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Unformatted text preview: Prof. Girardi Math 142 Fall 2011 09.22.11 Exam 1 MARK BOX PROBLEM _—
—E—
n_— K C
“— NAME:
___
“— class PIN:
__—
___—
__— 0
I
O INSTRUCTIONS: (1) To receive credit you must:
(a) work in a logical fashion, show all your work, indicate your reasoning;
no credit will be given for an answer that just appears;
such explanations help with partial credit
(b) if a. line/ box is provided, then:
— show you work BELOW the line/ box
— put your answer on / in the line/ box
(c) if no such line/ box is provided, then box your answer
(2) The MARK BOX indicates the problems along with their points.
Check that your copy of the exam has all of the problems. (3) You may not use an electronic device, a calculator, books, personal notes. (4) During this exam, do not leave your seat unless you have permission. If you have a question,
raise your hand. When you ﬁnish: turn your exam over, put your pencil down, and raise
your hand. (5) If you do not make at least 12.5 out of 25 points on Problem 1, then your score for the
entire exam will be whatever you made on Problem 1. (6) This exam covers (from Calculus (ET) by Stewart 6th ed.): Sections 7.1 — 7.5, 7.8, 11.1 . Hints:
(1) You can check your answers to the indeﬁnite integrals by diﬂ'erentiating.
(2) For more partial credit, box your u — du substitutions. ___—___——..——————— Honor Code Statement
I understand that it is the responsibility of every member of the Carolina community to uphold and maintain
the University of South Carolina’s Honor Code. As a Carolinian, I certify that I have neither given nor received unauthorized aid on this exam. I understand that if it is determined that I used any unauthorized assistance or otherwise violated the
University’s Honor Code then I will receive a failing grade for this course and be referred to the academic
Dean and the Ofﬁce of Academic Integrity for additional disciplinary actions. Furthermore, I have not only read but will also follow the above Instructions. Signature : ___/__ You Were Warned aha/Ct +h/L5 freight“; 1. Fill in the blanks (each worth 1 point).  1L
In. —“ = I" u+C 0. /
1b.Ifaisaconstantanda>0buta9él,thenfa"du= ha. +0
lc. fcosudu = Sln U +0
1d.fsec2udu = to.“ ll +C'
le. fsecutanudu = SCC u +C
1f. fsinudu = " COS U. +0
1g. fcscgudu = " 00+ UL +C
1h. fcscucotudu = ' C50 a +0 1i. ftanudu= h lCASill :C‘g'r; au lSCCULl likC— g5.
11. fcotudu = LE ISIYl kl ONE: Icscul W
1k.fsecudu= nguimu‘lﬂc’ g, 'Llsw.“ ‘MJEll‘Cge 11.jcscudu=" l u—‘l' Ml” 2.; ESCU—‘CDfMl‘l’Cié‘
. “
1m.Ifaisacontant anda > Othenfvzi—ugdu = '4 /a +0
‘ ‘
1n.Ifaisacontantanda>0thenfmdu = luv». ('7‘ +0 1. 10.Ifaisacontantanda>0thenfmdu = a. 5844 ”J a. + C
1p. Partial Fraction Decomposition. If one wants to integrate 19%; where f and g are polyonomials and [degree of f] 2 [degree ofg], then one must ﬁrst do long di VlSan lq. Integration by parts formula: f adv = E. "U— ‘ S 'U' du lr. 'Il'ig substitution: (recall that the integrand is the function you are integrating)
if the integrand involves a.2 —u2, then one makes the substitution u = A SI D 9 Is. Trig substitution:
0. inn 9 if the integrand involves a2+u2, then one makes the substitution u 11:. ’D‘ig substitution: 9
ifthe integrand involves u2—a2, then one makes the substitution u = 0. $60. In. trig formula your answer should involve trig functions of 0, and not of 20: sin(29) = 1 5i n 9 case . 1v. trig formula cos(26) should appear in the numerator: cos“(0) = + 2'9) . co 2.9) 1w.trig formula cos(20) should appear in the numerator: sin2 (0) = 1x. trig formula since cos2 0 + sin2 9 = l, we know that the corresponding relationship beween 7.
