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Unformatted text preview: TEST NUMBER 1 MA 261 FINAL EXAM December 10, 2007 Name:
Student ID Number:
Lecturer: _Recitation Instructor: Instructions:
1. This exam contains 22 problems worth 9 points each. 2. Please supply aﬂ information requested above and on the marksense sheet. Be sure
to ﬁll in the “test number” from the top of the page as well. 3. Work only in the space provided, or on the backside of the pages. Mark your answers
clearly on the scantron. Also circle your choice for each problem in this booklet. 4. No books, notes, or calculator, please. ao/lad Clo/C E ll/eyl beca
a bad L1,..." Name: 1. The curve r1(t) = (t2 — t, t3 — t, t) and the line r2(t) = (0,0,1) + t(2, 1, —1) both
intersect at the point (0,0,1). The acute angle 6 between the curve and the line at (0,0, 1) is A. 0:%
B. 0:;
(10:2
D.0=g
E. c9=lcos_1 2. If the acceleration of an object is a = (3t2 + 2t)i + 3 j and its initial velocity is v(0) = i +j, what is its speed at t = 1 ? A.\/§
B.\/ﬁ C. NE
D. 10
E. 5 Name: 3. The directional derivative of f (11;, y) = 1:2 —~ 3/4 + 1 at (1, 1) in the direction from (1, 1]
to (2,0) is 2
A. —
\/§
4
B. —
\/§
6
C. —
\/§
8
D. —
x/i
E. m
4. Let w = f(u,v) where g = 27162“ and 2—7]: = 62“. If w(:c,y) = f(sc2 — yz, 3y),
then at the point (170,310) = (17 1), we have (2—1” =
y
A. —9
B. —7
C. 0
D. —62 Name: 5. An equation of the tangent plane to‘the surface 2932 + y2 + 262 = 73 at the point
(1, —1,0) is
A. 2: —4x+2y+6
B. 2: —4x—2y+6
C. 2 = 42': + 2y — 6
D. 2 =.—4x—2y—6 E.z=4x—y+6 6. If the function u is deﬁned implicitly as a. function of a: and y by the equation $3y+xu2 =4sinu+8, A, _ M
23m — 451nu )
A) 23m — 4 cosu C. ( $3y+2u ) D. _ 3x2y +721,2
23m — 4 cos u E. _ (3x2y+u2)
25w Aunt"ax. Name: 7. Given that the function g has continuous ﬁrst and second partials and given the table
of values below, classify the'critical points (1,1) and (3,0) of g: I
In“
Inn“‘ A. A local minimum at (1, 1) and a saddle point at (3,0)
B. A local maximum at (1,1) and a saddle point at (3,0)
C. A local maximum at (1,1) and a local minimum at (3,0) D. A local minimum at (1, 1) and a local maximum at (3,0) E. A saddle point at (1, 1) and a local minimum at (3,0) 8. Find the minimum value of on the surface iyz = 4. Name: f(x,y,z) = 42:2 +1;2 +22 C. 10 nan1m”;uu.m~n Name: 9. Let E be the solid that lies within the cylinder 9:2 + y2 = 1, above the plane 2 = 0,
and below the paraboloid z = 4m2 + 4y2. Express ///Exzdv as an iterated integral in cylindrical coordinates. 21r l 472
A. f f / r3 cos2 6dzdrd0
0 0 0
21r 4 21'
B. / / / r3 cos2 0dzdrd0
' 0 0 0
21r 1 472
C. / f / r2 cos2 6dz‘drd6
' 0 o .0
21r 2 41'
D. f / f r3 cos2 6dzdrd6
0 0 o
21r 1 4
E. / / / r3 cos2 dedrdﬁ
0 0 0 10. Find the values a, b which give the correct change in order of integration /59 [Zﬂx’wdxdy = f(~’v, y)dyda: A. a=0 b=\/9—:1r:2 B. a=0 b=9:I:2 Name: _ 11. Find the volume of the solid below the plane 2 = a: and above the triangle with vertices
(0,0,0) (0,3,0) and (3,3,0). A. 1/2
B. 2/3
C. 14/3
D. 3/2 E. 9/2 12. Find the surface area of the portion of the paraboloid
z = 16 ~ 3:2 — y2
that lies above the disk :32 + y2 g 2. A. 7r/2
B. 77r/6
C. 137r/3 ' D. 27r/3 Name: 13. A lamina in the shape of the rectangle 0 S x S 77/ 2, 0 S y S 1, has density f (x, y) = w cos($y). Find the mass of the plate. 'D. 37r~ 5 E. 7r/4' 14. A plane contains the points P1(0, 1,0), P2(0, 3,3), P3(1, 1,4). One vector normal to the plane is: Ai—%+k
B.&+m—%
C.%+Bk
D.i+ﬁ+ﬂk E.n—m+% 15. Determine a so that the line Name: x=at+3
y=2t—5
z=4t—1 is parallel to the plane 23: + By — 5z = 14. 16. In spherical coordinates, the surface is a: p2(1 — cos2 ([5) = 16 A. a = —4
B. a = 3 C. a = 7
D. a = ——5
E. a = —14
A. half cone
B. plane '
C. sphere D. cylinder E. hyperboloid Name:
17. The traces of the surface z2 + 1 = x2 — y2 in the planes z = k are:
A. hyperbolas and lines
B. parabolas only
C. hyperbolas only
D. parabolas and lines
E. hyperbolas and the point (0,0) only 18. The arclength of the curve given by r(t) = t, e‘, e“) for 0 S t g 2 is: 19. Let C be the curve y z; 20. Let C be the segment from (0,2) to (2, 5). Compute fog Name: $3, 0 S x S 1. Compute / 1
~— mydm — zdy C 12yds. 8(23/2 — 1)
12(23/2 — 1)
2 g (23/2 — 1)
3 2 (23/2 — 1)
2(23/2 — 1)
2 E 2 5 2 4 3 Name: 21. Let C be the Closed curve pictured that is the boundary of the region in the ﬁrst
quadrant satisfying 1 g 1:2 + y2 S 4. Compute / (a: — x2y)d:z: + my2dy
C y A. 7%
° B. 15% ° ” c. a;
D. 1177 E; 7: 14 lultﬂrglr1.vamvmn1uhm .mxtn—b—I.” 7 e . .. . . ‘ .  r  ... Name: 22. Let S be the surface described by r(s, t) = (25—75, 3+t, 5—75) for 0 _<_ s S 2, 0 S t S 1. Compute
/ / (y + z)dS
S A. 2m
B. rm E. 4v 14 ...
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 Spring '08
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