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**Unformatted text preview: **Math 2263 Name (Print).____
Spring 2009
FINAL EXAM Signature Recitation InstructormSection.—_ 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 right—hand corner all the requested information, and sign your name.
Put your initials on the top of every page, in case the pages become separated. TURN
OFF CELL PHONES! Scientiﬁc calculators allowed, but not graphing calculators. One crib sheet 8.5” x 11” is allowed, written in your own handwriting and only on one side. NO OTHER TEXTBOOKS 0R NOTES ARE PERMITTED! Do your work in the blank
spaces and back of pages of this booklet. Show all your work. There are 10 multiple choice problems each worth 15 points, and a hand—graded part with 6 problems counting 25 points each, for a total score of 300 points. You have 3 hours to
work on this exam. INSTRUCTIONS FOR MACHINE—GRADED PART (Questions 1-10): You MUST use a soft pencil (N0. 1 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 HAND—GRADED PART (Questions 11-16):
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 . _ Hand—graded part Total Letter Grade 16
-l Name:__________ Multiple Choice Part 1. An equation for the tangent plane to the surface 9:2 — a: + 2.742 + 82 + z : 3 at the point
(1,1,0) is A.z:a:+4y—5
B.:c+4y+2z:5
C.x+4y—22=5
D.a:+4y+2z=0
E. (233—1)(.r —-1)+4y(y—1)+ (ez+1)z:0 N ame :___._.___-_______..._,___. 2.1fu = 3:4 + 23/3 and :10 : e“ and y = 654 then 28% at s = 2, t: 1 equals
A. 68 + 283
B. 866 — 663
C. 4e8 — 683
D. 868 — 663 E. 888 + 663 Name: 3. Let f(:c, y) : x2 ‘— 2y3 —— ny — 18y + 23: + 14. 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 minimum at (—7, —2), and no other
critical points. . D. f has a local maximum at (—4, —1) and a local minimum at (—7, —2), and no other
critical points. E. f has a saddle point at (—4, —1) and a local maximum at (—7, ~2), and no other
critical points. 4. Evaluate the integral A. ——2 cos 256 B. 2(1— cos 256)
05 256 (cos 256 ~ 1)
(1 — cos 256) (‘3 m U O
NH NH NH Name: 5. Let D be the circular region {:32 + y2 :<= 430}. Then, in terms of polar coordinates, ff (9:2 + y2)5/2d93 dy
D equals 1r/2 4
A./ / rsdr d9
—-7r/2 0
7r/2 4
B. / / r6d7‘ d6
—7r/2 O
271' 4
0/ / redr d6
0 0
27r 4cos€
D/ / red?" d9
0 O
7r/2 4cosO
E/ / rsdr d6
—1r/2 O N aine:_______n___.______.__.________ 6. Let C be the closed curve in R2 consisting of the line segment on the alt—axis joining
(2,0) to (—2, O), and of the semicircle a32+y2 : 4, (y < 0) (also joining (2,0) and (—2, 0)),
oriented counterclockwise. Then /(a:2 + 8y)d:c + (39: + 43/ + m2)dy
0 equals
A. —107r
Bl, 107r
C. —207r
D. 22W E. none of the above. N ame :_____..___._________ 7. Find a (1) Whose gradient is the vector ﬁeld < 243:2 — 5y2 ‘— 63:22, —10my+ 8Z3, —6:z:2z +
243122 >. A. < 83:3 —— 5$y2 —— 3x222, ~5xy2 + 83,123, —3a:222 + 83/23 >
B. 812:3 — 10513312 — 630222 + 16yz3 C. < 481:.— 622, ——10:c, ~6$2 + 48yz > D. 3830 —— 63:2 — 622 + 48yz E. 81:3 — 5my2 —— 333222 + 83/23. N ame :_____._______.___~____._.____. 8. Let C be the curve in R2 which goes from the point (9,3) to (1, 1) along the parabola
y = ﬂ and then continues on to (0, 0) along the line y 2 ac. Then the integral /C(y2 ‘+ 1)d:c + (23:3; — 2)dy equals A. 84
B. ——84
C. 165
D. ~165
E. 0 N ame:,___,_ 9. The domain of integration for the integral 2 «AT—”F? 8—m2—y2
my, z>dz dy dzc
£21m 2x/x2+y2 is A. a subset of R3 bounded from below by the any—plane, on the sides by a cylinder,
and from above by a paraboloid. B. a subset of R3 bounded from below by a cone and from above by a paraboloid.
C. a subset of R3 bounded from below by a cone and from above by a sphere.
D. a subset of R3 bounded from below by a cone and from above by a hyperboloid. E. a subset of R3 bounded from below by a sphere and from above by a paraboloid. 1V ame :________..____________~i__.w 10. The surface area of the part of the surface 2 = my situated inside the cylinder a32+y2 = 8
equals A. 471' N a1ne:___________.____~
Hand—Graded Part 11.(25 pts) The temperature at a point (:c,y, z) is given by
T(:c,y,z) : Stacy + yez + 26"". (a) Find the rate of change of the temperature at the point P(0, 0,0) in the direction
toward the point (5,1,——2). (13) Starting at the point P, in which direction does the temperature increase the
fastest? (Please give answer as a unit vector.) (c) What is the maximum rate of increase at P? lV' ame Z__—______._.._______.__.______._.__ 1.2.(25 pts) Find the minimum and maximum values of the function f (a), y, z) : myz subject
to the constraint x2 -+ 3y2 + 422 = 12. Find a point Where the maximum occurs, and a point where the minimum occurs. N ame:____________._.__ 13.(25 pts) Let —v 147:1}, y, z) : (sinz + 33y2);+ mgeBZj—F (c083 CE + mzzﬂc. Let T be the surface bounding the region R given by 1:2 +y2 § 2 § 6 — 51:2 + yz, oriented
outward. Use the divergence theorem to evaluate the ﬂux f/Fds
T N ame :.____ 14425 pts) (a) Use spherical coordinates to evaluate the triple integral f/fg COSZW + 92 + z2)3/2l dV, Where E is the region enclosed by the sphere x2 + y2 + 2:2 = 4 in the ﬁrst octant
{51:30, y:>=0, 2&0}, (b) What is the answer if x2 + y2 + 22 is replaced by 9:2 + 53,12 + 322 in both the integral
and the region? N ame :._____.____.____.________._ 15.(25 pts) Consider the portion P of the sphere 3:2 +y2 +22 2 1 located in the ﬁrst octant
{:5 ; O,y ; 0, z 3 0}. Calculate the work done by the force ﬁeld ﬁ($, y, z) = memii—l— (x4 + yeyﬁ-l— (2y + zsin3 )1; as a particle moves under the ﬁeld’s inﬂuence along the edge of P once in a counterclockwise
direction (as seen from very high above the asy~plane). N ame:__;_____.___.___________.___ 16.(25 pts) Let T be the portion of the surface 302 = y2 +22 lying between the planes at z: 0
and cc 2 2 and above the plane 2 : 0. Calculate the surface integral [/TQ + $2y2)dS (z'.e. the mass of surface T if its density is 2+ 932312). ...

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- Fall '08
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- Math, Multivariable Calculus