F08_Final_Exam-C.Teleman - Mathematics 53.2, Fall 2008...

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Mathematics 53.2, Fall 2008 Final Examination, 20 Decernber 114points total NAME: ID: GSI: INSTRUCTIONS: The examination has T.WO PAFUTS. of 90 rninutes each. Part I (54 pts) is MULTIPLE CHOICE and no justification is necessary. Record your ans\r'ers by circling the appropriate letters on the ANSWER SHEET (last page). Detach it from the exam paper, WRITE DOWN YOUR NAME, ID AND GSI on it and pass it on towards the aisle when so instructed. The answer sheets will be coliected 90 minutes into the exarrr. In Part II (60 pts), you must justify your answers. All the u'ork for a question must be on the respective sheet. You may start work on Part II before the 90 minutes for Part I elapse (but it is unq,ise to do so before you finish Part I). \bu need not turn in the last sheet for roueh work.
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Part I: 18 questions in three groups. 3 points for correct answers, 1 point penalty for wrong answers. However, you will not receive a negative total on any group. 1. A (continuously differentiable) parametric curve C is given by I * (r(t),0(t)), in polar coor- dinates, with 0 < t < 1. A formula for the arc length of C is 2. If the parametric curve t r* r(l), a 1t 1b, satisfies r(l).r'(t) > 0, it follows that (.) .fi \FTFT o@dt r") Jo'ffiat (a) lr(t)l is constant (c) jr(t)l is increasing (b) "6Wdt (d) } "[i r2 (t)rtt (b) lr'(t)l is increasing (d) None of the above. 3. The parametric curve (r, y): (f (t),9(l)) must have a horizorrtal tangent at (/(t6),9(t6)) if (a) //(*o) : s (b) e'(ts; : s (") /'(to):0 and s'(to) + 0 (d) f'(to) * 0 and g'(to) :0 4. If thethreevectorsu,v,winR3 satisfy :uxvf0anduxwf 0, but (uxv) '(uxw) :9. then it follows that (a) The plane spanned by {t,t} is orthogonal to that spanned by {u,w}. (b)vIw. (c)uIvanduIw. (d) r, v and w lie in the same plane. 5. If f (n,U):12+A3+z4,thenthetangeritplanetothelevelsurface f(r,A,z):3 atthepoint (1,1,1) is given by the equation (a)2r*3y*42:0 (c) 2i + 3j + 4k:9 6. The function defined on R2 by f (r,y) : if (r,y)
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F08_Final_Exam-C.Teleman - Mathematics 53.2, Fall 2008...

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