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Unformatted text preview: ‘ M M (o ; RT’ICK ANSWERS, NO JUSTIFICATION EQUIRED, NO PARTIAL OREDI l (2 rips/part, total 10 pts). Which of the following series converge and which diverge?
Circle your answer.
00 211+?
la) 2 Converges Diverges Converges Diverges
Converges Diver ges
Converges Diver ges Converges Diverges n! + 1)}
(272)! 11. 2 (Zpts). F ill in the blank: the radius of convergence of the power series Z
71:0
is R 2 3 (lpt/part: total 4pts. Note 1 blank = 1 part}. Fill in the blanks for the ﬁrst few
coefﬁcients in the Maclaurin series for the following integral: ' 2 sin I
/ dt = :
i=0 t 3 each for parts a—d? apts each for parts e—f, total lopts). The following picture
snows level curves for a function f (33: y}, and the value of the gradient \7f at some points. __JDATA‘ & I ,gSD i
ﬁx rgma) 4 '
; ae.
P2 P3 '3
. \ Q
\ an A PA. =9. 4a) Draw a vector on the figure at P1 pointing in the direction of maximum increase
of f. I ( 3 Circle the correct answer for 4b, 4c, 4d: 8f . i . . . . .
4b) —— is posrtive zero negative does not exist. 62:5 1392 . 6f . ' . . ‘ .  .
4c) 5— I 15 posrtive zero negative does not eXist.
4d) lim ﬁx, 3;} is positive zero negative does not exist. 4e) Fill in the blank: The equation of the tangent line to the level curve passing
through,}2(7,3)is ' 1 RE
(aylﬁ 41“ Fill in the blank: We start horn the oint P5 5 0 . and move a distance of d3 =
1' 7 . units in the direction of the vector \7 2 i+ Qj. Then the change in the value of f
approximately * ‘ _ , , .L . — 1 .  . a UL; psi/part, total 10.5pts. Note 1 blam: = 1 part). Consnder the snaded region R
below [R is common to the mm circle: q‘nnwn and lip: n‘hmm Hm T9Yi§ ) Fill in the blanks
for the following integrals in rectangular and polar coordinates: mg {same] (13? d9. I 6 (0.5 pita"part, total 9.5 pts. Notel blank 2 1 part}. In each of the pictures below, we
give a 3dimensional picture and a crosssection of a region D (which is always part of the
half—ball 2:2 f— 3/2 + 22 S l, 2 2 O). In each case, ﬁll in the blanks for integration in spherical
coordinates: 7 6a) MW: fz—j—M f—w Wig. 6b) 1W :l/ f / [sameldpdgodﬂ
9: (15': p: HOLLOWED Jogr HQLFSALL (wLINDER 01:
RADJUS .4 Ramon/go) 6C) l dV 2 / f / [some] {13p dgp d9.
6: ' .99: p: r I f (2' tits/part, total 8 pts. Now 1 blank = 1 part). The following picture shows two
curves Cl and C2 inthe plane. D; is the region inside C; . D2 is the region that is inside
Cg AN D outside Cl. We go around each of Cl and Cg counterclockwise, and the normal veCtor points outwards of 01 and Cg in each case, as drawn on the ﬁgure. W“ rang}; ﬁ "'T. ~ ‘I .
I‘yﬂr‘ﬂ  ‘. _{\ “
J I " i it .xr'ﬂ“
A ' _' _‘ .I ;\;’Mr
State Green’s theorem for F : xi + xyj = (as, 333;): L" Cl
D1 7 ,3  2 ".~ ———— i“ .
