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Unformatted text preview: Math 2263 Name (Print) Fall 2004
FINAL EXAM Signature___._____________.___—'——__ Recitation Instructor__+______Section___ I.D.# READ AND FOLLOW THESE INSTRUCTIONS
This booklet contains 17 pages including this cover page. Check to see if any are missing. PRINT on the upper righthand corner all the requested information, and signiyour name.
P t ' 'ti Won the to of every page, cinrcase the pages become separated. bc’) 1,... ‘ Do your work in the blank spaces an
of pages of this booklet. Show all your work.
There are 10 multiple choice problems each worth 15 points and 6 handgraded problems.
each worth 25 points, for a total score of 300 points.
INSTRUCTIONS FOR MACHINEGRADED PART (Questions 110):
You MUST use a soft pencil (No. l or No 2) to answer this part. Do not fold or tear the answer sheet, and carefully enter all the requested information according to the instruc—
tions you receive. DO NOT MAKE ANY STRAY MARKS ON THE ANSWER SHEET. When you have decided on a correct answer to a given question, circle the answer
in this booklet and blacken completely the corresponding circle in the answer sheet. If you
erase something, do so completely. Each question has a correct answer. If you give two
different answers, the question will be marked wrong. There is no penalty for guessing,
but if you don’t answer a question, skip the corresponding line in the answer sheet. Go on to the next question. INSTRUCTIONS FOR THE'HANDGRADED PART (Questions 1116):
SHOW ALL WORK. Unsupported answers will receive little credit. Notice regarding the machine graded sections of this exam: Either the student or the
School of Mathematics may for any reason request a regrade of the machine graded part.
All regrades will be based on responses in the test booklet, and not on the machine graded
response sheet. Any problem for which the answer is not indicated in the test booklet,
or which has no relevant accompanying calculations will be marked wrong on the regrade. Therefore work and answers must be clearly shown on the test booklet. AFTER YOU FINISH BOTH PARTS OF THE EXAM; Place the answer sheet
between two pages of this booklet (make a sandwich), with the side marked “GENERAL PURPOSE ANSWER SHEET” facing DOWN. Have your ID card in your hand when
turning in your exam.
Multiple choice part Handgraded part Total Letter Grade _______ Multiple Choice Part Name: 1. An equation for the tangent plane to the surface 2:22 — 2y3 + 1 = 0 at the point (1,1,1)
is (A) 22(z—l)—6y2(y—1)+2:I:z(z—1) =0
(B)z1=(m—1)—6(y—1>
(C)x—6y+22=—1
(D)x—6y+2z=——3 (E) (m—1)+6(y—1)+2(z—1)=0 Math 2263 Name:
Fall 2004 2. Let the curve F(t) be given by F(t) = (et + e" + 1);+ («'3t  e‘1t + 3).;+ (t2 — 4t + 5)];
A unit tangent vector at the point corresponding to t = 1 is (A) (e — 33+ (6 +931 21‘; (B) (e+§+1)?+(e—%+3)5‘+2E
(C) mm? “ éﬁ‘i‘ (6 + 53"— 2’3]
(D) mm + 55+ (e — bi" 213] (E) ﬁne  %>?+ (e + a: 21?] Math 2263 Name: Fall 2004 3. Ifu=as3+y3 andzrzets+1andy=et+s+1then % ats=0, t=1equals
(A) 8 + (e + 1)3 (B) 3e(e + 1)2 (C) 6 + 3(e + l)2 (D) 12 + 3e(e + 1)2 (E) 126 + 3e(e + 1)2 Math 2263 Name: Fall 2004 4. Let f(a:, y) = x2 + 23/3 — 6:1:y+ 18y — 2.7: + 11. Then (A) f has saddle points at both («2, —1) and (—5, —2) and no other critical points (B) f has a local minimum at (4, 1) and a local maximum at (7, 2) and no other critical
