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Unformatted text preview: MA 166 Page 1/11 FINAL EXAM Spring 2006 NAME 10DIGIT PUID #
RECITATION INSTRUCTOR
RECITATION TIME ' LECTURER INSTRUCTIONS 1. There are 11 different test pages (including this cover page). Make sure you have a
complete test. 2. Fill in the above items in print. Also write your name at the top of pages 2—11. 3. Do any necessary work for each problem on the space provided or on the back of
the pages of this test booklet. Circle your answers in this test booklet. No partial
credit will be given, but if you show your work on the test booklet, it may be used in
borderline cases. 4. No books, notes, calculators, or any electronic devices may be used on this exam. 5. Each problem is worth 8 points. The maximum possible score is 200 points. 6. Using a #2 pencil, ﬁll in each of the following items on your answer sheet: (a) On the top left side, write your name (last name, ﬁrst name), and ﬁll in the little
circles. (b) On the bottom left side, under SECTION, write in your division and section
number and ﬁll in the little circles. (For example, for division 9 section 1, write
0901. For example, for division 38 section 2, write 3802). (c) On the bottom, under STUDENT IDENTIFICATION NUMBER, write in your
10digit PUID, and ﬁll in the little circles. (d) Using a #2 pencil, put your answers to questions 1—25 on your answer sheet by
ﬁlling in the circle of the letter of your response. Double check that you have ﬁlled
in the circles you intended. If more than one circle is ﬁlled in for any question,
your response will be considered incorrect. Use a #2 pencil. 7. After you have ﬁnished the exam, hand in your answer sheet and your test booklet to
your recitation instructor. MA 166 FINAL EXAM Spring 2006 Name: —— Page 2/11 1. If (i = §+ 17+ I; and I; = —;+;+ 1;, ﬁnd a unit vector orthogonal to both (i and 1—; and
having negative kcomponent. A._f—E
1“ 1 “
C iii/Z
'«53 «a
rag“
D.\/§z ﬁk 2. The vertices ofa triangle are P = (——1,0, 1), Q = (1,1,3) and R = (2,1,0). If 0 is the
angle of the triangle at P, then cos 9 = 5
AW 1
REV—1:0—
CO 5
Um 2 3. The radius of the sphere $2 + y2 + 22 + 21: — 42 = 3 is 21:30.00?
“ems MA 166 FINAL EXAM Spring 2006 Name: ._ Page 3/11 . 1n(1 + n2)
4. 1 —— =
$122 1n(1 + n) A 0 9?“? .U
M: F11
IbIﬁ Nlﬁ :12‘;
6. / (sina: + cos x)2da: =
o ?> .U .0 .w
to]: Nil>1 r—lwlﬁ mm
+
Min— 
)— Nln—l F11
+ MA 166 FINAL EXAM Spring 2006 Name:—_____._ Page 4/11 3 1
7. / 2 d$= 2
2 $ _$ 111E 4
Bl—
n3 1
0.1—
n6 1
D.1—
I13 3
E. 1115 8 /$(1n$)4d$= _A. (1n$)4—4/$(ln$)3dm $2
B. 3(ln$)4~2/$(ln$)3d$ C. 4$(ln$)3—/(ln$)3dac D m—2(1n$)5—1/(1n$)5dm
' 10 5 1 1 E. 3(ln$)5—3/$(1n$)5dm 1 5
9. The trigonometric substitution 2:6 : sin0 converts the integral / (1 — 4$2)%d:1: into
1 4 1
A. cos5gsin0d0
2 6 1
B. 64/ cos6 0d6 1 2 C. ——%/2 c03563in6d6
% l D. g / cos6 0d6 7r 6 N .1 2
E. 64 / c0360d0 6 MA 166 FINAL EXAM Spring 2006 Name: —_ Page 5/11 10. The region in the ﬁrst quadrant bounded by the y—axis, the graph of x + y = 2 and
the graph of y = x2 is revolved about the x—axis. The volume V of the solid generated
is A. g
B. 3;
C. 471'
14
D. T" 11. The region bounded by the graph of y = sin x, 0 g x g 71', and the xaxis, is revolved
about the y—axis. The volume V of the solid generated is
71' A.§ B. 271'
C. 271'2
D. 71'2