tangent (i.e., tan) and secant (i.e., sec) is l Mtg : Sec 9
1y. arctan( \/3_ ) = "' T” 3 RADIANS. (your answer should be an angle)
1 AT
+0..“ . ISL : 6; n 7
c. Jr to S , E
n —; L (‘5784—5) *"C/ Sir/(IMF Jro an mwwjvjt 4mm class. sinalx
3 [sinzz cosazdz =
a, 5
g L— 1/ 4C,
3 5
3 905
: ($0100 , {51>— +C
/ 5 vasfnhy) ,7 vﬁs"
Au =lms(u)5h)4: e 5" \ ,5X
5e Aus‘ZNZn (2*)J‘f V 5 " e 5;n(u\':—73e“00$(7,ﬁ)3x ,. .33 3'20: (13):)» 4. 2 by 2 '37
fehcos(2z)dz = 236 5:0(Q>)* 29‘ (05(0)) Hint: bring to the other side idea. fa. 5' “557“). 4.7.." 3 49 ”1:95): W : coS (1‘3 6“! ‘
d¢.:5€5* JV Y: é9n(QY\ 1'  5y I E '5! Pf. _
v) 21' lge 5'n(2x\l‘2J‘e 5:0(2x) J)’ ' 47 r» \ w“ dv= «sandy,
5x
dw 5L 4" V‘~"3,605(Z¥)
\
g)! 5 5V ‘
,2 ZLQSf5(ﬂ(’Z~/> , if: (' “£2 (05(2x) — , 2 Kg (05(Qy\¢ly>/
" ’ ’ ‘1’ 5 s
« ie"31n(?»)— ‘36 2e (M0 + 2 j». YC05( my)
Lizj S 25
$ ' S Y _ 3‘! n
Se5x(°S[Zy\ as! 1 '27( v Saﬂ(1y)“’ ‘q C (03(1)?) L! 39.. (105(Lx‘) Jy
."‘ J L 5V S], >
T H: (05(2)!)dy 2cx5:r‘(:‘r‘)" ”C 6° “V
U
S L J ‘ys (ﬁx) J, 5 (syco>(’2_y> N .
Xe XCL‘JUXMX {'9 2 ’ ”' ° " ,mw
("0 5y 33“? “LU“ “hf r‘
Sie>ylsin(qx\.l E e, C°§(q)3 % riff “I‘lyn ‘ ”:1
2 g .‘V 
MI¥ ~V _'U ~— — —  _" ‘w‘ "" t! J n..'l.A‘ :,
Q 5 j. 3x h \ .. f U’ ”a
at, 5m<9x>+nq¢ C03(‘*")‘¢ 4.; ‘Y'wur {5);
—~‘ “ ' «J, :W )uJ‘I' ‘h h‘lLﬁfd ’ S‘vm" lﬂﬂ’ 4,0 i—O’WLCWQTE. ’Prob/(LM, S: ¥‘/—2HL ‘23 63. Complete the square by ﬁlling in each of the two lines with a (positive or negative) number. 2:2—6$+13=(a:+ '3 )2+ ‘1 m 24.t“6t+13=(t2—6t+9)+4=(t—3)2+22. \luswzz
Ii wt—3=2tan9,sodt=2sec20d0.Then 49'5”” _3
éq'ﬁ dt 1 t
= ——————2sec29d9
[m / (2mg)2+22 2 I 25x20
= zsec9d9= seCM"=llllsuec9+ttml¥l+01 [byFormula7.2.1] ‘/2_ _
t 6t+13+t 3’+C'1 =1’“ 2 2 =1n\/t§—6t+13+t—3+C whereC=Cl—ln2 7. / (332+ 1)4 a: C79 Warning: write your solution m proper form. J“? +C 8. Part 89. should help with part 8b. 8b. The functions y = e32 and y = 9:2 e”2 do not have elementary antiderivatives.
But the function y = (2x2 + 1) «3‘:2 does have an elementary antiderivative.
