234-assadi-07fafin

234-assadi-07fafin - MATH 234 (A. Assadi) - Fall 2007 -...

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Unformatted text preview: MATH 234 (A. Assadi) - Fall 2007 - FINAL Wednesday, December 19, 2007 NAME (print): Instructions: 1. Write your name on each page. 2. Circle the name of your TA: CHILDERS TONEJC 3. Closed book. Closed notes. Calculators allowed. No laptops. 4. 2 index cards of size 3” X 5” with formulas allowed. 5. Answer your questions on the exam paper. There are some extra pages in the back if you need them. 6. Show all your work. Partial credit is given Only if your work is clear. 7. Enclose your final answers clearly in a rectangle. 8. Time allowed: 120 minutes. 20 0 Problem 5 Problem 6 1+c0526 , 1—c0326 ' ' 71—1 _ I 0082 9 = M2 7 $1112 0 = *2 / sin” (13: d3: = —s—~————m ax COS ax + n 1 / sin’“2 ax da: na n sec2 0 = 1 + tam2 0, csc20 == 1 + cot20 COSn—l amsin ax n _ 1 2 n _ n— sin(A:l:B)=sinAcosBicosAsinB /COS afidm" na + n /C05 (lde cos(A:tB)=cosAcosBIFsinAsinB cos(a+b)m c0s(a—b):1: smamcos bxdx —— tan(AiB)_ tanAzlztanB 2(a+b) 2(a—b) _ 1 q: tanAtanB /Sin ax Sin bx d2: _. sin(a ~— 13):; _ sin(a + b):1; “ 2(a—b) 2(a+b) /_.——_—-—d$ : item—1 ax ‘ b /cos ax cos bx dx — Sin(a _ was + cos(a + b)“; xx/ax ~ I) x/B b 2(a — b) 2(‘1 + b) 2 / d3: : _2_1n Vax+ —\/l_7 /sinaa:cosaxdx= —CO: ax xx/am—H) x/E x/ax+b+\/I—) 1 a da: 1 x /c0t axdcc : —ln]sina:c| /2+ 2:6”“1-1” a a a: a 1 d3: 1 1 m+a /tanaxd:c= glnfsecaxl 02‘$2:2‘an5€“a dz 1 am ___~.__ = _t _ Lzlln 5” /1+cosaac a 2Ln2 v dam + b) b ax + 1) dx __ 1 cot ax dx x 1 - ac 1—cosam — a 2 ~———— = ———~ + —-tan 1 ~ (a2 +x2)2 2a2(a,2 +952) gas a ftang wdx : 1 tan” _$ (1.7: ' a . __.___= ‘/ 2 2 I V/W ln(xl+ a+x) V 2 1 cot axd$=—Ecotax—x 1/ 2' ‘ 2 ‘/ 2" 2 ' / a +117 dx:m_aln (1+ a +2: 1 cc :1: /secazdz=Elnlsecam+tanax| 1 \/ 2 2 1 ————d$—~=~—ln w cscaxdx': ———1nlcscax+c0taa:l xx/a2+a:2 a :8 (1 d3; _ Va2 +m2 / sec2 axdm: ltanaav x2x/a2+x2 #— (12$ a1 / dsc . _1 :c /csc2axd$= ~gcotaa: —~——— =sm ~— 1/012__a72 a x a2 x lnaxdx=m1nax—w x/a2~x2da:=— aZ—x2+~—sm"l— 2 2 a 1 arc __ _ ax d3: __ lln a+\/a2—$2 /e dwwrae ./ 2_ 2 —~_ am it a I a :6 f6“1 cos bx da: 2 (126+ b2 (a cos ban + bsin bx) fig: : 3111—1 (a: —a) ax . 66x . x/2ax—x a e smbxdxz 2+b2(as1nbaxc—bcosbx) a ac a x 2 / a2+m2dm=§ a2+x2+31n(z+ a2+x2) ' 2 /\/m2—a2dx=§\/m2—a —g2——ln!a:+\/x2—a2' NAME (print): Problem 1. The plane a: + yv+ 42 2 2 cuts the cone 22 = $2 + y2 in an ellipse. Using the method of Lagrange multipliers, find the greatest and the smallest values that the function f (x, y, z) = 22 takes on the ellipse. Write the equations that you need to solve in the box below. Put your final answer in the two boxes at the bottom. Greatest value: [:1 Smallest value: !::1 NAME (print): Problem 2 . (a) Compute the gradient of the function f (cc, y, z) = exy + cosy + 22x. (b) Let F = 62 i + sin yj + 4933/2 k. Find cur1(Vf + (c) Now let G : exyi + sinyj + 51322 k. Compute div (G + cur1(Vf + NAME (print): Problem 3. Consider the annular ring A = {1 g :1:2 + y2 < 4} and let M = y(cos(:cy) — 1), and N = x(1+ cos(xy)). (a) Sketch the region A and the vector field F = M i + N j at the points where the boundary of A intersects the coordinate axes. Make sure that your vectors have proper length! (b) Compute the circulation / M da: + N dy 8A of F around the boundary of A. HINT: Green’s theorem might be useful. NAME (print): Problem 4. Consider the sphere 3:2 +3;2 +22 = a2 andlthe cylinder 9:2 +312 = (12. Let 0 S hl < hg g a. Show that the surface area of the part of the sphere between the planes z = h and z = hg equals the surface area of the part of the cylinder between the same two planes. In other words, the surface area of the sphere over any vertical span is equal to the surface area of the circumscribing cylinder over that same span. (This was already known to Archimedes!) NAME (print): Problem 5. Consider the surface S, which is the upper half of the torus (see figure) and whose parameterization is r(u,v) =‘ ((2 + cosu) cosv, (2+cosu) sinv,sinu), 0 g u g 7r, 0 g u g 27r. Let F = <—y, x, e22 cos(x2)>. Compute f/curlF«nda, S Where n is the upward normal. NAME (print): Problem 6. Compute the outward flux of F = 2xi+3yj —4zk across the boundary of the upper half of the solid unit sphere, that is, across the boundary of the set D={($7y,z)t x2+y2+22<1,z>0}. ...
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234-assadi-07fafin - MATH 234 (A. Assadi) - Fall 2007 -...

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