This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: University of Maryland, Nam“ Lﬂ Baltimore County Id #:
Math 15211 Exam 2
Fall 2011 Monday, November 21, 2011 [2 points] Put your name on the test above, and put a check mark next to
your discussion section in the following chart: James Travis, 4:00  4:50 James Travis, 7:00  7:50 Instructions: Possible Score 0 Read each problem carefully. 0 Write legibly. 0 Show all your work on these sheets.
Feel free to use the back of the page. 0 We will not accept answers without
justiﬁcation. o This exam has 9 pages, and 8 questions.
Please make sure that all pages are in
cluded. 0 You may not use books, notes or calcu lators.
a You have 75 minutes to complete this exam. Good luck! Math 15211 Exam 2 Fall 2011 Page 2 of 9 Question 1. [15 points extra credit, 0.5 per blank] Complete the following statements. 1
(a) The pseries Z Z; converges if a? I , and diverges if E 4— l . co 00 Q
(h) The geometric series 2 411"“ or 2 ar” converges to ‘1 2." ,1 ‘ n=l n=0
if r'lé ‘ ,and diverges if Ir! 7, l . (c) The Integral Test: Let f (x) be a function that is Cam/ﬂaws , [gets ﬂ , and WC? .
Let an = £64 . Then 2 an and l 00] 1) (WM n=l either both converge or both diverge. (d) The Comparison Test: Let 2 an and 2 bn be series with e 0 st I'M terms.
0 If an 4 bn for all n, and 2 b" converges, then 2 an M 5 .
o If an a b" for all n, and 2 bn diverges, then 2 an AV—Ugé . (e) The Limit Comparison Test: Let 2 an and 2 b" be series with K as: W terms, A
and let c = $3,). “6 .
If c is a QQSXW and E “:1! number, then 2a,, and 2 bn either both converge or both diverge. (f) The Alternating Series Test: Let 2 an be an alternating series, i.e., an = (—1)"b,, or
an = (—1)"‘1b,,, with bn > 0. If 17,, has the following two properties: bun 4’ ‘3“ , and Nﬁu Ix ‘7 9 then ‘A “n" (g) The Ratio Test: Let L = “3,, l1:
0 IfL> l ,thenZan CAM) .
o If L < l , then 2a,, aLss‘tsk/lq mvtrgg . OIfL= ,then 51. rkﬂ'mi‘éi’ ‘3 NWLLUM __________.______—___——
LM ]
(h) The Root Test is exactly like the Ratio Test, except that L = .31., n l M . Math 15211 Exam 2 Fall 2011 Page 3 of 9 Question 2. [8 points each] 1 1
(a) Find the length of the curve g(y) = % — Zyz on the interval 1 s y S 2. (b) Find the surface area of the surface obtained by rotating the same g(y),
1 S y s 2, about the xaxis. A : 3%‘EM3/eu7 1 LFJI17{% “9J7
taffe‘é‘ﬂy = arkng]? : Lwﬂj+%)/%¢ Math 15211 Exam 2 Fall 2011 Page 4 of 9 Question 3. [8 points each] Find the sums of the following series, or show that they diverge. °° 1 1
— — ' : 'nk t 'al .
(a) ;2 (11101 + 1) Inn) (Hint Thl abou part1 sums) H; ”L
53’1"} ill Ayker 9—
Wm L3 M A; m M Math 152—11 Exam 2 Fail 2011 Page 5 of 9 Question 4. [8 points each] Determine whether the following series converge or diverge. Justify your answers. M
00 0' m i
"—3 7’ ‘__\
5" 3
(a) Zn4+2 < v:4 2: h
n=1 AM “A
00 n~3
i “bl4’1 OWN—‘89
AV} n=2
J00 \ 3 \Jzﬂkx
.Lxm‘Q (kw0 ikp
is I _
\‘ S vs , in t
95¢“ .2 7({Lx3‘x’f I, {r590 £1 (15% K
N it.“ :1, + ‘_\__ﬁ _ x ‘
“M 74m“ 20.231 7 (1&1
We j’L C6030” (mu059,» . Mil”); Math 15211 Exam 2 Fall 2011 Page 6 of 9 (_1)n
«7m Question 5. [12 points] Determine whether the series 2( is absolutely convergent, conditionally convergent, or divergent. mm») wan w L“: :57; L“ a WSW [M r l no M
M “Leo‘s 4;; i O 4 9“” C S
n24 {Sad/S“ \ gran to? :43: me ”‘9‘!" Wm (”if“) Math 15211 Exam 2 Fall 2011 Page 7 of 9 Question 6. [12 points] Find the radius of convergence and the interval of convergence for °° _ n+1 _ n
the power series 2 (—D—(x—i. "=0 (n2)22n—2
B ' V‘M ‘— I‘M «74“7, L
th) jal’i’ ll: m‘tlk \{[)y[% iL.‘ Lyman K
J A m ”\M MW LVN/l {L (J) 60.13)
zlklril. 4‘ IL. ALL—32‘; a x3
”394 11 sz'h‘n \/,J/ W \l W‘%+JT) “ t,‘ / Math 15211 Exam 2 Fall 2011 Page 8 of 9 Question 7. [10 points] Find the 3rd degree Taylor polynomial for f (x) = Vx + 6 about a=3.
4131.3
Il/lx): ﬁ 13/(3):'::
.,» V, v)
pox ‘; W 973)  mg
W  L, m L l
i M" um)” l (3) ma
1 '3
f ,. 4, ‘ '3 Cw” '3
LU”; gummy 33 WWW/[l + My.) Math 15211 Exam 2 Fall 2011 Page 9 of 9 ”x" '.
":0 11. Question 8. [16 points total] Recall that ex = (a) [8 points] Use the power series given above to ﬁnd I e"‘4 dx.
#1 h {mic
gk‘ﬂ A Z jw is J— 1‘50 :jnﬁQ/ C )
t X
av; 14W ICtzwfﬂt’C/r‘ (b) [8 points] Use the ﬁrst two nonzero terms of your answer for part (a) to estimate
1 f e"‘4 dx. How accurate should you expect this estimate to be?
0 x“ " “XV
: E, 0‘40 g X S—  , {\KH «v _/x:.7\ r. ——i :1
W XOCANVXYJ09\S\5\ 7h%?5a1‘*”*“‘3,s. 2+ stu ma"
1% 3s; rm 1 a 4 ...
View
Full Document
 Fall '08
 Tighe
 Calculus, Geometry

Click to edit the document details