**Unformatted text preview: **09/11/2002 WED 09:40 FAX 6434330 MOFFITT LIBRARY 001 Department of Mathematics, University of California, Berkeley Math 1B
Alan Weinstein, Spring, 2002
Final Examination, Tuesday, May 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 TA’s name (Tathagata Basak, Tameka Carter, Alex
Diesl, Clifton Ealy, Peter Gerdes, John Goodrick, Matt Harvey, George Kirkup, Andreas
Lin, 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 “=” Sign between things which are not equal.
When giving explanations, write complete sentences. You are more likely to
get credit for a solution if it is written clearly. 1. [10 points] The error bound for the trapezoidal rule applied to the integral of f(a:) on
an interval of length L, with n subintervals, is: M2133 <
JETI - 127:2 ’ where M2 is any estimate for the second derivative of f, i.e. |f”(m)| 3 M2 for all m in the
interval of integration. 7 Suppose that we want to integrate the function 37100 + 2 over the interval from 0 to 1, with
an error of at most 0.01. Find a real number N such that using a subintervals, where n is
an integer greater than N, will guarantee the required acauracy. 2. [10 points] Write each of the following complex numbers in the form a + bi.
(A) 1/(6'i + 4) (B) (1 —75)1° (C) e2+3i 3. [10 points] A particle moving along a straight line has velocity u(t) = 3tze_t meters per
second after t seconds. (A) How far does it move during the ﬁrst 10 seconds? (B) How far does it move during the next 10 seconds? (C) Where is the closest place to the origin that you could put a wall which will never be
reached by the particle, no matter how long it moves? (Assume that the particle starts at
the origin and moves toward the well.) 4. [10 points] For which values of the real number q is the improper integral
I100 3‘1 sin2(1/m) dm convergent? (You must justify your answer.) 5. [10 points] Find the average value of the function f = 4 + sin2 3w on the interval
[0,271']. CONTINUED ON THE OTHER SIDE 09/11/2002 WED 09:41 FAX 6434330 MOFFITT LIBRARY 002 6. [10 points] The half life of radium—226 is 1590 years. (A) Let m(t) be the mass at time t of the radium-226 in a sample. Write down a ﬁrst-order
differential equation which describes how m(t) changes with time. (B) How long does it take for the mass of the radium-226 in the sample to (imp to 1/8 times
its initial value? 7. [10 points] For which (integer) value(s) of the exponent It does the sequence _ 211’“ v- 3
_ 372’” w 2 an converge? Find the limit when it exists. (The answer may depend on 8. [10 points] Find two different functions which are solutions of the differential equation
43;” 4- 5y' + y = em. There should be no arbitrary constants in your answers. 9. [10 points] Find the ﬁrst four terms of the power series solution of the initial value
problem dgy/dhv2 + dy/dx + 562:1 m 3y 9(0) = 24:10) = 1- 10. [10 points] Find the sum of the series 23:3 [Hint: Expand the general term of
the series in partial fractions] 11. [10 points] Determine whether each of the following series converges or diverges. You
must justify your answer. (A) (3:13)“
(B) m n
E3177:
(0) i(~—1)”ln(l + l)
n=1 n 12. [10 points] Tell whether the following statement is-true or false, and explain your
reasoning: If a solution of the diﬂ‘erential equation dy/dt = (y — 1)(y —— 2)(y H 3)(y~5)(y 7 8)
has the value 3; = 4 when t = 0, then, for this solution, y > 4 when t = 17. [Note: your
explanation may include a ﬁgure] ...

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