Final_Fall-2003-2004_Makdisi

Final_Fall-2003-2004_Makdisi - ‘ M M (o ; RT’ICK...

Info iconThis preview shows pages 1–14. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 14
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ‘ M M (o ; RT’ICK ANSWERS, NO JUSTIFICATION EQUIRED, NO PARTIAL ORE-DI 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 T-9Yi§ ) 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. Note-l blank 2 1 part}. In each of the pictures below, we give a 3-dimensional picture and a cross-section 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 Jog-r HQLF-SALL (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‘j-fidaz/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 him-Juan l swims” n in. itilmARY :H' azFJIK-U'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: ($(t),y(t):z(t)) ={ , ' cost, 81-1115}: 0 < t < 2,7. -——__.._... IPART II. FULL'SOLUTIONS REQUIRED, PARTIAL CREDIT AVAILABLE. 9 (10 pts}. 93) Find the second degree Taylor approximation, centered at a. = ‘2, to the function fix] 2 lnz. (Your answer should have the form Pfiz] = c0+cl($—~ 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! {mu-1‘ ‘ ' '-. r : I! ‘ 3 '1, '1 in; ' “ '. i , i U:— " g 1 1“..--a»-- {O \ - 12 (jig-13:5}. 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 ...
View Full Document

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.

Page1 / 14

Final_Fall-2003-2004_Makdisi - ‘ M M (o ; RT’ICK...

This preview shows document pages 1 - 14. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online