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Unformatted text preview: Math 2263 Name (Print) Fall 2005
FINAL EXAM Recitation Instructor READ AND FOLLOW THESE INSTRUCTIONS This booklet contains 15 pages including this cover page. Check to see if any are missing.
PRINT on the upper righthand corner all the requested information, and sign your name.
Put your initials on the. top ofﬂeverylpage, in case the pages become separatedsf' 7* S t ~ 1: Do your work in the blank spaces an
pages of this booklet. Show all your work. This is a closed book exam. There are 12 multiple choice problems each worth 15 points, for a total of 180 points and
a handgraded part with 8 problems for a total score of 220 points. You have 3 hours to work on this exam. INSTRUCTIONS FOR MACHINEGRADED PART (Questions 112): 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 1320):
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. Signature Section___.__ I.D.# w .55.“. Multiple choice part Handgraded part __ Problem Total _ Letter Grade Subtotal w Baertr Math 2263 Name:
Fall 2005
Page 2 Multiple Choice Part 2. Let f(x,y) = me”. Then 687:9%(0,0) is Math 2263 Name: Fall 2005
Page 3 3. An equation for the tangent plane to the surface 11:22 — 23/3 = —l at the point (1, 1, 1) is 4. The function f(x,y) = :02 + y3 + 6(33 + yz) — 15y has, at the point (—3, —5),
A) A local minimum B A local maximum D A critical point Where the second derivative test is inconclusive ( ( > (C) A saddle point ( ) (E) No critical point of any type. Math 2263 Name:
Fall 2005
Page 4 5. f02 Oar/2 f (x, y)dyda: is, for every continuous function f (ac, y),
(A) 5/2 f: f(rv, y>dxdy (B) fez 0W2 f (w,y)dxdy (C) fol f2: “9”: y)d33dy (D) f3 [21 for, y)dzdy (E) If Om/2f<x,y>dxdy 6. The domain of integration for the integral ﬂ m m
/ f / f<x,y,z>dzdydx
—\/§ 42—35 ac 2+y2
is
A) The solid in R3 bounded below by a paraboloid and above by a sphere B The solid in R3 bounded below by a cone and above by a sphere. ( ( ) (C) The solid in R3 bounded below by a parabolid and above by a cone.
( D) The solid in R3 bounded below by the (x, y)—plane, on the sides by a cylinder and
above by a cone. (E) The solid in R3 bounded below by a cone and above by a paraboloid. Math 2263 Name: Fall 2005
Page 5 7. The volume of the solid in R3 consisting of points (m,y, z) lying inside the cylinder
:02 + y2 = 9 and outside the cone (z — 3)2 = x2 + 3/2 is 8, The vector ﬁeld 13(53, y) = (6azy + 453/2 + 102:2)“— (12:132 + 4mg — 9%); is conservative
when (A)a=1,ﬁ=0
(BM—5%:
(C)a=—4,,8=l/2 Mathe 2263 Name:
Fall 2005
Page 6 9. Evaluate fan dm — .732y2dy Where C is the triangle With vertices (0, 0), (1,1),(0,l) and
C' is oriented in the counterclockwise direction 10. The surface area of the portion of the sphere 3:2 + y2 + Z2 = 25 that lies above the
triangle with vertices (0, O), (3, 0), (0,2) is 3 —23 2 $2 1
(AND 0‘ / )“ «Wade: (13) f5” 0“”3)“2<25 ~ 9:2 — y2>3/2dydx 3 2
(C) f0 f0 ﬁdydﬂc 3 —23:€2
(D)f0(/)+ 5 0 25—m2—y2 2 (—3/2)y+3 25
(E) f0 0 mdmdy Math 2263 Name:
Fall 2005
Page 7 11. Let Fm, y, z) = x2yi~+ (—yz)5+ 2721;. Then Guru—7“ is
(A) 2333/5— 25+ x]; (B) —y{+ 25+ 1:21; (C) —y;+ zj— 2:2];
D) y?— z; 332]; (
(E) y5+ zj+ :32]; 12. Let E be the solid cone bounded below by z = \/$2 +y2 and above by the plane
2 = 2. Let 13(a:,y,2) = 1:274 343+ 21?. Then divF dV is
E A) )
(C) §
3
877' (
( Dd 167r
(D) 13677
(E) 0 Math 2263 Name:
Fall 2005
Page 8 Handgraded part. 13.(25 pts) Find the length of the parametrized curve FOE) = 1575+ @t3/25+ g9]; from
(0,0,0) to (15, [email protected]/2). Math 2263 Name: Fall 2005
Page 9 14.(25 pts) If 2 = x2eV $2+y2 and :1: = rcos 6, y = rsin6’, ﬁnd 3% and %. Math 2263 Name: Fall 2005
Page 10 l5.(30 pts) Using Lagrange multipliers ﬁnd the point on the line Where the planes
2: + y + z = 6 and 2m — 33/ + z = 0 meet which is closest to the origin. Math 2263 Name: Fall 2005
Page 11 16.(25 pts) Evaluate the double integral ff WdA Where D is the region
D {(xay)74 S 9:2 +312 S 9} Math 2263 Name:
Fall 2005
Page 12 l7.(30 pts) Evaluate the integral fff eV z2+92+z2 dV Where D is the solid in the ﬁrst octant
D determined by 4 _<_ x2 + y2 + 22 S 9. Math 2263 Name:
Fall 2005
Page 13 18.(30 pts) Using the change of variables x = “J2”, y = u — v evaluate the double integral 213]
f/e m+y dA
D Where D is the trapezoid in the ﬁrst quadrant with vertices (2,0), (4,0), (0,4), (0,8). Math 2263 N amezh—hhm
Fall 2005 Page 14 19.(25 pts) Evaluate the surface integral ff xz2dS Where S is the portion of the cone
S 2 = «$2 + y2 between the planes 2 = 2 and z = 4. ...
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