Emulate I (2x2 + 1) 6‘2 dz. Emma Lam yewamt: g 1.5 1* <2! 31. The function y = 2am:In does have an elementary antiderivative. so we’ll use this fact to help evaluate the integral. a ,2
f(2;cz+1)e‘2dz=f2xze’ dzvlfe d1 = fm(22:e‘2) da:+fe=2 dz , 36L {'13 56C
H w M  .
is; Parmd?5 Mill‘k ls: u = z, du: 2Z6”: (13,]
2
(in = dz 1;: e” “11:92”2 — f 812 {it} f e’2 da: iceI +0 “1) : M61212») + (my > i n'lveﬁrn— —_____—___———————
Prof. Girardi Math 142 Fall 2011 09.22.11 Exam 1  takehome MA 0
\
NAME: Mi PROBLEM POINT
CLASS PIN: ‘ 7’ E (I) II “——
INSTRUCTIONS: (1) To receive credit you must:
(a) work in a logical fashion, show all your work, indicate your reasoning;
no credit will be given for an answer that just appears;
such explanations help with partial credit
(b) if a line/box is provided, then:
— show you work BELOW the line/ box
— put your answer on/ in the line/box
(c) if no such line/box is provided, then box your answer
(2) The MARK Box indicates the problems along with their points.
Check that your copy of the exam has all of the problems. (3) You may not use a calculator, books, personal notes. (4) During this exam, do not leave your seat unless you have permission. If you have a question,
raise your hand. When you ﬁnish: turn your exam over, put your pencil down, and raise
your hand. (5) This exam covers (from Calculus (ET) Stewart 6th ed.): (a) Sections 7.1 — 7.5, 7 .8 for the inclass problems
(b) Section 11.1 for the take home part. Due Friday Sept. 23 at the beginning of class
in LC 102. Honor Code Statement
I understand that it is the responsibility of every member of the Carolina community to uphold and maintain
the University of South Carolina’s Honor Code. As a Carolinian, I certify that I have neither given nor received unauthorized aid on this exam. I understand that if it is determined that I used any unauthorized assistance or otherwise violated the
University’s Honor Code then I will receive a failing grade for this course and be referred to the academic
Dean and the Ofﬁce of Academic Integrity for additional disciplinary actions. Furthermore, I have not only read but will also follow the above Instructions. Signature : ____—_________________ “P:"il ©. For the following SEQUENCES in 1—5: 0 if the limit exists, ﬁnd it
o if the limit does not exist, then say that it DNE. Put your ANSWER IN the box and Show your WORK BELOW the box. ’5 1' 5 4f
‘ "2 + _
/. “ alga 6n3+7n2+1 _
2.
8 l'
L.“ niacin 6n3+7n2+1 _
3.
3  _ _
)Vl nh—{Jgo 6n3+7n2+1 5+ L
lim "5’1 3 5+0 _ S
“goo f 2 ’ ——
(”é+51“: (awm é; lim (—0.917991799917)" = 0 71—900 r —, —0.3l'+‘5‘3’+ 3‘33 I? lr\<\ . $11130 (—1.0000000000000017)" = D“ E (M 3 6. A sequence {an} has the limit L , written as
11111 a,1 = L , foreverys>0 +haL 4:9 Nil” $1»de
LG n>rJ Hm“, land—Viz (Finish ﬁlling in the box with the proper Deﬁnition 2 (not Def. 1) on page 677. I started you out) 7. Prove that lim (8+( 1)") = 8 n—voo n3
by using the deﬁnition of limit in the previous problem. An outline of the proof is provided, you just need to ﬁll in the blanks. . . I I_ 3
P——r°°me—L——>" “we @ 2 "l @
Pick a. natural number N E N so big that ,
which we can do by Archimedes Principle. Fixn>N. _ n _.
Thaw—Ll: (‘3 +( > g
__ n
I. .L 37’
__ H3 gg
. g g
< ’ 4 5 . 4. ...
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 Fall '11
 KUSTIN
 Calculus

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