E. 71' MA 166 FINAL EXAM Spring 2006 Name: ——_ Page 6/11 12. Consider the lamina bounded by the curves 2: + y = 1, a: = 0, and y = 0 and with
density p = 1. If (‘25, y) is the center of mass of the lamina, then 5 = A;
3.;
3
Cg
D;
2 0°
13 SuPpose that Zen 2 3. Let bn = 2% and sn 2 b1 + b2 +    + bn. 2
n=1
Which one of these statements is true? A. lim sn 2 7r and lim bn = 0
Tia—FOO 11—)“, B. lim sn 2 7r but lim bn cannot be determined
Tia—>00 n—)'00 C. lim sn 2 00 and lim bn = 7r
11—}00 77,—}00 D. lim bn = 0 but lim 5,, cannot be determined
77,—}00 n—‘FOO E. lim 5,, = 0 but lim bn cannot be determined
11—}00 n—FOO MA 166 FINAL EXAM Spring 2006 Name: — Page 7/11 °° 1
14. The series —n
n=1 6 A. diverges
1
B. =
e — 1
e
C. — e — 1
1
D. =
1 — e
1
E. =
1 + e 00
1
15. The series E (1 + 7)
n “:1 A. Converges by comparison with i 1 n=1 n2
B. Converges by the ratio test
C. Diverges by the ratio test D. Converges by the limit comparison test E. Diverges MA 166 FINAL EXAM Spring 2006 Name: ___— Page 8/11 16. Which of these series converge? °° 1 A. All
(I) Z —,
n=1 ” + V712 + 5 B. Only (III)
00 sinzn C. Only (II)
(II) Z —,
n=1 VH3 +5 D. Only (I) and (III) 00
n E. Only (II) and (III)
(III) “4:31 V715 + 272,2 + 1 17. Of the series (1) 2:11;", (II) Z<—1)"(1+%), (III) 2 n=1 n=2 A. (I) and (III) converge absolutely
B. (I) and (II) converge absolutely
C. Only (III) converges absolutely D. All converge but none converges absolutely E. (I) and (II) diverge 18. If f = tan 3:, the terms of the Maclaurin series of f up to the third power of :1: are A. 113—332—3337
B. 312133?3
C.1+$+%:—+:—?
D. 33+; E. :1: _ :123 MA 166‘ FINAL EXAM Spring 2006 Name: _——_ Page 9/11 2 00
19. The radius of convergence of the power series 1; 1)! x" is A. 00
B. 1
C. 2
D.
E. 0
20. Match the functions with their Maclaurin series.
00
(1) e" (a) Z (—1)"$", —1 < x < 1
n=0
1 x2 2:3
(2) 1+$ (b) 1+x+§+§+...,—oo<x<oo
x
(3) 1 x (c) $+xz+x3+...,—1<x<1
32 2 34 4
(4) xsina: (d) 1——2T—+—4f—— ,—oo<a:<oo
371‘i x6 I
(5) cos3a: (e) 1143—37 —5—'—— ,—oo<9:<oo A. 1b,2c,3a,4d,5e
B. 1e,2a,3c,4b,5d
C. 1a,2b,3d,4c,5e
D. 1b,20,3a,4e,5d
E. 1b,2a,3c,4e,5d MA 166 FINAL EXAM Spring 2006 Name: ——_ Page 10/11 21. The graph of the polar equation 7' = 1 + cos0 is
A. A circle with center at (x, y) = (0, 1) B. A circle with center at (x, y) = (1,0) C. A twoleaved rose D. A cardioid with the point farthest
from the origin at (x, y) = (2,0) E. A cardioid with the point farthest
from the origin at (x, y) = (0,2) 22. Convert the polar equation 7' = —2 cos (9 to rectangular coordinates
A. (x—1)2+y2=1
B. (x+1)2+y2 = 1
C. x2+(y—1)2=1
D. x2+(y+1)2=1
E. x2 + y2 = 2 23. The curve described parametrically by A 11.
. an e 1pse . 7r
x _ 2cos t’ y = _ smt’ 0 S t S 5 B. a quarter of a circle
is C. a half of a circle
D. a half of an ellipse E. a quarter of an ellipse MA 166 FINAL EXAM Spring 2006 Name: _—_ Page 11/11 24. The length L of the curve in problem 23 is given by A. [2 \/3sin2t+1dt
o B. /2 \/2cos2t+1dt
o
12" C. / \/—sin2t+1dt
o D. /2(—2sint+cost)dt
o E. /2 (x/Esint+ 1)dt
0 25. The polar form of the complex number with argument between 0 and 27r is Z.
1+x/52'
7T 71' A. — " — cos 3+zs1n 3 B 1( us 71) .—cos— 2m— 2 3 3
7r 7r C. — " — cos 6+zs1n 6 1
D. 5 (cos g+isin 7r ,_ 7r
E. «5 (cos 5 +zs1n ...
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