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Unformatted text preview: Millersville University Name Al‘sw K26
Department of Mathematics MATH 211, Calculus II, Final Examination May 10, 2007, 2:45PM4z45PM Please answer the following questions. Your answers will be evaluated on their correctness,
completeness, and use of mathematical concepts we have covered. Please show all work and
write out your work neatly. Answers without supporting work will receive no credit. The
point values of the problems are listed in parentheses. sin :6 . .
, the xax1s, and the lines 1. (5 points) The region bounded by the graph of f (x) = x
a: = 7r/ 2 and a: = 7r is revolved around the y—axis. Find the exact volume of the
resulting solid of revolution. v = 2“ \ka 4654 M (mam Gs 45mm “(Z
2 “ so
'~‘ ﬂ My.
“’2.
_ Z “
  «as;
7‘ “’2
= —2r\(cm<—c,a“/z) 2. (6 points each) Determine if the following inﬁnite series are divergent, conditionally
convergent, or absolutely convergent. You must justify your answers. °° +1 1 _ l 4.
(avg—1r 1w; Xv: mk Hm w by JR" , , 1:: °°
\Q‘M‘N 23—: JZIM =‘ )0. ‘5 '
22+.» be kw, H43: 23:} ‘9‘““8‘ (b) iii); W +9.2. Rat, 3m we M
_ (23&+7 k.‘ 5%
g‘“‘ Uh—rHt‘i'zﬂﬂ — £4,“ 2 ﬂ“)
a“. (23"EM 9? "w + +5
:E+ 5::
l+<£~5>5a
_ 2 92+!
k—sw 1+ (was
= 2/5 <' 3. (5 points) A spring has a natural length of 10 inches. A force of 800 pounds stretches
the spring to 14 inches. How much work is done in stretching the spring from 12 inches
to 14 inches? ﬁne: was?” we.“ mo: 92(4) => Re=2w‘ES/a\. \J = \ HAM f
'5?
1a
§ u
a
O
72 . (6 points each) Find the values of the following limits, if they exist. (a) limx—lm WW °/o (b) £5201”? sin (i) VKMQANWQ on . o
z 3;“ “WW Mum‘mm
’1’ >oo {'7
= M ' Vac" “10’le
’7» '9 9°  2/1}
= f: 1;: m m 8 = M  ’K. ‘90“. ‘IM 2. 1 9 Mg, 1. \ SZIIE t I o = M tM‘F, I; = W /O .P __, o _( 2. Ix”, 0+ Sum: (an: 211 . 1' O : lb!“ 2. = O :=‘> vQA/VN (tM$\ = C = ‘ 190" CaaﬁSin'ﬂ 4.. ‘50+ 5. (6 points) Find the area of the region inside of m the polar coordinate equation
7‘ = 4 + 2 cos 6. A= ‘ziff‘cwa
= a \TWZMW _ I 1“
‘ a Oib+lbcaa+4wfe Abe i\:llp&e +12"Z\\lo0on’a&e + ﬁftiufeae
u—W =0 an
43(23» 4 \ i+0&29&19
o = ((011 + 2.“ A= Vim 6. (5 points each) Evaluate the following indeﬁnite integrals. (a) /x21nacdm 1“:bi 8,0465: (“Emu roz 1313 and/1&4. dingM
3 3
= #31 4M. — \Jv icon _.L3 __3+
_3¥.J?m1. (1'16 CL (b) / hm TCCWCCL SUJDStL’tLCIm
()0 = Z‘tM—Q
aw «<19 21
Z
aseceztme .
—— —————— 2
\J ‘\ + II the (De __ & Zsecza'tm '99
 J l+ thB (c) /cos(sin_1x)dm .—
.— ‘
— V1 —a:2 XmMM VmM+c . 0“ \+0 m+C 3,5 —\<1L<. . (5 points each) The position of a moving point at time t is given by {3 (a) Find the speed of the point at t = 7r/3. fig: lmt ﬂ = 0&'t mom 4&«9 = Jl‘w‘mﬁ (ma “law/33‘
= W?
—‘ 42—, l + (Eg I\" :2 0.044645 (b) Find the distance the point travels for t in the interval [0, 7r / 2]. You may evaluate
this result numerically. g = Vial (lOnt\2+ (“but +tookt)2 M t: I.3'IZI‘Z> t—sint
1— tcost. (C) Find the slope of the tangent line to the path of the point at t = 7r/ 6. W8: ~oat+tcw$~t
3& = —— Wax = I cut
.. If 11' ' u _ 1L ‘
___. TV  (2313' W ’5 “45' If you are a MATH major, please print your name here: 8. (6 points) Use a partial fractions decomposition to compute the following indeﬁnite
integral. Simplify your answer. x—l
/x(x2+l)dx ’Yvl : L + ‘BquC—
¥(¥3‘+\\ $6 ¥z+l If you are a MATH major, please print your name here: 9. (6 points) Use the ratio test to determine the radius of convergence for the following
power series. Also determine the interval of convergence. —v0 W ﬁ—yw
W
= JENII < =5 %_l<2
on ( 9w 0: ,, (“33“
£421 ans 9:2, 0%}— ‘3—6 :3” k c :3
=l .
1” 6° QM 9* EL
6° _ a  _
:\ =(
= Z 1:63" 94%. ...
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 Spring '07
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