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Unformatted text preview: MA 1022 A Term 200 C SKILLS Students Name: Please Circle Your Class Section: 01 02 03 D4
Lecturer: C. Lee K. Lnrie J. Masemune K. Lurie
Time: 8:00 2:00 10:00 10:00 0 This Basic Skills exam has Seven (7) problems. It is your responsibility to make sure that you have
all the pages. 0 Your are given two hours to complete both the Basic Skills and the Final Exam. Budget your time
accordingly. 0 Work all the problems and write your solutions in the space provided. You need not simplify your
answers. 0 Only your answers count on the Basic Skills exam. No partial credit is given.
0 No notes or books are to be consulted. I Calculators are NOT allowed. I The solutions you submit must be your own work. You may not look at or copy the work of others. In 1. / (c052:c+sin2$) aim
‘: '1 ON ~n<+C ans. ' ans. 3.115. dz
I V1 — $2 :7 MLSM’X + 4 W‘MUo s 5d ‘m’d w ans . LC) : Oink/K Q . I .
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m arth @ + 8:me :TT, ans. ‘ MA 1022 — Calculus II Students Name: Final Exam A2007 . Section: Instructions: Do your work on the paper provided. Put your name on each page and label each problem clearly. Work neatly and show your work. Remember, your work and your explanations are graded, not
just the numerical answers. 1. Evaluate the foilow'mg integrals. Simplify your answers as much as possible. cla:
(a) / ﬁlm: + 2) 2 u: Sm) , ._ .J .
‘ S n+1 5L“ ‘ i _
S .egurp. ' 'JX—‘dx
(3pm: [’EX' 2:: \L\+A\ ~—'\— C :M/méetM +C (c)/m“1nzdm, naé—l ‘ .h
@ L99» ) U“ :x ,_
r J. _._._k_ MK
u — x; ‘U‘—nﬁm
 _ ‘ t .
‘ . _ . t _ rm cw
gKn/QMXOK‘L: SUN? 12%? g“ .
I “14‘ ﬂat— V\+“ __ ' WH ‘ _\_. “Cl
LA .ﬁ 93 _ M Sm >4
u. “H ~ _. “Hi
"" y‘ﬂ (K MPH h‘l'l {X C ‘
‘WH _ “AH—k... “:71? (X 2Q)" \h—Hf (A :— +Ji M; HZ +39
I “ "3: 4r éQm—E‘i +319
a. . y  2. Compute the following derivatives. d 0 dt i it : 
_ —— ____.— ~ \ 7"”
(3’) dx Lax 1 + t2 _'. . ' . x :32 'Ma' ._. (Hiram) 1: 6X12 _ 3“ : VQMA‘ $711+ 33‘ ax
‘  ‘H _
r: (M? Jm + 25‘ 99 dy ,
 _ —— 2
(c) dx, 1f cosy :r C036 ":2 37C
f—jMé. 3
ob —... .4.
AK ‘— 3. Sketch the region surrounded by the given curves, and ﬁnd its area, it“: Dev 5‘: Z vigiser‘XLt ‘EX L2 . ‘5 '3
e— 3 '— _ ,__.L ‘——D_ + (“2. ‘\ KCEL\)
— gnaw (BL v a>
7 . ., ._
10131 + 0r?)  (Q; ~ 193 :2 ~ :7; +(«30 : 9,be ~ 36
4. Let f and g be differentiable functions on the interval [(5, b} where a < b. Prove the following provided f(a)9(a) ”~= f(b)g(b) = 0: ~ «5 b
j f'(x)g(m)dm=—/ mgwmx. \ : Igorﬁm CL — 8Q SSW?)th
1—; 9300‘) 1 §—( ck) 0509— ﬁtxl C809 dX :2 v— gg‘maMdﬂ 5. Find the volume of the solid that is generated by revolving around the acaxis the plane region
surrounded by two curves y=9—$2, and y=0. hv‘) $6 _ 1 6. We want to ﬁnd the Riemann sum for the integral / (3362 + 29:) dx. 3
U (a) Partition the interval [0, 1] into an. equal parts. Label the grid points 0 [0, l] as mogr, ‘ . . ,osn 
where 220 m 0, an = 1. Give a formula for xi. ‘
 J— r
é : Y} “a 1: . ..
X: ~ XL L:O}\)=——/Y\‘ (b) Using the sigma (E) notation, write down the Riemann sum of the above integral with respect
to this partition, (c) Find the limit as n —> oo in the Riemann sum for part (b) by using the summation formulae: i
l r
I
I
l
l ” =”(”+1) “MW
gt , Z 6 . 2
I k‘MIW ~ ~€ ﬁfzxtr—L—
"I’akg 'r'LcBM'W'M WWJUV‘N— I, \~ . L _ “I
Ari/vex“ ‘ H \ r 1 x
E: 2. been eatiii“
('“l
ZQ‘L it} + ___ n Cn_{_\)(.1i/\+\) “Li/Hi) ‘ , . 2
"‘ nu, ’ e h; ' > 1+ 3. 9‘ p (d) Evaluate the integral by using Fundamental Theorem of Calculus and make sure that the result
matches that in part ‘  ' . ,  , ﬂ b'ﬁamﬁlt: ng+5ﬂL t: 1 ...
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This note was uploaded on 10/13/2009 for the course MA 1022 taught by Professor Abraham during the Spring '09 term at WPI.
 Spring '09
 Abraham
 Calculus

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