{bl //—————dA (+0r—T} le nds+ (+0r“?) fcn Dds U! af., ‘c tiptg‘part, 4 pts total}. Recall that Stokes: Theorem says: f/(curlf‘jﬁdaz/F‘odf,
_ IC
5 where the curve C is the boundary of the surface S, and we go around C in the direCLion
compatible 'with 13., according to the right hand rule. Below is a picture where S is the
part of the sphere of radius 2 (r2 —: 3:2 + 22 = 4) with x g the normal vector fi points
away from the origin. C is the circle which is the boundary of S. l..,..s_.um——w——— “:7 himJuan l swims” n in. itilmARY
:H' azFJIKU'f 8a) 011 the figure, indicate the direction that one has to go around C for Stokes’
Theorem. 8b) Parametrize C in the orientation that you have described. You need to enter a
constant number in each of the blanks below: ($(t),y(t):z(t)) ={ , ' cost, 811115}: 0 < t < 2,7. ——__.._... IPART II. FULL'SOLUTIONS REQUIRED, PARTIAL CREDIT AVAILABLE. 9 (10 pts}. 93) Find the second degree Taylor approximation, centered at a. = ‘2, to the
function ﬁx] 2 lnz. (Your answer should have the form Pﬁz] = c0+cl($—~ 2)+c;($—2)2 for appropriate co, (:1, C2.)
9b) Use the Remainder Theoremrfor Taylor series to show that: if1.9 g x g 2.1, then lln(r) — Pgml <
f ‘ i ‘— Useful information so you can avoid doing long calculations by hand :
L8 < (1.9)2 < 4,. 6 < (1.9)3 < 7, 4 < (2.1)2 < 5, 9 < (2.1)3 <10. 10 (’10 pus}. Evaluate the {UHOWing limit: “Brim.
Ll I11: p73) We are given a
V1" : 5i + 4j, [(7,3) Assume that :c 2 55(3, t) = 52 —i+j. ' l.(0,4) t2 and y = y(s.,t) 2 5t: —— 4. Find the value of ' as ' _ 'H.».—F“"" _ r , i . ‘ ‘ . :‘z ‘ ' 9!
{mu1‘ ‘ ' '. r : I! ‘ 3 '1, '1 in; ' “ '.
i ,
i U:— " g
1 1“..a» {O \  12 (jig13:5}. We are given the function ﬁzz, y, z; = :2 — 22 +Iy, and we restrict (or, y, z)»
to lie in the solid cylinder D : 2:2 + 3'2 g 2 (and z arbitrary). At, what poinﬁs) of D
does f attain its minimum value? ' (Hints: (i) remember to Check the interior and the boundary of D: (ii) remember that.
we are in three dimensions: (iii) do not try to do the second derivative test — we didn‘t cover it in class for three dimensions!) a.« ' . NISQEE) Let R be the region in the ﬁrst quadrant underneath the parabola y = 3—3552.
Pint? the average value of f(2:, y} 2 23 on the region R. I Let D be the region in the ﬁrst octant cut out by the planes y + z = 1 and :r: = 2.
(See the ﬁgure.) The density of D is given by 6(55, yrz) = 3:2. Find the total mass of D. 12. 4 15;?8 nts). Given the two vector ﬁelds in the plane: 1*: = 2:3: (0,33), G : (2333; + 1)i+ (3:2 —:— 59);: (2mg +17$2 ; 61’).
15a) For each of F and (‘3‘: either Show that the vector ﬁeld is. not conservative; or Show that the ﬁeld is Conservative by ﬁnding a potential function.
15b) Let C be the curve in the plane parametrized by (EU): W» = W) = (m2), Find the work integrals / 1:“  (if; / é . (if.
C’ C 15 $1? pts}. Let D be the solid cone with side z = ""2 i .t —.— 1/2, top 2 2 l1 and bottom
2 r: . Let 5 be the surface of D (S has two parts: a ﬂat circular lid 5'1 and the conical
side 52). The normal vector on S points outwards as shown in the ﬁgure. We are also
given the vector ﬁeld ' E = zzi= (1:2, 0, 163) Explain Why // E  13 do: 0. (This can be done Without detailed calculations.)
5'1 16b) Find  n do by directly computing the ﬂux integral. 52
16c) (Recall that the divergence theorem tells us that sag: ///(divf‘)dv. D
Here if 15 = Mi+ .vj'+ P12, then div 15 = M1 + Ny + P2.) Find div 1*: and integrate / f / (div E) dV directly, in order to verify that you get the
17 same result as the sum of the answers in parts a and b above. PLEASE START YOUR SOLUTION ON THE FOLLOWING PAGE ...
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This note was uploaded on 03/01/2010 for the course MATH 201 taught by Professor Variousteachers during the Spring '10 term at American University of Beirut.
 Spring '10
 VariousTeachers
 Calculus

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