final exam ans buch - Millersville University Name Al‘sw...

Info icon This preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 8
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
Image of page 11
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Millersville University Name Al‘sw K26 Department of Mathematics MATH 211, Calculus II, Final Examination May 10, 2007, 2:45PM-4z45PM 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 x-ax1s, 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—rH-t‘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$\ = C = ‘ 1-90" Caafi-Sin'fl 4.. ‘50+ 5. (6 points) Find the area of the region inside of m the polar coordinate equation 7‘ = 4 + 2 cos 6. A= ‘ziff‘cwa = a \TWZMW _ I 1“ ‘ a Oib+lbcaa+4wfe Abe i\:llp&e +12"Z\\lo0on-’a&e + fiftiufeae u—W =0 an 43(23» 4- \ i+0&2-9&19 o = ((011 + 2.“ A= Vim 6. (5 points each) Evaluate the following indefinite integrals. (a) /x21nacdm 1“:bi 8,0465: (“Emu ro-z 13-13 and/1&4. ding-M 3 3 = #31 4M. — \Jv icon _.L3 _|_3+ _3¥.J?m1. (1'16 CL (b) / hm TCCWCCL SUJDStL’tLCIm ()0 = Z‘tM—Q aw «<19 21 Z aseceztme . —— ——————- 2 \J ‘-\ + II th-e- (De __ & Zsecza'tm '99 - J l+ thB (c) /cos(sin_1x)dm .— .— ‘- — V1 —a:2 XmMM VmM+c . 0“ \+0 |m+C 3,5 —\<1L<|. . (5 points each) The position of a moving point at time t is given by {3 (a) Find the speed of the point at t = 7r/3. fig: l-mt fl = -0&'t mom 4&«9 = Jl‘w‘mfi (ma “law/33‘ = W? —‘ 42—, l + (Eg- -I\" :2 0.044645 (b) Find the distance the point travels for t in the interval [0, 7r / 2]. You may evaluate this result numerically. g = Vial (l-Ont\2+ (“but +tookt)2 M t: I.3'-IZI‘Z> t—sint 1— tcost. (C) Find the slope of the tangent line to the path of the point at t = 7r/ 6. W8: ~oat+tcw$~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 ’Yv-l : L + ‘Bqu-C— ¥(¥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. —v-0 W fi—yw W = JEN-II <| =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%. ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern