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Unformatted text preview: page 1 Your Name: Circle your TA’S name:
Ben Akers Adam Berliner Ben Ellison
James Hunter Peter Spaeth Mathematics 221, Fall 2003 Lecture 2 (Wilson)
Second Midterm Exam November 18, 2003 Write your answers to the eight problems in the spaces provided. If you must continue an answer
somewhere other than immediately after the problem statement, be sure (a) to tell where to look
for the answer, and (b) to label the answer wherever it winds up. In any case, be sure to make
clear what is your ﬁnal answer to each problem. Wherever applicable, leave your answers in exact forms (using %, x/g, cos(0.6), and similar num
bers) rather than using decimal approximations. If you use a calculator to evaluate your answer
be sure to show what you were evaluating! There is a problem on 'the back of this sheet: Be sure not to skip over it by accident! There is scratch paper at the end of this exam. If you need more scratch paper, please ask for it. You may refer to notes you have brought in on one or two index cards, as announced at the class
website. BE SURE TO SHOW YOUR WORK, AND EXPLAIN WHAT YOU DID. YOU MAY. RE
CEIVE REDUCED OR ZERO CREDIT FOR UNSUBSTANTIATED ANSWERS. (“I‘did‘ it on
my calculator” and “I used a formula from the book” (without more details) are not suﬁicient
substantiation...) page 2 Problem 1 (12 points)
Let f(x) = m3 — 6:52 ~ 15w + 1.
Find all local and global maxima and minima of f(m) on the interval [—3, 6]. For each answer, be sure to distinguish the a: value and the value the function takes there, and tell
how you know that this point is what you claim it is (local max, global max, local min, global min). page 3 Problem 2 (13 points)
Let f(a:) = :1; + 2. (a) Set up and evaluate the Riemann sum for f which results from partitioning the interval [~1, 1]
into 4 equalwidth subintervals and using the left end of each subinterval as the sample point. (b) Set up and evaluate the Riemann sum for f (:13) which results from partitioning the interval
[—1, 1] into n equal—width subintervals (not for a speciﬁc value of n), and using the left end of each subinterval as the sample point. (Your answer when evaluating this sum should be a
formula that involves n and possibly some summation signs 2.) (c) Evaluate the limit of the sum in (b) as 17. goes to 00. This should give the same answer as 1
/ (x + 2) dw, and you can check your answer that way, but you must show how to get it from
1 . the limit of Riemann sums in order to receive credit for this problem. page 4
Problem 3 (13 points)
Solve the initial value problem dy 3:2 —=—, f0 >0, ‘th 0:.
dm y r y W y( ) 3
Show explicitly the general solution to the differential equation and then how you pick the particular
solution meeting the initial condition. 1 l page 5 Problem 4 (12 points) Use a linear approximation to f = 3/57: to approximate {77.
Hint: The tangent line to the graph of f (m) at m = 8 is particularly easy to work with. l 1 page 6 Problem 5 (12 points) 0
(3.) Evaluate / sin2(2z) cos(2$)dm. ...."L
4 (b) Find the area of the region shown, bounded by
parts of the curve at = ~y2, the line 2y 2 1—3:,
and the artaxis. page 7 Problem 6 (12 points)
Let f(:1:)= 2:2 — 650 +7.
(This same function is used in both (a) and (b) below!) (a) Find the average (mean) value of f on the interval [2, 5]. (b) The Mean Value Theorem for Integrals guarantees the existence of a number c within the interval
[2, 5] (i.e. c is not either of the endpoints) such that f (c) is the average value. Find such a number
0. page 8 Problem 7 (13 points) The region between the graphs of y = ﬂ and y = $2 is shown
at the right. Set up and evaluate an integral to compute the
volume of the solid that results when this region is rotated
about the xaxis. ‘ page 9 . Problem 8 (13 points) _
Suppose f is a function with the following properties: 0 f(0)=2
o f’(x)<0for—3S$<0
«mm=0 o f’(:1;)<0f0r0§$<6
o f’(x)>0for6<m§10 o f"(:c)>0for—3$:r<0 ofwm=o
. f”(:r)<0for0<x<4
~fwe=o  f”(x)>0for4<a:§10 Draw on the axes below the graph of a function with these properties. f page 10 I‘M AN IRE WORu)
£38m GUY wm
m A CATCH
MULTWLE W. Copgright Q 2882 United Feature Sgndicate, Inc. SCRATCH PAPER ...
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 Summer '07
 Denissou
 Calculus, Geometry

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