222-kastermans-07fafin

222-kastermans-07fafin - Math 222 Final Dec 19, 2007 Show...

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Unformatted text preview: Math 222 Final Dec 19, 2007 Show all your work. A correct answer without adequate justification will receive zero credit. Please write your answers neatly. One of the things being tested on this exam is your ability to understand the questions. ‘ Name: I . ‘ Circle your TAs name: 321, 327: Luanlei Zhao . 323, 324: Christine Lien ' 325: Harris Nover 326, 328: Anakewitl(Tete) Boonkasalne 329, 322: Anne Candioto 330, 331: Diana Thomson Dec 19, 2007 ' Math 222 Final 1. (2 points per item, no partial credit. No explanation needed.) Decide for the following if they are true, false ‘ (a) Let f be a function (from the reals to the reals) and an = f If 233:0(17, converges, so does f0°° f True False (b) If |di| converges, so does d1. ' True False (c) If Eg:0(—1)kak converges, then so does Efioak. True False (d) .If yl and y2 are solutions to the equation y” + y' + y = 0, then any solution to this equation is of the form 0191 + 02342. True False (e) Let r = f (0) describe a curve in polar coordinates. If - (r, 0) satisfies this equation, but (1”, —0) does not, then the curve is not symmetric in the x-axis. True False (f) VAx2 + Basy + 03/2 + D3: + Ey + F = 0 can be turned into A’ m2 + C"y2 .2 O by rotation. True False — page 2 of 10 — Dec 19, 2007 1 Math 222 Final 2. (a) (5 points) Find the fourth order Taylor polynomial at :1; = 1 for sin(%). (b) (3 points) Use the Taylor polynomial in part (a) to approximate sin(0), and give the best possible estimate for the error of this approximation. ' ' — page 3 of 10 — Dec 19, 2007 Math 222 Final 3 3. (5 points) Determine Whether the series 220:1 film—5:131 converges or diverges. — pagev4 of 10 — I Dec 19, 2007 Math 222 Final M 4.. (a) (4 points) Determine whether 1'2 = cos(0) is symmetric in the ac—axis, the y—axis, and the origin (determine all that apply). I (b) (5 points) Determine the total area inside this curve. ~ page 5 of 10 — Dec 19, 2007 ' Math 222 Final 5. (5 points) Solve 635% + 26% = 1. — page 6 of 10 — 'Dec 19, 2007 v - Math 222 Final 6. (5 points) Given vectors v = (2, 1, 3) and u = (1,1,0) find vectors a and b such that v = a + b, and a parallel to u,'and b orthogonal to u. ~page 7 of 10 — Dec 19, 2007 . » ' Math 222 Final 7. (6 points) For the following three pairs of planes show Whether they intersect in a line, and if they do find the line of intersection: ’ o x+y+z=0and2(a:—1)+2(y+4)+2(z—2)=0, o x+y+z=0and2x+2y+2z_=0, o x+y+z20andx+y+>2z:0. — page 8 of 10 — Dec 19, 2007 > ,Math 222 Final 8. (6 points) Find an arc length parametrization for r(t) = 6‘ sin(t)i + at cos(t)j +’ etk where t 2 0. — page 9 of 10 — Dec 19, 2007 Math 222 Final 9. (5 points) Determine f 62:6 COS(5:E)d.’L’. — page 10 of 10 — ...
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This note was uploaded on 08/08/2008 for the course MATH 222 taught by Professor Wilson during the Fall '08 term at Wisconsin.

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222-kastermans-07fafin - Math 222 Final Dec 19, 2007 Show...

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