This preview shows pages 1–10. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 xaxis. 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 — ...
View
Full
Document
This note was uploaded on 08/08/2008 for the course MATH 222 taught by Professor Wilson during the Fall '08 term at Wisconsin.
 Fall '08
 Wilson
 Calculus, Geometry

Click to edit the document details