1220-90 Final Exam
Spring 2014
Name
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. (l5pts) Compute the following derivatives:
(sin’ (x
1
(a) (5pts) D
))
4
(b) (5pts) D(ln(cosx))
=
(-s)
os
(c) (5pts) D(x
’)
4
Lb(
:x
Z
.&X
N
x
(k))
2. (l5pts) Compute the following indefinite integrals: Remember: +0!
(a) (5pts)
I
tL
3x—
2
dx
(j_
-
3-
W- 3ct
(b) (5pts)
(c) (5pts)
f
2 x sec
tan
2 x dx
fcos2x
dx
x
-
-
—
1
30
3. (lOpts) Use integration by parts to compute the following indefinite integrals: Remember: +C!
(a) (5pts)
k
f
xsin (3x) dx
4
—
K
)
(b) (5pts)
f
V
inx dx
3
x
4. (lOpts) Evaluate
[
J
5
—
z+S
dx
x
—
2
x—2
x—
1°
X
-
1
(x—)
x -l
:
t
=
(‘
53
/_
-
i
x
4
-
(iiv-1
3+i\
0
23
(x+i)
((x-)
-
3k
j
A
3)(ul
9
-
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.
3
2
n
C
1
D1
(3)
D
c
(—1)’e
C
cos(n7r)
6. (l2pts) Perform the integral test in two steps:
(a) (6pts) Use substitution to find
I 2x+1
clx.
I
J
C
+x)
2
(x
5
C
(b) (6pts) Use the above and the integral test to determine whether the following series converges
or diverges
2n+1
(n + n)
-—
C
/‘
1’
/ zJ_
/L{4
(
+
lI
f
)
-
l1/
(
—
3
t
(x)
7. (l4pts) Let
f(x)
(a) (3pts) Find the following derivatives of
f(x)
=
e2
evaluated at x
e
f(1)=
=
1:
r
‘() 2f’(l)=
I
f”(l)
-
(b) (3pts) Find a formula (seeing the pattern in your computations above) for the 7lth derivative of
f(x)atz=1:
k 2.f(7o)(1)=
2- e
c7
(c) (2pts) Write out the full Taylor series for
above:
i
()(i)
2. e
n=O
n!
at x
=
1 using the formula you found for f()(1)
(x,)
A
n=O
f(z)
(d) (6pts) Use the Ratio Test to determine the interval of convergence of the Taylor series you found
in part (c) above.
2
‘4
I’
fx-,f
Li
2
&H
2—
j41
-2.-
e tx—il
—
k
‘2-’
,
(t4_flI
jx—iI
)
$
tc
8. (6pts) Use the fact that cosx is equal to its Maclaurin series
00
x
6
x
2
1)x
2
x
=1—+—+...
‘2)!
fl=
to find the 4th derivative of
Cs(3xj I
f(x)
=
) at x
2
cos (3x
=
3x
(9+
2-
z.’l
0.
(3x)
2-
Z
ii’.
—9
t,t
::.
f/o)
LJ(
-
41o)
4
)
9. (l2pts) Match the equation with the type of conic section it describes by writing the letter in the blank
provided.
F
E
A
+4x+y
x
—
2
2y=—6
A. circle
±4x+y
x
—
2
2y=—5
B. ellipse
+4x±y
x
—
2
2y=—4
C. parabola
j) 2
+2yr=—2
x
+
4x—y
C
D. hyperbola
+8x+y
2x
—
2
2y=—9
E. a point
—2y=—4
2
4x+y
F. the empty set (no solution)
10. (8pts) Match the equation of the conic section to its graph by writing the letter in the blank provided.
Every answer will be used exactly once.
2;2
I
Øx2
A ç+
p _x-j
22
A
I
33
tB
I
)[(
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. a circle centered at the origin
B. a circle centered on the y-axis
cos 6
•fl
V
C
A
/1
F
r=
2
sin 6 + cos 6
C. a horizontal line
r=-4
sin6
r=1
D. an angled line
r=r4sin6
F. a circle centered on the x-axis
r2cos6
C. a spiral
E. a vertical line
5
3L,L
12. (8pts) Convert between polar and Cartesian Coordinates as indicated.
(a) (4pts) Find the polar coordinates of the point with Cartesian coordinates (—1, 1).
r
)
Lf
( )
(b) (1pts) Find the Cartesian coordinates of the point with polar coordinates (3,
).
-rcOs&\- 3c()
q
rsIL9-
3(3,r)
—3
(_
13. (l4pts) Consider the curve determined by the polar equation
between 0 = 0 and 6 7t is given a the bottom of the page.
r
=
L.f,
62.
—
A graph of this polar curve,
(a) (7pts) Find the area of the region inside the curve and above the x-axis (between 6
6=).
=
i7)
I
—
__c77
0
(b) (Tpts) Find the length of the curve between 0
=
0 and 6
=
ir.
1
r
LJt
LA
tJ{
—10
—I
—6
—4
—2
)
‘t
4
—
I
-L9
(i4j
-
6
—
f /-
0 and