EXAM 3, MATH 233 FALL, 2003 This examination has 20 multiple choice questions, and two essay questions. Please check it over and if you find it to be incomplete, notify the proctor. Do all your supporting calculations in this booklet. In case of a doubtful mark on your answer card, your instructor can then check here. When you mark your card, use a soft lead pencil (#2). Erase fully any answers you want to change. ProbleIllS 1 through 20 are worth 3.5 points apiece for a total of 70 points. On probleills 21 and 22, show all your work and indicate clearly your answer to the problem. Partial credit will be given for partially completed solutions on these two problems. Each of these problems is worth 15 points for a total of 30 points. Please write your name on the indicated lines on the two pages at the end of this test booklet. These pages contain the two problems to be handgraded and may be separated from the rest of the booklet in the grading process. There is a total of 100 points for the whole examination. You may use a scientific calculator and a 3 x 5 note card. (1) The following MATLAB script calculates the Riemann sum of a double integral using the Midpoint Rule and m = 100, n = 50. Evaluate this double integral to find the approximate value of RS which will be returned by MATLAB. dx = 2/100; dy = 1/50; [x,y] = meshgrid(dx/2:dx:2dx/2, 1+dy/2:dy:2dy/2); RS = sum(sum(x.*y»)*dx*dy (A) 1..."'1. (B) 2 {«; )( :() (D) 4 (E) 5 (F) 6.(G) 7 {H) 8 (I) 9 (J) lu ; '=' )((#, (:\(1\ 3 Lt If''
2 EXAM 3, MATH 233 FALL, 2003 (3) Find the surface area of the part of the plane z = 3x+4y+2lying above the rectangle [0,2] x [1,3]. (8) 1/8 (C) 1/4 (D) 3/8 tj (F) 5/8 (0) 6V29 (H) 3V3f (J) 7/8 (J) 1 ::.., \ [:_.(Q; \ 1 "' J\;.4.,. ia
EXAM 3, MATH 233 FALL, 2003 (4) Find the center of of a lamina (thin plate) occupying the square R [1,1] in the xy plane and having the density function p(x, y) = x. {a, 2] x (A) (8) (0) (D) (E) (F) @ (H) (I) (J) (0, (0, (0, (0, (i, (, (, (, (0, 0) 2 ) "3 i) 1) 0) (1, 0) Ot :'=J 0) O} ) $, f ')Co """ R C( """ R 1. ) :\ y, =o /Co z. )Co =0 ;Z3(3 2"iJ.a. :: f"31. (5) Find the volume of the tetrahedron bounded by the coordinate planes and the plane 3x + By + 4z = 24. ",,0\ = (A) 4 (B) 6 3 (C) 8 (D) 10 (E) 12 (F) 14 (G) 16 (H) 18 (1) 22 ::. 24..1) 24
i EXAM 3, MATH 233 FALL, 2003 (6) If f (x, y) is any integrable function with values :2: 0, set up the integral expressing the volume under the surface z = f (x, y) and above the region in the xy plane bounded by the curves y = x and x = y2 y. 11 1 4 2 '1. r. s,sc. Co (A) fo f f(x,y)dydx JC,':' :. = 'j"2._ (B) 102 fz2 f(x, y)dydx 1.\ lj2::. 2'j 'to..,. "'", (C) Jo r2 Jl/2 rl/2+./ii+r«f ( x, y ) dydx y "" 'j :% (!) )Co (D) fo. f: f(x, y)dydx )C 'j 2. r.of 2 2 (E) f2fl/2v'i=i6f(x,y)dydx '1... (F) fo12 f f(x, y)dxdy 2: tta ",,0\ ': \ (0) f08 f(x, y)dxdy (H) f04 f'j f(x, y)dxdy (I).f3 r f(x, yldxdy lf't\4 (7) Let S be the surface with vector parameterization r(u,v)=(u+v,3uv,2u+v), ':'lul, Ov2 Find the surface area of S. >, (A) 6 'f' < \ :2) '\ 3, (B) 8V2I 4 (@Iy\ (D) 14V3 (E) 10V'i. T : (F) 36 u.( \') \) "J..,,", (0) 6vT4 40 l u {' oj \ :. (H) 8V32 (I) 27 {J)42Vf$ A(S\ : r1. \ L =0 ::n v; J;;. APt, = 4 J"":t"2..
