Unformatted text preview: 04/10/2001 TUE 17:01 FAX 6434330 MOFFITT LIBRARY .001 Department of Mathematics, University of California, Berkeley Math 1A
Alan Weinstein, Fall 2000 Final Examination, Thursday, December 14, 2000 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 (Eric Antokoletz, Victor Deletang, Matthieu
Hamel, Andre Henriques, Di—An Jan, Chu—Wee Lim, Russell O’Connor, Alf Onshuus, Em
manuel Py, Shahed Sharif, Dan Stevens, Karla Westphal, or Alexander Woo). 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. Note: In solving the problems on this exam, you may not use advanced techniques (such
as integration by parts) which you may have learned in a previous calculus course. If you have any question about what you are permitted to use on this swam, please ask one of the
proctors. 1. [15 points] A driver travelling the 60 mile stretch of highway from Bluesville to Greenville
computes the average speed in miles per hour for each 10 mile segment of the trip and ﬁnds
these averages to be 20, 30, 40, 60, 40, and 20. (a) What is the total time required for the trip?
(b) What is the average Speed for the entire trip? (This is the usual average speed, the “average with respect to time".) (c) The state highway department decides to post an electronic sign in Bluesville which will
tell drivers the expected time for a trip to Greenville. They place a large number of sensors
along the highway so that, at any moment, they can accurately estimate the ﬁmction 5(a)
which gives the speed (in miles per hour) of the trafﬁc which is currently at a distance :r
miles from Bluesville (and going in the direction of Greenville). Under the assumption that
traﬂic conditions are steady, so that the function 5(a) does not change during the time it
takes to make the trip, a computer uses this data to calculate the expected travel time from
Bluesville to Greenville as an integral, and the result is displayed on the sign. What is this integral? In other words, write an integral which expresses in terms of 3(a) the
expected travel time T (in hours) for the 60 mile journey. You must justify your answer.
[Hint Check the units in your answer. Also check to see that your answer is correct when
the function 3(a) is constant] 2. [15 points] Let f (:12) = 2:4 + %m + g(:s), where g(:c) is a differentiable function such
that g(——1) = 9(0) = 9(1) = 0. Using a theorem or theorems about continuous and/or
differentiable functions, prove that there is at least one value of m for which f ' (r) = O. 3. [15 points] (a) Use a linear approximation to f (as) 2 $1M around 3:0 2 16 to estimate
the 4th root of 16.32.
(b) Is this estimate greater than or less than the cxact 4th root. Why?
(c) Use a single iteration of Newton’s method to ﬁnd an approximate solution of the equa
tion m4 — 16.32 + 0, starting with the initial guess :3 = 2. 04/10/2001 TUE 17:02 FAX 6434330 MOFFITT LIBRARY 4. [15 points] Evaluate each of the following deﬁnite integrals. [Hint look for shortcuts on
some of these problems. Don’t just look for antiderivatives.] (a) f
jﬂ 7r assin(m2) do: (b) 1 _ smzL' d3:
_1 e37 + 6F":
(c) 1 _
f icmcw
0 :17 Jil— 1
(d)
[1 dt
0 1 +t2
(e) 3
V 9 — ”2 do 5. [15 points] (a) Find the area of the region bounded by the curve 3; z m4 and the line
y = 16 in two ways, ﬁrst by integrating with respect to :r, and then by integrating with
respect to y. Check that the two answers are equal. (b) Find the number 1 for which the vertical line .7: = 1' divides the region in part (a) into
two pieces of equal area. 6. [15 points] Do the following computations. (a) Find
50 _ ‘
2(3‘ _ 311).
i=1
(1)) Find
. 1 3+5
5113?) E 3 Int dt.
(c) Find
d .
@(Sln(29)).
(d) Find
h'm n2(—1‘+—1+—1——)
n—roo (11+ 1)3 (n+2)3 (n+n)3 ' [Hint write the expression as 1/7; times a sum. Be careful with the algebra] 7. [15 points} Let a > 1. Two horizontal lines are drawn in the (37,31) plane, one through
the point P = (O,a) and one through the two points on the parabola y = 9:2 which are
closest to P. What is the distance between the two lines? 8. [15 points] Find the volume of the solid of revolution obtained by revolving about the :c—axis the region bounded by the lines a: : 2, a: = 6, and y = 0 and the curve 3; = J1? (MATH if}! Alqﬂwtmﬂii‘vu F...“ Econ, ‘FMuJ? 6.1an I002 ...
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