Unformatted text preview: 09/11/2002 WED 09:31 FAX 6434330 MOFFITT LIBRARY .001 Department of Mathematics, University of California, Berkeley
Math 1B Alan Weinstein, Spring 2002
First Midterm Exam, Thursday, February 21, 2002 Instructions. Be sure to write on the front cover of your blue book: (1) your name, (2)
your Student ID Number, (3) your GSI’s name (Tathagata Basak, Tameka Garter, Alex
Diesl, Clifton Ealy, Peter Gerdes, John Goodrich, Matt Harvey, George Kirkup, Andreas
Liu, Rob Myers, or Kei Nakamura). Read the problems very carefully to be sure that you understand the statements. Show
all your work as clearly as possible, and circle each ﬁnal answer to each problem. When
doing a computation, don’t put an “2” sign between things which are not equal.
When giving explanations, write complete sentences. Remember: if we can’t
read it, we can’t grade it. 1 . [20 points] Evaluate each of the following. (A)
/ sin2 m da: + / cos2 (r dr
f sin 2.1." sin .r dr
f :62 + 2 rim
:1; — m 2 . [10 points] For which values of p is the integral 7’ sin :6
0 933’ improper? For which of these values of p is it convergent? You must justify your answers. 3 . [15 points] Suppose that a thin wire of uniform linear density A (i.e. the mass of
any segment of the wire is A times its arc length) is represented by the graph of a function
y : f(:r) deﬁned in the interval [(1,3)]. Putting together what you know about arc length and centers of mass, write a. formula for the mass m of the wire, and a pair of integral
formulas for the coordinates (E, g) of the center of mass of the wire. The integrands in the formulas should be expressed in terms of f (9:) and, possibly, its derivative(s).
You must give some justiﬁcation for your formulas (a limit argument or a “diﬁ‘erential” argument), but it does not have to be a formal proof. ...
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