EXAM 3, MATH 233 FALL, 2003 (8) Find JJ(X2 + Y + Z2) dv, where E is the region in the first octant lying below the sphere x + y2 + z2 = 4 (A) 321r /15 (B) 311r /14 (C) 301r /13 (D) 291r /12 (E) 281rj11 (F) 271r/10 (G) 261r /9 (H) 251r /8 QJ] (J) 23./6. a..s (J 8:0 "t\1oz.. 1\,... \ (J?o "1. () p 'tco '1../S cm '" co...( '321\/'0
12),., '3 (C) 2 (D) 1 (E) 0 (F) 1 (G) 2 (H) 3 (I) 4 (J) 11'2 EXAM 3, MATH 233 FALL, 2003 JiDsin(x)sin(y)dA, where D = {(x,y): 0.$ x.$?r,o.$ y.$ 1l'} 'If" i"i"",4;, I f 1"0\\j 0 f.., C: c.us Tr y... O )", (13) Evaluate.J J D yda, where D is the region bounded by the line y = x and the parabola y2 = x. "i1 /J. j J ''f/j: (A) 2 l.(..( A...t 't00'e.. ' 1 '\ 4X" b) to ((,(\ 1 12 'i,'"". '1 0 ':)C. ": '! s JC.: 2. (E) 27/12 (F) 1 (G) 3/2 (H) 7/4 (I) 2 (J) 9/4 (C,OJ ':c) I S=C) 1(3. '2 (l "f (14) Let l(x, 11) = xy, find the directional derivative Duf(l, 1), where u (A) 1 (..,('I"(t t' t t) (B) 0 "" T J,\) ) < v'2/2t.f2/z >
8 EXAM 3, MATH 233 FALL, 2003 (16) Find the tangent plane of the surfoce X2 3y2 + Z2 + 2 = 0 at (1,1,1). (A) (x 1) 2(y 1) +(z 1) = 0 l= (')C. \ 'y.,'z_ "'l+ "2 (B) 2(x 1) (y 1) + 2(z 1) = 0 J'i, "\ (C) 2(xl)(yl)(Z1)=O (V\ (l,(1\\ Q(\')1,\) (D) 2(x 1) 3(yl)+(z I):=. 0 (E) 4(x 1)+ 5(y 1) +(z 1) = 0 0
EXAM 3, MATH 233 (F) 3(x 1) 4(y 1) 2(z 1) = 0 (G) (x 1) 6(y 1) (z 1) = 0 FALL, 2003.9
Your Name: EXAM 3, MATH 233 FALL, 2003 (21) Write your answers to parts (a)(c) in the space provided and, if you need additional space, on the back of this page. Try to write legibly and neatly. H an answer from one part is needed to answer another part and you're unable to obtain the first answer, state the method you would use to answer the second part if you had the first answer. Find the absolute maximum and absolute minimum of the function f(x, y) = 2X2 + 3y2 4x 5 over the disk r + y2 4. (a) [6 points] First find all the local maximum and local minimum of f in the interior r + y2 < 4 of the disk; "':' P" ': b.f = 'j =""> ')(.Z\, o w +(\,b\ :: A.Jr d oqoq "l. > J (1,0) (b) [6 points] Find the absolute maximum and absolute minimum of f on the boundary x2 + y2 = 4 of the disk; (4'4, ')": V "':: 'V a.1.>0. <)C. '> z.,.'12.= \' 'j :. 0 q,i. x to 2. a c... OI"l", Adc..l\0.A4 ('2 0 '\ :2 ( " 'If \ S' :..:. 4.f..c. bi4.oin I 4"", f o""' 1;. (a 0", ': (J \(",\.. CI('J\ 5 =(ii) c:, i w..., oc +....{t.,. ".., (c) (3 points] Then find the absolute maximum and absolute minimum of the function f(x, y) = 2r + 3y2 4x 5 over the disk X2 + y2 $: 4. JC' O,t.oc; W\ t ;", If f ol.4:. \l.(' k ':
EXAM 3, MATH 233 FALL, 2003 13 Your Name: (22) Write your name in the line above. Set up BUT DO NOT EVALUATE integrals expressing the following quantities. As relevant, show your work: general formulas, sketches, coordinate changes. You can be brief provided it's clear how you arrive at your expression. Just giving an integral without explanation of any kind may mean!iz ('.0\ losing all credit. (a) The volume under the surface z = x2y and above the triangle in the xy plane with z.,&t\ vertices (0,0), (2,0), (2,4). \ ':='.J 0\ ": d J(. o l( (Z:.. \ 'j'=o '1"7, ( (b) The volume bounded by the cylinder X2 + y2 = 4, the xy plane, and the plane y + z = 3. Express the integral in either cylindrical or spherical coordinates depending on which is more suitable for computations. 'I.\t"' x \/0\ :" r :2' (3..e ') '<'"tl'f'" e '"'D e..o ("=0 2..3'c1,1!01 y2 X2 that lies (c) The surface area of the part of the hyperbolic paraboloid z between the cylinders X2 + y2 = 1 and X2 + y2 = 4. '" V \'at I 040 4'2."" 'J'). ) ':!'... If\"')....:.. tcli..,..s) u,. ""\ '" ca. 001' 5 A (5\ J(; (", (' co (;) 2."" e.'c) 'f": (d) P(X + Y + Z) 1 where X, Y, Z are independent which has an exponential distribution with mean 2. ;'j (,lilt \ :. t J"(\ "'"'1(,,\ ('I \ I. oj I l t R2.. )(! e (i;'z.. e (;? random variables each of (2..) t '1lt (j ) ":: ) \IjlSI 8' ()I..,'f,l") \/ 'E11 \ "'I "1,\ \ _:() c::> e 0{;