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Unformatted text preview: W Math 16C Sec 2 (Malkin) Name:
Midterm exam 2 Student ID:
Fri May 30th 2008 Signature: DO NOT TURN OVER THIS PAGE
UNTIL INSTRUCTED TO DO SO! Write your name, student ID, and signature NOW! NO NOTES, CALCULATORS, OR BOOKS ARE ALLOWED.
NO ASSISTANCE FROM CLASSMATES IS ALLOWED. Read directions to each problem carefully. Show all work for full
credit. In most cases, a correct answer with no supporting work
will NOT receive full credit. Be organized and neat, and use
notation appropriately. You will be graded on the proper use of derivative and integral notation. Please write legibly! 3 15
4 6
5 5
6 7
Total ' 50 Page 1 of 6 1. Consider the following integral: 1 1
/ / 6.27231 dydar.
0 x/E (a) (2 points) Sketch the region R over which we are integrating. (b) (5 points) Evaluate the double integral. \ l ‘
ll jﬁ try (lath : 0 Wﬂﬁ ax
: l; (WW (misc) le Page 2 of 6 2. (a) (5 points) Write down the double integral expressing the area of the region R
given in the diagram for both orders of integration. DO NOT EVALUATE THE INTEGRALS! 4+
ii ' ’2, H ‘i ‘L
(Mi 10( R r S Clgdl : down;
0 LHI oilj (b) (3 points) Write down a double integral expressing the volume of the solid under
the surface 2 = f (x,y) = 3x2y3 and above the triangular region R bounded by the curves as = 0, y = 0, and m + y = 1. DO NOT EVALUATE THE
INTEGRAL! ' l‘WL \ ‘43
= 238ij 6(de OR Ax k5 (idol;
0 O O O (c) (2 points) Write down a double integral expressing the average value of the function f (ac, y) = 2x2 + y2 over the rectangular region R with vertices (1,1),
(3,1), (3, 2), and (1, 2). DO NOT EVALUATE THE INTEGRAL! J— :L
Amiga mm 2 1 (QvL‘Hﬁdng
\ ‘ l L E
Page (50f 6 «it (31(31)de ,\\ 3. (a) Determine the nth term (starting with n = 1) of each of the following sequences: i. (2 points) 1,2,6, 24,120,720,5040,....
ii. (3 points) —%,g,—%, (b) Determine whether each series converges or diverges. You must justify your
answer and state which test you are USES. (2points) )2203—45. , Z 0% 6 35 CMQXUL S€F\€S\ 4&3“—
m% g S\Y\CL\V:] \5\<\, 1. (4 points)‘; ”:1 V4771? ..L “’w‘j W13 “haw hrs—3‘ ﬁ=h¢¢ M 9% NJme 116$!— iii (410mm): n=1<n—1>!' NH  Chm ‘ ' , f ’___‘,A
Jam \ mitt: m rim 1““ N hose: ”*3" nae.
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on“ \H\ to a
V900 4. (6 points) Find the radius of convergence and the interval of convergence of the following power series:
00 23%‘(H1V n=0
You do NOT need to say what happens at the end points of the interval. i\\—\ &\ Clue KE/ 0 ﬂ
mm a“ T3 : WW lx—W’ “a” “/3“ [sun {53 gm COGdo Xerﬁ UM $€H~€S (mueau . §l\ Mbcﬂ} l(/5 lDL‘L‘ V3 ("D *Z (304 <3 (:3  Raul
go‘ mg 0% m{9%‘QS \S 3, 0M) WUW) 6} WWng 1's [Lh'l)‘ \
5. (5 points) Suppose that a rubber ball, when dropped on a ﬂat concrete surface,
rebounds 90 percent of the distance it falls. Find the total vertical distance, both up and down, traveled by the ball if it is dropped from a height of 10 feet onto a ﬂat
concrete surface and allowed to bounce forever. V L093“) Detox/m 2 l0 + V;%\OX(OPQ
’0 my : )0 + 2051040 mm)  leoroq : lo 4 \_0 0‘ 6. (7 points) Use the method of Lagrange multipliers to minimize the function
f(x, 3/, z) = 3:2 + 2y2 + 322 subject to the constraint that :c — 4y + 32 = 12. twig) f scuba so; , A [M3 we A1) , @ )P'k: (Z'DKVA :0 6:3 (A22)g.
@ :3: (V3 ”('ng t. O '
Cg \F‘}; :— bL ~ 3‘) '7’ O
@V’). 1 “()Lth‘s’rgl’
RAH M3 80:11 'wh C7) 31%) . Lhi)’: H1133!) (C) {ft} *31‘: 0 L33 \3: ‘11,
9&1“ j ‘3;1x “Nb ® ﬂﬁms . ks, %('li\[email protected] (an; ~hDLILO (a) >9 :x,
“Aﬁrhs {ﬁt'11 01M ”biog \‘Vvév ® (BurLL . ‘ — >0
~(3Lv‘1{le)¥39(,’n)>0 e) 3L+9x+3>1 \1
(—5)le thug) 3L9\ . 50‘ ‘0“1 Wk Mk 7&6 minimum (7% g scams a} 03‘ng . ‘ END OF EXAM 1 a.
ﬂie NAM CW UWA WMCJ _g r; m“ 411(4) ‘30) : \4 8+3=YL, Page 6 Of 6 ...
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 Spring '11
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