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Unformatted text preview: Math 201 — Fall 2004—05
Calculus and Analytic Geometry III, sections 5—8
Quiz 2, December 2 —— Duration: 1 hour GRADES (each problem is worth 12 points): TOTAL/72 YOUR NAME: C) Meet.) 3“, QM}:\: YOUR AUB ID#: ‘PLEASE CIRCLE YOUR SECTION: Section 5 Section 6 Section 7' Section 8
Recitation M 1 Recitation Tu 12:30 Recitation Tu 2 Recitation Tu 3:30
Professor Makdisi Mr. Khatchadourian Mr. Khatchadourian Mr. Khatchadourian INSTRUCTIONS: 1. Write your NAME and AUB ID number, and circle your SECTION above. 2. Solve the problems inside the booklet. Explain your steps precisely and
clearly to ensure full credit. Partial solutions will receive partial credit.
Each problem is worth '12 points. 3. You may use the back of each page for scratchwork OR for solutions. There
are three extra blank sheets at the end, for extra scratchwork or solutions. If you need to continue a solution on another page, INDICATE CLEARLY
WHERE THE GRADER SHOULD CONTINUE READING. 4. No calculators, books, or notes allowed. Turn OFF and put away any cell
phones. GOOD LUCK! An overview of the exam problems. Each problem is worth 12 points.
Take a minute to look at all the questions, THEN
solve each problem on its corresponding page INSIDE the booklet. 1. Some multiplechoice questions on the next page: (i) relate surfaces and their equations,
(ii) approximate the values of a function, (iii) questions about level curves. Note that there
are SIX parts lawlf, with 2 points for each part. 2. Using polar coordinates, sketch and clearly label the curves 01,02 given by
Clzrmcosﬂ, Cg:r=1—cost9 and ﬁnd the area of the region that is inside Cl and outside 02. 2
3. Consider the function f(:c, y, z) = yex z and the point P0(—1,2, 1). a) Find the gradient 6 f P .
O
2
‘0) Find the equation of the tangent plane to the surface yex Z = 28 at the point P0. 0) Find the directional derivative of f" at P0 in the direction of the vector if = (1,2, 3) 4. Given the parametrized curve in space: P(t) = (:1:(t), y(t), z(t)) 2 (et, 94, \/§ ' t). a) Find the arclength of the part of the curve between the points P1 2 (1, 1,0)
(corresponding to 751 = 0) and P2 = (e, 1 / 6, ﬂ) (corresponding to 252 z 1). Note: you can,
and should, simplify the expression inside the integral to get rid of the square root. b) (UNRELATED to part (a)) Use the same curve P(t) as above. Given a function f(P) = f(:t,y, z) with 6f 2 (1,6,4), ﬁnd the derivative %[f (P(t))] (3,1/3,\/§ 1113) i=1n3' 5. a) Find the maximum and minimum values of the function f (P) = f (:rgy, z) =
:5 +1; + 2, under the constraint that the point P(:n, y, z) is restricted to lie on the ellipsoid
2:222 +y2 + 22 = 1/2. b) (UNRELATED to part (a)) Find the critical points of the ﬁmction f (93,11) 2 — cc + 2533; + yz, and classify each point as a local minimum, a local maximum, or a
saddle point. 563 ,.3
6. a.) Using the two~path test, show that lim L y
(:ayHULOJ :54 + 3/4
3 2
. 1 _ J: y
b Challen ‘111 ' fallow that hill 2 U.
) ( g g) ' (am—40,0) 3:4 + 3/4 does NOT exist. 5AM " Linux“) . Then its critical point is 0" ﬁlm i ‘ 7. Let f(x,y) = e3xy+5 A) a local min. B) a local max.
C) a saddle point. 8. Consider the paraboloid x2 + y2 —4z =1 and the sphere x2 + y2 + 22 = 3.
Then the tangent planes to both surfaces at the intersection point (1, 1, 1) are A) parallel
B) perpendicular
C) neither perpendicular nor parallel. 9 . Given thatF(x, y, z)——  .If the components of VF are never zero, then
é.£.ﬂ & g. g are
ﬂx 03! 072 03c 5y
0?:
A)—1&—— resp. B) +1&—éresp.
63/ 41
(9/: 01'
C) —1 & —— resp. D). +1 & — resp. E) None of the above 2 l
3
10. The value of the double integral J J3yex dxdy is
0 pa
A) 9/2 (61)
B) 2(e—1;
C) 8(e1)
D) 25/2 (e—l) PartII 50% Sub‘ective 11. (5 0/o) Find the area of the surface cut from the bottom of the paraboloid z = x2 + y2 by the
plane 2 = 9 . (Grading: 4pts for setting it up & changing it to polar) “ 12. (7 %) Use Green’s Theorem to ﬁnd (3 xy3dx + (2)62322 +1)dy
C where C (traversed counterclock wise) is the boundary of the “triangular region in the
15t quadrant enclosed by the xaxis, x=1 and y = x2 13. (5 %) Set up (but do not evaluate) the double integra1(s) in Eolar coordinates to ﬁnd the
area of the “triangular” region in the ﬁrst quadrant bounded by y=4x2 , x=0 & x+y : 5.. Hint:
The point (1, 4) is a corner point of the region. 14. (8 %)
(i) Show that F = (y — x2 )i +(x + yZ )j is a conservative vector ﬁeld
(ii) Find a potential function for F (iii) Evaluate ({0} — x2)dX +(x + y2 )dy where C is the line segment from (0, l) to ((3,0).
C‘ p  7......“
t... . h " i, l
“Min“:
1' LIBRARY ’3 15. (5 %) Set up (but do not evaluate) the triple integra1(s) in Spherical coordinates
to ﬁnd the volume and in the ﬁrst octant of the surface inside the cylinder x2 + y2 = 4 and inside the sphere .zc2 +y2 + z2 = 8 —_—__—__—___—_.u—.___——————— 16. (5%) Set up (but do not evaluate) the triple integra1(s) in Cylinderical coordinates
to ﬁnd the volume in the 1St octant common to the cylinders x2 + y2 = 4 and 4x2 «t—z2 =1. ;  l7. {5 %) Consider the transformation u = x — xy & v = xy
(50 x = u + v & y = ....... ) aw) = 1 (1) Show that the Jacobian J = .
604,1!) u + v , " (ii) Use the above transformation to ﬁnd ”xdydx where R is the region bounded by the curves
R x—xy=1,x—xy=2, xy:1,xy=3 ...
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