA’ Se-c
2 øL
ç
/0
=
x
(x
I
5. (lOpts) Find the anticlerivative
f
—
3+2
A(x-)
+
Blx-2-)
(A113:?)L4 (--2
Ai I
2
-A
-
dx.
x
iO
--(_
0
—=.1
-:=:)
-j
co
I
c&X
Jx2
: ) rji’<zzJz
6. (l4pts) 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.
Do
n= 1
4ri
C®oc 3n +9
n=1
D
n=1
Do
D
n=1
Do
6°
n=1
9
DO(7\fl1
n=1
Do
C
7. (lOpts) Use the integral test to determine whether the series
Do
2
n(lnn)
n=3
converges or diverges.
(
—
J
X
(u)
2I,U
4
J
ILkd
1
•)
L°
/
L
3)
&
3
(14t
)
Hx
1
(•)
x—
>(
N
(%.
It
H
HH
L.3 +
$
Co
C
Co
C
Co
Co
cJ
C
C
C
C
C
C
C
0
Co
Co
Co
aq
0
C
0-
Co
0
Co
H
Co
C
Co
C
C
cr
0
0
0
0
Cm
cI
—
%—
I
‘I
N
-4-
÷
‘I
\—
\_
-
-
o
C
0Co
a
C
Co
C-
C
0-
Ij
U)
C
+
H
Co
Co
A
0
$
/
NAJ
g.
L
$1
“-‘j
f%
1
H
\“
r
‘I
tJ’
!r
—
?<
—
a1
•
—
-4-
cn
(I)
Co
Co
Co
CD
C
CCo
0
Co
C
C
Co
Co
C
C
0
C
Co
C-
C
Co
10. (l2pts) Match the equation with the type of conic section it describes by writing the letter in the blank
provided.
A. circle
y
+
2
x
y
—
F
A
P
11.
B. ellipse
1
g
+
2
x
—
2y=—3
C. parabola
y
+
2
x
2
y—0
D. hyperbola
y
+
2
2x
—
2y=1
E. a point
—
x
—
2
2y=z0
y
F. the empty set
solution)
(no
(a) (4pts) Find the polar coordinates of the point with Cartesian coordinates
—
9-
(
(b) (4pts) Find the Cartesian coordinates of the point with polar coordinates (4,
XrSfr-to.c()
L
jV1M(9
:)
—).
7r
9
(—±).-2
‘(ç)
(xy)
(-a, zJ)
12. (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. a circle centered at the origin
sin 6
B. a circle centered on the y-axis
8
A
Th
.L
r=sec6
C. a horizontal line
r=8sjn9
D. an angled line
E. a vertical line
=71
1
r=
sin9—cos6
r = 4 cos 9
F. a circle centered on the x-axis
G. a spiral
5
13. (20pts) Consider the lirnacoo determined by the polar equation r
curve is given a the bottom of the page.
=
2 + sin
.
A graph of this polar
(a) (6pts) Find the slope of the tangent line to the curve at the point (2,0) (that is, when
fi
==
f to)
=
0).
2—
J
J’o
+ 2
.....
1
z
—
-
(b) (9pts) Find the area of the region inside the limacon.
A
ç
CZ’)
f(4
Cf
tw
-f
4
tfr
(c) (5pts) Set up an integral to find the perimeter of the limacon. Do not attempt to solve.
S
L
ç
2Tr
&toj
0
6
C