1220-90 Fiiial Exam
Spring 2013
Name
Y
Instructions. Show all work and include appropriate explanations when necessary. Answers unaccompa
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(ln
(2
+ x))
L
(b) (5pts) D(2’
(Z;(1i)
X)
(1z) zz 22. (lOpts) Compute the following indefinite integrals: Rememlr: +0!
(a) (5pts) fxe dx
—‘C
ç
(b) (5pts)fxlnxdx
Ix ø9C
—
3. (lOpts) Compute the following limits using L’Hôpital’s Rule. Note: Answers may be ±.
(a) (5pts) lim
5x 2
sinx
L.Il
(OK—LoSx
x
x-*O
(b) (5pts) lim
—
.
-
I
x2
x-*O 51111— I
-
+ Lo ‘C
1
—
—
$n6
xl
*
1
•4_j
“4
V
*
C
‘I
c1
1
i
x
1
U)
CD
CD
CD
n
fT I
C-
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.
c
n=1
c@i
: (lllfl)
00
Ti
n=1
00
9
(_1)11
n=1
00
cc
5Th
n=1
n=1
00
4 + 1
n
1
7. (l2pts) Use the integral test or the comparison test to determine whether the series
00
n=1
converges or diverges.
c
Ti 2
3 +
2n
5)2
Note: You can assume that this series satisfies the hypotheses of both tests.
I
e
t
2-
41
z;;:-’9’
—
2
L2-’1-s)
IAC
U.A4
.-4,o
h
.-
-
co
—
—
çte
if
24+ £
j1
/‘>I))
1f_
p-i
L7
tc-T
(3+
(L
q2
4
k;--”
1f)
2
v’
42Covt1,7e-S.
3
&)
8. (lOpts) Find the radius of convergence of the power series. Note: The answer may be a positive
number, zero, or oo.
3Th(x5)n
-
in!
3
v’k
1c’-c
31xsi
.
1
V’
3 hcS1
X.
9. (l4pts) Let
f(x) =
(a) (8pts) Find the following derivatives of
f(1)=
f(x)
evaluated at x
=
1:
I
I
f”(1)=
(x), the Taylor polynomial of degree 3 based
3
(b) (6pts) Use your computations above to write out P
at a = 1 for f(x).
3
r3
fr)
t’i
I
(,c)
/
—
()
(x’-i)
(x—i) ÷
4
10. (lopts) Consider the conic section
9
9
x—y —8x—2y=l3.
4
(a) (2pts) What type of conic section is this? Circle one answer below:
rbola
Ellipse
Parabola
(b) (6pts) The center of the conic section is the point
I
(
(c) (2pts) The shortest distance from the center to a point on the conic section is
LiLx x)
x)_
lL(?(_
.
2
z)
(y
13
13
ii-2j)
13
—
4
t
11. (l6pts) Match the polar equation to its graph by writing the letter in the blank provided. Every answer
will be used exactly once.
r=2
A
r=cos
H
r=sin6
__
i.2
sin 26
r=1+2cos6
P
r=2+lcos8
A
E
B
i—)
0
G
J
5
RED
24
12. (24pts) Consider the cardioid determined by the polar equation r
curve is given a the bottom of the page.
=
2
—
2 cos 9. A graph of this polar
(a) (6pts) Find the slope of the tangent line to the curve at the point (0,2) (that is, when 6
=
2.
f)
2-st&
fY-)
(b) (9pts) Find the area of the region inside the cardioid.
A
-
21
r
f(—
Jo
#toSZ9)ct
-
-ir
0
(c) (9pts) Find the perimeter of the cardioid (that is, find the arc length of r = 2 2 cos 9 between
1 cos 9.
9 0 and 9 = 2r). Hint: You will have to use the identity 2 sin
2 (9/2)
—
—
ltt%)
V
2-if
Jo
a
V
I
C3j0
2-c
.J1lkt%J
V
—0
LI
Jo
6
-
/-)4-o