points (C) f has a. saddle point at (4, 1) and a local maximum at (7, 2) and no other critical points (D) f has a saddle point at (4,1) and a local minimum at (7,2), and no other critical
points (E). f has a local maximum at (4, 1) and a local minimum at (7, 2), and no other critical
points Math 2263 ‘ Name:
Fall 2004 5. Changing the order of integration in the iterated integral 2 3511
/ f f (x, y)dydx
0 $ leads to (A) In5 fig—:1 f (95, y)da:dy
(B) f01 02” f(«% y)dxdy + If Mjm, y)dmdy
(c) f: 2,, 5‘1 ﬂx, wdzdy
(D) 101 1023’ f(a:, y)dzdy + If fag: m; y>dzdy (E) fol 0”” f (at, y)dwdy + ff 07m f (w, y)d$dy Math 2263 Name: Fall 2004 6. Let C be the closed curve in R2 consisting of the line segment from (—4, 0) to (4, 0),
and the semiCircle $2 + y2 = 16 (y > O). Orient C counterclockwise. Then the integral / (3:132 + 7y)d:z: + (49: + 4y)dy
0 equals
(A) 88%
(B). —247r
(C) 247r
(D) 327r
(E) 0 Math 2263 Name:
Fall 2004 7. Let D be the circular region {:122 + y2 .5— 4y}. Then, in terms of polar coordinates, / [D Wadi,
equals
(A) f0" fo“ rdrde
(B) f0” [04r2drde
(C) f02"f:r2drd9
(D) f027r fgsmerzdrdﬁ (E) f0" [fine errde Mathe 2263 Name: Fall 2004 ‘ 8. Let C' be the path which goes from (—2,0) to (2,0) along the semicircular arc y =
—\/4 — 292, then to (—2, 2) along the line at + 2y = 2. Then /(2xy + 3)dx + ($2 +1)dy
0 equals
(A) 18
(B), 10
(C) ~10
(D) —18
(E) 0 Math 2263 Name: Fall 2004
9. A potential for the vector ﬁeld 17“ =< 62:312 + 52:2 — 6:122, 6:323; — 62312, 10x2 — 23/3 >
is
(A) < 3.1323;2 + 522:1: — 22:3, 3932312 — 22313, 51:22 — 223/3 >
(B) 2:I:3 + 103m2 + 6393/2 — 423/3
(C) < 6y2 — 12x, 61:2 — 12zy, 10m >
(D) 6:122 + 63/2  2m — 12yz
(E) 3.11323;2 + 5x22 — 211:3 — 2y3z Math 2263 Name:
Fall 2004 10. Consider the region bounded by the lines y = 1 and y = 3, the hyperbola xy = 3, and
the y—axis. The centroid of this region (i.e., the centerofmass of this region for density
= 1) is the point (A) (£125,513)
(B) (1:13, F23)
(C) (1n3, %1n3)
(D) (2,2) 03) ($2) Name:
11. Suppose that over a certain region of space the electrical ﬁeld potential is given by ¢(:r, y, z) = x2 + 32:31 + xyzz. a) Find the rate of change of the potential at the point P(1, 1, —1) in the direction
toward the point Q(2, 2, ——3). b) Starting at P, in which direction does ¢ increase most rapidly? (Please give answer
as a unit vector.) c) What is the maximum rate of increase at P ? ' (25 pts) Name: 12. Find the point or points on the surface
22 = my + 32 which lie closest to the origin. (25 pts)  Name: 13. Calculate the outward ﬂux of the vector ﬁeld ?($, 3/, z) = (:1:2 + y2 + + y? — 42?) across S, the surface of the solid bounded by z = 0 and the paraboloid z = 4 — x2 — y2.
(25 pts) Name: 14. A solid metal sphere 2:2 +;z/2+z2 5 a2 has a conical piece in the shape of z = — V22 + y2
drilled out of it. If the metal has density 6(z, y, z) = 1+z2, ﬁnd the mass of the remaining
solid. (25 pts) Name: 15. Let T be the portion of the cylindrical surface 2:2 + 22 = 16 lying between y = 0 and
y = 2.(The planes are not; included as part of the surface.) Assume that the outside of
T is plated With a goldlike material Whose density is 1 +3122 per unit surface area. Find
the total mass of the the plating. (25 pts) Name: \ 16. Calculate the work done by the force ﬁeld ?(z, y, z) =< xcosz, 5zy3 + ysz'ny, zy2 + .262 > ...
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