Unformatted text preview: 09/11/2002 WED 09:34 FAX 6434330 MOFFITT LIBRARY .001 Department of Mathematics, University of California, Berkeley Math 1B
Alan Weinstein, Spring 2002
Second Midterm Exam, Thursday, April 11, 2002 Instructions. Be sure to write on the front cover of your blue book: (1) your name, (2)
your Student ll) Number, (3) your GSI’s name (Tathagata Basak, Tameka Carter, 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. [8 points] Find the general solution of the differential equation dy/clm = $2392, and ﬁnd
the particular solution for which y : 1 when a: = 1. 2. [5 points] For which values of p is the series 230:1 sin(1/n)np convergent?. 3. [5 points] Show that absolute value I sinr — w — $3 / 6 is less than 1/100 for all a: in the
interval (—1,1). 4. [9 points] For each of the following statements, tell whether it is true or false, and
give a justiﬁcation for your answer. In particular, the falsity of an “if/then” statement
should be justiﬁed by a counterexample. (A correct T or F gets no credit without a correct
justiﬁcation.) (A) If the sequence {on} converges to L, then (1100 is closer to L than 0.99 is.
(B) If the series 2,221 an is convergent, then the sequence {on} is also convergent.
(C) If an —> 0 as n —> 00, then the series 230:1(—1)”cn is convergent. 5. [11 points] The velocity v z 5% of a vehicle on a experimental highway is controlled by
signals transmitted from the roadway so that, when the vehicle is at the position :1: miles
from the beginning of the highway, its velocity is 20:13 + 10 miles per hour. (A) Find the position of the vehicle as a function of t if it starts at a: = 0 when t = 0. (B) How long does it take the vehicle to travel from a: m 0 to :1: : 3 if it starts at 1: = 0
when t = 0? (G) How long does it take the vehicle to travel from {r = 0 to :r = 3 if it starts at a: : 0
when t 4 1? 6. [7 points] Find an explicit formula for the function of as represented by the power series
230:0 (3” + l):t". What is the radius of convergence of the series? ...
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